Identifying the Genes of Unconventional High Temperature Superconductors

Jiangping Hu1, 2, 3, ∗

1Beijing National Laboratory for Condensed Matter Physics,and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

2Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA3Collaborative Innovation Center of Quantum Matter, Beijing, China

We elucidate a recently emergent framework in unifying the two families of high temperature (highTc) superconductors, cuprates and iron-based superconductors. The unification suggests that thelatter is simply the counterpart of the former to realize robust extended s-wave pairing symmetries ina square lattice. The unification identifies that the key ingredients (gene) of high Tc superconductorsis a quasi two dimensional electronic environment in which the d-orbitals of cations that participatein strong in-plane couplings to the p-orbitals of anions are isolated near Fermi energy. With this gene,the superexchange magnetic interactions mediated by anions could maximize their contributionsto superconductivity. Creating the gene requires special arrangements between local electronicstructures and crystal lattice structures. The speciality explains why high Tc superconductors areso rare. An explicit prediction is made to realize high Tc superconductivity in Co/Ni-based materialswith a quasi two dimensional hexagonal lattice structure formed by trigonal bipyramidal complexes.

I. INTRODUCTION

Almost three decades ago, the first family of unconven-tional high Tc superconductors, cuprates [1], was discov-ered. The discovery triggered intensive research and hasfundamentally altered the course of modern condensedmatter physics in many different ways. However, even to-day, after tens of thousands of papers devoted to the ma-terials have been published, understanding their super-conducting mechanism remains a major open challenge.Researchers in this field are sharply divided and disagreewith each other on many issues arranging from minimumstarting models to basic physical properties that are rel-evant to the cause of superconductivity. There is even agrowing skepticism whether there are right questions thatcan be asked to settle the debate on the superconductingmechanism.

Many reasons can be attributed to the failure of an-swering the question of how superconductivity arises incuprates. For example, material complexity makes theo-retical modeling difficult, rich physical phenomena blindus from distinguishing main causes from side ones, andinsufficient theoretical methods leave theoretical calcu-lation doubtable. However, beyond all these difficultiesand the absence of consensus, the lack of successfully re-alistic guiding principles to search for new high Tc super-conductors from theoretical studies is the major reason.The failure was witnessed in the surprising discovery ofthe second family of high Tc superconductors, iron-basedsuperconductors[2], in 2008. Today, those who are the-ory builders and those who are material synthesizers stillremain disentangled.

Can valuable leads be provided from the theoreticalside ahead of the potential discovery of the third familyof high Tc superconductors? It is conceivable that the

∗Electronic address: jphu@iphy.ac.cn

hope to settle high Tc mechanism relies on a positive an-swer to this question. Here, we believe that it is the timeto seek a positive answer based on the following two rea-sons. First, in the past seven years, the intensive researchon iron-based superconductors has brought much new in-formation. For those who believe that cuprates and iron-based superconductors should share a common high Tc

mechanism, an opportunity to settle the debate arisesas it is the first time that the traditional inductive rea-soning becomes available in research. On one side, iron-based superconductors and cuprates share many com-mon features, but on the other side they are not clonesof each other. The similarities and differences can thusspeak promising clues. Second, from the past massivesearching efforts, it has become increasingly clear thatunconventional high Tc superconductors are rare mate-rials. Moreover, for the two known families, their su-perconductivities are carried robustly on CuO2 layers incupates and on FeAs/Se layers in iron-based supercon-ductors respectively. The simultaneous existence of therareness and robustness suggests that the unconventionalhigh Tc superconductivity is tied to special ingredients inthe electronic world, which define the gene of unconven-tional high Tc superconductivity. Thus, using inductivereasoning to identify the gene can open a new window tosearch for high Tc superconductors.

In this article, by taking the assumption that acommon superconducting mechanism is shared by bothknown high Tc superconductors, we elucidate a recentlyemergent path to end the deadlock in solving high Tc

mechanism by implementing inductive reasoning to reex-amine the high Tc problem[3, 4]. This path stems froma simple framework that unifies cuprates and iron-basedsuperconductors based on previous understandings in re-pulsive interaction or magnetically driven high Tc mech-anisms. It suggests that iron-based superconductors aresimply the counterpart of cuprates to realize robust ex-tended s-wave pairing symmetries in a square lattice.Both materials share a key ingredient, the gene of un-

arX

iv:1

512.

0919

0v1

[co

nd-m

at.s

upr-

con]

31

Dec

201

5

2

B

A

B

B

B

~3.8Å ~3.8Å

~2.8Å

X

Y

(a) (b) Cu

O As/Se

Fe

FIG. 1: The comparison of the lattice unit cells betweencuprates and iron-based superconductors: (a) The unit celland lattice constant of the CuO2 layer in cuprates; (b) theunit cell and lattice constant of the FeAs/Se layer whichincludes two irons marked as A and B.

conventional high Tc superconductivity: a quasi two di-mensional electronic environment in which the d-orbitalsof cation atoms that participate in strong in-plane cou-plings to the p-orbitals of anion atoms are isolated nearFermi energy. This environment allows the antiferromag-netic (AFM) superexchange couplings mediated throughanions, the source of superconducting pairing, to maxi-mize their contributions to superconductivity. Creatingsuch a gene is tied to special arrangements between localelectronic structures and crystal lattice structures, whichexplains why cuprates and iron-based superconductorsare special and high Tc superconductors are so rare. Theframework can be explicitly tested in future experimentsas it leads to an explicit prediction to realize high Tc

superconductivity in the Co/Ni-based materials with aquasi two dimensional hexagonal lattice structure formedby trigonal bipyramidal (TBP) complexes[3]. The newmaterials are predicted to be high Tc superconductorswith a d ± id pairing symmetry. If verified, the predic-tion will establish powerful guiding principles to searchfor high Tc superconductor candidates, as well as to set-tle the debate on unconventional high Tc superconductingmechanism.

