n-bands hubbard models. iv. comparisons of electron- or hole-doped quaternary oxypictides laompn...

N-Bands Hubbard Models. IV.Comparisons of Electron- or Hole-Doped Quaternary Oxypictides LaOMPnSuperconductors With Cuprates

K. YAMAGUCHI,1 S. YAMANAKA,1 H. ISOBE,1 M. HAGIHARA,1

D. YAMAKI,2 M. NISHIHARA,2 Y. KITAGAWA,2 T. KAWAKAMI,2

M. OKUMURA2

1Center for Quantum Science and Technology Under Extreme Conditions (KYOKUGEN),Osaka University, Osaka 563-8531, Japan2Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka,Osaka 560-0043, Japan

Received 11 May 2008; accepted 5 June 2008Published online 6 August 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21840

ABSTRACT: A lot of theoretical and experimental results are examined to obtain aunified picture of magnetism and superconductivity in strongly correlated electronsystems. First principle computational results reported recently have been used toelucidate electronic structures of strongly correlated electron systems. In this series ofarticles theoretical efforts for the systems such as transition metal oxides have beenextended to search high-Tc superconductors (HTSC) other than cuprates in relation topossible mechanism(s) of high-Tc superconductivity. Very recently, Kamihara et al. havefound that quaternary oxypictides LaOMPn (M � V, Cr, Mn, Fe, Co, Ni, and Pn � P,As) are a new family of high-Tc superconductors, exhibiting high-transition temperatureTc � 26 K for the case of M � Fe and Pn � As. Since iron has been believed to bemagnetic in many systems, their finding raises extremely interesting question

Correspondence to: K. Yamaguchi; e-mail: [email protected]

Contract grant sponsor: Grant-in-Aid for Scientific Research.Contract grant numbers: 185008, 18205023.Contract grant sponsors: Ministry of Education, Culture,

Sports, Science, and Technology (MEXT); Japan Society for thepromotion of Science (JSPS).

Additional Supporting Information may be found in theonline version of this article.

International Journal of Quantum Chemistry, Vol 108, 3016–3041 (2008)© 2008 Wiley Periodicals, Inc.

concerning with relationships between magnetism and superconductivity. Moreover,coexistence of antiferromagnetism (AF) and superconductivity (SC) is also discovered inthe recently synthesized cuprates with five CuO2 planes. This may indicate theuniformly mixed phase of AF and SC in cuprates without disorder corresponds to spinglass, instead of other complex phases resulting from random potentials in disorderedcuprates. Coexistence of AF and SC phases and multiband effects are also suggested forquaternary oxypictides LaOMPn. Here, comparison between quaternary oxypictidesLaOMPn and cuprates is carried out from both theoretical and experimental grounds,leading to an extension of our magnetic (J) model for LaOMPn. Implications of theseanalyses are discussed in relation to possible candidates of HTSC with multi (N)-layersconsisted of iso-electronic �-d, �-R, and �-R systems, for which similar electronicphases are expected from the theoretical grounds. © 2008 Wiley Periodicals, Inc. Int JQuantum Chem 108: 3016–3041, 2008

Key words: spin-mediated mechanism; antiferromagnetism; ferromagnetism;LaOMPn systems; multi-layered system; cuprate; superconductivity; non-BCSmechanism

1. Introduction

A bout 40 years ago, Little [1] has presented anexciton [or charge fluctuation (CF) model]

for high-Tc superconductivity by replacing Debye(phonon) frequency with electron excitation energyin the Bardeen–Cooper–Schreiffer (BCS) theory [2].Little, Ladik and coworkers have proposed severalmodel compounds such as polymers with polariz-able substituents [1, 3]. However, Little-typehigh-Tc superconductor [1, 3] has not been discov-ered yet. On the other hand, Bednorz and Muller [4]have discovered the high-Tc cuprates on the basis oftheir working hypothesis, namely the strong elec-tron–phonon interaction via the Jahn-Teller distor-tion in copper (II) oxides. After their discovery [4],a lot of experimental and theoretical studies oncuprates have been carried out to elucidate origin(s)of high-Tc superconductivity of cuprates, which areregarded as Mott insulators [5–8] before hole orelectron doping. Since antiferromagnetism and su-perconductivity are characteristics of cuprates, sev-eral magnetic models [9–35] have been proposed toexplain the high-Tc superconductivity. For exam-ple, we have replaced charge excitation energy byLittle with magnetic excitation energy (given by J)in our magnetic model of high-Tc superconductiv-ity. Recently, magnetic excitation in cuprates hasbeen detected by imaging technique of the neutrondiffraction experiments and related techniques [36–39]. Since the observed maximum transition tem-perature (Tc) for HgBa2Ca2Cu3O8 is lower than 135K (160 K under pressure)[40], new superconductorswith more higher Tc are now interesting targets for

various theoretical and experimental investiga-tions.

Recently, Kamihara et al. [41] have found thatseveral quaternary oxypictides LaOMPn(M � Fe,Ni, Pn � P, As) undergo superconducting transi-tion under electron doping by partially replacingthe oxygen dianion (O2�) with fluorine anion (F�),though the superconducting transition temperature(Tc) is not so high as summarized in Table I.LaOMPn [41–43] has a layered crystal structurewith the tetrahedral P4/nmm space group symme-try, where the M2Pn2 layer is sandwiched betweenthe La2O2 layers (OLa2O2OM2Pn2OLa2O2O) as il-lustrated in Figure 1(A). The divalent transitionmetal ions M(II) construct two-dimensional latticeas shown in Figure 1(B), where the M(II)–M(II)interaction through p-orbitals of Pn [see Figs. 1(C)and (D)] is essential. The replacement of O2� withF� in the insulating phase (La2O2) provide electroncarriers (electron doping) in the conductive two-dimensional layer (M2Pn2), giving rise to the super-conductivity. Very recently, Kamihara et al. [44]have reported a surprising fact that La(O1�x

Fx)FeAs obtained by substitution of O2� with F� inthe parent compound LaOFeAs exhibits the high-Tc

� 26 K (Tonset � 30 K) at x � 0.11. Further, they havepointed out that supercondutivity was not observedfor Ca2�-doped samples, indicating an important roleof electron doping instead of hole doping. On theother hand, Wen et al. [45] have reported that holedoped (La1�xSrx)OFeAs achieved by substitutingLa3� with Sr2� indicates the high-Tc (� 25 K) super-conductivity at x � 0.13. This implies that LaOMPnfamily is similar to cuprates. Moreover, severalgroups [46–50] have reported quite high-Tc supercon-

N-BANDS HUBBARD MODELS IV

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ductors obtained by replacement of La with Ce, Pr,and so on (cond-mat papers) as summarized in TableII, where the numbers of d-electron and ion radious ofLa series are also cited. The reported compoundscorrespond to the electron-doping superconductors inLaOFeAs family. Since Tc of La(O1�xFx)FeAs has in-creased to 41 K under the external pressure [44], theorigin of high-Tc of Ce, Pr, and other related systemsin Table II may be ascribed to chemical pressure withdecrease of ion radius. According to these preliminarybut fascinating data [45–50], quaternary oxypictidesLaOMPn are regarded as a new superconductingfamily.

Interestingly, both undoped cuprates andLaOMPn are regarded as Mott insulators [5–8] ex-hibiting magnetic instability instead of charge(CDW) instability. Available evidence indicates thatelectron and spin correlations play crucial roles inmagnetism and superconductivity in these systems.Thus, the instability in chemical bonds [51] viastrong electron–electron interaction is now one ofthe important and crucial concepts even in materialscience as well as in quantum chemistry. Theoreti-cal investigations of these systems are indeed es-sential for elucidation of interrelationship betweenmagnetism and high-Tc superconductivity in gen-eral, since both characteristics have been commonlyobserved experimentally. About 22 years ago, our

TABLE I ______________________________________________________________________________________________LaOMPn-type superconductors and related species.

System Tc (K) System Tc (K)

LaOFeP 4 Sr2RuO4 1.4 (p-wave)CeOFeP hfa 7 UPt3 0.54 (p-wave)La(O1�xFx)FeP 3 NaxCoO2�1.3H2O 5LaONiP 4 LaOCoAs Ferromagnetic model (FM)La(O1�xFx)NiP 2.75 LaOMnAs Antiferromagnetic insulator (AFI)LaONiAs 3.7 Ce(O1�xFx)NiP Ferro(La1�xSrx)ONiAs 16.6 CeORuAs FerroLuNi2B2C 13.2 La(O1�xFx)RuAs Metal(Sr1�xKx)Fe2As2 38 Ce(O1�xFx)RuAs Metal(Ba1�xKx)Fe2As2 38 LaOCoPn FerroK Fe2As2 Metal

a Heavy fermion system.

FIGURE 1. (A) Layer structure of LaOMPn, (B) transferintegrals for the nearest and next nearest neighborsites, (C) sueperexchange interaction between M(II) viatrianion, and (D) 4p-3d orbital interaction in MPn plane.

TABLE II _____________________________________La(O1�xFx)FeAs-type superconductors.

Compounds Tc (K) La3� r

La(O1�xFx)FeAs 26 (43)a f0 1.03Ce(O1�xFx)FeAs 41 f1 1.01Pr(O1�xFx)FeAs 52 f2 0.99Nd(O1�xFx)FeAs 52 f3 0.98Pm(O1�xFx)FeAs — f4 0.97Sm(O1�xFx)FeAs 43 f5 0.96Eu(O1�xFx)FeAs — f6 0.95Gd(O1��)FeAs 53.5 f 7 0.94Gd(O1�xFx)FeAs 7 f7 0.94Tm(O1�xFx)FeAs — f12 0.88

r: Ion radius.a Under the pressure of 3GPa.

YAMAGUCHI ET AL.

3018 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 108, NO. 15

ab initio calculations indicated that the CuOCu unitexhibits very strong effective exchange interaction( �J� �� 0) [52]. But we could not imagine that suchfinding might be related to the high-Tc supercon-ductivity, because our main interest is to elucidateelectronic structure and reactivity of transitionmetal oxides. After the discovery of high-Tc cu-prates by Bednorz and Muller [4], we immediatelyproposed a magnetic (magnetic excitation-medi-ated) model (Tc � cJ(kB)) [15], where J is taken as ameasure of magnetic excitation. The constant c isnow determined by possible microscopic modelssuch as t-J, boson-fermion, magnon excitation, andspin fluctuation models [9–35]. It is noteworthythat our magnetic (J) model has been presented onthe basis of ab initio calculations before the Zhang-Rice t-J model [18] in the strong correlation region.On the other hand, the magnetic model [15] in thek-space is formulated as spin fluctuation (SF) the-ory [9, 10, 28, 29] and/or spin-bag theory [17, 19] inthe weak correlation regime. Since metallic andmagnetic phases have been observed, dependingon kinds of M and Pn in LaOMPn [41–50], magneticexcitation mechanisms have been received currentinterest in relation to the exotic superconductivity.As shown in this series of articles (I–III), the mag-netic model [15] enables us to propose isoelectronicp-d, �-d, �-R, and �-R conjugated systems on thebasis of N-band Hubbard model [33]. LaOMPn isone of p-d conjugated systems in our terminology.

