Subtle Features in Transport Properties Evidence for a Possible Coexistence of Holes and El

a r X i v :c o n d -m a t /0003074v 2 [c o n d -m a t .s u p r -c o n ] 8 M a r 2000

Subtle Features in Transport Properties:Evidence for a Possible Coexistence of Holes

and Electrons in Cuprate Superconductors

Nie Luo

Department of Physics and Astronomy,Northwestern University,Evanston,IL 60208

(February 6,2008)Transport properties of high transition temperature (high T c )cuprate superconductors are investi-gated within a two-band model.The doping dependent Hall coe?cients of La 2?x Sr x CuO 4(LSCO)and Nd 2?x Ce x CuO 4(NCCO)are explained by assuming the coexistence of two carriers with oppo-site charges,loosely speaking electrons (e )and holes (h ).Such a possible electron-hole coexistence (EHC)in other p -type cuprates is also inferred from subtle features in the Hall coe?cient R H and thermopower S .The EHC possibly relates to the pseudogap and sign reversals of transport coe?-cients near T c .It also corroborates the electronlike Fermi surface revealed in recent photoemission results.An experimental veri?cation is proposed.PACS numbers:74.25.Fy,74.25.Jb,74.62.Dh,71.35.-y

Superconductivity in high T c cuprates is strongly dop-ing dependent [1].The parent compounds are antifer-romagnetic insulators,which upon proper doping with charge carriers,become superconductive.The type of carriers can be inferred from dopant valences,and is rou-tinely checked by the Hall e?ect.Although it is natural to assume that a p -type sample carries only holes,as is widely believed in the literature,a positive R H does not in theory preclude electrons as the minority carriers.Because of the complicated defect chemistry and band structures [2]in cuprates,one dopant may have bipolar property (i.e.,it functions as both donor and acceptor).Before elaborating on this,however,we want to digress to two anomalies,which we think intimately relate to the possible electron-hole coexistence.

One such anomaly is the normal-state pseudogap.Ex-periments have revealed various non-Fermi-liquid behav-iors in the normal state,particularly the opening of a gap in both spin and charge excitation spectra at a char-acteristic temperature T ?above T c [3].The pseudogap appears in the underdoped regime and weakens as dop-ing level is increased.There is no consensus yet on its origin,but various models have emerged.One approach resorts to the spin-charge separation of Luttinger liquid,originally advocated by Anderson [4].It describes the gap in the spin degree of freedom.However relatively few have been done to the charge sector,where infrared (IR),transport and photoemission [3,5]experiments also reveal gaplike structures.We believe that EHC may help answer this question;electron (e )and hole (h )attract each other and form excitonic states,resulting in the loss of spectral weight in the charge excitation spectrum.The other anomaly is the Hall sign reversal near T c oc-curring to properly doped samples.It is often explained in terms of vortex dynamics [6],which explains some phenomena.Recent experiments however have raised se-rious questions not yet answered by these theories [7].We argue that vortex motion and pinning are not the

Holes per CuO 2

1

T c

S (μV /K )

c

Temperature T

S (a r b i t r a r y u n i t )

FIG.1.(a)Generic thermopower S versus temperature T for properly doped p -type samples near T c [8].The dot-dashed line is S =0.(b)S versus doping for p -type cuprates.S values of circles are from Ref.[9]while doping levels are inferred from a general T c dependence on doping,shown as the dashed line.T c are normalized to the maximal T c ,T cm .The solid line is the generic behavior of S in p -types although the S values are not to be taken as exact.

whole story because there exists a similar anomaly in the thermopower S ,usually measured without magnetic ?eld.The S anomaly is not widely discussed but it shows up clearly in S -T plots [8]like Fig.1(a).It occurs to major cuprate series,on samples from ceramics to single crys-tals,suggesting a nontrivial nature.This S anomaly is likely a general and doping dependent behavior of cuprate superconductors near T c ,just like that in R H .It then poses a problem to theories based on vortices;here we have no magnetic ?eld,then where do vortices come?The sign of normal-state S also depends on doping [9]as shown in Fig.1(b).We will see that both are explained naturally in terms of EHC.

The doping dependent R H in LSCO [10]is shown in Fig.2(a).The dashed line is the theoretical R H as-suming that each Sr gives one hole.The squares trace the theory relatively well when x <0.05.At higher x ,increasing deviation from the dashed line results in a

00.10.20.30.4

Sr content x

20

21

C o n

c e n t r a t i o n (c m ?3

)0.05

0.10.150.2

Ce content x

?4?2|R H | (c m 3

/C )La 2?x Sr x CuO 4

Nd 2?x Ce x CuO 4

FIG.2.R H in (a)LSCO and (b)NCCO.The squares are after Ref[10],while the dashed theoretical curves assume that each dopant atom gives one e or h .Calculated n e ,n p in (c)LSCO and (d)NCCO.Shaded bars indicate superconducting regimes.

change in sign.Such a behavior is not explained by a single parabolic band,which requires R H ∝1/n with n the carrier density.R H =0means n →∞,which is clearly unphysical.Microscopic models based on local density approximation (LDA)band theory and the Hub-bard model are only partially successful in ?tting the curve.The former failed to get an insulator at x =0[2],while the latter predicted an electronlike Fermi surface (FS)in NCCO [11],just opposite to experiments.