II. QUESTIONS FOR UNCONVENTIONALHIGH Tc SUPERCONDUCTIVITY

Implementing inductive reasoning to understand bothcuprates and iron-based superconductors, we lay outthe high Tc problem with the following three subsequentquestions:

(1) What is the common interaction responsible forhigh Tc superconductivity in both families?

(2) What are the key ingredients to make both familiesspecial to host high Tc superconductivity?

(3) Where and how can we search for new high Tc

superconductors?

The three questions are highly correlated. They forma self-contained unit to reveal high Tc superconductingmechanism.

In the past, the first question was the central ques-tion. Its answer was debated wildly. The second ques-tion was largely ignored. However, after the discoveryof iron-based superconductors, it becomes clearer thatthe second question should be the central piece. Whilemost researches have concentrated on these two fami-lies of high Tc superconductors, it is equally importantto answer why numerous materials, which are similar tocuprates or iron-based superconductors in many differentways, do not exhibit high Tc superconductivity. There-fore, the essential logic here is that whatever our answerto the first question is, the answer must provide an an-swer to the second question. The answer to the secondquestion can provide promising leads to answer the thirdquestion. An explicit theoretical prediction of new highTc superconductors and its experimental verification canfinally justify the answer of the first question to end thedebate on high Tc mechanism.

III. THE ANSATZ TO THE FIRST QUESTION

We start with the first question. Our proposed answerto the first question is that only the superexchangeantiferromagnetic(AFM) interactions mediated throughanions are responsible for generating superconductivityin both families of high Tc superconductors. We callthis ansatz as the selective magnetic pairing rule[4]in the repulsive interaction or magnetically drivensuperconducting mechanisms. One may argue that thisanswer is somewhat trivial as it has been accepted ina variety of models for cuprates[5, 6]. However, as wewill discuss below, the answer is highly non-trivial iniron-based superconductors because their magnetismsare involved with different microscopic origins. Threemain reasons to support this rule are:

(1) It naturally explains the robust d-wave pairingsymmetry in cuprates and the robust s-wave pairingsymmetry in iron-based superconductors;

(2) It is supported by a general argument that withoutthe existence of mediated anions in the middle, theshort-range Coulomb repulsive interactions between twocation atoms can not be sufficiently screened to allowsuperconducting pairing between them;

(3) It places strict regulations on electronic envi-ronments that can host high Tc superconductivity andthus results in a straightforward answer to the secondquestion.

3

A. The case of cuprates

As we have pointed out above, the rule is a familiaransatz in cuprates. It has provided a natural explana-tion to the d-wave pairing symmetry [7], arguably themost successful theoretical achievement in the studies ofcuprates. In fact, historically, in determining the pairingsymmetry of cuprates, the d-wave pairing symmetry wastheoretically predicted before the emergence of major ex-perimental evidence.[8–10].

Here we briefly review the main theoretical approachesin obtaining the d-wave pairing symmetry in curpates.There are two types of approaches to obtain the d-wavepairing symmetry based on effective models built in atwo-dimensional Cu square lattice. One is the traditionalweak coupling approach. This approach starts with aclosely nested Fermi surfaces in which the spin-densitywave (SDW) instability can take place by onsite electron-electron repulsive interaction ( the Hubbard interaction)[7, 8]. The other is the strong interaction approach. Itstarts directly with short-range magnetic exchange inter-actions. In cuprates, the magnetic exchange interactionsare the nearest neighbor(NN) AFM superexchange inter-actions mediated through oxygen atoms[5, 9, 10]. Bothapproaches consistently predict d-wave superconductingstates.

The consistency can be attributed to the following sim-ple pairing symmetry selection rule: the pairing symme-try is selected by the weight of its momentum space formfactor on Fermi surfaces[11]. This rule is based on thefollowing observation in repulsive interaction or magneti-cally driven high Tc superconducting mechanism: the su-perconducting pairings are dominated on bonds with thestrongest effective AFM exchange couplings. This rulehas been emphasized in the second type of models withlocal AFM superexchange interactions[10, 12]. In thecase of cuprates, the decoupling of the NN AFM superex-change interaction in the pairing channel results in twopossible pairing symmetries: an extended s-wave witha superconducting order in the reciprocal space ∆s(k) ∝coskX+coskY and a d-wave with ∆d(k) ∝ coskX−coskY .With the Fermi surface shown in Fig.2 (c), the d-waveform factor has a much larger amplitude on the Fermisurfaces than the extended s-wave. Thus, the d-wave isfavored by opening much larger superconducting gaps tosave more AFM exchange energy in the superconductingstate. This rule is also behind the weak coupling ap-proach based on the Hubbard model in cuprates[7]. Asthe Hubbard model only includes the onsite repulsive in-teractions and its kinetic part is dominated by the NNhopping, the leading effective AFM exchange couplingsare also generated on the NN bonds. In fact, consideringthe AFM fluctuations near half-filling in the Hubbardmodel, the effective electron-electron interaction medi-ated by the AFM fluctuations in the pairing channel hasthe following property[7]: it starts with a large repulsiveonsite interaction followed by an attractive interactionbetween two NN sites, and then oscillates between repul-

