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The Hubbard model and pressure effects of YBa2Cu3O7−δ superconductors
Physica C 383 (2002) 89–94 www.elsevier.com/locate/physc
The Hubbard model and pressure effects of YBa2Cu3O7d superconductors q E.S. Caixeiro *, E.V.L. de Mello Departamento de Fısica, Universidade Federal Fluminense, Av. Litor^ ania s/n, Gragoata, Niter oi, RJ 24210-340, Brazil Received 22 November 2001; received in revised form 19 February 2002; accepted 6 March 2002
Abstract We apply a mean field approach to the extended Hubbard model on a square lattice to the YBa2 Cu3 O7d family of superconductors under pressure. The parameters of the tight-binding band are taken from experiments, and the coupling strength U and V are estimated by the zero pressure phase diagram (Tc nh ). This scheme yields the non-trivial dependence of the superconductor critical temperature Tc as a function of the hole concentration nh in the CuO2 plane. With the assumption that the pressure P modifies the potential V and the on-plane hole content nh , we can distinguish the charge transfer and the intrinsic contribution to Tc ðP Þ. We show that the changes on Tc ðP Þ for the YBa2 Cu3 O7 optimally doped compound at low pressures are almost entirely due to the intrinsic term. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: BCS model; Hubbard Hamiltonian; Phase diagram; Pressure effects
1. Introduction Pressure experiments have played an important role in understanding high-Tc superconductors (HTSC). Several review papers have been written on this subject [1–3]. In short, the pressure investigation is a helpful tool to display the potential effects of the chemical pressure, thereby providing useful information for synthesis of related compounds and, more importantly to the present work, to understand which are the structural parameters that influence the microscopic superconducting mechanism responsible for the HTSC. q
Partially supported by the Brazilian agencies CAPES, FAPERJ and CNPq. * Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.S. Caixeiro).
It is well known that the pressure effects are twofold: (1) It is documented that the applied pressure P increases the hole concentration nh on the CuO2 planes [3], with a maximum rate of charge transfer approximately given by onh =oP ¼ 0:02 GPa1 . This is called the pressure induced charge transfer (PICT) [4] term. (2) Another pressure effect is the increase of Tc for the optimum doped compound (nop ) above the zero pressure value, which indicates another mechanism independent of the PICT, known as the ‘‘intrinsic’’ contribution term [5]. Recently, a very large intrinsic term of oTc =oP ¼ 6:8 K/GPa has been measured [6], however, its origin is still a matter of research. Usually these two effects, the PICT and the intrinsic term, have to be combined in order to account for the experimental data on a given family of compounds. It is very convenient to separate both contribution because the PICT can
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 2 ) 0 1 2 7 2 - 8
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be measured independently by thermal measurements, like the Hall coefficient. Some earlier works have estimated the magnitude of both effects through a phenomenological pressure expansion [5,7] applied to the YBa2 Cu3 O7d family of compounds. Different methods using the extended Hubbard Hamiltonian to study the HTSC phase diagram are well known [8]. In particular we apply a BCSlike mean field approach to derive the phase diagram Tc nh for singlet pairs with s- and d-wave symmetries [9,10]. This procedure was used to deal with the pressure effects and it was assumed that, besides the hole content nh , the pressure also changes the magnitude V of the attractive potential, and ultimately changes the zero temperature superconducting gap Dð0Þ [11]. Thus, the critical temperature Tc ðnh ; P Þ was obtained as an expansion in powers of P, for a determined nh value, and each contribution was singled out, which is very convenient to interpret the experimental results. This method was applied to the experimental results of Hg1201 and Hg1223, with a fixed set of parameters, with excellent agreement with the underdoped and overdoped compounds [11]. In a similar approach, also using the BCS method and the extended Hubbard Hamiltonian, Angilella et al. [9] used the isothermal compressibility tensor to estimate the change of nh with the pressure and, through the derived Tc nh , they obtained the variation of the attractive potential V with the pressure P. In their calculations they have also taken into account pressure dependence for the lattice parameter ai , so that the hopping integrals ti depends on the pressure too ðti ¼ ti ðai ðP ÞÞÞ. In this way, they could separate the charge transfer and the intrinsic contribution. Their calculations for the intrinsic term were in good agreement with the measurements of the Bi2212 [12]. Calculations based on the extended Hubbard Hamiltonian provide a possible microscopic interpretation for the intrinsic term since it is related with the superconducting interaction. In our approach it is related with the attractive potential V but, depending on the method, the superconducting interaction can be an effective interaction from the Coulomb repulsion U and the hopping parameters [13]. Through such calculations, one may
