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Correlation between the superconducting transition temperature and crystal structure of high-Tc cuprate compounds
PhysicaC 166 (1990) 133-139 North-Holland
CORRELATION BETWEEN THE SUPERCONDUCTING TRANSITION CRYSTAL STRUCTURE OF HIGH-T, CUPRATE COMPOUNDS D.M. de LEEUW,
W.A. GROEN,
L.F. FEINER
TEMPERATURE
AND
and E.E. HAVINGA
Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, The Netherlands Received 20 January 1990
For the various p-type cuprate superconductors we have calculated the formal valence for the copper and the oxygen ions in the central Cu02 planes from bond lengths according to Zachariasen rules. It is shown that in all structures these values correlate remarkably well with the maximum critical temperature. The correlation found shows that T,,,, increases the more the holes in the Cu02 planes prefer the oxygen sites over the copper sites. In a correlated electron picture this implies a higher value for U-A+ W/2.
1. Introduction Phase diagrams that show the superconducting critical temperature, T,, as a function of the density of holes in the CuOZ planes have been reported for YBa2Cu30, [ 11, LazCuOd [ 21, BiZSr2Cu06+6 [ 31 and BizSrzCaCuzOs+g [ 41. These phase diagrams show an apparently universal behaviour for all p-type high-T, cuprate superconductors, viz. the superconducting boundaries in the diagrams are broad domeshaped curves and the maximum T, is obtained at similar doping levels of about 0.2 holes per copper per CuOl plane in all cases. However, the maximum values for the critical temperatures themselves are significantly different, viz. 30 K for BiZSr2Cu06+6, 35 K for Lal.&,&u04, 80 K for Bi2Sr2CaCu208+6 and 93 K for YBa&&O,. The differences in the maximum values of T, can as yet not be understood from first principles, because detailed knowledge about the electronic and magnetic structure is lacking, and the mechanism of superconductivity is still unknown. For the time being rationalization of the data has to proceed by a semi-empirical approach. There is a general agreement about the key role of the nearly planar CuO, entities in the structure. Since the doping levels (i.e. the total charges in the CuOl planes) are similar in the various series of compounds, the differences in Tc,maxwill originate from difirent distributions of 0921-4534/90/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland )
charge within the CuOZ planes, i.e. from different electronic structures. A semi-empirical description will involve concepts like covalency, electronegativity, bond strength and polarizability. As changes in crystal structure are correlated with changes in electronic structure, one expects that T_= may be correlated with the crystal structure. To this end appropriate parameters that characterize the crystal structure must be defined, preferably such that connect naturally with the description of the electronic structure. Recently, various attempts for such a parametrization have been suggested. For instance, to explain the values of T,,,,, in the series and T12Ba2Can-1Cu,0Zn+4+6 with n=l, 2 and 3 it has Bi&Xa,&,OZ,,+~+~ been proposed that T,,,,, increases in a particular way with n, the number of Cu02 planes [ 51. Although in each series T,,,,, increases with n, this parametrization is not sufficient because it does not account for the difference between the n = 1 compounds T12Ba2CuOg+d with T,,,,,=SO K [6] and Bi2Sr2Cu06+6 with T,,,,, = 30 K [ 3,7 1. Another proposal concerns a correlation between the critical temperature and the in-plane CuO bond lengths [ 8 1. Three classes are distinguished, depending on which ion, La3+, Ba2+ or Sr2+ occupies the nine-fold coordination site connected with the apical oxygen ions of the CuOZ layers. Each class, then, has to be sub-
134
D.M. de Leeuw et al/T, and crystalstructure
divided again according to the number of CuO, planes per unit formula. The changes in T, reported by these authors actually reflect changes in the hole density, not changes in a structural parameter. The same holds for the proposed relation between T, and average electronegativity of the constituent ions of the unit cell [ 91. A clear correlation of T,,,,, with crystal structure has not yet been reported. A well-suited framework to discuss correlations with crystal structures, without implying the physical mechanism that produces them, is the bond valence model of Zachariasen [ IO- 121. This model is particularly useful to discuss bonding in cases where the coordination is very distorted such as for Cu2+ [ 13 1. It takes into account the number of bonds, weighing their importance according to bond length. This model can be used in assessments of the correctness of crystal structure determinations, for estimations of chemical activity and for predictions of charge transfer reaction mechanisms [ 13 1. Here, we apply the bond valence model to relate the crystal structure with T,,,,, for the series of ptype high-T, cuprate superconductors. The crystal structure is parametrized by bond valence sums for the copper and oxygen ions located in the Cu02 planes. We then show that these bond valence sums correlate unambiguously with T,,,,,. Finally, the implications of this relation are discussed.
2. Bond valence sums of copper and oxygen The bond valence, s,,, of a cation and anion at distance R,, is defined by [ l&l 1 ] : ~,,=ew(&-R,,IBl.
