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Physica C: Superconductivity A possible explanation of gap features in highTc cuprates: Tunneling density of states and...
A possible explanation of gap features in highTc cuprates: Tunneling density of states and infrared conductivity
Yasushi Shiina, Naoya Matsuda, Yoshiko Oi Nakamuraكم أعجبك هذا الكتاب؟
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المجلد:
212
عام:
1993
اللغة:
english
الصفحات:
12
DOI:
10.1016/09214534(93)90500p
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PHYSICA PhysicaC 212 (1993) 173184 NorthHolland A possible explanation of gap features in highTc cuprates. Tunneling density of states and infrared conductivity Y a s u s h i Shiina, N a o y a M a t s u d a a n d Y o s h i k o Oi N a k a m u r a Department of Applied Physics, Faculty of Science, Science University of Tokyo, Shinjukuku, Tokyo 162, Japan Received 16 April 1993 Within the Eliashberg theory, we have studied the paramagnetic pairbreaking effect on gap features in the tunneling density of states and infrared optical responses in highTo cuprates. Especially, the dependence of this effect on the shape of ot2F(to) is examined by using two types of ot2F(og), each of which is supposed to be relevant to Bi2212 and YBCO, respectively. The feature of gapfilling and the reduction of both Tc and the gap edge due to the pairbreaking effect are shown to depend significantly on the shape of ot2F(o~). From these analyses, a possible explanation of the discrepancy between the gap ratios estimated from tunneling measurements and infrared measurements is proposed. Calculations of the optical responses with the inclusion of both paramagnetic and normal impurities of various concentrations indicate that infrared responses in these materials can be described well by the Eliashberg theory including both kinds of impurities of certain concentrations. 1. Introduction To know the superconducting gap is of particular importance for the study of the highTo cuprate superconductors ( H T S C ) since it provides a clue to the nature of the pairing mechanism and the strength of the coupling. The tunneling measurements have for many years provided reliable gap determinations for these materials, which have given much larger gap values than the ones predicted from weakcoupling BCS theory [ 1 ]. As another useful means, infrared measurements also provide important information on the gap features and the excitation spectrum of the system in the superconducting state and in the normal state: twice the value of the e; nergy gap do of single particle excitations can be obtained from the observation of the reflectivity edge or the absorption edge in the infrared conductivity, and the nature of quasiparticles and excitations of the system is known by analyzing the frequency dependence of the optical conductivity. For this reason, many infrared experimental studies on the highTo copper oxides have been made, most of which are cited in the recent review of Tanner and Timusk [2 ], i To whom correspondence should be addressed. One of the characteristic features of the optical conductivity in the superconducting state measured in HTSC is a large and wide absorption peak around the frequency ~o~ 10Tc which can be explained by the socalled Holstein effect [3 ] due to the strong electronphonon coupling if the phonon mechanism is really responsible for the superconductivity in these materials. Therefore this feature is consistent with the result of large gaps observed in tunneling measurements. Another characteristic feature, as seemingly opposed to the above one, is that the values of the gap edge determined from optical measurements are, generally speaking, much smaller than, about a half of, those determined from tunneling experiments. Besides the smallness of the gap value, the obscure gap feature is also one of the characteristics of the optical conductivity in the superconducting state observed in HTSC [2]. To explain these characteristic features of the infrared optical responses, many theoretical works have been done on the basis of the Eliashberg theory [ 4  7 ] . Since cuprates are known to be clean superconductors whose mean free path l (order of 100 A) is much larger than the coherence length ~, we are required in calculations to take this situation into account. The classical work by N a m [8 ], which is an extention of the Mattis 09214534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved. 174 K Shiina et al. / Gap features in high 7~.euprate.s Bardeen (MB) theory [ 9] for a BCS superconductor to the strongcoupling Eliashberg theory, has a difficulty in incorporating arbitrary concentration of impurities. This problem was solved by recent work by Lee, Rainer, and Zimmermann (LRZ) [4 ] which has succeeded to provide the realfrequencyaxis formula of the optical conductivity which is valid for any normalimpurity concentration and at any frequency co. A little later, Bickers et al. [ 5 ] calculated the infrared conductivity in strongcoupling superconductors including normal impurities in a complementary way of calculation, where the imaginaryfrequencyaxis Matsubara formulation was employed and inevitably accompanied by a final analytic continuation technique by means of Pade approximants [ 10]. There, they discussed the dependence of the optical responses on the normalimpurity scattering effect