analytisprb2007

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Angle-dependent magnetoresistance measurements in Tl2Ba2CuO6+and the need for anisotropic scattering

J. G. Analytis,1 M. Abdel-Jawad,1 L. Balicas,1,2 M. M. J. French,1 and N. E. Hussey11H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom

2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA

Received 25 May 2007; revised manuscript received 13 August 2007; published 28 September 2007

The angle-dependent interlayer magnetoresistance of overdoped Tl2Ba2CuO6+has been measured in highmagnetic fields up to 45 T. A conventional Boltzmann transport analysis with no basal-plane anisotropy in thecyclotron frequency cor transport lifetimeis shown to be inadequate for explaining the data. We describein detail how the analysis can be modified to incorporate in-plane anisotropy in these two key quantities andextract the degree of anisotropy for each by assuming a simple fourfold symmetry. While anisotropy in candother Fermi surface parameters may improve the fit, we demonstrate that the most important anisotropy is thatin the transport lifetime, thus confirming its role in the physics of overdoped superconducting cuprates.

DOI:10.1103/PhysRevB.76.104523 PACS numbers: 74.72.Jt, 71.18.y, 72.10.Bg, 73.43.Qt

I. INTRODUCTION

There are many routes to investigating the mechanism ofhigh-temperature superconductivity, and naively one mightexpect the normal state to bethe simplest. Despite concertedexperimental effort, however,1 the normal-state properties ofcuprates remain a profound theoretical challenge.2 Indeed,even from the earliest transport measurements in these com-pounds it was clear that the normal state was far fromconventional.3 Arguably the most remarkable phenomena arethe distinct power laws of the in-plane resistivity ab andinverse Hall angle cot Htemperature dependences. In opti-mally doped YBa2Cu3O7 YBCO and La2xSrxCuO4LSCO, for example, abTvaries linearly with temperatureover a wide temperature range, whereas cot H maintains astrongT2 dependence.4,5 In other words, it is as if these ma-terials exhibit distinct scattering mechanisms which are sepa-rately manifested according to the experimental probe being

considered. Anderson coined the phrase lifetime separationto describe this anomalous behavior, and today its interpre-tation remains one of the greatest obstacles to the develop-ment of a coherent description of the normal-state quasipar-ticle dynamics in high-Tc cuprates.

Three contrasting approaches dominate the current think-ing on the transport problem in cuprates: Andersonstwo-lifetime picture,6 marginal Fermi-liquid MFLphenomenology,7 and models based on fermionic quasiparti-cles that invoke specificanisotropicscattering mechanismswithin the basal plane.812 In the two-lifetime approach, scat-tering processes involving momentum transfer perpendicularand parallel to the Fermi surface are governed by indepen-dent transport and Hall scattering rates 1/trand 1/H withdifferent Tdependences. The proponents of the MFL hypoth-esis assume a single T-linear scattering rate which naturallyaccounts for abT, but introduce an unconventional expan-sion in the magnetotransport response whereby the Hallangle, for example, is given by the square of the transportlifetime.13 This anomalous expansion is attributed to aniso-tropy in theelasticimpurity scattering rate, possibly due tosmall-angle scattering off impurities located away from theCuO2plane.13

Attempts to explain the anomalous behavior ofabTandcotHT in cuprates within a Fermi-liquid FL approach

have centered around the assumption of a single inelasticscattering rate that is strongly dependent on the quasiparticlewave number k. This anisotropy can arise either due to an-isotropic electron-electronpossibly umklappscattering12 or

coupling to a singular bosonic mode, be that of spin,8,9charge,10 ord-wave superconducting fluctuations.11 Generat-ing a clear separation of lifetimes within these single-lifetimescenarios, however, requires a subtle balancing act betweendifferent regions in kspace with distinct Tdependences.14

In order to test these various proposals and to proceedtowards a theoretical consensus, information on the momen-tum k and energy or T dependence of the transportlifetime at or near the Fermi level Fis urgently required.This is a nontrivial exercise, however, since the transportcoefficients themselves are angle-averaged quantities involv-ing differently weighted integrations around the Fermi sur-faceFS. While angle-resolved photoemission spectroscopy

ARPES can probe directly the in-plane quasiparticle life-time via the imaginary part of the self-energy Im k ,, itsrelevance to dc transport is still unclear.15 Moreover, thereremains some dispute as to the correct form of Im k ,even for samples with nominally the same composition.16,17

Measurements of interlayer magnetoresistance as a func-tion of angle have yielded important information about theFS topology size and shapein a variety of layered metalsincluding organic conductors18 and quasi-two-dimensionalQ2D oxides.1921 In a recent paper, we showed that thistechnique could be developed to extract information on thescattering rate anisotropy and applied the technique to over-doped Tl2Ba2CuO6+ Tl2201.

