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Superconductivity in doped two-leg ladder cuprates

Within the t-J ladder model, superconductivity with a modified d-wave symmetry in doped two-leg ladder cuprates is investigated based on the kinetic energy driven superconducting mechanism. It is shown that the spin-liquid ground-state at the half-filling evolves into the superconducting ground-state upon doping. In analogy to the doping dependence of the superconducting transition temperature in the planar cuprate superconductors, the superconducting transition temperature in doped two-leg ladder cuprates increases with increasing doping in the underdoped regime, and reaches a maximum in the optimal doping, then decreases in the overdoped regime.

In recent years two-leg ladder cuprates have attracted great interest since their ground state may be a spin liquid state with a finite spin gap 1-3 . This spin liquid state may play a crucial role in superconductivity of the planar cuprate superconductors as emphasized by Anderson 4 . When carriers are doped into two-leg ladder cuprates, a metal-insulator transition occurs [5][6][7] . Although the ambient pressure ladder superconductivity was not oberved until now, superconductivity in one of the doped twoleg ladder cuprate Sr 14−x Ca x Cu 24 O 41 has been observed under high pressure [5][6][7] , which is the only known superconducting (SC) copper oxide without a square lattice. It has been shown that most important role of pressure for realizing superconductivity is the doped hole redistribution between chains and ladders [5][6][7] , and then the number of charge carriers on the ladders is increased 8,9 . Moreover, the structure under high pressure remains the same as the case in ambient pressure 7 , and the spin background in this SC phase does not drastically alter its spin gap properties 2 . The particular geometrical arrangement of the Cu ions in two-leg ladder cuprates provides a playground for the normal-and SC-state studies of low-dimensional strongly correlated materials 1-3 . This follows from the fact that all planar cuprate superconductors found up now contain square CuO 2 planes 10 , whereas doped two-leg ladder cuprates consists of twoleg ladders of other Cu ions and edge-sharing CuO 2 chains 1-3 . By virtue of the nuclear magnetic resonance and nuclear quadrupole resonance, particularly inelastic neutron scattering measurements, it has been shown that there is a region of parameter space and doping where doped two-leg ladder cuprates in the normal state is an antiferromagnet with the commensurate short-range order 3, 11 . Moreover, transport measurements on doped two-leg ladder cuprates in the same region of parameter space and doping indicate that the resistivity is linear with temperatures 6 , one of the hallmarks of the exotic normal state properties found in the planar cuprate superconductors 10 . These experimental results have revealed some close analogies between the doped planar cuprates and doped two-leg ladder cuprates. The normalstate of doped two-leg ladder cuprates exhibits a number of anomalous properties which is due to the charge-spin separation (CSS), while the SC state may be characterized by the charge-spin recombination.
Theoretically, there is a general consensus that the charge carrier pair of doped two-leg ladder cuprates in the SC-state are created on the ladders 2 , i.e., superconductivity develops mainly within the ladders, with a minor role played by the interladder hopping amplitude. Within the t-J ladder model, many authors have shown that the charge carrier pair correlation is very robust 12 , clearly indicative of a ground-state dominated by strong SC tendencies. Moreover, it has been shown in the renormalized mean-field (MF) theory that superconductivity should exist in the d-wave channel 13 , which has been confirmed by variety of numerical simulations 14 . Within the framework of the CSS fermion-spin theory 15 , we have developed a kinetic energy driven SC mechanism 16 , where the dressed holons interact occurring directly through the kinetic energy by exchanging the spin excitations, leading to a net attractive force between the dressed holons, then the electron Cooper pairs originating from the dressed holon pairing state are due to the charge-spin recombination, and their condensation reveals the SC ground-state. This SC-state is controlled by both SC gap function and quasiparticle coherence, then the maximal SC transition temperature occurs around the optimal doping, and decreases in both underdoped and overdoped regimes 17 . In particular, this kinetic energy driven SC mechanism does not depend on the fine details of a lattice structure, and the main ingredient was identified into a charge carrier pairing mechanism involving the internal spin degree of freedom 16 . Therefore this SC mechanism shows that the strong electron correlation favors superconductivity 16,17 . Since there is a remarkable resemblance in the normal-state properties between the doped planar cuprates and doped two-leg ladder cuprates as mentioned above, and the strong electron correlation is common for both these cuprate materials, then two systems may have similar underlying SC mechanism, i.e., it is possible that superconductivity in doped two-leg ladder cuprates is also driven by the kinetic energy. In this paper, we discuss superconductivity in doped two-leg ladder cuprates along with the kinetic energy driven SC mechanism. We show that in analogy to the doping dependence of the SC transition temperature in the planar cuprate superconductors, the SC transition temperature in doped two-leg ladder cuprates increases with increasing doping in the underdoped regime, and reaches a maximum in the optimal doping, then decreases in the overdoped regime.