(1)
The constant B can be taken independent of the type of constituent ions and typically a value of 0.31 A is applied [ 111, while the reference distance R. does depend on the type of cation and anion that form the bond. It has been shown [ 121 that the values for R,, can be determined in such a way that the sum of bond valences around any ion is close to the expected formal oxidation state of that ion: v,= c s,,= 1 exp(R,-RJ0.37) I I In calculating
the bond valence sum of a cation, the
qfcuprate
cornpoundv
summation is over the anions and vice versa. Because of the strong dependence of the bond strength on the bond length only the first coordination sphere has to be taken into account. The parameter values for R. have been determined by Brown and Altermatt [ 111 for 750 ion pairs using the Inorganic Crystal Structure Database. Here we use eq. (2 ) to calculate the formal oxidation states of copper and oxygen in the p-type high-T, superconducting cuprates. The absolute values calculated for the oxidation states strongly depend on the parameter values Ro. Especially, the value of R. is different for different nominal oxidation states [ 1 1,141 (e.g. Cu’+ or Cu’+ ), and hence the calculated bond valence sum depends on the reference nominal oxidation state [ 141. However, the reported phase diagrams, presenting 7, as a function of hole density, show that for all superconducting cuprate compounds the maximum T, is obtained at similar hole densities of about 0.2 holes per copper per CuOz plane (with respect to cu2+ and 02-) [ l-41. Therefore, in the present calculations we use the fixed values for R. pertaining to cu2+ and 02- from ref. [ 111. As a consequence, the absolute values for the bond valence sums should be regarded with some reservation. The results of the calculations are intended to show trends. The calculated bond valence sums for copper and oxygen ions that are located in the CuOz planes of the high-ir, superconducting cuprate compounds are presented in table I. For copper the total bond valence sum is shown as well as the separate contributions from the in-plane and the apical oxygens. Table I also presents values for T,.,,,, the highest critical transition temperature of a certain type of compound. For YBa2Cu30,, La2Cu0,, Bi2Sr,CaCuzOs+~ and Bi2Sr2CuOb+,- values of T,.,,, are taken from phase diagrams that show Tc as a function of hole density [ l-41. For the other compounds T,.,,, is a typical value reported as the maximum T,. We note that until the complete phase diagrams are available, these values for T,.,,, are only approximate and may be too low. For instance the T, of YBazCu40s was increased recently from 80 K to 90 K by replacing 10% of the yttrium ions by calcium ions [IS]. Also in Y2Ba4Cu,0,4.3 the T, has been increased from 40 K to 93 K by partial replacement of yttrium by calcium [ 161.
D.M. de Leeuw et al./T, and crystal structure of cuprate compounds
135
Table I Summary of calculated bond valence sums of copper, VC,, including the contribution of the in-plane and apical oxygen ions, VCu,,n_pl~c respectively, calculated bond valence sums of oxygen, V,, and maximum values of the critical temperature, T,.,,,, for and VCu,oY1-pl.ne high-T, cuprate superconductors Compound YBa2Cu408
Vcu
VO
ref
2.03 1.99 2.04
[1X241
90
1.935
0.187
2.12
93
1.936
0.197
2.13
[16,251
2.04
YBa$&O, YBaSrCu,O, Lal.8Sr0.2Cu04
PbzSr,(Ca, Y)Cu308 Pb%8h2CU206+6 Pbo.sSr2.5Cao.sYo.sCu207-a a,
T12Ba2Cu06+d T12Ba2CaCu208+d Tl2BaKaKuG&+~ TlBa2CaCu20,_6 TlBa2Ca2Cu,0g_d TlBa2Ca3Cu40,, _-d (Tl, Pb)Sr2CaCu20,_d (Tl, Pb)Sr2Ca2Cu309_d Bi2Sr2CuOe+,r Bi2Sr2CaCu208+d (Bi, Pb)2Sr2Ca2Cu3010+6
93
1.958
0.190
2.15
80
2.049
0.197
2.25
36 70 26 50
2.264 2.047 2.142 2.142 2.142 2.013 2.042 2.055 2.085 2.077 2.057 2.185 2.141 2.201 2.119 2.131
0.263 0.194 0.326 0.172 0.265 0.122 0.063 0.0 0.124 0.0 0.0 0.127 0.0 0.175 0.073 0.0
2.53 2.24 2.47 2.31 2.41 2.13 2.11 2.06 2.2 1 2.08 2.06 2.31 2.14 2.38 2.19 2.13
80 110 125 70 125 125 85 122 30 85 110
2.03
2.01 2.13 2.11 2.23 2.07 2.07 2.07
[261 [201
1271 1281 iI81 [I71
2.16 2.01
161
1.87
1301 t311
2.09 1.87 2.08 1.78 2.25 2.08
1291
[321 1331 1341 1341 [6,71 1351 [361
‘) Assuming full occupancy of apical oxygen ions, 6=0.
In the calculations we have used crystal structures which were derived from samples exhibiting a critical temperature close to T,,,,,. The only exception is Bi2Sr2Cu06+6. According to the phase diagram of Bi2Sr2Cu06+6 13 I, a maximum T, of 30 K can be obtained upon replacing 25% of strontium by lanthanum. However, only the crystal structure of pure BiZSr2Cu06+6 has been reported [ 6 1, and this has been used in the calculation. We note however, that bond valence sums calculated for the undoped and such as YBa2Cu306 and doped compounds, YBa2Cu30,, typically agree within 0.05. The compounds which crystallize in an orthorhombic Bravais lattice, i.e. YBazCu,08, Y2Ba4Cu70L4.3, YBa2Cu30, and YBaSrCu307, exhibit two different oxygen sites in the CuO, planes. In table I the bond valence sums are given for both types of oxygen ions. The crystal structures of the compounds T1Ba2Ca2Cu309_6, TlT12Ra2Ca&uJOi0+~,
BazCa3Cu40,, _&, (Tl, Pb)Sr2CazCus09_6 and (Bi, Pb)zSr2CazCu30,0+6 contain inequivalent CuOz planes. The entries in table I pertain to the inner CuOz planes containing four-fold coordinated copper. Refinement of the crystal structure of Pbo.sSr2.,Cao.5Yo.,Cuz07_a yields a partial occupancy of the apical oxygen ions of 65Oh [ 171. For comparison bond valence sums were also calculated assuming full occupancy. The occupancy of the apical oxygen site is probably lower because the lone pair of the Pb *+ ion is situated near this site. Atomic positions and occupancy numbers for the constituent ions of T1Ba2Ca&u401 1_-6 and (Bi, have not yet been reported. We Pb)8rKaKu3010+6 therefore calculated the copper bond valence sum from the lattice constants assuming flat CuOz planes. The other cation-oxygen bond lengths cannot reliably be estimated; therefore the valence sums for oxygen in these compounds cannot be calculated.