In part I of this series [53], we have proposedseveral working hypotheses for high-Tc supercon-ductivity from both experimental and theoreticalstudies of cuprates: (1) low dimensionality (2D), (2)border of metal–insulator transition (intermediatecorrelation regime), (3) quantum spin (s � 1/2), (4)magnetic excitation, and (5) multi-band effects. N-band Hubbard models have been derived for apossible theoretical model of describing these re-quirements on the basis of quantum chemical cal-culations of cuprates and isoelectronic p-d, �-d,�-R, and �-R conjugated systems. Past decadesmagnetic models such as spin (SF) fluctuationmodel are possible working hypotheses for molec-ular design of high-Tc superconductors [33, 53].Indeed, several kinds of SF models have been pro-posed on the experimental and theoretical groundsin Part II [33]. However, a unified theory describingphase diagrams in an intermediate (near Mott tran-sition) regime as well as both weak and strongcorrelation limits is not established yet [54], partlybecause serious disorder and defects in each CuO2plane (BEC-BCS crossover problem in part III [35]).

Here (Part IV), as an extension of our theoreticalwork, electronic structures of MPn sheets inLaOMPn are theoretically examined as comparedwith cuprates with five CuO planes permitting co-existence of antiferromagnetism and superconduc-tivity. Possible p-d, �-d, �-R, and �-R isoelectroniccompounds with multi-layers have also been dis-cussed to clarify present stage of rational design ofnew materials, mainly from our theoretical view-points. Finally, implications of these theoreticalstudies are discussed in relation to the goal (roomtemperature superconductivity [1]).

2. Electronic Structures of LaAOMPn

2.1. LIGAND FIELDS OF MPn SHEETS

The ligand field of metal center M (� Fe, Ni, etc.)in LaOMPn is essentially regarded as distorted tet-rahedra (S4) in sharp contrast of the octahedralligand field of Cu(II) ion in cuprates. Therefore, fived-orbitals are splitted into lower eg (dz2, dx2�y2)and higher t2g (dxy, dxz, dyz) groups at Td config-uration, and further d-orbital splitting occurs in thedistorted tetrahedral as illustrated in Figure 2(A).Since formal charges of LaO and MPn in LaOMPnare regarded as �1 and �1, respectively [see Fig.1(A)], the transition metal ions Mm� is divalent(m � 2) under the assumption that Pn is trianion(Pn3�). The numbers of d-electrons are 3, 4, 5, 6, 7,and 8 for V(II), Cr(II), Mn(II), Fe(II), Co(II), andNi(II), respectively. The highest (HS), intermediate(IS), and lowest (LS) spin states are feasible, de-pending on the strength of ligand fields of trianionPn3� as summarized in Table III. The local HSconfigurations of M(II) should be realized under theweak ligand field (WLF) condition. Such situationhas been observed in the case of iron–sulfur (Fe–S)clusters [s � 5/2 for Fe(III) and s � 4/2 for Fe(II)] inferredoxin. However, superexchange interactionbetween Fe(X)(X � II,III) through S2� entails anti-ferromagnetic (AF) effective exchange interaction,leading to the AF-type ground state in Fe–S clusters[55]. Therefore, AF state (J is negative in sign) viasuper-exchange interaction through P3� is expectedfor MPn sheets consisted of early transition metals[M � V(II), Cr(II), Mn(II)]. On the other hand, tran-sition metal complexes consisted of late transitionmetals such as Co(II) is known to have the ferro-magnetic ground state, indicating that Coulombicexchange interaction (K in our notation instead of Jin solid state physics) plays a role. It is well known



that Cr and Mn in their element form usually takesantiferromagnetic (AF) state whereas Co and Ni areferromagnetic. Fe displays a variety of magneticphases: (a) strong ferromagnet in bcc form, (b) non-magnetic in fcc form, and (c) AF or spin density

wave (SDW) in fcc form with other local environ-ments. This implies that LaOFePn systems mayexhibit magnetic or nonmagnetic states dependingon combinations of component atoms.

The HS, IS, and LS configurations of Fe(II) inLaOFeAs are depicted in Figure 2(C). According toaforementioned experimental results, the relativestability of these configurations seems variable de-pending on the strength of the ligand field of Pn3�.The FeOPn bond length and deformation Td ligandfield are considered to be important factors sinced-orbital splittings are sensitive to them (see Fig. 2).Several possible models are therefore conceivableas starting points for theoretical understanding ofsuperconductivity of LaOMPn: (1) HS (S � 4/2)model, where all the d-orbitals are equally impor-tant, (2) IS (S � 1) model, where 3dxz and 3dyzorbitals are particularly important, and (3) LS (S �0) model, where electron correlation plays an im-portant role because of near orbital degeneracy. Infact, microscopic models proposed at the presentstage are easily classified on this orbital configura-tions, though the site (real space) models are trans-ferred into tight-binding and band (k-space) mod-els for LaOMPn with 4P/nmm space symmetry[41–50].

2.2. ELECTRON OR HOLE DOPING IN MPnSHEET

The extra electron generated by replacement ofO2� with F� may be doped into d-orbital of Fe(II)ion in the case of La(O1�xFx)FePn since p-orbitals ofAs3� is fully occupied as shown in Figure 3(A).Kamihara et al. [44] have pointed out that Fe(II)with d6 configuration is partly converted to Fe(I)with d7 configuration, which is isoelectronic toCo(II), in the electron-doped La(O1�xFx)FeAs. Theyhave pointed out a possible role of ferromagneticspin fluctuation in the superconductivity in analogywith the LaOCo(II)Pn ferromagnet. Substitution of

FIGURE 2. (A) crystal field splitting of d-orbitals of Tdand deformed structures of MPn4, Td like M4 configu-ration in LaOMPn, (B) �1 and �2 express the distortedtetrahedral structure and (C) HS, IS and LS electronconfigurations of Fe (II) ion in LaOFeAs.

TABLE III ____________________________________________________________________________________________Orbital configurations of M(II) ion and possible spin states.

V(II) Cr(II) Mn(II) Fe(II) Co(II) Ni(II)

d-electron d3 d4 d5 d6 d7 d8

HS s � 3/2 s � 4/2 s � 5/2 s � 4/2 s � 3/2 s � 2/2IS (LS) s � 1/2 s � 2/2 s � 3/2 s � 2/2 s � 1/2 s � 0LS s � 0 s � 1/2 s � 0State AF AF AF AF F NM,F

YAMAGUCHI ET AL.


Pn3� with dianions (Y2�) (Y � S, Se, Te, etc.) inLaOFe(Pn1�xYx) is interesting from the view pointof electron doping. LaOMPn system seems ex-tremely interesting since there are many possibili-ties of atom substitutions for carrier doping.

Two different situations are feasible for the FePnsheet in the hole-doped case as illustrated in Figure3. Analogy of CuO2 plane indicates one possibilitythat the p-level of Pn tri-anion could be higher thanthe d-level of Fe(II) [see Fig. 3(B)], though thecharge transfer (CT) from Pn3� to Fe(II) is forbid-den because of strong on-site Coulomb repulsion U[see Fig. 3(B)]

��CT� � ��dd � �pp� � �Udd � Upp�

� ��dp � Ueff 0 (1)

where �X and Ux denote, respectively, the orbitalenergy and on-site Coulomb repulsion of the orbitalX. In this situation, the hole-doping occurs at Pn3�

site: from Pn3�(Fe(II)Pn3�) to Pn2� (Fe(II) Pn2�), asillustrated in Figure 3(C). The orbital energy diagramis similar to that of cuprates. Therefore, the reportedhigh-Tc superconductivity of (La1�xSrx)OFeAs can bealso understood in the same picture of the hole-dopedcuprates. The other case is that Fe(II) ion is oxidizedinto Fe(III) ion if ��dp � 0 as illustrated in Figure 3(D).Since Fe(III) has the d5 configuration like Mn(II), the

hole-doped Fe(II)Pn layer may exhibit AF semicon-ducting character. No superconductivity in Ca2�

doped LaOFeAs may be explained by this senario.Theoretically substitution of Pn3� with tetraanions(Z4�) (Z � Si, Ge, Sn, etc.) is interesting from the viewpoint of hole doping, namely from Fe(II)Pn3� toFe(III)Z4� in LaOFe(Pn1�xZx). A lot of experimentalstudies are desirable from these theoretical viewpoints to elucidate the nature of electronic structuresof LaOMPn family.

2.3. N-BANDS HUBBARD MODELS FORLaOMPn FAMILY

Electronic, magnetic, and optical properties ofbulk materials are dependent on the crystal struc-tures. First principle calculations have been appliedfor elucidation of band structures of LaOMPn. Forexample, density functional theory (DFT) with gen-eralized gradient approximation (GGA) in the Per-dew–Burke–Ernzerhof (PBE) variant have been car-ried out for LaOMPn using WIN2K package andVASP (Vienna ab initio simulation package) [56–63]. Here, we consider possible tight-binding mod-els on the basis of electron configurations of M(II)ions in Figure 1 and Table III, and P4/nmm crystalstructure of LaOMPn. To this end, in this series ofarticles, N-bands Hubbard models have been usedfor cuprates and isoelectronic molecule-based ma-terials to elucidate important roles of electron andspin correlation effects [51]. For MPn sheets, wemust at least consider d-orbitals on M(II) thoughtheir interactions are mediated through p-orbitalsof Pn3� anion as illustrated in Figure 1(D). Theelectronic interactions of d-orbitals are expressedby

H � �i

�jk

tjkai, j�� ai,k� � Ueff

0 �i

� jni, j�1�ni, j�2�

� Ueff1 �

i

�jk

�ni, j�1� � ni,k�2� �ni, j�1� � ni,k�2�

� K �i

�jk

Si, jSi,k (2)

where tij denotes transfer integrals between M(II)via p-orbitals of Pn, and U0 and U1 denote on-siteintra- and interorbital Coulomb repulsion integrals,respectively. K means the interorbital Coulomb ex-change integral. Judging from the experiments forLaOMPn, relative importance of K and U1 is valu-able, depending on carrier concentrations. The

FIGURE 3. (A) electron doping for Fe(II) ion in FePnsheet, (B) Electron configuration and energy diagram ofM(II) and Pn, (C) hole doping at Pn site and (D) holedoping at M(II) site.



number density of the up (or down) spin is definedby creation and annihilation operators as

ni, j�1� � ai, j� �1�ai, j�1� (3)

and spin operators are also given by

Si, j �12 ai, j

� �1�� ai, j��1� (4)

where �� denotes the Pauli matrix. These parame-ters are often assumed to be empirical parametersdetermined on the experimental grounds, thoughthey are now determined by first-principle calcula-tions, for example, the general spin orbital (GSO)HF and DFT calculations as shown in Figure 4 [51].In this series of articles, the covalent bonding pa-rameter x and inonic parameter y are defined by

x � � t/Ueff � �/Ueff, y � ��dd � �pp�/Ueff (5)

where Ueff0 � Ueff

1 � Ueff is assumed. LaOMPn isregarded as the 4p–3d conjugated systems in ourN-bands Hubbard model [33].

