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The order disorder phenomenon in YBa2Cu3O7_ δ for varying δ and the possible existence of two ordering
temperatures
M.T. Béal-Monod
To cite this version:
M.T. Béal-Monod. The order disorder phenomenon in YBa2Cu3O7_ δ for varying δ and the possible existence of two ordering temperatures. Journal de Physique, 1988, 49 (1), pp.103-110.
�10.1051/jphys:01988004901010300�. �jpa-00210664�
The order disorder phenomenon in YBa2Cu3O7_ 03B4 for varying 03B4
and the possible existence of two ordering temperatures
M. T. Béal-Monod
Laboratoire de Physique des Solides, Bâtiment 510, Université de Paris-Sud, 91405 Orsay, France (Requ le 23 octobre 1987, accept6 le 3 novembre 1987)
Résumé. 2014 Nous examinons un modèle d’ordre atomique à longue distance dans YBa2Cu3O72014 03B4 (0 ~ 03B4 ~ 0,5 ), à température T variable, en particulier pour T inférieure à la température de transition
To séparant les phases tétragonale et orthorhombique. Une description simple en champ moyen est proposée
avec deux paramètres d’ordre et des interactions entre premiers et seconds voisins pour rendre compte d’un ordre à longue distance dans les plans de base CuO déficients en oxygène. Cette description est la transposition
à deux dimensions de celle donnée autrefois pour les alliages cubiques centrés Fe1-xAlx et semble appropriée
ici d’après des résultats expérimentaux récents. Selon un tel schéma, une deuxième température d’ordre T’o est attendue en dessous de To ; elle croît avec 03B4 tandis que To décroît. L’observation de T’o a pu passer
inaperçue dans les expériences à refroidissement rapide, puisque l’équilibre est très long à s’établir à une
température d’ordre. Récemment des expériences utilisant des cycles de recuit lents semblent en effet suggérer
l’existence d’une seconde température d’ordre, d’autant plus importante qu’elle s’accompagne, expérimentale-
ment, d’une augmentation de la température de supraconductivité Tc.
Abstract. 2014 We examine a model of long range atomic ordering in YBa2Cu3O7201403B4 (0 ~ 03B4 ~ 0.5 ), when the temperature T varies, in particular below the tetragonal-orthorhombic transition temperature To. A simple
mean field description with two order parameters and interactions between first and second neighbours, is proposed for the ordering in the oxygen deficient basal CuO planes. It is the two dimensional analogue of the
one given in the past for b.c.c. Fe1-xAlx, alloys and seems appropriate here according to recent experimental
results. According to this scheme, a second ordering temperature T’o is expected to occur below To and to increase with 03B4 while To decreases. The observation of T’o may have been missed in fast cooling experiments so far, since equilibrium requires a long time to be reached at ordering temperatures. Recently experiments performing slow annealings appear to indeed support the existence of that second transition, all the more important that it is experimentally accompanied by an increase of the superconducting temperature Tc.
Classification
Physics Abstracts
74.90 - 64.60C - 61.50K
1. Introduction.
The system YBa2Cu307 - 8 [11 ] has become within a few months the archetype one for reaching the highest (so far) superconducting temperature Tc.
The structure of such a system has been established
by various methods [2, 3]. For 0 -- 5 -- 0.5 these systems are metallic and superconducting (they
become insulators at 5 -- 0.5). A transition between
a high temperature tetragonal disordered phase and
an ordered orthorhombic one has, in particular,
been shown [3] ; at room temperature, typically, the systems are in the orthorhombic phase.
In the present paper we study for varying tempera-
ture a possible phase diagram for such systems in particular inside the orthorhombic phase. We em- phasize the possibility of having two ordering transi-
tions at To and T.’ (not to be confused with T,), in the oxygen-vacancy system when 5 varies from 0 to 0.5 by analogy with b.c.c. Fel-,,Al.,
metallic alloys. We examine more precisely the
Cu-0 planes, between the Ba ones, where it is believed that the action of 5 takes place [2, 3]. One
thus assumes that the other Cu-0 planes, between
the Ba and the Y ones, remain completely filled with oxygen, when 6 varies, and we ignore them in the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004901010300
104
following. The Cu-0 planes of interest here are then rather far apart and are considered to be indepen-
dent so that the problem reduces to a two dimension-
al one.