The basic element of two-leg ladder materials is the two-leg ladder, which is defined as two parallel chains of ions, with bonds among them such that the interchain coupling is comparable in strength to the couplings along the chains, while the coupling between the two chains that participates in this structure is through rungs 1,2 . In this case, the t-J ladder model on the two-leg ladder is expressed as, whereη = ±c 0x , c 0 is the lattice constant of the twoleg ladder lattice, which is set as unity hereafter, i runs over all rungs, σ(=↑, ↓) and a(= 1, 2) are spin and leg indices, respectively, C † iaσ (C iaσ ) are the electron creation (annihilation) operators, S ia = C † ia σC ia /2 are the spin operators with σ = (σ x , σ y , σ z ) as the Pauli matrices, and µ is the chemical potential. This t-J ladder Hamiltonian (1) is subject to an important on-site local constraint σ C † iaσ C iaσ ≤ 1 to avoid the double occupancy. In the materials of interest 5-7 , the exchange coupling J along the legs is close to the exchange coupling J ⊥ across a rung, and the same is true of the hopping t along the legs and the rung hopping strength t ⊥ . Therefore, in the following discussions, we will work with the isotropic system J ⊥ = J = J, t ⊥ = t = t. On the other hand, the single occupancy local constraint in the t-J ladder Hamiltonian (1) can be treated properly in analytical calculations within the CSS fermionspin theory 15 where the spinful fermion operator h iaσ = e −iΦiσ h ia describes the charge degree of freedom together with some effects of the spin configuration rearrangements due to the presence of the doped hole itself (dressed holon), while the spin operator S ia describes the spin degree of freedom (spin), then the electron local constraint for single occu- , is satisfied in analytical calculations, and the double spinful fermion occupancies h † iaσ h † ia−σ = e iΦiσ h † ia h † ia e iΦi−σ = 0 and h iaσ h ia−σ = e −iΦiσ h ia h ia e −iΦi−σ = 0, are ruled out automatically. Since these dressed holons and spins are gauge invariant, and then in this sense, they are real and can be interpreted as the physical excitations 18 . Although in common sense h iaσ is not a real spinful fermion, it behaves like a spinful fermion. In this CSS fermionspin representation, the low-energy behavior of the t-J ladder Hamiltonian (1) can be expressed as, In this CSS fermion-spin representation, the kinetic terms have been expressed as the dressed holon-spin interactions, which reflect that even the kinetic energy terms in the t-J ladder Hamiltonian have the strong Coulombic contribution due to the restriction of no doubly occupancy of a given site. As in the planar cuprate superconductors 19 , the SC state in doped two-leg ladder cuprates is also characterized by electron Cooper pairs 5-7 , forming SC quasiparticles. Because there are two coupled t-J chains in the two-leg ladder cuprates, therefore the order parameters for the electron Cooper pair is a matrix ∆ = ∆ L + σ x ∆ T , with the longitudinal and transverse order parameters are defined as, respectively, where the longitudinal and transverse dressed holon pairing order parameters are expressed as, In this case, the physical properties of doped two-leg ladder cuprates in the SC state are essentially determined by the dressed holon pairing state, i.e., the SC order parameters are determined by the dressed holon pairing amplitude, and are proportional to the number of doped holes, and not to the number of electrons. For discussions of superconductivity in the doped two-leg ladder cuprates, we now introduce the dressed holon normal and anomalous Green's functions and spin Green's functions as, , respectively, where the longitudinal and transverse parts of these Green's functions are defined as, with a ′ = a. In the MF level, the spin system is an anisotropic away from the half-filling 20 , therefore we have defined the two spin Green's functions D(k, ω) and D z (k, ω) to describe the spin propagations, while the longitudinal and transverse parts of the MF dressed holon normal Green's function and MF spin Green's functions have been obtained as 20 , , and the MF dressed holon and spin excitation spectra, where Within the Eliashberg's strong coupling theory 21 , it has been shown that the dressed holon-spin interaction in the doped planar cuprates can induce the dressed holon pairing state (then the electron Cooper pairing state) by exchanging the spin excitations in the higher power of the doping concentration 16 . Following their discussions 16 , the self-consistent equations that satisfied by the full dressed holon normal and anomalous Green's functions in doped two-leg ladder cuprates are expressed as 21 , where the self-energy functions Σ 2T (k, ω), with the longitudinal and transverse parts are evaluated as, where γ p+p ′ +k = 2tcos(p x + p ′ x + k x ), and the spin bubbles (k, ω) are even functions of ω. Now we define the quasiparticle coherent weights as Z 1T (k, ω). As in the case of the planar cuprate superconductor 17 , the retarded function ReΣ (k, ω) is a constant, independent of (k, ω), and just renormalizes the chemical potential, therefore it can be dropped. Since we only discuss the low-energy behavior of doped two-leg ladder cuprates, then the effective dressed holon pair gap functions and quasiparticle coherent weights can be discussed in the static limit, i.e.,∆ h (k) = Σ 1T (k, ω) | ω=0 , then the longitudinal and transverse parts of the dressed holon normal and anomalous Green's functions in Eq. (8) can be written explicitly as, where the dressed holon quasiparticle coherence factors U 2 νk = [1 +ξ νk /E νk ]/2 and V 2 νk = [1 −ξ νk /E νk ]/2, the renormalized dressed