B

A

B

B

B

X

Y

(a) (b)

Γ

(c) (d)

Γ

×

FIG. 2: The comparison of superconducting pairings betweencurpates and iron-based superconductors in both real and mo-mentum spaces: (a) the real space pairing configuration in thed-wave superconducting state of cuprates; (b) the real spacepairing configuration in the extended s-wave superconductingstate in iron-based superconductors ( the red multiplicationsign indicates the forbidden pairing between A and B sublat-tices); (c) the Fermi surfaces of cuprates and the weight distri-bution of the d-wave order parameter in the momentum space(red and blue colors represent regions with large positive andnegative values respectively); (d) the typical Fermi surfacesof iron-based superconductors and the weight distribution ofthe extended s-wave order parameter in the momentum space.The Fermi surfaces at Γ with dashed lines are hole pocketswhich can be absent in iron-chalcogendies[13].

sive and attractive with a rapid decay as increasing thespace distance. This property essentially tells us that thepairing is also dominated on the NN bonds.

B. The case of iron-based superconductors

Comparing an FeAs/Se layer with a CuO2 layer, asshown in Fig.1(a,b), we notice several important differ-ences: (1) the As/Se atoms in the former are locatedexactly below or above the middle points of the four-Fe squares; (2) the distance between two NN Fe atomsis very short, which is only about 2.8 A. This value isclose to the lattice constant of the body-centered cubicFe metal; (3) the distance between two next NN (NNN)Fe atoms is about 3.8A, which is close to the distance oftwo NN Cu atoms in the CuO2 layer. These differencessuggest that the magnetic exchange couplings betweentwo NNN Fe atoms, just like those between two NN Cuatoms, are mediated by the p-orbitals of As/Se atoms.Thus the magnetic couplings between two NNN sites aredominated by superexchange mechanism. However, twod-orbitals between two NN Fe atoms have large over-lap which causes direct hoppings and direct magnetic ex-

4

change couplings. Therefore, the NN exchange magneticcouplings have a different microscopic mechanism fromthe NNN ones. These differences explain why the effec-tive magnetic models in iron-based superconductors aremuch complex and exhibit both itinerant and local typesof magnetic characters[14].

The short NN distance and the existence of direct mag-netic exchange mechanism also have a profound effect onthe superconducting pairings. In cuprates, one can arguethat the repulsive interaction between two NN Cu atomscan be ignored because of the existence of oxygen atomsin the middle, which create a large local electric polar-ization to screen the effective Coulomb interaction. Thisallows pairing to take place on the NN bonds. However,if there is a direct hopping between two atoms, there isno local electronic polarization to screen the Coulombinteractions between them. Thus, in iron-based super-conductors, the repulsive interaction between two NNFe sites must be large so that the pairing between NNbonds is essentially forbidden. But the physics betweentwo NNN Fe sites are the same as those between two NNsites of cuprates. The effective Coulomb interaction be-tween two NNN sites is screened by the strong electronicpolarization created by As/Se atoms.

We can picture the above discussion in a simple man-ner. Considering the original two-iron unit cell as shownFig.1(b), we label the two Fe sites in the unit cell asA and B respectively so that the Fe square lattice com-poses of two square sublattices, A and B. Each sublatticecan be considered as an analogy of the Cu square latticeof cuprates, The pairings between the two lattices areforbidden due to the existence of strong repulsive inter-actions. The pairing exists only within each sublattice.Namely, as illustrated in Fig.2(b), the pairings are onlyallowed between different 2-Fe unit cells and are forbid-den within the unit cells. Such an analogy allows us toapply the same pairing symmetry selection rules to pre-dict the pairing symmetry of iron-based superconductors.If we draw the Fermi surfaces, as shown in Fig.2(d), inthe Brillouin zone of the two Fe unit cell, which is alsothe Brillouin zone with respect to each sublattice, theFermi surfaces are located either at the corner(M) or atthe center(Γ). As shown in Fig.2(d), the form factor ofthe extended s-wave ∆s(k) has a large weight on Fermisurfaces. Thus, the extended s-wave is clearly favored.The picture does not depend on the presence or absenceof hole pockets at Γ points.

