Three-Dimensional Multiband d-p Model of Superconductivity in Spin-Chain Ladder Cuprate
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Three-Dimensional Multiband d-p Model of Superconductivity inSpin-Chain Ladder Cuprate
Shigeru Koikegami ∗ and Takashi Yanagisawa Second Lab, LLC, 2-32-1 Umezono, Tsukuba, Ibaraki 305-0045 Nanoelectronics Research Institute, AIST Tsukuba Central 2, Tsukuba, Ibaraki 305-8568
We study the superconductivity in the three-dimensional multiband d-p model, in whicha Cu O -ladder layer and a CuO -chain layer are alternately stacked, as a model of thesuperconducting spin-chain ladder cuprate. p z -Wave-like triplet superconductivity is found tobe the most stable, and its dependence on interlayer coupling can explain the superconductingtransition temperature dependence on pressure in real superconducting spin-chain laddercuprates. The superconductivity may be enhanced if hole transfer from the chain layer tothe ladder layer can be promoted beyond the typical transfer rate. KEYWORDS: superconductivity, three-dimensional multiband d-p model, spin-chain laddercuprate, interlayer coupling
1. Introduction
The spin-chain ladder cuprate Sr − x Ca x Cu O (Sr14-24-41) has attracted much inter-est as a related material of high- T c cuprates for more than two decades. It has three typesof layers, known as the chain, spacer, and ladder. These layers are stacked in an alternat-ing manner along the layer axis b . The chain layer contains CuO chains, while the ladderlayer consists of two-leg Cu O ladders. Theoretically, the two-leg spin-1 / The hole-doped two-leg spin ladder has been investigated theoretically and confirmednumerically to possess an instability towards d -wave-like superconductivity on the basis of the t - J ladder model, one-band Hubbard ladder model, and three-band Hubbard lad-der model.
10, 20)
In 1996, Sr − x Ca x Cu O with x = 13 . P ≥ In Sr14-24-41, doped holes move from the chain layer tothe ladder layer by Ca substitution for Sr.
High pressures intensify this self-doping effectand afford the electrons in the ladder layer sufficient itinerancy.
23, 24)
However, the supercon-ducting transition temperature ( T c ) has a maximum at P ≈ The dependence of T c on pressure is common to other compositions aswell, such as Sr − x Ca x Cu O for x = 11 . and x = 10 , ,
27, 28)
Such superconductiv-ity behavior is similar to that of high- T c cuprates in the overdoped regime. Considering theexperimental results of the electronic properties, the superconductivity of Sr14-24-41 can be ∗ E-mail address : [email protected] 1/16 . Phys. Soc. Jpn.
Full Paper thought of as an extension of that of high- T c cuprates. In accord with the above novel results, the two-leg Hubbard ladders coupled via a weakinterladder hopping were investigated theoretically by Kishine and Yonemitsu.
Accordingto their perturbative renormalization group analysis, the system has a d -wave-like supercon-ducting ground state, and restores the interladder coherence within an increase in the extentof interladder hopping. Moreover, the superconducting state and its T c were evaluated onthe basis of the two-dimensional (2D) Trellis-lattice Hubbard model, i.e., the coupled two-legladder model. In their studies, a wide doping regime exists in which d -wave like singletsuperconductivity appears. In one of them, it is also shown that a certain doping regimeprefers p z -wave like triplet superconductivity. Thus, it is plausible that the superconductiv-ity may be intrinsically linked to the 2D two-leg ladder structure. Practically, however, thesuperconductivity in Sr14-24-41 appears for certain compounds only under high pressures. Inorder to understand this situation, we should find routes for enhancing superconductivity inreal compounds, that have not been considered yet in past theoretical studies. It is worthnoting that Isobe et al. have already pointed out the effects of the hybrid orbital betweenCu 3 d in the ladder layer and O 2 p in the chain layer on the superconductivity under highpressures.