employing a electronphonon spectral function c~2F(co) of a truncated Lorentzian form which neglects the low frequency spectra. Using the same model c~2F(co), Akis et al. [6] have presented, with use of the LRZ formula valid for any normaMmpurity concentration, the calculation of the optical conductivity at finite temperatures 7". There they employed the technique of Marsiglio, Schossman, and Carborre (MSC) [ l 1 ], which is a much more timesaving iterative method than the ordinary one to solve the realfrequencyaxis Eliashberg equations at finite temperatures. Although these works have been done in the consciousness of optical responses in HTSC, they did not aim to offer a conclusive argument on the interpretation of infrared measurements but limited them to analyze qualitatively the features of a conventional strongcoupling superconductor involving the normalimpurity effect. These works as well as the work done by Dolgov et al. [ 7 ], where the calculation has been carried out using a more realistic spectral function a2F(co) obtained from tunneling measurements for BiSrCaCuO systems, seem to succeed to explain qualitatively two of the characteristic features in the infrared conductivity; the feature above the gap by the effect of normal impurities and the feature in the midinfrared region by the Holstein effect [ 3 ] caused by strong electronphonon interaction. But the remaining characteristic features, that is, the obscure gap feature and the smallness of the gap value could not be explained by only the inclusion of normal impurities. These circumstances motivate us to examine a possibility of explaining the gap features observed in the infrared measurements by the pairbreaking effect due to paramagnetic impurities and reconciling seemingly discrepant gap values determined in both kinds of experiments, optical and tunneling measurements. Furthermore, if this pairbreaking effect exists, the common feature of gap smearing observed in both kinds of measurements will also be explained in the same context. In the present work we examine, within the Eliashberg theory, the effect of paramagnetic impurities as well as normal impurities on the tunneling density of states and on the infrared conductivity of the highT~, cuprates. The effect of the paramagnetic scattering on the optical conductivity has already been studied by Nicol and Carbotte [ 12 ] with use of the spectral function c~ZF(co) of Pb which scarcely has low frequency phonon spectra. However, the near gap feature of strongcoupling superconductors is known to depend significantly on the shape of c~2F(co), especially on the low frequency part of it [7]. Thereforc, if one wants to say something rather quantitatively about the gap features of HTSC, it is required to study them with use of o~2F(co) somewhat more realistic for HTSC. Another purpose of this paper is, therefore, to examine the dependence of the effect of paramagnetic impurities on the shape of c~2F(co) by calculating the above quantities with use of two types of model c~2F(co) which are constructed from the inelastic neutron scattering experiment (INS) data for Bibased cuprates [ 13]. Since we assume that the superconductivity in cuprates is a phononmediated one, possible candidates of paramagnetic impurities are the local moments of Cu ions in the CuO2 planes where the superconductivity takes place. In section 2 we present our model and show numerical results of Eliashberg calculations for the gap function A(co) and the normalized tunneling density of states Ns ( c o ) / N ( O ) at zero temperature to discuss the effect of paramagnetic impurities on the gap feature and its dependence on the shape of c~2F(co). In section 3, we present the numerical results for the optical conductivity and the reflectivity in the superconducting state with both the paramagnetic and the normal scattering effect. The results are dis 175 Y. Shiina et al. / Gap features in highT,, cuprates cussed in comparison with those obtained from experiments for HTSC. In section 4, summary and conclusions are to be found. 2. Paramagnetic impurity effect on gap features in HTSC Our theoretical basis here is the Eliashberg theory. Then the question is what is the paramagnetic impurity in HTSC. In an earlier [ 14 ], we have shown that high T¢ values as well as the gap values of these materials obtained in experiments are reproduced well from the isotropic Eliashberg theory using model spectral functions oflF(og) reasonably constructed from the information of the phonon density of states obtained in INS experiments [ 13 ]. Further, in a preceding paper [ 15 ], we examined the effect of Cu spins in the CuO2 planes standing on the same theoretical point of view that these materials are phononmediated strongcoupling superconductors. There, we have made an assumption that the spins of Cu 2+ in the CuO2 planes are harmless and scarcely give a net effect on the superconductivity in a qualified single crystal sample. This assumption is based on the recognition that the spins of Cu 2+ in a CuO2 plane are highly localized and form an antiferromagnetically correlated twodimensional spin system and this strong antiferromagnetic correlation survives even in the superconducting state. From these considerations we have assumed roughly that the net compensation of the Cu 2+ spins due to the antiferromagnetic correlation with their neighbours occurs unless