- Email: [email protected]
Thermal conductivity of polycrystalline YBa2Cu4O8 and the phonon transport model
Physica C 235-240 (1994) 1377-1378 North-Holland
PHYSICA
Thermal conductivity of polycrystalline YBa2Cu408 and the phonon transport model B. Sundqvist and B.M. Andersson Department of Experimental Physics, Ume~ University, S-901 87 Um~, Sweden* We have measured the thermal conductivity ~: of bulk YBa2Cu40 s (1-2-4) between 30 and 310 K and at pressures p up to 1 GPa. At p = 0, ~: = 10 Wm-IK-1 near 100 K, twice that of sintered YBa2Cu3OT.s (1-2-3). ~¢ also decreases with increasing T more rapidly than in 1-2-3, to 7.4 Wm'IK "1 at 300 K. Under pressure, ~: increases slowly with p above 150 K, but dvddp decreases at lower T. Our results are in excellent agreement with a semi-classical model for phonon thermal conductivity, provided we assume significant charge transfer with pressure. The calculated phonon mean free path is much larger than the lattice spacing at all T.
The thermal conductivity ~: is non-zero in both normal and superconducting states and thus a good probe for electron and phonon systems in both. Data for ~: are avaliable for most high temperature superconductors (HTSs), even in single crystal form, and over wide ranges in T both above and below T c. For YBa2Cu40 s (1-2-4) few data are available since it is difficult to produce this material in bulk form. We have recently studied K for bulk 1-2-4 produced by hot isostaUc pressing [ 1] a 1-2-3/CUO mixture. ~: was measured using three different methods: At p = 0, ~: was measured directly by a longitudinal method at T < 150 K. Above this radiation loss can play an important role and we found ~: indirectly by measuring the thermal diffusivity a using ,~ngstrOm's method, which eliminates radiation loss errors. ~: was then found by multiplying a by densi~5, and specific heat capacity. Under pressure, conduction heat loss dominates and we measured a by a dynamic two-frequency method [21. Our data for ~: at zero pressure [3] are shown in Figure 1. ~: is about twice as large in 1-2-4 as in sintered 1-2-3, and has a stronger T depeadence. Only a small enhancement of 1
cause the pressure medium vitrified. Although the experiment was nominally carried out as isobaric temperature runs, the drop in T caused changes in p, and far below the glass transition temperature (250-290 K) of the pressure medium thermal contraction probably gave rise to strain in the sample. During a final zero pressure run the medium crystallized, breaking the sample into pieces. We have analysed our data using a modified version of the semi-classical model for phonon heat transport in solids given by Tewordt and W~lkhausen I51. We have added a (dominating) phononphonot~ term with an exponential Umklapp cutoff at yen' low T: the resulting expressions are
12
.
'
""'
I ...... '
'
'
I
''
'
'
i
-I
4 0
l
0
'
'
I
100
I.
I
i
I
200
T (K) Figure 1. Thermal conductivity vs. T
* Work supported financially by NFR and TFR 0921-4534194/$07.00 © 1994- Elsevier Science B.V. AH rights reserved. SSD! 0921-4534(94)01252-0
I
I
I
1
300
8. Sundqoist, B.M. Andersson/physica C 235-240 (1994) 13 77-13 78
1378
Xp= AT3S r(x,T)x4eX(e x - l)-2dx, o
(I)
where A is a constant, t = O/T, x = hro/kT, O is the Debye temperature and x is a relaxation time
x.1 = Zb.l +
%.t+ zp.1.
(2)
Here b denotes scattering by boundaries, e by electrons, and p by other phonons. Further details are given in [31 and [61. We have neglected scattering by planar faults and also the anisotropy of the lattice, since we study a polycrystalline sample. After subtracting an electronic thermal conductivity calculated from the measured resistivity using Wiedemann-Franz' lag', we have fitted Eq. (1) to our data. The results can be summarized as follows: 1. The best fit was obtained without no point defect scattering term. For 1-2-3, this term dominates ~: at practically all T, even in single crystals. Since point defect scattering depresses i< very strongly [61, this is also the reason why the magnitude of ~: in polycrystalline 1-2-4 approaches the in-plane K of single crystal 1-2-3. As discussed previously [61 we assume that this difference is due to the stable oxygen stoichiometry of 1-2-4, and the virtual absence of oxygen defects. Conversely, the ox3,gen defects in 1-2-3 have a very important effect on K. 2. The best fit was found for values of ® very much lower than those found from cp, reflecting the fact that heat is carried mainly by acoustic modes. 3. The fitted relaxation times had magnitudes very close to estimates from simple models [6]: (i) from the boundary scattering term we calculate an average grain diameter of 6 ~tm, practically identical to the observed values 4 - 9 p.m; (it) the phononphonon term (which dominates the thermal resistivity) agrees well with an estimate based on a model given by Slack [71; (iii) and the electron scattering term gives a carrier density n in excellent agreement with band structure calculations. 4. Assuming that the electronic thermal conductivi~ scales wi.:th resistivity, the model .it~-, ,,,.,,,4~,,,,. ra] .~.,~.~=,,,~=,t., t T r j drddp = 0.05 GPa -1 at 300 K, in excellent agreement with experiment. The model also predicts a decreasing drddp below 200 K if n increases with p. Such charge transfer effects are known to occur in 1-2-4. The small [dK/dpi implies that K is practically the same at constant V as at constant p, and our data can thus be compared directly with theory [81.