136
D.M. de Leeuw et al./T, and crystal structure ofcuprate compounds
Also for Pb2Sr0.BLal.ZCu206+6 no atomic positions and occupancy numbers have been reported. The bond valence sum of copper has therefore been calculated using the tentative crystal structure proposed in ref. [ 181.
130 l
12” 110 100 t
90
5
3. Correlation between bond valence sums and T,,,,,
90 x
70
i
6”
c 50 40
In figs. 1 and 2 we show Tc,maxas a function of the bond valence sums for copper and oxygen respectively. For compounds containing inequivalent CuOz. planes, the values for the inner plane are taken. In addition, since a correlation has been suggested bebond tween T,,,,, and the in-plane copper-oxygen lengths [8], which scale with the in-plane copper bond valence sums, I’cu.in_piane,we also present r,,,,, as a function of Vcu,,n_planein fig. 3. The critical temperature could also depend on the formal coordination number of the copper ions. In fig. 1 therefore, data points for compounds with a formal coordination number for copper of 4, 5 or 6 are marked differently.
30 20 10 -~
-
0 1.7
1.8 BOND
19
20
VALENCE
SUM
2.1
22
OF OXYGEN
2.3
-
Fig. 2. Maximum values of the critical temperature. 7;,,,,, presented as a function of calculated bond valence sums for oxygen. VO. The data points are given in table I. Data points connected by straight lines corresponds to crystal structures containing CuOL planes with inequivalent oxygen sites.
1 130 120 110 T
T,m.. W
100
go 60 70 60 50 40 30
“I_‘-’
20
.-I
19
10
2.0 IN PLANE
0 1.9
2.0
2.1
BOND
VALENCE
2.2 SUM
2.3 OF COPPER
2.4
2.5 mt
Fig. I. Maximum values of the critical temperature, T,,,,,, presented as a function of calculated bond valence sums for copper, Vc,. The data points are given in table I. Data points for compounds with a formal coordination number for copper of 4, 5 or 6 are indicated by squares, triangles and lozenges respectively. The two data points connected by a straight line correspond to Pb0.sSrz.sCa0.5Y,.,Cu207_a with 65% and full occupancy of the apical oxygen ions.
2.6
21
22
BOND
VALENCE
SUM
OF COPPER
-
Fig. 3. Maximum values of the critical temperatures. I;,,,,, presented as a function of calculated in-plane bond valence sums for The data points are given in table 1. copper. VC,.,,.,~,,,.
Figs. 1 and 2 clearly show a definite correlation between T,,,,, and the bond valence sums for both the copper and the oxygen ions located in the CuO, planes. Lower bond valence sums correspond to higher values of T,,,,,. We note that these correla-
D.M. de Leeuw et al/T, andcrystalstructureof cupratecompounds
tions are quite pronounced, considering the uncertainties in crystal structures and values for the critical temperatures. We stress that the origin of the correlations cannot be found in the CuOz planes alone. As fig. 3 demonstrates, T,,,,, does not correlate with the in-plane bond valence sum of copper, and therefore not with in-plane copper-oxygen bond lengths, as proposed in ref. [ 8 1. These in-plane bond lengths apparently are not the appropriate parameter relating crystal structure with critical temperature. Comparison of figs. 1 and 3 proves that the definition of a structural parameter relating to the copper ions in the CuOz planes must include the presence of the apical oxygens. A similar conclusion can be drawn with regard to the oxygen ions in the CuOz planes. When completely isolated CuOz planes are considered, the bond valence sum for oxygen is by definition half the inplane bond valence sum of copper. The absence of correlation between T,,,,, and the in-plane copper bond valence sum demonstrated in fig. 3, therefore implies an absence of correlation between T,,,,, and the in-plane bond valence sum of oxygen. The correlation between T,,,,, and the total bond valence sum of oxygen found in fig. 2, therefore indicates that the presence of cations above and below the CuOz planes may not be disregarded. Consequently, theoretical models and mechanisms that aim at a quantitative explanation of superconductivity in the high-T, cuprates cannot focus on the Cu02 planes alone. The presence of the other constituents ions has to be taken into account explicitly. On the other hand, the mere presence or absence of apical oxygens is not a useful structural parameter either. Fig. 1 shows that there is no strict correlation number between T,,,,, and the formal coordination of copper: there are compounds with different coordination numbers that have the same maximum and critical temperature, e.g. T12BaZCu06+6 with the Bi2SrzCaCu208+J, as well as compounds same coordination number but different T,,,,,, e.g. Pbo.SSr2.5Cao.5Y0.5Cu207-- and T1,Ba,CaCuzO,++ It thus appears that it is the total bonding arrangement that matters, which is apparently expressed by the total bond valence sums in an adequate way. In fact, this finding shows up most directly if one com-
137
pares isostructural compounds. For example, the relatively high T,,,,, of 80 K which is observed in TlzBazCu06+, in comparison to 30 K in Bi2Sr2Cu06+6 is associated with considerably smaller values of V,, and Vi,. Also the decrease of T,,,, when Ba is replaced by Sr in T12BaZCaCu208+6 [ 191 and in YBa@_t~O, [20] is accompanied by an increase of V,,. This is readily explained if one notes that replacement of Ba by the smaller Sr will result in a decrease of the lattice constants and consequently in a decrease of the Cu-0 bond lengths. We further point out that our results imply that Tc,maxdoes not depend on the number of CuOz planes per unit formula as such. In fact they indicate that incorporation of additional CuO, planes, such as in and T12Ba2Can-1CU,02n+4+a with n > 2 will not lead to Bi2Sr2Ca,- LC@Zn+4+6 higher critical temperatures since the bond valence sums will not change. This is because for n> 3 the coordination of the ions in the inner CuO, planes as well as the a- and b-lattice constants, and consequently the Cu-0 bond lengths, remain essentially the same.