there exists some kind of the defect of the localized Cu 2+ spin which breaks this compensation mechanism. On this assumption, the results of the substitutional study for Cu ions in the CuOz planes were analyzed successfully to some extent. This assumption is, of course, too primitive and may be only approximately valid in real samples: samples would contain some defects in the CuO2 planes which break locally the antiferromagnetic correlation between neighbouring spins. Further, even in perfectly qualified samples, holes introduced on the oxygen sites in the CuO2 planes prevent the antiferromagnetic coupling between the neighbouring Cu spins and affect the compensation mechanism of local moments of Cu spins. Therefore, in real samples used in var ious experiments, there would be some local fluctuations of Cu spins which produce local moments and act as paramagnetic impurities. This is a rough sketch of paramagnetic impurities supposed in our model. We now turn back to the modeling of the system for Eliashberg calculation. Although HTSC have anisotropic (layered) crystal structures, the inplane (inCuO2plane) anisotropy in the superconducting properties is not believed to be so large as to alter the swave pairing. To examine the inplane gap feature, therefore, we solve the isotropic Eliashberg equations at zero temperature by considering a twodimensional system, which has an isotropic cylindrical Fermi surface, and supposing that all effects of this Fermi surface geometry are included in the shape of otZF(o)). In the present calculation, this situation is reflected on the model o?F(co) functions which are constructed from the inelastic neutron scattering (INS) data for B i  S r  C a  C u  O systems [ 13 ] by extracting the sum of phonon modes relevant to the atoms in the CuO2 plane [ 16 ], where the superconductivity takes place. In the inset of fig. 1, this prototype of the phonon density of states is shown, from which we produce two types of model a2F(og) functions by weighting the low frequency part (model A) or the high frequency part (model B). The re 6 5 4 0 40 model A LL 3 80 ,: o.I 2 ......... , !i 1 0 20 40 60 80 100 oJ (meV) Fig. l. Electronphonon coupling function o~2F(m)for model A (solid line) and for model B (dotted line ). The curve in the inset is a phonon densityof states from which two model functions are produced by weightingwith appropriate couplingfunctions. )2 Shiina et aL / Gapjeatures in high7~,cupratev 176 suiting ot2F(o)) of model A and model B are supposed to be relevant to the o~2F((o) functions for Bi2212 and YBCO respectively. In order to make a comparative study of the pairbreaking effect and their dependence on the shape of o/2F((o), the overall coupling strength of both model c~2F((o) functions in fig. 1 are adjusted to give the same value of )'~ of 80 K before the inclusion of paramagnetic impurities. Each model ~2F(oa) function gives the electronphonon coupling parameter 2, which is defined by the relation 2 = 2 ~ do)/co ol2F(~o), the values are 4.4 (model A) and 2.4 (model B), respectively. The zerotemperature realfrequencyaxis Eliashberg equations including the effect of the spinflip scattering are given by [ 14 ] 0 kN//(~) ((A)') z~ ((~°') 2 ) o' + ~o+$2+i5 0 i A(m)  2to ~ / o ~ ( ~ ) :  j ( { o ) ~ (1) and { o5(o') ekk/{ ( {,X ,;,(,o)=,,, ) 0 o)' + {o+ g2+ i6 0 1 )+ i  ~o'~o+~2ia_ ,v(0) dJ((o) 2rp ~/{7_)(oo)23((o) 2 " impurity effect on the gap feature in the tunneling density of states, we examine how the reduction of i/'~.due to this effect depends on the shape of c~2F((o ). Following the well established prescription [ 17 ], thai the critical temperature Y~.with the inclusion of paramagnetic impurities can be calculated by solving the linearized imaginaryfrequencyaxis Eliashberg equations. The numerical calculation gives the result thai T~, is 66 K for model A and 62 K for model B when l / r p = 8 meV. The Tc values are reduced to 83% for model A and 78% for model B compared to the value in the pure case. (Here after we call the case of 1/ rp=0 the pure case.) When l / r p = 16 meV, the reductions of Yc result in 66% for model A and 55% for model B, respectively. From these results we see that the reduction of T~ by the inclusion of paramagnetic impurities is more effective in the material with a highfrequencyweighted olZF((o) than that with a lowfrequencyweighted one. Therefore this reduction of Tc depends on the shape of o~2F(to) although the original Tc value does scarcely depend on the shape of a2F(~o) but does on the area under il [14,181. In order to examine the effect of the paramagnetic scattering on the gap feature of cuprates in the superconducting state and to illustrate how this effect depends on the shape of oz2F(o)), we calculate the gap function A(o)) and the normalized tunneling density of states N s ( ( o ) / N ( O ) including paramagnetic impurities with use of these specific types of model c~2F((o). The tunneling density of states is related to A((o) by the equation ( 2) where the pairing energy A(oJ) and the renormalized frequency o5(oJ) are related to the gap function A ( to ) and the renormalization function Z((o) by the relations A(o)) =A(o))Z({o) and (5(o))oJZ(oJ), IL* is a Coulomb pseudopotential with a cutoff frequency (o~, and rp is the scattering time related to paramagnetic impurities. In the following numerical calculations we take the value