Impurity scattering in the unitary limit produces low energy quasiparticles
with anisotropic spectrum in a two-dimensional $d$-wave superconductor. We
describe a new {\em quasi-one-dimensional } limit of the quasiparticle
scattering, which might occur in a superconductor with short coherence length
and with {\em finite} impurity potential range. The dc conductivity in a
$d$-wave superconductor is predicted to be proportional to the normal state
scattering rate and is impurity-{\em dependent}. We show that {\em
quasi-one-dimensional } regime might occur in high-$T_c$ superconductors with
Zn impurities at low temp\ eratures $T\lesssim 10$~KComment: 6 pages , Revtex 3, Los Alamos Preprint LA-UR-94-9

Balatsky, A. V.

English

arXiv

Impurity scattering in the unitary limit produces low energy quasiparticles with anisotropic spectrum in a two-dimensional d-wave superconductor. The authors describe a new quasi-one-dimensional limit of the quasiparticle scattering, which might occur in a superconductor with short coherence length and with finite impurity potential range. The dc conductivity in a d-wave superconductor is predicted to be proportional to the normal state scattering rate and is impurity-dependent. They show that quasi-one-dimensional regime might occur in high-{Tc} superconductors with Zn impurities at low temperatures T {approx_lt} 10 K

UNT Digital Library

Los _ NalionalLaboratoryis _ed bythe b_ ol Callornla forthe Unled Slates Departmento( Energyunderconlmd W-7405-ENG-36.TrrLE:IMPURITIES AND CONDUCTIVITY IN A D-WAVESUPERCONDUCTORDISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theirAtnSO_S): A. V. Balatsky, T-i I employees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-enee herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.su_rrrEoT_. Los Alamos Symposium 1993: Strongly Correlated Electrort_ MaterialsLos Alamos, NM 87545, Dea_mber i5-18, 1993By acceplanoeol thisarticle, the publlsherrecognizedthal the U S Govemmentretainsa nonexduslve,royalty-freellcenseto publshor reproducethe publlshedformol thlscontdbutlonor to alow others toclosoforU S Governmentpurposes.The LosAlamosNatlonalLaboraloryrequeststhatthe publisherIdenlly thisarticleas workperfommdundertheauspicosoftheUS Departmenlof Energy.Los Alamo" National LaboratoryLos Alamos, New Mexico87545FORM NO. 8361:t4 1;;i_ _:'4.-.-;_;__ _"' :i'_ (/_ST. NO. 26295/81 n:',_)_i_-_'_ __._._Oi_t_TDUTIONFTHISDOCUMENTIS UNLIMIt_|LA-UR-Impurities and Conductivityin a D-wave Superconductor*A. V. Balatsky tTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87595(December 28, 1993)AbstractImpurity scattering in the unitary limit produces low energy quasiparticleswith anisotropic spectrum in a two-dimensional d-wave superconductor. Wedescribe a new quasi-one-dimensional limit of the quasiparticle scattering,which might occur in a superconductor with short coherence length and withfinite impurity potential range. The dc conductivity in a d-wave supercon-ductor is predicted to be proportional to the normal state scattering rate andis impurity-dependent. We show that quasi-one-dimensional regime mightoccur in high-To superconductors with Zn impurities at low temperaturesT< 10g.PACS Nos. 74.25.Fy; 71.55.Jv; 74.20.MnTypeset using REVTFfiPaper presented at the Los Alamos Conference on Strongly Correlated Electron Systems, Decem-ber 19931In this short note I will address the role of a strongly scatterin9 impurities with finiterange on the dc conductivity in a short coherence length superconductor. This is report onthe work, done in collaboration with A. Rosengren and B. Altshuler [1].It is well known that scalar impurities are pair breakers in d-wave and any other nontrivialpairing state superconductor [2-4]. They produce a finite lifetime of the quasiparticles inthe nodes of the gap, a finite density of states at low energy, and a finite low frequencyconductivity at low temperatures, ignoring localization effects. For the special case of a 2Dsuperconductor with a d-wave gap, a straightforward calculation yields the surprising resultthat dc conductivity a(w _ 0) is a "universal" number [4], independent of the lifetime ofquasiparticle (but