22 In the present paper, wepresent a more thorough and detailed analysis of our angle-dependent magnetoresistance ADMR measurements onoverdoped Tl2201, focusing in particular on the procedureused to fit ADMR, and show how this analysis fails at highertemperatures unless one includes such anisotropy in . Weprogressively introduce anisotropy into the formalism andexplore the effects of this in both the cyclotron frequency cand the transport lifetime . The approach presented here issimilar to that described recently by Kennett and McKenziewho derived a generalized expression for ADMR in layeredmetals with basal-plane anisotropy.23 In this paper, we focuson issues pertinent to Tl2201, the importance of each param-

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eter in fitting the ADMR signal and their interdependence,and the issue of sample misalignment. Although it is difficultto isolate anisotropy in one from anisotropy in the other, thestrong temperature evolution of the ADMR signal and sub-sequent measurements of its doping dependence15suggeststhat the dominant anisotropy is in and not c. The paper isset out as follows. Section II describes the FS parametriza-tion of Tl2201 and the necessary symmetry considerationswith respect to the ADMR analysis. Section III briefly de-scribes the ADMR experiment itself. The Boltzmann formal-ism and the resulting analysis is described in Sec. IV for thecases where the parameters c and are both isotropic andanisotropicwithin the basal plane. Our conclusions are pre-sented in Sec. V.

II. THREE-DIMENSIONAL FERMI SURFACE

OF Tl2Ba2CuO6+

The present FS parametrization is identical to that usedpreviously20,22,23 and so shall be described here only briefly.The interested reader is referred to Ref. 19which details asimilar parametrization of the sheet of Sr2RuO4. Beingextended in the kz direction, the quasiparticle dispersioncontains a finite though small meV transfer integral

t, kz, where is the azimuthal angle in the kx-ky plane.The parameter t, kz is anisotropic in the plane, and theFermi wave vector kF is therefore modulated by both thein-plane dispersion andt, kz. The clearest way to expressthis is by expanding kFinto cylindrical harmonics19,20

kF,= m,n=0

kmn

cs cs

cos

sin n

cos

sin m. 1

where cs denotes coefficients corresponding to cosine andsine terms, =kzc/2, and c=23.2 is the interlayer dimen-sion of the unit cell. The symmetry of the Brillouin zonelimits the number of parameters of interest, and a pictorialillustration of this is shown in Fig. 1. In the kz direction,inversion symmetryrequires that the only terms con-taining be cosines. Three further symmetries restrict theparametrization: i the twofold rotational symmetry +, ii the mirror plane /2, and iii the screwsymmetry ,+,+/ 2. The transformations differfrom those in Ref. 19because of a different choice in coor-

dinate axes, but the operations are identical. The first sym-metry requires that all m be even. The next symmetry re-quires that all cosine terms have mmod40 and all sineterms have mmod4 2. For example, cos 4=cos 4/2 whereas cos 2=cos 2/2. The reverse is truefor the sine terms. The final symmetry requires that all of thecosine terms be accompanied byn that are even and the sineterms be accompanied by any n that are odd. For example,cossin 2=cos+sin 2/ 2+, but cos 2sin 2=cos 2+sin 2/ 2+. The converse is of course truefor the cosine terms that have mmod40. Equation1canthus be simplified to

kF,= m,n=0m mod 4=0

n even

kmncosncosm

+ m,n=0

m mod 4=2

n odd

kmncosnsinm . 2

We have shown previously that the minimum number ofparameters required to fit the data that simultaneously satisfythese symmetry constraints are k00, k04, k21, k61, and k101.20

Figure2shows the warping created by progressive inclusionof these parameters, beginning with a dispersionless t=0isotropic FS. Equation 2 has exact fourfold symmetrythough the modulation tgives rise to eight highly symmet-ric points where the transfer integral vanishes, as predictedby band-structure calculations.24 Figure3 shows the projec-tion of the three-dimensional FS as deduced by ADMR20

overlaid on that determined by ARPES25 see Eq. 15; thiscurve corresponds to the nominal doping level of this crys-tal. The agreement is very good, but most importantly thetwo experiments, to a good approximation, share the eightpoints of high symmetry. For ease of computation, the Fermisurface can be described by

.

FIG. 1. Color online The Brillouin zone stacking forTl2Ba2CuO6+. There is inversion symmetry about thekxkyplane:,,and three further symmetries about the axis along the

Xline. As described in the text they are ithe twofold symmetry+, iithe mirror symmetry / 2, and finally iiithe screw symmetry involving a translation in the kzdirection and arotation: ,+,+/2.

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F,= 2

2mkF

2 2tcos

a

k21sin 2+k61sin 6+k101sin 10 , 3

where kF =k00+ k04cos4 and a=3.866 is the in-

plane lattice parameter.

III. EXPERIMENT

Tl2201 is the most suitable cuprate system for ADMRstudies due to its single Fermi sheet,26 its strongtwo-dimensionality,27 its low residual resistivity,2830 and itsaccessibility to the whole overdoped region of the cupratephase diagram.31 Single crystals were fabricated using a self-flux method in alumina crucibles.32 As-grown crystals arenaturally overdoped, and the doping level and therefore thedesired Tc is set by annealing in oxygen or argon or in avacuum.32 The crystal used in this study 300m150m20m was annealed in oxygen at 600 K for200 min, resulting in Tc 17 K. Electrical contacts were at-tached using Dupont 6838 silver paste in a quasi-Montgomery four-wire configuration. ADMR measurementswere performed at 45 T in the hybrid magnet at the NationalHigh Magnetic Field Laboratory, Tallahassee, FL, using a

probe with a two-axis rotator. Initially, the platform on whichthe sample was mounted was rotated by an azimuthal angleexptand then the interplane resistivity zz was measured asthe polar angleexptwas swept at constant temperature andconstant fieldsee Fig.4.

