Acta Astronautica 63 (2008) 1372 – 1375
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ACADEMY TRANSACTIONS NOTE
Correlation between oxygen excess density and critical transition
temperature in superconducting Bi-2201, Bi-2212 and Bi-2223
H.P. Roesera,∗ , F. Hetfleischa , F.M. Huberb , M.F. von Schoenermarka , M. Steppera ,
A. Moritza , A.S. Nikoghosyanc
a Institute of Space Systems, Universitaet Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany
b Steinbeis-Transferzentrum Raumfahrt, Roetestr. 15, 71126 Gaeufelden, Germany
c Department of Microwave and Telecommunication, Yerevan State University, Alex Manoogian 1, Yerevan, 375025, Armenia
Received 19 May 2008; accepted 4 June 2008
Available online 21 July 2008
Abstract
It is suggested that oxygen excess in undoped material of the high temperature superconducting bismuth family creates periodic
two-dimensional Cu3+ -ion patches in the copper–oxygen planes. These nanostructures have a size of 3.6 nm square and show
a strong linear relationship to their critical transition temperatures Tc .
© 2008 Elsevier Ltd. All rights reserved.
Keywords: High temperature superconductor; Layered cuprate; Nanostructures
1. Introduction
Many high temperature superconductors (HTSC)
are layered cuprates consisting of one or more
copper–oxygen (CuO2 ) planes in the crystallographic
unit cell. They are separated by other planes of insulating rare-earth elements or other oxides. For more layerlike cuprates it has been demonstrated that Tc increases
when the number of CuO2 planes containing Cu3+ -ions
is getting larger at least up to 3 or 4 CuO2 layers per
chemical formula. As an example, the bismuth-based
copper oxide superconductors called Bi-family have a
transition temperature up to ∼ 110 K for three layers. In
this paper we will analyze single-layer Bi2 Sr 2 CuO6+
∗ Corresponding author. Tel.: +49 711 685 62375;
fax: +49 711 685 63596.
E-mail address: [email protected] (H.P. Roeser).
0094-5765/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.actaastro.2008.06.001
(Bi-2201), double-layer Bi2 Sr 2 CaCu2 O8+ (Bi-2212)
and triple-layer Bi2 Sr 2 Ca2 Cu3 O10+ (Bi-2223) materials without traces of additional atoms.
2. Bi2 Sr2 CuO6 (Bi-2201)
For Bi-2201 six O-atoms need 12 electrons which
are provided by 2Bi3+ + 2Sr 2+ + 1Cu2+ . This material is not superconducting. If an extra O-atom is
added in the overdoped case O6+ , residing between
the BiO planes, it needs two more electrons which are
provided by transforming 2Bi3+ into 2Bi4+ -ions because Bi has the lowest ionization energy compared
to the other ions. The electronic arrangement 2Bi4+ +
2Sr 2+ + 1Cu2+ + 7O2− is not superconducting either
because of the missing Cu3+ -ions in the CuO2 plane.
The best way to create Cu3+ -ions is achieved by replacing Sr 2+ -ions by trivalent ions like La3+ (Bi-2201-La
with Tc = 37 K) [1].
H.P. Roeser et al. / Acta Astronautica 63 (2008) 1372 – 1375
3. Bi2 Sr2 CaCu2 O8+ (Bi-2212)
The unit cell of Bi-2212 (Fig. 1) has an orthorhombic form with dimensions a ≈ b = 0.544 nm and c =
3.090 nm [2] which consists of 2·[Bi2 Sr 2 CaCu2 O8 ] and
has 15 planes. In one chemical formula eight O-atoms
(8O2− ) need in total 16 electrons which are provided by
2Bi3+ + 2Sr 2+ + 1Ca2+ + 2Cu2+ . This material is not
superconducting unless it is overdoped with O-atoms in
the range = 0.10.0.23 [3,4]. The extra O-atom needs
1373
two more electrons which are provided by transforming
2Cu2+ -ions into 2Cu3+ -ions. The electronic arrangement of the unit cell with an oxygen excess (2Bi3+ +
2Sr 2+ + 1Ca2+ + 2Cu3+ + 9O2− ) forms two superconducting CuO2 planes in which the hole doped cuprates
work with an effective mass of Meff = 2me . The highest
transition temperature published so far is Tc = 83 ± 1 K
[1,3–5] with an oxygen excess of = 0.18. A maximum
transition temperature in the range of 90–96 K for socalled “optimized doping” levels with a value of about
= 0.16 has been measured. But in these cases the material has been additionally doped with a small amount
of yttrium or other atoms [1,3–6]. These materials will
be discussed in a later paper.
