Charge transfer, phase separation and percolative superconductivity in YBa2Cu3O6+y

PHYSICA ELSEVIER

Physica C 246 (1995) 357-374

Charge transfer, phase separation and percolative superconductivity in YBa2Cu306+y Michihito Muroi *, Robert Street Department of Physics, The University of Western Australia, Nedlands, WA 6907, Australia Received 20 October 1994; revised manuscript received 13 March 1995

Abstract

The dependence of the superconducting properties of YBa2Cu306+y on the oxygen content y is discussed in terms of a two-phase model in which it is assumed that the optimal hole concentration, Popt = 0.25 per CuO 2 unit, is a local condition and that the average hole concentration p smaller than Popt results in phase separation into superconducting and insulating phases. For y = 1, P = P o p t is satisfied everywhere in the CuO 2 plane and the material is a homogeneous bulk superconductor. As p decreases with decreasing y, the fraction of the insulating phase increases and superconductivity takes place through a 2D percolative network. Macroscopic superconductivity is lost when the insulating phase forms an infinite cluster. It is argued that the oxygen ions in the CUOy plane affect the superconducting properties not only by controlling p but also by modifying the hole distribution in the CuO 2 plane through local structural changes and long-range Coulomb interaction. Explanations are given of the two-plateau structure in the Tc versus y curve and of the appearance of the seemingly homogeneous "60 K phase" at y ~ 0.5-0.6. The relevance of the two-phase model to other systems and the origin of the phase separation are also discussed.

1. Introduction

The superconducting properties o f YBa2Cu3Or+y (YBCO) depend strongly on the oxygen content y. As oxygen is removed from the basal CUOy plane, T~ decreases from ~ 92 K, following the well-known Tc versus y relationship with two plateaus occurring at ~ 9 0 K ( 0 . 8 < y < 1 ) and ~ 6 0 K ( 0 . 5 < y < 0.65). T~ becomes zero and the material turns into an insulator at y ~ 0.4, in near coincidence with an orthorhombic-tetragonal structural transition [1,2]. However, the oxygen content alone does not uniquely determine the superconducting properties; T~ de-

* Corresponding author.

pends also on the oxygen ordering in the CuOy plane, although this effect is much weaker than that of y. This has been demonstrated most clearly by the experimental observation that Tc of quenched samples increases with time as the oxygen ordering develops [3]. It is now widely believed that Tc is largely determined by the hole concentration in the CuO 2 plane ( p ) and that the variation of Tc reflects that of p, which in turn depends on both y and the degree of oxygen ordering. The consensus ends, however, at this somewhat vague conclusion. A comprehensive understanding of the above behaviours remains elusive because of the following problems. Firstly, the determination of p itself is not straightforward, since the holes provided by the added

0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

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358

M. Muroi, R. Street/Physica C 246 (1995) 357-374

O atoms could reside either in the CuO 2 plane or in the CuOy plane. Various attempts have been made to estimate p or equivalently the oxidation state of Cu in the CuO 2 plane. They include calculations of the bond-valence sum from structural data [2,4,5], measurements of the Hall constants [6], chemical substitution experiments [7], band-structure calculations [8] and Monte Carlo simulations [9,10]. However, the results are significantly different from one another; the derived relationship between p and y is nearly linear in some [5,7] while it is strongly nonlinear in others [2,8,9]. This problem has also made it difficult to answer the more general question whether the Tc versus p relationship is universal [7,11,12] among all the cuprate superconductors. Secondly, there is strong experimental evidence that the material is intrinsically inhomogeneous: (1) Except near y = 7 and 6.5, where O ions in the CuO 2 plane tend to order into Cu-O chains, the superconducting transition becomes broad and very sensitive to the current or applied magnetic field, indicative of granular superconductivity [2,13]. (2) Enhanced critical current densities in slightly oxygen-deficient samples suggest that oxygen vacancies work as pinning centres [14,15], that is, the order parameter is locally suppressed near the vacancies. (3) The specific-heat anomaly at Tc and the Meissner signal simultaneously decrease as y decreases, consistent with the picture that the superconducting volume fraction decreases with y [16]. (4) Measurements using various kinds of local probe techniques, such as neutron spectroscopy [17], nuclear quadrupole resonance [18,19] and M6ssbauer spectroscopy [20], clearly show that the electronic structure is spatially inhomogeneous except for y ~ 0 and 1. These observations cast doubt whether the superconducting properties can ever be well described using a single parameter p. In fact, it has been suggested that the material may be viewed as a mixture of two or more different phases and that the superconducting properties are better described in terms of a percolation model [15,21,22]. In this paper, we first discuss in detail the p versus y relationship in YBCO. We derive p as a function of y using a simple charge-transfer model and compare the result with other estimates of p. We

then discuss the dependence of the superconducting properties on y on the assumption that in the CuO 2 plane there always exist two phases (except at y = 0 and 1), namely, a superconducting phase with the optimal hole concentration and an insulating phase with no holes, the fraction of the former being proportional to p. Superconductivity then occurs through a 2D percolative network, the geometry of which is determined so as to minimise the sum of the condensation, magnetic and Coulomb energies. It is argued that this model provides consistent explanations of a number of experimental observations outlined above and that phase separation is a universal origin of the Tc variation with p in the whole range of cuprate superconductors. The basic ideas of the present work have been given in Ref. [23].

2. Relationship between p and y We first compare p versus y relationships estimated by various methods. Cava et al. [2] calculated the Cu(2) [24] bond-valence sum (BVS) from the structural data for various values of y and found that the variation with y of the effective Cu(2) valence is very similar to that of T~. (A typical phase diagram of YBCO, constructed by plotting published data of Tc [1,2] and TN (N6el temperature for the antiferromagnetic (AFM) ordering of Cu(2) ions) [25,26] as a function of y, is shown in Fig. l(a).) This led them to conclude that p is a strongly nonlinear function of y and that Tc is nearly proportional to p [2]. Brown [5] suggested, however, that the effect of internal strain should be considered to derive the correct Cu oxidation state from BVS. He applied a correction for the strain and obtained the p versus y relationship shown in Fig. l(b) (plot A), which was reproduced from Ref. [5]. Another way of correcting for strain is to take the difference between the Cu and O bond-valence sums. Tallon [4] has calculated 2 + Vcu(2) - Vo(2) - Vo(3), which he identifies as p. The results are also reproduced in Fig. l(b) (plot B). Both plots show similar features: a relatively quick increase of p for 0.4 < y < 0.5, a relatively slow increase of p for 0.5 < y < 0.6 and the resultant kink at y ~ 0.5. p may also be estimated from measurements of the Hall effect. The hole carrier number ( I / R n e )

