Daniel Impurezas Principal

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Residual absorption at zero temperature in d-wave superconductors

E. Schachinger1,* and J. P. Carbotte21Institut fur Theoretische Physik, Technische Universitat Graz, A-8010 Graz, Austria

2Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1

Received 27 January 2003; published 4 April 2003

In a d-wave superconductor with elastic impurity scattering, not all the available optical spectral weight goes

into the condensate at zero temperature, and this leads to residual absorption. We find that for a range ofimpurity parameters in the intermediate coupling regime between Born weak and unitary strong limit,

significant oscillator strength remains which exhibits a cusplike behavior of the real part of the optical con-

ductivity with upward curvature as a function of frequency, as well as a quasilinear temperature dependence of

the superfluid density. The calculations offer an explanation of recent data on ortho-II YBa2Cu3O6.5which has

been considered anomalous.

DOI: 10.1103/PhysRevB.67.134509 PACS numbers: 74.20.Mn, 74.25.Gz, 74.72.h

I. INTRODUCTION

Recent measurements1 of the microwave conductivity up

to 21 GHz at low temperatures in single crystal oxygen or-

dered ortho-II YBa2Cu3O6.5 YBCO have revealed thatthere remains significant absorption at T0 in the supercon-

ducting state, while at the same time the inverse square of

the penetration depth shows a linear in temperature behavior.

In the regime where this happens, the real part of the optical

conductivity 1( ) shows an unusual characteristic upward

curvature very different from a Drude-like form. Anomalous

residual absorption has also been seen in the past in THz

work on thin films of Bi2Sr2CaCu2O8 Bi2212 Ref. 2

and in this case was interpreted as due to coupling to a col-

lective mode, with the spectral weight of this mode tracking

the temperature dependence of the superfluid density.

In a pure d-wave BCS superconductor all of the optical

spectral weight will go into the condensate at T0 providing

a delta function to the real part of the conductivity at zero

frequency. This follows from the Ferrell-Glover-Tinkham

sum rule3,4 which holds when the kinetic energy change be-

tween normal and superconducting state is negligible. When

elastic impurity scattering is introduced into the system, this

no longer holds. Some optical spectral weight will remain at

finite frequency under the real part of the conductivity at T

0 and correspondingly, the superfluid density will be re-duced below its pure limit value. At the same time, however,in the unitary strong limit of impurity scattering which isthe most studied case so far, a switch from linear to quadraticbehavior in temperature is expected for the penetrationdepth. In this work we consider the case when the impurityscattering strength falls in the intermediate regime betweenBorn weak and unitary strong limit. We find that for arange of impurity parameters, i.e., concentration and strengthof the potential measured in terms of the width of the elec-tronic band, the residual, impurity induced absorption re-mains very significant, while at the same time the superfluiddensity obeys a quasilinear law over the entire low tempera-ture range and, more importantly, 1( ) shows anomalousbehavior in agreement with the experimental observation inortho-II YBCO.

II. FORMALISM

The general expression for the optical conductivity attemperatureT and frequency is well known and has beengiven in many places.5 9 It can be written as

T, p

2

4 0

dtanh 2J ,

dtanh 2

J,

,

1

wherep is the plasma frequency and the function J( , )takes on the form

2J , 1

E1E2 1N, N,

P, P, 1

E1*E2

1N*, N, P*,

P, . 2

In Eq.11/kBT, withkB the Boltzmann factor and Tthetemperature. In Eq. 2

E1, 2 i02 , ,

E2, , E1, ,

and

N, i0

E1, , P,

,

E1, .

Hereis the angle giving the direction of momentum on theFermi surface in the two-dimensional CuO2 Brillouin zone.In BCS the gap does not depend on frequency and has the

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form (, )(T)cos(2) with (T) a temperature de-pendent amplitude. The angular bracket indicates anaverage over angle .

The renormalized frequencies ( ) do not depend onangle but are changed because of the elastic impurity scat-tering which is the only scattering processes accounted for.

The equation for ( i0) i n a t-matrix approximation

is610

i0i N,

c2N, 2. 3

is proportional to the impurity concentration and c1/ 2 N(0 )Vimp with N(0 ) the normal band density ofstates at the Fermi surface and Vimp the impurity potential.

An important parameter is defined as Im ( i0) at0 which is shown in the bottom frame of Fig. 4. Forsmall values of Vimp , Borns approximation applies and c with t/c 2 and the impurity term in Eq. 3 be-comes proportional to N(, ) , its real part is the super-

conducting state density of states Ns( ) , while for largevalues of Vimp , (c0) it is inversely proportional to

N(, ) unitary limit. Both Born and unitary limit areextreme and a finite c is more realistic. As N(0) is roughlyinversely proportional to the electronic band width ( W), wehave VimpW/ (2 c). For c0.2 the corresponding Vimp is1.3 times the band width.

III. RESULTS AND DISCUSSION

The quasiparticle density of states Ns( ) for a d-wavesuperconductor in the case of a finite value of c0.2 isshown in Fig. 1 top frame. The gap amplitude was set at

242 meV and the impurity parameter was varied from0.01 meV solid, 0.05 meV dashed,

0.1 meV dotted, and 0.15 meV dash-dotted.Clearly, impurities affect most strongly the low energy re-gion near 0 up to, say, 10 meV. The pure limit, a straightline through the origin is approached as is reduced, buteven for the smallest value of 0.01 meV, changes inNs( ) persist up to 4 meV. It is the behavior ofNs( )in the low-energy range which determines low-temperatureproperties such as the specific heat. The behavior found for afinitec could not be inferred from a knowledge of the unitaryor Born limit. In the Born limit the deviation from linearitywould only occur at an exponentially small value of . Re-sults for the unitary limit are shown in the middle frame and

are seen to be very different from the top frame. The energydependence of the quasiparticle lifetime of Eq. 3 Im ( i0) is shown in the bottom frame of Fig. 1. Forc0, thescattering rate has its maximum at 0 and then dropsmonotonically as increases. For the larger values of cshown here this is no longer the case. A maximum in scat-tering rate occurs around 8 meV which is much largerthan its value at 0 denoted).