IV. FITTING OF THE ANGLE-DEPENDENT

MAGNETORESISTANCE IN Tl2201

In this section we review how the Boltzmann transportequation can be used to fit the ADMR data. We begin withthe simplest case whereby both cand are isotropic beforegoing on to discuss the more general case in which bothparameters are anisotropic within the conducting plane.

A. Isotropic and c

In the presence of a magnetic field a quasiparticletraverses the FS following the contours defined by the dis-persion. During this journey the quasiparticle will gain ve-locity from the electric field until it encounters a scatteringevent, after which it begins its journey again. As the angle of

the field with respect to the crystal axes is adjusted, the qua-siparticle will traverse different orbits and the average veloc-ity in the direction of the current can vary dramatically. Thispicture is formalized in the Chambers tube integral, which isthe solution to the Boltzmann transport equation in therelaxation-time approximation

ij= e2

432

FS

dkfkk

vik,0

0

vjk,tet/dt, 4

where fkis the mean occupation of state k and is assumedto be independent ofk or, equivalently, isotropic in the azi-

FIG. 2. Color online Quasi-2D Fermi surfaces described byEq. 2with progressive inclusion of cylindrical harmonicsa k00,b k21, c k04, andd k61and k101.

.

.

FIG. 3. Color onlineThe projection of the 3D dispersion foroverdoped Tl2201as determined by ADMR thin black lines, plot-ted over the ARPESthick green lineresults for a compound witha nominally similar doping Ref.25. The warping of the transferintegral is exaggerated by approximately 100 times for clarity.

FIG. 4. Color online Diagram describing the ADMR experi-mental technique, whereby the sample schematically shown inblue is rotated by an azimuthal angle expt with respect to thelaboratory framexp ,yp ,zp, and then data are continuously taken asa function of polar angle expt. In the actual experiment, the crys-

tallographic xc ,yc plane does not lie exactly in the laboratoryframe due to a slight misalignment and the corresponding polarangle crys taken as the angle between the field direction andthe normal to the plane on the samplediffers fromexpt. Moreover,the azimuthalcrysmay change as a function ofexptas explained inthe Appendix.

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muthal angle . The Chambers formula can be used in asituation where both closed and open orbits are present.33,34

For our particular interest only closed orbits are involvedthe FS is Q2D,20,25 and we are able to use the simplerShockley-Chambers tube integral. Furthermore, it is easier touse cylindrical coordinates, in line with our description of

kF,. The interplane conductivity is then given by

zz= e2

432 d f

0

dkB

0

2

dvz,kB,

c

0

dvz ,kB,

c

e/c, 5

where kB is the reciprocal-space direction parallel to themagnetic field H, d=cdt, and c is considered isotropic.In order to use Eq.5to analyze the ADMR data, we followYamaji35 and define a vector kz

0 as shown in Fig. 5b. The

projection of the magnetic field on the azimuthal plane rela-tive to the kxaxis corresponding to the Cu-O-Cu bond di-rectionis labeled cryssee Fig.5a. Each orbital plane isthen defined by three parameters: the polar angle crys ,the azimuthal angle crys, and kz

0. The former two are deter-mined during the experiment whereas the latter is an integra-tion variable in the fitting procedure described below.36 Theintersection of this plane with the FS gives the path of thequasiparticle in reciprocal space. This plane is given by theequation

kxsin+kzcos=kB=kz0 cos. 6

This is a convenient notation because kz can be uniquelydescribed in terms ofkz0 and the projection of the Fermi wave

vector onto the azimuthal plane as the quasiparticle traversesan orbit, k F

. In summary,

kz=kB=kz0 kF

cos crystan. 7

We make the replacement Fin Eq. 5 by approxi-mating f0/F for kTF. The periodicity ofvzand vzinand, respectively, is of somecomputational benefit. Taking the Fourier transform of vzRefs.37and38and writing it as a Fourier sum gives

vz=a0+0

ancosn+bnsinn,

vz =c0+0

cncosn +dnsinn , 8

where an, bn, cn, and dn are Fourier coefficients. Using aLaplace transform and after a few algebraic manipulations,

the conductivity is finally given by37,38

zz=e 30Bcos

222 dkz001

a0c0+12 n=1

ancn+bndn1 +00

2andnbncn00n

1 +002 ,

9

where c =0 and =0 to emphasize that these parametersare isotropic. Equation 9is used to calculate the resistivityin the transverse direction zz by taking the inverse of zz,which is correct to a good approximation due to the large

anisotropy of the in-plane and interplane resistivity. Becauseparameters such as the effective massm* are not well known,it is the usual practice to simulate the relative change inmagnetoresistivityzz/zz0, where zz0is the interplane re-sistivity at zero field, rather than zzdirectly. This normaliza-tion procedure means that the warping parameters in the kzdirection can only be determined as ratios. In other words,the ADMR can be used to obtain values for k61/k21 andk101/k21but not k21, k61, or k101directly.

