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Resistivity of high-Tc superconductors: Linear in T at constant P, non-linear at constant V
Solid State Cmmnications, Printed
Vol.
in Great Britain.
RESISTMTYOPEIGS-Tc
76, No. 8, pp. 1019-1022,
S-B:
LINEAR
1990.
IN T AT CONSTANT
00381098/90$3.00+.00 Pergamon Press plc
P, NON-LINEAR
AT CONSTAt?l'V
B. Sundqvist and B.M. Andersson Department of Experimental Physics, Umea University, S-90187 Umed, Sweden (Received 24 September 1990 by T.P. Martin)
The observed constant-pressureresistivity p of most high-T superconductors is linear in temperature T over lgrge ranges. We ghow that this is not true for the constant-volume resistivity p ; for l-2-3 YBCO the correction from p to P is -32% at 500 K relxtive to the low-T value. Since theory 'predi&s p rather than p , the observed linearity of p probably cannot be use8 as a tool to ghoose between theories for high-T= superconductivity.
theory with experiment. YBa2Cu30,_a (henceforth
The observed normal-state resistivity p of highTc superconductors usually has several anomalous
high-Tc
properties: - The measured p, or at least the measured abplane resistivity pab, is linear [l-4] in temperature
T between Tc and 300 K, with no
to this
rule is YBa2Cu408
indica-
(here denoted
with
Cu.
The
ratio
V(300 K)fl(O K) between the crystal volumes at 300 K and 0 K is about 1.0073 for l-2-3 [ll] and 1.0098 for Cu 1121, while the bulk moduli B are practically identical at 157 and 151 GPa, respectively [13,14]. In order to reduce V(300 K) to V(0 K) we must apply 1.15 GPa (11.5 kbar) to l2-3 and 1.48 GPa to Cu._pf~;;=uf; [ ;;ffi;; ents of p are -0.12 GPa t
tion of the stronger Tn (n = 2-5) dependence expected for normal metallic superconductors well below the Debye temperature s. The excep tion
superconductors,
To see this, we compare l-2-3), the archetype of
l-
2-4], in which p conforms to a Bloch-Griineisen type function of T [5,6]. - The magnitude of p is very large, but p is still linear at high T with no evidence for resistivity saturation [l-3]. - The volume dependence of p is very large 16-81 (dlnp/dlnv = 10-20). m addition, the anisotropy [1,3,4] in p is large-$ and the c-axis resistivity pc has an un-
-0.0186 GPa-’ for Cu [15], giving correction factors of -14% for l-2-3 but only -2.7% for Cu. While the latter value is usually negligible, the former is not, especially since these factors are practically zero up to = 100 K and increase almost linearly with T above 200 K. Carrying out the correction from pp to pv in
These usual (anomalous) temperature dependence. properties are often used as evidence for or against various theoretical models [l] for the resistivity and/or mechanisms for superconductivitv in these materials. WGpoint out here that the observed linear T dewndence of D should not in itself be used to ch-mse between’different theories. The guantity actually measured is always the isobaric (constant pressure) resistivity p , while theory pre-
simple nor straiqhtforward. Both the linear the-ml expansivity a and the compressibility K are anisotropic, and available data differ siqnificantly between different sources. In priiciple, we need to measure or calculate how each component of the resistivity tensor depends on changes in the length of each of the crystal axes. We should then use anisotropic thermal expansion and compressibility data to calculate the correction factors, assuming the dimensions of the crystal to be independent of T. Since the necessary data are not available we cannot carry out this procedure, and as a first approximation we have used the volume thermal expansivity (=3a) and the volume compressibility K = l/e to carry out the correction. Choosing reliable data for the correction is another problem. Below 300 K many sets of data for a exist in the literature, but due to the effect of anisotroPy and grain structure data from different sources or for different specimens rarely agree. To avoid these problems we have used single-crystal x-ray data [ll] in this range. Above 300 K data are more rare, and since the data of White et al. [16] and those of Salomans et al. [17] are in excellent agreement we have used an average of these. There are even
