Implications of the isotope effects on the magnetization, magnetic torque and susceptibilit

a r X i v :c o n d -m a t /0410547v 1 [c o n d -m a t .s u p r -c o n ] 21 O c t 2004

Implications of the isotope e?ects on the magnetization,magnetic torque and

susceptibility

T.Schneider

Physik-Institut der Universit¨a t Z¨u rich,Winterthurerstrasse 190,CH-8057,Switzerland

We analyze the magnetization,magnetic torque and susceptibility data of La 2?x Sr x Cu 16,18O 4and YBa 263,65Cu 3O 7?δnear T c in terms of the universal 3D-XY scaling relations.It is shown that the isotope e?ect on T c mirrors that on the anisotropy γ.Invoking the generic behavior of γthe doping dependence of the isotope e?ects on the critical properties,including T c ,correlation lengths and magnetic penetration depths are traced back to a change of the mobile carrier concentration.

A lot of measurements of the oxygen and copper isotope e?ect on the magnetization,susceptibility,magnetic torque etc .have been performed on a variety of cuprate superconductors [1–9].To obtain estimates for the shift of the transition temperature T c upon isotope exchange,the respective mean-?eld models [10,11]have been invoked.For the magnetization m they imply that the mean-?eld transition temperature T c 0can be estimated by extrapolating m (T )linearly.To illustrate this data of Batlogg et al.[1]for a nearly optimally doped La 1.85Sr 0.15Cu 16,18O 4T .The straight lines are parallel linear extrapolations,yielding the ≈34.12K and ?T c 0/T c 0≈?0.012,where ?Y = 18Y ?16Y /18Y .However,because the dominant role of critical ?uctuations near T c [12–21]and depth [22–25]etc.are neglected,the anisotropy is not taken into account,and appears arbitrary.

m ( 10

e m u )

FIG.1.16,18m (T )vs.T for a La 1.85Sr taken from Batlogg et al.[1].The parallel straight lines are linear T c 0≈34.56K,18T c 0≈34.12K and ?T c 0/T c 0≈?0.012.The black circles are the 16m (T )data rescaled according to Eq.(4)with a ?0.986.

Noting that the isotope e?ects in cuprate superconductors pose a fundamental challenge in its understanding,the shortcomings of the traditional interpretation of magnetization and related data call for a treatment that goes beyond mean-?eld and takes the anisotropy into account.In this work we concentrate on the critical regime of anisotropic extreme type II superconductors where 3D-XY ?uctuations dominate [12–21].Invoking the universal scaling functions for magnetization and magnetic torque we analyze the magnetization data of La 1.85Sr 0.15Cu 16,18O 4[1],magnetic torque data of La 1.92Sr 0.08Cu 16,18O 4[6]and the susceptibility data of YBa 263,65Cu 3O 7?δ[4]near T c .An essential additional relation emerges from the observation that data taken at ?xed magnetic ?eld and on samples with 16O or 18O,as well as with 63Cu or 65Cu,collapse near criticality within experimental error,when the temperature is rescaled appropriately.Although this property provides retrospectively partial support for the traditional approach [1–9],3D-XY scaling uncovers the essential role of the anisotropy γ.Indeed,the change of T c is found to mirror the shift of the anisotropy γ.As a consequence,the generic shift of the temperature dependent magnetization,susceptibility and magnetic torque upon isotope exchange at ?xed magnetic ?eld does not provide estimates for the change of the transition temperature only,as hitherto assumed [1–9].Together with the generic behavior of the anisotropy [18,26–31],the doping dependence of the isotope e?ects are then traced back to the change ?x u of the underdoped limit x u .It implies a shift of the phase diagram in the temperature-doping plane towards a

slightly higher dopant concentration x,along with a reduction of the mobile charge carrier concentration

T √

Φ3/2

γ?3/2(δ)

1

z

dG(z)

Φ0

?(δ),(1)

where?(δ)= cos2(δ)+sin2(δ)/γ2 1/2andγ=ξab/ξc denotes the anisotropy.Q3is a universal constant,G(z)a universal function of its argument,ξab,c the correlation lengths in the ab-plane and along the c-axis,H the mag-netic?eld andΦ0the?ux quantum.Close to the zero?eld transition temperature T c the correlation lengths diverge asξab,c=ξab0,c0|t|?νwhereν?2/3and t=T/T c?1.An essential implication is that in the plot m(T,δ,H)/ γ?3/2(δ)T√

H adopts the universal value [15,17–20]

m(T c)

H =?

k B Q3c3,∞

H vs.T,the data taken in di?erent

?elds cross at T c.In powder samples and su?ciently large anisotropy(γ>>1)this relation reduces to

m(T c)

H =?