M. Muroi, R. Street / Physica C 246 (1995) 357-374 500

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359

calculated from the data presented by Wang et al. [6] is plotted in Fig. l(c). The same features as in Fig. l(b) are seen but the increase of p near y ~ 0.5 and 1 is very steep and the kink at y ~ 0.5 is much more pronounced. In Fig. l(c), as well as in Fig. l(b), there is no plateau at y ~ 1, suggesting that Tc is not a linear function of y. The two sets of plots shown in Fig. l(d) are predictions of theoretical models. Plot A was derived by Poulsen et al. [9] from Monte Carlo (MC) simulations on the assumption that only 2D clusters of ordered O ions exceeding a certain size can contribute to the charge transfer that creates holes in the CuO 2 plane. The curve is similar to the Tc versus y curve (Fig. l(a)), meaning that Tc is proportional to p over the whole range of y. Zaanen et al. [8] obtained a similar curve from band-structure calculations. Plots B1 and B2 were derived by Selke and Uimin [10] also from MC simulations but on the different assumption that C u - O chains larger than a critical value contribute to the creation of holes in the CuO 2 plane. Plot B1, corresponding to the lower temperature (kT/V 1 = 0.2), is qualitatively similar to those in Figs. l(b) and (c). Using inelastic neutron scattering (INS), Mesot et al. [17] have studied the low-energy crystal-field excitations of Er 3+ in ErBa2Cu306+y and found that the energy spectra can be decomposed into three lines, the relative intensities of which vary with y. (We neglect the difference between YBa2Cu306+y and ErBa2Cu306+y, which is expected to be small because of the small difference in the ionic radius between Y (1.02 ~,) and Er (1.00 ,~).) They argue that the three transitions are associated with two local regions of metallic character and a local region of semiconducting character and that the relative intensity of each line equals the fraction of the corresponding region. Since the intensities of the three lines vary smoothly with y, they conclude that p is linearly related to the oxygenation process [17]. As we have mentioned earlier, the results obtained by various methods disagree and none of them appears to be conclusive. In the following, we derive p from the INS data, which are reproduced in Figs. 2(a-c) from Ref. [17], interpreting them in a different way. Mesot et al. attributed the transitions A 1 and A 2 to two different superconducting phases with Tc's of 90 K and 60 K, respectively, and the transi-

M. Muroi, R. Street / Physica C 246 (1995) 357-374

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tion A 3 to an insulating phase [17]. We instead assume that a CuO 2 plane consists of only two electronic phases, i.e., a hole-free insulating phase and a hole-rich metallic (superconducting) phase, and that the three transitions are associated with the three inequivalent local environments for Er 3+, as schematically shown in the inset of Figs. 2(a-c): In (a), the two CuO 2 units adjacent to an Er ion both belong to a metallic region. In (b), one of the two CuO 2 units belongs to a metallic region and the other to an insulating region. In (c), both CuO 2 units belong to an insulating region. We further assume that within the metallic phase the hole concentration is uniform. This assumption will be justified by the fact that the only three local environments are detected in the INS experiment. On these assumptions, the fraction of the metallic phase in each CuO 2 plane, which equals p normalized to the value at y = I (Po), is simply given by p 0 = I ( A l ) + I(A2)/2, where /(A t) and I(A 2) are the relative intensity of A1 and A 2, respectively. (Here p represents the hole concentration averaged over the whole CuO 2 plane.) P0 is plotted against y in Fig. 2(d). It can be seen that the P0 versus y plot strongly resembles curve B1 in Fig. l(d) and is qualitatively similar to those in Figs. l(b) and (c) although the concavity of the curve between the stoichiometric compositions (y = 0, 0.5 and 1) is more (less) pronounced in Fig. l(c) (Fig. l(b)). This seems to be reasonable in view of the inhomogeneous nature of the material in the intermediate composition: The structural data determined by diffraction techniques are intrinsically averaged information, and the BVS derived from them will also tend to be an average of the BVS for the end compositions, resulting in an overestimate of p for intermediate compositions. On

0.8

Fig. 2. (a-c) Variation with y of the relative intensities of the lowest-lying CEF transitions ( A 1 - A 3) observed in the INS spectra of ErBa2Cu306+y (reproduced from Ref. [17]). The inset figures show the local environment of an Er 3+ ion, assumed to correspond to each transition; the hatched and blank squares, respectively, indicate the CuO 2 unit adjacent to the Er 3+ ion belonging to a metallic region and that belonging to an insulating region. (d) Variation with y of I ( A 1 ) + 1(A2)/2, where I(A 1) and I(A z) are the relative intensities of transitions A 1 and A2, plotted in (a) and (b). The curves in (a-d) are the predictions of the charge-transfer model described in the text.

M. Muroi, R. Street / Physica C 246 (1995) 357-374

the other hand, the Hall carrier number will tend to be smaller under the influence of disorder. For example, the C u - O chains, which are known to make a significant contribution to the conductivity [27] and therefore tend to decrease R n for y ~ 0.5 and 1, are disrupted for intermediate compositions, resulting in an underestimate of p. We believe that the P0 versus y relationship derived here is more reliable than other estimates since it is based on direct, quantitative information [17]. The fact that it resembles plot B1 in Fig. l(d) but is inconsistent with curve A in Fig. l(d), suggests that a primary factor controlling p is the ordering of O(1) ions not into 2D clusters [9] but into 1D C u - O chains [10].

3. Charge-transfer model In this section, we discuss further the INS data (Fig. 2) in terms of the charge-transfer model proposed in Ref. [10] with minor modifications. On the assumption that Y, Ba and 0(4) ions have the formal ionic charges, i.e., y3+, Ba2+ and O 2-, p is determined by the average charge per CuOy in the basal plane (Qc) through the charge neutrality condition Qc + 2 p = 1. We start by considering the charge distribution within a C u - O chain of length l. (The chain length l ( l = 0, 1, 2 . . . . ) is defined as the number of O ions contained in the chain, i.e., the length is l for a chain ( C u - O - ) t C u . ) An isolated Cu ion (l = 0) is expected to have a valence of + 1 since the two-fold coordination is characteristic of Cu ÷. An addition of one O atom results in the formation of a chain with l = 1. The two holes provided by the O atom will naturally be used to oxidise the neighbouring two Cu ions, and the charge distribution in the chain will be C u 2 + - O 2 - - C u 2÷. When another O atom is added to form a chain with 1 = 2, one of the two holes provided by the O atom will be used to oxidise the neighbouring Cu ÷ ion that has just joined the chain. However, since the other Cu ions in the chain have already been oxidised, the second hole will either stay on the O site or be transferred to the CuO 2 plane. In this way, every time an O atom is added to the chain, one Cu ion at its end is oxidised and one extra hole arises. The charge distribution in a chain is then expressed as (Cu2+-O'~--)tCu2+ (l 2 . We assume a = 1 . 5 . In the