25, 34)
According to their arguments, the hybrid orbital accelerates the redistributionof holes and enhances the superconductivity.In this study, we investigate the superconductivity of the two-leg ladder layer coupledwith the chain layer. We adopt the three-dimensional (3D) d-p model with the quasi-one-dimensional (Q1D) structure in which a Cu O ladder layer and a CuO chain layer arealternately stacked. In our model, the ladder and chain layers are coupled via hybridizationbetween the Cu 3 d orbital in the ladder layer and the O 2 p orbital in the chain layer. Moreover,we introduce a sufficiently small on-site Coulomb interaction showing that the second-orderperturbation theory (SOPT) can be justified. We can treat the superconductivity using aweak coupling analysis because, in our model, the effective interaction for Cooper pairing isso small that only electrons on the Fermi surface are involved in the superconductivity. Inparticular, the weak coupling formulation by Kondo is applicable even in the case with a verysmall effective interaction. We previously applied Kondo’s formulation to the study of the3D d-p model with multilayer perovskite structure and investigated how the superconductinggap depends on the number of layers.
As in the study of multilayer cuprates, we show thatcalculation on the basis of a 3D model is practical for assessing the superconductivity in spin-chain ladder cuprate. Our results give a possible explanation as to why the superconductivityappears in Sr − x Ca x Cu O for x ≥
10 only under high pressures.
2. Formulation
Our 3D d-p model is schematically shown in Fig. 1. We consider two Cu sites and three Osites in the ladder layer, and one Cu site and two O sites in the chain layer. In the chain layer, . Phys. Soc. Jpn.
Full Paper p p p p p p p d d aa/2 a/2a ac t pp t dp p p d d p p p p d dp p
4v 33 33 aaa/2 a/2 ac t dp t dp t pp t pp t’ pp t’ pp t’ pp t’ pp b/2b/2 bc d d p p p p d d p p p p d d d d p p p p p p p p bc ac t tt tt tt t b/2b/2a/4 a/4 a/4 a/4 a/4 a/4 a/4 a/4a/4 a/4 a/4 a/4 a/4 a/4 a/4 a/4 (a)(b) (c) Fig. 1. (Color online) Transfer energies in the 3D chain-ladder d-p model: (a) within the ladder layer,(b) within the chain layer, and (c) for the interlayer hopping. . Phys. Soc. Jpn.
Full Paper the lattice constants along the c-axis and a-axis are √ / √ H = H + H + H − Chain + H ′ − µ X k σ h d † k σ d k σ + d † k σ d k σ + p z † k σ p z k σ + p x † k σ p x k σ + p z † k σ p z k σ + p x † k σ p x k σ + p x † k σ p x k σ + d † k σ d k σ + p w † k σ p w k σ + p v † k σ p v k σ + p w † k σ p w k σ + p v † k σ p v k σ i , (1)where d l k σ ( d † l k σ ) and p νm k σ ( p ν † m k σ ) are the annihilation (creation) operators for d-electrons inthe l -th site and for p ν -electrons in the m -th site, having a momentum k and spin σ = {↑ , ↓} ,respectively. The site indices l and m , and the orbital index ν are defined as shown in Fig. 1.In the following, we take both a and b as the unit of length and put a = b = 1. µ represents thechemical potential. The noninteracting parts in eq. (1), i.e., H , H , and H − Chain ,are represented by H = X k σ (cid:16) d † k σ d † k σ p z † k σ p x † k σ p z † k σ p x † k σ p x † k σ (cid:17) ε dp ζ z ∗ k ζ x ∗ k − ζ x k ε dp − ζ x k ζ z ∗ k ζ x ∗ k ζ z k ζ p ∗ k ζ p k − ζ x ∗ k ζ p ∗ k ζ z k ζ p k ζ p ∗ k ζ x k ζ p k − ζ x ∗ k ζ x k ζ p ∗ k ζ p k d k σ d k σ p z k σ p x k σ p z k σ p x k σ p x k σ , (2) H = X k σ (cid:16) d † k σ p w † k σ p v † k σ p w † k σ p v † k σ (cid:17) ε dp − ∆ V − ξ w k ξ v ∗ k ξ w ∗ k − ξ v k − ξ w ∗ k − ∆ V ξ z ∗ k ξ x ∗ k ξ v k