holon excitation spectrumξ νk = Z (ν) F A (k)ξ νk , the renormalized dressed holon pair gap function∆ still are a function of k, the wave vector dependence may be unimportant, since everything happens at the electron Fermi surface 21 . In this case, we need to estimate a special wave vector k 0 that guarantees Z (ν) F A (k 0 ) near the electron Fermi surface. Following the discussions in the case of the planar cuprate superconductors 17 , this special wave vector can be obtained as k 0 = k A − k F , with k A = π and k F ≈ (1 − x)π/2, then we only need to calculate Z (ν) F A (k 0 ). On the other hand, many authors have shown that superconductivity in doped two-leg ladder cuprates possesses a modified d-wave symmetry 13,14 , and the gap function in this modified d-wave symmetry can be expressed as, ∆ (1) for the antibonding and bonding cases, respectively. In this case, the dressed holon effective gap parameters and quasiparticle coherent weights in Eq. (9) satisfy following four equations, where the kernel functions Λ (1) νν ′ ν ′′ (k, q, p) and Λ (2) νν ′ ν ′′ (q, p) are evaluated as, . These four equations (11) must be solved together with other self-consistent equations, then all the above dressed holon effective gap parameters, quasiparticle coherent weights, dressed holon particlehole order parameters, decoupling parameter α, spin correlation functions, and chemical potential µ are determined by the self-consistent calculation 20 . With the helps of the above discussions, we now obtain the longitudinal and transverse parts of the dressed holon pair order parameter in Eq. (4) in terms of the dressed holon anomalous Green's functions (10c) and (10d) as, As in the case of the planar cuprate superconductors 16,17 , this dressed holon pairing state originating from the kinetic energy terms by exchanging the spin excitations in doped two-leg ladder cuprates also leads to form the electron Cooper pairing state 16 . For discussions of superconductivity in doped two-leg ladder cuprates, we need to calculate the electron anomalous Green's function Γ † (k, ω) = Γ † L (k, ω) + σ x Γ † T (k, ω), with the longitudinal and transverse parts are defined as, Γ † a). These longitudinal and transverse parts of the electron anomalous Green's func-tion are the convolutions of the corresponding longitudinal and transverse parts of the dressed holon anomalous Green's function and spin Green's function in the CSS fermion-spin theory, and reflect the charge-spin recombination 22 . In terms of the MF spin Green's functions in Eqs. (6c) and (6d) and dressed holon anomalous Green's functions (10c) and (10d), we obtain the longitudinal and transverse parts of the electron anomalous Green's function, then the longitudinal and transverse parts of the SC gap function ∆(k) = ∆ L (k) + σ x ∆ T (k) are evaluated as, which shows that the symmetry of the electron Cooper pair in doped two-leg ladder cuprates is essentially determined by the symmetry of the dressed holon pairs. In this case, the SC gap function is written as Our result in Eq. (15) also shows that the SC transition temperature T c occurring in the case of the SC gap parameter ∆ = 0 is identical to the dressed holon pair transition temperature occurring in the case of the dressed holon pairing gap parameter ∆ h = 0. In correspondence with the SC gap parameter, the SC transition temperature T c as a function of the hole doping concentration δ for t/J = 2.5 is plotted in Fig. 2. For comparison, the experimental result of the SC transition temperature T c in the doped two-leg ladder cuprate Sr 0.4 Ca 13.6 Cu 24 O 41 as a function of pressure 7 is also shown in Fig. 2 (inset). Experimentally, it has been shown that the main effect of pressure in doped two-leg ladder cuprates is to reduce the distance between the ladders and chains, which leads to the doped hole redistribution between chains and ladders [5][6][7] . However, the structure at ambient pressure with clearly defined ladders and chains remains stable under the pressure needed to induce the SC regime 7 , and the spin background in this pressure induced SC phase does not drastically alter its spin gap properties 2 . On the other hand, when Ca is doped upon the original Sr-based Ca-undoped phase, the interatomic distance ladder-chain was found to be reduced by Ca substitution, leading to a redistribution of holes originally present only on the chains 8,9 . These experimental results show that an increase of the pressure may corresponding to an increase in the number of charge carriers on the ladders [7][8][9] . In other words, the doping dependence of the SC transition temperature should be similar to the pressure dependence of the SC transition temperature. In this sense, our present result of the doping dependence of the SC transition temperature is qualitatively consistent with the experimental result 7 . The maximal SC transition temperature T c occurs around the optimal doping concentration x opt ≈ 0.07, and then decreases in both underdoped and overdoped regimes. In particular, this domed shape of the doping dependence of the SC transition temperature is same as that of the doping dependence of the longitudinal part of the SC gap parameter, which shows that superconductivity is mainly produced by the development of the pairing correlation along legs, and is consistent with the one-dimensional charge dynamics under high pressure 7,2 . Furthermore, T c in the underdoped regime is proportional to the hole doping concentration δ, and therefore T c in the underdoped regime is set by the hole doping concentration, which reflects that the density of the dressed holons directly determines the superfluid density in the underdoped regime.
