Chemical pressure analysis of high temperature superconductors: Anisotropic effects in YBa2Cu3O7−x and YBa2(Cu1−yFey)3O7−x

343

Materials Chemistry and Physics, 36 (1994) 343-349

Chemical pressure analysis of high temperature superconductors: anisotropic effects in YBa,Cu,O,_, and YBa,(Cu, -yFey)307_-x F. Hadley Cocks and Holly M. Hammarstrom (Received February 24, 1993; accepted June 24, 1993)

Abstract Quantitative evaluation of selected sets of critical temperature and lattice parameter data for the orthorhombic compounds YBazCu,07_, and YBaz(Cu,_,Fe,),O,_, reveals a strongly anisotropic dependence of the critical temperature on these parameters, especially for YBazCu307_,. This method of deriving the relationship between critical temperature and changes in lattice constants is termed chemical pressure analysis (CPA). The signs of the anisotropic chemical pressure coefficients, kj, for the critical temperature of YBa2Cu307_, are found to be opposite to those determined for anisotropic~~y applied physical (mechanical) pressure and of greatly increased magnitude. One explanation for these differences may be that physical pressures do not alter chemical bonds in the same manner as do chemical pressures. For pressures below 4 GPa, both CPA and physical pressure results indicate that increasing hydrostatic pressure increases T,, but chemical pressures are found to be more effective at increasing critical temperature than are physical pressures. Hydrostatic pressures above 4 GPa, however, decrease T,. Since the fracture strength of real crystals limits the magnitude of the anisotropic physical stresses that can be applied, only CPA appears capable of yielding the anisotropic dependence of critical temperature on indi~dual lattice parameters in the ul~ahigh stress regime above 4 GPa.

Introduction Hydrostatic pressure and atomic composition have long been known to affect the critical temperatures of high temperature superconductors. However, no definitive theory yet exists to quantitatively explain these effects, especially the influence of lattice parameter changes on critical temperatures, despite the fact that many lattice parameter studies have been carried out [l-4]. The results of the present work offer a new, phenomenological, chemical pressure analysis (CPA) approach toward predicting the effect of lattice parameter changes on superconducting critical temperatures. Recent work by Meingast et al. [5, 61 and by Welp et al. [7], based on both thermal expansion and direct physical pressure data, show results which are related to the results presented here, although differing in several important respects. The anisotropic stress dependence of the critical temperature of high temperature superconductors has been the subject of considerable recent work. Crommie et al. [8] appear to be the first to have applied uniaxial stress to a high temperature superconducting compound. They apphed compressive stress up to approximately 0.1 GPa along the c-axis of single-postal rectangular parallelepipeds of YBa,Cu,O, and found dT,/dcc to

02%-I-0584/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved

be about 0.8 K GPa-I; however, there exists some question as to whether their measurements represent the true bulk effect [9]. Koch et al. [lo] suggest that the pressure dependence of the critical temperature of single crystal YBa,Cu,O,., is highly anisotropic and negative for pressure applied parallel to the c-axis, but positive for pressure applied parallel to either the aaxis or the b-axis. They used this idea as a possible explanation for their finding of an anomalous hydrostatic pressure dependence for the critical temperature of YBa,Cu,O,.,. Meingast et al. [6] investigated the thermal expansion coefficient of YBa&u,O,_, as a function of crystal direction and predicted, from the observed anisotropy in this expansion, a similar anisotropy of the pressure dependence of the critical transition temperature. They predicted that dTJdu,= + 1.9 K GPa-‘, dT,/du, = -2.2 K GPa-I, and dT,/da,=O K GPa-‘. Welp et al. [7] have experimentally verified the Meingast et al. [6] prediction and find that dT,/dog = + 2.0 +0.2 K GPa-‘, dTJdtr,= -1.91f:O.2 K GPa-‘, and dT,/ da,= +0.3fO.l K GPa-‘. It should be noted that the high hydrostatic pressure regime (P > 4 GPa) for polycrystalline materials shows a significantly larger influence on T, than lower pressures. This ultrahigh pressure regime cannot be reached in uniaxial

