60K plateau of Tc (x) in YBa2Cu3O6+x, critical chain length 
Origin of the 60K plateau in YBa2Cu3O6+x
V. M. Matic and N. Dj. Lazarov 
Laboratory of Theoretical Physics,  
Institute of Nuclear Sciences “Vinca”, 11001 Belgrade, Serbia 
 Abstract 
 A model for charge transfer mechanism in YBa2Cu3O6+x high-Tic cuprate based on 
critical chain length concept is proposed to account for 60K and 90K plateaus in Tc(x) 
dependence. It has been shown, when the statistics of CuO chain formation was described 
in terms of two dimensional asymmetric next-to-nearest neighbor Ising (ASYNNNI) 
model, that at any constant temperature below the top of OII phase there exists a uniquely 
defined value of critical chain length lcr(T) that yields a constant doping p(x)≈const over 
the regime of OII phase (related to 60K plateau of Tc(x)), while 90K plateau coincides 
with the monotonously increasing p(x) over optimal doping level p=0.16 in the regime of 
OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2 holes in longer 
chains (l≥lcr(T)), are taken as not capable of attracting electrons from CuO2 planes. It is 
shown that only a part (≈41%) of the remaining l-lcr(T)+1 holes in the long chains can 
capture electrons. The results obtained indicate that the ASYNNNI model and two-
plateaus-like behavior of Tc(x) in YBa2Cu3O6+x are closely connected.  
PACS: 74.72Bk, 64.60.Cn, 81.30Dz 
 CuO2 layers are key ingredients in all high-Tc cuprates given the fact that 
superconductivity occurs in these materials when a part of 3d copper electrons, usually 
between 5% and 27%, is taken away from the layers. The missing electrons are 
commonly referred to as “holes” which can move throughout the layers and make the 
material superconducting if the temperature is low enough. The electrons are 
conventionally subtracted from the layers either by chemical substitution of interlayer 
metal atoms, as for example, substitution of La2+ by Sr3+ in La1-x(Sr)xCu2O4 and Y2+ by 
Ca3+ in Y1-b(Ca)bBa2Cu3O6+x, or by pumping oxygen into the material. Oxygen is 
introduced into separate layers in which it orders to form CuO chains and it is these 
chains that are known to act as efficient attractors of electrons from the CuO2 layers. The 
number of created holes per Cu atom is typically denoted as “doping” p and the notion 
that CuO2 layers have an unavoidable role in the onset of superconductivity is further 
corroborated by the fact that a number of important physical characteristics, as, for 
example, the pseudogap energy Eg and critical transition temperature Tc, are coupled to p 
by universal relations that are common to practically all high-Tc cuprates. Thus, it has 
been obtained empirically that the Tc is scaled with Tc,max (maximal transition 
temperature) through the following, approximately parabolic, function of the hole 
concentration  [2]  
( )[ ]2max, 16.06.821)( −−= pTpT cc  ,                                                            (1) 
which has its onset, maximum and termination at p=0.05, 0.16 (optimal doping) and 0.27, 
respectively.  
 YBa2Cu3O6+x superconductor has probably been the most thoroughly studied 
compound of all high-Tc cuprates because it has a relatively simple synthesis route and it 
was the first superconductor discovered with Tc above the liquid nitrogen temperature. 
The Tc changes in nonlinear manner with oxygen composition revealing two well-known 
plateaus at 60K and 90K. These features are clearly associated with the Ortho-II and 
Ortho-I phases, respectively [3]. While the 90K plateau is in fact a broad maximum at 
x≈0.92 that is associated with transition from underdoped to overdoped regime, the origin 
of 60K plateau at 0.5<x<0.7 is not yet well understood [3-5] (although, in fact, some 
advance has recently been made along this line [6]). One popular explanation is that 
when the oxygen content is increased over x=0.5, where Ortho-II phase emerges in the 
form of alternating columns of fully occupied and empty oxygen sites (directed along b 
axes), additional oxygen fills the empty columns making a relatively small contribution 
to hole doping, inasmuch as they are far apart from each other and only a small fraction 
of them are able to form CuO chains that are long enough to initiate the charge transfer 
process. It has often been guessed that there might be a certain minimal (critical) chain 
length lcr defined so that only chains of length that is equal to, or greater than, lcr can 
effectively attract electrons from CuO2 layers [3-6]. However, even though the existence 
of Ortho-II phase was resolved a long ago in terms of the classical two dimensional 
asymmetric next nearest neighbor Ising (ASYNNNI) model [7], no convincing 
explanation has been provided over the last 20 years as to why exactly the concentration 
p(x) of holes induced in CuO2 sheets would remain constant when x changes within the 
Ortho-II phase regime, nor, if the critical chain length concept is presupposed, what the 
value of lcr would be equal to and how it can be determined.  
 Although it is well known that high-Tc cuprates are complex quantum many-body 
systems with the pairing mechanism still remaining controversial, here we 
unambiguously grounds that it is the classical ASYNNNI model combined with the 
concept of minimal chain length (needed for charge transfer to take place) that accounts 
for constant doping at p≈0.094 in the region of 60K plateau, as well as for the broad 
maximum of Tc at optimal doping (p=0.16) at x≈0.92.   
 There is a general agreement that copper in the chain (basal) plane can be either 
Cu1+, which is the case when it is not coupled to the in-plane oxygen (but connected only 
to two apical O(4) ions and therefore 2-fold coordinated), or Cu2+, when it is inserted 
within a CuO chain (4-fold coordinated), or at the chain end (3-fold coordinated). If 
isolated oxygen is introduced into the basal plane it then takes away two electrons from 
the two nearest neighbor Cu1+ transferring them into Cu2+ state. Thus, isolated oxygen 
does not have a tendency to attract an additional electron from other the parts of the 
system. When another oxygen is added to make a chain with two O atoms (chain of 
length l=2) there is only one electron available from its nearest neighbor copper 
coordination and the absence of another electron, needed for oxygen to become O2-, is 
usually referred to as a “hole”. In this way, a chain of length l is seen as to have created l-
1 chain-holes, which in principle can attract electrons from the other parts of the system, 
presumably from CuO2 layers.  