dependent on the anisotropy ratio of the velocities of the quasiparticle inthe node of the gap) [5]. However, recent experiments on microwave absorption in YBCOcrystal _ with Zn impurities [6] show a linear temperature dependence of the conductivityfor pure samples, evolving to the quadratic behavior for higher impurity concentration, andlow-temperature conductivity, inversely proportional to the impurity concentration.i) We find a new quasi-one-dimertsional regime for dc conductivity in superconductorswith a short coherence length _, comparable to the range of impurity potential A. Thequasiparticle contribution to dc conductivity is governed by self-energy Z(w _ 0) = -i7and by the phase space available for low-energy quasiparticles. The quasiparticle dispersionis strongly anisotropic in the vicinity of the nodes in a 2D d-wave superconductor: Ek =_/v_k_ + V_FI_and vl/vF "_ Ao/eF. Here we linearized spectrum in the vicinity of thenodal point close to (_, _), so that kx is the momentum a!ong the Fermi surface and k3 isperpendicular. We find that the overall contribution to the conductivity depends on theratio of the energy of the quasiparticle to the scattering rate v_A-_/'_, A is the range of theimpurity potential, and we get at T = 0:2, h (1)For vlA-1/7 << 1 quasiparticle dynamics is essentially quasi-one-dimensional and conduc--1 Our model predicts that the dctivity depends on the impurity concentration crOwD--_ him p.conductivity at low temperature should be proportional to the scattering rate in the normalstate. This limit might occur in high-To superconductors, for which we estimate )_/a -.. 1 - 3and Ao/eF "-_ 10-1. In the hmit )_ --, 0 Eq. (1) gives the "universal" dc conductivitytr2n "_ vF/v,, found in [4].To explain this effective change of dirnensionalit v we note that transverse momentum islimited by kl < 2/A and quasiparticle dispersion on such a small scale is irrelevant, comparedto 7- The condition for this to occur is precisely via -x/7 << 1. The transverse (along theFermi surface) scattering does not contribute effectively to the conductivity; we call thiscase a quasi-orte-dirnettsiortal limit. The existence of this limit is the result of the finiteimpurity range A [7]. In the opposite limit vtA-x/7 >> 1, which holds for "zero" impurityrange, we recover standart unitary scattering results [8-10].ii) Here we will explain the assumptions we made to calculate conductivity. We assumethat impurities are strong scatterers with s-wave phase shift 60(q) -_ _r/2, for ]q] < A-1,where q is wavevector, counted from the Fermi wavevector. This assumption is well sup-ported by experiments on cuprates with Zn imputrities. The origin of strong potentialimpurity scattering in high-To superconductors is the highly correlated antiferromagneticnature of the normal state. The second assumption is about the finite range A of the impu-rity potential, which palys role of the momentum cut off in the momentum dependence ofphase shift 60(q). It is as well motivated by the fact that high-T, superconductors have asubstantial antiferromagnetic coherence length _aFvt "" 3a at the transition temperature. Ascalar impurity will produce distortions in magnetic correlations on the range of the _aFM.On the other hand superconducting coherence length _ --- 20 A is comparable to this scaleand thus, the range of the potential is finite on the scale relevant for superconductivity.This point should be contrasted to the case of heavy-fermion superconductors, where thecoherence length is _ 10 3 _, and therefore, any potential impurity will have its range sub-stantially shorter than the coherence length. We retain this cut off finite and on the orderof few h, ttice constants (_ _ 2a). This implies that impurities still are well screened ands-wave scattering is dominant.tiii) To calculate the quasiparticle conductivity we use lowest order bubble diagram withself-consistent Green functions with no vertez corrections, see for example [4]. For the dcconductivity we get [11]"0)= (-o,=(¢))(ia"(k,IF"(k,c)l'), (2)whcre,lineaxizingquasiparticlespectruminthevicinityofnodes,G"(k, w = O)= _/(_2+(v,k_) 2 + (vFk3)_), F"(k, w = 0) = 0. The