Phase diagram of high-temperature superconducting cuprates and pnictides

Phase diagram of high-temperature superconducting cuprates and iron pnictides is considered based on the theory emphasizing that the electronic state of superconductors can be described by the composed fermions. The theory is constructed with the Hamiltonian so modified by the unitary transformation as to apply the mean field approximation. It is indicated that the phase diagram of these superconductors can be explained from the viewpoint unifying the electronic states of superconducting cuprates and pnictides. The issues relevant to exotic phases are also discussed.


Introduction
Though there has been much study about high-temperature superconductors such as cuprates etc, its microscopic understanding still remains an insufficient stage [1][2][3][4][5][6][7][8]. Recently, the author proposed a unified theory to explain generally the properties of high-temperature superconducting cuprates and iron pnictides [9]. The theory is based on the extended d-p model emphasizing that the electronic state of superconductors can be described by the composed fermions constructed with d-p operators, and is indicated to provide a universal explanation about the superconducting and normal properties of these superconductors. In this paper the phase diagrams of high-temperature superconducting cuprates and iron pnictides will be investigated and then evaluated comparing with the experimental facts.

Basic equations
The Hamiltonian is assumed to be an extended d-p model for a single layer of square planar where the operator  Since the operator   inl p is not orthogonal between the neighboring M sites, it does not exactly satisfy anti-commutation relations. However it is approximated here that   inl p can be well-defined fermion operator.
To the second order in perturbation theory on the condition of , U U nn n nl let us find out the effective Hamiltonian of Eq. (3). For simplicity, Hund coupling and the interaction between different 3d orbitals are neglected. First in the second term of Eq. (3), the terms acting on the interaction U n as a perturbation can be selected as follow as where H' indicates the terms in the presence of fermions with the opposite spin at either d or p sites because the ground state is not assumed here to include the double occupancy states. Further, let us average the occupancy factors such as The Coulomb interaction U n in the ground state can be effectively eliminated from the starting Hamiltonian because double occupancy states at d-sites are inhibited by Coulomb repulsion. Considering the second term except H', Though the former part of the forth term in (5) corresponds to the kinetic energy due to the hopping between sites, it is neglected here due to the second order kinetic energy. The latter part corresponds to the effective anti-ferromagnetic interaction between d-p fermions. Thus, the effective Hamiltonian is finally obtained as Coulomb interaction between the nearest neighboring sites in (1), : Now the ground state of effective Hamiltonian (6) will be generally considered in two states depending on the doping conditions. One of these is the states in the neighborhood of the insulator (socalled Mott insulator) and the other is the superconducting state based on the band picture. First let us consider the region in the neighborhood of the insulator. The effective Hamiltonian (6) is appropriate in this case because its representation in real space is very useful in the treatment of this region. In order to enable to apply the mean field approximation, the operators where b i + , c i + are defined as the mixing operators of M and L fermions. For the condition of ), are omitted to avoid the complex representation. Since there appear many interaction terms in the transformed Hamiltonian (8), this transformation might make the problem somewhat difficult. However, the mixing of d and p fermions can be directly built in this Hamiltonian, which consists of two free fermions and the interaction terms between them. As a result this treatment has an advantage to enable to apply the mean field approximation because of the explicit representation of free fermions and their interaction terms. Next let us consider the case of the superconducting state based on the band picture. Since the representation in momentum space is appropriate in this region, the effective Hamiltonian of (6) is transformed into refers to the axis for the unit cell, and N is number of M sites in a single layer. The operators , are reconstructed with fermion operators defined by unitary transformation are omitted to avoid the complex representation. Based on these effective Hamiltonians, the phase diagram of cuprate and pnictide superconductors will be considered in next chapters.