As shown in this series of articles [33], chargedensity wave (CDW) type state becomes theground state if U0 U1, though it is eliminated inthe LaOFeAs system. On the other hand, the un-conventional superconductivity such as p-wave isfeasible if K � U1 in the case of two-band model[33], but spin density wave (SDW) state arises if K U1. In the strong correlation regime with magneticphases, the Hubbard model can be reduced to theHeisenberg spin Hamiltonian model [51]

H�Heisenberg� � � 2 � JabSaSb (6)

where Jab is the effective exchange integral. The Jabvalue for multi-center radicals is calculated by

Jab�Y� �LSEY � HSEY

HS�S2�Y � LS�S2�Y(7)

where XEY and X�S2�Y denote, respectively, the totalenergy and total spin angular momentum of thespin state X [X � the lowest spin (LS) and/or thehighest spin (HS) state] by a computational meth-ods such GSO DFT and hybrid GSO DFT [15, 64–66] as shown in Figure 4.

2.4. MAGNETIC MODEL FORSUPERCONDUCTIVITY

In this series of articles, magnetic excitation en-ergies have been formally expressed by the effec-tive exchange integrals (J). However, it is notewor-thy that our J value is defined by the total energydifference between HS and LS states as shown inEq. (7). For example, the magnetic excitation energyfrom the ground singlet (S) to the excited triplet (T)state is simply expressed by the 2 2 CI schemebased on the Hubbard model [51] as

��ST�/ 2 � Jab�CI� � Jab�Hubbard�

�14 �Ueff � �16t2 � Ueff

2 � (8)

Then spin excitation (fluctuation) energy is de-termined by both t and Ueff in general. However,the energy gap can be often expanded by the per-turbation series in the weak (1/x) and strong (x)correlation regimes as

Jab�Hubbard� � �t�2t/Ueff � higher term��Ueff �� t�

�Heisenberg Limit� (9a)

� �t�1 � Ueff/4t � �Ueff/4t�2�Ueff t�

�Band�Huckel� Limit� (9b)

The Jab values by first principle calculations canbe transformed to those of Hubbard model, whichare used to determine t and Ueff values in Eq. (2).Since Jab(CI) values involving correlation effects arereliable in the whole region of t/Ueff, the mappedJab(Hubbard) values are also useful even in an in-termediate regime as well as strong (Heisenberg J)and weak (band width t) limits as shown above.

FIGURE 4. First principle computational methods forstrongly correlated electron systems in quantum chem-istry and their applications to material science.

YAMAGUCHI ET AL.


The t-J model and spin fluctuation (SF) models areused in the former and latter regions effectively.Unfortunately, a reliable theory for superconduc-tivity in the intermediate (Mott transition) regionbetween these limits is not established yet in thefield of solid state physics [54], though the magneticexcitation energy can be determined by first-prin-ciple calculations of finite cluster models in quan-tum chemistry. So, we have constructed a possiblephenomenological magnetic (J) model on the basisof available experiments combined with varioustheoretical results accumulated [9–35]. Interest-ingly, several band calculations available now haveshown that LaOFePn family just on the border ofMott (metal–insulator) transition, suggesting thatreliable theoretical treatments of these systems arenot so easy as in the case of cuprates.

3. Magnetic Models forSuperconductivity of LaOFePn

3.1. CHARACTERISTICS OF LaOFeAs BASEDON THE BAND CALCULATIONS

Several band calculations [56–63] available atthe present stage have elucidated following charac-teristics of LaOFeAs:

A. The parent compound LaOFeAs is a nonmag-netic (NM) metal with a low density of carrierand its normal phase is located at the border-line of magnetic phases [56, 57]. There is littledispersion in the valence bands along thec-axis direction (see Fig. 1), corresponding toa nearly 2D electronic structure. The crystalfield effect on the Fe-3d orbitals is muchweaker than in transition metal oxides, whichis understandable because the electronegativ-ity of As atom is much smaller than that of Oatom. Therefore, all Fe-3d electrons are ex-pected to play a dominant role in conductiv-ity and related superconductivity. Indeed,the Fermi surfaces in the NM state (no spinpolarization) are made up of five sheets andexhibit the five bands crossing; two sheetsdue to two electron bands are forming twocylinder-like shapes centered around M-A.The two electron cylinders expand signifi-cantly after electron doping; but the densityof states at the Fermi energy heavily de-creases, suggesting that the doping in the NM

state may not favor the high-Tc superconduc-tivity.

B. LaOFeAs may become an antiferromagnetic(AF) rather than the NM metal because it liesthe border of the metal–insulator transition[60, 62]. In fact, the AF metal (AFM) state with2.3 �B on Fe site was calculated to be morestable (0.22 eV/cell) than the NM state be-cause of the Fermi surface nesting. There arethree bands crossing of the Fermi energy inthe AFM state. Unlike in the NM state, thedensity of states at the Fermi energy de-creases not so much with doping. This mayimply that AF spin fluctuation plays an im-portant role of the high-Tc superconductivity.

C. Inclusion of electron correlation effectsthrough U is inevitable even in the band cal-culation of LaOFeAs. The DFT � U calcula-tion [61] have shown an unambiguous anti-ferromagnetic (AF) ground state for undopedLaOFeAs, which is 84 meV per Fe lower thanparamagnetic (PM) and ferromagnetic (F)states. The Mott gap of about 1.0 eV is ob-served in the LDA � U calculation at U � 4.5eV. The degenerated dx(y)z bands are splittedbecause of the spin polarization in the DFT �U calculation. The U value on the border ofthe metal–insulator transition is estimated tobe about 3 eV. This in turn indicates that theeffective transfer integral for the border ofMott transition is about 0.38 eV by using the2D model (U � 8t).

D. The DFT � U method does not include thescreening effect of U at finite temperature,leading to the DFT plus dynamical mean fieldtheory (DMFT) calculation [58]. The DFT �DMFT approach [58] has shown thatLaOFeAs remains metallic [U(T) 3 eV] atfinite temperature, although the Fermi sur-face shrinking brings about the semiconduct-ing gap [U(0) � 3.0] at the zero temperature.Thus the correlations enhance the crystalfield splittings which lead to a state being aband-semiconductor acting as bad metal atfinite temperature. The DFT � DMFT calcu-lation predicts that the dominant states at theFermi level come from Fe 3d atomic statesextending roughly between �2 eV and 2 eV.However, a strong mixing with As is appar-ent at 3.2 �2.7 eV where As 4p band isstrongly peaked. Then there is a noticeablesimilarity between the Fe-As mixing in



LaOFeAs [see Fig. 3(B)] and oxygen (2p) andCu(3d) mixing in cuprates, though the Asbands are much more itinerant and broaderthan the O(2p)-bands. However, it is note-worthy that the cluster model extension ofDMFT [67, 68] to include nonlocal correla-tions seems crucial for much more exact dis-cussion of the correlation effect in both cu-prates and LaOFeAs.

3.2. CHARACTERISTICS OF LaOFeAs BASEDON THE BAND CALCULATIONS

The above ab initio computational results can bemapped into N-band Hubbard models as in thecase of cuprates [53]. For example, two-site (two-band) model considering both Fe(Cu) 3d and4p(2p) of As(O) (one of the p-d models) in given by

H � �j

�ddj��dj� � �

l

�Ppl��Pl�

� Ueff �j

dj1� dj1dj2

� dj2

� �j�

�l

�t�p � d�jldj��Pl� � H.C. (10)

where �x (x � d, p), Ueff, and t(p-d) denote, respec-tively, the orbital energy of site X, effective on-siteCoulomb repulsion (Ueff � Udd � Upp � 2.0–5.0eV). This d-p (p-d) model Hamiltonians for MAs(MO) clusters (M � Fe, Co, �� , Cu) can be diago-nalized to elucidate charge and spin populations,and effective exchange integrals (J) in these sys-tems. The several computational results for MOclusters are given in supplementary materials.

The two-site p-d model in Eq. (10) is often re-duced to single site Hubbard model (t,U), in Eq. (2),which has been used for construction of spin fluc-tuation model [U/t expansion in Eq. (9)]. On theother hand, the Hubbard model is further simpli-fied into Zhang and Rice [18] t-J Hamiltonian model[t/U expansion in Eq. (9)], assuming that hole (elec-tron) doped on the oxygen site forms the singletpair with the unpaired electron on the Cu(II) site

H � � �i, j�

tijdj�� di� � 2J �

i, j

Si � Sj (11)

where the double occupancy of electrons at theCu(II) site is completely removed by using the re-lation di� � di��1 � ni��:ni� is the density at the sitei. Here, we do not touch a reliability problem of

such reduction. The exact diagonalization of the t-Jmodel has indicated that the const. in the J-model(Tc � cJ) is about 0.10 [25]. The slave-boson approx-imation to the t-J model has revealed possible phasediagrams [21] for cupurates, which have been usedto obtain one of reasonable explanations of ourmagnetic model in the under-doped regime [28].The t-J model is also applicable to MPn sheets if thehole is mainly doped into Pn site. Probably, manyexperimental results will be published near future,and applicabilities of N-band Hubbard and t-J mod-els for LaOMPn systems will be clarified. In thispart IV, simple magnetic model is introduced inSection 3.4.

3.3. POSSIBLE MECHANISMS OFSUPERCONDUCTIVITY FROM BAND MODELS

Several possible mechanisms of superconductivityin LaOFeAs systems have already been proposed onthe basis of the band calculations and model Hamil-tonians constructed on the basis of available bandstructures [56–63, 69–71]. Here, underlying assump-tions and employed computational results are exam-ined to understand possible origins of proposedmechanisms of the superconductivity.

A. BCS (electron–phonon) mechanism. Bandcalculations of La(O1�xFx)FeAs have beenperformed to elucidate the possibility ofconventional BCS (electron-phonon; e-p)mechanism. The calculated e-p couplingcoefficient and vibrational frequency �were 0.21 and 206 K, respectively, givinga maximum Tc of 0.8 K, using thestandard Migdal-Eliashberg theory, thoughthe same technique has correctly repro-duced the Tc (� 39 K) of MgB2, a typical e-psuperconductor [59]. So, it was concludedthat the e-p mechanism is not sufficient toexplain the experimental Tc � 26 K forLa(O1�xFx)FeAs and Tc � 50 K for relatedmaterials in Table II.