To fix ideas and for reasons given shortly, we
present, in figures 1, 4 completely ordered (at
T = 0) plausible configurations corresponding to 4
different values of 6. For the time being let us
consider them just as common sense possibilities :
- for 8 = 0 (Fig. la), one has horizontal rows of
alternating « Cu » (.) and « 0 » (o) separated by
rows of empty sites (« e » sites (0)). These empty
site rows are supposed to remain unchanged all the
way from 5 = 0 to 6 = 0.5 (at T = 0 K), so that only
the CuO rows are alterated when 5 varies in the above range. We thus have, for 6=0, and if we ignore (for the moment) the Cu sites, a two dimen-
sional (2d) square [4] - centred lattice (S-C) of the
O-e system, with 50 % of « 0 » and 50 % of « e ».
We may also decompose this (S-C) lattice into two
interpenetrating simple squared sublattices (S-S),
one full or « 0 », the other one full of « e ». If N is the number of « 0 » plus « e » sites, there are
cN = N/2 «0» and (1 - c) N =N/2«e»onthe (S-C) lattice with c = 1/2. On the (S-S) oxygen sublattice alone, one thus have c’ N’ « 0 », with
N’ = N/2 and c’ = 2 c = 1, and, of course, (I - c’ ) = 0 oxygen vacancy « v ». c is related to 8
by c = (1 - 5 )/2 :
Fig. 1. - Four possible ordered states of the basal plane
of CuO : the small dots are the Cu atoms, the white circles
are the empty sites « e », the black circles are the oxygen atoms « 0 », the crossed circles are the oxygen vacancies
« v » ; (a) 8 = 0, C = 1/2 ; (b) 8 = 0.125, C = 7/16 ; (c)
5 = 0.25, C = 3/8 ; (d) 8 = 0.5, C = 1/4 (C is the oxygen concentration).
- when 6 increases from zero (Fig. Ib is for
6 = 0.125), and compared to the above state, one supposes that one horizontal row out of two starts to contain an increasing amount of oxygen vacancies
« v », regularly spaced (Q) (we will come back to
this point later in connection with the experiments of
Ref. [5]), the other rows of Cu-0 remaining filled
with « 0 » until 5 reaches 0.25. In the example of figure 1b (6 = 0.125), one has 7 N116 « 0 » (c = 7/16), N12 « e » and N116 « v » ; alternatively, on
the (S-S) sublattice of 0-v, one has a concentration c’=7I8 of «0» and (1-c’)= 1/8 of «v»;
- for 5 = 0.25 (Fig. Ic), the rows containing
« v » will exhibit the sequence 0-v-O-v-O-v. There
are 3 N18 0 >> (c = 3/8), N12 e >> and N18 v >>.
Obviously on the (S-S) sublattice of 0-v, one has
c’ = 3/4 « 0 » and (1 - c’) = 1/4 « v ». When
6 > 0.25, the O-v-0-v... rows are supposed to
remain unchanged but the other rows, so far filled with « 0 », start to exhibit regularly spaced « v »
until 5 = 0.5 ;
- for 6 = 0.5 (Fig. 1d), both kinds or rows
exhibit the sequence O-v-0-v. There are N14 « 0 » (c = 1/4), N/2 « e » and N/4 « v ». On the (S-S)
sublattice one of 0-v, one has concentration c’ = 1/2 of « 0 » and (I - c’ ) = 1/2 of « v ». One could as
well have the equivalent structure of alternating
rows of 0 and v (one row of Cu-O-Cu-0... with no
« v », the next one, of Cu-v-Cu-v... with no « 0 »
etc...).
A that stage, there are, a priori, several other possibilities for a regular arrangement of this 2d lattice. However a number of arguments appear to favor the one presented here. Electron diffraction data of reference [5] proves that the vacancies,
introduced when 5 =A 0, do order regularly ; actually figure 10 of reference [5], proposing a structure possibly compatible with the data is the one that we
give here as figure lb. On the other hand, let us
consider figure la and figure 1d of the present paper : these 2d structures are the exact 2d analogues
of the ordered structures in the 3d b.c.c. FeAl and
Fe3AI alloys, respectively (belonging to the so called
B2 and D03 types). More precisely, figure la and figure 1d identify with the planar projections of the
b.c.c. structures in FeAl and Fe3Al, with the « 0 »
and the « e » (or « v ») playing respectively the roles
of Al and Fe. However the phase diagram of Fel-.,Al,, is complicated by magnetic effects, so that
atomic as well as magnetic orderings, with inter-
dependence of both, are involved. A qualitative description of such alloys was given in reference [6],
within mean field theory. However an earlier paper
by Rudman [7] (hereafter referred to as (R)), ignored the magnetic effects and appears more
appropriate to compare with our present case.