The above discussion suggests that iron-based super-conductors are simply a counterpart of cuprates to real-ize the extended s-wave pairing symmetry in a square lat-tice. The extended s-wave in iron-based superconductorsendures the same robustness as the d-wave in cuprates.The robust s-wave symmetry in iron-based superconduc-tors has been supported by overwhelming experimentalevidence accumulated in the past several years[4, 15, 16].This understanding explains the missing part in the pre-vious theoretical studies which failed to obtain the ro-bust s-wave pairing. In the previous studies based on

weak coupling approaches[17], the repulsive interactionbetween the A and B sublattices is not seriously consid-ered and only onsite repulsive interactions are consideredin calculating pairing symmetries. With only onsite re-pulsive interaction, the effective attractive interactionsare generated in both NN and NNN bonds. In general,the NN bonds favor the d-wave pairing symmetry[18]and the NNN bonds favor the extended s-wave symme-try. Thus, pairing symmetries from these models becomevery sensitive to the detailed parameters and Fermi sur-face properties[17, 18]. The same sensitivity also existsin the models based on local AFM J1 − J2 exchangecouplings[12]. With the existence of both J1, the NNAFM exchange couplings, and J2, the NNN AFM ex-change couplings, the phase diagram is very rich[12, 19].The robust s-wave is only obtained when J1 is argued tobe inactive in providing pairing[20].

Summarizing above discussions, iron-based supercon-ductors and cuprates can be unified in one superconduct-ing mechanism. The former provides extreme valuableinformation to distinguish the roles of different magneticinteractions in providing superconducting pairing. Therobust s-wave pairing symmetry in iron-based supercon-ductors, just like the d-wave in cuprates, is a strong in-diction to support the AFM superexchange couplings asthe dominant sources for pairing.

IV. THE ANSWER TO THE SECONDQUESTION

As we have mentioned earlier, the challenge is that theanswer to the first question has to result in a naturalanswer to the second question. To show that this is thecase for the above ansatz, we first discuss explicit con-ditions posed by the answer to the first question. Then,we discuss how both cuprates and iron-based supercon-ductors fulfill these conditions. Finally, we address whyit is difficult to satisfy these conditions and explain whyunconventional high Tc superconductors are rare.

A. Conditions and rules for unconventional highTc superconductivity

In order to generate the strong AFM superexchangecouplings and maximize their contributions to high Tc

superconductivity, we can argue the following specificrequirements for potential high Tc candidates:

(1) The necessity of cation-anion complexes: Asthe AFM superexchange couplings are mediated throughanions, the potential candidates must include structuralunits constructed by cation-anion complexes. Withinthe units, there must be shared anions between twoneighboring complexes. Moreover, strong chemicalbondings between two anions should be forbidden asthey generally destroy the AFM exchange processes.

5

(2) The orbital selection rule: the orbitals of cationatoms that participate in strong chemical bondings withanion atoms to generate strong AFM superexchange cou-plings must play a dominant role near Fermi energy. Thebest electronic environment for high Tc superconductivityis achieved when these orbitals are isolated near Fermienergy. Namely, the band structures near Fermi energyshould be dominated by the orbitals of cation atomswhose kinematics are generated through the couplings toanions. We will show that this requirement essentiallyanswers why cuprates and iron-based superconductorsare special to host high Tc superconductivity. It is themost powerful rule to narrow our search for potentialhigh Tc candidates. Following this rule, we can combinesymmetry analysis and density functional theory(DFT)to search for new high Tc electronic environments.This rule has been implicated in cuprates as an orbitaldistillation effect based on the observation that thehigher Tc is achieved when dX2−Y 2-orbitals are dressedless by dZ2 orbitals in cuprates[21].

(3)The pairing symmetry selection rule: We haveexplicitly discussed this rule above. This rule allows usto link pairing configurations in real and momentumspaces directly. Following this rule, we may be able todesign structures to realize superconducting states withspecific pairing symmetries.

(4) Electron-electron correlation and half-filling: Theatomic orbitals in cation atoms that can produce strongAFM superexchange couplings require to balance theirspatial localization and extension. Moreover, in general,the strong AFM superexchange couplings are achievedwhen the orbitals are close to be half-filling. Thus, thehalf-filled 3d orbitals in transition metal elements areclearly the best choices.

(5) Dimensionality: For d-orbitals, due to theirtwo-dimensional nature in the spatial configuration, theorbital selection rule naturally demands a quasi twodimensional electronic environment. In an electronicband structure with strong three-dimensional banddispersions, it is difficult to maintain a purified orbitalcharacter. While one may argue that it is possibleto satisfy these requirements in quasi one dimensionalelectronic environments, finding such an example isextremely difficult.

Summarizing these conditions and rules for transi-tion metal based compounds, we can specifically definethe gene of high Tc superconductors as a quasi two di-mensional electronic structure in which the d-orbitals ofcation atoms that participate strong in-plane chemicalbonding with the p-orbitals of anion atoms are isolatednear Fermi energy. In the following two subsections, weshow that both cuprates and iron-based superconductorsare special materials to carry such a gene.

t2g

eg dX

2-Y

2, dZ2

dXY, dXZ, dYZ

dZ2

dX2-Y

2

d9-Cu2+

dX2-Y

2

(a)

(d)

(b)

(c)

FIG. 3: Local electronic environment and selected orbitalsin cuprates: (a) the sketch of an octahedral complex; (b)the coupling configuration of the selected dX2−Y 2 orbital inCuO2 layers; (c) the crystal field splitting of cation d-orbitalsin an octahedral complex; (d) the true local energy configura-tions at Cu sites in curpates to indicate that the blue orbital,dX2−Y 2 , is selected in the d9 filling configuration.