B. p-wave superconductivity. Both ferromag-netic (F) and antiferromagnetic (AF) fluctua-tions are apparently present in LaOFeAs, sug-gesting possible analogies to ternary rareearth, heavy fermion, ruthenate, and cupratessuperconductors in Table I. This entails theso-called p-wave mechanism based on thetriplet Cooper pair like 3He and Sr2RuO4,leading to an assumption that strong ferro-

YAMAGUCHI ET AL.


magnetic fluctuations suppress even parity s-or d-wave pairing. According to the bandcalculation of LaOFeAs, the carrier densitywas low at Fermi level, favoring B(BW)-phaselike state of 3He instead of it’s A(ABM) phase(note that A phase(gap less) is realized forSr2RuO4 because of high density [44, 70].

C. d-wave superconductivity. The spin-polar-ized DFT calculations have predicted a com-mensurate antiferromagnetic spin densitywave due to the Fermi surface nesting, whichis robust against the F-doping. This may en-tail d-wave superconductivity arising fromthe antiferromagnetic spin fluctuation like cu-prates [61]. The DFT � U calculations havepredicted an ordered antiferromagnets withstaggered moment about 2.3 �B on the borderwith the Mott insulating state. The coexist-ence of antiferromagnetism (AF) and super-conductivity (SC) was suggested like in theelectron-doped cuprates.

D. Unconventional s(�) and s(�) wave super-conductivity via the two-band model. In thismodel, the degenerated d-electron bands ofLa(O1�xFx)FeAs are utilized to construct atwo-band model as illustrated in Figure 5.The bands are expressed by the expandedk-space as

�k,1 � � 2t1coskx � 2t2cosky � 4t3coskxcosky

�k,1 � � 2t2coskx � 2t1cosky � 4t3coskxcosky.

(12)

where t �� t2 t1.The key concept is that two bands bring strong

pairing force, provided that the order-parameterson the two sets of Fermi surfaces have the opposite

sign [28, 29, 60]. This mechanism is based on strongbut broad antiferromagnetic (AF) spin fluctuationsnear the M point in the Brillouin zone and suggeststhe multigap superconductivity like in MgB2. Butthis is different from AF spin fluctuation based onthe superexchange interaction like cuprates (see thecase C mentioned earlier).

E. Even parity, orbital singlet and spin tripletsuperconductivity via the two-band model. Inthis model, the degenerated d-electron bandsof La(O1�xFx)FeAs in Eq. (12) are utilized toemphasize strong ferromagnetic fluctuationinduced by Hund rule coupling construct Kin Eq. (2).

Then the Hund coupling is an origin of spintriplet Cooper pair of the electrons on two differentbands [69]. However, this mechanism is differentfrom the simple p-wave one for Sr2RuO4 because itis robust against disorder like in the s-wave BCSsuperconductivity (see case A). The exchange inter-action K in Eq. (2) in real space is transformed intothe k-dependent contributions in the k-space, de-pending on the crystal structure as follows:

Kk � K0 � K1�cos�kx� � cos�ky�� . . . . . . (13)

If only the first term is retained, the orbital partof Cooper pair becomes s-wave and orbital singlet:it is noteworthy that the extended s- or d-waveshould be favored if the second term becomes pre-dominant [69].

F. Unconventional singlet superconductivity viathe five-band model. In this model, all thed-band of La(O1�xFx)FeAs are utilized to de-rive a possible unconventional singlet super-conductivity based on the multiple spin fluc-tuation modes arising from the nestingvectors across and within disconnected Fermisurfaces [71]. Therefore, this mechanism in-volves C- and D-type spin fluctuation (SF)mechanisms as two limiting cases discussedearlier. Random phase approximation (RPA)is applied to the model to calculate chargeand spin susceptibilities, which are pluggedinto the linearized Eliashberg equation to con-firm important contributions of both types ofspin fluctuations. Relatively small U value(about 1 eV) was employed for small orbitalsplitting in this model.

FIGURE 5. (A) real space description of MPn 2Dsheet, (B) Fermi surface for MPn 2D sheet, (QAF de-notes the nesting vector), and (C) expanded unit cell forMPn.



3.4. A MAGNETIC MODEL FORSUPERCONDUCTIVITY OF LaOFeAs

From results of above band calculations in Sec-tion 3.3, characteristic features of LaOMPn are (i)border of metal–insulator transitions, (ii) impor-tance of magnetic fluctuations, (iii) multi-band ef-fects, and (iv) possibility of non-BCS (electron–pho-non) mechanism. The possible models B-F inSection 3.3 are regarded as magnetic mechanisms ofsuperconductivity. The antiferromagnetic (AF) (in-traband) spin fluctuations are responsible for ex-tended s-wave without node or with node, or d-wave superconductivity. While ferromagnetic (F)(intraband) spin fluctuations are origins of spintriplet s-wave or p-wave superconductivity. In thisseries of articles [53], we have already discussedsimilar characteristics even for cuprates. The situa-tions are quite similar in both materials, leading toan extension of the simplest magnetic model, Jmodel, to LaOMPn family.

As shown in band calculations in Section 3.1, thenesting of Fermi surfaces provides the antiferro-magnetic (AF) ground state with spin structure I inFigure 6 if the nearest neighbor effective exchangeinteraction (J1) is predominant ( �J1� � 2� J2� ). On theother hand, a stripe-like AF spin structure II isfavorable if the next nearest neighbor effective ex-change interaction (J2) plays a significant role ( �J1�

2� J2� ). If the GSO-type band calculations in Fig-ure 4 were performed, the noncollinear spin struc-tures III and IV with more or less spin flustrationare also feasible, depending on the relative magni-tude of � J1� and � J2� .

H � � �ij

2J1Si � Sj � �ik

2J2Si � Sk (14)

where (i,j) and (i,k) denote, respectively, the nearestand next nearest neighbor pairs of spins as illus-trated in Figure 6.

The magnetic structures in Figure 6 enable us toextend the magnetic (J) model for pairing of elec-trons doped in MPn sheet as shown in Figure 7. Theone electron doping violates the magnetic interac-tions (4J1 � 4J2), and therefore loss of the magneticinteractions for separated electrons is given by�2 (4J1 � 4J2). Although it becomes �[7J1(8J2) �8J2(7J1)] for paired electrons as illustrated in Figure7. The net result, J1 (J2) � �[7J1(8J2) � 8J2(7J1)] � 2 (4J1 � 4J2), is regarded as a driving force for pairformation. This is the simplest explanation of mag-netic models. However, effective Coulombic repul-sion (denoted as I) is also operative for electronpairs. Moreover, the Cooper pair for superconduc-tivity is defined in k-space. Therefore, the I and Jvalues should be transformed into Fourier compo-nents like in Eq. (13). Types of Cooper pairs [d orp(f)] in k-space are determined by the conditionwhether AF(J 0) or F(J � 0) component becomesstronger than the corresponding electron repulsionterm However, the reliable calculations of their rel-

FIGURE 6. (A) AF spin structure I, (B) stripe-type AFspin structure II, and (C) and (D) noncollinear spinstructures for FeAs sheet.

FIGURE 7. (A) magnetic interaction model (Jeff), (B)destruction of J network by one-electron (or hole) dop-ing, and (C) pair formation via magnetic interaction.

YAMAGUCHI ET AL.


ative contributions are very difficult as discussed inSection 3.3.

4. Ginzburg–Landau Model VIA StrongEffective Exchange Interaction

4.1. GINZBURG–LANDAU MODEL FORSECOND-ORDER PHASE TRANSITION

After the discovery of LaOMPn-type supercon-ductors, a lot of experimental and theoretical stud-ies have already been carried out to elucidate pos-sible mechanism(s) of high-Tc superconductivity.However, the true microscopic mechanism has notbeen settled yet as shown in Section 3.3 since exactestimations of several interaction parameters forthe multi-band systems are not so easy, thoughnon-BCS (electron-phonon) magnetic theories havebeen possible and interesting candidates amongvarious theories as shown in Section 3.4. Veryrecent specific heat measurements [72] onLa(O1�xFx)FeAs have concluded a nonlinear mag-netic field dependence of the electronic specific heatcoefficient �(H), which is consistent with the pre-diction for a nodal superconductor: for example,the gap � � 3.4 � 0.5 meV has been obtained on thebasis of the d-wave assumption. The point contactspectroscopy [73] for junctions between a normalmetal and superconductor La(O1�xFx)FeAs haveshown a zero-bias conductance peak whose shapeand magnitude indicate the existence of Andreevbound states. This observation has provided astrong evidence of a nodal gap superconductivitysuch as p-wave or d-wave. Thus several experi-ments suggest non-BCS (electron–phonon) mecha-nisms proposed theoretically. The angle integratedphotoemission study [74] of Sm(O1�xFx)FeAs hasalso demonstrated the existence of large ungappedregions in the Brillouin zone, suggesting unconven-tional superconductivity. Therefore available ex-perimental results are not inconsistent with nonBCS picture, suggesting a magnetic mechanism inSection 3.4.

The Ginzburg–Landau (GL) model [75] can bederived for any kind of magnetic mechanism ofhigh-Tc superconductivity since magnetic field ex-periments are easily analyzed on this model. Thefree energy of nonuniform superconductors underthe magnetic field H is given by the GL model as[75, 76]

GSC � Gn � ��2 ��

2 ��4 �1

2m* �� i�� e*c A��2

��rotA�2

8�, �H � rotA� (15)

where � denotes the order parameter and is de-fined by the transition temperature

� ��T � Tc�. (16)

For magnetic materials, � is taken to be the mag-netization m, whereas it becomes density n in thecase of gas–liquid phase transition. In our magneticmodel for these phase transitions, Tc is expressedby effective exchange interactions (J) calculated byCI total energies in Eqs. (7)–(9) [15]

Tc/kB � const � �JJ� (17)

where const. is a phenomenological parameterwhich is determined by lattice dimensionality (e.g.,const � 0 for pure 1D) and other factors such asinterlayer coupling, and J and J� are the effectiveexchange integrals in the two-dimensional (2D) lat-tice, where pairing energy of holes (electrons) isgiven by J as illustrated Figure 7. It is noteworthythat our Jab values are determined by the first-principle calculations of model clusters, and there-fore reliable even in the intermediate correlationregime.