361

local-charge picture, this means that a hole resides in every other O ion in the chain. In other words, 0 2and O - alternate in the chain. This assumption is consistent with the analysis by Tokura et al. [7]. Further justifications for this choice of the parameter ( a = 1.5) include the following: (1) In Ref. [10], a = 1.7 is assumed based on the data of X-ray absorption spectroscopy (XAS) and electron-energy-loss spectroscopy (EELS) [28] and electric-field gradient (EFG) calculations [29], which show that the hole concentration on the O(1) site is 0.27-0.3 for y ~ 1. However, the X A S / E E L S and EFG results both suggest that 0.1-0.14 holes occupy the 0(4) site, which is neglected in Ref. [10], resulting in an unreasonably large value of p (0.35) for y = 1. Since the holes on the 0(4) site do not contribute to p, it is more reasonable to regard them as existing on the O(1) site rather than in the CuO 2 plane. (2) The stability (and hence the occupancy) of holes on the 0(4) site relative to those on the 0 ( 2 ) / 0 ( 3 ) sites is expected to decrease as y decreases, since the 0(4) site is closer to the CuOy plane than the 0 ( 2 ) / 0 ( 3 ) sites and the increase in the Madelung potential upon removal of O(1) atoms is larger for the former than for the latter. Thus, the occupancy of holes on the 0(4) site may effectively be taken into account, by identifying them as those on the O(1) site. There are two important consequences of the above assumptions: (1) the existence of the critical chain length Ic at or below which no holes are transferred to the CuO 2 plane, and (2) the variation with l of the "hole-doping efficiency". Let us compare the net charge of a chain of length l with that of I isolated Cu + ions. For ! = 1, the net charge is + 2 both for the former (Cu 2+O 2 - - C u 2+) and for the latter (Cu+-Cu+). This means that a chain with l = 1 has no influence on the CuO 2 plane. For l >_ 2, the net charge of the chain, which consists of l 0 1 5 - ions and ! + 1 Cu 2+ ions, is ( / / 2 ) + 2, whereas the net charge of l + 1 isolated Cu + ions is l + 1. The extra charge of the chain is therefore ( 2 - l)/2 and, in order to maintain the charge neutrality, ( l - 2 ) / 4 holes will appear in each CuO 2 plane. ( l - 2 ) / 4 is positive only for l >__3. Thus it is not until the length of the

M. Muroi, R. Street/Physica C 246 (1995) 357-374

362 0.25 I~ " ' O0 ~

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chain reaches three that holes are transferred to the CuO 2 plane, that is, Ic = 2. This value is the same as that in Ref. [10] and similar to those predicted by band-structure calculations (l c = 1 or 3 depending on the assumptions) [8]. The "hole doping efficiency" ~(l) is defined as the number of holes doped in a CuO 2 plane per O ion in a chain. A chain with l _< 2 makes no contribution to p and hence e(l) = 0. A chain with l > 3 provides the CuO 2 plane with ( l - 2 ) / 4 holes and hence e(l) = (l - 2)/(4l). e(l) increases with l and approaches 0.25 for large l, as can be seen in Fig. 3. (The l dependence of s ( l ) originates from the fact that l O ions in a chain must first oxidize l + 1 Cu(1) ions.) From the above considerations, all the ions in the CUOy plane are classified into four depending on the charge they have, as shown in the first three columns of Table 1, and we can calculate Qc for a given configuration of the CuOy plane. (A partial justification for this very simple relationship between the

ionic charge and local coordination comes from the fact that the lines corresponding to the Cu(1) site in the Cu nuclear quadrupole resonance (NQR) spectra can be assigned in terms of the coordination number of Cu(1) ions [18,19].) One way to derive p ( y ) is to calculate Q¢ for an oxygen configuration produced by MC simulations as has been done in Ref. [10]. Here we instead calculate Qc for idealised model configurations, which allow us to derive an approximate analytical expression of p(y). The configurations shown in Fig. 4 were produced using the procedure described by Kubo and Igarashi [21]. In these figures, an occupied O site is represented by a dot whose size is the same as the unit cell (a 0 X b0). Thus a Cu-O chain is represented by a vertical line with a width of a 0. Figs. 4(a-f) are obtained by removing an appropriate number of O ions randomly from the ideal orthorhombic(I) (O(I)) structure. Figs. 4(a'-f') are obtained by attaching (for y > 0.5) or removing (for y < 0.5) O ions randomly to or from the ideal orthorhombic(II) (O(II)) structure. Judging from the phase diagram [30,31], the configuration at room temperature is expected to change as a-b-c'-d'-e'-f as y decreases [21]. In the O(I)-like structure, e.g., Figs. 4(a-f), Ny 0 ions occupy randomly on the N O sites along the b-axis, where N is the number of Cu sites in the CuOy plane. Considering that the probability of an O site being occupied is y, the numbers (per unit cell) of the four types of ions (ni's) are easily derived, as listed in the fourth column of Table 1. Qc is then given by ~,,1=lniqi, f r o m which p is determined through the charge neutrality condition Qc + 2 p = 1. The result is p(y)=(1/4)y

3

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Table 1 The ionic charge (qi) and number per unit cell (hi) of the four types of ions in the basal CUOy plane, n i is shown for the O(I)-like structure, e.g., Figs. 4(a-f) and for the O(II)-like structure, e.g., Figs. 4(a'-f'). Qc is the total charge of the basal plane (per unit cell) i

Ion

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Qc = E4i=lniqi

M. Muroi, R. Street / Physica C 246 (1995) 357--374

In the ideal O(II) structure, fully occupied and empty Cu-O chains alternate along the a-axis (Fig. 4(d')). (We call the occupied and empty O sites in this structure O(o) and O(e) sites, respectively.) For y < 0.5, Ny 0 ions occupy randomly on the N/2 O(o) sites. Considering that the probability of an O(o) site being occupied is 2y, ni's and Qc are again easily calculated. The results are listed in the fifth column of Table 1. We then obtain p __y3 through the charge neutrality condition. For y > 0.5, the same consideration is applied to the O(e) sites, and we obtain p = (y - 0.5) 3 + 0.125. Thus for the O(II)-like structure,

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f o r 0 _
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The normalized function, po(y)=p(y)/p(1), is plotted in Fig. 2(d) for the O(I)-like and O(II)-like structures. It can be seen that the experimental data are well fitted by Eq. (2). As we have mentioned earlier, for small and large values of y the configuration is likely to be O(I)-like, to which Eq. (1) should be applied. However, the difference between Eqs. (1) and (2) is small near y---0 and 1, and to a first approximation Eq. (2) may be used over the whole range of y. On the basis of our interpretation of the three lines A1-A 3 observed in the INS spectra [17], their relative intensities are given by I(A1)={p0(y)} 2, I(A2) --- 2{p0(Y)}{1 -P0(Y)}, and I(A 3) = {1 p0(y)} 2, assuming that the hole distributions in the adjacent CuO 2 planes are independent. I(A1), I(A 2) and I(A 3) thus obtained from Eq. (2) are plotted in Figs. 2(a-c) (solid lines). The fit to the experimental data is good, demonstrating further the validity of the model described above. (The large deviation of the data points from the theoretical predictions near y 0.5 is reasonably explained by the dynamic correlation of the hole distributions in the two adjacent CuO 2 planes, discussed in the next section.) It is noted that the curves for the O(I)- and O(II)-like structures in Fig. 2(d) are qualitatively similar to plots B2 and B1 in Fig. l(d), respectively. This is easily understood from the magnitude of the parameters used in the MC simulations [10]: V2/V t = - 0 . 5 and V3/V 1 = 0.5, where Vt, V2 and V3 are