extract some clues to the behavior of the fundamental microscopic mechanism under the structural changes provided by the applied pressure, and some hints for the mechanism itself. Recently, we have combined both approaches of Refs. [9,10] in order to study the measured Tc P curves of the Tl0:5 Pb0:5 Sr2 Ca1x Yx Cu2 O7 series [14]. A characteristic of this method is that, if the phase diagram is experimentally known, Tc ðP Þ can be calculated for the entire family of different nh compounds, and the value of onh =oP calculated for this specific family. Our calculated Tc ðnh ; P Þ for the Tl0:5 Pb0:5 Sr2 Ca1x Yx Cu2 O7 series agreed well with the experiments for the extendeds-wave symmetry [15]. In a recent letter, Chen et al. [16] have calculated the pressure effects on the nearly optimum doped YBa2 Cu3 O7 and underdoped YBa2 Cu4 O8 using a BCS-type method. Using values of onh =oP averaged from different results of several groups, they obtained very good agreement with the experimental data. However, calculating oTc =oP (P 0) for different nh compounds of the YBa2 Cu3 O7d family, they found almost the same values for different potential parameters, which led them to conclude that the main contribution to Tc ðnh ; P Þ was from the PICT term. To check whether the YBa2 Cu3 O7 compound has an intrinsic contribution term or not we have performed calculations on the YBCO system using our method. In our method we use a BCS approach with the extended Hubbard model to calculate the phase diagram of the YBa2 Cu3 O7d system, which is compared with the experimental data. Then we take into account the effects produced by the pressure through a change in the intersite potential: as first order approximation we write V ðP Þ ¼ V þ DV ðP Þ. Here DV ðP Þ ¼ c1 P , with c1 being a constant independent of P that will be defined later. Since the structural changes for typical pressures are small, we neglect its effect on the hopping integrals. For the change in the density of hole carriers, we use the well known linear dependence of nh with P [7]. From these considerations, we obtain the critical temperature Tc ðnh ðP Þ; V ðP ÞÞ as an expansion in terms of the pressure P. Our results obtained from such expansion are in good agreement with the experimental data.
E.S. Caixeiro, E.V.L. de Mello / Physica C 383 (2002) 89–94
2. The phase diagram
Dk ¼
To develop the dynamics of the hole-type carriers in the Cu–O planes, we adopt a two dimension extended Hubbard Hamiltonian [9,11,17] in a square lattice of lattice parameter a X y X H ¼ tij cir cjr þ U ni" ni# h ijiir
þ
X
i
Vij cyir cyjr0 cjr0 cir ;
ð1Þ
k0
Vkk0
Dk0 E0 tanh k ; 2Ek0 2kB T
ð3Þ
with Ek ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2k þ D2k ;
ð4Þ
which contains the interaction potential Vkk0 that comes from the transformation to the momentum space of Eq. (1) [9,22] Vkk0 ¼ U þ 2V cosðkx aÞ cosðkx0 aÞ
hijirr0
where tij is the nearest-neighbor and next-nearestneighbor hopping integral between sites i and j; U is the Coulomb on-site correlated repulsion and Vij is the attractive interaction between nearestneighbor sites i and j. The transformation of Eq. (1) to the momentum space leads to the appearance of the dispersion relation. In order to compare with YBCO system, we used a dispersion relation within a tight-binding approximation, which may be compared with the one estimated from ARPES measurements [18]. Thus, ek ¼ 2t1 ðcosðkx aÞ þ cosðky aÞÞ þ 4t2 cosðkx aÞ cosðky aÞ 2t3 ðcosð2kx aÞ þ cosð2ky aÞÞ þ 4t4 cosð2kx aÞ cosð2ky aÞ 4t5 ðcosð2kx aÞ cosðky aÞ þ cosð2ky aÞ cosðkx aÞÞ l;
X
91
ð2Þ
where it was considered identical hopping integrals along both directions in the Cu–O planes for the nearest-neighbor (tx ¼ ty ¼ ti ), a different one for the next-nearest-neighbor (t2 ), and so on. Here it was considered a hopping from the first to the fifth neighbor. Eq. (2) is obtained from the ARPES estimation of Schabel et al. [18]; l is the chemical potential which controls the hole concentration. Like the low temperature superconductors, the HTSC exhibit an energy ‘‘gap’’ (order parameter) in the superconducting phase which separates the paired states from the single-particle states. Using a BCS-type mean-field approximation [21] to develop Eq. (1) in the momentum space, one obtains the self- consistent gap equation, at finite temperatures [21]
þ 2V cosðky aÞ cosðky0 aÞ:
ð5Þ
The substitution of Eq. (5) into Eq. (3) leads to appearance of a gap with two distinct symmetries [9]: Dk ðl; T Þ ¼ Dmax ðl; T Þ½cosðkx aÞ cosðky aÞ;
ð6Þ
where the plus sign is for extended-s-wave and the minus sign for d-wave symmetry. In accordance with Ref. [9] one observes that the d-wave part of the gap does not depend on the coupling constant U, depending only on V. The extended-s symmetry depends on both U and V. As it is well known [23,24] at the critical temperature there is no symmetry mixture, and the gap might be in the extended-s- or d-wave state when T ! Tc . Using the same BCS-type mean-field approximation used in the gap equation, one obtains the hole-content equation [25] 1X ek Ek nh ðl; T Þ ¼ 1 tanh ; ð7Þ 2 k Ek 2kB T where 0 6 nh 6 1. This equation, together with the gap equation and the dispersion relation, are solved numerically self-consistently, in the limit of T ! Tc , in order to obtain the phase diagrams for both gap symmetry. We found the hopping integrals which gave the best fitting to the Y123 system: t1 ¼ 0:32 eV as the nearest-neighbor value; for t2 it was adopted the ratio t2 =t1 ¼ 0:50, for the d-wave symmetry and t2 =t1 ¼ 0:57 for the extended-s; the remaining band parameters are t3 =t1 ¼ 0:16, t4 =t1 ¼ 0:11, and t5 =t1 ¼ 0:031. All these values are close to the ARPES measurements on Y123 system [18]. Different combinations for these ratios may also be used to obtain similar results for the phase
