Phase separation in high-Tc superconductors

PHYSICA

Physica C 235-240 (1994)253-256 North-Holland

Phase separation in high-Tc superconductors E. Sigmund a V. Hizhnyakov b and G. Seibold a aInstitut f. Physik, Technische Universit£t Cottbus, PBox 101344, 03013 Cottbus, Germany bInstitute of Physics, Estonian Academy of Sciences, Riia 142, Tartu, Estonia The mechanism responsible for phase separation in the cuprate superconductors is studied. It is argued that the doped holes are localized within spin-polarized clusters which at higher doping concentrations start to form a highly conducting metalic network. This picture allows us to understand the experimentally observed coexistence between antiferromagnetic and metalic regions.

1. I N T R O D U C T I O N Doping of the layered cuprate superconductors with holes leads to drasticM changes in their electrical and magnetic properties. The strong antiferromagnetic order is lost at low doping concentrations and the system turns from the insulating to a conducting state which below T¢ becomes superconducting. However, also in the metalic regime antiferromagnetic fluctuations are still observed [1]. This coexistence of both phenomena leads to the conclusion that at least on short space and time scMes the systems are strongly inhomogeneous due to a dynamical phase separation. When decreasing the carrier concentration, however, the time which is characteristic for the dynamics of the phase separation is increased. As result, the inhomogeneity becomes more pronounced and the phase separation exists on a time scale which can be resolved experimentally. In the second section of this paper we will present experimental evidence for phase separation based on the thermal quenching of moderately doped cuprate superconductors. The first section deals with the quasiparticles which built up the metalic phase in the CuOs planes, namely the formation of spin-polarized clusters. 2. S P I N - C L U S T E R

STATES

The concept of magnetic polarons has been introduced in connection with magnetic semiconductors quite early by de Gennes and Nagaev [2,3]. In the high Tc superconductors Schrieffer 0921-4534/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSEq 0921-4534(94)00701-2

et al [4] have proposed the formation of spin-bags which are polaron like particles in weakly correlated systems. We consider the opposite coupling limit where the magnetic polaron will be seen to be strongly localized in space with a nonsaturated ferromagnetic core for which we will use the term 'spin-cluster'. In the low doping regime, hints towards the formation of spin-cluster states in the CuO2 planes have been obtained by neutron spectroscopy [5] and E P R experiments [6,7]. The calculation of size and binding energy of these states are usually performed starting from the three band Hubbard model which contains the transfer between Cu d~_u~- and O Px/uorbitals and additionally strong on-site correlations on Cu as essential ingredients. An important difference of this model in contrast to the one band Hubbard model can be realized by considering the limit U -+ oo in the one-dimensional case where both models can be solved exactly. In a closed ring with N sites and N-1 particles for example, the motion of a hole can simply be described by the permutation of the whole spinconfiguration. In the three-band model this unrestricted permutation can only take place when the hole is located on a Cu-site (with probability Pcu). If it is located on an O-site (probability po), the neighbouring electrons on Cu can hop on this site only when they have the appropriate spin orientation (spin-conditional hopping). This mechanism immediatly favors a parallel alignement of Cu-spins adjacent to the O-hole (probapit

bility ptot). The fraction

Q is determined by the

PCu

254

E. S i g m u n d el a/..~tfln,xica (" _ .~ . . ~ 5 . "dO ~_.. (Iq04)

2.')3

2~(

hybridization matrix element t and (in the limit N --+ cxD) reads as p?o"t pc~

16t 2 --

n~

(A -- ~/A2 + 24t2)2

(li

where A = Ep - Ed. However, an important feature which is not contained in a one-dimensional consideration is the existence of closed loops in higher dimensions. According to Nagaokas theorem [8] these loops immediately lead to a ferromagnetic ground state for a square lattice with one hole in the U -4 oo limit. This is because of the fact that in the ferromagnetic case the hole can pass each closed path without changing the spin configuration of the system which results in a m a x i m a l gain in delocalization energy for the ferromagnetic background. Finite values of the Hubbard repulsion p a r a m e ter U lead to magnetic interactions between (hi and O spins [9,10]. At half filling the antiferromagnetic interaction between adjacent Cu spins reads as T4 ~[int

--

'~

sds d

sl'sa

n

~

Figure 1. Spin-polarization mean values on a given Cu site of the 3" sublattice. Solid lin< undoped antiferromagnet. Dashed line: ,.,m:' doped hole, one spin reversed. Dotted line: on, doped hole, two spins reversed (diagonally neighboured).

d !!

When a hole is doped on an oxygen site a much stronger antiferromagnetic interaction between the oxygen hole spin and the two nearest Cu spins starts to work 8T2

r

(~/

zJ

Hint

I

~. ~,I ¸

(a)

ij which leads to an effective ferromagnetic interaction between these two Cu spins. Together with the gain in delocalization energy due to the spinconditional hopping, these interactions favor a situation where the hole is located within a spinpolarized cluster with more or less parallel aligned Cu spins. Binding energy, size and polarization of these clusters have also been obtained using various mean-field approximations of the three-band H u b b a r d model (nartree-Fock, slave-boson)[ll]. Fig. 1,2 show the dependency of spin cluster polarization and binding energy on the repulsion p a r a m e t e r U in the Hartree-Foek approach. The upper curve in fig. l corresponds to the polarization of a Cu ]" spin in the homogeneous antiferromagnetically ordered system. The lower

no

U

Figure 2. Binding energies for the two spincluster configurations of fig. 2

curw~s show the potarizal;ion al. the same s~me site; with the differem:e that we now allow tbr a (self-consistently determined) spin fluctuataon when one hole is added to the system. As can be seen the resulting configuration consists of one (dashed line) or two (dotted line) non-saturated reversed spins. In the latter case these two flipped spins are diagonally neighboured forming a cigar shaped spin-cluster which for large U values ha~ a larger binding energy (fig. 2) than the (:luster with one turned spin only. Further increasing of U would end up with all spins ferromagnetically polarized, i.e. the Nagaoka state [8].