Since the state of quasi one dimensional electronic subsystem of a chain of a 
given length (say, l), which reflects the charge transfer effectiveness of the chain, is not 
expected to depend upon history of chain formation, one is free to assume that the chain 
has been formed by adding oxygen one by one as that would allow to shadow the 
evolution of charge transfer process as a function of l. Within the concept of critical chain 
length no charge transfer is supposed to occur unless l=lcr so that the first lcr-2 initially 
created holes will stay inactive in attracting electrons from the planes. As l further 
increases beyond lcr the transfer of charge is set on during which process the remaining l-
lcr+1 holes are created. Therefore, our strategy for counting doping is, first of all, to 
discard contribution not only of all holes in short chains (l<lcr), but also of the first lcr-2 
holes in long chains (l≥lcr), for they had been created before any charge transfer took 
place (we shortly denote these holes as passive holes). It then naturally evolves that the 
number of attracted electrons (transferred holes) from the chain of length l≥lcr should be 
as greater as more holes were created after the charge transfer process had been triggered 
(at l=lcr), i.e. it should be proportional to l-lcr+1 (we call these holes the active holes for 
their occurrence coincides with the development of charge transfer). In the case of Ortho-
II phase, the concentration of active holes h (the number of active holes per Cu) is given 
+−++−= ∑ ∑
=cr crll ll
crcr lfllnlfllnh )()1()()1(4
2211 αααα ,                       (2) 
where nα1 and nα2 denote the fractions of 3-fold coordinated Cu ions on two different 
sublattices of oxygen sites (usually denoted as α1 and α2), and fα1(l) and fα2(l) are 
corresponding fractions of CuO chains of the same length l.   
 If NCu and n=(nα1+nα2)/2 are total number and the fraction of 3-fold coordinated 
Cu in basal plane, then (n/2)NCu is the total number of CuO chains and the number of 
passive holes per Cu is surely not greater than (n/2)(lcr-2), given the fact that no one of 
chains with l<lcr has more than lcr-2 holes. At x=1 (OI stoichiometry) chains are very 
long, virtually infinite, and there are just a few chain ends in the system. This means that 
n tends to zero as x approaches to 1, so that the concentration of passive holes becomes 
negligible whatever the value of lcr. This in turn implies that practically each chain 
ordered oxygen has created one active hole, i.e. h≈1 at x≈1. Since, in the YBa2Cu3O6+x 
system, one chain plane supplies holes to two CuO2 layers, the doping would have been 
equal to p=0.5 if each active hole had succeeded to capture one electron. Experimental 
findings, however, clearly contradict such a scenario for it was found that p(x≈1) only 
slightly overshoots 19% [2]. Thus, at least at x≈1, it can be stated p=(χ/2)h where χ 
(≈40%), as defined by the number of effectively attracted electrons (transferred holes) per 
active hole, reflects the capability of an active hole to capture electron. On the other hand, 
long (infinite) chains also prevail at x=0.5 (OII stoichiometry) on every even column of 
oxygen sites. The concentrations of passive holes is also negligible here, bur h cannot be 
greater that 0.5; in fact, h can be only less than 0.5 due to thermally activated chain 
fragmentation (for example, one might expect that h≈0.48, as at x≈0.5, so in the region of 
the 60K plateau). Given the fact that 60K plateau of Tc, according to (1), corresponds to 
doping level p≈0.094 it appears that it is the same fraction (of nearly 40%) of active holes 
that is transferred to the layers not only at x≈1, but also at x≈0.5. The charge transfer 
model that we propose here assumes that the same percentage of active holes is 
transferred, not only at stoichiometries x=0.5 and x=1, at which long CuO chains are 
known to dominate, but as well at off-stoichiometry 0.5<x<1 (and also at x<0.5) where 
chain fragmentation is more intense. We therefore propose that doping is connected to the 
active hole concentration h, as given by (2), by p=(χ/2)h (χ≈40%) throughout the whole 
range of oxygen concentration 0<x<1. The quantity χ introduced in this way should be 
perceived as average capability of an active hole to capture an electron from the CuO2 
planes that lie above, or below, the basal plane (averaging is done over all chains in the 
system, or equivalently, over the whole volume of the material).  
To calculate the hole concentration h (eq. (2)) and doping p=χh/2, at a given point 
of (x,T) space, it is necessary to determine the fractions of the 3-fold coordinated Cu and 
length distributions of CuO chains fα1(l) and fα2(l). We used Monte Carlo (MC) method 
applied to the ASYNNNI model to calculate these quantities for it is known that the 
model stabilizes both structures Ortho-II and Ortho-I that are responsible for 60K and 
90K plateaus [9]. Although ASYNNNI model cannot stabilize other structures with 
longer periodicities along a axis, like Ortho-III, Ortho-IV and Ortho-V, their 
superstructure reflections have already been reported to be much weaker than those of the 
main phases [8,9], so that they are thought to appear only as small patches embedded in 
large domains of main phases [10]. Besides, since these structures were mainly observed 
at oxygen compositions that correspond to transition region between the two plateaus it 
implies that the ASYNNNI model alone should account for both plateaus of Tc(x), 
especially given the fact that none of these structures, except Ortho-II, was reported at 
x<0.62 [9]. The chain length distributions fα1(l) and fα2(l) were determined in the 
following way: In each MC step we counted the total numbers of chains Nα1 and Nα2, on 
sublattices α1 and α2 , respectively (Nα1 and Nα2 are in fact equal to one half of unlike V2 
bonds on the corresponding α sublattices), as well as the numbers of chains of the same 
length, Nα1(l) and Nα2(l), for lengths ranging from l=1 to l=300. The ratios Nα1(l)/Nα1 and 
Nα2(l)/Nα2 were then equilibrated through the MC process and the so obtained values 
were finally assigned to fα1(l) and fα2(l). The MC calculations were performed using 
single-spin-flip Glauber dynamics, where the oxygen concentration x is a functions of 
temperature T and chemical potential μ. We have studied lattices with periodic boundary 
conditions that consisted of 400x400 oxygen chain sites (O(1) sites, that split into two 
nonequivalent sublattices α1 and α2, in OII phase), and as many sites on β sublattice 
(O(5) sites). One MC step included flipping of all 2X(400X400) lattice spins and one MC 
run (at a particular point (x, T)) typically consisted of 3·104 to 5·104 MC steps, where only 
every tenth was used to calculate chain length distributions fα1(l) and fα2(l), l=1,2, …,300, 
and other relevant quantities (oxygen sublattice occupancies x1 and x2, 3-fold Cu fractions 
n1 and n2, etc.). At a certain number of points we have even used a really large number of 
MC steps, ranging from 105 to 3·105.  