l-2-3
dicts the isochoric (cons& volume) resistivibetween the isobaric spectY Pv. The difference ific heat capacity cp and the isochoric value cv are well
known, and
cv
from cp when a comparison
is routinely
computed
with theory must
be
made. For transport properties, such as p or the thembal conductivity, X, the correction is rarealthouqh the effect is readilv Iv imoortaht. o&se&able in metals (91 and sometimes very i& portant for X in soft materials [lo]. For highdifference superconductors, however, the Tc between pp and pv seems to be quite large and should be
taken into
account before
comparing 1019
(or
other high-Tc
materials)
is
neither
1020
RESISTIVITY OF HIGH-T c SUPERCONDUCTORS
larger differences between d a t a for K from dlfferent sources [13,18-20]. Since we are i n t e r e s t ed in true crystal volume changes we do not use the low values of B given by ultrasonic measurements [16], but even high-pressure x-ray data for B range from 95 [20] to almost 200 GPa [18]. We use the value 157 GPa given by Olsen et al. [13], because a) this value is close to the average value from all determinations, b) it agrees well with the value 155 GPa found for Gd [21] and Yb [22] based 1-2-3 materials, and c) it is based on measurements up to 52 GPa, significantly higher than in other experiments, implying a higher accuracy. We allow B to decrease with increasing T at a rate [21] of I0 GPa per 150 K; this is a small correction. Finally, it is well known from the literature that the pressure coefficient of p is always near dlnp/dp - -0.12
TABLE
I: C o r r e c t i o n
T (K)
pp -
ixl0-6T
Rcm
(dashed line),
which
is
400 450 500
0
//
I
/
0.4
/
/
/
////
0.2 I...if) W
I I I
n,.
0.0
i
200
,
T
I
(K)
400
Fig. i. Isobaric (dashed line) and isochoric (full curve) resistivity of 1-2-3 vs. T.
1.0
I
I
0 E
.C 0 E >I--
0.5
300
K, but in this range the results for C are surprisingly close to those found above using bulk properties (Table I). To what extent is the above calculation realistic? For the Y-based series Y2Ba4CU6+nOl4+n with n - 0,1,2 (1-2-3, 2-4-7, 1-2-4) it is well known that smell changes in the detailed crystal structure, brought about by substitution, oxidation/reduction, or pressure, result in large changes in T c. In particular, T c depends lineardistance d between the Cu
-
E
is basically a two-dimensional transport property. For single-crystal 1-2-3, Crommie et al. [24] have recently found that Pab is independent
ly [25] on the
-
E .C 0
The above result should be considered a first approximation only, since we have used bulk values for ~ and K to correct P=Pab' which
~ab from the same source [ii] as before. Unfort-
1.14(6) 1.13 1.13 1.12 1.11 1.10 1.08 1.04 1.000
I
linear in T anywhere above 150 K. In figure 2 we show similar data for dense sintered 1-2-4; details about this correction are given elsewhere [6]. We believe that similar results will be found for other high-T c materials.
of p under uniaxial c-axis compression. This indicates that we can, as an alternative, use a two-dimensional model, and we have repeated the calculation using an ab-plane compressibility [13,18,19,22] Kab - 3.2xi0-3 GPa -I and data for
Cab
1.16(2) 1.15 1.14 1.13 1.12 1.10 1.08 1.04 1.000 0.95 0.90 0.85 0.79
a
compared to theory; Pv is not even approximately
unately, Cab was studied only between 0 and
%
0 90 100 125 150 175 200 250 300 350
le crystals [23]. Also, dlnp/dp changes little with T: Most available data below 300 K agree well with the function dlnp/dp - (-0.10-8x10-5T) GPa -1 given earlier [7] for the range 300-500 K. We thus use this function to describe dlnp/dp vs. T. The correction factor C - pv/PpwaS calcula-
realistic value [1-4] for 1-2-3. (Alternatively, we could have chosen a set of experimental data from literature [2-4].) The full curve shows the corrected values for Pv' which is what should be
factor C - pv/Pp, calculated
relative to 300 K using either bulk values for = and K (Cv) or ab-plane values aab and Kab (Cab).
