We studied the effects of nonmagnetic impurities on high-temperature superconductors by solving the Bogoliubov-de Gennes equations on a two-dimensional lattice via exact diagonalization technique in a fully self-consistent way. We found that s-wave order parameter is almost unaffected by impurities at low concentrations while dx2-y2-wave order parameter exhibits a strong linear decrease with impurity concentration. We evaluated the critical impurity concentration nc i at which superconductivity ceases to be 0.1 which is in good agreement with experimental values. We also investigated how the orthorhombic nature of the crystal structure affects the suppression of superconductivity and found that anisotropy induces an additional s-wave component. Our results support dx2-y2-wave symmetry for tetragonal and s + dx2-y2-wave symmetry for orthorhombic structure

Bayindir, M.

Gedik, Z.

Bilkent University Institutional Repository

Eur. Phys. J. B 10, 287–291 (1999) THE EUROPEANPHYSICAL JOURNAL Bc©EDP SciencesSocietà Italiana di FisicaSpringer-Verlag 1999Suppression of superconductivity in high-Tc cupratesdue to nonmagnetic impurities: Implications for the orderparameter symmetryM. Bayindir and Z. GedikaDepartment of Physics, Bilkent University, Bilkent, 06533 Ankara, TurkeyReceived 23 November 1998Abstract. We studied the effects of nonmagnetic impurities on high-temperature superconductors by solv-ing the Bogoliubov-de Gennes equations on a two-dimensional lattice via exact diagonalization techniquein a fully self-consistent way. We found that s-wave order parameter is almost unaffected by impuritiesat low concentrations while dx2−y2-wave order parameter exhibits a strong linear decrease with impurityconcentration. We evaluated the critical impurity concentration nci at which superconductivity ceases tobe 0.1 which is in good agreement with experimental values. We also investigated how the orthorhombicnature of the crystal structure affects the suppression of superconductivity and found that anisotropyinduces an additional s-wave component. Our results support dx2−y2-wave symmetry for tetragonal ands+ dx2−y2-wave symmetry for orthorhombic structure.PACS. 74.72.-h High-Tc compounds – 74.62.Dh Effects of crystal defects, doping and substitution –74.20.-z Theories and models of superconducting stateThe symmetry of the order parameter (OP) in high Tccuprates is important both for understanding the mech-anism of superconductivity and also for technological ap-plications [1]. For example, d-wave symmetric OP effec-tively refutes phonon mechanism and for a device madeof a d-wave superconductor having no gap in energy spec-trum, no refinement would get rid of the dissipation at lowfrequencies, even at low temperatures. 3d metal (Zn, Ni,Al, Ga, Fe, ...) atom substitution for Cu atoms in high-Tccuprates may identify the symmetry of the OP [2]. It isa well known fact that for conventional superconductorshaving isotropic order parameter, nonmagnetic impuritieswith small concentrations have no effect on critical tem-perature [3–6] while magnetic impurities act as strong pairbreakers, and as a result of this superconductivity is sup-pressed very rapidly [7–9]. On the other hand, nonmag-netic impurities are very effective in anisotropic supercon-ductors [10–12]. For a pure superconductor, anisotropyleads to increase in Tc [10,13,14] and the critical tem-perature suppression rate with increasing impurity con-centration is proportional to the strength of anisotropy[10–12]. Unlike the conventional superconductors, in hole-doped [15] high-Tc cuprates both magnetic (Ni) and non-magnetic (Zn) impurities suppress Tc very effectively.Dependence of the superconducting properties (criti-cal temperature, order parameter, density of states, ...) onimpurity or point defect concentration is a subject of on-going research. So far, most of the experiments have beena e-mail: gedik@fen.bilkent.edu.trperformed to investigate effects of Zn and Ni substitutionin