 At all calculated points of (x,τ) space (τ is a quantity that scales with T according 
to τ=kBT/VB 1, where kBB is Boltzman constant and V1 the nearest neighbor O-O interaction 
of the ASYNNNI model) it was obtained [11]  
 2,1,
iaviav
α      ,                                                        (3) 
where lav,αi (i=1,2) denotes the average chain length on the corresponding sublattice (α1 or 
α2). Such a behavior of length distributions fα1(l) and fα2(l) was subsequently explained 
theoretically analyzing microscopic features of the ASYNNNI model lattice 
configurations [12]. In brief terms, the l dependence of probability of a chain to have 
particular length l can be derived in the following way: Consider a sequence of Ny  
oxygen chain sites that are aligned along b axis (Ny is a large number, and the sites are 
connected by copper mediated V2<0 bonds). Let xNy denotes the number of oxygen atoms 
on this column of α sites and let nNy stands for the number of unlike V2 bonds (n is 
therefore the fraction of 3-fold coordinated Cu along the column, and, consequently, 2x/n 
equals to the corresponding average chain length, lav). These xNy oxygen atoms are 
generally divided into (n/2)Ny groups (chains) that can have various lengths l=1,2,… . It 
is useful to recall that an each chain has two ends: one that is oriented towards positive 
side of b axes (the “positive” end) and the other one, oriented towards negative b axes 
(the “negative” end). Among these xNy oxygen atoms there are (n/2)Ny of them that are 
located at the positive chain end, and thus the probability for an oxygen to be lying at the 
positive chain end is equal to ω=n/2x={lav)-1. Consequently, 1-ω is probability for oxygen 
to be located either within the interior of the chain, or at the negative chain end. 
Assuming that chains are created by adding oxygen one by one, starting from the 
negative end, one arrives at the conclusion that the probability for obtaining chain of 
length l is equal to f(l)=ω(1-ω)l-1 [11,12] {such a form of f(l) dependence is known in the 
theory of probability as “geometric” probability distribution [13]). It should be noted, 
however, that a deeper analysis shows that the above reasoning applies only if 
fluctuations of energy of the ASYNNNI model are not too large [12]. Indeed, a certain 
deviation from linear behavior of ln[f(l)] versus l dependence has been found in the 
vicinity of the second order Ortho-I-to-Ortho-II phase transition curve (at x>0.5) [12] but, 
fortunately, such departures were observed only in a relatively narrow intervals Δx≈0.07 
around critical points. Furthermore, our extended analysis (not shown here) shows that in 
the critical regime these deviations, of calculated fα1(l) and fα2(l) dependences from the 
expected behavior, were in a certain way compensated by summations in (2), so that the 
calculated h(x) dependences were obtained to vary smoothly over the transition region at 
all τ=const.  
 The so obtained values of length distributions fα1(l) and fα2(l) for l=1,…,300 were 
inserted into (2) to calculate concentration h of active holes at different points (x,T). The 
geometric-like behavior of f(l) dependences ensures rather fast convergence of sums in 
(2). It should be mentioned, however, that the specific form of length distributions (3) 
makes it possible, instead of evaluating summations in (2) by first 300 terms, to transform 
each of sums into a closed analytical form, so that h would be connected through 
analytical expression with average oxygen occupancies, x1 and x2, the 3-fold Cu fractions 
n1 and n2, and the parameter lcr. Whatever the approach we used the calculated values of 
h were obtained to be practically indistinguishable one from the other (even in the critical 
region of the ortho-I/ortho-II transition), but we nevertheless gave advantage to 
calculating the first 300 terms in estimating sums in (2), as we wanted to keep under 
control the departures of length distributions from (3) that are known to occur in the 
critical regime [12]. In addition, at each point of (x,T) space h was calculated for the 
whole range of values of cutoff parameter lcr spanning from lcr=1 to lcr=50, so that h can 
be regarded as a function of three variables, i.e. h=h(x,τ,lcr) (aside from the fact that it 
also depends upon input parameters that define the ASYNNNI model, i.e. on O-O 
interactions V1>0 (nearest neighbor) and V2<0, V3>0 (next nearest neighbor)). There is 
one remarkable feature of the hole concentration h=h(x,τ,lcr), as defined by (2), that we 
have found while thoroughly analyzing its behavior: whatever the magnitudes of 
interactions V1, V2, and V3, when h is considered as a function of lcr at different points 
(xi,τ) that correspond to the same τ=const and oxygen concentrations xi, i=1,2,… 
spanning over the region of Ortho-II phase, all of these hxi(lcr), i=1,2,… functions 
intersect at a single, well defined value of lcr. This is shown in Figure 1 for interactions 
obtained by linear-muffin-tin orbital (LMTO) method [14] and at three different 
temperatures τ=0.45, 0.38 and 0.30, but the similar behavior we have also obtained (not 
shown here) for the so-called “canonical” interactions V2=-0.5V1, V3=0.5V1. From Figure 
1 it can be clearly seen that at a given τ=const, the value of lcr at which all hxi(lcr) curves 
intersect depends on temperature in the way that it increases with the temperature 
decrease. Thus, at three temperatures τ=0.45, 0.38, 0.30, the intersection values were 
found to be lcr=4(5), 6(7), 11(12), respectively. At a given temperature, the so obtained 
intersection value of lcr we name “the optimal minimal (critical) length” (denoted by 
lcr,opt(τ)) for it is the value at which h(x) stays constant over the regime of Ortho-II phase. 
Such behavior of h(x) at τ=const, for the corresponding lcr,opt(τ), is shown in Figure 2 at 
two temperatures: τ=0.45 (Figures 2a and 2b, for lcr,opt(τ)=4 and 5, respectively) and 
τ=0.30 (Figure 2c, for lcr,opt(τ)=12). From these results it can be seen that indeed h(x) 
demonstrates a constant section at x>0.5 that is even more pronounced at lower 
temperatures.  
 Calculated h(x) dependences were used to obtain doping versus x dependences, 
p(x)=χh(x)/2, that were then inserted into (1) to yield corresponding Tc(x)s (Figures 3a-c). 
The parameter χ was varied slightly around its expected value ≈40% [2] to achieve a 
better correlation between the so obtained Tc(x)’s and those from experiments [15] 
(shown by a solid line). We have indeed obtained h in the plateau regime to be slightly 
lower than 0.5: h=const=0.467 (0.450) at τ=0.45 for lcr=4 (lcr=5), h=const=0.483 at 
τ=0.38 for lcr=7, and h=const=0.495 at τ=0.30 for lcr=12. This gives χ=40.33%, 41.84%, 
38.98%, 3804% for h=0.467, 0.450, 0.483, 0.495, respectively. Thus, not only 
experimental data on doping at x≈1 [2], but also the analysis at x≈0.5 (that falls into the 
regime of 60K plateau) both infer that χ is lying at some point between ≈38% and ≈42%. 