GPa-I for both polycrystals and for pab in sing-
ted for 1-2-3 as a function of T up to 500 K. The results, normalized to unity at 300 K for practical reasons, are shown in Table I. (Choosing C - 1 at 0 K gives C - 0.681 at 500 K). The correction is clearly too large to be neglected. we illustrate this in Figure I, where we assume
VOI. 76, NO. 8
atoms
j-if) llJ
,
0.0 0
I
2OO
,
T (K)
I
4OO
Fig. 2. Isobaric (dots) and isochoric (curve) resistivity of 1-2-4 vs. T (from Ref. 8).
Vol. 76, NO. 8
RESISTIVITY OF HIGH-T SUPERCONDUCTORS 1021 c in the CuO planes and the apical 0 atoms. This be another unlikely coincidence if this would be is usually assumed to be a result of charge true for most high-Tc materials, and for 1-2-3 transfer between chains and planes, resulting in it is also contradicted by the results of Crcman increased carrier (hole) density as d decreamie et al. [24]. ses. Neutron scattering studies on 1-2-4 under Our results imply that a direct comparison pressure have indicated [26] a very large debetween theoretical predictions and experimental crease in d, which is believed [26,27] to be the data for p might be of little value for Y-based cause of the extremely rapid increase in T c with high-Tc superconductors, and probably also for p for this comp~m~ (and 2-4-7). It could thus most other high-Tc materials. We further conbe argued that the effects of T and p on V are clude that the true T dependence of Pv is not as qualitatively different: The distance d is almost [26] independent of T when the lattice exsimple as cc~uonly believed, and that the obserpands, while it is very sensitive to changes in ved linearity of p(T) must not be used as a V brought about by pressure. The carrier density general constraint on theories for the mechanism should thus be constant with T but increase with behind high-Tc superconductivity. Note, however, p, causing the observed rapid drop in p with p. that the correction mainly affects the high-T In such a model it is clearly not realistic to data, and that it does not rule out linear terms try to correct pp to Pv in the simple way indicor components in p: At ~ T (T_<100 K), the corated above. However, the model is not universalrection factor is approximately constant, prely accepted, and recent high-precision neutron serving the observed linear behaviour. The "most studies [28] question its validity. Also, there linear" materials at low T are the extremely anare some other inconsistencies in this picture: isotropic Bi-based [i] materials. The linearity 1 ) Hall effect measurements [29 ] on sintered Iof p in these materials might simply be due to 2-3 show the carrier density to be independent their two-dimensional (2D) structure, since it of pressure; on the other hand this density is has been suggested that the electron-phonon resproportional to T at normal pressure. istivity in 2D materials should remain linear in 2) Crommie et al. [24] find dlnPab/dC = 0. This T as T-~ [33]. We also note that an analysis of does not agree with the above model, in which the data for 1-2-4 in Figure 2 showed that Pv changes in c should lead to large changes in d was well described by a Bloch-GrUneisen type and thus in doping level and p. function PBG(T/SD) [6]. At a l l ~ T and p, a 3) p changes much more rapidly with V in 1-2-3 function • than in 1-2-4 [6], but dTc/d p is an order of magnitude larger in the latter material (and 2P-1 " (PBG + Po)-I + o, (1) 4-7 [27]) than in the former. Unless some other mechanism strongly decreases the scattering procould be fitted to the data with - 1 = 1500 ~ocm bability in 1-2-3 under pressure, these facts and an rms deviation similar to the experimental are incompatible. To explain the difference in scatter. Unfortunately, Eq. (i) cannot distingudTc/dp Tallon and Lusk [27] suggest that charge ish between resistivity saturation in Pab and transfer can only take place from double chains the effect of Pc in a polycrystalline material, under pressure, which still leaves the large since it is valid for both