4. Discussion So far we have used the bond valence sums as empirical quantities without further physical significance. According to the bond valence sum rule, however, the bond valence sums may be identified with the formal oxidation state or valence state of the corresponding ions in the following way. A calculated bond valence sum gives the valency that, on the basis of the experience from the huge body of crystal structure data from which the parameters have been determined, is expected to fit best with the particular arrangement of the ion. We may therefore try to interpret figs. 1 and 2 in terms of a correlation between maximum critical temperature and valence states of copper and oxygen. Although, as discussed above, the absolute values for the valence states should be regarded with reservation, the trend observed in figs. 1 and 2 is clear. It indicates that in the p-type cuprate compounds the maximum value of the critical temperature is higher, the more the crystal structure favours the charge distribution in the Cu02 planes to be “Cu2+01-” instead of “Cu3+02-“, or in other
138
D.M. de Leeuw et al./T, and qvstal structure rfcuprate cornpounds
words the stronger the tendency for holes to be on the oxygen ions (“0’-“) rather than on the copper ions ( “Cu3+“). It is of importance to establish the connection with the current theoretical picture of the (strongly correlated) electronic structure of the p-type cuprates [ 2 1 1, and the associated Hubbard-like many particle models. In the relevant situation of fairly strong doping the energy of a doped hole is, if covalency is neglected, higher by U-A+ W/2 when the hole is situated on a copper site (forming Cu3+ ) than when it goes into the oxygen p-band (corresponding to 0’ - ). Here U is the intra-atomic Coulomb repulsion energy on copper, A is the charge transfer energy which represents the difference in single particle energy between a hole on oxygen or on copper, and W is the oxygen bandwith. Therefore, recalling that T,,,,, is reached in the various compounds at similar hole densities in the CuOZ planes, our finding indicates in the language of this theory that the maximum value for the critical temperature increases with increasing U-A+ W/2. If U were constant, this would imply that T_,,,, is higher as the undoped compound is closer to the insulator-metal transition, since in the undoped situation the (charge transfer) bandgap is, again with covalency ignored, given by A- W/2. However, U also contains a polarization energy contribution, which is expected to be large in the highly anisotropic crystal structures [22] of the cuprates, and which may vary considerably between the various compounds. One would expect that the dominant variation in LJ- A comes from the difference in the Madelung potential between the copper and the oxygen site in the CuOz planes, because the polarization energy contributions to Uand A, though large, are equal in first approximation [ 2 11. Indeed, it was shown recently [ 231 that a correlation between the Madelung potential difference and T, exists, although it is less pronounced than the correlation with the bond valence sums presented here. However, the calculation of the Madelung potentials is not straightforward in the doped situations. Moreover, the bond valence sums, being to some extent empirical parameters, have the advantage that they may take covalency, i.e. band formation, and polarizability into account in the same effective way. Apparently they do rather
well in correlating superconductivity.
the
charge
distribution
with
5. Conclusion The results presented here demonstrate a definite correlation between the calculated bond valence sums of the copper and the oxygen sites in the central CuOZ planes on the one hand, and the maximum superconducting transition temperature, T,.,,,, on the other hand, for all p-type high-T, superconducting cuprate compounds. With this relation it is possible to rationalize the observation of the relatively high 7, of T12Ba2Cu06+6 compared to BizSr,CuO,+B, and the decrease of T, when Ba is substituted by Sr in TlzBazCaCuzO, +d and YBazCu307. Also the dependence of 7;,,,, on n in T1BaZCan-,C~nOZn+3, T1,BazCan_ ,CU,,O~~+~ and BizSrzCan_ ,CU,,O~,,+~+,~ is in agreement with this relation. The correlation found shows that T,,,,, increases the more the holes in the Cu02 planes prefer the oxygen sites over the copper sites. In a correlated electron picture this implies a higher value for U-A+ W/2.
