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Debye-temperature correlation in (La-M)2CuO4 superconductors
PHYSICA
Physica C 190 (1991) 129-130 North-Holland
Critical-temperature/Debye-temperature correlation in (La-M) 2CuO4 superconductors Hassel Ledbetter, Sudook Kim and Alexana Roshko National Institute of Standards and Technology, Boulder, CO 80303, USA
We consider three Lat.ssMoA5CuO4 compounds with M = Ca, Ba, Sr. By measuring ultrasonic sound velocities, we determined the acoustic Debye temperature OD, which at T = 0 K equals the specific-heat Debye temperature. Like conventional BCS materials, Tc depends regularly on Or,. Unlike BCS materials where Tc decreases with increasing Or, (lattice softening increases Tc), Tc increases with increasing OD. This difference reflects in how the electron-phonon parameter 2 depends on OD- In McMillan's analysis, 2~ O~ 2. Thus, for conventional materials, the Tc-OD curve increases, passes through a maximum at 2= 2, and then decreases. Keeping the usual Tc(Or,,2) models requires us to abandon the 3.~ O~ 2 relationship and adopt 2 ~ O~, where n > -2.
Because the BCS model remains disputed in its ability to explain the high Tcs found in the Bednorz-Miiller-discovered oxide superconductors, relationships between latticevibration (phonon) properties and Tc provide much interest. The quintessential phonon parameter is the Debye characteristic temperature OD- We can determine OD by measuring longitudinal and transverse sound velocities v~and vt, calculating the mean sound velocity 3Vm3 =V1-3 +2Vt-3,
( 1)
and using the well-known relationship
OD= (h/k) (3/4~ Va)1/3Vm.
(2)
Here, h and k take their usual meanings, and Va denotes average atomic volume. The present study determined OD for polycrystalline Lai.ssM0.1sCuO4, with M = C a , Ba, Sr. Hot-pressed specimens were prepared from powders obtained by a freezedrying acetate process. We obtained their Tcs from electricalresistance-temperature measurements, using the zero-resistance value. To get z4 and v,, we used a pulse-echo-overlap method described elsewhere [ 1,2 ]. A previous study [ 31 on La2CuO 4 and La,.ssSroAsCuO 4 showed that measurements on polyerystals agreed well with monocrystai measurements averaged over the directions. Table 1 shows the principal results for the void-free state [ 2 ]. Mass density p and average atomic volume Va were obtained by X-ray diffraction. The effective electron-phonon parameter it was calculated from Kresin's [4] approximation of the Eliashberg equations. Figure 1 shows the Tc-Or, diagram.
Table 1 Properties of studied Lal.ssMo.z5CUO4superconductors
P. . . . (g/cm3) Px-ray (g/cm 3) a (A) c (A) Va (-/k3) v~(cm/l.tS) v, (cm/las) OD (K) Tc (K) 2~tr
Ca
Ba
Sr
5.893 6.883 3.7863(7) 13.17(2) 13.49 0.471 0.261 364 19 0.50
6.493 7.077 3.7846(2) 13.288(4) 13.59 0.513 0.288 401 28 0.60
6.572 6.989 3.77814(7) 13.231(2) 13.49 0.568 0.310 432 36 0.68
The T¢-Oo relationship shown in fig. 1 is surprising. First, it shows an essentially linear dependence of Tc on Or,, the principal phonon parameter. Second, it shows that Tc increases with increasing OD. McMiltan's [ 5 ] analysis of conventional superconductors shows that the electron-phonon parameter 2 varies as Off 2 and, that for 2<2, Tc decreases with increasing OD. Simple analysis shows that if we retain the usual BCS-model relationship for Tc (O,, 2), we must abandon McMillan's harmonic relationship 2 ~ Off 2 and replace it by it ~ O~,, where n > -it. Thus, if it = 1, then n > - 1, contrary to McMillan's n = - 2. For these compounds, T~ increases monotonically with increasing Oo. Thus, the task of maximizing Tc becomes one of maximizing Oo. First, we can consider the Einstein [ 6 ] factorization of the characteristic temperature:
0921-4534/91/$03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
130
H. Ledbetter et aL / Critical-temperature/Debye-temperature correlation
3.792
40
ui re
~.