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diagram, i.e., these ratios are not unique in our calculations and the values above were used in order to reproduce the experimental nh T curve without deviating very much from the ARPES data. Therefore, we could also have used different band parameters based on others methods like the cell perturbation [19] or the mapping of the band structure [20], which have given somewhat different hopping ratios. With these band parameters we estimated the coupling constant V ¼ 0:1152 eV, for d-wave gap symmetry, which yielded the best fitting to the experimental phase diagram of Ref. [26]. For the case of the extended-s gap symmetry, the best value of V depends also on the intensity of U. Larger values of U will cause Tc to decrease, for a fixed V. Therefore, we need much larger couplings than V ¼ 0:1152 eV to have U 6 1 eV as it is appropriate for cuprates. However, just for comparison, we performed calculations for s-wave with the same V ¼ 0:1152 eV and an artificial weak value for U ¼ 0:0096 eV was obtained in order to reproduce a curve close to the experimental results. This can be another point against the extendeds-wave symmetry for these systems. To derive such phase diagrams, the density is calculated self-consistently through the chemical potential l. Fig. 1 shows the results of our numerical calculations together with the experimental data of Ref. [26] for the YBa2 Cu3 O7d series.
Fig. 1. Phase diagram Tc nh for the d-wave and extendeds-wave symmetries, with the experimental data of the YBa2 Cu3 O7d taken from Ref. [26] (open squares).
3. The method To calculate the variations of Tc for a given compound with a certain value nh , and under pressure P, we may use an expansion of Tc ðnh ; P Þ in powers of P [14]: X Pz ð8Þ Tc ðnh ; P Þ ¼ az ; z! z and the coefficients az are defined in Ref. [14]. The first coefficient (z ¼ 1) is given by oTc oTc a1 ¼ c 1 ; ð9Þ þ c2 oV onh where c1 ¼ ðoV =oP Þ and c2 ¼ ðonh =oP Þ are parameters usually determined from the experimental data; here they were obtained using the same procedure as in Ref. [14]. It is important to stress that the first term in a1 is the intrinsic contribution, and the second one is the PICT contribution. Therefore, both contribution can be singled out. In this work the derivative oTc =oV was estimated as in Ref. [14]. As in this earlier work, the derivative oTc =onh was obtained from the direct numerical differentiation of the curves of Fig. 2. One can also derive an analytical expression for this last derivative through the phenomenological universal parabolic relation between Tc and nh [14,15].
Fig. 2. Phase diagram Tc nh for the d-wave showing the effect of a change on the coupling constant V, together with the experimental data of the YBa2 Cu3 O7d taken from Ref. [26] (open squares).
E.S. Caixeiro, E.V.L. de Mello / Physica C 383 (2002) 89–94
93
The expression for the coefficient a2 , using numerical calculations to obtain oTc =onh , may be written as a2 c22
o2 Tc ; on2h
ð10Þ
which became different from the one in Ref. [14]; it was considered only first order dependence of Tc on V.
4. The pressure experimental data First of all, one observes that, at low pressures only the linear term (a1 ) of the expansion Tc ðnh ; P Þ comes into play. Therefore, we can approximate a1 as the slope of the initial points of the Tc P curve, for a given nh compound. Starting with nh ¼ nop , we determine c1 using the estimated a1 in Eq. (9), as long as at nop the charge transfer term vanishes. To determine c2 we obtain a1 from the Tc P curve of an nh 6¼ nop compound, and use again Eq. (9). Once these two constants are determined, the az coefficients for any nh value can be calculated. For the YBa2 Cu3 O7d system we only have the non-zero pressure data for the nh ¼ nop ¼ 0:165 (d 0) and nh ¼ 0:15 (d 0:15) compounds [27,28]. Thus, from the Tc P curves of Refs. [27,28] we estimate a1 ¼ 0:75 and 1.4 K/GPa for nop and nh ¼ 0:15, respectively. The phase diagram parameters DTc =DV taken from Fig. 2 and used in the calculations were 1652 and 1475 K/eV for the d-wave and 1164 and 1099 K/eV for the extended-s-wave for nop and nh ¼ 0:165 compounds, respectively. Therefore, the resulting constants for the d-wave were c1 ¼ 4:54 104 eV/GPa which furnishes an intrinsic contribution of 0.8 K/GPa, and c2 ¼ 3:15 103 GPa1 which is the charge transfer term ðc2 ¼ onh =oP Þ. For the extended-s we obtained c1 ¼ 6:4 104 eV/GPa and c2 ¼ 2:2 103 GPa1 . It is important to mention that our values for oTci =oP (the intrinsic contribution; see Eq. (9)) were in the same order of magnitude of Neumeier and Zimmermann [5], despite the difference in the approaches. In Fig. 3 we compare the Tc ðnh ; P Þ expansion with the experimental data. We observe a very good agreement for both compounds, especially
Fig. 3. Numerical results: the solid and dashed lines are the d-wave results, while the long-dashed lines are the extendeds-wave results. The open squares and filled squares are experimental data YBa2 Cu3 O7d taken from Refs. [27,28], respectively.