References ( I ] Y. Tokura, J.B. Torrance, T.C. Huang and AI. Nazzal, Phys. Rev. B 38 (1988) 7156. [2] J.B. Torrance, Y. Tokura, A.I. Nazzal, A. Bezinge. T.C. Huang and SK. Parkin, Phys. Rev. Lett. 6 1 ( 1988) I 127. ] W..4. Groen, D.M. de Leeuw and G.P.J. Geelen, Physica C 165 (1990) 305. ] W..4. Groen, D.M. de Leeuw and L.F. Feiner, Physica C 165 (1990) 55. ] P.M. Grant, preprint. ] C.C. Torardi, M.A. Subramanian. J.C. Calabrese, J. Gopalakrishnan, E.M. McCarron, K.J. Morrissey, T.R. Askey, R.B. Flippen, U. Chowdry and A.W. Sleight. Phys. Rev.B38(1988)225. [7] T. Den, A. Yamazaki and J. Akimitsu, Jpn. J. Appl. Phys. 27 (1988) L1620. [ 81 M.H. Wangbo, D.B. Kang and C.C. Torardi. Physica Cl 58 (1989) 371. [9] J. Gopalakrishnan, M.A. Subramanian and A.W. Sleight, J. Sol. St. Chem. 80 ( 1989) 156. lo] W.H. Zachariasen, J. Less-Comm. Metals 62 (1978) I. 111 I.D. Brown and D. Altermatt, Acta Cryst. B41 ( 1985) 244. 121 I.D. Brown and R.D. Shannon. Acta Cryst. A29 (1973) 266. 131 I.D. Brown, Chem. Sot. Rev. 7 ( 1978) 359. 141 I.D. Brown, J. Sol. StateChem. 82 (1989) 122.
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CORRELATION BETWEEN THE SUPERCONDUCTING TRANSITION CRYSTAL STRUCTURE OF HIGH-T, CUPRATE COMPOUNDS D.M. de LEEUW,
W.A. GROEN,
L.F. FEINER
TEMPERATURE
AND
and E.E. HAVINGA
Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, The Netherlands Received 20 January 1990
For the various p-type cuprate superconductors we have calculated the formal valence for the copper and the oxygen ions in the central Cu02 planes from bond lengths according to Zachariasen rules. It is shown that in all structures these values correlate remarkably well with the maximum critical temperature. The correlation found shows that T,,,, increases the more the holes in the Cu02 planes prefer the oxygen sites over the copper sites. In a correlated electron picture this implies a higher value for U-A+ W/2.
1. Introduction Phase diagrams that show the superconducting critical temperature, T,, as a function of the density of holes in the CuOZ planes have been reported for YBa2Cu30, [ 11, LazCuOd [ 21, BiZSr2Cu06+6 [ 31 and BizSrzCaCuzOs+g [ 41. These phase diagrams show an apparently universal behaviour for all p-type high-T, cuprate superconductors, viz. the superconducting boundaries in the diagrams are broad domeshaped curves and the maximum T, is obtained at similar doping levels of about 0.2 holes per copper per CuOl plane in all cases. However, the maximum values for the critical temperatures themselves are significantly different, viz. 30 K for BiZSr2Cu06+6, 35 K for Lal.&,&u04, 80 K for Bi2Sr2CaCu208+6 and 93 K for YBa&&O,. The differences in the maximum values of T, can as yet not be understood from first principles, because detailed knowledge about the electronic and magnetic structure is lacking, and the mechanism of superconductivity is still unknown. For the time being rationalization of the data has to proceed by a semi-empirical approach. There is a general agreement about the key role of the nearly planar CuO, entities in the structure. Since the doping levels (i.e. the total charges in the CuOl planes) are similar in the various series of compounds, the differences in Tc,maxwill originate from difirent distributions of 0921-4534/90/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland )
charge within the CuOZ planes, i.e. from different electronic structures. A semi-empirical description will involve concepts like covalency, electronegativity, bond strength and polarizability. As changes in crystal structure are correlated with changes in electronic structure, one expects that T_= may be correlated with the crystal structure. To this end appropriate parameters that characterize the crystal structure must be defined, preferably such that connect naturally with the description of the electronic structure. Recently, various attempts for such a parametrization have been suggested. For instance, to explain the values of T,,,,, in the series and T12Ba2Can-1Cu,0Zn+4+6 with n=l, 2 and 3 it has Bi&Xa,&,OZ,,+~+~ been proposed that T,,,,, increases in a particular way with n, the number of Cu02 planes [ 51. Although in each series T,,,,, increases with n, this parametrization is not sufficient because it does not account for the difference between the n = 1 compounds T12Ba2CuOg+d with T,,,,,=SO K [6] and Bi2Sr2Cu06+6 with T,,,,, = 30 K [ 3,7 1. Another proposal concerns a correlation between the critical temperature and the in-plane CuO bond lengths [ 8 1. Three classes are distinguished, depending on which ion, La3+, Ba2+ or Sr2+ occupies the nine-fold coordination site connected with the apical oxygen ions of the CuOZ layers. Each class, then, has to be sub-
134
D.M. de Leeuw et al/T, and crystalstructure
divided again according to the number of CuO, planes per unit formula. The changes in T, reported by these authors actually reflect changes in the hole density, not changes in a structural parameter. The same holds for the proposed relation between T, and average electronegativity of the constituent ions of the unit cell [ 91. A clear correlation of T,,,,, with crystal structure has not yet been reported. A well-suited framework to discuss correlations with crystal structures, without implying the physical mechanism that produces them, is the bond valence model of Zachariasen [ IO- 121. This model is particularly useful to discuss bonding in cases where the coordination is very distorted such as for Cu2+ [ 13 1. It takes into account the number of bonds, weighing their importance according to bond length. This model can be used in assessments of the correctness of crystal structure determinations, for estimations of chemical activity and for predictions of charge transfer reaction mechanisms [ 13 1. Here, we apply the bond valence model to relate the crystal structure with T,,,,, for the series of ptype high-T, cuprate superconductors. The crystal structure is parametrized by bond valence sums for the copper and oxygen ions located in the Cu02 planes. We then show that these bond valence sums correlate unambiguously with T,,,,,. Finally, the implications of this relation are discussed.
