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Appearance of Antiferromagnetism and Superconductivity in Superconducting Cuprates

We represent the superconducting ceramic compounds by the single band extended Hubbard model. We solve this model for the simultaneous presence of antiferromagnetism and the d-wave superconductivity in the Hartree-Fock (H-F) and in the coherent potential (CP) approximation, which is applied to the on-site Coulomb repulsion U. The hopping interaction used in addition to the Coulomb repulsion causes rapid expansion of the band at carrier concentrations departing from unity. This expansion shifts the d-wave superconductivity away from the half filled point reducing its occupation range approximately to the experimental range in the YBaCuO compound. The CP approximation used for the Coulomb repulsion is justified by the large ratio of Coulomb repulsion to the effective width of perturbed density of states being reduced by the hopping interaction. Fast disappearance of antiferromagnetism observed experimentally is supported by the relatively large value of the total width of unperturbed density of states. It is shown that our model is capable of describing the electron doped compounds as well.


I. Introduction
The coexistence of superconductivity (SC) and antiferromagnetism (AF) phenomena was observed in many types of superconducting layered cuprates like Nd 2-x Ce x CuO 4 , La 2-x Sr x CuO 4 [1,2] or YBa 2 Cu 3 O 6+x , in organic superconductors [3], in Uranium-and Ceriumbased heavy-fermion systems [4], in Fe-based oxygen pnictide [5] and MFe 2 As 2 family [6,7]. Transition between these two phases depends on carrier doping (by holes or electrons) and on external pressure. In layered cuprates, Fe-based oxygen pnictides, and iron arsenide family the superconducting and antiferromagnetic state can be suppressed by chemical doping. Almost all high-temperature superconducting cuprates at half-filling are the antiferromagnetic insulator. Hole doping (in La 2-x Sr x CuO 4 and YBa 2 Cu 3 O 6+x ) or electron doping (Nd 2-x Ce x CuO 4 ) causes vanishing of the antiferromagnetic state and appearance of superconducting state in the metallic state. The main difference between the hole-doped and electron-doped cuprates is in the range of existing antiferromagnetic state. In the hole-doped cuprates AF vanishes already at small doping (~0.03) and before appearance of SC state. In the electron-doped cuprates the AF state exists up to the relatively high values of doping (~0. 15), and the SC state rises before the AF state is extinguish. Schematic diagram for the hole-doped and electron-doped superconducting cuprates is shown in Fig. 1. Phase diagram of YBa 2 Cu 3 O 6+x (as a function of carrier concentration) compound is very similar to the diagram of La 2-x Sr x CuO 4 compound after interchanging the dopant concentration to the hole number by the relation: 1 x n = − (see e.g. [8]), and is shown in Fig. 6 below. There are also numerous theoretical papers devoted to the problem of competition between AF and SC. Several methods are used to describe this phenomenon. The first attempts were done by use of the mean-field approximation ( [9][10][11]). Some authors carry out their computations using the dynamical mean-field theory (DMFT), which is based on the transition from the Hubbard model to the simple impurity Anderson model (SIAM) [12].
An early version of cluster DMFT was used by Lichtenstein [13]. The authors found that d-wave superconductivity and antiferromagnetism coexist over most of the doping range.
In the paper [14] the cellular dynamical mean-field theory (CDMFT) was used to compute zero-temperature properties of the two-dimensional square lattice Hubbard model (using the exact-diagonalization method). The authors have calculated the phase diagram that describes the competition between antiferromagnetism and superconductivity for both hole and electron doping at various values of U. They found, in general, homogeneous coexistence between AF and d-wave SC in the underdoped region (that feature is quite generally observed in quantum cluster methods, variational approaches, and mean-field theories that do not allow for spin or charge density modulations on large length scales). There is also another fully causal selfconsistent method i.e. the dynamical cluster approximation (DCA) (see e.g. [15]). Using this method one can reduce the complexity of the lattice problem by mapping it to a finite-size cluster self-consistently embedded in a mean field. The main difference with their classical counterparts arises from the presence of quantum fluctuations. Mean-field theories for quantum systems with itinerant degrees of freedom cut off spatial fluctuations but take full account of temporal fluctuations. As a result the mean field is a time-or frequency dependent quantity.
Another methods use the spectral density approach [16]. The results obtained by the help of this method show the growing range of AF state with increasing U (the opposite behavior compared to the methods described earlier). Despite the large progress in theoretical analysis of coexistence between SC and AF there is still the need for description of experimental results. Particular attention should be devoted to decreasing the concentration range of coexistence between those two phases. Important is also explaining the strong electron-hole asymmetry especially visible for the AF ordering (see Fig. 1).
The main objective of this paper is to explain simultaneous appearance of AF and the d-type SC as shown by the diagram in Fig. 1, using the itinerant extended Hubbard model described by Hamiltonian of Eq. (1) below.