LaOMPn, cupurates, and related molecule-basedsuperconductors are characterized as the type IIsuperconductors. The critical external magneticfield (Hc) for destruction of superconductivity isdetermined by two parameters as

Hc

8��

2

2�(18)

The GL equations are obtained by derivations ofGSC with � and A as

� � ��2� �1

2m* � � i�� e*c A�2

� � 0 (19)

n � � � i�� e*c A� � 0 (20a)

�A � �4�

c js (20b)



js � �ie*h2m* ��

e*2

m*c ��2A (20c)

where js is the current density. The GL Eq. (19) isreduced to the linear form under the assumptionthat A � 0 and � is small. The solution of the linearequation provides the coherence length �, which iscrucial for estimation of pairing mechanism,

� ��2

2m*�� . (21)

The penetration length 0 of the magnetic field iscalculated by Eqs. (20b) and (20c), giving the fol-lowing relation

0 �m*c2�

4�e*2�� (22)

Then the GL parameter � is defined as the secondimportant criterion as

� � 0

�. (23)

Indeed the �-values for heavy fermion, cupu-rates, and molecule-based superconductors areusually larger than 1/�2 . For example, k is in theorder of 102 for cuprates such as YBCO. This meansthat spatial structures such as intrinsic inhomoge-neity [77, 78] can be realized even in the supercon-ducting state because of short coherence length (�)in case of cupurates (� � 20 Å) and related species.

Abrikosov and Eksperim [76] have solved the GLequations in Eq. (16) to elucidate the vortex state ofsuperconductors and vortex lattice such as squareplaner and triangular ones. Thus the gauge field (A)also induces spatial structures. The lower and up-per critical magnetic fields for the state are obtainedby

Hc1 ��c

2 0*e* log� (24)

Hc2 � �2�Hc (25)

where Hc is given by Eq. (18). However, this simplepicture suffers several anomalies in the case of cu-prates. The mixed state of superconducting andvortex states are realized in the region: Hc1 � H �Hc2, but the situation is really complex in cuprates

because of several reasons discussed later. Thecomplex behaviors in the mixed state of cupurateshave been considered to be closely related to shortcoherence length (� � 20 Å), namely large J in ourmodel [see Eq. (17)]. Therefore thermal fluctuationeffect for the mean field theory is very serious forthe species like for low-dimensional magnetic ma-terials [53].

Very recently, several experiments [79–82] haveshown that � values in LaOFeAs family in Table IIare 35–80 Å in the ab plane but 8 Å in the c-direction. This data indicate that the 2D character ofLaOFeAs family. The in-plane � values locate inbetween cuprates and organic superconductors(BEDT-TTF group in part III [35]), supporting the-oretical proposals based on strong electron correla-tions such as magnetic excitations. The LaOFeAsfamily seem very interesting even from the scien-tific view point since medium coherence lengthdoes not suffer the disorder problem in cuprates,but electron correlation is still crucial for supercon-ductivity.

5. Possible Models forSuperconductivity in IntermediateCorrelation Regime

5.1. POSSIBLE MECHANISM FROM THEWEAK CORRELATION LIMIT

Square planer lattice is common for cuprates,LaOMPn, and Sr2RuO4. The Hubbard model [8] forsquare lattice involving parameters determined bythe first principle calculations is one of the startingpoints for elucidation of the electronic mechanismof superconductivity [15], which can be treated byseveral quantum statistical methods. However, theexact estimation of Ueff is still very difficult. Forsimple theoretical treatment of magnetic and super-conductivity fluctuations, electron-repulsion effectsarising from Ueff are usually assumed to be weak inthe case of metallic view points for over-dopedcuprates [20, 22] (see also supplementary materi-als). Therefore Ueff may be regarded as a perturba-tion to metallic states, leading to the perturbationexpansion of Ueff: note that J is essentially �t in thisregime as shown in Eq. (9). A lot of theoreticalstudies such as spin fluctuation (SF)[9, 10, 22], spinbag [17, 19], FLEX [22], and SCR [24] calculationshave been performed to elucidate possible mecha-nisms of the high-Tc superconductivity on the basisof this assumption. In these theories, the general-

YAMAGUCHI ET AL.


ized Eliashberg equation for the superconducting-gap � is given by [29, 33]

��k,i�n� � �1�

�n�

�k�

Veff�k � k�,i�n � i�n��

� G�k�,i�n��G� � k�, � i�n��k�,i�n�� (26)

where G�k�,i�n� denotes the Green function [28, 83].The effective interaction Veff is not the electron–phonon interaction in the classical BCS model [2],but the electronic origin [9–35] given by the pertur-bation expansion by Ueff [see Eq. (9)] as

Veff�k � k�,0� � Ueff � Ueff�0

�32 Ueff

2 �0� 11 � Ueff�0

� 1��

12 Ueff

2 �0� 11 � Ueff�0

� 1� � . . . . . . (27)

where third and fourth terms denote the spin andcharge fluctuation effects, and �0 is susceptibility at

Ueff � 0. The first repulsive term in Eq. (27) pre-vents conventional s-wave BCS-type superconduc-tivity. Very recent cluster DFT calculations [67–69]have indicated a predominant role of the thirdterm, supporting the spin fluctuation model.

Since Veff is positive in the case of Coulombicinteraction, the gap function ��q � k � k�,0�should change the sign on the fermi surface, givingrise to the non s-wave superconductivity such asthe dx2 � y2 wave function for cuprates [33, 53].Such a picture remains correct even if the higher-order terms in Eq. (27) are taken into account [84].The high Tc superconductivity is expected to beresponsible for strong spin fluctuations (namelylarge �0) in the k-space [22].

The phase diagram obtained by various spinfluctuation theories [53] and perturbation expan-sions [84] of Ueff is qualitatively depicted in Figure8(A). According to Yamada [84], the triplet super-conductivity of Sr2RuO4 cannot be explained by thesecond-order (FLEX) theory, though third-orderperturbation theory works well even for the spe-cies. The key point is that spin (SF)-fluctuation is

FIGURE 8. Possible electronic phases after carrier doping to strongly correlated systems; (A) and (B) phase dia-grams for cuprates with disorder, which are described by the boson-fermion (BF) model, (C) coexistence state ofAFM and SC for cuprates without disorder, (D) phase diagram for electron-doped cuprates and several organic sys-tems, where the first-order transition appears.



not so effective for Sr2RuO4, leading to a electron-correlation (EC) mechnism [35, 84, 85]. It is note-worthy that the magnitude of antiferromagnetic Jvalues by the spin-polarized UHF is often too smallas compared with the experiments, indicating thenecessity of correlation corrections by UMP andUCC treatments in Eq. (7). This implies that ECeffects are really important for quantitative under-standing of superconductivity in cuprates and re-lated species such as LaOMPn.

The phase diagrams in the underdoped regimehas been investigated by using the spin fluctuation-exchange (FLEX) approximation. However, FLEXdoes not include phase fluctuations in the regime.So, Bennemann and his collaborators [26, 30, 31, 32]have improved FLEX by including phase fluctua-tions [86] of Cooper pair. According to this modi-fied FLEX (MFLEX) calculations, three differenttemperatures resulted: (1) T*, where the density ofstates at the Fermi energy starts to be suppressed,(2) Tc*, where preformed Cooper pair is formed,and (3) Tc, where superconductivity is realized be-cause of phase coherence of Cooper pair. The originof Cooper-pair phase fluctuations [86] is assumedto be due to the occurrence of vortices, which istreated within Kosterilitz–Thouless (KT) theory [87,88]. These results are consistent with available ex-periments for hole-doped cuprates, though MFLEXstill fails to treat the on-set region of Tc, namely theMott transition region and AF insulator [4–9]. TheSC fluctuation responsible for pseudo gap has beeninvestigated by the FLEX plus T-matrix theory bythe use of the Hubbard model [84]. Since the T-matrix theory is also based on the perturbationexpansion with respect to the SC fluctuation, it ishardly applicable to the critical region. These situ-ations are also similar for LaOFeAs in the border ofthe metal–insulator transition.

5.2. POSSIBLE MECHANISM FROM THESTRONG CORRELATION LIMIT

Theoretical approaches from AF side are alsofeasible for cuprates. The carrier-doped LaOFeAs isalso considered to be AF under the condition thatcarrier concentration does not exceeds a certainlimit. The two-band Hubbard (d-p) model in Eq.(10) can be reduced to the Heisenberg model re-sulted by the t/Ueff expansion as shown in Eq. (9).The t-J model is resulted at the strong correlationlimit (U3 �) in Eq. (9). To obtain a mean-fieldsolution of the t-J model, the operator dj� in Eq. (11)[18] is rewritten by introducing spinon and holon

operators under the slave-boson approximation as[11, 14, 21]

dj� � fj�hi� (28)

where fj� and hi� denotes, respectively, the annihila-

tion operator of fermion (spinon) and creation op-erator of boson (holon). The spin-charge separationhas been confirmed for 1D system such as SrCuO2[35]. These exotic particles exist in the confinementstate because of the following condition [21]

hj�hj � �

�

fj�� fj� � 1 (29)

There are a lot of theoretical studies based on thet-J model [18]. However, quantum fluctuation cor-rections for the mean-field slave-boson solution arehardly solved yet. Various theories based on t-Jmodel (for example resonating valence bond(RVB)[14]) have been presented, showing that mag-nitude of J is a useful scaling parameter to estimateTc. The exact diagonalizations of finite clusters onthis model indeed suggest the relation: Tc� J, sup-porting our J model [53] in Table IV. The large Jvalue is responsible for the high-Tc in the model.The phase diagram obtained by several theories inthe strong correlation regime is qualitatively de-picted in Figure 8(B). The on-set region of super-conductivity and magnetic insulator phase are welldescribed by the t-J model [18].

Kivelson et al. [54] have discussed possible rolesof fluctuation stripes in cuprates, which are re-garded as an important bridge between Mott insu-lator [4–8] and the more metallic state [20, 22] atheavy doping. They have examined charge and

TABLE IV ____________________________________Effective exchange integrals (Jab) and estimatedtransition temperature (Tc), using ab initio UHF andJ-model.

SystemJab

(cm�1) Tc (K)Experiment

(K)

CuOCu �1,110 159.6 160 (HgCuO2)NiONi �174 25.0 —CoOCo �89 12.8 5 (NaCoO2)MnOMn �18 2.59 —CuFCu �148 21.3 —NiFNi �43 6.18 —MnFMn �5 0.719 —

YAMAGUCHI ET AL.


spin stripes [35] revealed by STM [89–91] and neu-tron diffraction technique [92, 93]. They have usedthe J value to define the maximum spin flip excita-tion energy in superconducting cuprates as in ourmagnetic (J) model [15]. The J value can be obtainedexperimentally by two-magnon signal of Ramanscattering [54]. Therefore, smooth connection be-tween RVB (t-J) and SF models in the intermediateregime is a remaining difficult problem.