2. Bond valence sums of copper and oxygen The bond valence, s,,, of a cation and anion at distance R,, is defined by [ l&l 1 ] : ~,,=ew(&-R,,IBl.
(1)
The constant B can be taken independent of the type of constituent ions and typically a value of 0.31 A is applied [ 111, while the reference distance R. does depend on the type of cation and anion that form the bond. It has been shown [ 121 that the values for R,, can be determined in such a way that the sum of bond valences around any ion is close to the expected formal oxidation state of that ion: v,= c s,,= 1 exp(R,-RJ0.37) I I In calculating
the bond valence sum of a cation, the
qfcuprate
cornpoundv
summation is over the anions and vice versa. Because of the strong dependence of the bond strength on the bond length only the first coordination sphere has to be taken into account. The parameter values for R. have been determined by Brown and Altermatt [ 111 for 750 ion pairs using the Inorganic Crystal Structure Database. Here we use eq. (2 ) to calculate the formal oxidation states of copper and oxygen in the p-type high-T, superconducting cuprates. The absolute values calculated for the oxidation states strongly depend on the parameter values Ro. Especially, the value of R. is different for different nominal oxidation states [ 1 1,141 (e.g. Cu’+ or Cu’+ ), and hence the calculated bond valence sum depends on the reference nominal oxidation state [ 141. However, the reported phase diagrams, presenting 7, as a function of hole density, show that for all superconducting cuprate compounds the maximum T, is obtained at similar hole densities of about 0.2 holes per copper per CuOz plane (with respect to cu2+ and 02-) [ l-41. Therefore, in the present calculations we use the fixed values for R. pertaining to cu2+ and 02- from ref. [ 111. As a consequence, the absolute values for the bond valence sums should be regarded with some reservation. The results of the calculations are intended to show trends. The calculated bond valence sums for copper and oxygen ions that are located in the CuOz planes of the high-ir, superconducting cuprate compounds are presented in table I. For copper the total bond valence sum is shown as well as the separate contributions from the in-plane and the apical oxygens. Table I also presents values for T,.,,,, the highest critical transition temperature of a certain type of compound. For YBa2Cu30,, La2Cu0,, Bi2Sr,CaCuzOs+~ and Bi2Sr2CuOb+,- values of T,.,,, are taken from phase diagrams that show Tc as a function of hole density [ l-41. For the other compounds T,.,,, is a typical value reported as the maximum T,. We note that until the complete phase diagrams are available, these values for T,.,,, are only approximate and may be too low. For instance the T, of YBazCu40s was increased recently from 80 K to 90 K by replacing 10% of the yttrium ions by calcium ions [IS]. Also in Y2Ba4Cu,0,4.3 the T, has been increased from 40 K to 93 K by partial replacement of yttrium by calcium [ 161.
D.M. de Leeuw et al./T, and crystal structure of cuprate compounds
135
Table I Summary of calculated bond valence sums of copper, VC,, including the contribution of the in-plane and apical oxygen ions, VCu,,n_pl~c respectively, calculated bond valence sums of oxygen, V,, and maximum values of the critical temperature, T,.,,,, for and VCu,oY1-pl.ne high-T, cuprate superconductors Compound YBa2Cu408
Vcu
VO
ref
2.03 1.99 2.04
[1X241
90
1.935
0.187
2.12
93
1.936
0.197
2.13
[16,251
2.04
YBa$&O, YBaSrCu,O, Lal.8Sr0.2Cu04
PbzSr,(Ca, Y)Cu308 Pb%8h2CU206+6 Pbo.sSr2.5Cao.sYo.sCu207-a a,
T12Ba2Cu06+d T12Ba2CaCu208+d Tl2BaKaKuG&+~ TlBa2CaCu20,_6 TlBa2Ca2Cu,0g_d TlBa2Ca3Cu40,, _-d (Tl, Pb)Sr2CaCu20,_d (Tl, Pb)Sr2Ca2Cu309_d Bi2Sr2CuOe+,r Bi2Sr2CaCu208+d (Bi, Pb)2Sr2Ca2Cu3010+6
93
1.958
0.190
2.15
80
2.049
0.197
2.25
36 70 26 50
2.264 2.047 2.142 2.142 2.142 2.013 2.042 2.055 2.085 2.077 2.057 2.185 2.141 2.201 2.119 2.131
0.263 0.194 0.326 0.172 0.265 0.122 0.063 0.0 0.124 0.0 0.0 0.127 0.0 0.175 0.073 0.0
2.53 2.24 2.47 2.31 2.41 2.13 2.11 2.06 2.2 1 2.08 2.06 2.31 2.14 2.38 2.19 2.13
80 110 125 70 125 125 85 122 30 85 110
2.03
2.01 2.13 2.11 2.23 2.07 2.07 2.07
[261 [201
1271 1281 iI81 [I71
2.16 2.01
161
1.87
1301 t311
2.09 1.87 2.08 1.78 2.25 2.08
1291
[321 1331 1341 1341 [6,71 1351 [361
‘) Assuming full occupancy of apical oxygen ions, 6=0.
