aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Standard behaviour of Bi Sr C aC u O δ overdoped G.A. Ummarino
E-mail: [email protected]
Istituto di Ingegneria e Fisica dei Materiali, Dipartimento di Scienza Applicata eTecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;National Research Nuclear University MEPhI (Moscow Engineering PhysicsInstitute), Kashira Hwy 31, Moskva 115409, Russia
Abstract.
I calculated the critical temperature and superconducting gap in theframework of one band d wave Eliashberg’s theory with only one free parameter inorder to reproduce the experimental data relative to Bi Sr CaCu O δ ( BSCCO )in the overdoped regime. The theoretical calculations are in excellent agreementwith the experimental data and indicate that the cuprates in the overdoped regimeare well described by standard d-wave Eliashberg theory with antiferromagnetic spinfluctuations. tandard behaviour of Bi Sr CaCu O δ overdoped
1. INTRODUCTION
The properties of cuprates, and of Bi Sr CaCu O δ ( BSCCO ) in particular, dependstrongly on their oxygen content [1, 2]. This tunability allows to study the dopingdependence of superconductivity, an approach that has been used to investigate thefundamental properties of several unconventional materials [3, 4, 5]. For more thanthirty years there has been a major debate in the scientific community about whichmechanism is responsible for superconductivity in cuprates. However, most of theresearch has been focused on the underdoped regime, where there is a variety ofcompeting mechanisms that probably do not specifically concern the superconductingstate but that contribute to confuse and hide the true mechanism of superconductivity.The studies further underline all the differences between these new high criticaltemperature superconductors and the old low critical temperature superconductorswhose behavior is perfectly explained by the BCS theory or its natural generalization:the Eliashberg’s theory. In this article, however, I have set myself the goal of underliningthe aspects in common with the old superconductors not so much in the mechanism as inthe theoretical framework. In a recent paper [9] the authors experimentally investigatethe behavior of
BSCCO in the overdoped regime by measuring the critical temperature( T c ), the superconducting gap value (∆ ), the electron boson coupling constant ( λ Z ) andthe representative energy (Ω ) of the mechanism responsible for superconductivity. Suchan experimental study is very useful for theoreticians trying to clarify the mechanismresponsible for superconductivity in these materials. Moreover, in the overdoped regimethere are superconductors with very high critical temperatures (in our case T c ≤ BSCCO , in the overdopedregion until to non-superconducting phase. They find that the coupling strength λ Z in the antinodal region of the Fermi surface weakens with doping and at the criticalvalue λ Zc ≃ . p (the doping away from the half-filling) where the doping is expressedas p = 2 A F S − A F S is area enclosed by the Fermi contour [9] that are used asinput parameters in the Eliashberg equations. It is possible to see that both quantities:the electron boson coupling constant ( λ Z ) and the representative energy (Ω decreasewith the increasing of doping. For comparison with λ Z , the calculated coupling λ ∆ inthe gap channel, as it will be explained after, is also shown in the figure. tandard behaviour of Bi Sr CaCu O δ overdoped
2. MODEL: one band d-wave ELIASHBERG EQUATIONS
I calculated the experimental critical temperatures and the superconductive gaps shownin figure 1 by solving the one band d-wave Eliashberg equations [10, 11, 12, 13, 14, 15, 16,17]. In this case two coupled equations for the gap ∆( iω n ) and renormalization functions Z ( iω n ) have to be solved ( ω n denotes the Matsubara frequencies). The d-wave one-bandEliashberg equations in the imaginary axis representation are: ω n Z ( ω n , φ ) = ω n + πT X m Z π dφ ′ π Λ( ω n , ω m , φ, φ ′ ) N Z ( ω m , φ ′ ) (1) Z ( ω n , φ )∆( ω n , φ ) = πT X m Z π dφ ′ π [Λ( ω n , ω m , φ, φ ′ ) − µ ∗ ( φ, φ ′ )] ×× Θ( ω c − | ω m | ) N ∆ ( ω m , φ ′ ) (2)where Θ( ω c − ω m ) is the Heaviside function, ω c is a cut-off energy andΛ( ω n , ω m , φ, φ ′ ) = 2 Z + ∞ Ω d Ω α F (Ω , φ, φ ′ ) / [( ω n − ω m ) + Ω ] (3) N Z ( ω m , φ ) = ω m p ω m + ∆( ω m , φ ) (4) N ∆ ( ω m , φ ) = ∆( ω m , φ ) p ω m + ∆( ω m , φ ) (5)I assume [10, 11, 12, 13, 14, 15, 16, 17] that the electron boson spectral function α (Ω) F (Ω , φ, φ ′ ) and the Coulomb pseudopotential µ ∗ ( φ, φ ′ ) at the lowest order tocontain separated s and d-wave contributions, α F (Ω , φ, φ ′ ) = λ s α F s (Ω) + λ d α F d (Ω) √ cos (2 φ ) √ cos (2 φ ′ ) (6) µ ∗ ( φ, φ ′ ) = µ ∗ s + µ ∗ d √ cos (2 φ ) √ cos (2 φ ′ ) (7)as well as the self energy functions: Z ( ω n , φ ) = Z s ( ω n ) + Z d ( ω n ) cos (2 φ ) (8)∆( ω n , φ ) = ∆ s ( ω n ) + ∆ d ( ω n ) cos (2 φ ) (9)The spectral functions α F s,d (Ω) are normalized in the way that 2 R + ∞ α F s,d (Ω)Ω d Ω = 1and, of course, in this model λ Z = λ s and λ ∆ = λ d because in this case I searchfor solutions of the Eliashberg equations a pure d-wave form, as indicated by theexperimental data, for the gap function ∆( ω, φ ′ ) = ∆ d ( ω ) cos (2 φ ) (the s componentis zero and this happens for example when [18] µ ∗ s >> µ ∗ d ). In the more generalcase λ ∆ has d and s components. The renormalization function Z ( ω, φ ′ ) = Z s ( ω )has just the s component because. the equation for Z d ( ω ) is a homogeneous integralequation whose only solution in the weak-coupling regime is Z d ( ω ) = 0 [21]. For tandard behaviour of Bi Sr CaCu O δ overdoped α F s (Ω) = α F d (Ω) and that the spectral functions isthe difference of two Lorentzian, i.e. α F s,d (Ω) = C [ L (Ω + Ω , Υ) − L (Ω − Ω , Υ)] where L (Ω ± Ω , Y ) = [(Ω ± Ω ) +(Υ) ] − , C is the normalization constant necessary to obtainthe proper values of λ s , Ω and Υ are the peak energies and half-width, respectively.The half-width is Υ = Ω /