momentum integral in Eq. (2) is cut off atIk! < 2/A and it yields the final formula Eq. (1) for T = 0 with O(T _) corrections.For the particular case of strong disorder vzA-1/'r << 1, considered in [1], relation betweenscattering rate in the superconducting state 7 = iE(w --, 0) and scattering rate in the normalstate F = n,,,,p/_rNo is "r = r/8 pFA F. Note that the scattering rate at low temperaturesis linearly proportional to r, as opposed to the F_ dependence in the standart unitaryscattering case for vz)W-_/7 >> 1. The assumption va)_-_/'y << 1 is consistent at A --- 2afor r > 20K. This estimate shows that the quasi-one-dimensional regime of quasiparticlescattering should occur in not too clean samples at T < 7- In this limit scattering rate in thesuperconducting state is of the same order as the normal state scattering rate _r"_2F -.- 40 Kfor pFA -_ 6 and is similar to the scattering rate in the _D limit: "r/_ _ vt_A0 pFA "" 1.The finite density of states N(w _ O)/No = F/A0 "_n,,,p, linear in impurity concentration,is generated as well.Using the above estimates we find the conductivity in quasi-one-dimensional regimee_ 16 h_(_, --, 0) - xh _ mawr (3)It is smaller than the normal state conductivity o(w --, 0)=o_m_ = (e2/_rh) (_t,/F) due tosmall factor h/(mA_cf) < 1 with A > a. Conductivity is also impurity-dependent, aQ_D _.F -_ _ n,,,,p.-zThis model predicts that the dc cr at low temperatures should be i=verstz/proportional to the scatterirzg rate irt the normal state and to the impurity concentration.We emphasize that both a higher value of the conductivity in the superconducting stateas well as strong impurity dependence at low temperatures are observed experimentally inQmicrowave absorption of YBCO [fi]. It should b.,-.pointed out that we are interested in onlyelastic scattering and strong inelastic contribution to the scattering rate above Tc is notconsidered in this model.iv) One can be almost certain that for dirty enough superconductors the qu_i-or_e-dimensional regime will occur, since impurities will lead eventually to a very high scatteringrate. The question remains about the competing phenomena, such as the localization ofquasiparticles, which rrfight occur earlier than quasi-one-dimeasional regime. It is interestingto apply this model to the Zn impurities in the YBCO. The applicability of the resultspresented here to the different high-To materials depends on the ratio of scattering rate tothe relevant quasiparticle energy. For the same impurity concentration different cupratesmay be in different regimes, depending on _AFt_, Ao/ev, and F.Acknowledgements I would like to thank A. Rosengren and B. Altshuler for collaborationand discussions. I am grateful to E. Abrahams, P. Lee, P. Littlewood, D. Pines and D. J.Scalapino for discussions.REFERENCESt Also at the Landau Institute for Theoretical Physics, Moscow, Russia.[1] A.V. Balatsky, A. Rosengren and B.L. Altshuler, preprint, 1993.[2] K. Ueda and T. M. Rice, Theory of Heavy Fermions and Valence Fluctuations, Ed. T.Kasuya and T. Saso, (Springer, Berlin, 1985), p. 267.[3] L. P. Gor'kov and P. A. Kalugin, Pisma ZhETP, 41,208, (1985); JETP Lett., 41,253( 985).[4] P. Lee, Phys. Rev. Lett. 71, 1887, (1993).[5] For the semiconductor with Dirac quasiparticle spectrum this result was obtained pre-viously: E. Fradkin, Phys. Rev. B 33, 3263 (1986).[6] D. Bonn et al., preprint, 1993.[7] The fact that impurity scattering is unitary is crucial for this quasi-one-dimensionallimit to occur. Indeed the simple estimate shows that for the Born limit with weakscattering vx)_-a/7 >> 1.[8] P. J. Hirschfeld, P. WSlfle and D. Einzel, Phys. Rev. B 37, 83 (1988).[9] P. J. Hirschfeld, W. O. Putikka and D. J. Scalapino, preprint, 1993; unpublished.[10] P. J. Hirschfeld and N. Goldenfeld, Phys. Rev. B 48, 4219 (1993).[11] G. Mahan, Many Particle Physics, (Plenum, New York, 1990), p. 614.P

Impurities and conductivity in a D-wave superconductor

https://digital.library.unt.edu/ark:/67531/metadc1273128/m2/1/high_res_d/10119720.pdf

Impurities and Conductivity in a D-wave Superconductor

Abstract

Similar works

Full text

Available Versions

UNT Digital Library