D-wave overlapping band model for cuprate superconductors
Susana Orozco, Rosa María Méndez-Moreno, María de los Angeles Ortiz, Gabriela Murguía
aa r X i v : . [ c ond - m a t . s up r- c on ] J un D-wave overlapping band model for cuprate superconductors
Susana Orozco, Rosa Mar´ıa M´endez-Moreno, Mar´ıa de los Angeles Ortiz, and Gabriela Murgu´ıa ∗ Departamento de F´ısica, Facultad de Ciencias,Universidad Nacional Aut´onoma de M´exico,Apartado Postal 21-092, 04021 M´exico, D. F., M´exico
Within the BCS framework a multiband model with d-wave symmetry is considered. GeneralizedFermi surface topologies via band overlapping are introduced. The band overlap scale is of the orderof the Debye energy. The order parameters and the pairing have d-wave symmetry. Experimentalvalues reported for the critical temperatures T c ( x ) and the order parameters, ∆ ( x ), in terms ofdopping x are used. Numerical results for the coupling and the band overlapping parameters interms of the doping are obtained for the cuprate superconductor La − x Sr x CuO . PACS numbers: 74.20.Fg, 74.62.-c, 74.20.-z, 74.72.DnKeywords: High T c , Cuprate, d-wave, Band overlap I. INTRODUCTION
Measurements of angle-resolved photoemission spec-troscopy (ARPES)[1] and tunneling[2], provide enoughevidence for the relevant role of phonons in high- T c super-conductivity (HTSC). Experimental data accumulated sofar for the high- T c copper-oxide superconductors havegiven some useful clues to unravel the fundamental in-gredients responsible for the high transition temperature T c . However, the underlying physical process remainsunknown. In this context, it seems crucial to study newideas that use simplified schematic models to isolate themechanism(s) that generate HTSC.Pairing symmetry is an important element towardunderstanding the mechanism of high- T c superconduc-tivity. Although early experiments were consistentwith s-wave pairing symmetry, recent experiments sug-gest an anisotropic pairing behavior[3]. For manycuprate superconductors it is generally accepted thatthe pairing symmetry is d-wave for hole-doped cupratesuperconductors[4] as for electron doped cuprates[5]. Onthe other hand, recent experiments with Raman scatter-ing and ARPES[6, 7] have shown that the gap struc-ture on high- T c cuprate superconductors, as a func-tion of the angle, is similar to a d-wave gap[8, 9].The small but non-vanishing isotope effects in high- T c cuprates have been shown compatible with d-wavesuperconductivity[10]. A phonon-mediated d-wave BCSlike model has recently been presented to describe layeredcuprated superconductors[11]. The last model accountwell for the magnitudes of T c and the oxygen isotopeexponent of the superconductor cuprates. Calculationswith BCS theory and van Hove scenario have also beendone with d-wave pairing[12]. The validity of d-waveBCS formalism in high- T c superconductor cuprates hasbeen supported by measurements of transport propertiesand ARPES[13].Numerous indications point to the multiband nature ∗ Electronic address: [email protected] of the superconductivity in doped cuprates. The agree-ment of the multiband model with experimental findings,suggests that a multiband pairing is an essential aspectof cuprate superconductivity[14].First principle calculations show overlapping energybands at the Fermi level[15]. The short coherence lengthobserved in high- T c superconductors, has been relatedto the presence of overlapping energy bands[16, 17]. Asimple model with generalized Fermi surface topologiesvia band overlapping has been proposed based on indi-rect experimental evidence. That confirms the idea thatthe tendency toward superconductivity can be enhancedwhen the Fermi level lies at or close to the energy of a sin-gularity in the density of states (DOS)[18]. This modelthat can be taken as a minimal singularity in the densityof states and the BCS framework, can lead to higher T c values than those expected from the traditional phononbarrier. In our model, the energy band