The model used thus agrees extremely well with the experimental data. Small differences are observed between the globally "best" fitted function and the data, especially near T c, but with small changes in the parameters we find almost perfect agreement [61 over any "small" range in T, and we believe that the main reason for these discrepancies is that the Dcbye model is not a good model for 1-2-4, or for any I-ITS. Several recent papers [9,101 claim that more exotic models are needed, and even [10] that the concept of well defined phonons breaks down in HTSs. The latter seems not to be the case for 1-2-4: From the fitted parameters and (2) we can calculate the total x at any T which, combined with the sound velocity, will give an approximation to the mean free path A [ 11] of acoustic phonons. Carrying out this calculation we find that even at 300 K, Ami n = 430 A (at the Debye limit O/T), far larger than the nearest-neighbour distance usually considered the fundamental limit for A [7,101. The average A, weighted with the contribution to the total heat current, is about 750 A at 300 K and >1100 A at 100 K. Finally, since boundary scattering is too strong in our sample we cannot use the present data to settle the question of whether the peak in ~: below T c is due to the electrons or the phonons. To do this, studies on single crystal 1-2-4 are needed.
REFERENCES: 1. J. Niska et al., J. Mater. Sci. Lett. 9 (1990) 770. 2. B. Sundqvist and G. B/ickstrOm, Rev. Sci. Instr. 47 (1976) 177. 3. B.M. Andersson and B. Sundqvist, Phys. Rev. B 48 (1993) 3575. 4. B.M. Andersson and B. Sundqvist, Physica C 216 (1993) 187. 5. L. Tewordt and Th. W61khausen, Solid State Commun. 70 (1989) 839. 6. B.M. Andersson et al., Phys. Rev. B 49 (1994). 7. G.A. Slack, Solid State Physics 34 (1979) 1. 8. B. Sundqvist and B.M. Andersson, Solid State Commun. 76 (1090) ! 0 ! 9. 9. R.C. Yu et al., Phys. Rev. Lett. 69 (1992) 1431; A.S. Alexandrov and N.F. Mott, Phys. Rev. Lett. 71 (1993) 1075. 10. P.B. Allen et al., Phys. Rev. B 49 (1994) 9073. 11. R.A. Richardson et al., J. Appl. Phys. 72 (1992) 4788.
PHYSICA
Thermal conductivity of polycrystalline YBa2Cu408 and the phonon transport model B. Sundqvist and B.M. Andersson Department of Experimental Physics, Ume~ University, S-901 87 Um~, Sweden* We have measured the thermal conductivity ~: of bulk YBa2Cu40 s (1-2-4) between 30 and 310 K and at pressures p up to 1 GPa. At p = 0, ~: = 10 Wm-IK-1 near 100 K, twice that of sintered YBa2Cu3OT.s (1-2-3). ~¢ also decreases with increasing T more rapidly than in 1-2-3, to 7.4 Wm'IK "1 at 300 K. Under pressure, ~: increases slowly with p above 150 K, but dvddp decreases at lower T. Our results are in excellent agreement with a semi-classical model for phonon thermal conductivity, provided we assume significant charge transfer with pressure. The calculated phonon mean free path is much larger than the lattice spacing at all T.
The thermal conductivity ~: is non-zero in both normal and superconducting states and thus a good probe for electron and phonon systems in both. Data for ~: are avaliable for most high temperature superconductors (HTSs), even in single crystal form, and over wide ranges in T both above and below T c. For YBa2Cu40 s (1-2-4) few data are available since it is difficult to produce this material in bulk form. We have recently studied K for bulk 1-2-4 produced by hot isostaUc pressing [ 1] a 1-2-3/CUO mixture. ~: was measured using three different methods: At p = 0, ~: was measured directly by a longitudinal method at T < 150 K. Above this radiation loss can play an important role and we found ~: indirectly by measuring the thermal diffusivity a using ,~ngstrOm's method, which eliminates radiation loss errors. ~: was then found by multiplying a by densi~5, and specific heat capacity. Under pressure, conduction heat loss dominates and we measured a by a dynamic two-frequency method [21. Our data for ~: at zero pressure [3] are shown in Figure 1. ~: is about twice as large in 1-2-4 as in sintered 1-2-3, and has a stronger T depeadence. Only a small enhancement of 1
cause the pressure medium vitrified. Although the experiment was nominally carried out as isobaric temperature runs, the drop in T caused changes in p, and far below the glass transition temperature (250-290 K) of the pressure medium thermal contraction probably gave rise to strain in the sample. During a final zero pressure run the medium crystallized, breaking the sample into pieces. We have analysed our data using a modified version of the semi-classical model for phonon heat transport in solids given by Tewordt and W~lkhausen I51. We have added a (dominating) phononphonot~ term with an exponential Umklapp cutoff at yen' low T: the resulting expressions are
12
.
