Effect of inhomogeneity on the transport in underdoped superconductors

Journal of Physics and Chemistry of Solids 67 (2006) 464–467 www.elsevier.com/locate/jpcs

Effect of inhomogeneity on the transport in underdoped superconductors Laura Romano` a,*, Valerio Dallacasa b b

a Department of Physics, University of Parma, INFM, 43100 Parma, Italy Scientific and Technological Department, University of Verona, INFM, Verona, Italy

Abstract The in-plane normal state resistivity of a variety of underdoped cuprates is analysed in terms of a percolation model where carriers tunnel between metallic islands distributed in an insulating background. Exponential laws of the form rZr0exp(T0/T)a where the exponent a changes from 1/4 at low temperature to 1/2 at higher temperatures are found to fit accurately the experimental data. The results are shown to be consistent with superconducting islands of size !100 A and high Tcy100 K, together with larger islands in the normal state embedded in an insulating matrix. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Superconductors; D. Electrical conductivity; D. Transport properties

1. Introduction One interesting feature of the cuprate materials is the charge distribution on the CuO planes. The temperature dependence of the resistivity of the normal state rZ r0exp(T0/T)a turns out to be an interesting indicator of such a distribution. Typically fractional exponents 1/2 and 1/4 at sufficiently low number of carriers are found [1–3] with a progressive decrease of the characteristic temperatures T0 as this number increases. These results are usually interpreted in terms of homogeneous localization of charge induced by disorder and transport via variable-range hopping without (exponent 1/4, high T) or with (exponent 1/2, low T) coulomb interactions, with the variation of T0 being related to the variation of the localization length. However, these laws can arise from and indicate a percolative arrangement of the type encountered in granular metals. The fitting of data with localization and granular models gives evidence for localiz˚ in the more ation dimensions ranging from some 10 A ˚ in the optimal and overdoped insulating region up to 100 A regions [4–6]. Recently, STM images in Bi2SrCaCu2O8Cd? [7] have confirmed previous results on the coexistence of hole rich and

* Corresponding author. E-mail address: [email protected] (L. Romano`).

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.10.016

poor zones in different regions of the same sample. A strong support to the hypothesis of the presence of superconducting droplets above the bulk Tc was given by the direct observation by means of scanning SQUID microscopy [8] of inhomogeneous magnetic domains showing diamagnetic activity above Tc in underdoped La 1.9Sr 0.1CuO4. Moreover, an irreversible anomalous diamagnetism, evidenced in cuprates of YBCO and LSCO family, was interpreted by considering superconducting ‘islands’ [9,10]. These studies, as well as many others indicate that either inhomogeneites or phase separation are intrinsic to the microscopic structure of the cuprates La2CuO4Cd, La2K xSrxCuO4 and Yba2Cu3O6Cx and that superconductivity can occur in microscopically small regions of space. From a theoretical point of view different scenarios can be sketched to describe this picture. Charge segregation may be expected due to the existence of a quantum critical point [11]. Percolation can be assumed provided localization processes are important leading to formation of clusters of carriers whose dimension increases with the carrier number [5]. A model of metallic islands of radius r separated by an insulating background of thickness s with the transport occurring through a tunnelling process between neutral islands has been suggested by Sheng et al. [6] for dielectric metals. With the assumption of the homogeneity rule (r/s)Zconst they found a law of the form for the conductivity sZs0exp[K(T0/ T)1/2] at low temperatures. The appearance of this law in high Tc materials prompted us to suggest this model in cuprates [12]. It was found that by relaxing the homogeneity condition and

L. Romano`, V. Dallacasa / Journal of Physics and Chemistry of Solids 67 (2006) 464–467

considering both r and s as random independent variables to account for all possible fluctuations of the grain configurations it was possible to explain the appearance of fractional exponential laws sZs0exp[K(T0/T)a] with aZ1/2 at moderately low temperatures and aZ1/4 at the lowest temperatures in agreement with the transport data. In this paper, the consequences of granularity on the Dc resistivity are discussed. 2. The model for Dc transport The transport of charge is assumed to occur via tunnelling between pairs of metallic droplets of size r across an insulating barrier of width s, the tunnelling probability [12] per unit time and area being obtained by means of the Fermi golden rule: ð PG Z NðEKEc ÞD
ðEÞf ðEÞ½1Kf ðEGeV KEc Þ (1) where EcZe2/p3sc2 is the activation energy of the order of the Coulomb charging energy, 3 the dielectric constant, N(EKEc) the density of final states, eV the energy due to the electric field, D
ðEÞZ D
0 expðK2csÞ the transmission coefficient where c accounts for the height of the barrier and the sign G corresponds to motion along or opposite to the electric field. The arguments of the Fermi functions are related to the initial and final energies of the tunnelling process. The total transition probability becomes: PZPCKPK and is a function of r and s. In the case of metallic droplets in the normal state D*(E)Z D* and N(E)ZN can be taken as constants as evaluated at the Fermi level. Here P is related to the diffusion coefficient D as P fDeV [13], and by using the Einstein relation mZeD/KT and the Drude formula sZneem, the electrical conductivity is obtained as an average over the configurations: 1 hsi z KT