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363

364

M. Muroi, R. Street/Physica C 246 (1995) 357-374

the repulsive interaction between nearest-neighbour O ions, the attractive interaction between nextnearest-neighbour O ions with a Cu ion in between, and the repulsive interaction between next-nearestneighbour O ions with no intermediate Cu ion, respectively. At the lower temperature ( k T / V l = 0.2) near y = 0.5, O ions tend to order to form the O(II) structure since both IV2 I and 113, the driving force for the formation of Cu-O chains, are much larger than the thermal energy (I V2 I = V3 = 2.5kT). Long Cu-O chains are thus formed and a large number of holes are transferred to the CuO: plane. (The longer the Cu-O chains, the more efficient they are in providing holes to the CuO 2 planes; see Fig. 3). At the higher temperature ( k T / V 1 = 0.325), Iv21 and V3 become comparable to the thermal energy (I V2 I = V3 ~ 1.5kT), which tends to randomise the arrangement of O ions, as in the O(I)-like structures (Figs. 4(a-f)). (The temperature dependence of p is small near y = 1 since V1 is much larger than k T even at the higher temperature.) Eq. (1) and plot B2 in Fig. l(d) probably describe the p versus y relationship for RBa2Cu306+y with larger R, e.g., NdBa2Cu306+y, in which the interactions between O ions are expected to be weaker than those in YBCO due to the larger lattice constants. (An estimated value of V1 for NdBa2Cu306+y (0.124 eV) [32] is about half that for YBCO (0.2 eV) [33], meaning that the temperature scale used in MC simulations for NdBa2Cu306+y is about twice that for YBCO.) The following are a few more comments on the model configurations in Fig. 4 on which the above calculations are based: (1) In these configurations, the clustering tendency of O ions (vacancies) and disorder (disruption of Cu-O chains by defects, stacking faults of Cu-O chains along the a-axis, etc.) are neglected. The influence of these factors is expected to be fairly small since the two effects counteract; the former tends to increase the Cu-O chain length while the latter tends to decrease it. (These effects are fully taken into account in MC simulations [10].) (2) For a low value of y (Figs. 4(0 and (f')), the actual structures should be tetragonal. However, this does not seem to significantly affect the results since tetragonal structures are obtained simply by moving half of the Cu-O chains to the a-axes, which can be

carried out for low values of y without forming an unfavourable Cu coordination. In conclusion, qualitative features of the p versus y relationship are well described by the above model.

4. Hole distribution and superconducting properties Having established the p versus y relationship, we next discuss the y dependence of the superconducting properties in terms of a two-phase model, in which the following basic assumptions are made: (1) The optimal hole concentration, Popt = 0.25 per CuO 2 unit, is a condition to be satisfied within a local region whose dimension is of the order of the in-plane coherence length ~ab ( ~ 10 ,~). In other words, there is a gain in condensation energy only in the regions where p = P opt is satisfied on a microscopic scale. (2) When the average hole concentration Pay is less than Popt, the system separates into a superconducting phase (S phase) with p = Popt and an insulating phase (I phase) with p = 0. (The fractions of the S phase (fs) and I phase (fl) are Pav/Popt and ( 1 Pav)/Popt, respectively.) The superconducting properties then depend on the distribution of the two phases. Shown in the left column of Fig. 5 are the simulated O(1) configurations for various values of y, reproduced from Fig. 4. (Following the argument in Ref. [21], a, b, c', d', e' and f in Fig. 4 are adopted for y = 0.9, 0.8, 0.6, 0.5, 0.4 and 0.3, respectively.) In these figures, the Cu-O chains with l < 3, in which the charge redistribution occurs only between the Cu(1) and O(1) sites, are omitted. Thus, they show the charge distribution in the basal plane, with the black regions representing extra negative charges. We first discuss the hole distribution for y = 0.5, which highlights the central idea of the two-phase model. From Fig. 2(d), Pav/Popt ~ 0.5 for y = 0.5, and hence fs ~ f l ~ 0.5, i.e., the S and I phases occupy nearly equal areas in the CuO 2 plane. The hole distribution will be determined so as to minimise the sum of the following three energies: the superconducting condensation energy in the S phase (Econ) , magnetic energy in the I phase (Er, ag), and Coulomb energy associated with the inhomogeneous

M. Muroi, R. Street / Physica C 246 (1995) 357-374

charge distribution (Ecoul). Both Econ and Emag are minimised by complete phase separation, thereby minimising the S-I interface where the order parameter is suppressed [34] and the exchange energies arising from the antiferromagnetic coupling of Cu spins are lost. Such macroscopic phase separation is, however, most unfavourable in terms of Ecoul, which is minimised by a homogeneous hole distribution. The most stable hole distribution will then be the one in which the S and I phases intermingle in the CuO 2 plane, thereby compromising the above conflicting requirements. A possible hole distribution for y = 0.5 is schematically depicted on the right of Fig. 5(d). The S phase runs through islands of the I phase, forming a 2D percolative network. The asymmetry in the distribution (islands of the I phase embedded in the S phase, and not vice versa) is due to the Coulomb interaction between holes, which tend to pull apart the S phase. The size of the islands, determined by balancing the gain in Econ and Emag against the cost in Ecou~, is estimated at a few to several tens of .~ (see the Appendix). Since the electrostatic potential within the CuO 2 plane is uniform, due to the oxygen order in the basal plane, there is no particular region in which the holes prefer to reside, and the S-I boundaries will not be fixed. In other words, the order parameter is not uniform in space but its time average is uniform. This is a possible identity of the "60 K phase". The above picture could account for various apparently contradicting experimental observations: (1) Since the distribution of the S phase is not restricted by external perturbation (no weak links), the properties observed in resistivity or magnetic measurements will be those of a bulk superconductor [2,13]. The reduced Tc is explained by the suppression of the order parameter near S-I interfaces [23,34]. We note that the estimated inhomogeneity scale (on the order of ten ,~) is comparable to that in Y0.7Pr0.3Ba2Cu307 (see Fig. 4 in Ref. [23]), which also has Tc ~ 60 K. (2) The suppressed specific-heat jump at T¢, comparable to those for other compositions (0.4 < y < 0.8) and much smaller than that for y ~ 7 [35], is expected from the existence of a large fraction of the I phase. (3) INS [17], MiSssbauer [20] and NQR [18] measurements all detect unambiguously the existence of

365

local regions of insulating character, i.e., the electronic structure is inhomogeneous within the CuO 2 plane. From these considerations, we suggest that the so called "60 K phase" is an artifact; it is not a single superconducting phase, although it is a single crystallographic phase (O(II) phase). As y increases to 0.6, the added O ions occupy some of the O(e) sites to form CuO chains (Fig. 4(c')). However, Cu-O chains with l > 3, which contribute to charge transfers, are rare as can be seen in Fig. 5(c). Thus, the electrostatic potential in the CuO 2 plane, as well as p, are essentially the same as those for y = 0.5, and the superconducting properties hardly change, corresponding to the plateau of Tc at 60 K. As y increases further, the Cu-O chains in the O(e) sites become longer and p increases. However, the increase of p with y in the region 0.5 < y < 0.8 is rather slow, since many chains are not long enough to be efficient in transferring holes to the CuO 2 plane (see Figs. 2(d) and 3). (As y increases from 0.6 to 0.8, fs increases from ~ 0.5 only to ~ 0.6.) Although the increase of p is small, there is a significant change in the electronic structure in the CuO 2 plane. For y = 0.8, there appear local regions having the O(I) structure, which coexist with those having the O(II) structure (Fig. 4(b)). As a consequence, the charge distribution in the basal plane is no more uniform; the density of negative charges varies in space with periods comparable to or larger than ~ab, as can be seen in Fig. 5(b). This is expected to affec the hole distribution in the CuO 2 plane in the following two ways. Firstly, the variation of the electrostatic potential in the CuO 2 plane will directly reflect that of the charge distribution in the basal plane through long-range Coulomb interaction. Secondly, the extra negative charges in the basal plane will polarise the adjacent BaO plane; the Ba ions move towards the basal plane and the 0(4) ions towards the CuO 2 plane. Such movements, indeed observed as an average structural change by neutron diffraction [1,2], are likely to occur locally since the spatial variation of the charge density is fairly slow, allowing the strain energy to release. Both effects decrease the electrostatic potential for a hole in the regions where the basal plane is more negatively charged. A possible hole distribution in the CuO 2 plane for y = 0.8 is shown on the right of