It should be noted that this result disagrees with estimations of Gawiec et al. [16,17] that 
long chains release up to 70% of their holes.  
 Although we used here the ASYNNNI model interactions V1, V2, and V3 as 
obtained by Sterne and Wille [14] the issue of magnitudes of these interactions is still 
open so that new values were subsequently suggested [10]. Regardless of that, it seems 
very well established that nearest neighbor interaction V1 should be ranking around 
6.9mRy [10,14], which fixes scaling between T and τ to Δτ≈0.1⇔ ΔT≈100K. On the 
other hand, as one of the most important features of YBa2Cu3O6+x phase diagram is the 
location of the top of Ortho-II phase along τ axis (at x≈0.5) and that it may well be 
affected by the magnitudes of V2 and V3 (which in fact are not known precisely), perhaps 
the best strategy to estimate the reduced temperature τ corresponding to room 
temperature is to determine the “distance” (in units of τ) between room temperature and 
the top of Ortho-II phase. According to experimental data the top of Ortho-II phase 
corresponds to ≈125-140ºC [5,9], while theoretically obtained phase diagram for the 
LMTO interactions [18] points at τ≈0.58, thus making τ≈0.45 a fairly reliable estimation 
of room temperature.  
The established correlation between room temperature and τ=0.45 renders lcr,opt(τ) 
(that plays the role of lcr in (2)) 4, or 5 (Figure 1a). A better estimation seems to be 5 for 
it yields a somewhat more pronounced 60K plateau than lcr=4 (Figures 3a and 3b), 
despite the fact that lcr=4 would be closer to lcr=3 that was proposed in some theoretical 
studies [16,17,19]. Besides, lcr=5 appears to be well correlated with χ≈2/5 and with the 
basic idea lying in the background of the minimal (critical) chain length concept 
suggesting that one isolated hole cannot efficiently attract an electron, but only a 
combined effect of several holes can achieve this goal. Both lcr=5 and χ≈2/5 imply that 3 
chain-holes are still not enough to effectuate charge transfer, but that the joint impact of 5 
holes suffices to attract two electrons (one per each CuO2 layer).  
 In summary, despite the well known fact that in such highly correlated electron 
systems, as are the high-Tc cuprates, the nature of the controversial pairing mechanism is 
genuinely quantum mechanical, we have shown here that certain aspects of their behavior 
can be explained in terms of classical models. Such is the classical ASYNNNI model that 
successfully accounts for unusual two-plateaus-like behavior of Tc(x) in YBa2Cu3O6+x. 
The obtained Tc(x) dependence is in a remarkable correlation with experiment [15] for 
both lcr=4 and lcr=5, although we believe the later value is more realistic. It should be also 
pointed out that the presented results on Tc(x) dependence are in a qualitative agreement 
with some previous results on the same topic [20,21]. Generally, it can be expected that 
the capability of an active hole to attract an unpaired 3d electron, as expressed by the 
value of χ, should depend upon the density ρe of available electrons immediately above 
(below) the chain (aside from the fact that χ should also depend on a certain coupling 
between chains and planes). Our recently obtained results on Tc(x) in Y1-
b(Ca)bBa2Cu3O6+x (b=0.2) system [22] seem to be lying along this line, since obtained 
χ≈33% can be understood in the light of the fact that introduction of ≈20% of Ca has 
additionally increased doping, and therefore reduced ρe, so that χ attained a lesser value 
than in the parent YBa2Cu3O6+x compound.  
Acknowledgements  
 This work has been funded by the Serbian Ministry of Science and Technology 
through the Project 141014.  
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Figure Captions    
Figure 1. Calculated h(lcr) dependences at τ=const for several values of oxygen 
composition x that span the range of OII phase: a) τ=0.45=const; 0.52<x<0.61 (shown at 
the top), b) τ=0.38=const; 0.52<x<0.62 (shown in the middle section), and c) 
τ=0.30=const; 0.52<x<0.67 (shown at the bottom).  
Figure 2. Calculated values of h as a function of x at τ=const: a) τ=0.45=const, 
for lcr=4 (shown at the top), b) τ=0.45=const, for lcr=5 (shown in the middle section), and 
c) τ=0.30=const, for lcr=12 (shown at the bottom).  
Figure 3, Calculated Tc(x) dependences, using h(x) dependences from the Figure 
2 and thereafter obtained p(x)=χh(x)/2 dependences, that were then inserted into (1): a) 
τ=0.45=const, for lcr=4 and χ=39.4% (shown at the top), b) τ=0.45=const, for lcr=5 and 
χ=41% (shown in the middle section), and c) τ=0.30=const, for lcr=12 and χ=37.6% 
(shown at the bottom).  
-2 0 2 4 6 8 10 12 14
τ=0.30, 0.52<x<0.67
x=0.52
x=0.54
x=0.56
x=0.57
x=0.59
x=0.62
x=0.67
x=0.52
x=0.53
x=0.55
x=0.56
x=0.58
x=0.59
x=0.61
τ=0.45, 0.52<x<0.61
τ=0.38, 0.52<x<0.62
x=0.52
x=0.53
x=0.54
x=0.55
x=0.57
x=0.59
x=0.62
Figure 1  
0.0 0.2 0.4 0.6 0.8 1.0
τ=0.45,  l
τ=0.45,  l
τ=0.30,  l
Figure 2   
0.0 0.2 0.4 0.6 0.8 1.0
τ=0.45,  l
χ=0.394
τ=0.45,  l
χ=0.410
τ=0.45,  l
χ=0.376
Figure 3

  A model for charge transfer mechanism in YBa2Cu3O6+x high-Tc cuprate based on
critical chain length concept is proposed to account for 60K and 90K plateaus
in Tc(x) dependence. It has been shown, when the statistics of CuO chain
formation was described in terms of two dimensional asymmetric next-to-nearest
neighbor Ising (ASYNNNI) model, that at any constant temperature below the top
of OII phase there exists a uniquely defined value of critical chain length
lcr(T) that yields a constant doping p(x)=const over the regime of OII phase
(related to 60K plateau of Tc(x)), while 90K plateau coincides with the
monotonously increasing p(x) over optimal doping level p=0.16 in the regime of
OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2
holes in longer chains (l>lcr(T)), are taken as not capable of attracting
electrons from CuO2 planes.. It is shown that only a part (41%) of the
remaining l-lcr(T)+1 holes in the long chains can capture electrons. The
results obtained indicate that the ASYNNNI model and two-plateaus-like behavior
of Tc(x) in YBa2Cu3O6+x are closely connected.