B. The case of curpates

Cuprates belong to perovskite-related structural ma-terials. The perovskite-related structures are the mostpopular and stable structures in nature. In a perovskite-related structure, the basic building block is the cation-anion octahedral complex shown in Fig.3(a). In cuprates,the CuO6 octahedral complexes form two dimensionalCuO2 layers to provide a quasi two dimensional electronicstructure. In a pure CuO6 octahedral complex, the fived-orbitals of the Cu atom are split into two groups bycrystal fields, t2g and eg, as shown in Fig.3(c). The ener-gies of the two eg orbitals, dZ2 and dX2−Y 2 , due to theirstrong couplings to the surrounding oxygen atoms, arehigher. Moreover, in the CuO2 layer, the energy of dZ2

orbital is lowered either by the Jahn-Teller effect or bythe absence of apical oxygen atoms. Thus, the local en-ergy configuration at cation sites is described accordingto Fig.3(d) in which the dX2−Y 2 orbital sits alone at thetop.

It is easy to notice that only the dX2−Y 2 orbital hasstrong in-plane couplings to the p-orbitals of oxygens tomediate strong AFM superexchange couplings. Namely,only the electronic band attributed to the dX2−Y 2 or-bital can support high Tc superconductivity. To isolatethe dX2−Y 2 orbital near Fermi energy, nine electrons onthe d shell are required. Thus, the gene of high Tc su-perconductivity can only be satisfied in a d9 filling con-figuration at cation sites, which explains why Cu2+ is

6

d*xy dxy

d*x2-y

2

d6-Fe2+

B

B

B

B

d*xy(dxz/yz)

(a)

t2g

eg dz2

dxy, (dxz,dyz)

(b)

(c) (d)

dxy

dx2-y

2

dz2 dx

2-y

2

y

x

FIG. 4: Local electronic environment and selected orbitals iniron-based superconductors: (a) the sketch of an tetrahedralcomplex; (b) the coupling configurations of the selected dxy-type of orbitals to anion atoms in FeAs/Se layers; (c) thecrystal field splitting of cation d-orbitals in an tetrahedralcomplex; (d) the local energy configurations at Fe sites iniron-based superconductors ( the blue orbitals are isolated ind6 filling configuration to dominate electronic physics nearFermi energy) .

a natural choice. As a matter of fact, in the past sev-eral decades, numerous transition metal compounds withperovskite-related structures were discovered. Exceptcurpates, none of them exhibits high Tc superconduc-tivity.

C. The case of iron-based superconductors

The electronic physics of iron-based superconduc-tors locates on the two dimensional FeAs/Se layers.The layers are constructed by edge-shared tetrahedralFeAs4(Se4) complexes shown in Fig.4(a). The four co-ordination tetrahedral complex, just slightly less popularthan the octahedral complex, is another important struc-ture unit to form crystal lattices.

In a tetrahedral complex, as shown in Fig.4(c), the t2gorbitals have higher energy than the eg orbitals becauseof their strong couplings to anions. Under such a config-uration, one may jump to argue that a d7 filling configu-ration can make all t2g orbitals near half-filling to satisfythe gene requirements. However, the argument is mis-leading because of the following two major reasons. First,the crystal field energy splitting in a tetrahedral complexbetween the t2g and the eg orbitals is much smaller thanthe one in the octahedral complex. Second, the dx2−y2

eg orbital has very large dispersion due to the short NN

Fe−Fe distance in FeAs/Se layers. Therefore, the sim-ple argument can not exclude dx2−y2 eg orbitals near theFermi energy.

However, if we carefully examine the 2-Fe unit cell,because of the short distance between two NN Fe atoms,the local electronic environment of an Fe atom is notonly affected by the four surrounding As/Se atoms inthe tetrahedral complex but also the four neighboringFe atoms. In fact, the dxz and dyz orbitals are stronglycoupled to the dx2−y2 eg-orbitals of the neighboring Featoms. Thus, a more complete picture is that the dxzand dyz orbitals form two molecular orbitals. One ofthem, which has dx2−y2 symmetry character, stronglycouples to the dx2−y2 eg-orbitals of the neighboring Featoms. The coupling pushes this orbital to higher energy.The other, which has dxy symmetry character, remains toa pure orbital with strong couplings to the surroundingAs/Se atoms. Therefore, the more accurate local energyconfiguration is given by Fig.4(d), in which there are twodxy type of orbitals in the middle in which one of themis formed by dxz/yz orbitals. These two orbitals can hostpossible high Tc superconductivity. With this configura-tion, we immediately determine that the 3d6 configura-tion of Fe2+ is special to satisfy the gene requirements.

The above energy configuration has been hidden be-hind the simplified effective two-orbital models con-structed for iron-based superconductors[22]. Near Fermienergy, the two-orbital effective model was shown to cap-ture the band dispersions of the five-orbital models thatwas derived by fitting DFT calculations[23, 24]. If wecheck the symmetry characters of the two orbitals inthe two-orbital model, both of them have dxy symmetrycharacters rather than dxz/yz interpreted in the originalpaper[22].

The above analysis suggests that the electronic struc-ture in iron-based superconductors realizes the high Tc

gene. As a matter of fact, we also notice that there area variety of materials based on other transition metalelements with identical structures to iron-based super-conductors. However, none of them exhibits high Tc su-perconductivity.