~ L a l " 8 s M 0 " 1 5 C u O ~ ~4 / ~ ' ~
3.788
30
re
w n
2o
,3
S
3.784
~- --~
I-z I-
re b-
A °5
re I/J l-
ee < a. .d ..I
3.780
10 3a
0 I 360
.
,I 400
380
E z
<3 Sr ,[
Ba
, 420
i/d O
1.776 440
DEBYE TEMPERATURE, O D (K) Fig. 1. Dependence of critical temperature Tc and unit-cell a-parameter on Debye characteristic temperature OD. Note that both properties behave contrary to intuition, departing from the periodic-table Ca, St, Ba sequence.
100 90 m" ee "--s F< tw LU o w
vle%cu3ox
80
stiffness-increase sequence is Ba, St, Ca, La, Cu. This sequence suggests that Ca should effect the largest increase in OD. Yet, it effects the smallest, and thus eq. (3) fails here. Second, we note a force-constant-model-based correlation by Nakamura [ 7 ]: OomKM-l/2rff2.
(4)
Here, K denotes a constant, M atomic mass, and ro cationanion nearest-neighbor spacing. Considering the three doping elements themselves, Ca possesses the highest M - 1 / 2 V ~ 2 / 3 value; again wrong for explaining a high Oo in the (La-M)2CuO4 compounds. It is intriguing that substituting Ca, Ba, Sr for La produces peculiar effects in other basic physical properties. For example, the unit-cell a-parameter decreases in the sequence Ca, Ba, Sr instead of the expected sequence Ba, Sr, Ca, (see fig. 1 ). Another peculiar substitution effect is that both Ba and Sr, larger atoms than La, decrease the a-parameter. YiBa2Cu3Ox shows a similar dependence of Tc on 6)0, as is shown in fig. 2. We took Tc values from Poulsen and coworkers [ 8 ] and the specific-heat OD values from Nakazawa and Ishikawa [9]. We fit a linear equation to their nine OD measurements. (For Y~Ba2Cu3Ox compounds, the elastic OD cannot be determined accurately because of defects, perhaps microcracks, that reduce the apparent ultrasound velocities [101.
7o References
60
z
_o ffJ z
b-
50 4o 300
i
...........
350
DEBYE TEMPERATURE,
i 400
450 O D (K)
Fig. 2. Companion figure for YtBa2Cu3Ox showing results for various oxygen contents in the orthorhombic-I phase. OD = KVIa/6 B I / 2 M - I/2 .
(3)
Here, K denotes a constant, V~ atomic volume, B bulk modulus, and M atomic mass. Compared with Cu, all four other cations are large and highly compressible. The elemental
[ 1] H. Ledbetter, N. Frederick and M. Austin, J. Appl. Phys. 51 (1980) 305. [2] H. Ledbetter, M. Austin, S. Kim and M. Lei, J. Mater. Res. 2 (1987) 786. [3] H. Ledbetter, S. Kim, C. Violet and J. Thompson, Physica C 162-164 (1989) 460. [4] V. Kresin, Phys. Lett. A122 (1987) 434. [5] W. McMillan, Phys. Rev. 167 (1968) 331. [6] A. Einstein, Ann. Phys. 39 (1910) 170. [ 7 ] T. Nakamura, Jpn. J. Appl. Phys. 20 ( 1981 ) L653. [8] H. Poulsen, N. Andersen, J. Andersen, H. Bohr and O. Mouritsen, Nature 349 ( 1991 ) 594. [ 9 ] N. Nakazawa and M. Ishikawa, Physica C 162-164 (1989) 83. [ 10 ] H. Ledbetter, J. Mater. Res., forthcoming.