60K plateau of Tc (x) in YBa2Cu3O6+x, critical chain length 
Origin of the 60K plateau in YBa2Cu3O6+x
V. M. Matic and N. Dj. Lazarov 
Laboratory of Theoretical Physics,  
Institute of Nuclear Sciences “Vinca”, 11001 Belgrade, Serbia 
 Abstract 
 A model for charge transfer mechanism in YBa2Cu3O6+x high-Tic cuprate based on 
critical chain length concept is proposed to account for 60K and 90K plateaus in Tc(x) 
dependence. It has been shown, when the statistics of CuO chain formation was described 
in terms of two dimensional asymmetric next-to-nearest neighbor Ising (ASYNNNI) 
model, that at any constant temperature below the top of OII phase there exists a uniquely 
defined value of critical chain length lcr(T) that yields a constant doping p(x)≈const over 
the regime of OII phase (related to 60K plateau of Tc(x)), while 90K plateau coincides 
with the monotonously increasing p(x) over optimal doping level p=0.16 in the regime of 
OI phase. Short length chains (l<lcr(T)), together with the first lcr(T)-2 holes in longer 
chains (l≥lcr(T)), are taken as not capable of attracting electrons from CuO2 planes. It is 
shown that only a part (≈41%) of the remaining l-lcr(T)+1 holes in the long chains can 
capture electrons. The results obtained indicate that the ASYNNNI model and two-
plateaus-like behavior of Tc(x) in YBa2Cu3O6+x are closely connected.  
PACS: 74.72Bk, 64.60.Cn, 81.30Dz 
 CuO2 layers are key ingredients in all high-Tc cuprates given the fact that 
superconductivity occurs in these materials when a part of 3d copper electrons, usually 
between 5% and 27%, is taken away from the layers. The missing electrons are 
commonly referred to as “holes” which can move throughout the layers and make the 
material superconducting if the temperature is low enough. The electrons are 
conventionally subtracted from the layers either by chemical substitution of interlayer 
metal atoms, as for example, substitution of La2+ by Sr3+ in La1-x(Sr)xCu2O4 and Y2+ by 
Ca3+ in Y1-b(Ca)bBa2Cu3O6+x, or by pumping oxygen into the material. Oxygen is 
introduced into separate layers in which it orders to form CuO chains and it is these 
chains that are known to act as efficient attractors of electrons from the CuO2 layers. The 
number of created holes per Cu atom is typically denoted as “doping” p and the notion 
that CuO2 layers have an unavoidable role in the onset of superconductivity is further 
corroborated by the fact that a number of important physical characteristics, as, for 
example, the pseudogap energy Eg and critical transition temperature Tc, are coupled to p 
by universal relations that are common to practically all high-Tc cuprates. Thus, it has 
been obtained empirically that the Tc is scaled with Tc,max (maximal transition 
temperature) through the following, approximately parabolic, function of the hole 
concentration  [2]  
( )[ ]2max, 16.06.821)( −−= pTpT cc  ,                                                            (1) 
which has its onset, maximum and termination at p=0.05, 0.16 (optimal doping) and 0.27, 
respectively.  
 YBa2Cu3O6+x superconductor has probably been the most thoroughly studied 
compound of all high-Tc cuprates because it has a relatively simple synthesis route and it 
was the first superconductor discovered with Tc above the liquid nitrogen temperature. 
The Tc changes in nonlinear manner with oxygen composition revealing two well-known 
plateaus at 60K and 90K. These features are clearly associated with the Ortho-II and 
Ortho-I phases, respectively [3]. While the 90K plateau is in fact a broad maximum at 
x≈0.92 that is associated with transition from underdoped to overdoped regime, the origin 
of 60K plateau at 0.5<x<0.7 is not yet well understood [3-5] (although, in fact, some 
advance has recently been made along this line [6]). One popular explanation is that 
when the oxygen content is increased over x=0.5, where Ortho-II phase emerges in the 
form of alternating columns of fully occupied and empty oxygen sites (directed along b 
axes), additional oxygen fills the empty columns making a relatively small contribution 
to hole doping, inasmuch as they are far apart from each other and only a small fraction 
of them are able to form CuO chains that are long enough to initiate the charge transfer 
process. It has often been guessed that there might be a certain minimal (critical) chain 
length lcr defined so that only chains of length that is equal to, or greater than, lcr can 
effectively attract electrons from CuO2 layers [3-6]. However, even though the existence 
of Ortho-II phase was resolved a long ago in terms of the classical two dimensional 
asymmetric next nearest neighbor Ising (ASYNNNI) model [7], no convincing 
explanation has been provided over the last 20 years as to why exactly the concentration 
p(x) of holes induced in CuO2 sheets would remain constant when x changes within the 
Ortho-II phase regime, nor, if the critical chain length concept is presupposed, what the 
value of lcr would be equal to and how it can be determined.  
 Although it is well known that high-Tc cuprates are complex quantum many-body 
systems with the pairing mechanism still remaining controversial, here we 
unambiguously grounds that it is the classical ASYNNNI model combined with the 
concept of minimal chain length (needed for charge transfer to take place) that accounts 
for constant doping at p≈0.094 in the region of 60K plateau, as well as for the broad 
maximum of Tc at optimal doping (p=0.16) at x≈0.92.   
 There is a general agreement that copper in the chain (basal) plane can be either 
Cu1+, which is the case when it is not coupled to the in-plane oxygen (but connected only 
to two apical O(4) ions and therefore 2-fold coordinated), or Cu2+, when it is inserted 
within a CuO chain (4-fold coordinated), or at the chain end (3-fold coordinated). If 
isolated oxygen is introduced into the basal plane it then takes away two electrons from 
the two nearest neighbor Cu1+ transferring them into Cu2+ state. Thus, isolated oxygen 
does not have a tendency to attract an additional electron from other the parts of the 
system. When another oxygen is added to make a chain with two O atoms (chain of 
length l=2) there is only one electron available from its nearest neighbor copper 
coordination and the absence of another electron, needed for oxygen to become O2-, is 
usually referred to as a “hole”. In this way, a chain of length l is seen as to have created l-
1 chain-holes, which in principle can attract electrons from the other parts of the system, 
presumably from CuO2 layers.  