cases [6]. The data dlnp/dp of 1-2-3 unexplained. for 1-2-3 in Figure i are in general agreement 4) Strong support for the model comes from the with the same model; since the isobaric Pab is high pressure neutron data of Kaldis et al. [26] for 1-2-4. However, from their data we calculate known to be linear in T, the isochoric Pab must B = 1.5 TPa, three times that of diamond, and 13 saturate at high T in 1-2-3. This fact would times larger than the recent result of Ludwig et drastically increase previous estimates [2] of al. [30]. This is surprising in view of the high the strength of the electron-phonon interaction accuracy claimed for the shift in Cu-O distance in 1-2-3, in general agreement with the results with pressure. of recent calculations [34]. (The runaway incre5) Finally, pa,mn measurements correlate T c with ase in , at very high T [2] is caused by other a mode near 500 cm-I at normal pressure [31], mechanisms.) Other models [i] also predict a but not under higher pressures [32]. negative curvature in p for high-T c superconducIn the light of these inconsistencies we tors, however, and pp data for other groups of conclude that our simple correction is probably as realistic as is presently possible, in spite materials should be reduced to constant v before of the conflicts between the model and the we try to draw final conclusions about p(T). results. Finally we note that we still cannot rule The linearity of p vs. T in high-Tc materiout the standard model, in which the carrier density varies with p but not with T, as the als is at present an accepted truth. However, source of the large apparent correction factor. according to our results, above about I00 K this We suggest that further high pressure experiis merely a coincidence arising from a fortuitiments must be carried out to finally choose ous balance between the T dependence of Pv and between the models: Hall effect data on single thermal expansion effects, we do not understand crystals are needed, as are more resistivity how such a balance can be so finely tuned as to data under uniaxial load, and in particular we produce a linear T dependence for almost all believe that further neutron scattering studies h i g h T c materials over large ranges in T. Since should be performed on 1-2-3, 2-4-7, and 1-2-4 to higher pressures, in order to obtain more the ratios =c/=ab (= 1.2) and Kc/Kab (-0.8) are accurate results for d vs. p. very different in 1-2-3, it is possible that Acknowl~tsThis work was financially thermal expansion effects might cancel, giving supported by the Swedish Natural Science Pp = "v' if dPab/d(ab) and dPab/dC were both Research Council (NFR) and by the Swedish large but of different signs. However, it would National Board for Technical Development (S~'J).
1022
RESISTIVITY OF HIGH-T
SUPERCONDUCTORS
Vol. 76, NO. 8
C
1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
13. 14. 15. 16.
P.B. Allen, Z. Fisk & A. Migliori, in Physlcal Properties of High Te~xerature Superconductors, vol I (Edited by D.M. Ginsberg), p. 213, World Scientific, Singapore (1989). M. Gurvlch & A.T. Fiory, Phys. Rev. Lett. 59, 1337 (1987). S. Martin et al., Phys. Rev. Lett. 60, 2194 (1988). S.W. Tozer et al., Phys. P~v. Lett. 59, 1768 (1987). S. Martin et al., Phys. Rev. B 39, 9611 (1989). B.M. Andersson et al., Physica C 170, 521 (1990). B. Sunckjvlst, Solid State Ccmmun. 66, 623 (1988). B.M. Andersson et al., High Pressure Res. 3, 123 (1990). R.A. Matula & P.G. Klemens, High Temp. High Pressures 10, 105 (1978). R.G. Ross et al., Rep. Progr. Phys. 47, 1347 (1984). P.M. Horn et al., Phys. Rev. Lett. 59, 2772 (1987); H. Yuo et al., Phys. Rev. B 38, 9213 (1988). Y.S. Touloukian et al.: Thermophysical Properties of Matter, vol 12: Thermal Expansion - Metals and Alloys (IPI/Plenum, New York 1975) p. 77. J.S. 01sen et al., Z. Phys. B 72, 165 (1988). S.N. Vaidya & G.C. Kennedy, J. Phys. Chem. Solids 31, 2329 (1970). P.W. Brldgman, Proc. Am. Acad. Arts Sci. 72, 157 (1938). G.K. White et al., J. Phys. C 21, L631 (1988).