The parameters we wish to determine therefore are k00,k04, k61/k21, k101/k21, and 00 the cyclotron frequency andthe scattering time always appear as a product in the sum ofEq.9and thus behave as a single parameter. In a number

of earlier studies on different Tl2201crystals, in which allparameters were allowed to vary, a consistent set of FS pa-rameters were obtained.15,20,22 This enables us to refine ourparametrization and minimize the number of free parameterswithout losing confidence in their relative magnitudes. Wefix k00, for example, by first obtaining the doping level pusing the universal phenomenological relation39 between pand the critical temperature Tc,

Tcp

Tcmax

1 82.6p 0.162 , 10

and then adopting the simple hole-counting procedure

k00

2 /2/a2 =1 +p/2. 11

Our next simplifying assumption is that t vanishes ateight symmetry points on the FS see Fig. 3 as expectedfrom band-structure calculations24 and revealed by earlierADMR measurements.20 For this to be the case, we require

1 k61

k21+

k101

k21= 0, 12

which fixes k101/k21 to whatever value k61/k21 is given.Hence, only three parameters k04, k61/k21, and the product00, are used to fit simultaneouslyfive polar angle sweeps

FIG. 5. Color onlineSchematic representation of a single qua-siparticle orbit with an oriented magnetic field. Panel agives theprojected view of the Fermi surface, defining the azimuthal anglecrys, while panel bgives the definition of the parameters kB, kz

0,and crys.


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at different azimuthal angles in other words, the data aretreated as a single data set, not five separate curves. It isimportant to realize that these constraints could be relaxedwithout a significant effect on the other key parameters.

The fitting procedure begins by evaluating zzfor a givenpolar anglecrysand azimuthal anglecrys. The c-axis veloc-ity is evaluated, vz =1k/kz, as a function offor agivenkz

0, where is defined by Eq.3. Thekzdependence isdetermined by Eq.7and substituted into vz. For the givenkz

0, the Fourier transform is taken and the sum in Eq.9isevaluated. This is then integrated over kz

0 across the wholeBrillouin zone and the result inverted to give zzcrys ,crys.This is calculated for all crysand crysin a single data set.

This process is repeated for different parameter values until abest fit is achieved using standard minimization procedures.The solid lines in Fig. 6a are ADMR data taken at T

=4.2 K and0H=45 T, normalized to the zero-field resistiv-ity value zz0. Each color represents a different azimuthalangle at which the individual polar ADMR sweeps weretaken. Despite the fact that 00 is less than 0.5 in thissample, the variations in the c-axis magnetoresistance aresignificant, with both azimuthal and polar angles, thus tightlyconstraining our parametrization. Note that these data wereobtained on a different crystal to those reported in Refs.20and22though the resulting parametrizationk00=0.729

1,k04=0.022

1,k61/k21=0.7is very similar. The best least-squares fits to Eq. 9 are shown as black dashed lines andappear quite adequate for the full range of azimuthal andpolar angles studied.Data at larger angles were not taken atthis temperature in order to avoid the large torque forces thataccompany a transition to the superconducting state, whichoccurs here when Hc2 surpasses 45 T.

Corresponding data and fits for T=14 and 50 K are shownin panelsband c, respectively. For the fits at higher tem-peratureswhere a larger angular range can be swept, all FSparameters are fixed to their 4.2 K values and only the prod-uct 00is allowed to vary. The fits rapidly deteriorate as thetemperature is raised and are clearly no longer a reliable

representation of the data. In fact, even if we allow k00, k04,and k61/k21to vary with temperature, the fits do not signifi-cantly improve. Furthermore, if k00 is allowed to be a freeparameter, the fitting procedure tends to minimize at valueswhere the Fermi surface is larger that the first Brillouin zone,which is clearly unphysical. In response to this failing, weabandon our naive picture of isotropic cand and proceedto incorporate anisotropy into the formalism.

B. Isotropic and anisotropic c

To illustrate how significant anisotropy in c can be, weconsider first the most elementary tight-binding description

of an isotropic square 2D lattice. The dispersion of such asystem can be described by the equation

k 0= 2tcoskxa+ coskya , 13

where 0describes the quasiparticle dispersion taken rela-tive to some referencefor example, the nonbonding energy0. Quasiparticles complete orbits with a frequency c thatdepends on the scalar product kF vFvia the expression

c,=eBcoskF vF

kF2 . 14

Near the bottom of the band, the quasiparticle orbits in amagnetic field appear almost circular and vFis isotropic andnearly parallel to the crystal momentum k. AsFapproachesthe Van Hove singularity VHS, however, anisotropy in vFbecomes significant33 and c develops fourfold anisotropythat essentially becomes infinite at the VHS.

Let us now turn to consider the analogous situation inTl2201. According to recent ARPES experiments,25 the FScan be fitted by a tight-binding dispersion relation

FIG. 6. Color onlineThe solid lines are c-axis magnetoresistivity data zz =zzHzz0, normalized to the zero-field resistivityzz0,taken at different azimuthal rotations crys= 8 red, crys=18 orange, crys=32 green, crys=46 blue, and crys=56 violetrelative to the Cu-O-Cu bond directionat three different temperatures a4.2 K,b14 K, and c50 K. The azimuthal angles given herestrictly apply only toexpt = 90 due to misalignment of the crystal with respect to the platform axessee the Appendix for details. The blackdashed lines are the best least-squares fits obtained assuming that c=00is independent ofand that the parameters k04and k61/k21are fixed to their values at T=4.2 K. Thus only 00is allowed to vary with temperature.