compression

tests, however, since the material

344

fracture strength, even in compression, is too low. In this ultrahigh pressure regime Tissen and Nefedova ill], Klotz et d. [12] and Braithwaite et al. [13] find a decrease in critical temperature with increasing hydrostatic pressure that varies from about - 4 K GPa-’ to about -1.5 K GPa-*. In the present work, we have proceeded along an entirely new route to develop quantitative expressions relating critical temperature to the lattice parameters of the orthorhombic unit cells of YBa,Cu,O,_, and YBa,(Cu,_,FeJ,O,_,. As will be seen, these calculations, although based on an independent premise, lead to anisotropic results analogous to that reached by Meingast et al. [6] and Welp ec al. 171.

Chemical pressure analysis (CPA) results To determine the quantitative relationship that can be interpreted in terms of chemical pressure, a leastsquares analysis has been carried out to relate the critical transition temperature, T,, and room temperature lattice parameters, a, b and c, for the compounds YBa,Cu,O,_, and YBa,(Cu,_,Fe,),O,_,, from data that reveal the variation in these quantities as functions Physical pressure results for of composition. YBa,Cu,O,_, show that, up to approximately 4 GPa, T, increases with increasing pressure while above 4 GPa, T, decreases with increasing pressure. As we will show, however, our data can be considered to relate to chemical pressures only up to about 1 GPa, and therefore a linear model of the following form is assumed: T,-T,=k,(a-a,f+k,~b--b,)+k,(c-c,)

(1)

where Tco is the mean critical temperature of the data set, and a,, b0 and c, are the mean values of the corresponding lattice parameters a, b and c. It should be noted that the choice of Td and a,, b. and co does not, of course, affect the predictions which can be deduced from the resulting least-squares equation. Usingvalues other than the mean values for these quantities will simply cause an additional constant term in eqn. (1), and through algebraic manipulation the resulting equation can be shown to be equivalent to that calculated here. The data selected for the analysis of orthorhombic YBa,Cu,O,_, were those of Cava et al. [14], and consist of eight data points within 0.0 ~x~0.7. The seven data points used in the case of orthorhombic YBa,(Cu1_,FeJ30,_, were taken from Xu et al. [15]. In both cases, the critical temperature data were read from the reported graphical data. All concomitant numerical data used in the regression analyses are shown in Table 1 and 2. We have assigned lattice directions

TABLE I. Data used in regression taken from Lava et al. [I43

analysis for YBa2Cu307_r,

a

b

C

(4

(a

(4

3.822 3.827 3.830 3.835 3.831 3.838 3.845 3.851

3.891 3.895 3.898 3.890 3.889 3.887 3.887 3.883

11.677 11.722 11.728 11.716 11.736 11.747 11.768 11.789

TABLE 2. Data YBaz(Cu,-xFer)307_yr

regression used in taken from Xu et nl. [15]

a

b

C

(4

(4

(A)

3.820 3.822 3.825 3.829 3.834 3.842 3.852

3.885 3.885 3.885 3.881 3.879 3.868 3.861

11.676 11.680 11.672 11.678 11.677 11.678 11.686

92 86 77 60 58 55 44 28

analysis

for

90 90 89.5 86.5 86 86 83

in order of increasing lattice parameter magnitude such that the smallest parameter is termed a and the largest parameter is c. Choosing a
(2)

while the data from Table 2 for YBa,(Cu,_,Fe,),O,_, yield: T, - 87.3 = - 428.8@ - 3.832) - 299.7@ - 3.877) - 113.6(c - 11.678)

(3)

In these equations the temperatures are in Kelvin, and the lattice parameters are in angstroms. Because of the limited amount of data available for determining each least-squares line, many statistical measures are not effective. However, it can be said that the closeness of fit for these two equations is substantially different.