This crossover,shown as a sharp dip,is however not strange to the two-band,or two-carrier model,which ex-plains reasonably well similar behaviors of R H as a func-tion of temperature or composition in some chalcogenides like Bi 2Te 3[12].The term two-band here,should not be taken too literally;it is not needed to have two bands across the FS.The two carriers may simply come from di?erent parts of a single band of complex shape,or just originate from electronlike or holelike portion of the FS.In the two-band model [13],R H is given by

R H =

n p μ2p ?n e μ2

e

e c (n p +n e b )2

,(2)

where b =μe /μp is the mobility ratio.

We want to calculate n e using Eq.(2).To this end,we assume:(1)n p =2x/v u .(2)b does not vary with doping.The last assumption is crude;b in fact strongly varies with x ,but we have no better choice because only R H is known;we have to ?x b ?rst in order to get n e .Now we estimate b .As the zeroth order approximation,we take R H as coming from majority carriers only and then mobility μ=R H /ρwhere ρis the resistivity.This gives μe if R H <0and μp if R H >0.We surveyed data in the literature [14,15]from some of the best samples and got b =1.25.Details of this survey will be published elsewhere.This b is used for both LSCO and NCCO.Only data at 80K are taken to reduce errors from R H variation with T .n e is then directly calculated from the experimental R H .The results are in Fig.2(c)as solid circles.n e are then ?tted by power series shown as the solid curve that reads n e (x )=9.0×1018+7.3×1020x +6.3×1022x 3+2.3×1025x 10in cm ?3.We then plug n e (x )back in Eq.(2)to verify R H ,resulting in the solid line in Fig.2(a).Similar result for NCCO is shown in Figs.2(b)and 2(d)with n p (x )=?2.1×1020+1.0×1021x 0.5+5.3×1026x 7in cm ?3,which holds if x ≥0.05.(The triangle is a ?t using b =0.95.The ?t cannot be carried out if b >0.96.This shows b dependence on x .)

The presence of electron in LSCO is signi?cant.In the superconducting regime,n e ～0.17?0.5n p .As for NCCO,n p ～0.5?1.5n e .Thus EHC e?ect is better manifested in NCCO,which might explain why it has long been suggested by various groups [15].Our result this far con?rms their works and also gives evidence for EHC in LSCO,a p -type superconductor.

One may ask if EHC extends into other p -type cuprates where normal-state R H are positive regardless of doping.However,a sign reversal in S still occurs at high doping levels and near T cm ,as shown in Fig.1(b).This strongly suggests an EHC.Also R H is somewhat small.The op-timal doping is around 0.15holes per CuO 2(HPC)for most p -type cuprates,inferred from the Cu valence ob-tained from methods like iodometric titration [16].How-ever n p calculated from 1/e c R H is nearly 2times as large.The presence of e ,as in LSCO,is thus suggested.

We now want to ?nd out n e but b has to be deter-mined ?rst.μp in these cuprates is much higher than that in LSCO or NCCO,so b <1is likely.However,μe is not known in these materials;and we have to esti-mate.Because of the possible EHC,transport relaxation now involves multiple processes.There are 3processes from the inter-carrier (IC)relaxation channel:hole-hole

TABLE I.The e-h coexistence in YBa2Cu3O7?δ(Y-123), Bi2Sr2CaCu2O8(Bi-2212),Tl2Ba2CuO6+δ(Tl-2201)and HgBa2Ca n?1Cu n O2n+2+δ[Hg-12(n-1)n].T c and T?are in K,R H in10?3cm3/C,the assumed n p and calculated n e in 1021cm?3.T?is high in Hg-series,which are often under-doped.O:optimal.NO:near-optimal.SC:single crystal.PC: polycrystal.In-plane data for SC samples.

Series T c T?R H HPC n p n e Notes

FIG.3.Schematic of experimental veri?cation

loss in IR at T slightly above T c might be similarly ex-plained.

As for the stability of electron-hole liquids,we suggest two scenarios.First,e and h are spatially separated, as the result of layered cuprate structure or the stripe phase.Second,most cuprates have electronlike and hole-like band-edges well separated in the momentum space, which further reduces the possibility of annihilation. How this possible EHC relates to the excitonic mech-anism of high T c superconductivity[29]is interesting al-though‘excitonic’here is not limited to that of MW or Frenkel excitons.Moreover the question of electronlike FS needs evidence besides that from ARPES.The ver-i?cation of EHC is thus important and we suggest an experiment to probe the drift direction of carriers under an electric?eld E[30].Suppose that a laser pulse is fed through the?ber probe of a near-?eld scanning op-tical microscope(NSOM)as shown in Fig.3.The pulse excites the cuprate sample in a small spot,say100nm across,generating carriers,which drift under E and upon reaching the metal probe induce a voltage pulse there. By probing the direction at which carriers move relative to E,their signs are found.The distance between two probes needs to be small,say<1μm to cope with the fast recombination and small mobility of inequilibrium carriers.The conductance,the doping level of samples, and temperature should be chosen carefully.

The author is grateful for helpful discussions with Yoshimi Kubo.

Note Added.—J.E.Hirsch informs the author of similar conclusions from a di?erent perspective.See J.E.Hirsch and F.Marsiglio,Phys.Rev.B43,424(1991).