2. The two dimensional long range ordering in the
CuO planes.
In this section we show how the perfectly ordered
structures proposed in the introduction evolve at finite T. We follow closely the 3d description of the long range order given in (R) for Fel-,Al,, and
indicate the modifications required by the 2d charac- ter of the CuO planes, when needed.
In (R) a mean field (Bragg-Williams [8]) calcu-
lation was performed for the ordering in b.c.c.
Fel-,,Al,,, with interactions between first, second
and third neighbours and two ordering parameters
(Sll and S3,). Actually the only difference between the 2d case of interest here and the 3d one studied in
(R), arises from the different numbers of neighbours
on the various shells around a given atom, so that
different linear combinations of the first, second and
third neighbour interactions are involved. We then follow (R) step by step. Those readers who are
familiar with order-disorder phenomena in alloys
may escape most of this section which would be trivial for them and jump to formulas (15) and (20).
But for those who are not expert in that field, we
find it pedagogical to present the main lines of the
(R) paper adapted here to 2d. (R) decomposed the Fel -.,Al., lattice into 4 interpenetrating sublattices,
two of them being always identical, the « sites in (R), identified here with the empty sites « e » (see Fig. 2). The other two sublattices are the j6 and y
Fig. 2. - The three types of sites a, /3, y, as described in the Appendix (following Ref. [6]) and necessary to de- scribe the structure of the 0-(v, e) system.
sites, identified in our case with the « 0 » and « v » of the NaCI type of structure of the cubic (here squared) lattice of f3 + y sites. The analysis of the composition of the a, f3, y lattices is identical to the
one given in (R) : we too have Na = N/2 sites a,
sites respectively (see Fig. 2 and Fig. Id). Like in the
Fe} - xAlx alloys the change of order at T = 0 and varying x (or 8 here) is equivalent to the disordering Np = N /4 sites (3 and Ny = N /4 sites y. The composition of a particular lattice is defined as the number of oxygens on the particular lattice divided by the number of lattice sites on that lattice :
[a ], [J3] and [y ] represent the composition of the a, ,8 and y lattices. Like in (R) we write :
= nb. of oxygens (1)
(1) and (2) imply :
Then, following (R), we introduce two long range order parameters Sll and S31 as follows :
One thus gets :
Then one computes the energy of the system accord- ing to the zeroth approximation of the quasi chemical
bond theory (we will discuss shortly more elaborate methods). Instead of having like in (R) in 3d, 8 first neighbours, 6 second neighbours and 12 third
neighbours, we have in 2d, 4 first neighbours,
4 second neighbours and 4 third neighbours (note
that these will be followed by 8 fourth neighbours etc...).
With Eoo, E’O, E"O..., the interaction energies
between two 0 >> respectively first, second neighbours and third neighbours, one has :
where f refers indistinctly to « e » or « v ». Then the bonding energies due to first, second and third
neighbour interactions are in our particular case of
106
the CuO planes :
Then the free energy is :
The combinatorial problem involved in the calcu- lation of the entropy S is given as usual [8] in the
zeroth approximation and using Stirling’s formula (aside from a constant term of no interest in the
following) :
Then the order parameter Sil and 531 are determined by the minimization of the free energy :
We get :
(T stands for kB 7) -
S31 = 0 is a solution of (12) in which case (11)
reduces to :
which gives the variation of Sll with T. As pointed
out in (R) this is the solution for a CsCI structure
(the B2 type) which undergoes a second order transition for all compositions, with an ordering
temperature To obtained from :
which in our 2d case gives :
For To to exist one must have a « repulsive »
combination :
To is in (R) the disorder-B2 transition temperature, identified here with the tetra-ortho transition tem-
perature observed in reference [3].