'
""'
I ...... '
'
'
I
''
'
'
i
-I
4 0
l
0
'
'
I
100
I.
I
i
I
200
T (K) Figure 1. Thermal conductivity vs. T
* Work supported financially by NFR and TFR 0921-4534194/$07.00 © 1994- Elsevier Science B.V. AH rights reserved. SSD! 0921-4534(94)01252-0
I
I
I
1
300
8. Sundqoist, B.M. Andersson/physica C 235-240 (1994) 13 77-13 78
1378
Xp= AT3S r(x,T)x4eX(e x - l)-2dx, o
(I)
where A is a constant, t = O/T, x = hro/kT, O is the Debye temperature and x is a relaxation time
x.1 = Zb.l +
%.t+ zp.1.
(2)
Here b denotes scattering by boundaries, e by electrons, and p by other phonons. Further details are given in [31 and [61. We have neglected scattering by planar faults and also the anisotropy of the lattice, since we study a polycrystalline sample. After subtracting an electronic thermal conductivity calculated from the measured resistivity using Wiedemann-Franz' lag', we have fitted Eq. (1) to our data. The results can be summarized as follows: 1. The best fit was obtained without no point defect scattering term. For 1-2-3, this term dominates ~: at practically all T, even in single crystals. Since point defect scattering depresses i< very strongly [61, this is also the reason why the magnitude of ~: in polycrystalline 1-2-4 approaches the in-plane K of single crystal 1-2-3. As discussed previously [61 we assume that this difference is due to the stable oxygen stoichiometry of 1-2-4, and the virtual absence of oxygen defects. Conversely, the ox3,gen defects in 1-2-3 have a very important effect on K. 2. The best fit was found for values of ® very much lower than those found from cp, reflecting the fact that heat is carried mainly by acoustic modes. 3. The fitted relaxation times had magnitudes very close to estimates from simple models [6]: (i) from the boundary scattering term we calculate an average grain diameter of 6 ~tm, practically identical to the observed values 4 - 9 p.m; (it) the phononphonon term (which dominates the thermal resistivity) agrees well with an estimate based on a model given by Slack [71; (iii) and the electron scattering term gives a carrier density n in excellent agreement with band structure calculations. 4. Assuming that the electronic thermal conductivi~ scales wi.:th resistivity, the model .it~-, ,,,.,,,4~,,,,. ra] .~.,~.~=,,,~=,t., t T r j drddp = 0.05 GPa -1 at 300 K, in excellent agreement with experiment. The model also predicts a decreasing drddp below 200 K if n increases with p. Such charge transfer effects are known to occur in 1-2-4. The small [dK/dpi implies that K is practically the same at constant V as at constant p, and our data can thus be compared directly with theory [81.
The model used thus agrees extremely well with the experimental data. Small differences are observed between the globally "best" fitted function and the data, especially near T c, but with small changes in the parameters we find almost perfect agreement [61 over any "small" range in T, and we believe that the main reason for these discrepancies is that the Dcbye model is not a good model for 1-2-4, or for any I-ITS. Several recent papers [9,101 claim that more exotic models are needed, and even [10] that the concept of well defined phonons breaks down in HTSs. The latter seems not to be the case for 1-2-4: From the fitted parameters and (2) we can calculate the total x at any T which, combined with the sound velocity, will give an approximation to the mean free path A [ 11] of acoustic phonons. Carrying out this calculation we find that even at 300 K, Ami n = 430 A (at the Debye limit O/T), far larger than the nearest-neighbour distance usually considered the fundamental limit for A [7,101. The average A, weighted with the contribution to the total heat current, is about 750 A at 300 K and >1100 A at 100 K. Finally, since boundary scattering is too strong in our sample we cannot use the present data to settle the question of whether the peak in ~: below T c is due to the electrons or the phonons. To do this, studies on single crystal 1-2-4 are needed.
REFERENCES: 1. J. Niska et al., J. Mater. Sci. Lett. 9 (1990) 770. 2. B. Sundqvist and G. B/ickstrOm, Rev. Sci. Instr. 47 (1976) 177. 3. B.M. Andersson and B. Sundqvist, Phys. Rev. B 48 (1993) 3575. 4. B.M. Andersson and B. Sundqvist, Physica C 216 (1993) 187. 5. L. Tewordt and Th. W61khausen, Solid State Commun. 70 (1989) 839. 6. B.M. Andersson et al., Phys. Rev. B 49 (1994). 7. G.A. Slack, Solid State Physics 34 (1979) 1. 8. B. Sundqvist and B.M. Andersson, Solid State Commun. 76 (1090) ! 0 ! 9. 9. R.C. Yu et al., Phys. Rev. Lett. 69 (1992) 1431; A.S. Alexandrov and N.F. Mott, Phys. Rev. Lett. 71 (1993) 1075. 10. P.B. Allen et al., Phys. Rev. B 49 (1994) 9073. 11. R.A. Richardson et al., J. Appl. Phys. 72 (1992) 4788.