for nop , using the d-wave gap symmetry, where the data can be fitted up to 20 GPa. The increase of nop with the pressure is attributed to the intrinsic contribution in our theory, and the maximum Tc at P 7 GPa and subsequent decrease is due to the negative second order term of the expansion, which reflects the effect of the PICT. The extendeds-wave results did not show good results indicating a preference for the d-wave pairing mechanism for the Y123 system.
5. Conclusions We conclude this letter emphasizing that our method based on the BCS approach with the extended Hubbard model, with both the attractive potential V and the density of carriers nh depending on the pressure P, predicted an intrinsic term and a PICT contribution similar to those predicted earlier [5,7]. In our method we propose a direct pressure dependence for the attractive potential V, which is different from other works [9] that use V as an adjustable parameter in order to fit the known experimental dependence of Tc on P. Our results for the d-wave are in excellent agreement with the Y123 pressure data and the intrinsic term being the most important contribution to the low pressure data on the optimum compound.
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Concerning the underdoped compound the intrinsic and PICT contributions are both present and must, as previously known by several experiments and theories [3,5], be taken into account. Such results support the d-wave symmetry in these compounds.
Acknowledgements We thank CNPq, CAPES and FAPERJ for the finacial support.
References [1] J.S. Schilling, S. Klotz, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors, vol. 3, World Scientific, Singapore, 1992, p. 59. [2] H. Takahashi, N. M^ ori, in: A.V. Narlikar (Ed.), Studies of High Temperature Superconductors, vol. 16, Nova Science Publishers, New York, 1997, p. 1. [3] R.J. Wijngaarden, D.T. Jover, R. Griessen, Physica B 265 (1999) 128. [4] C.C. Almasan, S.H. Han, B.W. Lee, L.M. Paulius, M.B. Maple, B.W. Veal, J.W. Downey, A.P. Paulikas, Z. Fisk, J.E. Schirber, Phys. Rev. Lett. 69 (1992) 680. [5] J.J. Neumeier, H.A. Zimmermann, Phys. Rev. B 47 (1993) 8385. [6] M.T.D. Orlando, A.G. Cunha, E.V.L. de Mello, H. Belich, E. Baggio-Saitovitch, A. Sin, X. Obradors, T. Burghardt, A. Eichler, Phys. Rev. B 61 (2000) 15454. [7] R.P. Gupta, M. Gupta, Phys. Rev. B 51 (1995) 11760.
[8] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [9] G.G.N. Angilella, R. Pucci, F. Siringo, Phys. Rev. B 54 (1996) 15471. [10] E.V.L. de Mello, Physica C 259 (1996) 109. [11] E.V.L. de Mello, C. Acha, Phys. Rev. B 56 (1997) 466; E.V.L. de Mello, C. Acha, Physica C 282–287 (1997) 1819. [12] T. Huang, M. Itoh, J. Yu, Y. Inaguma, T. Nakamura, Phys. Rev. B 48 (1993) 7712. [13] A.N. Das, J. Konior, D.K. Ray, A.M. Oles, Phys. Rev. B 44 (1991) 7680. [14] E.S. Caixeiro, E.V. de Mello, Physica C 353 (2001) 103. [15] D.T. Jover, H. Wilhelm, R.J. Wijngaarden, R.S. Liu, Phys. Rev. B 55 (1997) 11832. [16] X.J. Chen, H.Q. Lin, C.D. Gong, Phys. Rev. Lett. 85 (2000) 2180. [17] E.V.L. de Mello, Braz. J. Phys. 29 (1999) 551; E.V.L. de Mello, Physica B 265 (1999) 142. [18] M.C. Schabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D.A. Bonn, R. Liang, W.N. Hardy, Phys. Rev. B 57 (1998) 6090. [19] R. Raimondi, J.H. Jefferson, L.E. Feiner, Phys. Rev. 53 (1996) 8774; L.E. Feiner, J.H. Jefferson, R. Raimondi, Phys. Rev. Lett. 76 (1996) 4939. [20] O.K. Andersen, A.I. Liechtenstein, O. Jepsen, F. Paulsen, J. Phys. Chem. Solids 56 (1995) 1573. [21] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, New York, 1966. [22] T. Schneider, M.P. S€ orensen, Z. Phys. B 80 (1990) 331. [23] E.V.L. de Mello, Physica C 328 (1999) 88. [24] E.V.L. de Mello, Physica C 341–345 (2000) 1701. [25] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [26] J.L. Tallon, Physica C 168 (1990) 85. [27] S. Klotz, W. Reith, J.S. Schilling, Physica C 172 (1991) 423. [28] V.G. Tissen, M.V. Nefedova, Sverkhprovodimost’ Fiz. Khim Tekh. 3 (1990) 999.