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
E. Salommm e t a l . , l~rslca B 145, 253 (1987). S. Block e t a l . , Adv. Ceram. Mater. 2, 601 (1987). W.H. Fietz, M.R. D i e t r i c h & J. Ecke, Z. Phys. B 69, 17 (1987). N. V i c t o r Jaya et a l . , So lid State Cr~m,n. 67, 51 (1988). J. Ecke et al., Physica C 153-155, 954 (1988). W.H. Fletz et al., J. Low Temp. Phys. (1990, accepted for publication). M. Konczykowski, F. Rulller-Albenque & G. Co111n, Physlca C 153-155, 1365 (1988). M.F. Crc~m1_e et al., Phys. Rev. B 39, 4231 (1989). R.J. Cava et al., Physica C 156, 523 (1988). E. Kaldls et al., Physlca C 159, 668 (1989). J.L. Tallon & J. Lusk, Physica C 167, 236 (1990). G.H. Kwel et al., Physica C 169, 217 (1990). I.D. Parker & R.H. Friend, J. Phys. C 21, L345 (1988). H.A. Ludwig et al., Physlca C 167, 335 (1990). M. Kaklhana, L. BSrjesson, and S. Erlksson, Physica B 163-166, 1245 (1990). K. Syassen et al., physlca C 153-155, 264 (1988). R. Micnas, J. Rannlnger, & S. Robaszkiewicz, Phys. Rev. B 36, 4051 (1987). J.H. Kim et al., Phys. Rev. B 40, 11378 (1989); R.E. Cohen, W.E. Pickett & H. Krakauer, Phys. Rev. Lett. 64, 2575 (1990).
Vol.
in Great Britain.
RESISTMTYOPEIGS-Tc
76, No. 8, pp. 1019-1022,
S-B:
LINEAR
1990.
IN T AT CONSTANT
00381098/90$3.00+.00 Pergamon Press plc
P, NON-LINEAR
AT CONSTAt?l'V
B. Sundqvist and B.M. Andersson Department of Experimental Physics, Umea University, S-90187 Umed, Sweden (Received 24 September 1990 by T.P. Martin)
The observed constant-pressureresistivity p of most high-T superconductors is linear in temperature T over lgrge ranges. We ghow that this is not true for the constant-volume resistivity p ; for l-2-3 YBCO the correction from p to P is -32% at 500 K relxtive to the low-T value. Since theory 'predi&s p rather than p , the observed linearity of p probably cannot be use8 as a tool to ghoose between theories for high-T= superconductivity.
theory with experiment. YBa2Cu30,_a (henceforth
The observed normal-state resistivity p of highTc superconductors usually has several anomalous
high-Tc
properties: - The measured p, or at least the measured abplane resistivity pab, is linear [l-4] in temperature
T between Tc and 300 K, with no
to this
rule is YBa2Cu408
indica-
(here denoted
with
Cu.