II. Model Hamiltonian
Our single band Hamiltonian has to represent the complex situation in the CuO 2 plane of layered compounds. Therefore it includes the on-site Coulomb repulsion U and the intersite charge-charge interaction V . In addition we have the hopping interaction t ∆ (see [17][18][19]). This interaction causes the rapid growth of the bandwidth when the electron occupation n departs from unity. It stimulates the s type superconductivity (see [17,18]), which in our model is suppressed by the relatively strong effective Coulomb repulsion. The existing dwave superconductivity (see [20][21][22] ( ) In the above equation 0 µ is the chemical potential, ˆi is the probability of finding the electron with spin σ in a band (we assume for simplicity, that the band is fully degenerated, i.e. the single band is composed of identical orbitals). Parameter t is the hopping amplitude for an electron of spin σ when both sites i and j are empty, and t ∆ is the hopping interaction. This interaction is defined as hopping amplitude for an electron of spin σ when one of the sites i or j is occupied by an electron with opposite spin [20]. The single site Hund's type exchange interaction in F added in the Hamiltonian can be interpreted as the interaction between different orbitals on the same lattice site in the multi-orbital single band model. By the reduction of the three-band extended Hubbard model to the effective singleband model used by us, we obtain the negative value of inter-site charge-charge interaction V . In the three-band extended Hubbard model the effective inter-site charge-charge interaction can take the negative value as according to Weber [22] at relatively strong 0 U D ( 0 D is the half-bandwidth) it can be expressed by the formula where z V and x V are the interactions between the copper where the unperturbed part of the Hamiltonian is given by the perturbed part is equal to and the superconducting part has the form The effective hopping interaction appearing in Eq. (4) is given by and the effective chemical potential is equal to m n n n n and the superconducting order parameters have the form The self-energy σ γ Σ which appears in Eqs. (4) and (5) is calculated by treating expression (5) as a perturbation. Subsequently we will use the equation of motion for the Green functions to analyze the Hamiltonian (3). In the Appendix A we derive the formalism for the interaction U (see Eq. (5)