The Hubbard model and pressure effects of YBa2Cu3O7d superconductors q E.S. Caixeiro *, E.V.L. de Mello Departamento de Fısica, Universidade Federal Fluminense, Av. Litor^ ania s/n, Gragoata, Niter oi, RJ 24210-340, Brazil Received 22 November 2001; received in revised form 19 February 2002; accepted 6 March 2002
Abstract We apply a mean field approach to the extended Hubbard model on a square lattice to the YBa2 Cu3 O7d family of superconductors under pressure. The parameters of the tight-binding band are taken from experiments, and the coupling strength U and V are estimated by the zero pressure phase diagram (Tc nh ). This scheme yields the non-trivial dependence of the superconductor critical temperature Tc as a function of the hole concentration nh in the CuO2 plane. With the assumption that the pressure P modifies the potential V and the on-plane hole content nh , we can distinguish the charge transfer and the intrinsic contribution to Tc ðP Þ. We show that the changes on Tc ðP Þ for the YBa2 Cu3 O7 optimally doped compound at low pressures are almost entirely due to the intrinsic term. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: BCS model; Hubbard Hamiltonian; Phase diagram; Pressure effects
1. Introduction Pressure experiments have played an important role in understanding high-Tc superconductors (HTSC). Several review papers have been written on this subject [1–3]. In short, the pressure investigation is a helpful tool to display the potential effects of the chemical pressure, thereby providing useful information for synthesis of related compounds and, more importantly to the present work, to understand which are the structural parameters that influence the microscopic superconducting mechanism responsible for the HTSC. q
Partially supported by the Brazilian agencies CAPES, FAPERJ and CNPq. * Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.S. Caixeiro).
It is well known that the pressure effects are twofold: (1) It is documented that the applied pressure P increases the hole concentration nh on the CuO2 planes [3], with a maximum rate of charge transfer approximately given by onh =oP ¼ 0:02 GPa1 . This is called the pressure induced charge transfer (PICT) [4] term. (2) Another pressure effect is the increase of Tc for the optimum doped compound (nop ) above the zero pressure value, which indicates another mechanism independent of the PICT, known as the ‘‘intrinsic’’ contribution term [5]. Recently, a very large intrinsic term of oTc =oP ¼ 6:8 K/GPa has been measured [6], however, its origin is still a matter of research. Usually these two effects, the PICT and the intrinsic term, have to be combined in order to account for the experimental data on a given family of compounds. It is very convenient to separate both contribution because the PICT can
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 2 ) 0 1 2 7 2 - 8
90
E.S. Caixeiro, E.V.L. de Mello / Physica C 383 (2002) 89–94
be measured independently by thermal measurements, like the Hall coefficient. Some earlier works have estimated the magnitude of both effects through a phenomenological pressure expansion [5,7] applied to the YBa2 Cu3 O7d family of compounds. Different methods using the extended Hubbard Hamiltonian to study the HTSC phase diagram are well known [8]. In particular we apply a BCSlike mean field approach to derive the phase diagram Tc nh for singlet pairs with s- and d-wave symmetries [9,10]. This procedure was used to deal with the pressure effects and it was assumed that, besides the hole content nh , the pressure also changes the magnitude V of the attractive potential, and ultimately changes the zero temperature superconducting gap Dð0Þ [11]. Thus, the critical temperature Tc ðnh ; P Þ was obtained as an expansion in powers of P, for a determined nh value, and each contribution was singled out, which is very convenient to interpret the experimental results. This method was applied to the experimental results of Hg1201 and Hg1223, with a fixed set of parameters, with excellent agreement with the underdoped and overdoped compounds [11]. In a similar approach, also using the BCS method and the extended Hubbard Hamiltonian, Angilella et al. [9] used the isothermal compressibility tensor to estimate the change of nh with the pressure and, through the derived Tc nh , they obtained the variation of the attractive potential V with the pressure P. In their calculations they have also taken into account pressure dependence for the lattice parameter ai , so that the hopping integrals ti depends on the pressure too ðti ¼ ti ðai ðP ÞÞÞ. In this way, they could separate the charge transfer and the intrinsic contribution. Their calculations for the intrinsic term were in good agreement with the measurements of the Bi2212 [12]. Calculations based on the extended Hubbard Hamiltonian provide a possible microscopic interpretation for the intrinsic term since it is related with the superconducting interaction. In our approach it is related with the attractive potential V but, depending on the method, the superconducting interaction can be an effective interaction from the Coulomb repulsion U and the hopping parameters [13]. Through such calculations, one may