The
ratio
V(300 K)fl(O K) between the crystal volumes at 300 K and 0 K is about 1.0073 for l-2-3 [ll] and 1.0098 for Cu 1121, while the bulk moduli B are practically identical at 157 and 151 GPa, respectively [13,14]. In order to reduce V(300 K) to V(0 K) we must apply 1.15 GPa (11.5 kbar) to l2-3 and 1.48 GPa to Cu._pf~;;=uf; [ ;;ffi;; ents of p are -0.12 GPa t
tion of the stronger Tn (n = 2-5) dependence expected for normal metallic superconductors well below the Debye temperature s. The excep tion
superconductors,
To see this, we compare l-2-3), the archetype of
l-
2-4], in which p conforms to a Bloch-Griineisen type function of T [5,6]. - The magnitude of p is very large, but p is still linear at high T with no evidence for resistivity saturation [l-3]. - The volume dependence of p is very large 16-81 (dlnp/dlnv = 10-20). m addition, the anisotropy [1,3,4] in p is large-$ and the c-axis resistivity pc has an un-
-0.0186 GPa-’ for Cu [15], giving correction factors of -14% for l-2-3 but only -2.7% for Cu. While the latter value is usually negligible, the former is not, especially since these factors are practically zero up to = 100 K and increase almost linearly with T above 200 K. Carrying out the correction from pp to pv in
These usual (anomalous) temperature dependence. properties are often used as evidence for or against various theoretical models [l] for the resistivity and/or mechanisms for superconductivitv in these materials. WGpoint out here that the observed linear T dewndence of D should not in itself be used to ch-mse between’different theories. The guantity actually measured is always the isobaric (constant pressure) resistivity p , while theory pre-
simple nor straiqhtforward. Both the linear the-ml expansivity a and the compressibility K are anisotropic, and available data differ siqnificantly between different sources. In priiciple, we need to measure or calculate how each component of the resistivity tensor depends on changes in the length of each of the crystal axes. We should then use anisotropic thermal expansion and compressibility data to calculate the correction factors, assuming the dimensions of the crystal to be independent of T. Since the necessary data are not available we cannot carry out this procedure, and as a first approximation we have used the volume thermal expansivity (=3a) and the volume compressibility K = l/e to carry out the correction. Choosing reliable data for the correction is another problem. Below 300 K many sets of data for a exist in the literature, but due to the effect of anisotroPy and grain structure data from different sources or for different specimens rarely agree. To avoid these problems we have used single-crystal x-ray data [ll] in this range. Above 300 K data are more rare, and since the data of White et al. [16] and those of Salomans et al. [17] are in excellent agreement we have used an average of these. There are even
l-2-3
dicts the isochoric (cons& volume) resistivibetween the isobaric spectY Pv. The difference ific heat capacity cp and the isochoric value cv are well
known, and
cv
from cp when a comparison
is routinely
computed
with theory must
be
made. For transport properties, such as p or the thembal conductivity, X, the correction is rarealthouqh the effect is readilv Iv imoortaht. o&se&able in metals (91 and sometimes very i& portant for X in soft materials [lo]. For highdifference superconductors, however, the Tc between pp and pv seems to be quite large and should be
taken into
account before
comparing 1019
(or
other high-Tc
materials)
is
neither
1020
RESISTIVITY OF HIGH-T c SUPERCONDUCTORS
larger differences between d a t a for K from dlfferent sources [13,18-20]. Since we are i n t e r e s t ed in true crystal volume changes we do not use the low values of B given by ultrasonic measurements [16], but even high-pressure x-ray data for B range from 95 [20] to almost 200 GPa [18]. We use the value 157 GPa given by Olsen et al. [13], because a) this value is close to the average value from all determinations, b) it agrees well with the value 155 GPa found for Gd [21] and Yb [22] based 1-2-3 materials, and c) it is based on measurements up to 52 GPa, significantly higher than in other experiments, implying a higher accuracy. We allow B to decrease with increasing T at a rate [21] of I0 GPa per 150 K; this is a small correction. Finally, it is well known from the literature that the pressure coefficient of p is always near dlnp/dp - -0.12
TABLE
I: C o r r e c t i o n
T (K)
pp -
ixl0-6T
Rcm
(dashed line),
which
is
400 450 500
0
//
I
/
0.4
/
/
/
////
0.2 I...if) W
I I I
n,.
0.0
i
200
,
T
I
(K)
400
Fig. i. Isobaric (dashed line) and isochoric (full curve) resistivity of 1-2-3 vs. T.