overlapping, mod-ifies the DOS near the Fermi level allowing the high T c values observed. A similar effect can be obtained withother mechanisms as a van Hove singularity in the den-sity of states[19].The high- T c copper-oxide superconductors have a char-acteristic layered structure: the CuO planes. Thecharge carriers in these materials are confined to the twodimensional (2D) CuO layers[20]. These layering struc-tures of high- T c cuprates suggest that two-dimensionalphysics is important for these materials[11].In this work, within the BCS framework, a phononmediated d-wave model is proposed. The gap equation(with d-wave symmetry) and two-dimensional general-ized Fermi surface topologies via band overlapping areused as a model for HTSC. A two overlapping band modelis considered as a prototype of multiband superconduc-tors. For physical consistency, an important requirementof the model is that the band overlapping parameter isnot larger than the cutoff Debye energy, E D . The modelhere proposed will be used to describe some properties ofthe cuprate superconductor La − x Sr x CuO in terms ofthe doping and the parameters of the model. II. THE MODEL
We begin with the famous gap equation∆( k ′ ) = X k V ( k, k ′ )∆( k ) tanh( E k / k B T )2 E k , (1)in the weak coupling limit, with V ( k, k ′ ) the pairinginteraction, k B is the Boltzman constant, and E k = ǫ k + ∆ k , where ǫ k = ~ k / m are the self-consistentsingle-particle energies.For the electron-phonon interaction, we have consid-ered, with V a constant, V ( k, k ′ ) = V ψ ( k ) ψ ( k ′ ) when | ǫ k | and | ǫ k ′ | ≤ E D = k B T D and 0 elsewhere. As usualthe attractive BCS interaction is nonzero only for unoc-cupied orbitals in the neighborhood of the Fermi level E F . In the last equation, ψ ( k ) = cos(2 φ k ) for d x − y pairing. Here φ k = tan − ( k y /k x ) is the angular directionof the momentum in the ab plane. The superconductingorder parameter, ∆( k ) = ∆( T ) ψ ( k ) if | ǫ k | ≤ E D and 0elsewhere.With these considerations we propose a generalizedFermi surface. The generalized Fermi sea proposed con-sists of two overlapping bands. As a particular distri-bution with anomalous occupancy in momentum spacethe following form for the generalized Fermi sea has beenconsidered n k = Θ( γk F − k ) + Θ( γk F − k )Θ( k − βk F ) , (2)with k F the Fermi momentum and 0 < β < γ < γ − β = 1 , (3)then only one of the relevant parameters is independent.The distribution in momentum induces one in energy, E β < E γ where E β = β E F and E γ = γ E F . We requirethat the band overlapping be of the order or smaller thanthe cutoff (Debye) energy, which means (1 − γ ) E F ≤ E D .The last expression can be written as(1 − γ ) E F = ηE D , (4)where η is in the range 0 < η < E F / (2 E D ). Equations(3) and (4) together will give the minimum γ value con-sistent with our model.In the last framework the summation in Eq. (1) ischanged to an integration which is done over the ( sym-metric ) generalized Fermi surface defined above. Onegets1 = λ π Z E γ + E D E γ − E D Z π dφ cos (2 φ ) tanh (cid:18) √ Ξ k k B T (cid:19) dǫ k √ Ξ k + λ π Z E F E β Z π dφ cos (2 φ ) tanh (cid:18) √ Ξ k k B T (cid:19) dǫ k √ Ξ k . (5) In this equation Ξ k = ( ǫ k − E F ) + ∆( T ) cos (2 φ ),the coupling parameter is λ = V D ( E ), with D ( E ) theelectronic density of states, which will be taken as aconstant for the 2 D system in the integration range. E F = ~ πm n D , with n D the carriers density per CuO layer. The two integrals correspond to the bands pro-posed by Eq. (2).The integration over the surface at E γ in the firstband, is restricted to states in the interval E γ − E D ≤ E k ≤ E γ + E D . In the second band, in order to con-serve the particle number, the integration is restrictedto the interval E β ≤ E k ≤ E F , if E γ + E D > E F ,with E β = (2 γ − E F , according to Eq. (3)in our model. While E F − E γ ≤ E D , implies thatthe energy difference between the anomalously occupiedstates must be provided by the material