V. THE ANSWER TO THE THIRD QUESTION

A clear message from above discussion is that the genesof high Tc superconductivity stem from very special col-laborations between the local electronic physics of cation-anion complexes and crystal structures. We can arguethat symmetry play the key role behind the collaboration.In fact, we can argue that it is the symmetry collabora-tion between local complex and global crystal structuresto make it possible to realize high Tc genes.

7

A. Octahedral/tetrahedral complexes and squarelattice symmetry

Both octahedral and tetrahedral complexes have afour-fold rotation principal axis. Their d-orbitals areclassified locally by C4 and S4 rotation symmetries re-spectively. If a d orbital can be isolated in a band struc-ture, it should have a similar classification in constructedcrystal structures. This argument suggests that a squarelattice symmetry is required to fulfill the gene conditionsfor materials constructed by octahedral and tetrahedralcomplexes. Both cuprates and iron-based superconduc-tors indeed have square lattice symmetry. The selectedorbitals that produce high Tc genes are classified identi-cally in the symmetry groups of the crystal lattices andtheir local complexes. This correspondence allows themto be isolated in the electronic structures near Fermi en-ergy without messing up with other orbitals.

The octahedral or the tetrahedral complexes are themost common structures in nature. They can form manydifferent two dimensional crystal lattices. If we considercrystal structures formed by these complexes beyond thesquare lattice symmetry, such a correspondence is ab-sent and different orbital characters generally get mixed.Thus, it is difficult to make the targeted orbitals to beisolated in band structures of non-square lattices formedby these two complexes, such as trigonal or hexagonallattice structures, to fulfill the gene conditions. This ex-plains why cuprates and iron-based superconductors areclose to be unique systems to host the high Tc genesin materials constructed by octahedral and tetrahedralcomplexes.

B. Prediction of trigonal/hexagonal high Tc

electronic environments

The symmetry collaboration argument suggests that ifwe want to create a high Tc gene in trigonal/hexagonallattice structures, we may have to search lattices builtby cation-anion complexes with three or six fold princi-pal rotation axis. Thus, we examine trigonal bipyrami-dal complexes (TBP) shown in Fig.5(a), which is a fivecoordination complex and carries a three fold principalrotation axis. The two dimensional hexagonal structureformed through conner-shared TBP, shown in Fig.5(b),has appeared in Mn-based YMnO3[25, 26] and Fe-BasedLu1−xScxFeO3[27].

The explicit prediction is that a d7 filling configura-tion, which can be realized by Co2+ or Ni3+ cations,fulfills the gene conditions of high Tc superconductivityin a material that carries above two dimensional hexago-nal layers. Moreover, the pairing symmetry selection rulepredicts that the superconducting states in these mate-rials close to the d7 filling configuration have a d ± idpairing symmetry.

The crystal field energy splitting of the d-orbitals inTBP is shown in Fig.5. The dz2 orbital has the high-

dX2-Y2, dXY

dZ2

dXZ, dYZ

d*XY/X2-Y2

d7-Co2+/Ni3+

dZ2

(a) (c) (b)

(d) (e) (f)

FIG. 5: Predicted Co/Ni-based hexagonal lattices con-structed by trigonal bipyramidal (TBP) complexes: (a) thesketch of a TBP complex; (b) the two-dimensional hexagonallayer formed by TBP complexes; (c) the weight distributionof an extended s-wave and Fermi surfaces (red color indicateslarge absolute values); (d) the crystal field splitting of cationd-orbitals in a TBP complex; (e) the local energy configura-tions at cation Co/Ni sites in an hexagonal layer; (f) similarto (c), the weight distribution of d ± id-wave and Fermi sur-faces.

est energy due to its strong couplings to the two apicalanions. The double degenerate dx2−y2 and dxy orbitalsare strongly coupled to the in-plane anions. The doubledegenerate dxz and dyz orbitals have the lowest energyand are only weakly coupled to anions. As the hexag-onal lattice is formed by three corner-shared TBPs, thedx2−y2 and dxy orbitals form two molecular orbitals. Oneof them can strongly couple to the dz2 orbital so that thedegeneracy is lifted. As the dz2 orbital has higher energy,the coupling lowers the energy level of the coupled molec-ular orbital. The other is completely isolated from otherorbitals and can be selected to provide the desired highTc electronic environment. A local energy configurationis described by Fig.5(e). The d7 filling configuration canfulfill the gene conditions. The DFT calculation on sucha structure confirms this picture[3].

Around the d7 filling configuration, a quasi two dimen-sional band structure is formed and electronic physicsnear Fermi energy is dominated by a single band at-tributed to the selected orbital. The band has a Fermisurface shown in Fig.5(c,f). If we apply the pairingsymmetry selection rule, as the pairing should be dom-inated on the NN bonds in the cation trigonal lattice,for the extended s-wave pairing, the form factor of thegap function in the momentum space is given by ∆s ∝cosky + 2cos

√32 kxcos

12ky, and for the d ± id-wave pair-

ing, the factor is given by ∆d ∝ cosky−cos√32 kxcos

12ky±

i√

3sin√32 kxsin

12ky. Fig.5 (c,f) illustrate the overlap be-

tween the amplitude of the two form factors with Fermisurfaces. The degenerate d ± id-waves collaborate wellwith Fermi surfaces near half filling. Therefore, the sys-

8

tem supports a robust d± id-wave pairing superconduct-ing state.