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0=t1

2coskx+ cosky+t2coskxcosky+

t3

2cos 2kx

+ cos 2ky+t4

2cos 2kxcosky+ cos 2kycoskx

+t5cos 2kxcos 2ky, 15

with t1 =0.725, t2 =0.302, t3 =0.0159, t4 =0.0805, and t5=0.0034eV.

In order to visualize the doping evolution of the FS pa-rameters according to Eq.15, we show in Fig.7the varia-tion ofkF, vF, and c left, center, and right pan-els, respectively for different values of the chemicalpotential assuming a simple rigid band shift. The energy con-tours have been centered on the X point of the Brillouinzone. Though the band structure may change with holedoping,40 this approximation scheme serves as a good illus-tration of how the anisotropy in c varies in a comparableway to that in vF. As expected, the anisotropy grows as theFS at,0approaches the VHS, though in the doping rangerelevant to Tl2201, it never exceeds 30%. Interestingly, asthe chemical potential is raised, the anisotropy ofcchangessign, so that the cyclotron frequency goes from being maxi-mal to being minimal along the zone diagonal, but retainingfourfold symmetry throughout. We approximate this withoutmaking any assumptions as to the sign of c using the ex-pression

c1 0

11 + cos4 . 16

The curves in the right-sided panel of Fig.7correspond toa range of values from 0.3 for p =0.3 to 0 atoptimal doping.

The addition of this extra parameter causes only minormodifications to the fitting procedure. The conductivity isnow replaced by the equation

zz= e2

432 Bcosdkz0

0

2

d0

dvz,kz

0,F

c

vz ,kz0,F

c

exphh , 17

where h= dc0

. Under isotropic circumstances h

=/00 as in Eq. 5. However, in the case where csatisfies Eq.16, this becomes

h= 1

00+1

4sin 4 . 18

We can now define two new periodic functions pz andpz whereby

pz vz,kz0,F

c e/4sin 4/00 . 19

The Fourier transform of each function is given by Eq.8. The form of Eq. 9 is identical, only that 0 is inter-preted as the average ofc within the plane see Eq. 16.The fitting procedure proceeds as described in Sec. IV Aexcept our fitted parameters are now k04, k61/k21, ,and 00. As before, k00=0.729

1, k04=0.022 1 and

k61/k21=0.71 are fixed at low T4.2 K, and only 00andare allowed to vary as a function of temperature. Figures8a8cshow the best least-squares fits of the same ADMRdata under this new parameterization scheme. While the fits

are closer to the real data than in the corresponding isotropiccase, there is still a clear problem with the higher-temperature fits. If we choose to allow k61/k21to vary, how-ever, the fits become reasonable at all temperatures, as shownin Figs.8d8f. The mathematical reason for this is that has two competing roles: it appears in the exponent handin the ratio vz/c. In the latter, plays a similar roleto k61/k21, as can be seen with an expansion using elemen-tary trigonometric identities

sin 2+ k61k21

sin 6+k101

k21sin 101 + cos 4

=1 /2 +k61

2k21sin 2+k61

k21 + /2 +

k101

2k21 sin 6+k101

k21+k61

2k21sin 10+ k101

2k21sin 14. 20

The k61/k21, k101/k21, and terms can compensate eachother as long as remains smallthat is, as long as productssuch as k101/2k21are negligible. The only noncompensat-ing contribution ofin this expansion is in the multiplica-tion of the sin 2and sin 14terms, though perturbations ofthe former would be noticeable first. In other words, the fit-ting procedure tends to keep the sum /2+ k61/k21constantas a function of temperature and so the T-dependent changesare contained in the behavior ofh. We return to this pointlater in our discussion of Fig. 10.

The changes ink61/k21and required to satisfactorily fitthe data are significant20% change ink61/k21and a factorof 10 increase in and, if correct, would imply pronouncedFS reconstruction with increasing temperature. Between 4and 50 K, one may expect the Fermi distribution to broadenby around 2% of the bandwidth about the chemical potential.At p =0.26 this is approximately equivalent to a change innominal doping of 0.02, allowing a change in of at most0.05, as is evident from Fig. 7. This is significantly lessthan is required to quantitatively account for the evolution of

.

.

FIG. 7. Color online The parameters kF, vF, and c for fourdifferent doping levels p =0.17, 0.22, 0.26, and 0.3 based on thedispersion relation given by Eq. 15 and assuming a rigid bandshift. The black dashed line corresponds to the doping level of ourTc = 17 K sample.


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the ADMR. To our knowledge, the FS restructuring requiredto fit the present data has never been reported in cuprates andso justifying it would require some very subtle physical ar-guments. Indeed, photoemission studies have reported insig-nificant changes as a function of temperature on overdopedcompounds.41 Moreover, in a recent doping dependencestudy,15 we found that the overall anisotropy decreaseswithincreasing carrier concentrationi.e., as one approaches theVHSin marked contrast to the band-structure picture dis-cussed above and illustrated in Fig.7. In the following sec-

tion, therefore, we turn to consider the effect of anisotropicscattering, which not only allows the data to be fitted accu-rately, but also avoids the physical and mathematical diffi-culties we have encountered when considering anisotropy inc alone.