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M. Muroi, R. Street / Physica C 246 (1995) 357-374

ioo A

(fs-0.76)~

0.8 (fs=0.6 )

Fig. 5(b). The "(positively charged) S phase preferentially occupies those regions in which the basal plane is more negatively charged to take advantage of the low electrostatic potential. The S - I boundaries are now fixed and the system is well described as a random array of weak links, i.e, larger regions of the S phase with stronger superconductivity (higher T¢) connected by narrow channels with weaker superconductivity (lower To). The superconducting transition is then a percolative process involving the coupling of larger S regions through the weak links owing to the proximity effect. This picture is consistent with the experimental observations that the superconducting transitions are broader and that the transition width is very sensitive to the magnetic field for y ~ 0.8 [2,13]. Thus the increase of T~ with y in the region ~ 0.6 < y < ~ 0.8, attributable to the appearance of larger S regions, accompanies the formation of weak links, resulting in granular superconductivity. It is important to note that the hole distribution in the CuO 2 plane does not coincide strictly with the distribution of O ions (negative charges) in the basal plane; S-I boundaries with irregular shape or very small regions of the S / I phase are unlikely to form since they cost a large surface energy (see Fig. 7(c)). The above argument is based on the assumption that the lowering of the Coulomb potential due to the extra O(1) atoms (through direct Coulomb interaction and through local structural changes) is large enough to dominate the hole distribution in the CuO 2 plane. The validity of this assumption is checked as follows: On the basis of the charge-transfer model described in Section 3, an O(1) atom added to the Cu-O chain effectively creates an extra negative charge 0.5e (e is the electron charge). In a rigid, purely ionic crystal, this would reduce the energy of a hole in the CuO 2 plane in the same unit cell by (1/41reoXO.5e2/r) ~ 1.8 eV, where ~0 is the permittivity of free space and r ( ~ 4 A) is the separation between the CuO 2 and CUOy planes. This value is reduced by a factor of e s, the static dielectric constant, because of the polarisation of the surrounding ions. (We believe that screening by conduction electrons is unimportant in this problem, since we are considering the influence of O(I) atoms on the n e a r e s t CuO 2 plane to which conduction electrons are largely confined. In a usual sense, the phase

1[]1k]11[]ti11111[[ ]1]![1[1[]1[I]q[I]1]1[] (fs=0-5) ~

~,

(d)

l;ill.,!"i: Ilk,Ill!

(e)

'!'I11',,IIII,I,II,'II It I 1 ilt it

I,,, ..,lh.h1,11o.4 1 I1111li [ l I

ila,
I

'ii,,,"II IIi'I',I

lll,hl llll,lllll!,[h, ii till

II I l

lil

I

ii

I

III

tl il

[ ,,

' Ill

Ii I

(f) 0.3

I

''(fs=0.11) i

e

I

Fig. 5. Left-hand column: Simulated 0(1) configurations in the basal plane for various values of y. a, b, c, d, e and f are respectively the same as a, b, c', d', e' and f in Fig. 4, except that the O(1) ions in a chain with l < 3 are omitted. Right-hand column: Possible distributions of the S phase (hatched areas) in the CuO 2 plane corresponding to the 0(1) configurations to the left. The fraction of the S phase, fs, estimated from Eq. (2), is indicated in each case.

M. Muroi, R. Street/ Physica C 246 0995) 357-374

separation in the above-mentioned way itself is a kind of screening; it reduces the influence of the 0(1) ions on further distant planes by attracting holes in the 0(1) rich regions.) Using a value e s = 30, we estimate the actual energy lowering at 0.06 eV. Similarly, the energy lowering arising from the shortening of the Cu(2)-O(4) distance from r 1 to r 2 is given b~Y ( 1 / 4 ~ 6 s e o X 2 eo2 X 1 / r 2 - 1 / r 1)"o With r I ----2.47 A (for y -- 0.09 (A) and r 2 = 2.30 A (for y = 0.93) [1] and e s = 30, this yields 0.03 eV. The total energy lowering is thus estimated at 0.06 + 0.03 = 0.09 eV. (We regard this as a lower limit of the Coulomb energy gained by a hole in the O(1) rich region, since we have considered the influence of just one O(1) or 0(4) ion in the above calculation.) In a region where the O(1) atoms form a duster, the lowering of the Coulomb potential in the CuO 2 plane will be much larger [36]. On the other hand, energy is needed to change the inhomogeneity scale (e.g., size of the I phase islands) of the hole distribution. A rough estimate of this energy cost may be obtained from Fig. 7(c) (in the Appendix): For a hole distribution shown in Fig. 7(a), the total energy Etotal takes a minimum 0.019 J / m 2 (0.14 eV per hole) at x = xmin ~ 49 ,~. With increasing x to 2Xm,t/n ( ~ 98 A) or with decreasing x to Xmin/2 ( ~ 25 A), Etota1 increases only by 4.75 × 10 -3 J / m E (0.035 eV per hole). This energy increase (0.035 eV) is much smaller than the estimated gain in Coulomb energy due to the 0(1) ions (0.09 eV), in spite of the fact that the latter value is most probably a significant underestimate as we mentioned earlier. Although the above calculation is very crude, we may safely conclude that the 0(1) distribution in the basal plane has a substantial influence on the hole distribution in the nearest CuO 2 plane. With further increasing y, p increases fairly quickly and fs increases accordingly (Fig. 2(d)). The islands of the I phase become smaller (Fig. 5(a)) and the coupling between larger S regions improves, resulting in a narrower superconducting transition [1,2,13]. However, Tc does not change significantly as y increases beyond ~ 0.8, since some of the S regions have become large enough (>> ~ab) to have T¢ close to the bulk value ~ 90 K already at y ~ 0.8. Note that more than 20% of the CuO 2 plane is still insulating even for y = 0.9. This is again consistent with the INS [17] and NQR [18] experiments, which