πk B Q3c3,∞ |cos(δ)|3/2

T c =

?γ(T c)

ξab0?

?ξc0

with 3D-XY critical behavior,allows to determine the rescaling factor a around T c ,and with Eq.(5)to estimate the shifts ?T c /T c and ?γ(T c )/γ(T c )rather accurately.

Before turning to the implications of these results,revealing that the isotope e?ect on T c mirrors that on the anisotropy γ,it is essential to explore the e?ect of the dopant concentration.Since su?ciently dense data appears to be available for underdoped La 1.92Sr 0.08Cu 16,18O 4only,we are left with the reversible magnetic torque data of Hofer et al.[6]shown in Fig.2in terms of τvs.T .At T c ,?xed orientation and magnitude of the applied ?eld τscales as τ(T c )=?constT c γ(T c )H 3/2[15,17–20]and at ?xed magnetic ?eld the shifts are related by ?τ(T c )/τ(T c )+?γ(T c )/γ(T c )+?T c /?T c =0.From Fig.2it is seen that with 18τ(T )=16τ(aT )and a =0.936near coincidence is achieved within experimental accuracy.Since the universal scaling law for the magnetic torque [15,17–19]is essentially analogous to Eq.(2)the near coincidence implies ?τ(T c )/τ(T c )?0and (5)holds in this case as well,so that ?T c /T c ???γ(T c )/γ(T c )??0.07.

141618202224

( 10

( n m )

T ( K )

FIG.2.Reversible magnetic torque τ(δ)vs.T at H =0.1T and δ=45?for La 1.92Sr 0.08Cu 16,18O 4taken from Hofer et al.[6].The black circles are 18m (T )?16m (aT )with a =0.936.

To check the generic signi?cance of this scenario we also analyzed the susceptibility data for the copper isotope e?ect in YBa 263,65Cu 3O 7?δof Zhao et al .[4].For all four dopant concentrations,extending from the underdoped to the optimally doped regime,we ?nd that 65χ(T )=63χ(aT )is satis?ed within experimental error,as discussed below.Since χ=m/H this strongly suggests that the scaling relation (5)holds for both,copper and oxygen isotope exchange,irrespective of the doping level.

Having established the consistency with 3D-XY critical behavior,together with the experimental facts that at T c and ?xed magnetic ?eld ?m/m ,?τ/τand ?χ/χvanish within experimental error,the isotope e?ect on T c does not mirror that of the anisotropy only (Eq.(5)),but is also subject to the other universal relations of the 3D-XY universality class.In particular,T c ,ξc 0and λab 0are not independent but related by the universal relation [13,15,19,20,23,32]T c =Bξc 0/λ2ab 0,where B is a universal constant and λab 0the critical amplitude of the in-plane penetration depth.This leads for the respective relative shifts upon isotope exchange to the additional relation ?T c /T c =?ξc 0/ξc 0?2?λab 0/λab 0.The lesson is,whenever 3D-XY ?uctuations dominate,the isotope e?ects,e.g.on T c ,γand λab 0are not independent.These relations are particularly useful to open a door towards the understanding of the common origin of these e?ects.For example,given a generic relationship between anisotropy γand mobile

carrier concentration δ

=x ?x u at ?xed x ,where x u ,is the underdoped limit,the isotope e?ects in the cuprates would arise from a shift of x u and the associated change of δ

.As shown in Fig.3for La 2?x Sr x CuO 4,the generic doping dependence of γis well established in a rich variety of cuprates in terms of the empirical relation [20,31,33]

γ(T c )=

γ0

x ?x u

,

(6)

where γ0is material dependent constant.Approaching the underdoped limit,where the cuprates correspond to an independent stack of sheets with thickness d s [20,31,33],this relation follows from the doping dependence of the critical amplitudes of the correlation lengths.Since ξc 0tends to d s ,while ξab 0diverges as ξab 0=

ξab 0,which is the critical amplitude of the anisotropy at the quantum superconductor to insulator

transition at x u .The doping dependence of the relative isotope shifts is then traced back to a change of γ0and the

shift ?x u of the underdoped limit and with that to a change of the mobile carrier concentration δ

=x ?x u according to

?γ(T c )

γ0

+

?x u

γ0

+

?x u

1?T c /T c (x m )

,

(7)

where we invoked the empirical relation between the hole concentration x and T c due to Presland et al .[34].x m ?0.16denotes optimum doping.This leads to the important conclusion that the doping dependence of the isotope e?ects in the cuprates stem from the shift of the underdoped limit.Finally,combined with Eq.(5)we obtain

?γ(T c )

γ0

+?x u

T c

=?ξab 0

ξc 0

=?