YBa2Cu3O7−δ compounds [16–39]. Alternatively, dis-order can also be introduced by creating defects with ionirradiation [40–44], but in this case affected region is oftenuncontrollable.In spite of the complexity of the high-Tc cuprates(boundaries, defects, ...), insufficient control of the actualimpurity or point defect concentration, solubility, and ho-mogeneity of the distribution of the dopants which maylead to contradictory data, we can summarize some of theexperimental results as follows:– For YBa2(Cu1−xZnx)3O7−δ compounds, at small con-centrations, x < 0.04, Zn ions occupy preferably Cu-sites in the CuO2 planes, however for x > 0.04 thesubstituent starts to occupy Cu-sites in the chain[16,19,45–47]. Since this compound has two planes andone chain in a unit cell, for x < 0.04 the actual (ef-fective) impurity concentration ni, i.e. the number ofimpurities per unit cell per CuO2 plane, becomes 3x/2.– The critical temperature decreases linearly with in-creasing impurity concentration in substitution [16,29](for x > 0.04 the drop rate decreases due to partialoccupancy of Zn at chain sites) and point defect con-centration in irradiation [44] experiments.– Due to orthorhombicity of YBCO material, CuO2planes exhibit an anisotropic behavior and admixtureof d- and s-wave is possible [35,48–50].– Zn substitution does not alter the carrier concentrationin CuO2 planes [32].288 The European Physical Journal BOn the theoretical side, the existing pair-breakingmodels overestimate the suppression of critical temper-ature and predict an increasing slope with increasing im-purity concentration [51–54] which contradicts the ob-served linear dependence of Tc on ni. The effects ofneither magnetic (Ni) nor nonmagnetic (Zn) impurities onthe superconducting properties of the cuprates have beenexplained clearly. The main reason for the discrepancybetween theory and experiment is that the conventionalAbrikosov-Gor’kov (AG)-type pair-breaking models ig-nore the position dependence of the order parameter nearimpurity sites. Recently, Franz and his coworkers [55], andZhitomirsky and Walker [56] argued that spatial variationof the order parameter must be taken into account forshort coherence length superconductors.In the present paper, we investigate effects of nonmag-netic impurities on high-Tc cuprates for both tetragonaland orthorhombic phases by solving the Bogoliubov-deGennes (BdG) equations [57,58] in a fully self-consistentway. In particular, we address the possibility of extractingthe OP symmetry by examining the effects of nonmagneticimpurities. Our results support dx2−y2-wave pairing sym-metry for tetragonal and s+dx2−y2-wave for orthorhombicstructure. The possibility of admixture of s- and d-wavesymmetries have already been proposed in various exper-imental [49,50] and theoretical works [59–61,51].The BdG equations on two-dimensional lattice havethe following form [58,62]∑j(Hij ∆ij∆?ij −H?ij)(un(j)vn(j))= En(un(i)vn(i)), (1)where un(i) and vn(i) are quasiparticle amplitudes at sitei with eigenvalue En, and ∆ij is the pairing potential. Thenormal-state part of the Hamiltonian can be written asHij = (tij + Uijnij/2)(1− δij)+(V impi − µ+ Uiinii/2)δij , (2)where tij is the hopping amplitude, µ is the chemical po-tential, Uijnij/2 and Uiinii/2 are the Hartree-Fock poten-tials with on-site interaction Uii and off-site interactionUij , respectively. Finally Vimpi is the impurity potential.The pairing potentials are defined by∆ij = −UijFij . (3)The charge density nij in the Hartree-Fock potentials andthe anomalous density Fij in the pairing potential are de-termined fromnij =∑σ〈Ψ†σ(i)Ψσ(j)〉, (4)Fij = 〈Ψ↑(i)Ψ↓(j)〉, (5)where σ is spin index, and Ψ†σ(i) and Ψσ(i) are relatedto the quasiparticle creation (γ†nσ) and annihilation (γnσ)operators[Ψ↑(i)Ψ†↓ (i)]=∑n[γn↑(un(i)vn(i))+ γ†n↓(−v?n(i)u?n(i))], (6)where γ and γ† satisfy the Fermi commutation relations.The self-consistency conditions can be written in terms ofun, vn, and Ennij = 2∑nu?n(i)un(j)f(En)+vn(i)v?n(j)[1− f(En)], (7)Fij =∑nun(i)v?n(j)[1− f(En)]−v?n(i)un(j)f(En), (8)where f(En) = 1/[exp (En/kBT ) − 1] is the Fermi distri-bution function.We solve the BdG equations on a 20×20 square lattice(hence, we diagonalize a 1600 × 1600 matrix) with peri-odic boundary conditions by exact diagonalization tech-nique using IMSL subroutines. After choosing a suitableinitial guess for OP, we solve equation (1). Next, we cal-culate the new charge density nij and anomalous densityFij via equations (7, 8) and iterate this procedure un-til a reasonable convergence is achieved. The BdG equa-tions are solved self-consistently. Self-consistency condi-tions (Eqs. (7, 8)) lead to 10 separate equations. Thefirst five (obtained from Eq. (7)) renormalize on-site en-ergies and hopping matrix elements while the last five(obtained from Eq. (8)) affect the on-site and nearest-neighbor interaction terms. Although, the first five of theseself-consistency conditions can be neglected for conven-tional superconductors where U/t 1, for strong interac-tion case we should keep them, since they play importantrole especially in the presence of impurities. The impu-rity potential V impi is treated in the unitary limit, i.e.V impi  t, and taken nonzero for randomly chosen latticesites.We first solve the BdG equations for tetragonal casetx = ty = t where tx and ty are nearest-neighbor hoppingamplitudes along x and y directions, respectively. For s-wave OP symmetry we assume that on-site (attractive) in-teraction is Uii = −1.7t and there is no nearest-neighborinteraction. In the case of d-wave OP symmetry on-site(repulsive) interaction is Uii = 1.4t and nearest-neighbor(attractive) interaction is Uij = −1.4t. With this choiceof parameters we fix the chemical potential µ so that theband filling factor is 〈n〉 ' 0.8 and the zero temperaturecoherence length is ξ0 ' 4a. These values are in goodagreement with the commonly accepted experimentalvalues.Figure 1 shows that, at low impurity concentrationss-wave OP symmetry is almost unaffected by the impu-rities or point defects. Although we use several on-siteinteraction values by keeping the band filling factor andzero temperature coherence length constant, we do notget any qualitative change. This result is consistent withAnderson theorem [5] and AG theory [8]. However, exper-imental data for high-Tc cuprates exhibit a much strongersuppression of superconductivity with increasing disorder.On the other hand, our d-wave calculations give re-sults similar to the behavior observed in experiments.For d-wave symmetry we find that on-site pairing poten-tial is negligibly small. In Figure 1, ∆d is amplitude ofthe nearest-neighbor pairing potential. We obtain a linearM. Bayindir and Z. Gedik: Nonmagnetic impurities and symmetry of the order parameter in cuprates 289Table 1. The critical temperature Tc0 and the initial drop χ =[(Tc0 − Tc(x))/Tc0]/x in various Zn doped YBCO compounds.In constructing the table, we used Tc(x) values at x < 0.04 forwhich χ is almost x independent.Material Tc0 [K] χ ReferenceYBa2(Cu1−xZnx)3O7 92 −13 [16]YBa2(Cu1−xZnx)3O7−δ 90 −12.3 [18]YBa2(Cu1−xZnx)3O7 90 −10.5 [19]YBa2(Cu1−xZnx)3O7 92 −12.3 [23]YBa2(Cu1−xZnx)3O7 92 −15 [25]YBa2(Cu1−xZnx)3O7 87 −8.7 [29]YBa2(Cu1−xZnx)3O6.9 93.6 −6.8 [34]YBa2(Cu1−xZnx)3O6.9 93 −15 [35]0.00 0.02 0.04 0.06 0.08 0.10 0.12ni0.00.20.40.60.81.0<∆ d>/∆d0, <∆ s>/∆s0s−waved−waveFig. 1. Normalized s- and d-wave order parameters, 〈∆d〉/∆d0and 〈∆s〉/∆s0, versus impurity concentration ni for tetragonalstructure. ∆d0 and ∆s0 are the magnitudes of the order pa-rameters in the absence of the impurities, and 〈· · · 〉 is takenover 20 different impurity distributions. Solid lines representthe best linear fit to the data.decrease in the mean OP, which is assumed to be propor-tional to the critical temperature Tc [67], and the slope ofthe straight line is in good agreement with the experimen-tal data summarized in Table 1. The critical impurity orpoint defect concentration nci at which