345

The percent average deviation between the experimentally measured temperatures shown in Table 1 and those calculated using eqn. (2) for YBa,Cu,O,_, is 6.16%. For the compound YBaz(Cu,_,FeJ30,_,, the percent average deviation is 0.63% as calculated using eqn. (3). This difference is associated at least in part with the fact that in the first case the experimentally measured critical temperatures range from 28 to 92 K, while in the second case the temperatures range only from 83 to 90 K. Several inferences may be drawn from these equations. The presence of both positive and negative signs in eqn. (2) immediately implies a strong anisotropic dependence of critical temperature on lattice parameter. In particular, the change in sign between k, and kz shows that an orthorhombic, not a tetragonal effect is being measured since by definition a crystal possessing tetragonal symmetry must have k, equal to k2. The presence of the same sign before each of the terms in eqn. (3) implies a weaker anisotropy for the iron-doped compound; however, the fact that the coefficients are not equal shows that anisotropy is still present. These relationships between lattice parameters and critical temperatures result from data derived from changes in composition, and hence this method of investigation may be termed chemical pressure analysis (CPA), since chemical bonding interactions, not applied stress, cause the lattice parameters to change. Chemical pressure effects were one of the reasons why yttrium was substituted for lanthanum to make YBa,Cu,O,_, initially [19], and the concept of chemical pressure has been discussed in a number of papers [ZO, 211. With respect to the results predicted using CPA, in comparison with the physical pressure data of Welp et al. f7], we first calculate the dependence of critical temperature produced by changes in physical pressure, assuming the relationship shown in eqn. (1). In applying this equation, the values for kI, k2 and k, that are used will depend upon whether chemical pressure or physical pressure is being considered, but in either case if we assume constant values for ki when differentiating eqn. (1) the result is:

(4) which may be rewritten

as

-db -da -dc dT --z =ak, $ +bk, $ +ck, L dP

dP

=ak, 2

+bk, 2

+ck, 3

where -d&V’, - d+,ldP and - de..ldP are the linear elastic compressibilities, p,, &, and &., in the a, b and

c directions. The negative signs are needed because it is general practice to report compressibilities as positive values even though strains are negative under compressive stress. Ludwig er al. [22] predict the linear compressibilities for orthorhombic YBa,Cu,O-I to be /3,= + 1.94X low3 GPa-“, )(a= +1.37~10-~ GPa-I, PC= +2.7X10p3 GPa-“. Upon substituting these values for Pa, fib and pc and the chemical pressure constants k,, k, and k3 of eqn. (2) into eqn. (5), we are, of course, combining properties derived from physical pressure (the elastic constants) with chemical pressure quantities (k,, kz and k3). In this case, eqn. (5) gives a pressure dependence of T, for YBa&u,O,_, equal to + 7.3 K GPa- I, which is substantially higher than measured physical pressure results. From hydrostatic physical pressure experiments, Zheng et al. [23] have given dTJd.2 for YBa&u,O, as +0.5 K GPa-l, Huber et al. [24] estimated it to be - 0.1 ~fi0.2 K GPa-l, and Stankowski et al. [25] reported + 1.06 K GPa- ’ for YBa,Cu,O,_,. The general consensus is that YBaZCu307 --xhas a dependence of critical temperature on pressure which is small It should be noted here that hydrostatic pressures are generally reported as positive even though the resulting strains are negative, as was the case for compressibility values. In subsequent calculations involving anisotropic stress, we will use the normal convention that positive stress causes positive strain and conversely that negative stress causes negative strain. In any event, the result calculated above enables us to conclude that the increases in critical temperature by chemical ‘hydrostatic’ pressures are substantially higher than those caused by physical hydrostatic pressures, and presumably therefore result from di~erent mechanisms. There do not appear to be available any elastic data for the iron-doped compound, but it is reasonable to assume that the moduli of YBa,(Cu,_,FeJ,O,_, are similar to those of YBa,Cu,O,_,, since moduli are generally an approximately linear function of composition. Because the linear compressibility is a function of the moduli, it is therefore reasonable to use the linear compressibilities of YBa,Cu,O, to approximate the linear compressibilities of YBa,(Cu,_,Fe,),O,_,. If one then calculates the hydrostatic pressure dependence, making the same assumptions as in the first case of YBa,Cu,O,_, the result is t-8.4 K GPa-I, which is again much higher than that reported for this compound. Kurisu et al. [26] report that dT,/dP is greatly enhanced by iron doping in the tetragonal phase (large doping content), whereas it is not appreciably enhanced in the orthorhombic case, to which our data belong. Meingast et al. [27], Huber et al. [24] and Kubiak and Westerholt [28] concur with this conclusion. Based on the above calculations, the ratio of the hydrostatic pressure dependence of the critical temperatures of