extract some clues to the behavior of the fundamental microscopic mechanism under the structural changes provided by the applied pressure, and some hints for the mechanism itself. Recently, we have combined both approaches of Refs. [9,10] in order to study the measured Tc P curves of the Tl0:5 Pb0:5 Sr2 Ca1x Yx Cu2 O7 series [14]. A characteristic of this method is that, if the phase diagram is experimentally known, Tc ðP Þ can be calculated for the entire family of different nh compounds, and the value of onh =oP calculated for this specific family. Our calculated Tc ðnh ; P Þ for the Tl0:5 Pb0:5 Sr2 Ca1x Yx Cu2 O7 series agreed well with the experiments for the extendeds-wave symmetry [15]. In a recent letter, Chen et al. [16] have calculated the pressure effects on the nearly optimum doped YBa2 Cu3 O7 and underdoped YBa2 Cu4 O8 using a BCS-type method. Using values of onh =oP averaged from different results of several groups, they obtained very good agreement with the experimental data. However, calculating oTc =oP (P 0) for different nh compounds of the YBa2 Cu3 O7d family, they found almost the same values for different potential parameters, which led them to conclude that the main contribution to Tc ðnh ; P Þ was from the PICT term. To check whether the YBa2 Cu3 O7 compound has an intrinsic contribution term or not we have performed calculations on the YBCO system using our method. In our method we use a BCS approach with the extended Hubbard model to calculate the phase diagram of the YBa2 Cu3 O7d system, which is compared with the experimental data. Then we take into account the effects produced by the pressure through a change in the intersite potential: as first order approximation we write V ðP Þ ¼ V þ DV ðP Þ. Here DV ðP Þ ¼ c1 P , with c1 being a constant independent of P that will be defined later. Since the structural changes for typical pressures are small, we neglect its effect on the hopping integrals. For the change in the density of hole carriers, we use the well known linear dependence of nh with P [7]. From these considerations, we obtain the critical temperature Tc ðnh ðP Þ; V ðP ÞÞ as an expansion in terms of the pressure P. Our results obtained from such expansion are in good agreement with the experimental data.
E.S. Caixeiro, E.V.L. de Mello / Physica C 383 (2002) 89–94
2. The phase diagram
Dk ¼
To develop the dynamics of the hole-type carriers in the Cu–O planes, we adopt a two dimension extended Hubbard Hamiltonian [9,11,17] in a square lattice of lattice parameter a X y X H ¼ tij cir cjr þ U ni" ni# h ijiir
þ
X
i
Vij cyir cyjr0 cjr0 cir ;
ð1Þ
k0
Vkk0
Dk0 E0 tanh k ; 2Ek0 2kB T
ð3Þ
with Ek ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2k þ D2k ;
ð4Þ
which contains the interaction potential Vkk0 that comes from the transformation to the momentum space of Eq. (1) [9,22] Vkk0 ¼ U þ 2V cosðkx aÞ cosðkx0 aÞ
hijirr0
where tij is the nearest-neighbor and next-nearestneighbor hopping integral between sites i and j; U is the Coulomb on-site correlated repulsion and Vij is the attractive interaction between nearestneighbor sites i and j. The transformation of Eq. (1) to the momentum space leads to the appearance of the dispersion relation. In order to compare with YBCO system, we used a dispersion relation within a tight-binding approximation, which may be compared with the one estimated from ARPES measurements [18]. Thus, ek ¼ 2t1 ðcosðkx aÞ þ cosðky aÞÞ þ 4t2 cosðkx aÞ cosðky aÞ 2t3 ðcosð2kx aÞ þ cosð2ky aÞÞ þ 4t4 cosð2kx aÞ cosð2ky aÞ 4t5 ðcosð2kx aÞ cosðky aÞ þ cosð2ky aÞ cosðkx aÞÞ l;
X
91
ð2Þ
where it was considered identical hopping integrals along both directions in the Cu–O planes for the nearest-neighbor (tx ¼ ty ¼ ti ), a different one for the next-nearest-neighbor (t2 ), and so on. Here it was considered a hopping from the first to the fifth neighbor. Eq. (2) is obtained from the ARPES estimation of Schabel et al. [18]; l is the chemical potential which controls the hole concentration. Like the low temperature superconductors, the HTSC exhibit an energy ‘‘gap’’ (order parameter) in the superconducting phase which separates the paired states from the single-particle states. Using a BCS-type mean-field approximation [21] to develop Eq. (1) in the momentum space, one obtains the self- consistent gap equation, at finite temperatures [21]
þ 2V cosðky aÞ cosðky0 aÞ:
ð5Þ
The substitution of Eq. (5) into Eq. (3) leads to appearance of a gap with two distinct symmetries [9]: Dk ðl; T Þ ¼ Dmax ðl; T Þ½cosðkx aÞ cosðky aÞ;
ð6Þ
where the plus sign is for extended-s-wave and the minus sign for d-wave symmetry. In accordance with Ref. [9] one observes that the d-wave part of the gap does not depend on the coupling constant U, depending only on V. The extended-s symmetry depends on both U and V. As it is well known [23,24] at the critical temperature there is no symmetry mixture, and the gap might be in the extended-s- or d-wave state when T ! Tc . Using the same BCS-type mean-field approximation used in the gap equation, one obtains the hole-content equation [25] 1X ek Ek nh ðl; T Þ ¼ 1 tanh ; ð7Þ 2 k Ek 2kB T where 0 6 nh 6 1. This equation, together with the gap equation and the dispersion relation, are solved numerically self-consistently, in the limit of T ! Tc , in order to obtain the phase diagrams for both gap symmetry. We found the hopping integrals which gave the best fitting to the Y123 system: t1 ¼ 0:32 eV as the nearest-neighbor value; for t2 it was adopted the ratio t2 =t1 ¼ 0:50, for the d-wave symmetry and t2 =t1 ¼ 0:57 for the extended-s; the remaining band parameters are t3 =t1 ¼ 0:16, t4 =t1 ¼ 0:11, and t5 =t1 ¼ 0:031. All these values are close to the ARPES measurements on Y123 system [18]. Different combinations for these ratios may also be used to obtain similar results for the phase