The real part of the optical conductivity for ad-wave BCS

superconductor is shown in Fig. 2 in units of p2/8. Five

values of temperature are considered, namely T0 Kshort

dotted line T5 K solid, T10 K dashed, T15 Kdotted, andT20 Kdash-dotted. Thed-wave gap ampli-

tude was set at 242 meV. The top frame applies to anintermediate scattering case with 0.01 meV and c0.3. The bottom frame is for comparison and shows uni-tary limit results ( c0). The real part of the optical conduc-tivity 1(T, ) vs up to 1 meV is radically different in thetwo cases. For c0.3 the finite Tcurves are concave up anddrop rapidly with increasing and show a very narrow half-width in qualitative agreement with the experiments byTurner et al.1 and Hosseini et al.11 The concave upward na-ture of these curves is due to the energy dependence of thequasiparticle scattering rate obtained from Eq. 3. Thesecurves all correspond to the temperature dominated regimeT see Fig. 4, bottom frame. Their half width is notrelated to the value of but rather to temperature which

samples the important energy dependence in Im

( i0). The T0 curve is impurity dominated and, conse-quently, shows a qualitatively different behavior. It cannot beobtained from an extrapolation to T0 of the finiteTcurvesshown. These conductivity curves are in qualitative agree-ment with the experimental findings.1 In the unitary limit thecurves show instead a concave downward behavior at smalland a broad half-width. This can be traced to the fact thatthe zero frequency scattering rate , is a rapidly increasingfunction of decreasing c as is seen in the bottom frame ofFig. 4. For the unitary limit is greater than its normal state

FIG. 1. The quasiparticle density of states Ns( )/N(0) vs ,

0242 meV. The top frame is for c0.2, and the middle frame

is for the unitary limit c0. The bottom frame gives Im (

0) vs for 0.15 meV and various values ofc.

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-dependent quasiparticle scattering rate in the exact expres-

sion by its

0 value, i.e.: by (c) . Here

0.15 meV. Comparison with the full resultssolid squaresreveals that the agreement is reasonable for c0.1, i.e., theregion for which the quasiparticle scattering rate has itsmaximum at 0, but fails when its maximum falls insteadat finite see Fig. 1, bottom frame.

In their analysis of optical conductivity of thin films ofBi2212 Corson et al.2 found a remaining spectral weight atT0 of about 30% of the corresponding value of the super-fluid density. In our language this corresponds to a value ofA(0)0.5. Corson et al.2 also found that the remaining op-tical weight was distributed over a width of order 12 K in-

dependent of temperature. Reference to the theoretical curvesin Fig. 4 shows that this value ofA (0) top frame is ratherhigh for the observed value of the width bottom frame;the data suggests c0.4 and of the same order as wasrequired13 to fit the single crystal infrared optical data of Tuet al.12 in Bi2212, 0.61 meV. While Corson et al. in-terpret their data in terms of extra absorption due to a col-lective mode, part if not all is due instead to the impurity

induced absorption discussed in this paper.We mention the previous work on films of

YBa 2Cu 3O 7by Hensen et al.14 in which the sizable sur-

face resistance observed at low temperatures and attendantquasilinear temperature dependent superfluid density is seento be consistent with d-wave and impurity scattering with aphase shift intermediate between Born and unitary limit. Thiswork, however, is limited to the consideration of a singlemicrowave frequency, namely, 87 GHz, and thus remainssilent on the unusual frequency dependence of 1( ) vs found also to be the characteristic of this regime in our the-oretical work and in the experimental data of Turner et al.1

IV. CONCLUSION

For impurity scattering with a phase shift intermediatebetween 0 and /2 we have found a regime in which con-siderable residual absorption can remain at zero temperaturewhile at the same time the superfluid density is quasilinear intemperature. In addition, and more importantly, in this re-gime, the real part of the optical conductivity 1( ) exhibitsa distinctly non-Drude dependence on frequency in the mi-crowave region. At low, but finite, temperatures 1( ) vs has a cusplike dependence on out of 0, followed by arapid drop concave upward curvature as increases. Thewidth at half maximum of 1( ), which can be very small,

varies strongly with temperature. This behavior is traced tothe strong energy dependence of quasiparticle lifetime whichis sampled in the temperature dominated regime discussed inthis work. All aspects of this work are in good qualitativeagreement with recent data in ortho-II YBa2Cu3O6.5 .

ACKNOWLEDGMENTS

Research supported by the Natural Sciences and Engi-neering Research Council of Canada NSERC and by theCanadian Institute for Advanced Research CIAR. J.P.C.thanks D.M. Broun for discussions.

*Electronic address: [email protected]; URL:

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FIG. 4. Top frame: remaining spectral weight A(0 )

0 d 1(0,) as a function of the impurity potential strength c.

The bottom frame gives the zero frequency value of the effective

scattering in the superconducting state (c) as a function ofc. The

solid gray line represents the results of the approximate formula for

0.15 meV.

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