1.0
I
I
0 E
.C 0 E >I--
0.5
300
K, but in this range the results for C are surprisingly close to those found above using bulk properties (Table I). To what extent is the above calculation realistic? For the Y-based series Y2Ba4CU6+nOl4+n with n - 0,1,2 (1-2-3, 2-4-7, 1-2-4) it is well known that smell changes in the detailed crystal structure, brought about by substitution, oxidation/reduction, or pressure, result in large changes in T c. In particular, T c depends lineardistance d between the Cu
-
E
is basically a two-dimensional transport property. For single-crystal 1-2-3, Crommie et al. [24] have recently found that Pab is independent
ly [25] on the
-
E .C 0
The above result should be considered a first approximation only, since we have used bulk values for ~ and K to correct P=Pab' which
~ab from the same source [ii] as before. Unfort-
1.14(6) 1.13 1.13 1.12 1.11 1.10 1.08 1.04 1.000
I
linear in T anywhere above 150 K. In figure 2 we show similar data for dense sintered 1-2-4; details about this correction are given elsewhere [6]. We believe that similar results will be found for other high-T c materials.
of p under uniaxial c-axis compression. This indicates that we can, as an alternative, use a two-dimensional model, and we have repeated the calculation using an ab-plane compressibility [13,18,19,22] Kab - 3.2xi0-3 GPa -I and data for
Cab
1.16(2) 1.15 1.14 1.13 1.12 1.10 1.08 1.04 1.000 0.95 0.90 0.85 0.79
a
compared to theory; Pv is not even approximately
unately, Cab was studied only between 0 and
%
0 90 100 125 150 175 200 250 300 350
le crystals [23]. Also, dlnp/dp changes little with T: Most available data below 300 K agree well with the function dlnp/dp - (-0.10-8x10-5T) GPa -1 given earlier [7] for the range 300-500 K. We thus use this function to describe dlnp/dp vs. T. The correction factor C - pv/PpwaS calcula-
realistic value [1-4] for 1-2-3. (Alternatively, we could have chosen a set of experimental data from literature [2-4].) The full curve shows the corrected values for Pv' which is what should be
factor C - pv/Pp, calculated
relative to 300 K using either bulk values for = and K (Cv) or ab-plane values aab and Kab (Cab).
GPa-I for both polycrystals and for pab in sing-
ted for 1-2-3 as a function of T up to 500 K. The results, normalized to unity at 300 K for practical reasons, are shown in Table I. (Choosing C - 1 at 0 K gives C - 0.681 at 500 K). The correction is clearly too large to be neglected. we illustrate this in Figure I, where we assume
VOI. 76, NO. 8
atoms
j-if) llJ
,
0.0 0
I
2OO
,
T (K)
I
4OO
Fig. 2. Isobaric (dots) and isochoric (curve) resistivity of 1-2-4 vs. T (from Ref. 8).
Vol. 76, NO. 8
RESISTIVITY OF HIGH-T SUPERCONDUCTORS 1021 c in the CuO planes and the apical 0 atoms. This be another unlikely coincidence if this would be is usually assumed to be a result of charge true for most high-Tc materials, and for 1-2-3 transfer between chains and planes, resulting in it is also contradicted by the results of Crcman increased carrier (hole) density as d decreamie et al. [24]. ses. Neutron scattering studies on 1-2-4 under Our results imply that a direct comparison pressure have indicated [26] a very large debetween theoretical predictions and experimental crease in d, which is believed [26,27] to be the data for p might be of little value for Y-based cause of the extremely rapid increase in T c with high-Tc superconductors, and probably also for p for this comp~m~ (and 2-4-7). It could thus most other high-Tc materials. We further conbe argued that the effects of T and p on V are clude that the true T dependence of Pv is not as qualitatively different: The distance d is almost [26] independent of T when the lattice exsimple as cc~uonly believed, and that the obserpands, while it is very sensitive to changes in ved linearity of p(T) must not be used as a V brought about by pressure. The carrier density general constraint on theories for the mechanism should thus be constant with T but increase with behind high-Tc superconductivity. Note, however, p, causing the observed rapid drop in p with p. that the correction mainly affects the high-T In such a model it is clearly not realistic to data, and that it does not rule out linear terms try to correct pp to Pv in the simple way indicor components in p: At ~ T (T_<100 K), the corated above. However, the model is not universalrection factor is approximately constant, prely accepted, and recent high-precision neutron serving the observed linear behaviour. The "most