ξ z k − ∆ V ξ x ∗ k ξ w k ξ x k − ∆ V ξ z k − ξ v ∗ k ξ x k ξ z ∗ k − ∆ V d k σ p w k σ p v k σ p w k σ p v k σ , (3)and H − Chain = X k σ h η + k (cid:16) p v † k σ d k σ − p w † k σ d k σ (cid:17) + η − k (cid:16) p w † k σ d k σ − p v † k σ d k σ (cid:17) + H . c . i , (4)respectively. In eqs. (2)–(4), we use the abbreviations ζ z k = 2i t dp sin k z , ζ x k = t dp e − i k x / , ζ p k =2i t pp e − i k x / sin k z , ξ w k = t dp e i k z / e i k x / , ξ v k = t dp e i k z / e − i k x / , ξ z k = t pp e i k z , ξ x k = − t pp e i k x / − . Phys. Soc. Jpn. Full Paper t ′ pp e − i k x cos k z , and η ± k = 2i t ⊥ sin (cid:16) k y ± k z (cid:17) . The transfer energies, t dp , t pp , t ′ pp , and t ⊥ aredefined as shown in Fig. 1. The increase in t ⊥ in our model is considered to represent theincrease in pressure in real Sr14-24-41. ε dp is the level difference between d- and p-electrons.Moreover, we use ∆ V to control the charge imbalance between the ladder and chain layers.The change in ∆ V can represent the change in Madelung energy due to the Ca doping of Sr14-24-41. Considering only the on-site Coulomb repulsion among d-electrons, the interacting part H ′ in eq. (1) is described as H ′ = UN X l =1 X kk ′ q d † l k + q ↑ d † l k ′ − q ↓ d l k ′ ↓ d l k ↑ . (5)In eq. (5), N is the number of k -space lattice points in the first Brillouin zone (FBZ).In the following analysis, we assume that only electrons on the Fermi surface of the sameband can have pair instability. For our 3D d-p model, 2 or 3 d -like bands intersect with theFermi level. Thus, according to the Bardeen-Cooper-Schrieffer (BCS) theory, we have thefollowing self-consistent equation for the pair function on the λ -th d -like band, Φ λ k :Φ λ k = − N X ijν k ′ V ij ( k + k ′ ) z λi ( k ) z νj ( k ′ ) q(cid:0) ε ν k ′ − µ (cid:1) + (cid:0) Φ ν k ′ (cid:1) Φ ν k ′ , (6)where i, j = 1 , , λ, ν = 1 , ,
3) ( d -like band indices). This equationis valid for both the spin-singlet and spin-triplet pair functions if we define V ij ( q ) differentlyas the need arises. Thus, hereafter, we omit the spin indices. V ij ( q ) represents the effectivepair scattering between a d-electron on the i -th site and one on the j -th site. The term ε ν k represents the energy dispersion of the ν -th d -like band, and z λi ( k ) represents the matrixelement of unitary transformation. These variables are obtained by solving the eigen-equationfor the noninteracting part H + H + H − Chain in eq. (1). We set Φ λ k = ∆ sc · Ψ λ k ,where ∆ sc denotes the magnitude of Φ λ k , and Ψ λ k represents its k dependence on the λ -th d -like band. On the basis of Kondo’s argument, retaining only the divergent term, we canrewrite eq. (6) asΨ λ k = log e ∆ sc · N X ijν k ′ V ij ( k + k ′ ) z λi ( k ) z νj ( k ′ ) δ ( ε ν k ′ − µ )Ψ ν k ′ , (7)for a very small ∆ sc . Equation (7) is a homogeneous integral equation for Ψ λ k with an eigen-value of 1 / log ∆ sc . We are interested in obtaining the most stable pairing state, so we mustfind the eigenvector Ψ λ k with the smallest eigenvalue 1 / log ∆ sc using eq. (7) when ∆ sc ismaximum. Given the quasi-two-dimensionality of our 3D d-p model, i.e., t ⊥ ≪ t dp , we assumefive functions as candidates for the most stable pairing state: a λ : s -wave-like singlet , (8) . Phys. Soc. Jpn. Full Paper M X m z =1 a λz ( m z ) cos m z k z + M X m z =1 2 m z X m x =1 a λx ( m z m x ) cos 2 m z − m x k z cos 3 m x k x + M X m y =1 a λy ( m y ) cos m y k y : d z − x -wave-like singlet , (9) M X m z =1 2 m z − X m x =1 a λ ( m z m x ) sin 2 m z − m x k z sin 3 m x k