367

clearly show that the electronic structure continues to change up to y ,,, 1. The S phase fills the whole CuO 2 plane when p reaches 0.25 at y ~ 1. As y decreases from 0.5, p decreases abruptly because of the disruption of the Cu-O chains. For y = 0.4, p ~ 0.26, which is only half of that for y = 0.5. A possible hole distribution for y = 0.4, just above the point corresponding to the metal-insulator transition, is shown on the right of Fig. 5(e). There are large islands of the I phase through which the S phase manages to percolate. The remaining S phase tends to follow the regions where the basal plane is more negatively charged, thereby reducing Ecour Tc is low and superconductivity is weak since supercurrents are restricted by the narrowest part of a percolating path. (By analogy with the case of Yl_xPrxBa2Cu307 [23,37], just before Tc becomes zero, the weakest part of the surviving percolating paths may be a channel one-unit-cell wide or even a Cu ion shared by two CuO 2 cells.) As p decreases with further decreasing y, a point is reached where the surface energy makes it unfavourable for percolating paths of the S phase to form. The S phase breaks up into small islands and macroscopic superconductivity disappears (Fig. 5(f)). Since the I phase now forms an infinite cluster, the AFM ordering of Cu(2) ions appears in turn. It is noted that superconducting regions locally exist even if macroscopic superconductivity is lost, until p becomes zero at y ,--0. The existence of local regions of metallic character is demonstrated clearly by the INS experiment [17] (Fig. 2) and is also indicated by the fact that Tr~ changes with y even in the region y < 0.3 (Fig. 1(a)). We next consider the influence of the holes in the adjacent CuO 2 plane, which we have neglected so far. Because of the Coulomb repulsion, the holes in the neighbouring CuO 2 planes will try to avoid occupying the same regions in the a - b plane. (Except for y = 1, Ecou, could be reduced by increasing the configurations shown in the inset of Fig. 2(b), which is more favourable than that of Fig. 2(a).) This is not possible however except near y ~ 0.5, because the hole distribution is largely determined by the O(1) configuration in the basal plane, as we have just argued. Since the 0(1) configurations in the neighbouring basal planes are expected to be uncorrelated for nonstoichiometric compositions, the hole distil-

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M. Muroi, R. Street / Physica C 246 (1995) 357-374

butions in the neighbouring CuO 2 planes will be independent of each other. Near y ~ 0.5, however, the ordered O(1) ions have no influence on the hole distribution in the CuO 2 plane. The holes in the neighbouring CuO 2 planes will then avoid each other's way to reduce Ecoul, thereby increasing the configurations shown in the inset of Fig. 2(b) and decreasing that of Fig. 2(a). This is a possible reason why in the INS data (Fig. 2) the intensity corresponding to transition A 1 is much lower and that corresponding to transition A 2 is much higher than the theoretical predictions near y ~ 0.5. The observation that slightly oxygen-deficient samples, with higher normal-state resistivity and reduced specificheat anomaly at To, have a slightly higher Tc than fully oxygenated samples [35,38] may also be explained by considering the interplane Coulomb interaction; with a small amount of oxygen deficiency (e.g., Fig. 5(a)), Ecoul is reduced by the presence of the I phase and yet many regions of the S phase are large enough not to be subject to the boundary effect. Before concluding this section, we make several comments on the relevance of the two-phase model to other systems. As we have argued in a previous paper [23], the above picture (phase separation and the resultant percolative superconductivity) is likely to be relevant to the whole range of high-T~ oxides. (1) The effects of various kinds of substitutions could be consistently explained by considering the hole distribution in the CuO 2 plane, which is determined not only by the total number of holes but also by the spatial variation of the Coulomb potential due to the substituted ions. (2) In general, cation substitutions near the CuO 2 plane have a stronger influence on the hole distribution because of the larger Coulomb interaction for a smaller distance. The best example is Yl_xPrxBa2Cu307 (YPBCO), in which holes are localised on O sites in the CuO 2 plane by Pr ions. (The hole localisation (not hole filling) is given evidence for by electron-energy-loss spectroscopy measurements [39] and is possibly explained by the similarity between the energy of a hole on a Pr site and that in the CuO 2 plane [37,40].) Since the on-site Coulomb repulsion, estimated at 3-8 eV [41], is very large, the distribution of the remaining mobile holes (S phase) is totally governed by the distribution of the localized holes (or Pr ions). In fact, a number of experimental

observations reported on YPBCO are consistently explained by a percolation model [23,37]. (3) The two-phase model may be applicable also to the overdoped region ( p > 0.25). In this case, we may consider a normal metallic phase (N phase) as a second phase, which coexists with the S phase. The fraction of the N phase increases with p, and the material becomes a normal metal when the N phase forms an infinite cluster. Since the N phase has a higher hole concentration than the S phase, the latter tends to form islands and the former tends to envelop them, that is, the positions of the two phases are opposite to those in the underdoped regions. (In Fig. 5(e), for instance, the S phase envelops islands of the I phase and, as a result, the S phase forms an infinite cluster despite the fact that it is the minority phase (fs ~ 0.26).) This is a possible reason why Tc decreases more quickly in the overdoped region than in the underdoped region [12,42]. (4) When the spatial variation of the Coulomb potential in the CuO 2 plane is large, the S, I and N phases may coexist. The variation with x of the superconducting properties, e.g., in L a e _ x S r x C u O 4 [42] could be accounted for by the coexistence of the three phases [23].

5. Phase separation and high-Tc superconductivity In this final section, we discuss the microscopic meaning of the phase separation assumed above in terms of the local pair resonance (LPR) model [43,44], which we have recently proposed as a possible mechanism of high-Tc superconductivity. The core of the model is the assumption that a hole introduced in the CuO 2 plane induces charge transfer, Cu2+--l-O2----~Cu+--[--O-, to create a pair of oxygen holes within a CuO 2 cell. This allows pairs of holes with an optimal concentration (0.25 per CuO 2 unit) to arrange themselves so that all the neighbouring spins achieve favourable antiferromagnetic coupling and to move coherently without causing frustration among the spins, i.e., without increasing the magnetic energy of the system. (One of the possible spin configurations is shown in Fig. 6(a), where + and - indicate the direction of spins (up and down) on Cu 2÷ and O - ions and the dots ( Q )

M. Muroi, R. Street / Physica C 246 (1995) 357-374 !

~v

(a)

(b)

Fig. 6. (a) One of the possible spin configurations in the two-hole state with the optimal hole concentration p = 0.25. The symbols + and - indicate the direction of spins (up and down) on Cu 2+ and O - ions and the dots ( O ) denote Cu ÷ ions. (b) Spin configuration after addition of an extra electron to the optimized system shown in (a).

denote nonmagnetic Cu ÷ ions.) Detailed discussions on the model have already been made in Refs. [43] and [44], and we make just a few additional comments: (1) In the LPR model, the condensation energy arises from the energy difference between the two-hole state (in which externally doped holes and the same number of holes created by the charge-transfer process mentioned above participate in conduction) and the one-hole state (in which only externally doped holes participate in conduction), and is expressed as A E = A E m + A Ect + A E r , where AEm, A E c y and A E r represent the magnetic, charge-transfer and resonance energies, respectively [43,44]. To see if the two-hole (superconducting) state could be the ground state (i.e., AE < 0), we look further into each term. AE m is given by AE m = --3Jdd $2 --Jpd$2, where Jdd (Jpd) is the exchange interaction between neighbouring Cu 2÷ spins (neighbouring Cu 2+ and O spins) [43]. (In this section, we consider the energy per one pair, i.e., per four CuO 2 units; thus the coefficients in this and following equations are four times those in the corresponding equations in Ref. [43].) Using numerical values Jdd = 0.1 eV, Jpd = 1 eV and S = ~1 we obtain AE m = - 0 . 3 2 5 eV. A E c t is represented by AEct = A, where A is the energy gap for the charge-transfer process, Cu2++ O 2 - ~ Cu ÷ + O - [43]. With experimentally derived values of A (A~xp) [45-47], AEct---- 1.4-2.0 eV, which is significantly larger than [A E m I- However, the effective energy gap A in the present situation is probably