superconductivityceases is also near to experimental value ' 0.1. In compar-ing our results with experimental data we should keep thefollowing point in our mind. For x < 0.04, substitutionalimpurities go preferentially to CuO2 planes [16,19,45–47]and hence the actual concentration is 3x/2. However forhigher concentrations some of the Zn atoms occupy thechain sites, and in this case we cannot relate the in planeconcentration to the actual one. Therefore, we used theinitial points, i.e. x < 0.04, to evaluate the initial dropin Table 1. To obtain the experimental value for criti-cal impurity concentration nci , we extrapolated the linearparts of the experimental curves to intersect the impurityconcentration axes.0.00 0.02 0.04 0.06 0.08 0.10ni0.00.20.40.60.81.0<∆ dx>/∆dx0Fig. 2. Normalized order parameter versus impurity concen-tration for orthorhombic structure. ∆dx0 is x component ofthe order parameter in the absence of impurities. Solid linerepresents the best linear fit to the data.0.00 0.02 0.04 0.06 0.08 0.10ni0.00.20.40.60.81.0<∆ dy>/∆dy0Fig. 3. Normalized order parameter versus impurity concen-tration for orthorhombic structure. ∆dy0 is y component of theorder parameter in the absence of impurities. Solid line repre-sents the best linear fit to the data.Similar equations have already been solved by Xiangand Wheatley [63], however our additional self-consistencyconditions and choice of parameters lead to a correct pre-diction for the critical impurity concentration. It is im-portant to note that we can not have a self-consistentsolution of the OP for extended s-wave by using anyphysical values for the above model parameters. This factwas pointed out by Wang and MacDonald [64], and theyfound that extended s-wave component is smaller thand-wave component by about two orders of magnitude.When we introduce an orthorhombic distortion by tak-ing ty = 1.5tx, as suggested by experimental data [35,48],we observe that an s-wave component (of approximatelyten percent of d-wave components) is induced. Figures 2and 3 show the variation of ∆dx and ∆dy components,290 The European Physical Journal B0.00 0.02 0.04 0.06 0.08 0.10ni0.00.20.40.60.81.0<∆ s>/∆s0Fig. 4. Normalized s-wave component of the order parame-ter versus impurity concentration for orthorhombic structure.∆s0 is the magnitude of the order parameter in the absence ofimpurities.respectively. In the absence of disorder, ∆dy/∆dx ' 1.5with ∆dx = 0.082t. With increasing disorder, the largerone, i.e. 〈∆dy〉, is suppressed faster. When we reach thecritical impurity concentration both components vanishsimultaneously. Moreover, d-wave components of the OPdecrease linearly with ni and vanish at ni ' 0.1 as in thecase of pure d-wave symmetry. Here 〈· · · 〉 denotes averag-ing over 20 different impurity configurations.As can be seen from Figure 4, s-wave componentalso decreases with ni, however while d-wave componentsexhibit a linear dependence on impurity concentrations-wave component shows a downward curvature similarto prediction of the AG theory.In conclusion, we investigated the effects of nonmag-netic impurities and point defects within a BCS mean-field framework by means of BdG equations. For tetrago-nal structure, we found out that the observed suppressionof superconductivity, when impurities are substituted orpoint defects are introduced, can be explained only if theOP is dx2−y2-wave symmetric. 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It is important to note that near Tc the coherence lengthdiverges and may become larger than the system size,therefore, due to finite-size effects, determination of Tc maybe difficult.

Suppression of superconductivity in high-Tc cuprates due to nonmagnetic impurities: Implications for the order parameter symmetry

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Suppression of superconductivity in high-Tc cuprates due to nonmagnetic impurities: Implications for the order parameter symmetry

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