itself. Finally∆( T ) ψ ( k ) = ∆( T ) cos(2 φ ) at the two bands.The critical temperature is introduced via the Eq. (5)at T = T c , where the gap becomes ∆( T c ) = 0. At thistemperature Eq. (5) is reduced to1 = λ Z E γ + E D E γ − E D tanh (cid:18) ǫ k − E F k B T c (cid:19) dǫ k ǫ k − E F + λ Z E F E β tanh (cid:18) ǫ k − E F k B T c (cid:19) dǫ k ǫ k − E F , (6)which will be numerically evaluated. The last equationrelates T c to the coupling constant λ and to the anoma-lous occupancy parameter γ . This relationship deter-mines the γ values which reproduces the critical tem-perature of several cuprates in the weak coupling region.At T = 0K, Eq. (5) will also be evaluated and γ valuesconsistent with the numerical results of Eq. (6) will beobtained:1 = λ π Z π dφ cos (2 φ ) × (cid:20) sinh − k B T D − (1 − γ ) k B T F ∆ | cos (2 φ ) | + sinh − (1 − γ ) k B T F + k B T D ∆ | cos(2 φ ) | + sinh − k B (1 − γ ) T F ∆ | cos (2 φ ) | (cid:21) , (7)where ∆(0) = ∆ .The model presented in this section can be used to de-scribe high- T c cuprate superconductors, the band over-lapping 1 − γ and relevant parameters are determined.In any case a specific material must be selected to in-troduce the available experimental data. Ranges for thecoupling parameter λ in the weak coupling region, andthe overlapping parameter γ , consistent with the modeland the experimental data, can be obtained for each ma-terial. The relationship between the characteristic pa-rameters will be obtained for La -based compounds atseveral doping concentrations x , ranging from the under-doped to the overdoped regime. Different values of thecoupling constant and the overlapping parameter consis-tent with the model, are obtained using the experimentalvalues of ∆ and T c .The single layer cuprate superconductor La − x Sr x CuO ( La − T c superconductors.This fact makes this cuprate very attractive for boththeoretical and experimental studies. High quality singlecrystals of this material are available with several dopingconcentrations which are required for experimentalstudies. Even the determination of charge carrierconcentration in the cuprate superconductors is quitedifficult, the La −
214 is a system where the carrierconcentration is nearly unambiguously determined. Forthis material, the hole concentration for
CuO plane, n D , is equal to the x value, i.e. to the Sr concen-tration, as long as the oxygen is stoichiometric[21, 22].Additionally, there are reliable data for the T c andthe superconducting gap ∆ for several samples in thesuperconducting region. III. RESULTS AND DISCUSSION
In order to get numerical results, with our overlap-ping band model with d-wave symmetry, the cuprate La − x Sr x CuO was selected. The values for ∆ aretaken in the interval 2 ≤ ∆ ≤
12 meV which includesexperimental results[22]. The behavior of λ as functionof x and γ at T = T c is obtained from Eq. (6); and λ asfunction of ∆ , x and γ at T = 0K is given by Eq. (7).To have coupled solutions of these equations the same λ value for T = T c and T = 0K is proposed. These so-lutions correspond to different overlap values 1 − γ , ateach equation. With this model and s-wave symmetry,the band overlapping 1 − γ was higher at T = 0K thanat T = T c [23]. We consider the same behavior with d-wave symmetry. The maximum T c for cuprate supercon-ductors is obtained at optimal doping. With the model λ ( x ) values are obtained, including at optimal doping λ ( x op )[24].In Fig. 1. values of the coupling parameter λ in termsof the overlapping parameter γ are shown in the weakcoupling region. The experimental results of T c and ∆ from Refs. [20] and [22] were introduced. The curves at T = 0K (broken curve) and at T = T c = 40K (con-tinuous curve) for La − x Sr x CuO , with optimal doping x op = 0 .
16 are shown. The minimum γ value of 0 . γ is larger for T = 0K thanfor T = T c . In order to use the same λ for T = T c and T = 0K, the λ values must be restricted i.e. , the λ valueat each γ must be larger than λ min = 0 .