of/l* and ~o~ as 0.15 and 500 meV, respectively. Before going into the study of the paramagnetic t..x/~,., 3(o2) / 1 In fig. 2, (a) for model A and (b) for model B, we show the numerical results for the normalized tunneling density of states in cases involving various concentrations of the paramagnetic impurity; the values of I / r p are 0 (solid curve), 4 meV (shortdashed one), 8 meV (longdashed one), and 16 meV (dotdashed one). These curves are calculated by using the gap function A((o), which follows from the solution of eqs. (1) and (2) for each value of l/rp. In fig. 2, in the pure case, we can clearly see phonon structures in the region above the superconducting gap A0o (Ao in the pure case) at which the density of states diverges due to its square root singularity. In 177 Y. Shiina et al. / Gap features in highTo cuprates 2 ~ "l','!',, ..... !i~:'i,~  I'~1",!~.~,,. . . . v, .%~ L 1/:p=0meV 1/Zp=4 meV 1/tp = 8 meV 1/tp= 16 maY ] i ] ] ! 2 ',, / 'i,',' & 1b:,ll~ L,j[ 1/tp = 0 meV 1/zp = 4 meV  .... i 1/Zp=8meV ~~ . . . . ~ ~ I : i ode,,, 1 ': i ] 100 (meV) 200 t 0 ' 100 ~ (meV) ~ia) 200 (b) Fig. 2. Normalized tunneling density of states N, ( to )/N( 0 ) at zero temperature. (a) For model A and ( b ) for model B. Curves are shown in the cases of l/zp = 0 meV (solid line ), 4 meV (dotted line ), 8 meV (dashed line ), and 16 meV (dotdashed line ). any way the points we wish to show in these figures are as follows. First, the p h o n o n structures are smeared out with the increase o f paramagnetic impurities. Second, the square root singularity o f the tunneling density of states is smeared gradually as 1 / Zp increases and is eventually eliminated. Third, the gap edge tog can be read off in fig. 2 by looking for the energy below which the density o f states vanishes, and we see that as 1 / % increases the gap edge becomes smaller a n d the density o f states fills the inside o f the original gap (the gap in the pure case). This reduction o f the gap edge tog larger for the highfrequencyweighted a E F ( t o ) ( m o d e l B) than for the lowfrequencyweighted one ( m o d e l A ) if c o m p a r e d to each other for the same i m p u r i t y concentration. In the pure case, the gap edges are 20.6 meV for model A and 17.2 meV for model B. As l / z p increases from zero to 4, 8, and 16 meV, the gap edge tog decreases from 20.6 to 14.5, 11.3 a n d 6.9 meV for model A, and decreases from 17.2 to 11.0, 8.0 and 2.8 meV for m o d e l B, respectively. In the pure case, the gap edge tog is equal to the superconducting gap Aoo, at which Ns(to) has the square root singularity; that is, Aoo is determined as a solution o f the equation states. This situation will be explained later in the discussion in relation to the discrepancy between gap ratios d e t e r m i n e d from different kinds o f measurements. We present the renormalization function Z(co) and the gap function A(to) in figs. 3 and 4 respectively, only for model A to see qualitative features o f them. In each figure, the top frame is in the pure case ( 1 / (4) with the gap function d ( t o ) as 1 / z p = 0 . However, for the case o f 1 / z p # 0 , the superconducting gap Ao det e r m i n e d by eq. ( 4 ) is not equal to the gap edge tog, nor coincides with the energy toM, the energy at the peak position o f the n o r m a l i z e d tunneling density o f ] I Re Z({o ) 4/ 2 S N Im Z(o~ ) ........ ' y 0 ", l/z p = 0 meV I , I , I , I 4 1/t p = 16 meV 2 _,  _ , Re[A(to=Ao)]=Ao, I I 6 0 I __, 2LO0 L , , 400 (meV) Fig. 3. Renormalization function Z(to) at zero temperature for model A. The values of 1/ zp are 0 meV (top frame) and l / ~p= 16 meV (lower frame). In each frame, the solid line indicates Re Z(to) and the dotted line Im Z(oJ). 178 Y. Shiina et al. / Gap.[i'atures in htghT, cuprate.~ 6O R e _ ~ l ,~ 40I 20 . . . . >: ~A,' ' r ' : I , I ',:! J' 0 > 10t d) E ,EE  2 0 g "1 20r Im±l..) 1/ c D  0 m e V 40 40 .q 0! 1 / r D = 16 m e V 2O O 20 20k (meV) 200 400 ) (meV) Fig. 4. Gap function A(oJ) at zero temperature in the cases 1/ rp=0 meV (top frame) and l/rp=16 meV (lower frame) for model A, In each frame, the solid line indicates Re A(o)) and the dotted line Im A(oJ). r p = 0 ) while the lower frame is in the case of 1/ rp= 16 meV. In the pure case, the imaginary parts of Z(o)) and A(~o) are strictly zero below the gap Joo. As l / r p increases from zero to 16 meV, p h o n o n structures in Z(~o) and A(~o) are smeared and shifts to the left due to the decrease of the gap edge. In fig. 3 we find that, in the lower frame, the peak of Re Z(co) corresponding to the lowest energy peak in the top frame shifts to the left and a new peak appears left of it above the gap edge. This new peak in ReZ(~u) and the corresponding dip in ReA((o) at the same position originate from the paramagnetic scattering. These features in A((o) and Z((o) are characteristics of the inclusion of the paramagnetic scattering. Now we would like to c o m m e n t on the problem about the difference a m o n g the gap edge e)g, the superconducting gap Ao and the energy ~OM. In fig. 5, we show again the low energy part of the curves for ReA(co) and I m A ( o ) ) drawn in the lower frame of fig. 4 by magnifying them, where the position of oJg, Ao and O)M are indicated• There, the position of Ao is illustrated as the intersection of the curve of Re A (oJ) and a linear line, the position of (og is indicated by