In the calculations we have used crystal structures which were derived from samples exhibiting a critical temperature close to T,,,,,. The only exception is Bi2Sr2Cu06+6. According to the phase diagram of Bi2Sr2Cu06+6 13 I, a maximum T, of 30 K can be obtained upon replacing 25% of strontium by lanthanum. However, only the crystal structure of pure BiZSr2Cu06+6 has been reported [ 6 1, and this has been used in the calculation. We note however, that bond valence sums calculated for the undoped and such as YBa2Cu306 and doped compounds, YBa2Cu30,, typically agree within 0.05. The compounds which crystallize in an orthorhombic Bravais lattice, i.e. YBazCu,08, Y2Ba4Cu70L4.3, YBa2Cu30, and YBaSrCu307, exhibit two different oxygen sites in the CuO, planes. In table I the bond valence sums are given for both types of oxygen ions. The crystal structures of the compounds T1Ba2Ca2Cu309_6, TlT12Ra2Ca&uJOi0+~,
BazCa3Cu40,, _&, (Tl, Pb)Sr2CazCus09_6 and (Bi, Pb)zSr2CazCu30,0+6 contain inequivalent CuOz planes. The entries in table I pertain to the inner CuOz planes containing four-fold coordinated copper. Refinement of the crystal structure of Pbo.sSr2.,Cao.5Yo.,Cuz07_a yields a partial occupancy of the apical oxygen ions of 65Oh [ 171. For comparison bond valence sums were also calculated assuming full occupancy. The occupancy of the apical oxygen site is probably lower because the lone pair of the Pb *+ ion is situated near this site. Atomic positions and occupancy numbers for the constituent ions of T1Ba2Ca&u401 1_-6 and (Bi, have not yet been reported. We Pb)8rKaKu3010+6 therefore calculated the copper bond valence sum from the lattice constants assuming flat CuOz planes. The other cation-oxygen bond lengths cannot reliably be estimated; therefore the valence sums for oxygen in these compounds cannot be calculated.
136
D.M. de Leeuw et al./T, and crystal structure ofcuprate compounds
Also for Pb2Sr0.BLal.ZCu206+6 no atomic positions and occupancy numbers have been reported. The bond valence sum of copper has therefore been calculated using the tentative crystal structure proposed in ref. [ 181.
130 l
12” 110 100 t
90
5
3. Correlation between bond valence sums and T,,,,,
90 x
70
i
6”
c 50 40
In figs. 1 and 2 we show Tc,maxas a function of the bond valence sums for copper and oxygen respectively. For compounds containing inequivalent CuOz. planes, the values for the inner plane are taken. In addition, since a correlation has been suggested bebond tween T,,,,, and the in-plane copper-oxygen lengths [8], which scale with the in-plane copper bond valence sums, I’cu.in_piane,we also present r,,,,, as a function of Vcu,,n_planein fig. 3. The critical temperature could also depend on the formal coordination number of the copper ions. In fig. 1 therefore, data points for compounds with a formal coordination number for copper of 4, 5 or 6 are marked differently.
30 20 10 -~
-
0 1.7
1.8 BOND
19
20
VALENCE
SUM
2.1
22
OF OXYGEN
2.3
-
Fig. 2. Maximum values of the critical temperature. 7;,,,,, presented as a function of calculated bond valence sums for oxygen. VO. The data points are given in table I. Data points connected by straight lines corresponds to crystal structures containing CuOL planes with inequivalent oxygen sites.
1 130 120 110 T
T,m.. W
100
go 60 70 60 50 40 30
“I_‘-’
20
.-I
19
10
2.0 IN PLANE
0 1.9
2.0
2.1
BOND
VALENCE
2.2 SUM
2.3 OF COPPER
2.4
2.5 mt
Fig. I. Maximum values of the critical temperature, T,,,,,, presented as a function of calculated bond valence sums for copper, Vc,. The data points are given in table I. Data points for compounds with a formal coordination number for copper of 4, 5 or 6 are indicated by squares, triangles and lozenges respectively. The two data points connected by a straight line correspond to Pb0.sSrz.sCa0.5Y,.,Cu207_a with 65% and full occupancy of the apical oxygen ions.
2.6
21
22
BOND
VALENCE
SUM
OF COPPER
-
Fig. 3. Maximum values of the critical temperatures. I;,,,,, presented as a function of calculated in-plane bond valence sums for The data points are given in table 1. copper. VC,.,,.,~,,,.
Figs. 1 and 2 clearly show a definite correlation between T,,,,, and the bond valence sums for both the copper and the oxygen ions located in the CuO, planes. Lower bond valence sums correspond to higher values of T,,,,,. We note that these correla-
D.M. de Leeuw et al/T, andcrystalstructureof cupratecompounds
tions are quite pronounced, considering the uncertainties in crystal structures and values for the critical temperatures. We stress that the origin of the correlations cannot be found in the CuOz planes alone. As fig. 3 demonstrates, T,,,,, does not correlate with the in-plane bond valence sum of copper, and therefore not with in-plane copper-oxygen bond lengths, as proposed in ref. [ 8 1. These in-plane bond lengths apparently are not the appropriate parameter relating crystal structure with critical temperature. Comparison of figs. 1 and 3 proves that the definition of a structural parameter relating to the copper ions in the CuOz planes must include the presence of the apical oxygens. A similar conclusion can be drawn with regard to the oxygen ions in the CuOz planes. When completely isolated CuOz planes are considered, the bond valence sum for oxygen is by definition half the inplane bond valence sum of copper. The absence of correlation between T,,,,, and the in-plane copper bond valence sum demonstrated in fig. 3, therefore implies an absence of correlation between T,,,,, and the in-plane bond valence sum of oxygen. The correlation between T,,,,, and the total bond valence sum of oxygen found in fig. 2, therefore indicates that the presence of cations above and below the CuOz planes may not be disregarded. Consequently, theoretical models and mechanisms that aim at a quantitative explanation of superconductivity in the high-T, cuprates cannot focus on the Cu02 planes alone. The presence of the other constituents ions has to be taken into account explicitly. On the other hand, the mere presence or absence of apical oxygens is not a useful structural parameter either. Fig. 1 shows that there is no strict correlation number between T,,,,, and the formal coordination of copper: there are compounds with different coordination numbers that have the same maximum and critical temperature, e.g. T12BaZCu06+6 with the Bi2SrzCaCu208+J, as well as compounds same coordination number but different T,,,,,, e.g. Pbo.SSr2.5Cao.5Y0.5Cu207-- and T1,Ba,CaCuzO,++ It thus appears that it is the total bonding arrangement that matters, which is apparently expressed by the total bond valence sums in an adequate way. In fact, this finding shows up most directly if one com-