C. Anisotropic and anisotropic c

In the most elementary description, the scattering lifetimeis the average time between collisions of an electron trav-eling in a metal. In a FL picture, however, this is taken to bethe mean lifetime of an electron excitation, giving the decay

time of a quasiparticle to its ground state near the chemicalpotential. The rate of change of occupation of a state at kis related to the intrinsic transition rate between two arbitrarystatesk andk, weighted by the occupation ofkand the lackof occupation of state k.

The transition rate cannot be calculated without a prioriknowledge of the scattering processes that are present. In thelimit of elastic scattering, however, both k and kare on thesame energy surface and this function is simply coskk,wherekk is the angle between kand k. We then use the

relaxation-time approximation, whereby Pkk=Pkk is afunction ofkkonly, and this allows a natural definition forthe scattering time :

1

1 coskkPkksinkkdkk. 21

The relaxation-time approximation is often an excellentstarting point for interpreting transport data, and is usuallyconsidered to be independent of momentum, of both the ini-tial and final states. A more general theory, however, wouldallow Pkk to be a function of the initial state k. In this

FIG. 8. Color onlineThe raw datasolid curves with colors corresponding to crysas defined in Fig.6and best least-squares fitsblackdashed linesfor ADMR taken at three different temperatures T=4.2 Kpanels aand d, 14 K band e, and 50 K cand f. In

panelsa,b, andc, the parametersk04and k61/k21are fixed at their 4.2 K values while00and are allowed to vary. In panels d,e,andf, k61/k21is also allowed to vary with temperature.


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instance, it may be assumed that the form of Eq. 21 staysvery similar,42 except that Pkk is now k dependent, andhence the replacement k needs to be made. In thiscase, it can be shown42 that

gkk gk

k gkgkPkkdk, 22

where gk =fk fk0

, and fk and fk0

are the probabilities of anelectron occupying a state k in the presence of a field and inequilibrium, respectively. k is the scattering rate. Equa-tion 22 is general enough to include inelastic scatteringmechanisms too involving energy transfers kBT for anygiven scattering event, and this is a direct consequence ofthe relaxation-time approximation. Such details would benormally be contained in gk, and the Boltzmann equationwould be very difficult to solve. In the relaxation-time ap-proximation, however, these details are deliberately ignoredand all that is required for Eq. 22to hold is that the statis-tical ensemble of quasiparticles return to equilibrium be-tween collision events.43

Under these circumstances, we are able to define an an-isotropic scattering time that will enter all of our calculationsof the conductivity. Since the scattering time alwaysappearsin the product cin the sum of Eq. 26, it is clear that theprocedure for incorporating anisotropicwill be identical toSec. IV B where we introduced anisotropy in c. The sim-plest model would involve a fourfold anisotropy, and in asimilar fashion to Sandeman and Schofield14 or Ioffe andMillis11 we write

= 01 + cos4 , 23

where 0 1/0.Figure 9 illustrates the effect of this form of scattering

rate anisotropy on the mean free path l for the FS derived inEq. 15 assuming p =0.26. If=0, the scattering rate isisotropic and lk simply follows the form of vFk panela. If0 panel b, k is maximal in the directionparallel to the zone axes and competes with vFk. If, on theother hand,0, kis maximal along the zone diagonals,the anisotropy in lkis enhanced in this direction.

With this definition of, the conductivity is identicalto Eq.17, but now we have h = d

c and

h= 1

001 +

2+1

4+ sin 4

+

16sin 8 , 24with the periodic functions pz and pz now rede-fined as

pz vz,kz

0,F

c exp

+

4 sin 4+

16sin 8

00 .

25

The conductivity is then given by

zz=e 30Bcos

222 dkz01

a0c0+12 n=1

ancn+bndn1 +0

2

andnbncn0n

1 +02 , 26

where =0/1+/2. Equipped with Eq. 26 we cannow follow the procedure described above. The variable pa-rameters are now k04, k61/k21, , , and 00. As before wefix k00 =0.729

1 and k04 =0.0221 and fit the low-

temperature data with all other parameters free to vary. Inthis instance, however, we are now overparametrized sinceand can compensate each other to within a factor of/ 4. As a consequence, one cannot accurately quote ab-solute values for each parameter individually, but rather thesum +see insetbof Fig.10.

We can parametrize the quality of the fits by the sum ofsquared differences of the data from the fitted curve, which isdenoted 2. As becomes large, the terms and k61/k21

are no longer able to compensate and the fits decline in qual-ity. However, as shown in Fig. 10,there is also a broad flatregion over which 2 is minimized and one cannot pinpointthe exact value of. The axes in inset aare shifted so thatit is apparent that the sum / 2+ k61/k21is pinned to a valueof about 0.72, a fact which continues to be true at highertemperatures no matter what one forces to be. Similarly,the value of+is pinned to nearly zero at low tempera-ture. From the low-Tparametrization used previously to setthe FS parameters, we can settle on a value of 0.10.1 which is comparable to that estimated from theARPES-derived dispersion despite being opposite in sign.

Panelsa,b, andcof Fig.11show the resulting fits tothe new parametrization scheme, in which only and

0

0are allowed to vary with temperature, for T=4.2, 14, and50 K, respectively. In contrast to previous schemes, the qual-ity of the fits are comparable at all temperatures, without theneed for any variation in the other parameters. Hence, byintroducing T-dependent anisotropy in the scattering rate,there is no longer any need to invoke FS reconstruction toaccount for the evolution of the ADMR data. We thereforeconclude that this is the most elegant and physically realisticparametrization scheme of all those considered here.