Fig. 5. Low energy part of the curves for ReA(oJ) and Im J(o) ) and Ndo))/N(O) for model A with I/rp= 16 meV. The position of the superconducting gap Jo, the gap edge OJg,and the energ~ ~OMare indicated for comparison. The position of Aois illustrated as an intersection of the curve of Re d (o)) and a linear line. the energy at which the curve of lmA(~o) grows up from zero and the energy O)M is read off from the m a x i m u m position of the curve of Ns(~o)/N(O). In this way, the values of~% Ao and O)M are determined in the cases of 1 / t o = 8 and 16 meV, and the resulting ratios 2OJg/kJ'~, 2Ao/k~T~ and 20)M/kBT~ for each model are tabulated in table 1. There, we can see that the reductions of all "'gap ratios" caused by paramagnetic impurities are larger for the highfrequencyweighted c~2F(o)) than for lowfrequencyweighted one. Since (UM does not shift appreciably from the original superconducting gap Ao0 and this (OM is often identified as the energy gap in tunneling experiments, the gap ratio obtained in the tunneling experiment, if the gap is determined in such a way, would differ scarcely from the value of 2Jno/kJ~, even if the sample contains some kind of paramagnetic impurities of low concentration or if the Cu spins in the C u O : planes act really as paramagnetic impurities. However if the "gap" value is identified with the gap edge eJg, as it is the case in the optical measurements, then the "'gap ratio" 2o)g/kBT,, is generally much smaller than the value of the gap ratio obtained in tunneling experiments, if the sample is considered to contain any kind of paramagnetic im 179 Y. Shiina et al. / Gapfeatures in highTocuprates Table 1 Reduction of various typesof"gap ratios" due to the pararnagneticpairbreakingeffect l/'gp (meV) Tc (K) o9s( m e V ) Model A 0 2oJ,/kBTc Ao ( m e V ) 2Ao/kBT~ o~M( m e V ) 2~M/kBTc 20.6 19.3 19.0 6.0 6.8 8.3 17.2 16.8 16.4 5.0 6.3 8.6 80 20.6 6.0 20.6 8 66 11.3 4.0 14.8 16 53 6.9 3.1 10.5 6.0 5.2 4.7 Model B 0 8 16 80 62 44 17.2 8.0 2.8 5.0 3.0 1.5 17.2 10.9 6.1 5.0 4.1 3.2 purities stated above. This result coincides with the situation occurring in the highTo cuprate superconductors, where the "gap ratios" obtained from the optical measurements are generally much smaller than those obtained from tunneling measurements. Until now the curve of Ns(og)/N(O) for the highTc cuprates has not been drawn by taking the paramagnetic scattering effect into account with use of realistic model a2F(og) functions. The similar curve for Pb have already been obtained by Daams et al. [ 19 ] in the realfrequencyaxis formulation, and recently by Nicol and Carbotte [ 12 ] in the imaginaryfrequencyaxis formalism employing the analytic continuation technique developed by Marsiglio, Schossmann and Carbotte [ 11 ]. As already seen above, the pairbreaking effect of paramagnetic impurities on the gap feature depends significantly on the detailed shape of ot2F(og), therefore, the present study of this effect on these quantities in highTo cuprates gives new informations on these materials which have ot2F(og) functions greatly differing in their shape from that of Pb, especially in a point that they have low frequency part of spectra significantly. 3. Numerical results for optical conductivity and reflectivity o(og, T )  N(O)e2v~ 4g 20) dr2 tanh M(o9 co) X [g(og' )g(og' +o9) + h(og' )h(og+og' ) + n2 l tanh [o9+o9'\ , (o9, ,o9 X [g* (o9')g*(og' +o9)+h*(og' )h*(og+ o9' ) + n 2 ] CO 09 o9 , + [ t a n h ( 2;~B T )  tanh(2kB T ) ] L (o9 , o9' ) X [g* (o9')g(og' +o9) + h*(og' )h(og+og' ) + n 2 l ~ , (5) where the functions g(og), h(og), M(og', o9) and L(og', o9) are expressed in terms of the pairing energy A(og) and the renormalized frequency o5(o9) by the relations n3(o9' ) g(og) =  x/z]'(o 9, )2_ O5(O9,)2' (6) nos(og') h(og) =  x/3(o9' )~o5(o9' )~ ' (7) M(O9', 09) = (x/A(O9+ o9' )2o5(o9+o9' )2 1 Now we will present the calculation of the optical conductivity. Following the LRZ realfrequencyaxis formulation, the optical conductivity a (co) in the superconducting state, involving the normalimpurity scattering with the scattering time T, is provided by the formula [ 4 ] (8) and L(Og', o9) = (x/,J(o9+ o9' )2~h(o9 + o9' )2 )'. Shiina et al. / (;apji, atures m high l~ cuprate.s 180 I (9) In eq. (5), e is the electronic charge, z'v is the Fermi velocity, kB is the Boltzmann constant, and N(0) is the singlespin electronic density of states at the Fermi surface which is related to the effective plasma frequency ~ov for quasiparticles moving along the CuO2 planes by the relation oJ2p= 4x ~ . . .tu;~t,~. . . )~ In order to calculate the optical conductivity including the effect of paramagnetic impurities as well as normal impurities from the LRZ formula (5), it is only required for us to calculate the pairing energy .3(w) and the renormalized frequency cO(~o) from eqs. ( 1 ) and (2) and to bring them into eqs. ( 5 )  ( 9 ) , since all the effect of paramagnetic impurities are already contained in 3 ( w ) and cb(~o). For c~2F(w) of model A and model B, the optical conductivities in the superconducting state are calculated for various concentrations of the paramagnetic impurity as well as the normal impurity. In fig. 6, (a) for model A and (b) for model B, curves of the real part of the optical conductivity ~ (w) are shown only in the cases with 1 / t o = 0 and 8 mev. The top frames are for l / / ' p = 0 and the lower frames