Electronic state
The characteristics of cuprate superconductors is that it is a single d-band, p-band (n = 1, l =1), superconductive in hole doping, and insulator in non-doping state. Though p-band is multi-orbitals, a single band of l = 1 will be only considered here. First let us consider the case of the nearly non-doped region. Assuming that there is no p-holes in half-filling, the relations of Here N v (the valence ratio of d to p) = 2 and ) 5 .
For the parameters of cuprates, it will be selected that eV. 2 1 eV, 1 eV, 5 eV, 1 eV, It is well known that the ground state of superconducting cuprates shows antiferromagnets (so-called Mott insulator) in the non-doped region [10,11]. Using the composite operators defined here, the non-doped wave function corresponds to where A or B shows the sub-space of antiferromagnetic lattice and N is number of Cu sites in a single layer. Since there exists no p-hole in half-filling Mott insulator, it will not be necessary to consider the interaction V in this case. However, in the existence of p-holes, since is interpreted to be attractive due to d-p exchange antiferromagnetic interaction, the case of 0  V will play an important role for determining the ground state. Considering the experimental facts that the doping can almost supply the p-holes, doping holes are expected to occupy the quantum state corresponding to this situation. For the condition α 2 < β 2 which allows the reliable value of  p and , the term can be the most dominant interaction. This means that b-c or c-b pair will mainly contribute to determine the ground state. Thus, for the equivalency of sites, the normalized wave function is assumed to be where the coefficient t indicates the probability of local-antiferro pair state. On the approximation using the relation of (δ: the doping ratio relative to half-filling), the ground-state energy is given by The energy of the excited states will be evaluated by using the presumed excited state wavefunction as follow as This is orthogonal to the ground state function and corresponds to breaking up a b-c (c-b) pair in i´, the spin-up member going to The excited energy is obtained as This indicates that there is the energy gap between the ground and the excited states. Thus, the wave function (14) will be identified as the pseudogap (PG) state in the nearly non-doped region. The PG energy is then estimated to be Δ p = |V| and the PG temperature T p is approximated as the  p (< 1). What the excited state (16) (14) and (16) will form the narrow bands reflecting the wavefunction overlap of the neighboring sites.
Next let us consider the case in the neighborhood of optimally doped region. In this doping region the relation of    5 0 will be generally reasonable. On the condition of ), )( ( Then, the gap equation is obtained as where the relation is defined, measuring the energy relative to the Fermi level εF. Replacing the sum in gap equation by an integral, the solution which is even in k is given by Δ k = Δ 0 α k β k (cosk x ±cosk y ). The solution decreasing the Coulomb interaction is Δ k ∝(cosk x -cosk y ) which agrees with the experimental fact about the anisotropic gap of high-temperature cuprate superconductors [12]. Thus Δ 0 is determined by the relation k .
Using several approximation on the band structure, the superconductive maximum gap energy and the ground-state energy are given by where N F is the density of states at Fermi level and ).
Denote that the energy relative to Cu 3d level is measured and the contribution of the   k c fermions to the total system energy is considered.

Phase diagram
The antiferromagnetic long-range order between copper sites disappears by holes doping, but there still exists the short-range order interaction which plays an important role in the presence of holes. As shown in the preceding section, the ground states in the neighborhood of the insulator and the superconducting region can be then calculated. However, the doping dependency of these states is not clear. There may be possible intermediate cases constructed with these mixed states. It will be then needed to consider the over-all phase diagrams. Now, an overall wavefunction is assumed to be .
Here δ p , δ s are the doping quantity of PG and superconductive states and the total doping quantity δ is defined to be δ = f δ p + (1-f) δ s . The total free energy is then given by G = f E p (δ p )+ (1-f) E s (δ s ), (0 < f < 1) and the intermediate region is determined by minimizing G with respect to δ, δ p , δ s . Let us evaluate the doping quantity δ pm , δ sm by minimizing E p (δ p ) and E s (δ s ) and the minimum point of the total free energy in the condition of δ pm < δ sm . As the case of δ < δ pm means δ p = δ, δ s = 0, the system will show a pure PG state. At δ = δ pm superconductivity appears in the PG state and the mixed state will be maintained until the PG state disappear at the doping quantity δ sm satisfying δ = δ s . In more increasing of δ there exists a pure superconductor. Thus, the overall wave function will be represented as three regions of f = 1 (δ < δ pm ), f = (δ sm -δ)(δ sm -δ pm ) -1 (δ pm < δ < δ sm ), and f = 0 (δ > δ sm ), respectively. For both regions of δ < δ pm and δ pm < δ < δ sm , Δ p and T p will decrease with increase of δ and vanish at δ = δ sm . Since the superconductive region appearing at δ = δ pm can be identified to be composed of very small superconductive region, the sufficient coherency to detect the net superconductivity will be less formed. Therefore, the superconductive state for δ pm < δ < δ sm could be given by the effective gap Δ eff = (δ-δ pm ) (δ s -δ pm ) -1 Δ s (δ s ). This also indicates that the superfluid density δ sf changes as δ-δ pm in the region of δ pm < δ < δ sm . The superconductor will be the optimally doped state at the neighborhood of δ = δ sm , and in the over-doped region Δ s and T c will decrease with δ according to doping dependency. Thus, an overall phase diagram is theoretically obtained by considering three types of electronic states.