Since the state of quasi one dimensional electronic subsystem of a chain of a 
given length (say, l), which reflects the charge transfer effectiveness of the chain, is not 
expected to depend upon history of chain formation, one is free to assume that the chain 
has been formed by adding oxygen one by one as that would allow to shadow the 
evolution of charge transfer process as a function of l. Within the concept of critical chain 
length no charge transfer is supposed to occur unless l=lcr so that the first lcr-2 initially 
created holes will stay inactive in attracting electrons from the planes. As l further 
increases beyond lcr the transfer of charge is set on during which process the remaining l-
lcr+1 holes are created. Therefore, our strategy for counting doping is, first of all, to 
discard contribution not only of all holes in short chains (l<lcr), but also of the first lcr-2 
holes in long chains (l≥lcr), for they had been created before any charge transfer took 
place (we shortly denote these holes as passive holes). It then naturally evolves that the 
number of attracted electrons (transferred holes) from the chain of length l≥lcr should be 
as greater as more holes were created after the charge transfer process had been triggered 
(at l=lcr), i.e. it should be proportional to l-lcr+1 (we call these holes the active holes for 
their occurrence coincides with the development of charge transfer). In the case of Ortho-
II phase, the concentration of active holes h (the number of active holes per Cu) is given 
+−++−= ∑ ∑
=cr crll ll
crcr lfllnlfllnh )()1()()1(4
2211 αααα ,                       (2) 
where nα1 and nα2 denote the fractions of 3-fold coordinated Cu ions on two different 
sublattices of oxygen sites (usually denoted as α1 and α2), and fα1(l) and fα2(l) are 
corresponding fractions of CuO chains of the same length l.   
 If NCu and n=(nα1+nα2)/2 are total number and the fraction of 3-fold coordinated 
Cu in basal plane, then (n/2)NCu is the total number of CuO chains and the number of 
passive holes per Cu is surely not greater than (n/2)(lcr-2), given the fact that no one of 
chains with l<lcr has more than lcr-2 holes. At x=1 (OI stoichiometry) chains are very 
long, virtually infinite, and there are just a few chain ends in the system. This means that 
n tends to zero as x approaches to 1, so that the concentration of passive holes becomes 
negligible whatever the value of lcr. This in turn implies that practically each chain 
ordered oxygen has created one active hole, i.e. h≈1 at x≈1. Since, in the YBa2Cu3O6+x 
system, one chain plane supplies holes to two CuO2 layers, the doping would have been 
equal to p=0.5 if each active hole had succeeded to capture one electron. Experimental 
findings, however, clearly contradict such a scenario for it was found that p(x≈1) only 
slightly overshoots 19% [2]. Thus, at least at x≈1, it can be stated p=(χ/2)h where χ 
(≈40%), as defined by the number of effectively attracted electrons (transferred holes) per 
active hole, reflects the capability of an active hole to capture electron. On the other hand, 
long (infinite) chains also prevail at x=0.5 (OII stoichiometry) on every even column of 
oxygen sites. The concentrations of passive holes is also negligible here, bur h cannot be 
greater that 0.5; in fact, h can be only less than 0.5 due to thermally activated chain 
fragmentation (for example, one might expect that h≈0.48, as at x≈0.5, so in the region of 
the 60K plateau). Given the fact that 60K plateau of Tc, according to (1), corresponds to 
doping level p≈0.094 it appears that it is the same fraction (of nearly 40%) of active holes 
that is transferred to the layers not only at x≈1, but also at x≈0.5. The charge transfer 
model that we propose here assumes that the same percentage of active holes is 
transferred, not only at stoichiometries x=0.5 and x=1, at which long CuO chains are 
known to dominate, but as well at off-stoichiometry 0.5<x<1 (and also at x<0.5) where 
chain fragmentation is more intense. We therefore propose that doping is connected to the 
active hole concentration h, as given by (2), by p=(χ/2)h (χ≈40%) throughout the whole 
range of oxygen concentration 0<x<1. The quantity χ introduced in this way should be 
perceived as average capability of an active hole to capture an electron from the CuO2 
planes that lie above, or below, the basal plane (averaging is done over all chains in the 
system, or equivalently, over the whole volume of the material).  
To calculate the hole concentration h (eq. (2)) and doping p=χh/2, at a given point 
of (x,T) space, it is necessary to determine the fractions of the 3-fold coordinated Cu and 
length distributions of CuO chains fα1(l) and fα2(l). We used Monte Carlo (MC) method 
applied to the ASYNNNI model to calculate these quantities for it is known that the 
model stabilizes both structures Ortho-II and Ortho-I that are responsible for 60K and 
90K plateaus [9]. Although ASYNNNI model cannot stabilize other structures with 
longer periodicities along a axis, like Ortho-III, Ortho-IV and Ortho-V, their 
superstructure reflections have already been reported to be much weaker than those of the 
main phases [8,9], so that they are thought to appear only as small patches embedded in 
large domains of main phases [10]. Besides, since these structures were mainly observed 
at oxygen compositions that correspond to transition region between the two plateaus it 
implies that the ASYNNNI model alone should account for both plateaus of Tc(x), 
especially given the fact that none of these structures, except Ortho-II, was reported at 
x<0.62 [9]. The chain length distributions fα1(l) and fα2(l) were determined in the 
following way: In each MC step we counted the total numbers of chains Nα1 and Nα2, on 
sublattices α1 and α2 , respectively (Nα1 and Nα2 are in fact equal to one half of unlike V2 
bonds on the corresponding α sublattices), as well as the numbers of chains of the same 
length, Nα1(l) and Nα2(l), for lengths ranging from l=1 to l=300. The ratios Nα1(l)/Nα1 and 
Nα2(l)/Nα2 were then equilibrated through the MC process and the so obtained values 
were finally assigned to fα1(l) and fα2(l). The MC calculations were performed using 
single-spin-flip Glauber dynamics, where the oxygen concentration x is a functions of 
temperature T and chemical potential μ. We have studied lattices with periodic boundary 
conditions that consisted of 400x400 oxygen chain sites (O(1) sites, that split into two 
nonequivalent sublattices α1 and α2, in OII phase), and as many sites on β sublattice 
(O(5) sites). One MC step included flipping of all 2X(400X400) lattice spins and one MC 
run (at a particular point (x, T)) typically consisted of 3·104 to 5·104 MC steps, where only 
every tenth was used to calculate chain length distributions fα1(l) and fα2(l), l=1,2, …,300, 
and other relevant quantities (oxygen sublattice occupancies x1 and x2, 3-fold Cu fractions 
n1 and n2, etc.). At a certain number of points we have even used a really large number of 
MC steps, ranging from 105 to 3·105.  