4. Cu3+ -ion and oxygen excess positions in the
CuO2 plane
Fig. 1. The unit cell of Bi-2212 consists of 2·[Bi2 Sr 2 CaCu2 O8 ] and
has 15 planes with four CuO2 planes. Note the alternate displacement
of the stacking sequence for pairs of CuO2 planes.
For cuprates, it is believed that the Coulomb potential
caused by excess O-atoms pins the doped holes which
will distribute themselves in the CuO2 planes so as to
minimize their total energy. And it is stated that the excess oxygen is responsible for superconducting in the
CuO2 plane. In a previous paper [7], oxygen deficiency
values have been transformed into distance values assuming a uniform density of deficiency positions in the
CuO2 plane by introducing a unit area for one doping
element. Now the same analysis will be used for an
oxygen excess case. The excess doping density
is given
√
by −1 and the doping distance is x = · a. Here
the value gives the number of Bi-2212 unit cells per
unit area for one oxygen excess atom. For an oxygen
excess value of = 0.18 we achieve a doping density of
−1 = (1 − 8.00/8.18) ≡ 2.2% which leads to a doping
distance of x = 6.74 · a = 3.67 nm in each CuO2 plane.
Fig. 2 shows a planar configuration of the Bi-family
crystal with a cross section in a CuO2 plane. The position of an excess O-atom has been projected into the
CuO2 structure and might be above or below the CuO2
plane as nicely illustrated in [1], which describes a
classification of HTSCs, including the Bi-family by a
disorder effect. It is likely that the O-atom is sitting
between two adjacent CuO2 planes attracting one electron from each plane. This makes it possible for the
electron flow to be in the direction of the connecting
line between two oxygen excess atoms and the hole flow
to be in the opposite direction. The distance between
two neighboring Cu-atoms in the designated
direction
√
is given by z = [(2a)2 + a 2 ]1/2 = a · 5 so that the
above determined
√ √ doping distance leads to x =6.74·a ≈
3·z=3·a· 5= 45·a=3.65
nm with an integer number
√
times the distance a · 5. This means that the distance
1374
H.P. Roeser et al. / Acta Astronautica 63 (2008) 1372 – 1375
Fig. 2. Superconducting CuO2 plane of Bi-2212 (a≈b=0.544 nm). The distance between oxygen excess atoms is the superconducting resonance
length x=3z. The superconducting unit cell has the size of 3z · 3z and covers (62 +32 )=45 CuO2 unit cells resulting in a density of 2.2%.
x is given by x 2 = (z12 + z22 ) · a 2 = 45a 2 = (62 + 32 ) · a 2
with z1 , z2 ∈ N. The pattern of the Bi-2212 unit cells
containing an excess O-atom is in agreement with the
suggestion [1] that a nanoscale electronic inhomogeneity exists as a self-organization process leading to twodimensional “patches” in Bi-2212 being 1–3 nm across:
The vertical stripes (Fig. 2) are separated by six unit
cells equal to 3.2 nm and the horizontal stripes by three
unit cells equal to 1.6 nm, respectively.
5. Bi2 Sr2 Ca2 Cu3 O10+ (Bi-2223)
The bismuth compounds Bi-2201and Bi-2212 have
been intensively studied, because they can be prepared
as stable phases and in relatively pure form ([1,2] and
references therein). Crystals of the Bi-2223 phase on
the other hand have not been prepared as an absolute
pure single phase, but to our knowledge only in the form
of an intergrowth in Bi-2212. The larger the number
of CuO2 layers the more difficult it is to produce a
single phase that is free of intergrowths. Therefore, it
is easy to understand that published values of Tc vary
between 106 and 110 K. The lower range is a result
of a mixture of Bi-2212 and Bi-2223 phases, and the
upper range by additionally doping Bi-2223 with crystal
stabilizing atoms like yttrium. Therefore, we estimate
that 108 ± 2 K is most likely the correct temperature
for pure Bi-2223 with the same doping concentration as
Bi-2212 resulting in a value for = 0.22. This results
in the same distance x between two excess O-atoms as
Bi-2212 and Fig. 2 also applies to Bi-2223.