x : d zx -wave-like singlet , M X m z =1 a λ ( m z ) sin m z k z : p z -wave-like triplet , (10) M X m z =1 2 m z X m x =1 a λ ( m z m x ) cos 2 m z − m x k z sin 3 m x k x : p x -wave-like triplet . (11)In order to solve eq. (7), we substitute these candidates for Ψ λ k and Ψ ν k ′ , and integrate for k z , k ′ z , k x , k ′ x , k y , and k ′ y . Then, we can safely reduce our original eigenvalue problem for Ψ λ k toan eigenvalue problem for a λ [ a λz ( m z ), a λx ( m z m x ), and a λy ( m y )] in order to obtain the moststable pairing state. When we solve it numerically by the standard method, we obtain boththe eigenvalue 1 / log e ∆ sc and the eigenvector a λ [ a λz ( m z ), a λx ( m z m x ), and a λy ( m y )]. Here, tosolve eq. (7) within SOPT for singlet-pairing states, we have V ij ( q ) = U δ ij + U N X κρ k z κi ( q + k ) z ρj ( k ) (cid:16) − f κ q + k (cid:17) f ρ k ε κ q + k − ε ρ k . (12)Meanwhile, for triplet-pairing states, V ij ( q ) = − U N X κρ k z κi ( q + k ) z ρj ( k ) (cid:16) − f κ q + k (cid:17) f ρ k ε κ q + k − ε ρ k , (13)where f ρ k = 12 (cid:20) − tanh (cid:18) ε ρ k − µ T (cid:19)(cid:21) , (14)and T denotes the temperature.
3. Results and Discussion
In our present analyses, all ε ν k and z λi ( k ) in eq. (7) are first calculated for the k − points onan equally spaced mesh in FBZ for each band. The mesh size along k z is 108, and the sizes along k x and k y are both 64. Then, we calculate V ij ( k + k ′ ) in eq. (7) only for k − and k ′ − pointssatisfying the conditions ε λ k = µ and ε ν k ′ = µ , respectively. When we calculate V ij ( k + k ′ )according to eqs. (12) and (14), we set the temperature T = 0 .
001 eV ≈
10 K, at which oursystem can be considered to behave similarly to the ground state. These calculations havebeen performed at U = 0 . t dp = 1 .
00 eV, t pp = − .
50 eV, t ′ pp = − .
10 eV, and ε dp = 2 .
60 eV. Theseparameters are determined using examples from studies of the three-band Hubbard ladder . Phys. Soc. Jpn.
Full Paper model,
10, 20) local-density approximation, and angle-resolved photo-emission spectroscopy(ARPES).
As mentioned in the last section, varying the pressure and Ca doping in real Sr14-24-41can be reproduced by changing t ⊥ and ∆ V , respectively, in our model. Thus, in order tocomprehensively understand the superconductivity of spin-chain ladder cuprate, we calculatethe log e ∆ sc of the most stable superconducting state for various t ⊥ and ∆ V values. All throughthese calculations, we fix average holes for Cu [ n h (Cu )] to 0 .
25 as well as the value of realSr14-24-41. Here, n h (Cu ) is defined as n h (Cu ) ≡ [2 n hLadder (Cu ) + n hChain (Cu )] / n hLadder (Cu ) and n hChain (Cu ) are average holes per Cu in the ladder and chainlayers, respectively. These values should be estimated as the total holes in our model in orderto be compared with those of real Sr14-24-41. We find that the most stable superconductingstate of our five candidates is always the p z -wave-like triplet, represented by eq. (10). Weconfirm that log e ∆ sc estimated for this state with M = 16 does not differ from that with M = 15 by more than 8%. The following discussion is therefore restricted to the p z -wave-liketriplet state with M = 16.In Fig. 2, we summarize how log e ∆ sc , n hLadder (Cu ) and n hChain (Cu ) depend on t ⊥ forvarious ∆ V values. Figure 2(a) shows the log e ∆ sc dependence on t ⊥ for ∆ V = − .
12 eV and∆ V = − .
11 eV, as well as the T c dependence on pressure in real Sr14-24-41, i.e., it has amaximum at an intermediate t ⊥ . Since the temperature T = 0 .