Physica C 190 (1991) 129-130 North-Holland
Critical-temperature/Debye-temperature correlation in (La-M) 2CuO4 superconductors Hassel Ledbetter, Sudook Kim and Alexana Roshko National Institute of Standards and Technology, Boulder, CO 80303, USA
We consider three Lat.ssMoA5CuO4 compounds with M = Ca, Ba, Sr. By measuring ultrasonic sound velocities, we determined the acoustic Debye temperature OD, which at T = 0 K equals the specific-heat Debye temperature. Like conventional BCS materials, Tc depends regularly on Or,. Unlike BCS materials where Tc decreases with increasing Or, (lattice softening increases Tc), Tc increases with increasing OD. This difference reflects in how the electron-phonon parameter 2 depends on OD- In McMillan's analysis, 2~ O~ 2. Thus, for conventional materials, the Tc-OD curve increases, passes through a maximum at 2= 2, and then decreases. Keeping the usual Tc(Or,,2) models requires us to abandon the 3.~ O~ 2 relationship and adopt 2 ~ O~, where n > -2.
Because the BCS model remains disputed in its ability to explain the high Tcs found in the Bednorz-Miiller-discovered oxide superconductors, relationships between latticevibration (phonon) properties and Tc provide much interest. The quintessential phonon parameter is the Debye characteristic temperature OD- We can determine OD by measuring longitudinal and transverse sound velocities v~and vt, calculating the mean sound velocity 3Vm3 =V1-3 +2Vt-3,
( 1)
and using the well-known relationship
OD= (h/k) (3/4~ Va)1/3Vm.
(2)
Here, h and k take their usual meanings, and Va denotes average atomic volume. The present study determined OD for polycrystalline Lai.ssM0.1sCuO4, with M = C a , Ba, Sr. Hot-pressed specimens were prepared from powders obtained by a freezedrying acetate process. We obtained their Tcs from electricalresistance-temperature measurements, using the zero-resistance value. To get z4 and v,, we used a pulse-echo-overlap method described elsewhere [ 1,2 ]. A previous study [ 31 on La2CuO 4 and La,.ssSroAsCuO 4 showed that measurements on polyerystals agreed well with monocrystai measurements averaged over the directions. Table 1 shows the principal results for the void-free state [ 2 ]. Mass density p and average atomic volume Va were obtained by X-ray diffraction. The effective electron-phonon parameter it was calculated from Kresin's [4] approximation of the Eliashberg equations. Figure 1 shows the Tc-Or, diagram.
Table 1 Properties of studied Lal.ssMo.z5CUO4superconductors
P. . . . (g/cm3) Px-ray (g/cm 3) a (A) c (A) Va (-/k3) v~(cm/l.tS) v, (cm/las) OD (K) Tc (K) 2~tr
Ca
Ba
Sr
5.893 6.883 3.7863(7) 13.17(2) 13.49 0.471 0.261 364 19 0.50
6.493 7.077 3.7846(2) 13.288(4) 13.59 0.513 0.288 401 28 0.60
6.572 6.989 3.77814(7) 13.231(2) 13.49 0.568 0.310 432 36 0.68
The T¢-Oo relationship shown in fig. 1 is surprising. First, it shows an essentially linear dependence of Tc on Or,, the principal phonon parameter. Second, it shows that Tc increases with increasing OD. McMiltan's [ 5 ] analysis of conventional superconductors shows that the electron-phonon parameter 2 varies as Off 2 and, that for 2<2, Tc decreases with increasing OD. Simple analysis shows that if we retain the usual BCS-model relationship for Tc (O,, 2), we must abandon McMillan's harmonic relationship 2 ~ Off 2 and replace it by it ~ O~,, where n > -it. Thus, if it = 1, then n > - 1, contrary to McMillan's n = - 2. For these compounds, T~ increases monotonically with increasing Oo. Thus, the task of maximizing Tc becomes one of maximizing Oo. First, we can consider the Einstein [ 6 ] factorization of the characteristic temperature:
0921-4534/91/$03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
130
H. Ledbetter et aL / Critical-temperature/Debye-temperature correlation
3.792
40
ui re
~.