?ξc 0

λab 0

,

(8)

relating the various relative shifts to the scaling factor a .With our estimates ?T c /T c ??0.014(x =0.15)and ?T c /T c ??0.07(x =0.08)for La 2?x Sr x CuO 4and relations (7)and (8)we obtain with x u =0.05for the essential,but material dependent parameters,determining the doping and T c dependence of αT c =?(M/?M )?T c /T c the values ?γ0/γ0??0.01and ?x u ?0.0024.To illustrate this feature and to check the generic signi?cance of this scenario further we consider the copper isotope e?ect on T c in YBa 2Cu 3O 7?δ.As aforementioned,our scaling analysis of the susceptibility data of Zhao et al .[4]reveals full consistency with 65χ(T )=63χ(aT )for all doping concentrations within experimental error.Since χ=m/H the implications are equivalent to those derived for the magnetization and Eq.(8)applies as well.The resulting estimates for αT c =?(M/?M )?T c /T c are included in Fig.3and compared with those obtained from the traditional extrapolation approach [4].More importantly,given ?γ0/γ0and ?x u the T c dependence of αT c is readily calculated with the aid of Eqs.(7)and (8).As shown in Fig.3in terms of the dashed line,agreement is achieved with ?γ0/γ0??0.0082and ?x u ?0.0012.In comparison with Y 1?y Pr y Ba 2Cu 316,18O 7?δthe data of Franck et al.[35]yields ?γ0/γ0??0.0060and ?x u ?0.0019.

0.050.100.150.200.250.30

T

/ T

(x

)

( T

)x

FIG.3.γ(T c )vs.x for La 2?x Sr x CuO 4taken from [18,26–30](?)and αT c vs.T c /T c (x m )for YBa 263,65Cu 3O 7?δtaken from Zhao et al.[4]( ).The dash-dot line is Eq.(6)with x u =0.05and γ0=2.The arrow indicates the ?ow to the superconductor to insulator transition.The open squares result from the scaling analysis of the susceptibility data for the samples with 7?δ=6.94,6.75,6.63and 6.48in terms of 65χ(T )=63χ(aT )yielding a ?1.0006,0.994,0.990and 0.988,respectively.The dashed and dotted curves result from Eqs.(7)and (8)with ?γ0/γ0=?0.0082,?x u =0.0012and T c (x m )=92.37K.

In summary we have seen that near T c ,where 3D-XY ?uctuations are essential,the isotope e?ects on various critical properties are not independent but related by universal relations.Together with the observation,that data taken at ?xed magnetic ?eld and on samples with 16O or 18O,as well as with 63Cu or 65Cu,collapse near criticality within experimental error,when the temperature is rescaled appropriately,we derived an additional relationship.It reveals the essential relevance of the anisotropy γ.Indeed,the relative shift of T c was found to mirror that of the anisotropy γ.As a consequence,the temperature shift of the magnetization,susceptibility and the magnetic torque at ?xed magnetic ?eld does not provide estimates for the change of the transition temperature only,as hitherto assumed [1–9].Together with the generic behavior of the anisotropy [18,26–30],the doping dependence of the isotope e?ects was traced back to a change of the underdoped limit ?x u ,or in other words,to a shift of the phase diagram in the temperature-doping plane towards a slightly higher dopant concentration,along with a reduction of the mobile charge carrier concentration.This contribution leads to a negative shift of T c .We identi?ed a positive shift as well.It stems from the change of the critical amplitude of the anisotropy γ0at the quantum superconductor to insulator transition.The magnitude and proportion of these contributions is controlled by ?x u and ?γ0.Their values turned out to

be material dependent.In any case,they control the isotope e?ects and remain to be understood microscopically. However,the emerging essential role of the anisotropy represents a serious problem for two dimensional models as candidates to explain superconductivity in the cuprates,and serves as a constraint on future work towards a complete understanding.In addition,isotope exchange leads unavoidably to lattice distortions.Their coupling with the in-plane penetration depth was recently established[24].

ACKNOWLEDGMENTS

The author is grateful to S.Kohout and J.Roos for useful comments and suggestions on the subject matter.