5.3. POSSIBLE MECHANISM FROM THEINTERMEDIATE CORRELATION REGIME

The ab initio and Hubbard-model calculations[15, 94, 95] of cupurates indicate that these specieslie in the intermediate regime between the weakand strong correlation limits (see supplementarymaterials). Unfortunately, neither the weak-cou-pling nor the strong-coupling approaches is welljustified in this regime [54], though recent develop-ments in this direction [67, 68] are remarkable. Thenone of possible phenomenological theories is to ex-trapolate results beyond their regime of validity.We have constructed a phenomenological phasediagram for qualitative purpose, which exhibitscharacteristics even at both limits [28, 33, 35]. Tothis end, available experimental results [96–99]have been used as several key constraints.

1. The carrier concentration for superconductiv-ity is expressed by doping concentration (x) inthe under-doped region, whereas it is givenby (1 � x) in the over-doped region. The su-perfluid density (ns) increases with the in-creasing doping (x) and is proportional to Tc

in the former region.2. The superconductivity energy gap (�) is not

so different for many cuprates in the under-doped region [100]: Tc is parallel to x in theunder-doped region, though the maximum Tc

becomes different for them because of severalreasons such as disorder: �/kBTc 3.54 .

3. The angle-resolved-photoemission spectros-copy (ALPES) experiments [92, 93] indicatethe continuous generation of the Fermi sur-face from Fermi point through Fermi arc withthe increase of doping concentration (x).

4. The pseudo gap (PG) appears in the undopedregion because of preformed pair formation.The ratio between inter- and intraplane resist-ibilities increase sharply with the decrease of

temperature (T) to the PG temperature (TPG),indicating importance of c-axis tunneling.

The slave-boson model [11, 14, 21] would be apossible route to understand the experiments fromthe AF side. However, to explain (3) and (4), theholon in the slave-boson t-J model may be replacedwith the Cooper pair (boson) with a short coherencelength (�) in Figure 7, and spinon can be regardedas true fermion (hole or electron) in the spin fluc-tuation (SF) model [35]. This picture may be respon-sible for two component models [101]. This picturealso leads to the so-called boson-fermion (BF)model proposed by Ranninger and coworkers [102–104] as discussed in part III [35]. However, thebinding energy for holes (electrons) arises from thestrong effective exchange interaction ( J ) in mag-netic model at the strong correlation region. Al-though it is assumed to be the spin fluctuation inthe weak correlation regime, the impurity effects byZn(II) and Ni(II) substitutions in CuO2 plane sup-port an important role of magnetic model [105],though cooperation of electron-phonon interactionscannot be neglected [35, 106–109]. The BF modelmay be responsible for intermediate region be-tween under- and over-doped domains of cuprateswith disorder.

We may obtain the phase diagram based on themagnetic (magnetic excitation) model, where mag-non excitation energy (J) is taken as a scaling pa-rameter though its reading terms are variable ac-cording to doping concentration of hole (orelectron) as shown in Eq. (9). The antiferromagnetic(AF) long-range order is realized because of large� J � for cuprates without holes (carrier) (x) but withweak three-dimension (3D) character. However, theAF state is destroyed with the increase of x as

TAF/J � const.�AF��xAF � x� (30)

where xAF denotes the critical concentration for AF.Here, J is the energy scale of a characteristic micro-scopic coupling in the magnetic theory, and weassume the linear variation of Tx with x like Kos-taliz-Thouless (KS) transition in 2D plane: kBTKT �const x V0 (coupling parameter in 2D XY model) �const x ns (superfluid density) [87, 88]. This as-sumption does not entail any problem for qualita-tive purpose since cuprates have 2D CuO2 planescoupled weakly. The spin glass state in LSCO andYBCO (see Fig. 8) is replaced by AF metal in themultilayered cuprates (see Fig. 9), where disorder



and other factors to stabilize spin glass phase isremoved out. The Eq. (30) is also applicable to AFmetal without disorder.

There are several kinds of pseudo gaps (PG).Here we focus attention to the pseudo gap (PG)arising from the formation of preformed Cooperpair like in the t-J model in Figure 8(A). Therefore,the PG temperature is decreased with x because thebinding energy is also weakened

TPG/J � const.�PG��xPG � x� (31)

where xPG denotes the hole concentration. TPG be-comes equivalent to TSC at xPG (note that pseudogap responsible for charge fluctuation is out of ourconcern here). On the other hand, the Bose–Einsteincondensation (BEC) temperature for the preformedCooper pair is dependent on the hole concentrationin the under doped region as

TBEC/J � const.�BEC��x � xBEC� (32)

This relation could be rationalized by the as-sumption of Kostalitz-Thouless theory [87, 88] and

experimental results (1) and (2). However, we can-not distinguish the BEC state from percolated SCstate in the case of cuprates with inhomogenity,disorder, and other origins [77, 78]. Therefore, con-struction of true microscopic theories is not easy inthis regime. The pseudo gap (Ecross) increasing withx observed for LACO and YBCO by the neutrondiffraction technique might be related to the con-centration of BEC.

On the other hand, the superconducting (SC)state is closely related to the stability of Fermi sur-face in the over-doped region: namely, the elec-tronic conductivity becomes the maximum becauseof its stability, but the pairing force (spin fluctua-tion) for SC in turn becomes weak in this region.This supports the following equation given by theSF theory in Figure 8(B).

TBCS/J � const.�BCS��xBCS � x� (33)

where xBCS denotes the critical concentration ofhole. The above BEC and BCS conditions for cross-over phenomena should be satisfied in the phase-coherent superconducting (SC) state, leading to thefollowing relation responsible for the weak cou-pling between the copper–oxygen planes along thec-axis.

TSC/J � const.�SC��xBCS � x��x � xBEC� (34)

Both BEC-type attractive interaction via J andBCS-type interaction via SF are assumed to be co-operative in this theoretical model

V�total� � CiJ�BEC� � C2

32 Ueff

2 �s�SF��BCS� (35)

where �s denotes the spin susceptibility and theirweight Ci are variable, depending on the magnitudeof doping. We can depict possible phase diagramsby using Eqs. (30)–(34) as shown in Figure 8. As canbe recognized from Eq. (8), both t and Ueff playroles from weak to strong through intermediatecorrelation regime. The magnetic excitation energy(J) by the first principle calculations is a practicalmeasure of Tc as shown in Table IV. The electron-correlation (EC) mechanism including charge-fluc-tuation (part I [53]) may be considered to be trueorigin in the intermediate regime [15], though J isused for the effective parameter in our model.

Judging from many experimental results for cu-purates, the critical concentrations of AF, PG, SC,

FIGURE 9. The AF, SC, and (AF � SC) phases of cu-prates with five CuO2 planes; (A) two-band model with-out AF, (B) coexistence of AF and SC in a unit cell, (C)coexistence of AF and SC in a outer plane (OP), and(D) itinerant AF and SC state. The diagrams (C) and (D)are qualitatively understood by the t-J (A) and SF(B)models in Figure 8, respectively. Then the magneticmodel is applicable for all the cases.

YAMAGUCHI ET AL.


and BEC are highly dependent on types of cupu-rates : La (Tc � 38 K)-, Y (Tc � 90 K)-, Bi (Tc � 110K)-, Tl (Tc � 125 K)-, and Hg (Tc � 150 K)-systems[54], where the maximum Tc is given in parenthe-ses. Figure 8 illustrates several phase diagrams withdifferent critical concentrations, which are probablyrelated to defect, disorder and other effects such asthe many-band effects. For example, the spin glassphase [54, 85] appears in the under-doped region ofLa2�xSrxCuO4 because of disorder as shown in Fig-ure 8(A). On the other hand, the first-order transi-tion might be observed even for cupurates withoutdefect like in the case of �-phase of the (BEDT-TTF)2X systems [106] as illustrated in Figure 8(D).The stripe and other local structures observed in theCuO2 phase by STM and STS [77, 78] might berelated to the coexistence of several electronicphases (namely characteristics of intermediate cor-relation regime) in Figure 8(B). The phase diagramfor cuprates with five CuO2 planes in Figure 8(C)may be effective to consider possible phases of elec-tron-doped LaOMPn family.

Thus Tc for SC is dependent on disorderand/or defect which are related to the short co-herence length (�)[41, 42]. Complex behaviors ob-served for cuprates are explained by strong effec-tive exchange interaction (� J � �� 0), like clusterformations of water with strong hydrogen bond-ing. The inhomogenuity observed by STM andSTS [77, 78] could be ascribed to strong electroncorrelation and short �. The magnetic field effectsfor SC given by the GL model in Section 4 [35] arealso important windows to elucidate its charac-teristics as shown in Figure 8(A). For example,the resistibility (�) measurements under the mag-netic field have been performed extensively,showing that preformed pair is formed in thepseudogap regime. The similar experiments havealso revealed that the resistibility of underdopedcuprates is semiconductive (d�/dT � 0), thoughthe superconductivity is observed even in thisregime. Although it becomes metallic (d�/dT 0)if the doping concentration exceeds a certain limit[109]. This may support the phase diagram inFigure 8, where the concentration at TPG � TSC isregarded as the BEC-BCS crossover point in partIII [35].

The phase diagrams in Figure 8 suggest possibleanalogs for hole-doped LaOMPn. The low spin state(S � 1/2) of Mn(II) and Co(II) in LaOMPn (M � Mnand Co) is equivalent to those of Fe(III) and Ni(III) infully hole-doped putative SrOMPn (M � Fe and Ni)and Cu(II) in undoped cuprates. These transition

metal compounds are usually magnetic like in AFLaOMnAs (high spin Mn), indicating that many vari-ations are conceivable on the basis of appropriatecombinations of component atoms to obtain ferro-magnetic, noncollinear magnetic (see Fig. 6), and non-magnetic heavy fermion states. A lot of physicochem-ical experiments on hole- or electron-doped thesespecies may elucidate interrelationships betweenmagnetism and high-Tc superconductivity.