001 eV ≈
10 K is thought to besufficiently low, T c should be proportional to ∆ sc ( T ) within the BCS theory. Thus, our resultsof the log e ∆ sc dependence on t ⊥ for ∆ V = − .
12 eV and ∆ V = − .
11 eV can qualitativelyreproduce the T c dependence on pressure in real Sr14-24-41. Moreover, as shown in Figs. 2(b)and 2(c), Cu holes are transferred from the chain layer to the ladder layer with an increasein t ⊥ , which also qualitatively agrees with the experimental results under high pressures.
25, 34)
On the other hand, for ∆ V = − .
10 eV, log e ∆ sc increases with an increase in t ⊥ . For∆ V = − .
09 eV, log e ∆ sc decreases with an increase in t ⊥ , but it tends to be much largerthan those in the other cases. The fact that the behavior of log e ∆ sc differs depending on∆ V can be explained by the configuration of the Fermi surface and the density of state(DOS) on it. Let us show the DOS’s on the Fermi surface for ∆ V = − . , − . , − . , and − .
09 eV in Figs. 3–6, respectively. For ∆ V = − . , − . , and − .
10 eV, we have twoquasi-one-dimensional (Q1D) branches of the Fermi surface, which we call “Branch 1” and“Branch 2”, as shown in Figs. 3–5, respectively. Branch 2 becomes increasingly warped as t ⊥ increases, and simultaneously the DOS on the branch is enhanced. A large DOS is favorablefor superconductivity. However, the warped Fermi surface makes Fermi surface nesting worse.The effective pair scattering V ij ( q ), defined by eqs. (12) and (14), is mainly enhanced byFermi surface nesting; thus, the warped Fermi surface suppresses pair instability. Owing tothese two conflicting effects on the superconductivity, the log e ∆ sc values for ∆ V = − . . Phys. Soc. Jpn. Full Paper T n C ha i nh ( C u ) + ∆ V=-0.12 ∆ V=-0.11 ∆ V=-0.10 ∆ V=-0.09 -7-6-5-4-3-2-1 0 0.01 0.02 0.03 0.04 0.05 ∆ V=-0.12 ∆ V=-0.11 ∆ V=-0.10 ∆ V=-0.09 l og ∆ e S C t (eV) T (a)-0.125-0.12-0.115-0.11-0.105 0.01 0.02 0.03 0.04 0.05t (eV) T n Ladde r h ( C u ) + ∆ V=-0.12 ∆ V=-0.11 ∆ V=-0.10 ∆ V=-0.09 (b)(c)
Fig. 2. (Color online) (a) log ∆ sc , (b) n hLadder (Cu ), and (c) n hChain (Cu ) for ∆ V = − . − . − .
10, and − .
09 eV. and − .
11 eV have maxima at approximately t ⊥ ≈ .
035 and 0 .
030 eV, respectively, when t ⊥ varies. Although this situation on the Fermi surface is common for ∆ V = − .
10 eV, the DOSon Branch 2 is enhanced more rapidly with t ⊥ , as shown in Fig. 5 (note that the color bar scalein Fig. 5 is about twice as large as those in Figs. 3 and 4), and such a large DOS surpassesthe other effect due to the warped Fermi surface in this case. This is the reason why log e ∆ sc for ∆ V = − .
10 eV increases with an increase in t ⊥ . On the other hand, for ∆ V = − .
09 eV,we have three branches of the Fermi surface. Of these, two are Q1D branches and are calledsimilarly to those for ∆ V = − .
12 eV. The other branch of the Fermi surface is a quasi-two-dimensional (Q2D) branch, newly labeled “Branch 3” in Figs. 6(a) and 6(b). Noting that the . Phys. Soc. Jpn.
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Fig. 3. (Color online) Densities of states (DOS’s) on Fermi surface for ∆ V = − .
12 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV. color bar scale in Fig. 6 is about 100 / t ⊥ . This is why log e ∆ sc for ∆ V = − .
09 eV always becomes large. However, as shown in Figs. 6(b) and 6(c), Branch 3becomes smaller and finally disappears as t ⊥ increases. This leads to a decrease in total DOSand the degradation of superconductivity.When we compare the Fermi surface obtained by our present calculation with that ob-served by ARPES, we find that the results for ∆ V = − . − .
11, and − .