 At all calculated points of (x,τ) space (τ is a quantity that scales with T according 
to τ=kBT/VB 1, where kBB is Boltzman constant and V1 the nearest neighbor O-O interaction 
of the ASYNNNI model) it was obtained [11]  
 2,1,
iaviav
α      ,                                                        (3) 
where lav,αi (i=1,2) denotes the average chain length on the corresponding sublattice (α1 or 
α2). Such a behavior of length distributions fα1(l) and fα2(l) was subsequently explained 
theoretically analyzing microscopic features of the ASYNNNI model lattice 
configurations [12]. In brief terms, the l dependence of probability of a chain to have 
particular length l can be derived in the following way: Consider a sequence of Ny  
oxygen chain sites that are aligned along b axis (Ny is a large number, and the sites are 
connected by copper mediated V2<0 bonds). Let xNy denotes the number of oxygen atoms 
on this column of α sites and let nNy stands for the number of unlike V2 bonds (n is 
therefore the fraction of 3-fold coordinated Cu along the column, and, consequently, 2x/n 
equals to the corresponding average chain length, lav). These xNy oxygen atoms are 
generally divided into (n/2)Ny groups (chains) that can have various lengths l=1,2,… . It 
is useful to recall that an each chain has two ends: one that is oriented towards positive 
side of b axes (the “positive” end) and the other one, oriented towards negative b axes 
(the “negative” end). Among these xNy oxygen atoms there are (n/2)Ny of them that are 
located at the positive chain end, and thus the probability for an oxygen to be lying at the 
positive chain end is equal to ω=n/2x={lav)-1. Consequently, 1-ω is probability for oxygen 
to be located either within the interior of the chain, or at the negative chain end. 
Assuming that chains are created by adding oxygen one by one, starting from the 
negative end, one arrives at the conclusion that the probability for obtaining chain of 
length l is equal to f(l)=ω(1-ω)l-1 [11,12] {such a form of f(l) dependence is known in the 
theory of probability as “geometric” probability distribution [13]). It should be noted, 
however, that a deeper analysis shows that the above reasoning applies only if 
fluctuations of energy of the ASYNNNI model are not too large [12]. Indeed, a certain 
deviation from linear behavior of ln[f(l)] versus l dependence has been found in the 
vicinity of the second order Ortho-I-to-Ortho-II phase transition curve (at x>0.5) [12] but, 
fortunately, such departures were observed only in a relatively narrow intervals Δx≈0.07 
around critical points. Furthermore, our extended analysis (not shown here) shows that in 
the critical regime these deviations, of calculated fα1(l) and fα2(l) dependences from the 
expected behavior, were in a certain way compensated by summations in (2), so that the 
calculated h(x) dependences were obtained to vary smoothly over the transition region at 
all τ=const.  
 The so obtained values of length distributions fα1(l) and fα2(l) for l=1,…,300 were 
inserted into (2) to calculate concentration h of active holes at different points (x,T). The 
geometric-like behavior of f(l) dependences ensures rather fast convergence of sums in 
(2). It should be mentioned, however, that the specific form of length distributions (3) 
makes it possible, instead of evaluating summations in (2) by first 300 terms, to transform 
each of sums into a closed analytical form, so that h would be connected through 
analytical expression with average oxygen occupancies, x1 and x2, the 3-fold Cu fractions 
n1 and n2, and the parameter lcr. Whatever the approach we used the calculated values of 
h were obtained to be practically indistinguishable one from the other (even in the critical 
region of the ortho-I/ortho-II transition), but we nevertheless gave advantage to 
calculating the first 300 terms in estimating sums in (2), as we wanted to keep under 
control the departures of length distributions from (3) that are known to occur in the 
critical regime [12]. In addition, at each point of (x,T) space h was calculated for the 
whole range of values of cutoff parameter lcr spanning from lcr=1 to lcr=50, so that h can 
be regarded as a function of three variables, i.e. h=h(x,τ,lcr) (aside from the fact that it 
also depends upon input parameters that define the ASYNNNI model, i.e. on O-O 
interactions V1>0 (nearest neighbor) and V2<0, V3>0 (next nearest neighbor)). There is 
one remarkable feature of the hole concentration h=h(x,τ,lcr), as defined by (2), that we 
have found while thoroughly analyzing its behavior: whatever the magnitudes of 
interactions V1, V2, and V3, when h is considered as a function of lcr at different points 
(xi,τ) that correspond to the same τ=const and oxygen concentrations xi, i=1,2,… 
spanning over the region of Ortho-II phase, all of these hxi(lcr), i=1,2,… functions 
intersect at a single, well defined value of lcr. This is shown in Figure 1 for interactions 
obtained by linear-muffin-tin orbital (LMTO) method [14] and at three different 
temperatures τ=0.45, 0.38 and 0.30, but the similar behavior we have also obtained (not 
shown here) for the so-called “canonical” interactions V2=-0.5V1, V3=0.5V1. From Figure 
1 it can be clearly seen that at a given τ=const, the value of lcr at which all hxi(lcr) curves 
intersect depends on temperature in the way that it increases with the temperature 
decrease. Thus, at three temperatures τ=0.45, 0.38, 0.30, the intersection values were 
found to be lcr=4(5), 6(7), 11(12), respectively. At a given temperature, the so obtained 
intersection value of lcr we name “the optimal minimal (critical) length” (denoted by 
lcr,opt(τ)) for it is the value at which h(x) stays constant over the regime of Ortho-II phase. 
Such behavior of h(x) at τ=const, for the corresponding lcr,opt(τ), is shown in Figure 2 at 
two temperatures: τ=0.45 (Figures 2a and 2b, for lcr,opt(τ)=4 and 5, respectively) and 
τ=0.30 (Figure 2c, for lcr,opt(τ)=12). From these results it can be seen that indeed h(x) 
demonstrates a constant section at x>0.5 that is even more pronounced at lower 
temperatures.  