2. This choice of the shape of spectral function is a goodapproximation of the true spectral function [19] connected with antiferromagnetic spinfluctuations. The same thing also happens in the case of iron pnictides [20]. In any case,even making different choices for Υ, it can be verified that as Υ increases, the value of λ d decreases and there are no large λ d variations for reasonable Υ choices. For exampleif Υ = Ω the reduction of λ d is of the order of four percent respect to Υ = Ω /
2. Thetrend as a function of λ d doping remains the same, just the coefficients in the functionfit ( λ d versus λ s ) change a bit. The cut-off energy is ω c = 1000 meV and the maximumquasiparticle energy is ω max = 1100 meV. In the first approximation we put µ ∗ d = 0 (ifthe s component of the gap is zero the value of µ ∗ s is irrelevant).In our model the experimental input parameters are Ω and λ s while there is justa free parameter: λ d . We solve the imaginary axis d-wave Eliashberg equations for anyvalue of Ω and λ s and I seek the value of λ d for obtaining the correct critical temperaturethat is the most reliable experimental data, more precisely measured than the value ofsuperconductive gap. After, via Pad`e approximants [22], I calculate the low-temperaturevalue ( T = 4 K) of the gap because, in presence of a strong coupling interaction, thevalue of ∆ d ( iω n =0 ) obtained by solving the imaginary-axis Eliashberg equations can bevery different from the value of ∆ d obtained from the real-axis Eliashberg equations[6, 13]. This approach to reproduce experimental data has proved to be very efficientand successful for several materials [7, 8].
3. RESULTS AND DISCUSSION
In Fig. 2 is shown the results. From Fig. 2 it is possible see that the critical temperaturesare perfectly reproduced and the behaviour of the gap with the doping well enough: allin the framework of a very simple model without nothing of ”exotic”. What may createsome perplexity are the large values of coupling constant for doping values close tooptimal doping Probably the values of the coupling constants are effective values [23]because in our model I do not take into account the violation of the Migdal theorem [24]that almost certainly happens. In fact, it has been shown that using an Eliashberg theorygeneralized to the case in which the Migdal theorem is not valid, we obtain values of thecoupling constants λ ∆ and λ Z that are much smaller than those used in the standardEliashberg theory to produce the same critical temperatures [25]. I find that the linkbetween λ d (free parameter, determined by the calculus of critical temperature exactlyequal to experimental one) and λ s (experimental data, input parameter) (see figure 1)is reproduced very well by the equation λ d = 1 . λ s − . . (10) tandard behaviour of Bi Sr CaCu O δ overdoped λ s is λ sc = 1 .
325 as Isee from experimental data [9]. The calculus the critical temperature from the solutionof Eliashbeg equations in this case depends strongly from the values of λ d and λ s andalso small differences in the values of coupling constants produce large variation in thecritical temperature. For this reason in the formula of λ d there are three values afterthe point. In Fig 3 the the values of ∆ d ( iω n =0 ) in function of temperature are shown.
4. CONCLUSIONS
I have shown through that the experimental data ( T c and ∆ ) in the overdoped regimefor the BSCCO can be reproduced a very simple model: the standard one band dwave Eliashberg equations with antiferromagnetic spin fluctuations which indicates thatthe superconducting state has no particular characteristics in the overdoped regime.This calculation that allows to explain very well the experimental data could be a newstimulus for further theoretical investigations in the underdoped regime that neglects alot of exotic competing orders present in the normal state that probably are not directlyconnected with the superconducting phase.
Acknowledgment
The author acknowledges support from the MEPhI Academic Excellence Project (Con-tract No. 02.a03.21.0005) and the dr. D. Torsello for useful suggestions. [1] T. Watanabe, T. Fujii, A. and Matsuda,
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26 is shown. tandard behaviour of Bi Sr CaCu O δ overdoped T c ( K ) , ( m e V ) p Figure 2. (Color online) Calculated values of critical temperature (full red squares)and superconductive gap (full black circles) compared with experimental data (openred squares for the critical temperature and open black circles for the superconductivegap) in function of the doping p . The lines are guides for the eye. tandard behaviour of Bi Sr CaCu O δ overdoped D d ( i w n = ) ( m e V ) T/T c Figure 3. (Color online) Calculated values of ∆ d ( iω n =0 ) in function of normalizedtemperature ( T /T c ) for the four cases examinated: p = 0 .
20 black solid line, p = 0 . p = 0 .
25 green dotted line and p = 0 ..