are for l / r p = 8 meV. In each frame, curves drawn together represent those in cases of the normal scattering rate l / r = 0 meV (solid), 10 meV (dotted), 100 meV (dashed), and 1000 meV (dotdashed). If the system has phonon spectra starting from zero frequency, the absorption edge in the optical conductivity will be j u s t a t twice the gap edge, 2Wg, in the case where normal impurities are not included in the system. But, since our model c~2k'(w) function grows up at com~ 5 meV. the minimum phonon frequenc~ of the system, the absorption spectrum starts at ( O : 2[Ug~ (O m even in this case• Looking at the top frames ( l / r v : 0 ) in fig. 6 (a) and (b), we note the following characteristic leatures• First, the solid curves ( 1 / r = 0 ) start really at w:2w~+~q,, and grow up gradually exhibiting phonon structures which extend over about 8 meV. The Holstein's absorption begins to exhibit itself at the energy about 2wg + 2(~ which is the energy for an absorbed photon to break up a Cooper pair and to excite two quasiparticles with the emission of real phonons and here ~;) is the average phonon energy. This phononassisted absorption exhibits a wide and large peak around 150 meV in each solid curve for model A and model B, except for the slight difference that grows up at lower frequency for model A than for model B, because the former model c~:F((o ) has a lowfrequencyweighted spectrum. Second, when the normal impurities are introduced, the absorption starts exactly at twice the gap edge, 2wg, because of occurrence of the impurityassisted absorp ~u • ,:> 1, 1 6J m o d e l /", 1 12' I i 7 : ,i) mcV ~ 0 me', :  : 100 me',,' i,]00 r>,/ 2 ; qc; " 4! 1' ~ (] m e V 2 ; J  0 " . . . . . "'4 . .,•] 8[ t 4L 0 .. 200 .~0,~ ;t Fig. 6. Real part o f the c o n d u c t i v i t y at ( w ) in the s u p e r c o n d u c t i n g state at zero t e m p e r a t u r e ( a ) for model A and (b) tbr model B. The top frames are for 1/rp = 0 meV, and lower frames are for 1 / r o = 8 meV. The values of 1 / r are 0 meV ( solid line ), 10 meV (dotted line ), 100 meV ( d a s h e d line), and 1000 meV ( d o t  d a s h e d line). Y. Shiina et al, / Gap features in highT~ cuprates tion, and it grows up abruptly as the impurity concentration increases. By the inclusion of normal impurities, the position of Holstein peak is scarcely affected, but the peak height as well as the phonon structures are smeared slightly. Third, in the dirty limit ( I / r = 1000 meV), the curves (dotdashed) are close to those calculated from Nam's formula [ 8 ] as expected because N a m ' s formula is applicable to the superconductor in the dirty limit or in the Pippeard limit. Further it should be noted that our curves exhibit the Holstein's absorption even in the dirty limit in contrast with those from MB theory [9]. We turn to look at the lower frames in fig. 6. As stated in section 2, by the inclusion of the paramagnetic scattering, the gap is filled up gradually as the impurity concentration increases. As a consequence the absorption edge of each solid curve ( l / r = 0 ) shifts its position to lower frequency side, and the smearing of the absorption edge occurs. When normal impurities are introduced in addition, the absorption starts clearly at twice the gap edge although it grows up more slowly than the corresponding curve in the top frames does. From these circumstances we understand that it is impossible to read off the absorption edge in the optical conductivity if the material is clean. Further, if it contains paramagnetic impurities, the gap identification becomes harder because of the additional smearing of the absorption edge, and the value of the absorption edge 2 % itself becomes much smaller than the one in the pure case. Therefore, if our model for Cu spins in cuprates is correct, the gap edge would be hardly seen in optical measurements and the "gap" value, even if observed, would be reduced greatly from values obtained in most of the tunneling experiments. (This situation really occurs in the cuprate superconductors. ) So if our model for Cu spins really works in cuprates, it gives a possible explanation of controversial gap features in cuprates observed in different types of measurements, the tunneling experiment and the optical one. In fig. 7, we show the imaginary part of the conductivity cr2(og) with 1 / r p = 0 for model A. In the case 1 / r = 0 , a2(o9) is positive at all positive frequencies, but in the dirty limit ( 1 / r = 1000 meV), az (co) becomes negative in the Holstein region. A similar result was obtained by Swihart and Shaw for Pb [ 20 ]. In infrared studies, experiments generally measure 24 T 181 ~ I i) I > (2) 20 ..... E 16 > I 1/r .... I : 0 meV 1 / v = 10 m e V 1/T =lOOmeV 1/z = 1000 m e V model A 12 °JO.  i \ 1/Vp=OmeV 8 3 b 4 1< 0 , ~. 