57 at T = 0K.In the region γ ≥ . λ value, a largerband overlapping 1 − γ is obtained for T = 0K than for T = TcT = = Γ Λ FIG. 1: The coupling parameter λ in terms of the overlappingparameter γ , with optimal doping x = 0 .
16. The T = 0K(broken curve) and T = T c = 40K (continuous curve) for La − x Sr x CuO , are shown. The horizontal line at λ = 0 . λ selected for T = T c with γ = 0 . T = T c in agreement with our assumption. For instance,the maximum λ for T = T c with γ = 0 .
95, is shown bythe horizontal line at λ = 0 .
68, and the intersection ofthis line and the T = 0K curve is at γ = 0 .
94. The samerestrictions over λ are considered at any other doping inthe superconducting phase. However, for any x = x op ,the λ value must be smaller than λ = 0 . x op , are com-pared with the underdoped x = 0 .
13 and the overdoped x = 0 . and T c for each doping, were introduced. The continuous curvescorrespond to x op , the small dashed curves show the un-derdoped behavior and the large dashed ones the over-doped results. In the optimal doped and underdopedcases, the T = 0K curves are above the corresponding T = T c ones. In the overdoped case, the behavior isdifferent i.e. , the T c curve is above the T = 0K one.In the three cases, the values of the coupling param-eter are in the weak coupling region for the γ valueswhich satisfy the conditions of our model. All the γ values which satisfy the λ restrictions are allowed. How-ever, as an example, we have selected extreme λ valuesin the three cases. The three horizontal lines show these λ values.As in Fig. 1 the maximum λ value selected at optimaldoping is λ = 0 .
68. In the underdoped case λ = 0 .
65 isselected. This value corresponds to the overlapping pa-rameter γ = 0 . T = 0K curve,and γ = 0 .
941 at the T = T c curve. In the overdopedcase, the selected λ value is 0 . i.e. the minimum of thecurve T = T c . With this λ , the overlapping parametersare γ = 0 .
599 for T = 0K and γ = 0 .
76 for T = T c .With numerical solutions of Eq. (7) we may obtain thegap ∆ in terms of the parameters of our model. The Γ Λ FIG. 2: The results for optimal doping (continuous curves)are compared with the underdoped (small dashed curves) x =0 .
13 and the overdoped x = 0 . λ values: λ = 0 .
68 at optimal doping, λ = 0 .
65 in the underdoped case,and λ = 0 .
51 in the overdoped case. underdoped material is considered in Fig. 3 because theadvantage of our model is easily shown. The gap ∆ isshown in terms of the coupling parameter λ . The gap∆ always increases with the coupling parameter λ . Thecurves are drawn for γ = 0 . , . . γ = 0 . λ value for any ∆ and for any λ the maximum ∆ value. D MF = k B T C (cid:144) = D = ΛD FIG. 3: The gap ∆ obtained in terms of the couplingparameter λ for the underdoped sample. The curves aredrawn for γ = 0 . , . . = 10 . MF = 6 . The continuous horizontal line shows the experimen-tal ∆ = 10 . MF = 6 . γ values in the range considered. The bandoverlapping model also allows higher ∆ values for theunderdoped system and lower ∆ for the overdoped one,than the ∆ MF = 2 . k B T c .In Fig. 4 the behavior between ∆ and λ for optimaldoping is compared with the underdoped and the over-doped cases. The γ values introduced are those selectedin Fig. 2 for T = 0K. The horizontal lines are the λ valuesalso selected in Fig. 2. All the continuous curves corre-spond to optimal doping. The large and small dashedcurves correspond to the overdoped and the underdopedsystems respectively. The curves show the interestingrelationship between these parameters. As for optimaldoping, the coupling parameter increases with ∆ for anydoping. The vertical lines are the experimental ∆ val-ues. It is possible to reproduce the experimental ∆ inthe range 0 . ≤ x ≤ .