6. Discussion
The calculation of the oxygen excess positions is only
applicable for one single (n=1) superconducting CuO2
plane. In a previous paper [7], it has been suggested
comparing the linear correlation function in Fig. 3 with
the energy of quasi-free moving particles like the Fermi
energy EF . In this case, the energy is connected to
the carrier density Nc by EF ∼(Nc )2/3 . Increasing the
number n of superconducting CuO2 planes means increasing the carrier density by the same factor so that
the energy increases by EF ∼(n · Nc )2/3 . Because the
expression (Nc )2/3 has the dimension of (area)−1 the
correlation will be extended by plotting (2x)2 · n−2/3
versus 1/Tc . With this assumption, the critical transition temperature should increase with the number of
CuO2 planes containing Cu3+ -ions by Tc ∼n2/3 . The
results of the Bi-family are summarized in Table 1
and illustrated in Fig. 3 together with data on other
HTSCs [7]. Results on very recently discovered ironbased HTSCs are also plotted, but are analyzed in detail
in [8]. The data points fit on a straight line with slope of
m1 =2.77×10−15 m2 K, and the correlation for HTSCs
[7] could be written in the form
(2x)2 · n−2/3 · 2Meff · kT c = h2
(1)
The result supports the idea of a resonance effect
between the de Broglie wavelength DB of the paired
superconducting particles and the structure of the crystal given by DB =2x [7]. Fig. 3 could be considered
a phase diagram with the superconducting state below
the straight line. But more HTSCs need to be investigated to see if the same correlation and analysis can be
verified for HTSCs in general [8].
It is interesting to note that Eq. (1) with n = 1 is very
similar to the basic Bose–Einstein–Condensate (BEC)
equation. The difference is a form factor which is given
by (2.612)2/3 = (DB )2 · N 2/3 , where N is the density of the atoms and Meff is replaced by the atomic
H.P. Roeser et al. / Acta Astronautica 63 (2008) 1372 – 1375
140
PLCCO
m1 = 2.77 · 10-15 [m2K]
NCCO
z = m1 · y + 0.3 · 10-18 [m2]
120
1375
z = (2x)2 · 10-18 [m2 ] · n-2/3
GOFFA
100
LSCO
LOFFA
LBCO
Bi-2201-La
80
COFFA
POFFA
GAFO NOFFA
60
Y123-0.45
Tl-1212
Bi-2212-Y91
40
Bi-2212-Y95
Bi-2212
Bi-2223
20
Y123-0.04
Tl-1234
0
Tl-1201
SOFFA
0
5
10
15
20
25
30
35
40
45
50
y = Tc-1 ·10-3 [K-1]
Fig. 3. Crystal geometry factor (2x)2 of the superconducting plane versus the inverse critical transition temperature of different HTSCs. The
value n represents the number of superconducting planes per chemical formula in the unit cell containing Cu3+ -ions in cuprates or Fe2+ -ions
in iron-based HTSCs.
Table 1
Correlation data for cuprates of the Bi-family with n = 1, 2, 3 CuO2 layers
Material
Tc (K)
Lattice structure (nm)
Carrier distance x (nm)
(2x)2 · n−2/3 10−18 (m2 )
No Cu3+ -ions in pure
Bi-2201
3.65
3.65
33.5 (n=2)
25.6 (n=3)
Bi2 Sr 2 CuO6+
Bi-2201
NO
a≈b=0.544, c=2.5
Bi2 Sr 2 CaCu2 O8.18
Bi2 Sr 2 Ca2 Cu3 O10.22
Bi-2212
Bi-2223
83 ± 1
108 ± 2
a≈b=0.544, c=3.1
a≈b=0.544, c=3.7
mass. The different form factor might come from the
fact that in HTSCs we have a motion of paired particles
in “solid” state devices instead of an atomic gas with a
self-forming “soft” lattice structure.
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