The essential physics of superconductivity in the present doped two-leg ladder cuprate superconductors is the same as that in the planar cuprate superconductors 16,17 , i.e., the SC-order in doped twoleg ladder cuprate superconductors is controlled by both gap function and quasiparticle coherence, which is reflected explicitly in the self-consistent equations (11). The dressed holons (then electrons) interact by exchanging the spin excitations and that this interaction is attractive. This attractive interaction leads to form the dressed holon pairs (then electron Cooper pairs). The parent compound of doped two-leg ladder cuprates is a Mott insulator, when holes are doped into this insulator, there is a gain in the kinetic energy per hole proportional to t due to hopping, but at the same time, the spin correlation is destroyed, costing an energy of approximately J per site, therefore the doped holes into the Mott insulator can be considered as a competition between the kinetic energy (δt) and magnetic energy (J), and the magnetic energy decreases with increasing doping. In the underdoped and optimally doped regimes, the magnetic energy is rather large, then the dressed holon (then electron) attractive interaction by exchanging the spin excitations is also rather strong to form the dressed holon pairs (then electron Cooper pairs) for the most dressed holons (then electrons), therefore the number of the dressed holon pairs (then electron Cooper pairs) and SC transition temperature are proportional to the hole doping concentration. However, in the overdoped regime, the magnetic energy is relatively small, then the dressed holon (then electron) attractive interaction by exchanging the spin excitations is also relatively weak, in this case, not all dressed holons (then electrons) can be bounden as dressed holon pairs (then electron Cooper pairs) by this weak attractive interaction, and therefore the number of the dressed holon pairs (then electron Cooper pairs) and SC transition temperature decrease with increasing doping.
In summary, we have discussed superconductivity with the modified d-wave symmetry in doped two-leg ladder cuprates based on the kinetic energy driven SC mechanism. It is shown that the spin-liquid ground-state at the half-filling evolves into the SC ground-state upon doping. In analogy to the doping dependence of the SC transition temperature in the planar cuprate superconductors, the SC transition temperature in doped two-leg ladder cuprates increases with increasing doping in the underdoped regime, and reaches a maximum in the optimal doping, then decreases in the overdoped regime.
When this work was finished we became aware of the discovery of the analogous quasi-one dimensional cuprate superconductor 23 Pr 2 Ba 4 Cu 7 O 15−δ at ambient pressure. It has been shown 23 that in addition to the CuO 2 planes, the cuprate superconductor Pr 2 Ba 4 Cu 7 O 15−δ contains two CuO chains. In particular, these two single chains bound together like the two-leg ladder in two-leg ladder cuprates. This cuprate superconductor Pr 2 Ba 4 Cu 7 O 15−δ is called a double chain SC system, since the quasi-one dimensional double chain turns into the SC-state at ambient pressure, while the planes remain insulating even below the SC transition temperature. Although the Cu ions in the double chain do not form the two-leg ladder structure but a zigzag chain, the quasi-one dimensional nature of the double chain in the cuprate superconductor Pr 2 Ba 4 Cu 7 O 15−δ is the same as that of the twoleg ladder in two-leg ladder cuprates. This experimental measurement 23 on Pr 2 Ba 4 Cu 7 O 15−δ provides an evidence that the doped quasi-one dimensional cuprates can become SC at ambient pressure when the doped charge carriers distribute properly along the chains. These and related issue is under investigation now.