346

YBaz(Cu,_,FeX),O,_, to that of YBa,Cu,O,_, is predicted to be approximately 1.1, which is consistent with the reported physical pressure ratio of essentially unity.

Effect of uniaxial physical stress Meingast et aE. {6] and Welp et al. [7] have reported the critical temperature dependence of YBa,Cu,O,_, on uniaxial physical pressure in each of the three c~stallographic directions, and it is possible to transfo~ their pressure dependences into equations of the form of eqn. (1) to make a direct comparison between physical and chemical pressures. To solve these equations from the uniaxial physical pressure dependence predictions made by both Meingast et al. [6] and Welp et al. [7], it is necessary to derive expressions for dT,/do==, dT,/ do, and dT,/da, in terms of the three constants k,, kz and k3 and the elastic parameters of YBa,Cu,O,; the resulting equations comprise a system of three linear equations with three unknowns, the constants kI, k2 and k3. In the derivation of the dT,/da;: equations, the variation with stress of the constants kj will again be neglected and a set of elastic compliance for the tetragonal4/mmm structure, where .T~~=s~~and s, =s13, will be assumed, since the original works by Meingast et al. [6] and Welp et aE. [7] make this same tetragonal elastic constant assumption in their calculations. Equation (1) may then be differentiated with respect to the uniaxial stress, a, in the a direction (not hydrostatic pressure, P, as was the case in eqn. (4)) to give:

which may as before be rewritten as:

dT,

=&

da,

dz +bk dEb +ck % ’ da, 3 dcr, * da,

da,

=klas,,

+k,&s,z+k3cs,,

(7)

(9)

Followjng the same procedure, similar relations may be found for the other two crystallographic directions:

3

da,

=k as +k bs +k 3 cs 13 ’ I’, ’ ”

13 +k 2bs 13 +k 3 cs 33

(11)

Using the average lattice parameter values from the data used in the regression analysis for II, b and c above, substituting the sij values from Reichardt ef al. [29], and using the Meingast et al. [6] and Welp et al. [7] uniaxial stress dependences, this system of equations then may be solved, first for the case of the Meingast et al. [6] values, to give: T,-T,=

+ 103.7(a -a,) - 121.9(b -b,)

- 2.l(c -co)

(12)

while the result from the Welp et al. [7] is: T, - Tco= + lM.O(a - a,> - 96.9(b - bo) + 4.4(c - c,)

(13)

In these equations, as before, the lattice parameters are in angstroms and the temperature in kelvin. Comparison of eqns. (12) and (13), which are based on physical pressure results, with eqn. (2) shows that the physical pressure constants, k,, k2 and k3, are of greatly reduced magnitude, and that the k, and k2 values are of opposite sign relative to the chemical pressure constants. The values of k3 are seen to be of similar small magnitude but of opposite sign. The values predicted, using eqns. (5), (12) and (13) for the hydrostatic physical pressure dependence of T, for YBa,Cu,O,_, together with the linear compressibili~ data of Ludwig, are both +0.05 K GPa-‘. As expected, these values are approximately an order of magnitude less than those predicted from eqn. (2) and come within the range of reported experimental values [23-251.