10 eV aresimilar. In other words, in real Sr14-24-41, the charge imbalance between the ladder andchain is too large to have three branches of the Fermi surface, as observed in our results for∆ V = − .
09 eV. Thus, the superconductivity in spin-chain ladder cuprate may be enhanced . Phys. Soc. Jpn.
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Fig. 4. (Color online) DOS’s on Fermi surface for ∆ V = − .
11 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV. if the large charge imbalance between the ladder and the chain can be resolved, for example,by varying the elements of the spacer layer without changing their valences.Hereafter, we discuss the gap functions of the p z -wave-like superconducting state in detail.Ψ λ k on the Fermi surface for ∆ V = − .
12 eV and for ∆ V = − .
09 eV are shown in Figs. 7and 8, respectively. For ∆ V = − .
12 eV, Ψ λ k changes its sign on the disconnected parts ofBranch 1, and it does so on those of Branch 2, as shown in Fig. 7. This is similar to thetriplet state derived by Sasaki et al. Furthermore, Ψ λ k has no nodes on the connected partsof each branch. Thus, this superconducting state is expected to behave as a fully gapped statefor the microscopically experimental probe. Actually, the relaxation rate T − of the NMRmeasurement suggests that the superconducting state of Sr14-24-41 at pressures of 3 . . . Phys. Soc. Jpn. Full Paper
Fig. 5. (Color online) DOS’s on Fermi surface for ∆ V = − .
10 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV. has an s -wave-like character.
41, 42)
Moreover, in their works, the Knight shift of Cu nuclei forthe ladder derived from high fields shows no change below T c . This fact strongly suggests thatthe superconducting state in real Sr14-24-41 can be a singlet-Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state or a triplet superconducting state, and the latter is consistent with our resultfor ∆ V = − .
12 eV.However, the fully gapped superconductivity for ∆ V = − .
12 eV does not remain for∆ V = − .
09 eV. As shown in Fig. 8, Ψ λ k has nodes on Branch 3 or 2. These results indicatethat our p z -wave-like gap function is not robust in terms of the absence of nodes. If the NMRmeasurement can proceed in Sr14-24-41 under different conditions, we may observe T − forthe gapless superconductivity. . Phys. Soc. Jpn. Full Paper
Fig. 6. (Color online) DOS’s on Fermi surface for ∆ V = − .
09 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV.
Finally, note that the amplitude of Ψ λ k on Branch 3 is smaller than that on Branch 1 or 2,as shown in Fig. 8. The large DOS on Branch 3 mainly enhances the superconductivity on theother branches, not on itself. This synergistic effect is caused by the interband interaction,i.e., V ij ( k + k ′ ) z λi ( k ) z νj ( k ′ ) for λ = ν in eq. (6), originating from the mixing between d-and p-orbitals. Thus, the hybridization effect also plays a significant role in enhancing thesuperconductivity in this material.
4. Conclusions
We have demonstrated that the 3D d-p model with the Q1D structure similar to Sr14-24-41can represent p z -wave-like triplet superconductivity up to the second order in the perturbationtheory framework. On the basis of this model, we can reproduce the T c dependence on pressure . Phys. Soc. Jpn. Full Paper
Fig. 7. (Color online) Ψ λ k on the Fermi surface for ∆ V = − .
12 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV. by changing the interlayer coupling. The calculated results on the Fermi surface configura-tion and the superconducting state can give a comprehensive picture to explain the ARPESand NMR experimental results. Moreover, our results show the possibility of enhancing thesuperconductivity if the charge imbalance between the ladder and the chain can be decreasedby varying the elements of the spacer layer. This speculation is based on the fact that thehybridization effect due to interlayer coupling and the other mixing between d- and p-orbitalsis crucial to enhance the superconductivity. . Phys. Soc. Jpn.
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Fig. 8. (Color online) Ψ λ k on the Fermi surface for ∆ V = − .
09 eV: (a) for t ⊥ = 0 .
010 eV, (b) for t ⊥ = 0 .
030 eV, and (c) for t ⊥ = 0 .
050 eV.
Acknowledgments
The authors are grateful to Professors K. Yamaji and I. Hase for helpful discussions. Theauthors also thank the referee for invaluable comments. . Phys. Soc. Jpn.
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