The sign ofis found to be positive, indicating that scat-tering is weakest along the zone diagonals, as determined

FIG. 9. The mean free path l for three different cases of theanisotropy parameter for Tl2201 with p=0.26: a the isotropic

case dashedline = 0, b =0.1,0.2,0.3 solidline, movinginwards along the zone diagonal, andc=0.1,0.2,0.3solidline,moving outwards along,.


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previously by azimuthal ADMR measurements.29 As thetemperature is raised, increases markedly. This implies thatthe anisotropy resides in the inelastic, rather than the elastic,scattering channel. As with anisotropy in c, anisotropy inthe scattering rate can be FS derivede.g., due to FS insta-

bilities such as charge-density waves, spin-density waves, orantiferromagnetic fluctuations. Spin, charge, or indeed super-conducting fluctuations all have a specific momentum and

frequencydependence that is peaked at or, in some cases,confined toparticular regions in k space. Anisotropy in 1

can also signify additional physics due, for example, tostrong electron correlations near a Mott insulating state oranisotropic electron-impurity scattering.13 The present analy-sis cannot of course reveal the microscopic mechanism of theanisotropic scattering itself, but can identify some importantcharacteristics of the scattering mechanism, such as its mag-

nitude or its symmetry. Systematic measurementse.g., as afunction of doping and or pressurewould then allow a de-tailed comparison with the various theoretical proposals andthus help to reveal important hints as to its microscopic ori-gin.

V. CONCLUDING REMARKS

In this paper we have set out a detailed formalism forincorporating in-plane anisotropy, both in the cyclotron fre-quency and in the transport lifetime, into the analysis of in-terlayer magnetoresistance of a Q2D metal. The focus of thepresent paper has been to illustrate the need to introduce an

anisotropic scattering rate in order to explain the evolution ofthe ADMR data in overdoped superconducting Tl2201within a Boltzmann framework. An anisotropic cyclotronfrequency c can fit the data, but only if we allow theparameters describing the Fermi surface itself to change as afunction of temperature. Given the absence of evidence forsuch a reconstruction, this hypothesis seems unlikely. If, onthe other hand, an anisotropic scattering time is introduced,all the FS parameters can remain constant and only 00andthe anisotropy in are adjusted. Such a simple param-etrization is both elegant and experimentally accurate, andwe therefore believe it to be the most likely explanation ofthe observed ADMR data.

A cautionary note is perhaps appropriate here. In the pre-ceding calculations we have assumed the relaxation-time ap-proximation and so the microscopic relaxation dynamics

.

.

.

.

. ..

.

.

.

.

.

FIG. 10. Color onlineThe quality of fit parameterised by thesum 2 as a function of, which is given a specific value between0.4 and 0.3. Conversely, the parameters 00 ,and k61/k21 arefree to vary. The parameter 2 has a broad minimum, indicatingthat a good fit can be achieved for a broad range of. Insetsaandb show respectively the interdependence of k61/k21 and on see also Eqs. 20 and 24. The axes in a are shifted as de-scribed in the text.

FIG. 11. Color online Raw ADMR data solid curves with colors corresponding to crys as defined in Fig.6 plotted with the bestleast-squares fitsblack dashed linesfor three different temperatures a4.2 K, b14 K, andc50 K. Here both cand are consideredanisotropic in the azimuthal angle. The parametersk04, k61/k21, and are fixed to their values determined at T= 4.2 K, while 00and are allowed to vary as a function of temperature.


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have been ignored. The concept of anisotropic scattering re-mains valid as long as is interpreted as the exponentialdecay of the distribution between scattering events.43,44

However, we cannot rule out more exotic relaxation dynam-ics that depends on the presence of the magnetic field andhence may be manifested differently if probed by a differentmeans for example, photoemission. We have discoveredthat such exotic dynamics does not need to be invoked to

explain our ADMR data and conclude that its evolution withtemperature, when viewed from a Boltzmann framework us-ing the relaxation-time approximation, is best explained by ascattering rate with a temperature-dependent anisotropy.

At high doping levels, the lifetime separation in cupratesis less apparent,28 leading some researchers to consider theproblem from this perspective. This route has the added ad-vantage of allowing the limits of the conventional Boltzmanntransport theory to be explored as one moves across thephase diagram towards to a more exotic and potentiallynon-FL ground state on the underdoped side. The key mes-sage here is that by generalizing the theory to include ananisotropic scattering rate k 1kone can continue to

apply the Boltzmann approach and successfully account notonly for the evolution of the ADMR with temperature, butalso the distinct Tdependences of ab and cotH found inoverdoped Tl2201.22 Furthermore, initial measurements ofthe doping dependence of in Tl2201 suggest a significantincrease in anisotropy in 1kas one moves towards opti-mal doping,15 consistent with the observed increase in life-time separation as manifest in the temperature dependenceof the Hall coefficientwith decreasing doping.5,31,45 The in-troduction of such anisotropy has proven a fruitful model tounderstand the normal state of high-temperaturesuperconductors,812,4648 though clearly more work isneeded to parametrize 1kfully and to identify the origin