E. Sigmund et al./Physica C 235-240 (1994) 253-256

3. P H A S E

255

SEPARATION

3.0 At higher doping concentrations the cluster wave-functions will start to overlap leading to a network of percolative type. The concentration at which long antiferromagnetic order is lost is determined by the percolation of clusters with different spin orientation [9]. Simple application of perco2.2 where N is the numlation theory gives cl - ~-~ ber of Cu spins per cluster. The threshold of conduction is mainly determined by the percolation of clusters with the same polarization direction. 4.4 In The corresponding concentration is c2 = ~W" the calculations cited above we obtained clusters with 5 < N < 10 which for La2_~(Sr,Ba)~CuO4 leads to xl ~ 0.02, x2 ~-- 0.05 in agreement with experiment. Further doping results in a metalization of the network which in addition is supported by the influence of charge compensators. Therefore the CuO2 plane is subdivided in two kinds of areas: insulating regions where antiferromagnetic correlations are still persistant and the metalic network which becomes superconducting below To. There is strong evidence for this type of electronic phase separation by thermal quenching experiments [12-14] which allow for a separation of the time scales of quasiparticle and charge compensator motion in the planes. In Fig. 3a the resistivities of the single crystal measured in perpendicular and parallel direction to [001] are shown. The data have been collected after different thermal treatments of the crystal. The curves 3a and 3d have been measured after fast quenching of the crystal from room temperature to 5 K. Curves 3b and 3c correspond to the slowly cooled samples. la all cases and at low enough temperatures we observe bulk superconductivity as prooved by zero resistance (fig. 3a) and almost complete diamagnetic shielding (fig. 3b). In addition, the crystal clearly displays long range antiferromagnetic order. It manifests itself in the occurence of the sharp peak at the Noel temperature TN 260 K (fig. 3b), which corresponds to antiferromagnetic ordering of slightly tilted Cu spins. This magnetic phase transition also shows up in the resistivity data for the [001] direction. Together with a sharp change of resistivity at TN

E

2.0

v

""--4 ~0.0 "7

~ ~ o:

E

~ -o~.-

¢0

:

..._),._ :/"

::" o':"

.41~,

T qK~

o'I.,

:

"-o

o

0

b

"~-

.,c~-- , " oc~

,

..."

~.

~i r ' ~ i ' ¢ - ' ' ' ~ i

r

loo

200

300

T (K)

Figure 3. a Resistivity of a La2CuO4+~ single crystal measured perpendicular (a and b) and parallel (c and d) to [001] direction, respectively. Curves a and d were obtained on heating the rapidly quenched sample. Curves b and c were measured while slowly cooling and heating the sample, b Diamagnetic shielding of La~CuO4+z measured parallel to [001]. o refers to the susceptibility after quenching and • to the susceptibility obtained after slowly cooling the sample from room temperature to 5 K, respectively.

we observe a hysteresis loop accompanying the magnetic phase transition. All these features displayed in the resistivity and susceptibility data have been found in ceramic materials too [12,13]. The existence of antiferromagnetic ordering at 260 K in these materials has been observed independently in NMR experiments [1,17]. Random field induced by doping may lead to the destruction of complete long range order by creation of locally ordered mesoscopic antiferromagnetic domains [18,19]. In our crystal with TN ~ 260 K these domains have rather large size (see e.g. [19]) and can be considered as crystal-

256

E. Sigmundet al./Phvsica C 235 240 (1994) 253 256

lites with long range antiferromagnetic order. The experimental data clearly show the coexistence of bulk superconductivity and strong antiferromagnetic order. Such coexistence of two macroscopic phases, however, is only possible if an inhomogeneous structure with threedimensional percolation is established and thus can occur already at low doping when the main part of the crystal still is in the insulating phase. This is in agreement with our considerations above where the percolative type electronic phase separation is caused by a fractal alignement of spin clusters. The observed resistivity indeed displays the existence of a three-dimensional percolation structure. While the resistivities are extremely anisotropic at higher temperatures this anisotropy is lessened at lower temperatures and vanishes near Te; then the resistivities drop to zero between 30 K and 34 K respectively., indicating bulk superconductivity. In a sample with the main part of the volume being insulating and antiferromagnetically ordered the significant reduction by a factor of 3 between 250 K and 50 K of the anisotropy of the resistivity near T~ is only possible if the superconducting subphase is formed by establishing a three-dimensional percolation network. This intrinsic inhomogeneous percolative structure of the sample also shows up when performing different thermal treatments. Slow cooling procedure allows for a diffusional growth of the three-dimensional percolation net and leads to a more isotropic conductivity then fast cooling. 4. C O N C L U S I O N We thus have seen that there is strong theoretical and experimental evidence for a microscopic phase separation mechanism in the cuprate superconductors on a nanometer scale. The metalic phase is built up by the percolative overlap of spin clusters of which each contains about 5 - 10 Cu spins aligned in the same direction. The insulating phase consists of regions in which the spins are still antiferromagnetically correlated. So the coexistence of bulk superconductivity and long range antiferromagnetic order is a generic feature

of our model. For a review on experiments on phase separation see [15,16]. REFERENCES

1.

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