 Calculated h(x) dependences were used to obtain doping versus x dependences, 
p(x)=χh(x)/2, that were then inserted into (1) to yield corresponding Tc(x)s (Figures 3a-c). 
The parameter χ was varied slightly around its expected value ≈40% [2] to achieve a 
better correlation between the so obtained Tc(x)’s and those from experiments [15] 
(shown by a solid line). We have indeed obtained h in the plateau regime to be slightly 
lower than 0.5: h=const=0.467 (0.450) at τ=0.45 for lcr=4 (lcr=5), h=const=0.483 at 
τ=0.38 for lcr=7, and h=const=0.495 at τ=0.30 for lcr=12. This gives χ=40.33%, 41.84%, 
38.98%, 3804% for h=0.467, 0.450, 0.483, 0.495, respectively. Thus, not only 
experimental data on doping at x≈1 [2], but also the analysis at x≈0.5 (that falls into the 
regime of 60K plateau) both infer that χ is lying at some point between ≈38% and ≈42%. 
It should be noted that this result disagrees with estimations of Gawiec et al. [16,17] that 
long chains release up to 70% of their holes.  
 Although we used here the ASYNNNI model interactions V1, V2, and V3 as 
obtained by Sterne and Wille [14] the issue of magnitudes of these interactions is still 
open so that new values were subsequently suggested [10]. Regardless of that, it seems 
very well established that nearest neighbor interaction V1 should be ranking around 
6.9mRy [10,14], which fixes scaling between T and τ to Δτ≈0.1⇔ ΔT≈100K. On the 
other hand, as one of the most important features of YBa2Cu3O6+x phase diagram is the 
location of the top of Ortho-II phase along τ axis (at x≈0.5) and that it may well be 
affected by the magnitudes of V2 and V3 (which in fact are not known precisely), perhaps 
the best strategy to estimate the reduced temperature τ corresponding to room 
temperature is to determine the “distance” (in units of τ) between room temperature and 
the top of Ortho-II phase. According to experimental data the top of Ortho-II phase 
corresponds to ≈125-140ºC [5,9], while theoretically obtained phase diagram for the 
LMTO interactions [18] points at τ≈0.58, thus making τ≈0.45 a fairly reliable estimation 
of room temperature.  
The established correlation between room temperature and τ=0.45 renders lcr,opt(τ) 
(that plays the role of lcr in (2)) 4, or 5 (Figure 1a). A better estimation seems to be 5 for 
it yields a somewhat more pronounced 60K plateau than lcr=4 (Figures 3a and 3b), 
despite the fact that lcr=4 would be closer to lcr=3 that was proposed in some theoretical 
studies [16,17,19]. Besides, lcr=5 appears to be well correlated with χ≈2/5 and with the 
basic idea lying in the background of the minimal (critical) chain length concept 
suggesting that one isolated hole cannot efficiently attract an electron, but only a 
combined effect of several holes can achieve this goal. Both lcr=5 and χ≈2/5 imply that 3 
chain-holes are still not enough to effectuate charge transfer, but that the joint impact of 5 
holes suffices to attract two electrons (one per each CuO2 layer).  
 In summary, despite the well known fact that in such highly correlated electron 
systems, as are the high-Tc cuprates, the nature of the controversial pairing mechanism is 
genuinely quantum mechanical, we have shown here that certain aspects of their behavior 
can be explained in terms of classical models. Such is the classical ASYNNNI model that 
successfully accounts for unusual two-plateaus-like behavior of Tc(x) in YBa2Cu3O6+x. 
The obtained Tc(x) dependence is in a remarkable correlation with experiment [15] for 
both lcr=4 and lcr=5, although we believe the later value is more realistic. It should be also 
pointed out that the presented results on Tc(x) dependence are in a qualitative agreement 
with some previous results on the same topic [20,21]. Generally, it can be expected that 
the capability of an active hole to attract an unpaired 3d electron, as expressed by the 
value of χ, should depend upon the density ρe of available electrons immediately above 
(below) the chain (aside from the fact that χ should also depend on a certain coupling 
between chains and planes). Our recently obtained results on Tc(x) in Y1-
b(Ca)bBa2Cu3O6+x (b=0.2) system [22] seem to be lying along this line, since obtained 
χ≈33% can be understood in the light of the fact that introduction of ≈20% of Ca has 
additionally increased doping, and therefore reduced ρe, so that χ attained a lesser value 
than in the parent YBa2Cu3O6+x compound.  
Acknowledgements  
 This work has been funded by the Serbian Ministry of Science and Technology 
through the Project 141014.  
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Figure Captions    
Figure 1. Calculated h(lcr) dependences at τ=const for several values of oxygen 
composition x that span the range of OII phase: a) τ=0.45=const; 0.52<x<0.61 (shown at 
the top), b) τ=0.38=const; 0.52<x<0.62 (shown in the middle section), and c) 
τ=0.30=const; 0.52<x<0.67 (shown at the bottom).  
Figure 2. Calculated values of h as a function of x at τ=const: a) τ=0.45=const, 
for lcr=4 (shown at the top), b) τ=0.45=const, for lcr=5 (shown in the middle section), and 
c) τ=0.30=const, for lcr=12 (shown at the bottom).  
Figure 3, Calculated Tc(x) dependences, using h(x) dependences from the Figure 
2 and thereafter obtained p(x)=χh(x)/2 dependences, that were then inserted into (1): a) 
τ=0.45=const, for lcr=4 and χ=39.4% (shown at the top), b) τ=0.45=const, for lcr=5 and 
χ=41% (shown in the middle section), and c) τ=0.30=const, for lcr=12 and χ=37.6% 
(shown at the bottom).  
-2 0 2 4 6 8 10 12 14
τ=0.30, 0.52<x<0.67
x=0.52
x=0.54
x=0.56
x=0.57
x=0.59
x=0.62
x=0.67
x=0.52
x=0.53
x=0.55
x=0.56
x=0.58
x=0.59
x=0.61
τ=0.45, 0.52<x<0.61
τ=0.38, 0.52<x<0.62
x=0.52
x=0.53
x=0.54
x=0.55
x=0.57
x=0.59
x=0.62
Figure 1  
0.0 0.2 0.4 0.6 0.8 1.0
τ=0.45,  l
τ=0.45,  l
τ=0.30,  l
Figure 2   
0.0 0.2 0.4 0.6 0.8 1.0
τ=0.45,  l
χ=0.394
τ=0.45,  l
χ=0.410
τ=0.45,  l
χ=0.376
Figure 3