of the anisotropy.Finally, although the focus of this paper has been a systemwith body-centered-tetragonal symmetry, the analysis couldvery easily be generalized to layered systems of other crys-tallographic symmetries as already pointed out in Ref. 23and perhaps also one-dimensional systems with anisotropicscattering.49 ADMR experiments on BEDT-TTF-based or-ganic superconductors have already been performed at lowtemperature and explained in a Boltzmann framework with-out the need to invoke an anisotropic scattering rate.34 A fullazimuthal and temperature dependence on other organic saltsmay, however, require the introduction of such a parametri-zation. Similarly the same ideas may also apply to layeredcharge-density-wave compounds such as the rare-earth tritel-luridesRTe3.50 The Boltzmann equation, though simple in itsassumptions, thus remains a powerful paradigm whose ex-planatory power is still to be explored. ADMR is an idealprobe for just such an exploration.

ACKNOWLEDGMENTS

We thank R. H. McKenzie, M. P. Kennett, A. Ardavan,and J. A. Wilson for helpful discussions. This work is sup-ported by the EPSRC and a cooperative agreement betweenthe State of Florida and the NSF. J.A. would like to thank theLloyds Tercentenary Foundation.

APPENDIX: ACCOUNTING FOR SAMPLE

MISALIGNMENT IN THE ADMR

FITTING PROCEDURE

In this section we illustrate how sample misalignment canbe accounted for. The effect of sample misalignment onADMR has been considered by several authors before thisstudy, in particular Goddard et al.51 and Abdel-Jawad.37 Be-fore we begin, let us define two frames of reference: that ofthe laboratory, xp, yp, zp, in which the field is parallel to thezpdirection, and that of the crystal, xc, yc, z c. The ypaxis istaken as the axis of polar rotation, and xpis perpendicular tothis. The normal to the crystallographic plane shall be de-fined as zc, and the in-plane directions xc and yc shall betaken to be parallel and perpendicular to the copper-oxidebonds, respectively. The angle between the field directionand the crystallographic normal z cgives the crystallographicpolar angle. The projection of the field onto the xc-ycplanegives the crystallographic azimuthal angle , taken from thexc axis.

There are two important differences between this studyand that of Goddard et al.51 First, instead of correcting ex-

perimentalexptandexptfor misalignment to find the appro-priate crystallographic and , we fit the experimental databy including the misalignment in the fitting procedure. Sec-ond, in the analysis of Goddard et al.,51 the crystallographicaxes xc and yc can fall anywhere in the plane of the crystaland do not have assigned direction with respect to the crystalbonds. This gives the misalignment one fewer parameter, andone of the crystal axes can always fall somewhere in thexp-yp plane of the laboratory frame. In the present analysisthis is not the case and so three rotations need to be includedin the fitting procedure in order to account for every possiblemisalignment.

iBeginning with the sample aligned with the laboratory

frame, there is a misalignment in azimuthal angle denoted byasym, rotated about zpas shown in Fig.12a.

FIG. 12. Color onlineThe reference frame of the laboratory,xp,yp,zp, and that of the crystal,xc,y c,zcwherexcis parallel to thecopper-oxide bonds. a The first misalignment considered is thatwhere the experimental zpand crystal zcdirections coincide but theaxes on the azimuthal plane are offset by an amount asym. Thesecond misalignment considered is that where the experimental andcrystal z directions do not coincide. Two rotations are responsible:one about the platform ypaxis by an angle asym

y shown inbandanother about xp by an angle asym

x shown in c. This gives acompletely general description of the crystal misalignment with re-spect to the platform axes.


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ii A rotation about the yp axis denoted by asymy as

shown in Fig.12b.iii A rotation about the xp axis denoted by asym

x asshown in Fig.12c.

These transformations are elegantly described algebra-ically. We follow the notation whereby a rotation Ris arotation of a vector by angleabout an axis . In particu-lar the laboratory axis zp is tranformed relative to the crys-

tallographiczc0,0 before any azimuthal or polar rotationaxis to

zp=Rxpasym

x Rypasym

y zc0,0 , A1

due to the misalignment of the sample. In an ADMR experi-ment, the sample is then rotated about the laboratory zpazi-muthally by an angle exptand then rotated about the xpaxisa polar angle expt. The position of the crystallographiczcexpt,expt after these azimuthal and polar rotations axisis given by

zcexpt,expt=RxpexptRz

pexptzc. A2

With reference to Fig. 13 it is elementary to see that theprojection of the field parallel to zp on the crystallographiczcexpt,exptwill give crys, which should be used to calcu-late the value of the magnetoresistance in the analysis. Simi-larly, the projection on the xcexpt,exptyccrys ,exptplane will yield crys. Algebraically we have

coscrys=zpzcexpt,expt A3

and

tancrys=z

p

xc

expt

,expt

zpycexpt,expt . A4

The asymmetries for the sample considered in the presentpaper were estimated to be asym=8, asym

y =0, andasym

x =3. The parameter asymwas approximately equal tovalues estimated from diffractometry performed after theADMR experiment.

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FIG. 13. Color onlineThe polar crysand azimuthal crysangles that enter Eq.17depend only on the projection ofzpon thecrystal axes. This projection will change as a function ofexpt andexptand so must be calculated at each point.


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