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diagram, i.e., these ratios are not unique in our calculations and the values above were used in order to reproduce the experimental nh T curve without deviating very much from the ARPES data. Therefore, we could also have used different band parameters based on others methods like the cell perturbation [19] or the mapping of the band structure [20], which have given somewhat different hopping ratios. With these band parameters we estimated the coupling constant V ¼ 0:1152 eV, for d-wave gap symmetry, which yielded the best fitting to the experimental phase diagram of Ref. [26]. For the case of the extended-s gap symmetry, the best value of V depends also on the intensity of U. Larger values of U will cause Tc to decrease, for a fixed V. Therefore, we need much larger couplings than V ¼ 0:1152 eV to have U 6 1 eV as it is appropriate for cuprates. However, just for comparison, we performed calculations for s-wave with the same V ¼ 0:1152 eV and an artificial weak value for U ¼ 0:0096 eV was obtained in order to reproduce a curve close to the experimental results. This can be another point against the extendeds-wave symmetry for these systems. To derive such phase diagrams, the density is calculated self-consistently through the chemical potential l. Fig. 1 shows the results of our numerical calculations together with the experimental data of Ref. [26] for the YBa2 Cu3 O7d series.
Fig. 1. Phase diagram Tc nh for the d-wave and extendeds-wave symmetries, with the experimental data of the YBa2 Cu3 O7d taken from Ref. [26] (open squares).
3. The method To calculate the variations of Tc for a given compound with a certain value nh , and under pressure P, we may use an expansion of Tc ðnh ; P Þ in powers of P [14]: X Pz ð8Þ Tc ðnh ; P Þ ¼ az ; z! z and the coefficients az are defined in Ref. [14]. The first coefficient (z ¼ 1) is given by oTc oTc a1 ¼ c 1 ; ð9Þ þ c2 oV onh where c1 ¼ ðoV =oP Þ and c2 ¼ ðonh =oP Þ are parameters usually determined from the experimental data; here they were obtained using the same procedure as in Ref. [14]. It is important to stress that the first term in a1 is the intrinsic contribution, and the second one is the PICT contribution. Therefore, both contribution can be singled out. In this work the derivative oTc =oV was estimated as in Ref. [14]. As in this earlier work, the derivative oTc =onh was obtained from the direct numerical differentiation of the curves of Fig. 2. One can also derive an analytical expression for this last derivative through the phenomenological universal parabolic relation between Tc and nh [14,15].
Fig. 2. Phase diagram Tc nh for the d-wave showing the effect of a change on the coupling constant V, together with the experimental data of the YBa2 Cu3 O7d taken from Ref. [26] (open squares).
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The expression for the coefficient a2 , using numerical calculations to obtain oTc =onh , may be written as a2 c22
o2 Tc ; on2h
ð10Þ
which became different from the one in Ref. [14]; it was considered only first order dependence of Tc on V.
4. The pressure experimental data First of all, one observes that, at low pressures only the linear term (a1 ) of the expansion Tc ðnh ; P Þ comes into play. Therefore, we can approximate a1 as the slope of the initial points of the Tc P curve, for a given nh compound. Starting with nh ¼ nop , we determine c1 using the estimated a1 in Eq. (9), as long as at nop the charge transfer term vanishes. To determine c2 we obtain a1 from the Tc P curve of an nh 6¼ nop compound, and use again Eq. (9). Once these two constants are determined, the az coefficients for any nh value can be calculated. For the YBa2 Cu3 O7d system we only have the non-zero pressure data for the nh ¼ nop ¼ 0:165 (d 0) and nh ¼ 0:15 (d 0:15) compounds [27,28]. Thus, from the Tc P curves of Refs. [27,28] we estimate a1 ¼ 0:75 and 1.4 K/GPa for nop and nh ¼ 0:15, respectively. The phase diagram parameters DTc =DV taken from Fig. 2 and used in the calculations were 1652 and 1475 K/eV for the d-wave and 1164 and 1099 K/eV for the extended-s-wave for nop and nh ¼ 0:165 compounds, respectively. Therefore, the resulting constants for the d-wave were c1 ¼ 4:54 104 eV/GPa which furnishes an intrinsic contribution of 0.8 K/GPa, and c2 ¼ 3:15 103 GPa1 which is the charge transfer term ðc2 ¼ onh =oP Þ. For the extended-s we obtained c1 ¼ 6:4 104 eV/GPa and c2 ¼ 2:2 103 GPa1 . It is important to mention that our values for oTci =oP (the intrinsic contribution; see Eq. (9)) were in the same order of magnitude of Neumeier and Zimmermann [5], despite the difference in the approaches. In Fig. 3 we compare the Tc ðnh ; P Þ expansion with the experimental data. We observe a very good agreement for both compounds, especially
Fig. 3. Numerical results: the solid and dashed lines are the d-wave results, while the long-dashed lines are the extendeds-wave results. The open squares and filled squares are experimental data YBa2 Cu3 O7d taken from Refs. [27,28], respectively.