200 _z_ , ~ ~o (meV) 460 Fig. 7. Imaginary part of the conductivity a2 (to) in the superconducting state at zero temperature for model A. The curves are drawn for 1/ zp = 0 meV varying the value of 1/ r as 0 meV (solid line), 10 meV (dotted line), 100 meV (dashed line), and 1000 meV (dotdashed line). the reflectivity R(og). It is therefore interesting to calculate R ( o9) from the numerical results for a (o9). For normal incidence, the reflectivity is written as 11 R(og) = (lO) in terms of ~(to), the frequency dependent complex dielectric function, which can be written as ~(o9)  1 + i 4~(T(O9_____)), ( 11 ) o9 The calculation of the reflectivity requires another parameter top, the effective plasma frequency, which is assumed to be 2 eV in the present calculation. In figs. 8(a) and (b), we show results for R(o9) with various values of 1/Vp and 1/T in the same style as for ~r~(o9). Looking through figures, we notice the following. In the clean case although the value of R(og) begins to drop from unity at og=2ogg+COm, R(og) does not exhibit any appreciable drop at this reflectivity edge in comparison with the one in the dirtylimit case. With the inclusion of paramagnetic impurities, the gap is reduced and the reflectivity edge becomes more vague. In both curves in (a) and (b), we can see a minimum around o9~ 150 meV in R(og) which corresponds to the Holstein peak in trt (o9). 182 Y. Shiina et al. / Gap features in high T~ cuprate.~ This m i n i m u m is most clearly seen in each solid curve ( l / r = 0 ) in the top frames. In these curves, we find also that R(o~) has a " s h o u l d e r " just left to this minimum. By c o m p a r i n g the curves in ( a ) and ( b ) , we note the following difference between them. Let us look at the solid curves in the top frames ( 1 / r = 0 and 1/ r p = 0 ) . The drop in R(co) due to the Holstein peak is larger in ( a ) than in ( b ) . This corresponds to the situation that the ~q(~o) curve in fig. 6 ( a ) has a higher Holstein peak than the one in fig. 6 ( b ) . On the contrary, the " s h o u l d e r " in the R(~o) curve in ( b ) is more p r o m i n e n t than in ( a ) . This "'shoulder" in the R ( o J ) curve comes from the dip s t r u c t u r e o f the aj (~o) curve a r o u n d ~o=2o9g +o), and this dip is seen more distinctly in the cq (~o) curve in ( b ) than in ( a ) because o f the highfrequencyweighted shape o f a 2 F ( o J ) in model B. This distinct structure o f a "'shoulder" in the R(co) curve followed by a large drop looks like an edge and is apt to be misinterpreted as the reflectivity edge. This stepwise structure remains even if the n o r m a l i m p u r i t y o f finite concentration is included especially in ( b ) . Although the structures in the R(~o) curves are smeared by additional inclusion o f p a r a m a g n e t i c impurities, we can find a r e m n a n t o f this shoulder in the curves with a finite value o f 1 / r in the lower frame o f fig. 8(b). 1 ~,,, In reflectivity measurements on the YBCO system [2,2125 ] this characteristic stepwise drop around 500 cm~ followed by a shallow m i n i m u m around 8 0 0 ~ 1000 cm~ has been observed in the superconducting state, and our calculation with model B reproduces the observed feature qualitatively. On the other hand, the reflectivity measured for Bi2212 [2,26,27] exhibits scarcely the corresponding stepwise structure around 500 c m  ~ and this agrees also with our results calculated with model A. As a whole, by including both types of impurities with a p p r o p r i a t e concentrations ( 1 / r p ~ 8 meV, 1/ r ~ 10 m e V ) , the characteristic features in infrared optical responses in the superconducting state in cuprates (typically Bi2212 and Y B C O ) c a n be explained by our results calculated with two specific models o f o~2F(oJ). The smallness o f the absorption edge or the reflectivity edge is explained by the inclusion o f the paramagnetic scattering. (The gap edge reduces to four seventh o f the value in the pure case for model A and to one half of the value in the pure case for model B. ) The smearing o f absorption edge and reflectivity edge may he due to the effect of the paramagnetic impurities as well as to the cleanliness in cuprates so as the clean limit ~o/l<< I to be realized. The large and wide absorption peak in a~ (~o) at 800 ~ 1200 c m  ~ seen in experiments is explained by the Holstein peak, and the shallow m i n i m u m o f model A • 0.8 ". I, p 0 me\,' "i':'":~'"TTi'7"7"Z.T............. i ~S 0~3 moael B  0 r r ~ o ', ................................... t 04i 7 • . 7,. l'7,~=8meV )z  i 0.8 z o > i~ ,~ . . . . L_'I" "" "'__'. . . . . . . . . . . . . . . . . . . 0.6 0.6' 04i ,_] 4 [. . . . . . . . . . . . . . . . . . . 0 20o . (meV) 400 7 200 .!.0f riqov Fig. 8. Reflectivity in the superconducting state (a) for model A and (b) for model B. The top frames are for 1/to=0 meV and the lower frames are for 1/r o= 8 meV. In each frame, the values of 1/r are 0 meV (solid line), 4 meV (dotted line), 8 meV (dashed line), and 16 meV (dotdashed line ). Y. Shiina et al. / Gapfeatures in highTocuprates R(to) in the same region is of the same origin. These features are explained by our model regardless of the specific shape of a2F(to). The details of the curves of al (to) and R(to), however, depend on the shape of o~2F(to); the magnitude of the Holstein peak, the size of the dip structure in 0"1(to) or the shoulder in R(to) mentioned above depend on the type of aZF(to). Because of the stepwise drop at this shoulder the shoulder has been misinterpreted sometimes as the reflectivity edge and, as a consequence, this misleads repeatedly to a large gap ratio ~ 8 [ 2,22 ]. 