studies [28] question its validity. Also, there linear" materials at low T are the extremely anare some other inconsistencies in this picture: isotropic Bi-based [i] materials. The linearity 1 ) Hall effect measurements [29 ] on sintered Iof p in these materials might simply be due to 2-3 show the carrier density to be independent their two-dimensional (2D) structure, since it of pressure; on the other hand this density is has been suggested that the electron-phonon resproportional to T at normal pressure. istivity in 2D materials should remain linear in 2) Crommie et al. [24] find dlnPab/dC = 0. This T as T-~ [33]. We also note that an analysis of does not agree with the above model, in which the data for 1-2-4 in Figure 2 showed that Pv changes in c should lead to large changes in d was well described by a Bloch-GrUneisen type and thus in doping level and p. function PBG(T/SD) [6]. At a l l ~ T and p, a 3) p changes much more rapidly with V in 1-2-3 function • than in 1-2-4 [6], but dTc/d p is an order of magnitude larger in the latter material (and 2P-1 " (PBG + Po)-I + o, (1) 4-7 [27]) than in the former. Unless some other mechanism strongly decreases the scattering procould be fitted to the data with - 1 = 1500 ~ocm bability in 1-2-3 under pressure, these facts and an rms deviation similar to the experimental are incompatible. To explain the difference in scatter. Unfortunately, Eq. (i) cannot distingudTc/dp Tallon and Lusk [27] suggest that charge ish between resistivity saturation in Pab and transfer can only take place from double chains the effect of Pc in a polycrystalline material, under pressure, which still leaves the large since it is valid for both cases [6]. The data dlnp/dp of 1-2-3 unexplained. for 1-2-3 in Figure i are in general agreement 4) Strong support for the model comes from the with the same model; since the isobaric Pab is high pressure neutron data of Kaldis et al. [26] for 1-2-4. However, from their data we calculate known to be linear in T, the isochoric Pab must B = 1.5 TPa, three times that of diamond, and 13 saturate at high T in 1-2-3. This fact would times larger than the recent result of Ludwig et drastically increase previous estimates [2] of al. [30]. This is surprising in view of the high the strength of the electron-phonon interaction accuracy claimed for the shift in Cu-O distance in 1-2-3, in general agreement with the results with pressure. of recent calculations [34]. (The runaway incre5) Finally, pa,mn measurements correlate T c with ase in , at very high T [2] is caused by other a mode near 500 cm-I at normal pressure [31], mechanisms.) Other models [i] also predict a but not under higher pressures [32]. negative curvature in p for high-T c superconducIn the light of these inconsistencies we tors, however, and pp data for other groups of conclude that our simple correction is probably as realistic as is presently possible, in spite materials should be reduced to constant v before of the conflicts between the model and the we try to draw final conclusions about p(T). results. Finally we note that we still cannot rule The linearity of p vs. T in high-Tc materiout the standard model, in which the carrier density varies with p but not with T, as the als is at present an accepted truth. However, source of the large apparent correction factor. according to our results, above about I00 K this We suggest that further high pressure experiis merely a coincidence arising from a fortuitiments must be carried out to finally choose ous balance between the T dependence of Pv and between the models: Hall effect data on single thermal expansion effects, we do not understand crystals are needed, as are more resistivity how such a balance can be so finely tuned as to data under uniaxial load, and in particular we produce a linear T dependence for almost all believe that further neutron scattering studies h i g h T c materials over large ranges in T. Since should be performed on 1-2-3, 2-4-7, and 1-2-4 to higher pressures, in order to obtain more the ratios =c/=ab (= 1.2) and Kc/Kab (-0.8) are accurate results for d vs. p. very different in 1-2-3, it is possible that Acknowl~tsThis work was financially thermal expansion effects might cancel, giving supported by the Swedish Natural Science Pp = "v' if dPab/d(ab) and dPab/dC were both Research Council (NFR) and by the Swedish large but of different signs. However, it would National Board for Technical Development (S~'J).
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