~ L a l " 8 s M 0 " 1 5 C u O ~ ~4 / ~ ' ~
3.788
30
re
w n
2o
,3
S
3.784
~- --~
I-z I-
re b-
A °5
re I/J l-
ee < a. .d ..I
3.780
10 3a
0 I 360
.
,I 400
380
E z
<3 Sr ,[
Ba
, 420
i/d O
1.776 440
DEBYE TEMPERATURE, O D (K) Fig. 1. Dependence of critical temperature Tc and unit-cell a-parameter on Debye characteristic temperature OD. Note that both properties behave contrary to intuition, departing from the periodic-table Ca, St, Ba sequence.
100 90 m" ee "--s F< tw LU o w
vle%cu3ox
80
stiffness-increase sequence is Ba, St, Ca, La, Cu. This sequence suggests that Ca should effect the largest increase in OD. Yet, it effects the smallest, and thus eq. (3) fails here. Second, we note a force-constant-model-based correlation by Nakamura [ 7 ]: OomKM-l/2rff2.
(4)
Here, K denotes a constant, M atomic mass, and ro cationanion nearest-neighbor spacing. Considering the three doping elements themselves, Ca possesses the highest M - 1 / 2 V ~ 2 / 3 value; again wrong for explaining a high Oo in the (La-M)2CuO4 compounds. It is intriguing that substituting Ca, Ba, Sr for La produces peculiar effects in other basic physical properties. For example, the unit-cell a-parameter decreases in the sequence Ca, Ba, Sr instead of the expected sequence Ba, Sr, Ca, (see fig. 1 ). Another peculiar substitution effect is that both Ba and Sr, larger atoms than La, decrease the a-parameter. YiBa2Cu3Ox shows a similar dependence of Tc on 6)0, as is shown in fig. 2. We took Tc values from Poulsen and coworkers [ 8 ] and the specific-heat OD values from Nakazawa and Ishikawa [9]. We fit a linear equation to their nine OD measurements. (For Y~Ba2Cu3Ox compounds, the elastic OD cannot be determined accurately because of defects, perhaps microcracks, that reduce the apparent ultrasound velocities [101.
7o References
60
z
_o ffJ z
b-
50 4o 300
i
...........
350
DEBYE TEMPERATURE,
i 400
450 O D (K)
Fig. 2. Companion figure for YtBa2Cu3Ox showing results for various oxygen contents in the orthorhombic-I phase. OD = KVIa/6 B I / 2 M - I/2 .
(3)
Here, K denotes a constant, V~ atomic volume, B bulk modulus, and M atomic mass. Compared with Cu, all four other cations are large and highly compressible. The elemental
[ 1] H. Ledbetter, N. Frederick and M. Austin, J. Appl. Phys. 51 (1980) 305. [2] H. Ledbetter, M. Austin, S. Kim and M. Lei, J. Mater. Res. 2 (1987) 786. [3] H. Ledbetter, S. Kim, C. Violet and J. Thompson, Physica C 162-164 (1989) 460. [4] V. Kresin, Phys. Lett. A122 (1987) 434. [5] W. McMillan, Phys. Rev. 167 (1968) 331. [6] A. Einstein, Ann. Phys. 39 (1910) 170. [ 7 ] T. Nakamura, Jpn. J. Appl. Phys. 20 ( 1981 ) L653. [8] H. Poulsen, N. Andersen, J. Andersen, H. Bohr and O. Mouritsen, Nature 349 ( 1991 ) 594. [ 9 ] N. Nakazawa and M. Ishikawa, Physica C 162-164 (1989) 83. [ 10 ] H. Ledbetter, J. Mater. Res., forthcoming.