6. Coexistence of Magnetism andSuperconductivity

6.1. THEORY OF COEXISTENCEMECHANISMS OF MAGNETISM ANDSUPERCONDUCTIVITY

Coexistence of antiferromagnetism and super-conductivity has been proposed for cuprates as wellas newly found LaOFeAs family (see Section 3).This in turn indicates common mechanisms mightbe operative for both species, suggesting an impor-tant role of comparison between them. The experi-ments on cuprates with five CuO2 planes [110–113]clearly demonstrated the coexistence of AF and SCeven in a single plane. The free energy of the Ginz-burg–Landau (GL) model can be expressed by fiveparameters for coexistence of antiferromagnetism(AF) and d-wave superconductivity (SC), leading toa conventional SO(3)xU(1) symmetry group. Zhang[114, 115] has proposed a more flexible SO(5) modelwith 10 freedoms, which incorporates the connec-tion between AF and SC observed in cuprates. Bur-gess et al. [116] have performed a microscopic der-ivation of the SO(5) symmetry from a microscopicpicture of electron dynamics. The AF and SC existnear each other in many underdoped cuprates. Thecompetition between AF and SC phases is deter-mined by the low-energy content of the Hubbardand t-J models of order of the gap energy � �� J/10.Zhang proposed a unified theory based on theSO(5) symmetry of SC and AF, in which the threecomponents of a spin and the real and imaginaryparts of the SC order parameter compose a fivecomponent superspin. Hu et al. [117] have investi-gated the following classical SO(5) Hamiltonian onthe simple cubic lattice

H � J �ij

cos�icos�jcos��i � �j� � �ij

sin�isin�jSi � Sj

� g �i

sin2�i (36)



where in their definition, J 0 and Si2 � 1. The first

term is for the SC order parameters, where �’s arethe phase variables, and the second term is for theAF components. The g-field is a renormalizedchemical potential which plays the role of a sym-metry-breaking field. They have elucidated thephase diagrams of normal, AF, SC, separationphase of AF and SC and coexistence phase of AFand SC (see Figs. 8 and 9). Thus J is a commonparameter describing several phases of cuprates inunder-doped region.

Nagao et al. [29] have extended the SO(5) modelfor coexisting states of ferromagnetism and tripletsuperconductivity. The coexistence of ferromag-netism and triplet superconductivity has been ob-served in UGe2 [118] and UReGe3 [119], thoughorganic analogs have not been observed yet. Real-ization of p- and/or f-wave superconductivity is avery interesting problem in relation to new direc-tion of molecular magnetic materials [120, 121].Similarly, Co and Ni analogs in few LaOMPn fam-ily are very interesting from the view point of co-existence of ferromagnetism and superconductiv-ity.

Demler et al. [122] have summarized varioustheoretical aspects of a quantum SO(5) nonlinear �model with anisotropic couplings. The GL free en-ergy is defined by two competing order parameters(AF and SC) [123, 124] as

F � 1�12 � 2�2

2 �12 �1�1

4 �12 �2�2

4 � �12�12�2

2

(37)

where 1 and 2 are responsible for AF and SC, re-spectively. These order parameters are determinedby minimizing the free energy F. The SO(5) theory[122] have revealed three possible transitions: (1)direct first-order phase transition between AF andSC as a function of the chemical potential [see Fig.8(D)], (2) first-order AF to SC transition as a func-tion of doping, where the phase separation appearsin the intermediate region, and (3) two second-order phase transitions with a uniform AF and SCmixing phase in between. The coexistence of AFand SC in a single CuO2 plane was really discov-ered very recently [110–113]. The same situationappears in multi-layer unit cells. The LaOMPn fam-ily is also interesting from the view point of coex-istence of magnetism and superconductivity.

6.2. EXPERIMENTAL EVIDENCE OFCOEXISTENCE OF AF AND SC STATES INCUPRATES WITH MULTILAYERS

In relation to newly found LaOFeAs family, cu-prates with multilayers have received a renewedinterest in relation to the coexistence of AF and SC.The interlayer hopping interaction (tinter) in the bi-layer cuprates, Bi2Ba2CaCu2Oy, has been observedby ARPES [125, 126]. The splitting of Fermi surfaceis clear in the overdoped regime of the species,though it is not observed in the underdoped region[119, 122]. The interlayer hopping has the disper-sion relation [127] as

�inter�k� � ��tinter/4��coskx � cosky)2 (38)

where tinter is taken to be the order of J [see Eq. (9)][128]. The situation is similar even in the trilayercuprates (n � 3) since the next nearest neighborlayer interaction is negligible. However, it becomescomplex for n � 3 because of two kinds of planesare realized.

Very recently Kitaoka and coworkers [110–113]have performed the NMR experiments of several cu-prates with five CuO2 planes as illustrated in Figure 9.Tokunaga et al. [110] have reported Cu-NMR studieson (Cui�xCx)Ba2Ca3Cu4O12�y (Cu1234) which com-prises two innerplanes (IP) and the outerplanes (OP).The SC gap in the underdoped IP fully developsbelow Tc � 117 K, but that in the heavily overdopedOPs increase linearly down to Tc2 � 60 K below Tc.The spin-gap behavior is observed in the IP but not inthe OP. Similarly, the outer CuO2 plane is found to beover-doped (OVD) (Tc � 65 K) in the Cu1245,whereas the inner three CuO2 planes are optimal-doped (ODP) (Tc � 90 K) as shown in Figure 9(A).However, the gap functions indicate the one Tc asillustrated in Figure 10(B) because of the intermediateinterband interaction in the case of Cu1245. Thismeans that at least two different Fermi surfaces ofthese systems should be observed by ARPES.

Kotegawa et al. [111] have demonstrated a coex-istence of SC and AF in the five-layered unit cell inoptimally doped HgBa2Ca4Cu5Oy (Hg1245-OPT). Itis composed of undoped three IPs in the AF (TN �60 K) state with the Cu moment of (0.3�0.4) �B andoptimally doped two OPs in the SC state (Tc � 108K) state, which exhibits 0.02 �B on Cu via the AFspin polarization by IPs [see Fig. 9(B)]. Mukuda etal. [113] have reported the Cu-NMR study on un-derdoped HgBa2Ca4Cu5Oy (Hg1245-UD), whichhas three AF IPs (TN � 290 K) with large AF mo-

YAMAGUCHI ET AL.


ment of 0.67�0.69 �B. The AF order is also detectedwith MAF(OP) � 0.1 �B even at two OPs, which areSC with Tc � 72 K [Fig. 9(C)]. The AF and SC canuniformly coexist on a microscopic level in thissystem. The phase diagram of Tl1245 in Figure 9(D)is similar to that of Hg1245-OPT in Figure 9(B),though the magnetic moment is very small in theinner planes.

Thus, the phase diagrams in Figure 9 clearlyshow the existence of the uniform mixed phase ofSC and AF without any vortex lattice and stripeorder. This in turn implies that complex behaviorsof cuprates with single or double CuO2 plane areclosely related to its low dimensionality and severaldisorders. In fact, spin glass phase, and spin andcharge stripe phases in the plane may be reduced tothe coexistence state of AF and SC in the case ofmulti-layer cuprates without defect. The phase di-agrams in Figure 9 are qualitatively explained bythe aforementioned SO(5) models by the use of t-Jmodel Hamiltonian, particularly for Hg1245-UDwith large AF moment of 0.67�0.69 �B in the unitcell. On the other hand, the SCR theory [22] by theuse of Hubbard model with well-renormalized Uvalue is rather appropriate for Tl1245 with antifer-romagnetic metal (AFM) phase with very small (0.1�B) magnetic moment in the unit cell.

7. Ginzburg–Landau Model forMultiband Effects for Cuprates andLaOFePn

Magneto-resistance is a very powerful tool toinvestigate the electronic scattering process and thestructures of the Fermi surface. For example, a largemagnetoresistance (MR) was found for MgB2 [129–

131], which is closely related to the multiband prop-erty. Similarly, MR experiments [79–81] on newlyfound LaOFeAs have also indicated a possibility ofmultiband effect even for this family. A single bandmetal with a symmetric Fermi surface should ex-hibit the Kohler’s low, which predicts that MR ef-fect measured at different temperatures should bescalable with the variable H/resitivity. For MgB2,the Kohler’s law was not obeyed because of themultiband property. The same situation was foundfor La(O1�xFx)FePn. Very recently, very high fieldresistance measurements up to 45 T [82] was carriedout to demonstrate the multiband effect. Indeed,the observed deduced critical magnetic field Bc2(0) � 63–65 T exceeds the paramagnetic limit in Eq.(25), consistent with strong coupling and importanttwo-band effects for La(O1�xFx)FePn. Thus multi-band effects are common characteristics for MgB2,LaOFeAs, and cuprates, though relatively longercoherence length for LaOFeAs may not entail thecomplex behavior in superconductivity. This is con-sistent with the multi-band theories in Section 3.

As is well known, cuprates with multilayers[110–113] indicate that electronic states of the spe-cies are variable from the AF insulator (AFI) toparamagnetic metal (PMM) through antiferromag-netic metal (AFM), coexistence of AFM and d-waveSC, and SC states unless disorder, defect, and otherstructural factors are not operative. Judging fromthese experiments, the BEC(or RVB)-BCS crossoverpicture [or boson-fermion (BF) model] presented inPart III [35] is therefore regarded as one of effectivetheoretical models for cuprates with disorderand/or cuprates under random potentials likeAnderson localization [132, 133]. Under the situa-tion, bond alternation and phonon modes wouldplay multiple roles to stabilize several competingphases as revealed by ARPES. The boson-fermion(BF) model also works well in the underdoped re-gion since ARPES for CuO2 planes have shown thatfree carriers (fermion) are first generated in the(�/2, �/2) region, and then they enter into themagnetic (�, 0) region, where preformed pair (bo-son) is formed via the coupling with magnetic in-teraction (J)[see also Eq. (35)]. The pseudo gap (PG)is still observed in the underdoped CuO2 planeseven in the multi-layer cuprates [110–113]. Theseresults indicate that detailed experiments by chang-ing dopant contribution in LaOFeAs systems arealso crucial for comparison with cuprates.

N-bands Hubbard model [33, 35, 53] is a naturalstarting point for cuprates with N-layers withoutdefect under the assumption that each layer can be

FIGURE 10. The SC gaps for multi-band systems; (A)uncoupled case, (B) intermediate-coupled case, and (C)strongly coupled case. The GL models for these casesare constructed (see text). The two gaps and two Fermisurfaces are observed in these multi-layer systems.



grasped by one-band Hubbard model. However,exact diagonalization of the N-band models becomemore and more difficult because of interlayer cou-plings. So, the GL model in Eq. (15) can be gener-alized for N-bands systems as

F � �j

�j�T � Tj��j2 �

12 �j�j

4 � �j, j�1�j�j�1

� �j, j�1�j2�j�1

2 (39)

where j denotes the j-th band, and third term isadded to retain the phase coherence in a unit cell.The last term express as the coupling effect. Thecoupling terms are semiempirically determined soas to reproduce the experiments as shown in Eqs.(30)–(34). To this end, the interplane coupling (tinter)is also expressed by the interplane J value in ourmagnetic (J) model. The generalized GL model canbe also applicable to the SC state of MgB2 [134, 135]for which one Tc has been observed because ofspecific condition, though two gaps are also de-tected as shown in Figure 10. Two gaps behaviorshave been observed for several cuprates with multi-layers by ALPES [136]. Judging from available ex-periments for LaOFePn, the multiband GL modelcan be applicable for this family.