N ð

N ð

pðcÞ pðsÞPðs; cÞdsdc 0

(2)

0

For convenience the variables in Eq. (2) are cZr/s and s. Performing the double integration in Eq. (2) one obtains two regimes separated by a temperature T Z q=l2 ðmC 2lÞ2 (a) at T / T the electrical conductivity is given by: 
1=4 
ððmK3=2Þ=4Þ s0 T0 T exp K 0 hsi Z T ðkTÞ2 T with T0Z64q  (b) while at T / Tone obtains: 
1=2  s00 T exp K 00 hsi Z ðmC3Þ=2 T ðkTÞ where q e2 ð2c C 1=s0 Þ  and q Z : T00 Z  1 p3kc20 a aK 2

(3)

(4)

465

The parameter m is related to the distribution functions assumed of the form p(x) fxmexp(Kx/x0), l and a are determined by the maximum of the distributions and turn out to be of order 1 [12]. Eq. (1) can still be used to include the case in which droplets are superconducting. Experimental and theoretical results [14,15,16] on tunnelling barriers indicate that the tunnelling current obtained by Eq. (1) should be proportional to the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi superconducting density of states Ns ðEÞZ ffi Nn ðEÞE= E2 KD2 as predicted by BCS theory when at least one of the films is superconducting. In the ohmic region Vw0 the factor f(EKEcCeV)Kf(EKEcKeV) in the integrand of the net transition probability PCKPK at small temperatures such that Ec/kT[1 will be 2eV(df(EKEc/dE)) with a narrow peak of width kT around EZEc. Therefore one can take other factors out of the integrand and in particular the density of states calculated at EKEc(kT. The result will be Eq. (1) with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi now N containing an extra factor ykT= ðkTÞ2 KD2 . The change will be small if Ec[kT[( whereas as kT approaches ( the conductivity will diverge leaving to a zero-resistivity state. Since, the local gap ( of the droplets will be a function of their size r the inequality kTO( will provide restrictions to the integrations over cOc*(T) while the integral over s is left unchanged. We note that at small temperatures Ec[kT the integration in Eq. (2) is determined by the upper limit cðTÞZ fe2 ð2cC 1=s0 Þ=½aðaK1=2p3kTg1=2 : So, when kT[D, we will have c(T)[c*(T) and c*(T) can be set to zero, leaving the results Eqs. (3) and (4) unchanged. We note that this requires that T[Tmin where Tmin will be defined by the actual form of DZD(c). The meaning of this is that only islands whose gap is sufficiently smaller than kT contribute to the normal value of the conductivity, which in other words means that the value of the conductivity is established by islands whose radius is sufficiently large. When however T!Tmin superconducting fluctuations will be important and the resistivity will decrease. An expression for D to discuss an order of magnitude of Tmin, can be obtained on solving the BCS equations in a finite volume. For a system of length LO L0 Z ð2pZkf Þ=ð2muÞwhere kf is the Fermi wave vector, u the phonon frequency, containing n cells, one finds [17]: DZE  F/n. With  the aid of the definition of c one finds DZ A=c3 AZ EF ro3 . In such a case we get kTmin Z ðe2 ð2cC 1=s0 Þ=3aðaK1=2ÞÞ3 =A2 . For ˚, typical values of metallic droplets r 0Z100 A cC 1=s0 Z ð10 AÞK1 , (Z1, 3Z10 and a gap value corresponding to TZ300 K, i.e AZ30 eV we have TminZ10 K. Such values of Tmin are consistently smaller than the interval at which the exponential laws are observed. Islands with sufficiently small radius can therefore coexist with those of larger radius yet not contributing to the current and this means the existence of locally dispersed domain in which sufficiently high Tc superconductivity exists, therefore leaving the possibility of a gap behaviour at temperature larger than the bulk Tc.

L. Romano`, V. Dallacasa / Journal of Physics and Chemistry of Solids 67 (2006) 464–467

466

Table 1 A summary of values of the significant temperatures of the percolation model as a function of the number of carriers x. Sample

x

T0 (K)

T00 (K)

q (K)?

 TðKÞ

Ref.