The superconducting transition temperature can be es-timated by comparing the energy scales of the couplingsbetween cations and anions in complexes. The Cu − Ocouplings in the octahedral complex of cuprates are morethan twice stronger than the Fe−As/Se couplings in thetetrahedral complex of iron-based superconductors. Theratio of the maximum Tcs observed in these two familiesis in the similar order. In the TBP complex, the couplingstrength sits between them and is about 2/3 of those incuprates. Thus, the maximum Tc that can be realized inthe TBP structure is expected to be around 100k as themaximum Tc in cuprates can reach 160k.

Materials constructed by the TBP complexes are verylimited. The Co/Ni based materials described abovehave not been synthesized. Thus, it is an explicit pre-diction to be tested in future experiments.

VI. DISCUSSION

The above prediction, if verified, justifies our answerto the first question. But most importantly, the verifica-tion can open the door to theoretically design and searchfor new unconventional high Tc superconductors. A gen-eral search procedure can be: (1) design a possible latticestructure that can be constructed by certain cation-anioncomplexes; (2) use symmetry tools to understand localelectronic physics; (3) perform standard DFT calcula-tions to obtain band structures and its orbital characters;(4) apply the gene requirements to determine conditionsand likelihood on the existence of high Tc superconduct-ing environment; (5) design realistic material in whichthe lattice structure can be stabilized.

In designing electronic environments for high Tc super-conductivity, there are helpful clues and possible direc-tions. For example, we can ask whether we can designcrystal structures for all 3d transition elements to real-ize high Tc superconductivity. As the d-orbitals whichare responsible for high Tc superconductivity must makestrong couplings to anion atoms, they typically gain en-ergy in the crystal field environment, which explains whyall high Tc superconductors, including predicted Co/Ni-based materials, are involved with the second half part ofthe 3d transition elements in the periodic table. Whetherwe can overcome this limitation to make specific designsfor the first half 3d transition elements, in particular,Mn and Cr, is a very intriguing question. Another ex-ample is to ask whether we can design superconductingstates with particular pairing symmetries as we have ex-

plicit rules to determine pairing symmetries. We havenoted that cuprates and iron-based superconductors areexamples of the d-wave and the extended s-wave pair-ing symmetry in a square lattice. Our predicted mate-rial is a realization of the d-wave pairing symmetry intrigonal/hexagonal lattice structures. Thus, a specificquestion is how to realize an extended s-wave in the trig-onal/hexagonal lattice structures.

We have mainly focused on the 3d orbitals which areknown to produce the strongest correlation effect. How-ever, even if carrying less electron-electron correlation ef-fect, we can also consider other type of orbitals at cationsites, including 4d, 5d, 4f, 5f and even higher level s-orbitals. We can search for materials as potential un-conventional superconductors in which the kinematics ofthese orbitals near Fermi energy can be isolated and isgenerated through strong couplings to the p-orbitals ofanions. In general, as long as there is a charge trans-fer energy gap between orbitals of cations and anions,the AFM superexchange coupling should be generated.Thus, moderate high Tc may be achieved. For 5d and5f orbitals, because of large spin orbital couplings, theorbitals can be reconfigured to have drastically differ-ent real space configurations. This may result in morepossible designs on crystal lattice structures to gener-ate superconducting states. One example is Sr2IrO4[28],which can be considered as a lower-energy scale clone ofcuprates[29, 30]. For the s-orbitals, as they are symmetricin space, we may design a cubic-type three dimensionallattice structure to achieve the conditions.

In summary, cuprates and iron-based superconductorscan be unified in a framework based on repulsive inter-action or magnetically driven high Tc mechanisms. Thisunification leads to important rules to regulate electronicenvironments required for unconventional high Tc su-perconductivity. The rules can guide us to search fornew high Tc superconductors. Following these rules, wemade an explicit prediction about the existence of highTc superconductivity in the Co/Ni-based two dimen-sional hexagonal lattice structure constructed by trigo-nal bipyramidal complexes. Verifying this prediction canpave a way to establish unconventional high Tc mecha-nism.Acknowledgement: The author acknowledges help-

ful discussions with D.L. Feng, X. H. Chen, H. Ding andD. Scalapino,. The author thanks X.X. Wu, C.C. Le andJ. Yuan for carrying out numerical calculations. Thework is supported by the National Basic Research Pro-gram of China, National Natural Science Foundation ofChina(NSFC) and the Strategic Priority Research Pro-gram of Chinese Academy of Sciences.

[1] Bednorz, J. G. & Muller, K. A. Possible high tc super-conductivity in the ba-la-cu-o system. Z. Phys. B 64,189 (1986).

[2] Kamihara, Y., Watanabe, T., Hirano, M. & Hosono, H.

Iron-based layered superconductor lao1−xfxfeas (x= 0.05-0.12) with tc 26 k. JACS 130, 3296–3296 (2008).

[3] Hu, J. P., Le, C. C. & Wu, X. X. Predicting un-conventional high temperature superconductors in trigo-

9

nal bipyramidal coordinations. Phys. Rev. X 5, 041012(2015).