369

much smaller than Aexp because of the following two factors. First, Aexp is obtained from optical conductivity measurements and therefore related to the excitation (charge transfer) at an optical frequency, whereas in the present case the charge transfer takes place at a lower frequency characterizing the motion of holes. Since the displacement polarisation, arising from the motion of the background ions, contributes to screening only at lower frequencies, the effective A in the present case should be smaller than A~xp. (Neutron-scattering [48,49] and Raman scattering [50] experiments have revealed dynamical local lattice distortion, suggesting that the lattice vibration is able to follow the holes as they move.) Second, Aexp was derived from the measurements on undoped insulating cuprates [45-47] while, in establishing the two-hole state (Fig. 6(a)), the charge transfer takes place in the presence of externally doped holes residing mainly in O sites. This reduces AVM (the difference in the Madelung potentials between the copper and in-plane oxygen) and hence A [51]. It is difficult to evaluate AE r, but in view of the resonance energy in benzene (C6H 6) [52] ( - 3 5 . 9 7 kcal/mol = - 1 . 5 6 2 eV per molecule = - 0 . 2 6 0 eV per C atom), I A E r [ could be on the order of 0.1-1 eV, comparable to [AE m [ or [AEct [. In conclusion, it is not unreasonable to expect AE < 0. (2) The material dependence of A E may also be decomposed into three terms: ~ ( A E ) - - ~ ( A E m) + ~ ( A E c t ) + ~ ( A E r ) , where 15 stands for the variation due to the material species. From the experimentally observed material dependence of A and Jdd (1.4 < Aexp < 2.0 eV, 0.1
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M. Muroi, R. Street/Physica C 246 (1995) 357-374

changes without changing d or AVr~, e.g., when the dp-AV M map (Fig. 1 of Refs. [43] and [44]) is traversed vertically. (Note that a decrease of dp does not necessarily result in an increase of AVr~ because of the difference in the out-of-plane structures.) In that case, Jad will strongly depend on dp since 4 Jdd (Xtpo at dp 14 (e.g., Jdd increases by ~ 44% as dp decreases from 1.95 to 1.9 ,~). More experimental work on a wider variety of materials is desirable to clear this point. Third, Jpd, which also contributes to AEm, will have a large dp dependence; from the expression Jpd = 2t2d[1/(Ua -- A) + 1/(Up + A)] [53], it is expected that Jp0 is only weakly dependent on A and Jpd Ot t2o ~ dp 7 (e.g., Jpd increases by 20% as d o decreases from 1.95 to 1.9 ,&). Assumo ing Jod = 0.1 eV and Jpd = 1 eV for dp= 1.95 A, we estimate the change in A Ern with decreasing dp from 1.95 to 1.9 A without changing AVu at 0.083 eV (i.e., ~(AEm)-- -0.083 eV). Thus I~(AEr,)I may not to be negligibly small compared with 8(AE~t) << 0.6 eV. (3) With the present simple model, it is hard to tell much about AE r, and we simply assumed [ ~(AEr) I << 18(AEm)I, l S(AEct)l [43]. However, the LPR model does predict one thing; AE r and hence Tc will decrease as the CuO 2 plane is subject to orthorhombic distortion, which makes the pairs with different orientations (labelled H and V in Fig. 6(a)) inequivalent, thereby weakening the resonance. This is consistent with the large negative (positive) effect on Tc of uniaxial compression along the a-axis (b-axis) observed in fully-oxygenated YBCO [54,55]. (As has been argued in Refs. [54] and [55], pressure-induced charge transfer is an unlikely explanation of these effects; the dependence of T¢ on the hole concentration is small for nearly optimised systems.) Similarly, the increase of the orthorhombicity is a possible reason for the decrease of T¢ with decreasing the R ionic radius in RBa2Cu307 [56]. We now consider an unoptimised system with p < 0.25. Adding one electron to the optimised system fills one hole in one of the CuO 2 cells. This results in annihilation of the pair in that cell, since an extra oxygen hole that causes frustration among the four Cu spins no more exists and the charge transfer ( C u 2 + - I - 0 2-.--4 Cu+--~ - 0 - ) does not occur. There thus appears a local region where the spins of the Cu 2+ ions are antiferromagnetically coupled, as

shown in Fig. 6(b). This hole-free region does not participate in the resonance, resulting in a local loss of condensation energy. Adding more electrons simply increases the fraction of hole-free regions, i.e, the system separates into the S and I phases. (The phase separation is a natural consequence of the short-range nature of both the interpair interaction in the S phase and the magnetic exchange interaction in the I phase. If the holes were distributed uniformly, there would be no gain in either condensation energy or magnetic energy.) The effect of hole localisation, e.g., by Pr in YPBCO, is expected to be essentially the same; the regions where a hole is localised simply do not participate in the resonance, locally destroying the superconductivity [23,37]. A major difference between hole filling and localisation is that the localised holes have a much stronger influence on the distribution of the remaining mobile holes than the ions that provide electrons to the CuO 2 plane, as we have argued in the previous section. A similar argument also applies to the overdoped region (p > 0.25). In this case, addition of extra holes results in phase separation into the S and N phases, thereby allowing the system to gain the condensation energy in the S phase and the kinetic energy in the N phase. The LPR model is thus consistent with the two-phase model. With the LPR model, the decrease of Tc by excess electrons (holes) or by doped impurities are understood as resulting from the local suppression of the resonance or equivalently the local loss of shortrange interpair interactions. The situation is analogous to the decrease of Tc (TN) by nonmagnetic substitutions in (anti)ferromagnets, which results from the local loss of exchange energy [57]. The model also provides a clear physical meaning of SCab not only in optimised systems [43,44] but also in unoptimised systems. It has been reported that ~ab increases as the composition deviates from the optimal one, e.g., with decreasing y in YBa2Cu306+. [58] or with increasing x in Yl-xPrxBa2Cu307 [59~ ~ab in these cases may be identified as the percolation correlation length [60], which roughly scales with the size of the largest cluster of the I phase (see Fig. 5 in this paper and Fig. 4 in Ref. [23]) and diverges at the percolation threshold where the I phase forms an infinite cluster and a metal-insulator transition takes place.

M. Muroi, R. Street/Physica C 246 (1995) 357-374

6. Conclusions In this paper, the relationship between the oxygen content y, the hole concentration in the CuO 2 plane p and the superconducting properties in YBa2Cu 3O6+y (YBCO) is discussed. We have derived a p versus y relationship from the data of inelastic neutron-scattering experiments [17] and compared it with various theoretical predictions, as well as with p versus y relationships estimated by other methods. The results indicate that a primary factor controlling the transfer of holes from the basal plane to the CuO 2 plane is the ordering of O(1) ions not into 2D clusters [9] but into 1D Cu-O chains [10] and that T~ is not proportional to p. The variation of the superconducting properties with y is discussed in terms of the two-phase model in which it is assumed that the optimal hole concentration Popt is a local condition and that p
371

Er~ and Ect depend on the crystal structure and therefore Tc max is intrinsic to a particular system. The other mechanism involves local suppression of the order parameter due to various kinds of perturbation such as chemical substitutions and oxygen nonstoichiometry. The superconducting properties including Tc are determined by the connectivity of the surviving superconducting regions. This effect is extrinsic and depends on the types of perturbation, as we have argued in this paper and in Ref. [23]. It is clear that the LPR model is just a starting point towards the comprehensive understanding of high-Tc superconductivity. The model, at present, is qualitative and descriptive. We nevertheless believe that it contains the essential physics of high-To superconductivity, which has been demonstrated by its ability to consistently describe the variation of Tc in cuprate superconductors. We hope that a more sophisticated, quantitative theory will be constructed based on the LPR model.