for nop , using the d-wave gap symmetry, where the data can be fitted up to 20 GPa. The increase of nop with the pressure is attributed to the intrinsic contribution in our theory, and the maximum Tc at P 7 GPa and subsequent decrease is due to the negative second order term of the expansion, which reflects the effect of the PICT. The extendeds-wave results did not show good results indicating a preference for the d-wave pairing mechanism for the Y123 system.
5. Conclusions We conclude this letter emphasizing that our method based on the BCS approach with the extended Hubbard model, with both the attractive potential V and the density of carriers nh depending on the pressure P, predicted an intrinsic term and a PICT contribution similar to those predicted earlier [5,7]. In our method we propose a direct pressure dependence for the attractive potential V, which is different from other works [9] that use V as an adjustable parameter in order to fit the known experimental dependence of Tc on P. Our results for the d-wave are in excellent agreement with the Y123 pressure data and the intrinsic term being the most important contribution to the low pressure data on the optimum compound.
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Concerning the underdoped compound the intrinsic and PICT contributions are both present and must, as previously known by several experiments and theories [3,5], be taken into account. Such results support the d-wave symmetry in these compounds.
Acknowledgements We thank CNPq, CAPES and FAPERJ for the finacial support.
References [1] J.S. Schilling, S. Klotz, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors, vol. 3, World Scientific, Singapore, 1992, p. 59. [2] H. Takahashi, N. M^ ori, in: A.V. Narlikar (Ed.), Studies of High Temperature Superconductors, vol. 16, Nova Science Publishers, New York, 1997, p. 1. [3] R.J. Wijngaarden, D.T. Jover, R. Griessen, Physica B 265 (1999) 128. [4] C.C. Almasan, S.H. Han, B.W. Lee, L.M. Paulius, M.B. Maple, B.W. Veal, J.W. Downey, A.P. Paulikas, Z. Fisk, J.E. Schirber, Phys. Rev. Lett. 69 (1992) 680. [5] J.J. Neumeier, H.A. Zimmermann, Phys. Rev. B 47 (1993) 8385. [6] M.T.D. Orlando, A.G. Cunha, E.V.L. de Mello, H. Belich, E. Baggio-Saitovitch, A. Sin, X. Obradors, T. Burghardt, A. Eichler, Phys. Rev. B 61 (2000) 15454. [7] R.P. Gupta, M. Gupta, Phys. Rev. B 51 (1995) 11760.
[8] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [9] G.G.N. Angilella, R. Pucci, F. Siringo, Phys. Rev. B 54 (1996) 15471. [10] E.V.L. de Mello, Physica C 259 (1996) 109. [11] E.V.L. de Mello, C. Acha, Phys. Rev. B 56 (1997) 466; E.V.L. de Mello, C. Acha, Physica C 282–287 (1997) 1819. [12] T. Huang, M. Itoh, J. Yu, Y. Inaguma, T. Nakamura, Phys. Rev. B 48 (1993) 7712. [13] A.N. Das, J. Konior, D.K. Ray, A.M. Oles, Phys. Rev. B 44 (1991) 7680. [14] E.S. Caixeiro, E.V. de Mello, Physica C 353 (2001) 103. [15] D.T. Jover, H. Wilhelm, R.J. Wijngaarden, R.S. Liu, Phys. Rev. B 55 (1997) 11832. [16] X.J. Chen, H.Q. Lin, C.D. Gong, Phys. Rev. Lett. 85 (2000) 2180. [17] E.V.L. de Mello, Braz. J. Phys. 29 (1999) 551; E.V.L. de Mello, Physica B 265 (1999) 142. [18] M.C. Schabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D.A. Bonn, R. Liang, W.N. Hardy, Phys. Rev. B 57 (1998) 6090. [19] R. Raimondi, J.H. Jefferson, L.E. Feiner, Phys. Rev. 53 (1996) 8774; L.E. Feiner, J.H. Jefferson, R. Raimondi, Phys. Rev. Lett. 76 (1996) 4939. [20] O.K. Andersen, A.I. Liechtenstein, O. Jepsen, F. Paulsen, J. Phys. Chem. Solids 56 (1995) 1573. [21] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, New York, 1966. [22] T. Schneider, M.P. S€ orensen, Z. Phys. B 80 (1990) 331. [23] E.V.L. de Mello, Physica C 328 (1999) 88. [24] E.V.L. de Mello, Physica C 341–345 (2000) 1701. [25] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [26] J.L. Tallon, Physica C 168 (1990) 85. [27] S. Klotz, W. Reith, J.S. Schilling, Physica C 172 (1991) 423. [28] V.G. Tissen, M.V. Nefedova, Sverkhprovodimost’ Fiz. Khim Tekh. 3 (1990) 999.