Discussion

where the sjj are the elastic compliances. By substituting these relations in eqn. (7), we arrive at the first of the three desired equations. dT

dmc

in terms of strain

One then obtains:

-f

EC=k 1as

Comparison of eqn. (2) with eqns. (12) and (13) shows that nonhydrostatic stressing, whether by chemical pressure or physical pressure, could increase the critical temperature of YBa2Cu307_x more than hydrostatic pressure, per unit of stress applied. No matter how it is applied, however, eqns. (12) and (13) for YBa,Cu,O,_, show that physical stress is not likely to produce a superconductor with a T, approaching room temperature. Varying estimates of the theoretical strength of solids are available 1301,and these estimates suggest that the theoretical shear strength of solids lies between G/l0 and G/50, where G is the shear modulus [31]. Estimates of the theoretical ‘cohesive’ strength are higher and are on the order of E/IO, where E is Young’s modulus [30]. The cohesive strength estimates the maximum force needed to completely break atomic bonds in solids, not merely shear them, and is thus

347

greater than the theoretical shear strength. The shear modulus will depend upon crystal direction. Taking the shear modulus as approximately 50 GPa (the most accurate reported values for polycrystalline material range from 34.9 to 59.0 GPa [32]) gives a maximum possible applied shear stress of between 1 and 5 GPa. However, for highly perfect, dislocation-free crystals stressed in more than one direction in such a way that the shear stress is reduced (becoming zero under either hydrostatic tension or hydrostatic compression), it is the cohesive strength, not the shear strength, that becomes limiting. In such a case the maximum stress increases to E/10 which is between about 9 and 15 GPa [32]. It is worth noting too, that in the case of either unequal triaxial tension or unequal triaxial compression, the shear stresses which are produced are progressively reduced as the magnitudes and signs of the triaxial stresses approach one another, until in the result of either hydrostatic tension or hydrostatic compression no shear stress at all is produced, in which case dislocations, even if they were present, would be sessile so that material failure can only occur through the complete cleavage of all bonds simultaneously. Triaxial stresses whether tensile or compressive, although they may be unequal, can still have a hydrostatic stress component, provided that these unequal stresses are all of the same sign, and in this event higher absolute stress levels than the maximum possible shear stress are, in fact, theoretically possible, up to the limit of the maximum cohesive stress, even in the case of materials which contain dislocations. Of course, if stresses of opposite sign are applied along the three crystal axes, then via a Mohr’s circle argument it is easily seen that the resulting shear stress would be increased, not decreased, as the magnitudes of the three stresses approach one another. These considerations demonstrate that with respect to physical pressure even non-hydrostatic stressing will be able to raise the critical temperature substantially only with extraordinarily high stress levels. From eqn. (12), using an applied tensile (physical) stress of 10 GPa, which would give the exceedingly high strain of +5.49% in the udirection, the increase in critical temperature will still be only about 20 K. Of course, this value assumes linear behaviour and this assumption is unlikely to be valid over a large pressure range, and such a uniaxial strain is not in any event reasonably to be expected. Assuming instead an applied strain of l%, taken as positive in the u-direction and negative in the b- and c-directions, gives an increase in critical temperature of only about 9 K_ Furthermore, the preparation of either crystals or films that are free of domains in which the a and b axes are shifted and which are dislocationfree would be difficult, to say the least. Domains have

been shown to be present in particular in thin epitaxial films, in which case the selective stressing of either axis individually would be essentially impossible. The preparation of dislocation-free films or crystals has not yet been demonstrated. For these reasons, it would seem that only by means of chemical pressure is it potentially possible to achieve room temperature superconductivity with the compounds considered here. As we have seen, the constants relating critical temperature to lattice parameter for YBa,Cu,O,_, are approximately an order of magnitude greater for the chemical pressure case (eqn. (2)) than for the physical pressure case (eqns. (12) and (13)). Also, unlike the uniaxial pressure case, there is no easily discernible upper bound to the limit of chemical pressure effects. In the present case, the unit cell volume may be calculated from the information given in Table 1, and the cell volume may be shown to vary from 173.6 to 176.3 cubic angstroms as the critical temperature decreases from 92 K to 28 K. This change in volume can be converted to a chemical pressure via the bulk modulus, K, of crystalline YBa,Cu,O,_,. The bulk modulus is given by: K=[s,,+s,+s,,+2(s,,+s,,+S~1)]-l