Appendix In this Appendix, we make a more quantitative discussion on the hole distribution for y = 0.5 (YBa2Cu306.5). As we have argued in Section 4, the hole distribution for y = 0.5 is likely to be the one in which islands of the I phase are embedded in the S phase (Fig. 5(d)) and the size of the islands is determined so as to minimise the sum of the three energies: the superconducting condensation energy in the S phase (Eeon), magnetic energy in the I phase (Emag), and Coulomb energy associated with the inhomogeneous charge distribution (Ecoul). To estimate the size of the I phase islands, we calculate the above three energies for the hole distribution shown in Fig. 7(a); circular islands of the I phase with a diameter of are embedded in the S phase matrix, forming an array with a period of x in both vertical and horizontal directions. (The total areas occupied by the S and I phases are the same,) We first consider E~on and Emag in terms of the surface energy associated with S - I interfaces. At an S-I interface, the order parameter ~ is suppressed over a distance of ~ ~ab, the in-plane coherence length, as schematically shown in Fig. 7(b). In other

~/-f/Trx

M. Muroi, R. Street / Physica C 246 (1995) 357-374

372

S phase

~-E~ 2x

~- x -~1

/2U/2U/22UA2U/ ) S

I

number for Cu 2+. The total surface energy per unit length is therefore y = ")/conq- "~rnag = PAso ~ab/a2 qJddS2/a. Using numerical values, 2Asco = 0.06 eV ( ~ 8kTc), Cab = 10 ,~, p = 0.25, a = 3.9 A, Jdd = 0.1 eV and S = ½, we obtain %on = 1.03 × 10 -11 J / m , Ym~g = 7.89 × 10 - 1 2 J / m and y = 1.82 X 1 0 -11 J / m . (It is interesting that %o° and Ymag are comparable.) Considering that the area of the fundamental unit of the hole distribution (Fig. 7(a)) is x 2 and the length of the S - I interface contained in it is 2v'2~wx (the circumference of an I phase island), we obtain the expression for the surface energy (per unit area) as a function of x:

E,..=

x)/x 2 4.56 × 10 11

0.1

....

~ ....

(c) 0.05

×rain= 4 9 ~

.9.0 ILl

0

' 0

'

'

,

,

(J/m2).

,

50

,

I

1 O0

i

,

(3)

x

~ ....

,

,

150

x (A) Fig. 7. (a) Model distribution of the S and I phases for the calculation of Emag, Econ and Ecour Circular islands of the l phase are embedded in the S phase (shaded area), forming a 2D array with a period of x. The S phase is regarded as a sheet of uniform charge density o-. (b) A schematic picture showing an S - I interface. Near the interface, the order parameter ~b is suppressed over a distance of ~ ~ b , the in-plane coherence length. (c) The total e n e r g y (Etota I = Emag + Eco n + gcoul) plotted against the period x of the distribution shown in (a).

To derive the expression for Eco,l, we regard the S phase (shaded area in Fig. 7(a)) as a uniform sheet of charge (charge density o') and refer to the calculation by Kittel [61] of the magnetostatic energy for the free pole distribution similar to the one shown in Fig. 7(a). The two problems (electrostatic and magnetostatic) are formally the same and, following the procedure described in Ref. [61], we obtain E~oul = 8.41 × 108o. 2x (in MKS). (The prime on Eco,J indicates that the polarisation effect has not been considered.) Considering that 0. = p e / a 2 ( e is the electron charge) in the above uniform-charge approximation and taking account of the polarization effect, which decreases E~ouJ by a factor of the dielectric constant es, we have the expression for Ecuui 5.82 × 107 Ecou, =

words, the pairs contained in this region do not contribute to Eco.. Since the area occupied by one pair is 2a2/p (a is the in-plane lattice constant, p the hole concentration in the S phase), the cost in Eco n per unit length of an S - I interface (Ycu,) is 2 Asc ~ab/(2 a 2 / p ) = pAsc ~ b / a 2, where 2 A~c is the pair potential. Similarly, the exchange energy arising from C u - C u antiferromagnetic bonds is lost at an interface. The number of C u - C u bonds lying at an interface of unit length is l / a , and the cost in Emag ('~mag) is J a d S Z ( 1 / a ) = J a a S 2 / a , where Jdd is the exchange interaction and S is the spin quantum

× (J/m2).

(4)

The total energy for the hole distribution shown in Fig. 7(a) is thus given, as a function of x, by Etutal = E~.rf + Ecoul ot

= -

x

+ ~x,

(5)

where c~ = 4.56 × 10 -11, /3 = 5.82 × 1 0 7 / / 6 s . Etota 1 becomes minimum at x = Xmi n = ]f'~-///~ = 8 . 8 6 × 10-1°~s-s . For e s = 30, Etota I takes a minimum value 1.9 × 10 -2 J / m ( = 0.14 eV per hole) at Xmin = 4.85 × 1 0 - 9 m = 48.5 .~, that is, the diameter of the I phase islands corresponding to the minimum Etota I

M. Muroi, R. Street / Physica C 246 (1995) 357-374 is ~2-/-~Xrnin ~ 39 A. Etota I is p l o t t e d a g a i n s t x in Fig. 7(c) for e s = 30. T h e a b o v e c a l c u l a t i o n is c r u d e m a i n l y b e c a u s e o f the n u m e r i c a l u n c e r t a i n t y in the p a r a m e t e r s u s e d (2 Asc a n d e~ in particular), b u t t h e i r i n f l u e n c e o n the result is e x p e c t e d to b e fairly s m a l l s i n c e xmi a h a s only a square-root dependence on these parameters. T h u s the size o f the I p h a s e o i s l a n d s is m o s t likely to b e a f e w to s e v e r a l tens o f A. A n i n t e r e s t i n g f e a t u r e s e e n in Fig. 7(c) is that the m i n i m u m is v e r y b r o a d , i.e., the v a r i a t i o n o f Etota l w i t h x is v e r y s l o w a r o u n d x~i .. T h i s i m p l i e s that t h e r e is a s i g n i f i c a n t f l u c t u a t i o n in size o f the I p h a s e i s l a n d s at finite t e m p e r a t u r e o w i n g to the e n t r o p y c o n t r i b u t i o n a n d that the hole d i s t r i b u t i o n is susceptible to e x t e r n a l p e r t u r b a t i o n , e.g., t h e spatial v a r i a t i o n o f the C o u l o m b p o t e n t i a l d u e to c o m p o s i t i o n a l v a r i a t i o n s o n a m i c r o s c o p i c scale.

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[61]

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