Next we make the same assumption that (R) did of
a second order transition between the FeAl(B2) and
the Fe3Al(DO3), types of ordering ; then :
which defines a second ordering temperature To separating the B2 structure (T.’ -- T To ) from the D03 one (T TO):
where Sll is obtained from (13) when T = T.’ given by (18).
Again for T.’ to exist, we need a « repulsive »
combination :
« repulsive » has the usual sense of unlike atoms
attracting each other while atoms of the same kind
repell each other [9].
In (R) a typical variation of Sll and S31 with T was
drawn (Fig. 6 of (R)), for Fe3Al. We would have a similar one here. (Note that both curves may or not interact depending on the concentration.)
(R) remarked that for C :> 0.25,
S’, - max Sll = 2 C. Since we are interested here by 0 -_-- 5 -- 0.5, i.e. 0.25 --::: C = 0.5 we may reasonably
use this approximation. Then we get :
and :
Now, according to (R), including or not the third neighbour interactions did not change much the results, whereas the second neighbour interactions
are crucial in order to get the D03 type of ordering
aside the B2 one. Similarly here we retain, in the following, only V1 1 and V2, and putting :
we have :
In order to have T.’ To (to account for recent experiments as will be discussed in the next section)
we need max
This yields:
We are then able to draw a phase diagram from To and T.’ versus C (0.25 , C , 0.5 ) analogous to figure 5 of (R) and discuss the above results.
3. The phase diagram of the 2d oxygen-vacancy system in the CuO planes.
The overall qualitative picture is similar to the one in
(R) for 3d b.c.c. Fel -,,Al., alloys : a typical phase diagram, for particular ratios of the various interac-
tions, is given in figure 3, and similar to figure 5 of (R). For decreasing temperatures, and fixed concen- tration (i.e. for a given 5), one first goes from the disordered state, at temperature T>- To, to a B2
type of order at T To ; To is here associated with the tetra-ortho transition of the YBa2CU307-,,. At
still lower temperature, one then finds a second transition at T.’ separating the B2 phase from a D03 one, at T = To. This second transition is the
key point of the present paper.
According to such a simple minded view point,
one could expect two ordering temperatures in
YBa2CU307-,, for varying 5, To and To with T.’ To [10] and T.’ being the highest for 5 closest to 0.5. The
existence of such a second transition temperature smaller than the tetra-ortho one is indeed suggested
in recent experiments [11]. These experiments are particularly exciting since they exhibit an increase of the superconducting temperature Tc typically from
95 K to 130 K. The experimentalists of reference [11]
associated such an increase of Tc with the occurrence
of a particular ordering occurring around 240 K.
They quote previous experiments which mentioned
an increase of Tc but which did not seem reproduc-
ible. In reference [11] a slow temperature cycling
appears to be crucial to allow such observations. If
our model is correct, i.e. if To indeed exists, then it is quite understandable that fast cooling experiments
so far, have missed the effects associated with
T.’, since equilibrium requires a long time to be
reached at ordering temperatures, as would be the
Fig. 3. - A typical possible phase diagram for 0.25 -- C * 0.5 exhibiting two ordering temperatures, with To T,,. The upper curve is the tetra-ortho transition temperature To(C ) determined in the text from the data of reference [3]. The other curves are the To ones for
different values of the ratio V2/Vl = A : (a) A = 0.4 ; (b)
À = 0.33 ; (c) A = 0.315 ; (d) À = 0.25 ; (e) A = 0.20.
The region T"> To is the disordered phase, T.:::. T>.
T.’ is the proposed B2 type ordered phase, To’ :. T is the proposed DO3 ordered phase. The dotted lines correspond
to To = 240 K (the structural transition suggested in
Ref. [11]), for C = 7/16 whose ordered structure is given
in figure 1b and mentioned in reference [5] (as noted in the text this attribution is only tentative).
case at To. Instead, the slow temperature cycling of
reference [11] is much more favorable. Of course, the main problem would then be to relate the increase of Tc with the existence of To. This point
will not be discussed here, where we confine to the
first step of the scheme, namely the existence (or absence) of To.
In the crude phase diagram of figure 3 (analogous
to Fig. 5 in (R)), we have been able to draw the upper parabola (the tetra-ortho transition or the