137
pares isostructural compounds. For example, the relatively high T,,,,, of 80 K which is observed in TlzBazCu06+, in comparison to 30 K in Bi2Sr2Cu06+6 is associated with considerably smaller values of V,, and Vi,. Also the decrease of T,,,, when Ba is replaced by Sr in T12BaZCaCu208+6 [ 191 and in YBa@_t~O, [20] is accompanied by an increase of V,,. This is readily explained if one notes that replacement of Ba by the smaller Sr will result in a decrease of the lattice constants and consequently in a decrease of the Cu-0 bond lengths. We further point out that our results imply that Tc,maxdoes not depend on the number of CuOz planes per unit formula as such. In fact they indicate that incorporation of additional CuO, planes, such as in and T12Ba2Can-1CU,02n+4+a with n > 2 will not lead to Bi2Sr2Ca,- LC@Zn+4+6 higher critical temperatures since the bond valence sums will not change. This is because for n> 3 the coordination of the ions in the inner CuO, planes as well as the a- and b-lattice constants, and consequently the Cu-0 bond lengths, remain essentially the same.
4. Discussion So far we have used the bond valence sums as empirical quantities without further physical significance. According to the bond valence sum rule, however, the bond valence sums may be identified with the formal oxidation state or valence state of the corresponding ions in the following way. A calculated bond valence sum gives the valency that, on the basis of the experience from the huge body of crystal structure data from which the parameters have been determined, is expected to fit best with the particular arrangement of the ion. We may therefore try to interpret figs. 1 and 2 in terms of a correlation between maximum critical temperature and valence states of copper and oxygen. Although, as discussed above, the absolute values for the valence states should be regarded with reservation, the trend observed in figs. 1 and 2 is clear. It indicates that in the p-type cuprate compounds the maximum value of the critical temperature is higher, the more the crystal structure favours the charge distribution in the Cu02 planes to be “Cu2+01-” instead of “Cu3+02-“, or in other
138
D.M. de Leeuw et al./T, and qvstal structure rfcuprate cornpounds
words the stronger the tendency for holes to be on the oxygen ions (“0’-“) rather than on the copper ions ( “Cu3+“). It is of importance to establish the connection with the current theoretical picture of the (strongly correlated) electronic structure of the p-type cuprates [ 2 1 1, and the associated Hubbard-like many particle models. In the relevant situation of fairly strong doping the energy of a doped hole is, if covalency is neglected, higher by U-A+ W/2 when the hole is situated on a copper site (forming Cu3+ ) than when it goes into the oxygen p-band (corresponding to 0’ - ). Here U is the intra-atomic Coulomb repulsion energy on copper, A is the charge transfer energy which represents the difference in single particle energy between a hole on oxygen or on copper, and W is the oxygen bandwith. Therefore, recalling that T,,,,, is reached in the various compounds at similar hole densities in the CuOZ planes, our finding indicates in the language of this theory that the maximum value for the critical temperature increases with increasing U-A+ W/2. If U were constant, this would imply that T_,,,, is higher as the undoped compound is closer to the insulator-metal transition, since in the undoped situation the (charge transfer) bandgap is, again with covalency ignored, given by A- W/2. However, U also contains a polarization energy contribution, which is expected to be large in the highly anisotropic crystal structures [22] of the cuprates, and which may vary considerably between the various compounds. One would expect that the dominant variation in LJ- A comes from the difference in the Madelung potential between the copper and the oxygen site in the CuOz planes, because the polarization energy contributions to Uand A, though large, are equal in first approximation [ 2 11. Indeed, it was shown recently [ 231 that a correlation between the Madelung potential difference and T, exists, although it is less pronounced than the correlation with the bond valence sums presented here. However, the calculation of the Madelung potentials is not straightforward in the doped situations. Moreover, the bond valence sums, being to some extent empirical parameters, have the advantage that they may take covalency, i.e. band formation, and polarizability into account in the same effective way. Apparently they do rather
well in correlating superconductivity.
the
charge
distribution
with
5. Conclusion The results presented here demonstrate a definite correlation between the calculated bond valence sums of the copper and the oxygen sites in the central CuOZ planes on the one hand, and the maximum superconducting transition temperature, T,.,,,, on the other hand, for all p-type high-T, superconducting cuprate compounds. With this relation it is possible to rationalize the observation of the relatively high 7, of T12Ba2Cu06+6 compared to BizSr,CuO,+B, and the decrease of T, when Ba is substituted by Sr in TlzBazCaCuzO, +d and YBazCu307. Also the dependence of 7;,,,, on n in T1BaZCan-,C~nOZn+3, T1,BazCan_ ,CU,,O~~+~ and BizSrzCan_ ,CU,,O~,,+~+,~ is in agreement with this relation. The correlation found shows that T,,,,, increases the more the holes in the Cu02 planes prefer the oxygen sites over the copper sites. In a correlated electron picture this implies a higher value for U-A+ W/2.
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