8. Discussions and ConcludingRemarks

8.1. UNIFIED PICTURE OFUNCONVENTIONAL SUPERCONDUCTIVITY

Recent experiments on the multi-layer cuprates[110–113] have clearly demonstrated that spin-free-dom plays an essential role in both antiferromag-netism (AF) and superconductivity (SC) since thecoexistence of AF and SC is indeed realized in oneCuO2 sheet with appropriate doping. Therefore,magnetic theories via strong electron correlations inthis series of articles [15, 33, 35, 53] are reasonable atleast as guiding principles for theoretical under-standing of both AF and SC. However, instead ofthe boson-fermion (BF) model in part III [35], theSO(5) theory for AF and SC [122] is applicable as afirst step to uniform multi-layer cuprates withoutdisorder. Both real space and k-space descriptionsof the freedom are useful for cuprates since thecopper oxygen bonds are regarded as typical chem-ical bonds in the intermediate correlation regime

from the view point of instability in chemical bonds[53]. Since reliable theories in this regime are notestablished yet in the solid state physics, phenom-enological phase diagrams are constructed semiem-pirically as shown in Figure 8. The experiments onmulti-layers cuprates suggest that the maximumtemperature 133 K is realized in the three layer (n �3–4) with a uniform hole concentration. This in turnindicates that multiband effects can be expectedeven for systems with n � 4 layers if optimumdoping is realized. However, the reported experi-mental results indicate that such effective controlsof doping concentration on each layer are not soeasy. We may expect much more higher Tc after theoptimal control of all the layers in cuprates.

Very recently, ARPES experiments [136] haverevealed that electron- and hole-doped Fermi sur-faces exist in F-substituted four-layered cuprates(n � 4) without formal doping. The multi-layerstructures constructed of alternating hole and elec-tron doped superconductors would be desirable tocontrol doping concentration and multi-layer stack-ing structures. Recent experiments on multi-layercuprates [110–113] indicate that cuprates, heavyfermion systems, related organic systems [120, 121]and newly found LaOFeAs family may be under-stood in a unified picture based on intermediate orstrong electron correlations, which are main originsof both magnetism and superconductivity. In thissense, quantitative theories of electron correlationsdeveloped in Figure 4 would be promising even inthe intersection area of material science and quan-tum chemistry. Extensions of first principle nonper-turbative methods [137–142] to strongly correlatedsystems are desirable for developments of micro-scopic theories to rationalize the phase diagrams inFigures 8 and 9.

8.2. MOLECULAR DESIGN OF NEW p-d,�-d, �-R, AND �-R CONJUGATED SYSTEMS

Previously [143, 144], we have examined possi-bilities of 1D and 2D lattices constructed of iron–sulfur (Fe-S) complexes with tetrahedral ligandfields, which could become triangular spin latticesresponsible for spin flustrations. In part I [53] of thisseries, we have pointed out that CuO unit in cu-prates is isoelectronic to 4B(C)-3B(B) or 4B(C)-5B(N)unit in periodic tables. The boron-doped diamondsuperconductor is one of the 4B(C)-3B(B) systems.The discovery of LaOFeAs family [44] suggest anew avenue to high-Tc superconductivity using tet-rahedral lattice constructed of multilayers of iso-

YAMAGUCHI ET AL.


electronic p-d, �-d, �-R, and �-R conjugated sys-tems. The diamond structures are regarded asstating point as illustrated in Figure 11(A). Substi-tutions of one carbon (4B group) tetragonal sheetwith another components such as boron (3B) ornitrogen (5B) groups provide electron deficient orrich 2D sheet, respectively, as illustrated in Figures11(B) and (C). The carrier doping into these sheetswould be accomplished by chemical and physical(for example FET) methods. The self-doping sys-tems are also conceivable by combinations of 3B,4B, and 5B groups as illustrated in Figure 11(D).Search of such light atom systems is interestingfrom cooperation of electron–phonon interactionand electron correlation as discussed in Part III [35].To this end, defect and disorder of crystals of theseisoelectronic systems should be suppressed by mo-lecular design of component molecules (atoms) andcrystal engineering. Possible candidates of suchp-d, �-d, �-R, and �-R conjugated systems fromthis view points will be examined elsewhere.

8.3. NECESSITY OF A UNIFIED THEORY

Accumulated experimental and theoretical re-sults indicate that a unified microscopic theory isrequired for systematic descriptions of severalphases appeared before and after carrier doping instrongly correlated electron systems as illustratedin Figure 12. Stripes and related charge orders arealso appeared in the intermediate correlation re-gime. However, they are not touched here, sinceonly magnetic phases such as SDW and SC ofLaOMPn family, together with cuprates are of ourconcern in part IV. As discussed in this series ofarticles [35], charge fluctuation (CF) as well as elec-tron–phonon (e–p) interaction would play signifi-cant roles to suppress the electron repulsion effect,though the J-parameter in Eq. (9) is used as anempirical parameter in Figures 8 and 9. As dis-

cussed in part III [33], cooperation of CF, e-p, andSF seems effective toward the goal (room tempera-ture superconductivity). Very recently, Nomura etal. [145] have emphasized an important role of co-operation of charge and spin fluctuations induced byelectron-lattice interaction in F-doped LaOFeAs. Sev-eral experimental results for Sm(O1�xFx)FeAs [146]have indicated existence of quantum critical point(QCP) in Figure 12 like in case of heavy fermionsytems.

Here, we did not touch the LaOFeP system.Bruning et al. [147] have experimentally demon-strated that CeOFeP exhibits a heavy fermion me-tallic behavior with ferromagnetic correlations(Kondo temperature � 10 K and m* � 200), thoughLaOFeP is a superconductor (see Table I). It wasfound that the magnetism in CeOFeP is completelydominated by the Ce 4f electrons and the absenceof any contribution to the effective moment. This inturn indicates that roles of f-electrons in high-TcRe(O1�x Fx)FeAs (Re � Gd, Sm, etc.) should be inves-tigated carefully, and (La1�xCex)OFeP is an interest-ing system to examine a heavy fermion to supercon-ductor transition. Very recently, GdO1�xFeAs (Gdwith 4f7) was found to be a high-Tc superconductorwith Tc � 53.5 K. Accumulated results for cuprates,LaOMPn, heavy fermion, and organic systems indi-cate a necessity of a unified theory to understand

FIGURE 11. Possible Diamond-type lattices: (A) un-doped structure, (B) IIIB group substituted structure, (C)VB group substituted structure, and (D) IIIB and VB sub-stituted structure.

T

δ

HeisenbergModel

SF Theory

AFMAFI

t-J model

SF Theory

TN

Quantum critical (QC) line

AFM

AFI

M-Itransition

paramegneticmetal(PMM)

paramegneticInsulator(PMI)

cuprates Multi-layer cupratesLaOFeAs

FIGURE 12. A unified picture of phase diagrams[temperature (T) vs. interaction parameter x) from strongcorrelation to weak correlation regime through the inter-mediate one. The solid-state theory in the intermediatemetal–insulator (M–I) transition region is still insufficient,though numerical calculations of small cluster modelsare feasible by several methods in Figure 4.



whole region depicted in Figure 12, particularly nearthe Mott (metal–insulator) transition point (see dottedline) [148, 149], though the spin-fluctuation theorynear the quantum critical point (QCP) is well estab-lished. In this sense, theoretical studies on molecularmagnets (insulator) and molecular itinerant magnets(metal) are also interesting. We may expect that co-operation of SF, CF, and e–p interactions near M-Itransition discussed in part III [33] become feasible toraise Tc for superconductivity of LaOMPn family,where a lot of combinations of atoms are feasible. It isnoteworthy that several theoretical approaches to ex-amine possible cooperations of these interactions tohigh Tc (goal is over the room temperature) are pre-sented recently [150, 151].

8.4. CONCLUDING REMARKS

Comparison between LaOMPn and cuprates hasbeen performed on both experimental and theoret-ical grounds, for the purpose to elucidate interrela-tionship between magnetism and non BCS super-conductivity. It is found that these families havesimilar characteristics from the view points of elec-tronic structures such as the border of metal–insu-lator transition, intermediate correlation regimeand multi-band systems. These characteristics havebeen employed as key working concepts for mate-rial design in this series of papers [33, 35, 53], whichindeed enable us to propose isoelectronic p-d, �-d,�-R, and �-R conjugated systems. The discovery ofnew high-Tc LaOMPn systems in Tables I and II hasindicated surprising fact that materials constructedof magnetic metals such Fe, Ni, etc. can exhibitsuperconductivity as well as magnetism. Our phe-nomenological magnetic (J) model [15] of phasediagrams from magnetic (Mott) insulator to con-ventional metal through superconductive state isuseful for qualitative understanding of nature ofstrongly correlated electron systems. This in turnsuggest both experimental and theoretical effortsbased on the above concepts are fruitful toward thegoal: a possible rout for the room temperature su-perconductivity different from the exciton modelby Little [1].

Note added in proof

After the submission of this paper, new hole-doped Fe2As2 superconductors, (A1�xKx) Fe2As2,(A � Sr, Ba) were discovered by several groups (seeTable I) [152, 153]. The transition temperature (Tc)

for the species were found to be 38 K. The presentcompounds AFe2As2 with the tetragonal Th Cr2Si2–type structure exhibited the SDW anomaly (itiner-ant magnetism) at 210 K (A � Sr) [152] and 130 K (A� Ba) [153] like the LaOFeAs family, though KFe2As2 was concluded to be a good metal withoutthe SDW anomaly. Very recently spectroscopicstudies on the LaOFeAs family have been carriedout [154–158], providing key constraints for possi-ble theoretical models. The 19F [154], 75As and 139LaNMR [154] in the La (O1�xFx) FeAs system havedemonstrated that the temperature dependence of1/T1 decrease suddenly without showing Hebel-Slichter (coherence) peak, indicating unconven-tional superconductivity with line nodes. Thetemperature-dependent laser photoemission spec-troscopy of the species (x � 0, 11) have elucidatedthe pseudo gap feature arising from the magneticfluctation [156].

The NMR measurements [157] in Pr(O1�xFx)FeAs (Tc � 45 K) (x � 0.11) have concluded that theCooper pair is in the spin singlet with two energygaps opening below Tc, in addition to the nodes inthe gap and the T3-like variation of 1/T1 just belowTc. The high resolution photoemission measure-ments for R(O1�xFx) FeAs (R � La, Ce, and Pr) havealso indicated the pseudo gap [157]. On the otherhand, the NMR of the undoped LaOFeAs [153]have demonstrated an itinerant antiferromagnetism(AFM) with TNE142K. The 57Fe Mossbauer spec-troscopy and muon spin relaxation of the species[158] have further elucidated the strongly reducedordered moment of 0.25 �B at the iron site belowTNE138K. Available spectroscopic results are there-fore not inconsistent with the present magneticmodel of superconductivity of the LaOFeAs andAFe2As2 families. Further discussions will be givenelsewhere.

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