(A) The Hartree-Fock calculations
In this Section we apply the H-F approximation to the on-site Coulomb repulsion U .
In addition we introduce the inter-site charge-charge interaction V (creating the SC of the d type) to which we also apply the H-F approximation. In the H-F approximation the antiferromagnetism is created by the sum of the intra-atomic exchange field, in F , and the onsite Coulomb repulsion U . In our single band model the intra-atomic exchange field can be originated by the inter-orbital exchange interaction on the same site of the multi orbital model.
The influence of the hopping interaction t ∆ for critical interaction creating the AF state in the H-F approximation, based on the 2D tight binding initial density of states (DOS), is shown in Fig. 2. This interaction causes the bandwidth change with occupation. When we depart from the half-filled point the exchange field is becoming less capable of creating AF since the critical field necessary to sustain AF grows. We may compare these results with the CP approach in Fig. 9.
The results shown in Fig. 2 indicate that the calculated range of AF state is larger than the experimental range in the doped YBaCuO compound; moreover the hopping interaction causes increase of this theoretical range.
The intersections with critical curves determine the carrier occupation range of existing AF state in these two cases.
In  Néel's temperature N T (dashed line) on carrier concentration n for the SC energy gap of the d-type. The H-F approximation was applied to interactions U and V . The interactions are: and n is the carrier concentration. The hopping interaction As one can see from Fig. 3, in the H-F approximation there is a broad range of carrier concentration at which the AF state coexists with SC state, without repelling each other. The SC state disappears merely at concentration 1 n → . This is the consequence of using the low level approximation for the interactions. As can be seen from Fig. 2  (B) Coherent potential calculations, the bandwidth independent of carrier The phase diagram obtained in the H-F approximation shows the broad range of coexistence between SC and AF phases. This result disagree with the experimental evidence for cuprates, where the range of coexistence is either very narrow (e.g. for Nd 2-x Ce x CuO 4 ) or it does not exist (e.g. for YBa 2 Cu 3 O 6+x ). Wysokiński and Domański [21] have shown that in the higher order than the H-F approximations the maximum of the critical SC temperature is shifted to the concentrations below the half-filling. At large enough value of U they obtained disappearance of SC phase at 1 n = . Therefore, we will apply the coherent potential (CP) approximation to the Coulomb interaction (see e.g. [20,23]) to analyze coexistence of SC and AF phases (see Appendix B). The inter-site charge-charge interaction V we still treat in the H-F approximation. It is important to note here, that in this stage our computations are still made with 0 t ∆ = , i.e. with the bandwidth independent on carrier concentration. The interaction 0 t ∆ ≠ will be considered in the next step.  As we can see, the influence of the Coulomb repulsion U on the range of the AF state is relatively small comparing to the influence of the initial unperturbed bandwidth. Therefore the mechanism controlling the range of AF state is the increase in 0 D , and (as we will see in the next Section) we can control the range of SC state by using the proper U values (which do not strongly influence the AF state).
In Fig. 5 below we show the dependence of Néel's temperature on electron concentration for different U values at 0 2eV D = (interaction 0 t ∆ = ). One can see the growing range of AF state with increasing U . This effect is observed also in the spectral density approach [16]. Opposite effect was obtained by Kancharla et. al [14] [25]. To obtain a Mott insulator at 1 n = , we will employ in the next chapter the interaction 0 t ∆ ≠ which will cause the band-split, due to the decrease of effective D and increase of U D , specially for carrier concentrations close to unity. Fig. 5 The dependence of critical AF temperature on electron concentration for ). Experimental Néel's temperature at 1 n = was obtained by fitting the in F parameter. The green curve represents the experimental data for YBCO.
In Fig. 6 we show phase diagram for the antiferromagnetic and superconducting states.   To rectify the situation of the d-wave SC persisting almost to the half filled band we include now the hopping interaction t ∆ . This interaction was broadly used to the description of superconducting state [17,20,26,27], metal-insulator transition [28] and in magnetism [29][30][31]. Its influence on superconductivity goes in two ways. It creates the pairing potential for In Fig. 9 below we compare (as in the case of H-F approximation) the influence of t ∆ interaction on the range of existence of the AF state.
In Fig. 12 we show the results for SC and AF in a full model of Eq. (1), i.e. with all interactions U , V , in F and t ∆ being nonzero, and the on-site repulsion U treated in the CP approximation. Our results (black and red curves) are compared with experiment (green curves). We can see that the agreement between theory and experiment is significantly improved as compared to Fig. 6 which was obtained with 0 t ∆ = .  The figure shows that the stimulation of AF and SC by the hopping interaction is stronger for electron-doped ( 1 n > ) compounds. At smaller charge-charge interactions (the red curves) it is even possible for a given compound not to have a SC at the hole-doped concentrations ( 1 n < ), and to obtain a good fit to the experimental results at 1 n > for the case of Nd 2-x Ce x CuO 4-y compound, see Fig. 1. The detailed numerical results with included coexistence between AF and SC ordering will be presented in the future.

IV. Conclusions
The results for the SC and AF critical temperatures after applying the CP approximation are close to the experimental data. Fast decrease of AF observed experimentally is obtained for relatively high initial bandwidth and the effective bandwidth increasing with carrier concentration departing from the half filled point (due to nonzero t ∆ interaction). Mutual dumping of AF and SC states is strong, specially at relatively strong / U D , which is achieved at nonzero t ∆ .
In The increase of the initial bandwidth causes the range of AF state to shift towards larger concentrations closer to 1 n = . As mentioned just above the ( ) U D n ratio, increased by t ∆ interaction, repels the SC to smaller concentrations. As a result at relatively high values of t ∆ interaction there is no region of coexistence between SC and AF states in the CP approximation (see e.g. Fig. 10a). In this approximation, and when 0 t ∆ ≠ , both the d-wave SC critical curve and AF critical curve can be well adjusted to the experimental data for YBCO ceramics. The self-energy described by above equation is energy independent. It depends on carrier concentration and magnetization only. Taking into account Eqs (9) and (10) The nesting vector Q has the property: tanh tanh The equation for the d-wave superconducting state can be obtained by using a Fourier transform of the superconducting order parameter given by Eq. (11) Next we use the moments method (see [17,20]), which in this case is reduced to comparing coefficients at terms with the second power of k where the second-order moment is given by The critical temperature is found from equation (A. 19) in the limit of 0 k ∆ → .

Appendix B: Coherent Potential Approximation for SC coexisting with AF
We treat the interaction U in the coherent potential approximation. The interaction part of the Hamiltonian (5) takes on the following form with the stochastic potentials and the corresponding probabilities given by   where the moment of the second order, 2 L , is given by 1  2  2  1  1  1  2  2  1  2  2  1  2  1  2   tanh  2 tanh  2  1 , , ,