La2xSrxCuO4

0.04 0.08 0.07 0.08

14429.2 13909.7 12200.0 9417.0

341.5 217.3 260.5 300.4

225.4 217.3 190.6 148.1

123.4 95.3 53.2 123.4

[18]

0.08 0.84

3209.7 7163.9

225.0 372.9

50.2 111.9

25.0 69.5

[20] [21]

0.11

–

169.0

69.4

[22]

0.42 0.44 0.47

– 8371.0 10898.3

324.0 361.2 400.5

3. Results A large variety of cuprates, reported in Table 1 with their doping level, has been analysed in terms of this theoretical model. The exponent m in Eqs. (3) and (4) is an adjustable parameter and in all cases its best value is evaluated mZ3/2. Fig. 1 shows the normalized plot of log(rab/roT2) vs. (T0/T)1/4at small T for all the samples at different values of the doping level while Fig. 2 shows log(rab/roT2.25) vs. (T00/T)1/2at high T. We note that the powers 2 and 2.25 of T in the pre-exponentials are a direct result of the calculation. The fitting values of T0 and T00 are reported in Table 1. The straight lines, whose slope is one, confirm the universal behaviours described by Eqs. (3) and  q are also reported. It (4). In Table 1 the evaluated values of T, is interesting to note that the temperatures T0, T00 and q all increase with the decrease of the number of carriers which in turn corresponds to a decrease of the droplet dimensions. The exponential laws discussed here have been previously [12] applied to underdoped La-compounds where they have been found successful in describing the pseudogap region. The results obtained here for a variety of systems at various doping levels and conditions suggest a more general validity of the fractional exponents of the behaviour of the resistivity as a function of temperature. In particular the exponents 1/4 at low temperatures and 1/2 at higher temperatures are characteristics of granular systems, since quantum localization alone produces the same exponents but in the reverse order, i.e. 1/4 as typical of Mott’s variable range behaviour at higher temperatures and 1/2 as required by Coulomb gap behaviour at lower temperatures. 4. Discussion and conclusions The electrical conductivity at low temperatures has been calculated by means of a percolative model where electrical charges (electrons or holes) undergo to activated tunnel process between metallic bubbles. A characteristic temperature q, dependent on the droplet size and on the separation between two neutral bubbles, arising from the charging energy, is a most significant parameter of the model, playing the role of a gap in a free carrier system.

[19]

[23] 130.8 170.3

This model, applied with success to the granular metals, shows consistency with transport properties of the cuprate high Tc superconductors. The obtained electrical conductivity describes both a ‘semiconducting’ behaviour s fexp[K (1/T)1/4] in a temperature range decreasing with the increase of the doping and a non linear trend s fexp[K(1/T)1/2] usually ascribed to the presence of a pseudo gap. A preliminary analysis about the behaviour at high temperature indicates that this model can be able to describe also the expected linear trend in the rab(T) at higher temperatures and larger doping. In this way the existence of the linear contribution to the resistivity vs. temperature in optimally or overdoped materials can be explained. In this description q plays the role of a gap in the normal state. The meaning of this gap can be traced back to the –1 ref. 18, x=0.04 ref. 18, x=0.05 ref. 18, x=0.07 –2

ref. 19 ref. 20

log(ρ/ρ0T2) (K–2)

La1.6KxNd 0.4SrxCuO4 La2KxSrxCuO4 Bi2Sr2KxLaxCuO6Cd Bi2Sr2Ca1KxYxCu2Oy Bi2Sr2Ca1KxCu2O8

–3

–4

ref. 21 ref. 23, x=0.47 ref. 23, x=0.44

–5

2

3

4

5

6

1/4

(To/T) 1/4

Fig. 1. Resisivity as a function of (T0/T) for the samples reported in Table 1. The plots are shifted so that they overlap for convenience. In any case the slope of the straigth line is one.

L. Romano`, V. Dallacasa / Journal of Physics and Chemistry of Solids 67 (2006) 464–467

467

The resistivity data turn out to be consistent with both pictures of a gap around the Fermi level without precursor superconductivity in the normal state (coulomb gap) and of a true superconducting gap, with local origin within sufficiently small droplets.

ref.18, x=0.04 ref.18 x=0.05 ref. 18, x=0.07

log(ρ/ρ0T2.25) (K–2.25)

ref. 19 –4

References

ref. 20 ref. 21 ref.22

ref. 23, x=0.47 ref. 23, x=0.44 ref. 23, x=0.42 –5 0.8

1.2

1.6

2

(Too/T)1/2 Fig. 2. Resisivity as a function of (T00/T)1/2 for the samples reported in Table 1 The plots are shifted so that they overlap for convenience. In any case the slope of the straigth line is one.