(14)

which yields K=69 GPa for the sij values of Reichardt et al. [29]. With the volume change given above, the effective chemical pressure can be calculated to be approximately 1 GPa for YBa,Cu,O,_,. In the irondoped case, this value is less, being approximately 0.3 GPa. However, there appears to be no reason why the CPA method cannot be applied to the ultrahigh pressure regime above 4 GPa, unlike the case for nonhydrostatic physical pressure, in which the fracture stress sets an upper limit to the anisotropic stress which may be applied. It will be of interest to see how in the high pressure regime the signs of the chemical pressure constants, ki, change from these low pressure values. The interpretation of these anisotropic lattice parameter effects, whether derived from chemical pressure TABLE 3. Elastic stiffness (C,) in GPa for tetragonal YBaeCu@_, Reference

C,,

C12

Cl3

c33

c‘s4

G6

29 5

230 234

100 21.3

100 83.4

150 14.5

50 -

85 -

TABLE 4. Compliance (S,) for tetragonal YBa,Cu,O,_, in GPa-’ calculated from the data given in the cited references Reference

S,,

S,*

29 5

- 0.00339 0.01119 0.002 0.01765 0.00639 - 0.0013 0.000787 -0.00361 0.01105 0.00549 -

348

or physical pressure analysis, may possibly offer important insights into the role played by lattice dimensions in determining critical temperature. One conceivable avenue for rationalizing the observed differences between physical and chemical pressure effects may extend from the observation that physical pressure effects do not in general change the qu~itative character of the chemical bonds, provided a lattice phase change does not occur, while chemical pressure effects will inevitably change the detailed nature of the resulting chemical bonding orbitals. Chemical pressure changes, brought on by changes in composition, can bring about differences in the copper mixed-valency and oxygen content. In addition, the charge-carrier (hole) concentration may be altered. These are all recognized as critical factors in the mechanism of superconductivity, and physical pressure will not yield similar changes. Furthermore, others have speculated that the extent of oxygen-vacancy ionization or the appearance of acceptor centers are significant in the physics of superconductors [33], and chemical pressure, but not physical pressure, would affect these parameters as well. For YBa,Cu,O,_,, a given change in lattice parameter produced by chemical pressure is approximately an order of magnitude more effective in changing critical temperature than this same change in lattice parameter when produced by physical pressure, for pressures less than one gigapascal.

Conclusions The relationship between lattice parameter and critical temperature for YBa,Cu,O,_,, as calculated for physical pressure and stress and as calculated for chemical pressure and stress, shows in each case very strong orthorhombic anisotropy. However, in the chemical pressure case, the magnitude of the constants relating these two properties are approximately an order of magnitude higher, and of opposite sign in the chemical case compared to the physical case. The chemicai pressure analysis method appears to be an interesting adjunct to the physical pressure method of evaluating the anisotropic dependence of critical temperature on lattice parameter. Both methods offer possible insights into the control of critical temperature, and the comparison of their separate effects may also provide insight into the role played by bonding in dete~in~g critical temperature. Because the fracture strength of real crystals sets a limit to the anisotropic physical stresses that can be applied, only chemical pressure analysis will be capable of giving info~ation on the anisotropic dependence of critical temperature on lattice parameter in this ultrahigh stress regime.

Acknowledgements The many helpful comments of Professors Ulrich Goesele and Teh Tan in the preparation of this paper are gratefully acknowledged.

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