[4] Hu, J. P. & Yuan, J. Robustness of s-wave pairing sym-metry in iron-based superconductors and its implicationsto fundamentals on magnetically-driven high tempera-ture superconductivity. arXiv:1506.05791 (2015).

[5] Anderson, P. W. et al. The physics behind high-temperature superconducting cuprates: the ‘plainvanilla’ version of rvb. J. Phys.-Cond. Matt. 16, R755–R769 (2004).

[6] Scalapino, D. J. Condensed matter physics - the cupratepairing mechanism. Science 284, 1282–1283 (1999).

[7] Scalapino, D. The case for dx2−y2 pairing in the cupratesuperconductors. Phys. Rep. 250, 329 (1995).

[8] Bickers, N., Scalapino, D. J. & Scalettar, R. T. Cdw andsdw mediated pairing interactions. Int. J. Mod. Phys. B1, 687 (1987).

[9] Gros, C., Poilblanc, D., Rice, T. M. & Zhang, F. C. Su-perconductivity in correlated wavefunctions. Physica C.153, 543 (1988).

[10] Kotliar, G. & Liu, J. Superexchange mechanism and d-wave superconductivity. Phys. Rev. B 38, 5142 (1988).

[11] Hu, J. P. & Ding, H. Local antiferromagnetic exchangeand collaborative fermi surface as key ingredients of hightemperature superconductors. Scientific Reports 2, 381(2012).

[12] Seo, K. J., Bernevig, B. A. & Hu, J. P. Pairing symmetryin a two-orbital exchange coupling model of oxypnictides.Phys. Rev. Lett. 101, 206404 (2008).

[13] Dagotto, E. Colloquium: The unexpected properties ofalkali metal iron selenide superconductors. Rev. Mod.Phys. 85, 849–867 (2013).

[14] Dai, P., Hu, J. & Dagotto, E. Magnetism and its micro-scopic origin in iron-based high-temperature supercon-ductors. Nature Physics 8, 709 (2012).

[15] Richard, P., Qian, T. & Ding, H. Arpes measure-ments of the superconducting gap of fe-based supercon-ductors and their implications to the pairing mechanism.ArXiv:1503.07269 (2015).

[16] Q., F. et al. Plain s-wave superconductivity in singlelayer fese and sto probed by stm. Nature Physics 10,1038 (2015).

[17] Hirschfeld, P. J., Korshunov, M. M. & Mazin, I. I.Gap symmetry and structure of fe-based superconduc-tors. Rep. Prog. Phys. 74, 4508 (2011).

[18] Maier, T. A., Graser, S., Hirschfeld, P. J. & Scalapino,D. J. d-wave pairing from spin fluctuations in the kxfe2-yse2 superconductors. Phys. Rev. B 83, 100515–100515(2011).

[19] Yu, R., Zhu, J. & Si, Q. Orbital-selective superconduc-tivity, gap anisotropy and spin resonance excitations in amultiorbital t-j1-j2 model for iron pnictides. Phys. Rev.B 89, 024509 (2014).

[20] Fang, C., Wu, Y. L., Thomale, R., Bernevig, B. A. & Hu,J. P. Robustness of s-wave pairing in electron-overdopeda(1−y)fe(2−x)se2 (a = k, cs). Phys. Rev. X 1 (2011).

[21] Sakakibaraa, H. et al. Three-orbital study on the orbitaldistillation effect in the high tc cuprates. Phys. Proc. 45,13 (2013).

[22] Hu, J. & Hao, N. S4 symmetric microscopic modelfor iron-based superconductors. Phys. Rev X 2, 021009(2012).

[23] Kuroki, K. et al. Unconventional pairing originatingfrom the disconnected fermi surfaces of superconduct-ing lafeaso1−xfx. Phys. Rev. Lett. 101, 087004–087004(2008).

[24] Graser, S., Maier, T. A., Hirschfeld, P. J. & Scalapino,D. J. Near-degeneracy of several pairing channels in mul-tiorbital models for the fe pnictides. New J. Phys. 11,5016–5016 (2009).

[25] Yakel, H. L., Koehler, W. C., Bertaut, E. F. & Forrat,E. F. On the crystal structure of the manganese iii triox-ides of the heavy lanthanides and yttrium. Acta Cryst.16, 957 (1963).

[26] Smolenskii, G. A. & Bokov, V. A. Coexistence of mag-netic and electric ordering in crystals. J. Appl. Phys. 35,915 (1964).

[27] Masuno, A. et al. Expansion of the hexagonal phase-forming region of luscfeo by containerless processing. In-org Chem 54, 9432 (2015).

[28] Crawford, M. K. et al. Structural and magnetic studiesof sr2iro4. Phys. Rev. B 49, 9198–9201 (1994).

[29] Wang, F. & Senthil, T. Twisted hubbard model forsr2iro4: Magnetism and possible high temperature su-perconductivity. Phys. Rev. Lett. 106, 136402 (2011).

[30] Yan, Y. J. et al. Electron-doped sr2iro4: An analogueof hole-doped cuprate superconductors demonstrated byscanning tunneling microscopy. Phys. Rev. X 5, 041018(2015).