Coulomb energy required to charge a cluster. Efros and Shklovskii [4] have shown that when a Coulomb energy is involved in a hopping process between any two sites, the density of states has a gap as a function of energy around the Fermi level. Therefore q can be interpreted as a true gap, which can be an explanation of the gap observed in angle-resolved photoemission spectroscopy measurements [24], without the need of invoking a pre-pairing mechanism in the normal state. However results indicate that for TOTmin the possibility exist that data of the resistivity as a function of temperature be consistent with local high Tc superconductivity developing in the smallest droplets, coexisting with non-superconducting larger ones. The occurrence of superconductivity in small metallic clusters embedded in insulating matrix may be a particular case of supercondutivity of small metallic particles. There is a well established experimental evidence for the occurrence of superconductivity in various systems of reduced space [25,26]. The predicted existence of lower limit of size around 20– ˚ for the superconductivity to occur in too small systems 25 A due to the strong fluctuations expected for the order parameter, allows a concrete possibility for superconductivity to develop for larger sizes.

[1] Y. Iye, Transport properties of high Tc cuprates, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors III, vol. 285, World Scientific, 1992. [2] B. Dabrowski, D.G. Hinks, J.D. Jorgensen, R.K. Kalia, P. Vashishta, D.R. Richards, D.T. Marx, A.W. Mitchell, Physica C 156 (1988) 24–26. [3] D.M. Leew, C.A.H.A. Mutsaers, R.A. Steeman, E. Frikkee, H.W. Zandbergen, Physica C 158 (1989) 391–396. [4] A.L. Efros, B.I. Shklovskii, J. Phys. C: Solid State Phys. 8 (1975) L49–L51. [5] V. Dallacasa, R. Feduzi, J. Alloys Compd 195 (1993) 531–534. [6] P. Sheng, B. Abeles, Y. Arie, Phys. Rev. Lett. 31 (1973) 44–47. [7] K.M. Lang, V. Madhavan, J.E. Hoffmann, E.W. Hudson, H. Eisaki, U. Uchida, J.C. Davis, Nature 415 (2001) 412. [8] I. Iguchgi, T. Yamaguchi, A. Sugimoto, Nature 412 (2001) 420. [9] A. Lascialfari, A. Rigamonti, L. Romano’, P. Tedesco, A. Varlamov, D. Embriaco, Phys. Rev. B 65 (2002) 144523. [10] A. Lascialfari, A. Rigamonti, L. Romano’, A. Varlamov, I. Zucca, Phys. Rev. B 68 (2003) 100505-R. [11] C. Castellani, C. Di Castro, M. Grilli, J. Phys. Chem. Solids 59 (1998) 1694. [12] V. Dallacasa, L. Romano`, Intern, Int. J. Mod. Phy. B 17 (2003) 636. [13] C.A. Neugerbauer, M.B. Webb, J. Appl. Phys. 33 (1962) 74. [14] I. Giaever, Phys. Rev. Lett. 5 (1960) 464. [15] J. Bardeen, Phys. Rev. Lett. 6 (1961) 57. [16] M.H. Cohen, L.M. Falicov, J.C. Phillips, Phys. Rev. Lett. 8 (1962) 316. [17] V. Dallacasa, Phase transitions and self-organization in electronic and molecular networks, in: J.C. Phillips, M-F Thorpe (Eds.), Quantum Percolation in High Tc Superconductors, Kluwer, 2001, p. 357. [18] H. Takagi, B. Battlog, H.L. Kao, J. Kwo, R.J. Cava, W.F. Peck Jr., Phys. Rev. Lett. 69 (1992) 2975. [19] N. Ichikawa, S. Uchida, J.M. Tranquada, T. Niemoller, P.M. Gehring, S.H. Lee, J.R. Schneider, Phys. Rev. Lett. 85 (2000) 1738. [20] G.S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, S. Uchida, Phys. Rev. Lett. 77 (1996) 5417. [21] S. Ono, Y. Ando, T. Murayama, F.F. Balakirev, J.B. Betts, G.S. Boebinger, Phys. Rev. Lett. 85 (2000) 638. [22] K.Q. Ruan, Q. Cao, S.Y. Li, G.G. Qian, C.Y. Wang, X.H. Chen, L.Z. Cao, Physica C 351 (2001) 402. [23] D. Mandrus, L. Forro, C. Kendziora, L. Milhay, Phys. Rev. B 44 (1991) 2418. [24] T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1991) 61. [25] H.R. Zeller, I. Giaever, Phys. Rev. 181 (1969) 789. [26] B. Wentzel, H. Micklitz, Phys. Rev. Lett. 66 (1991) 385.