Exotic phases
Recently there has been much interest in various exotic phases of cuprates superconductors etc. [13]. Here let us discuss briefly the charge density wave (CDW), electron-nematic order and strange metal in normal state. First, it has been reported that CDW can coexist with superconductivity in underdoped region [14]. It has been confirmed that the observed CDW is static and short-range order. According to this paper, the state in underdoped region is constructed with PG and superconductive mixed phases, and the mixing state is evaluated by minimizing simply the free energy of these phases. However, more exactly, it is necessary to consider the space-like change of charge distribution due to the mixing of phases. Then, though PG ground state in mixing phase is not equivalent to CDW, it will possibly contain the charge order such as CDW. It has been also reported that the electron-nematic order can exist in the normal state of underdoped region [15]. Such orders have been interpreted as phases breaking the fourfold symmetry, which are translational invariant, but break rotational symmetry. The symmetry breaking suggests that the two oxygen ions in the copper-oxide unit are inequivalent. Since the electron-nematic order occurs in the PG phase, it may be related with the excited PG state. The reason is that there are the excitations of O holes in the excited PG state. Last let us discuss the strange metal. This has been discussed in the relation with quantum critical point (QCP) [13]. The strange metal region in phase diagram has been explained by interpretation that in temperature plane there is a quantum critical wedge opening up from the QCP. Here, since the optimally doped point is the same as minimum doping of superconductivity without PG state, this point will be interpreted as the QCP.

Electronic state
The characteristics of pnictide superconductors is that it is multi d band, p band (n >1, l >1) metal in non-doping, and superconductive in electron doping [16]. For the parameters of pnictides, it will be selected that eV.
Since the parent materials of pnictide are metal in non-doping, it seems to be appropriate to treat nearly nondoped region in the momentum representation. However, considering the antiferromagnetism (AFM) in non-doping, it is considered here to start from Eq. (8) in the real space presentation. The metallic state in nearly non-doped region can be interpreted to originate from the nonexistence of PG state due to the relation of .
This situation may correspond to the metallic state. In the momentum representation, V c can be neglected due to the screening effect of carriers, then .
In similarity with the cuprates, the superconductive gap equation derived from Eq. (9) is given by Since Fe-3d electrons are multi orbitals, it is needed to select the dominant orbitals of Fe contributing to superconductivity [17].
where the half of Fe-Fe distance is used as the length unit. In the case of d xz (n = 1), replacing the sum in (24) by an integral, the solution which is even in k is given by . cos (1111 system). The interesting thing is that the optimal doping level indicating the highest T c is decreased with decreasing the size of rare earth ion [21]. According to the theory, ].
Since doping dependence of T c is determined by V(1-cδ), it will be needed to consider this variance relating with the size of RE ion. On the condition of fixed V (d-p AF interaction), decreasing the ion size will weaken the correlation of fermions in the system. As a result, V will be inclined to increase, and at the same time Fe-Fe AF interaction will weaken by carrier doping. Since the value of V and c increase with decreasing the size of RE ion, the optimal doping level decreases with increasing of T c . Considering the above relations, the fitting curves of doping dependency of T c are shown in figure 2. Theoretical doping dependency of T c is consistent with the data of the highest T c in REFeAsO 1-x H x superconductors.

Conclusions
Phase diagrams of high-temperature superconducting cuprates and iron pnictides are investigated by extended d-p theory. In cuprates, PG and superconductive gap energy are evaluated, emphasizing on the minimum and optimal doping points in the superconductive state which are indispensable for determining the phase diagrams. It is then found that the phase diagram can be constructed with pure pseudo gap and superconductive states, and their mixed state. In pnictides, the phases constructed with the anti-ferromagnetism and superconductor can be explained in similar situation with the case of cuprates. The theory can also explains how the superconductive phases of REFeAsO 1-x H x change with the size of RE ion.