2. The band overlapping in-troduced in this model allows the reproduction of thebehavior of ∆ with doping. x = = = D Λ FIG. 4: ∆ as function of λ for optimal doping, underdoped,and the overdoped cases. The horizontal lines are the λ valuesselected in Fig. 2. The vertical lines show the experimental∆ values. The continuous lines correspond to x = 0 .
16, thesmall dashed ones to x = 0 .
13, and the large dashed ones to x = 0 . In conclusion, we presented an overlapping band modelwith d-wave symmetry, to describe high- T c cuprate su-perconductors, within the BCS framework. We have useda model with anomalous Fermi Occupancy and d-wavepairing in the 2D fermion gas. The anomaly is introducedvia a generalized Fermi surface with two bands as a pro-totype of bands overlapping. We report the behavior ofthe coupling parameter λ as function of the gap ∆ andthe overlapping parameter γ , for different doping sam-ples. The λ values consistent with the model are in theweak coupling region. The behavior of ∆ as functionof λ shows that for several band overlapping parametersit is possible to reproduce the experimental ∆ valuesnear the optimal doping, for the cuprate La − x Sr x CuO .The band overlapping allows the improvement of the re-sults obtained with a d-wave mean-field approximation,in a scheme in which the electron-phonon interaction isthe relevant high- T c mechanism. The energy scale of the anomaly (1 − γ ) E F is of the order of the Debye energy.The Debye energy is then the overall scale that deter-mines the highest T c and gives credibility to the modelbecause it requires an energy scale accessible to the lat-tice. The enhancing of the DOS with this model simu-lates quite well intermediate and strong coupling correc-tions to the BCS framework. [1] X. J. Zhou et al. , Phys. Rev. Lett. , 117001 (2005).[2] J. Lee et al. , Nature , 546 (2006).[3] G. Deutscher, Rev. Mod. Phys. , 109 (2005).[4] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. , 969(2000).[5] C. S. Liu and W. C. Wu, Phys. Rev. B , 014513 (2007).[6] G. Blumberg et al. , Phys. Rev. Lett. , 107002 (2002).[7] M. M. Qazilbash et al. , Phys. Rev. B , 214510 (2005).[8] D. G. Hawthorn et al. , Phys. Rev. B , 104518 (2007).[9] M. Le Tacon, A. Sacuto, and D. Colson, Phys. Rev. B , 100504 (2005).[10] J. P. Franck, Physical Properties of High TemperatureSuperconductors IV (World Scientific Publi. Co., Singa-pure, 1994), p. 184.[11] X.-J. Chen et al. , Phys. Rev. B , 134504 (2007).[12] Z. Hassan, R. Abd-Shukor, and H. A. Alwi, Int. J. Mod.Phys. B , 4923 (2002).[13] K. Nakayama et al. , Phys. Rev. B , 014513 (2007).[14] N. Kristoffel, P. Robin, and T. Ord, J. Phys.: Conf. Ser. , 012034 (2008). [15] T. Thonhauser, H. Auer, E. Y. Sherman, andC. Ambrosch-Draxl, Phys. Rev. B , 104508 (2004).[16] C. M. I. Okoye, Physica C , 197 (1999).[17] S. A. Saleh, S. A. Ahmed, and E. M. M. Elsheikh, J.Supercond. Nov. Magn. , 187 (2008).[18] M. Moreno, R. M. M´endez-Moreno, M. A. Ort´ız, andS. Orozco, Mod. Phys. Lett. B , 1483 (1996).[19] T. M. Mishonov, S. I. Klenov, and E. S. Penev, Phys.Rev. B , 024520 (2005).[20] D. R. Harshman and A. P. Mills, Phys. Rev. B , 10684(1992).[21] Y. Ando et al. , Phys. Rev. B , R14956 (2000).[22] A. Ino et al. , Phys. Rev. B , 094504 (2002).[23] S. Orozco, M. Ortiz, R. Mendez-Moreno, and M. Moreno,Appl. Surf. Sci. , 65 (2007).[24] S. Orozco, M. Ortiz, R. Mndez-Moreno, and M. Moreno,Physica B , 4209 (2008).[25] H. Won and K. Maki, Phys. Rev. B49