4. Summary and conclusions We have studied, within the Eliashberg theory, the effect of the paramagnetic impurity on the gap features in highTc cuprates, and have examined how the effect depends on the shape of ot2F(to) by making all calculations using two types of realistic model a2F(to), the lowfrequencyweighted one (model A) and the highfrequencyweighted one (model B) which are constructed from INS data for cuprates. As a result, it is revealed that reductions of the superconducting temperature Tc and the gap edge tog due to the paramagnetic pairbreaking effect depend significantly on the shape of ot2F(to); both of these reductions are more prominent for a highfrequencyweighted o~2F(to) (model B) than for a lowfrequencyweighted aEF(to) (model A). Further, the "gap ratio" 2tog/kBTc calculated from the gap edge tog is also reduced more for model B than for model A. These results caution us that, if one wants to say something quantitatively about the paramagnetic pairbreaking effect on the gap features in HTSC, one must choose a2F(to) carefully so that it models the characteristics of the system concerned. From the calculation of the normalized tunneling density of states Ns(to)IN(O) with the inclusion of paramagnetic impurities using two types of realistic model spectral functions a2F(to) for cuprates, it has been clarified that the feature of the gap filling and the details of the smearing and shift of phonon structures due to this effect depend on the shape of otEF(to). In the process of these analyses, it has been pointed out that the discrepancy between the gap ratios determined from tunneling experiments and optical measurements can be understood if the system 183 is supposed to contain "paramagnetic impurities", possibly originating from Cu spins; the discrepancy will be traced to the difference of the definition of the "gap". The calculation of the optical conductivity and the reflectivity in the superconducting state has been made for various concentrations of the paramagnetic as well as normal impurities. From these results, the characteristic features of the optical responses measured in cuprates are shown to be explained well by the Eliashberg theory with the inclusion of both types of impurities: The smallness of the gap edge and the obscure features of the absorption edge and the reflectivity edge are explained by the inclusion of the paramagnetic impurity, and the abovegap features in the farinfrared region are done by the inclusion of the normal impurity. The calculated optical conductivity also shows the existence of a large Holstein absorption due to the strong electronphonon interaction. This large and wide Holstein peak fits the observed enhancement extending over 800 ~ 1200 c m  1 in the optical conductivity in cuprates. On the other hand, this Holstein absorption causes a wide and shallow drop in the reflectivity in the same region, and this result fits also the feature of the reflectivity in the superconducting state observed in cuprates at low temperatures. The slow drop of the refiectivity from its value of unity at the reflectivity edge observed in cuprates originates from the fact that cuprates are clean enough as the clean limit ~o/l<< 1 is realized. Although some of characteristic features in infrared responses are derived regardless of the shape of ot2F(to), the details of optical responses seem to depend on the shape of otEF(to); for example, the details in the reflectivity, such as the size of the reflectivity edge, the magnitude of the Holstein peak and whether there exists a distinct "shoulder" or not, depend on the shape of a2F(to). It is likely from our analysis that the optical responses in Bi2212 and YBCO are explained by the strongcoupling theory with use of a2F(to) of model A and model B respectively and with the inclusion of impurities of such concentrations as 1/ zp = 8 meV and l / r = 10 meV. Our program in the nextcoming paper is to calculate the temperature dependence of the optical conductivity both in the superconducting and normal state in the cuprate superconductors, by using the same model proposed here and including 184 Y. Shiina et al. / Gap features in high 7~, cuprate.~ b o t h k i n d s o f i m p u r i t i e s o f t h e s e c o n c e n t r a t i o n s . It will b e s h o w n t h a t " u n u s u a l " f e a t u r e s o f t h e o p t i c a l c o n d u c t i v i t y o f c u p r a t e s in b o t h t h e s u p e r c o n d u c t i n g a n d t h e n o r m a l state, as well as t h e l i n e a r d e p e n d e n c e o f D C r e s i s t i v i t y o n t e m p e r a t u r e T in t h e s e m a t e r i a l s , c a n b e successfully e x p l a i n e d by o u r m o d e l . T h e a b s e n c e o f t h e c o h e r e n c e p e a k in t h e N M R ratio, t h e b e h a v i o u r o f t h e s p e c i f i c h e a t a n d t h e r m o d y n a m i c p r o p e r t i e s in H T S C will b e c a l c u l a t e d in the s a m e t h e o r e t i c a l basis. T h e f i n a l goal o f o u r p r o g r a m is to e x a m i n e w h e t h e r all t h e s e u n u s u a l a n d c o n t r o v e r s i a l f e a t u r e s o b s e r v e d in v a r i o u s k i n d s o f e x p e r i m e n t s for c u p r a t e s u p e r c o n d u c t o r s c a n b e s y s t e m atically explained by our model that the strong e l e c t r o n  p h o n o n c o u p l i n g is r e s p o n s i b l e for t h e sup e r c o n d u c t i v i t y in c u p r a t e s a n d t h e f l u c t u a t i o n s o f C u s p i n s w h i c h fail to c o u p l e w i t h t h e i r n e i g h b o u r s a n t i f e r r o m a g n e t i c a l l y act as p a r a m a g n e t i c i m p u r i ties o f low c o n c e n t r a t i o n . 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