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Phonon scattering in HTSC cuprate crystals
Physica B 263—264 (1999) 674—677
Phonon scattering in HTSC cuprate crystals V.B. Efimov, L.P. Mezhov-Deglin* Institute of Solid State Physics RAS, Chernogolovka, Moscow distr. 142432, Russia
Abstract From results of measurements of thermal conductivity (i and i ) and electric resistivity (o and o ) of the HTSC ?@ A ?@ A YBa Cu O and Bi Sr CaCu O single crystals, made on the same samples with transition temperatures ¹ close \V >W to 90 K, we have estimated that above ¹ the phonon contribution i to the total thermal conductivity in the ab-plane i can be comparable or higher than the electronic contribution i in Y-crystals, though in all studied Bi-crystals ?@ i
in contrast to predictions of the theories of electronic heat transport in anisotropic HTSC cuprates. Taking into account the estimations of the electronic contributions i (¹) from the known experimental behaviour of conductivity of the normal carriers below ¹ and recent observations of the thermal Hall conductivity in Y-crystals in the mixed state we can conclude that: (a) the nature of the observed maxima should be attributed to the increase in both the phonon i and electron i contributions below ¹ , and (b) in perfect HTSC crystals the mean free paths of phonons l and normal electron excitations l , restricted at ¹&¹ by the mutual phonon-carrier scattering, should grow quickly the temper ature is lowered below ¹ to overcome the decrease of the appropriate heat capacity terms C and C . In the c-direction i ;i in all the samples, and the total thermal conductivity i (¹) decreases monotonously at temperatures 200—7 K. In A addition to the strong anisotropy of the layered cuprate crystals one should take into account the scattering of phonons by interlayer defects and the sample boundaries when discussing the i (¹) dependence. 1999 Elsevier Science B.V. All rights reserved. Keywords: Thermal conductivity; Superconductor; Electric resistivity; Cuprate crystals
1. Introduction The behaviour of the thermal conductivity j of a superconductor is determined by the sum of the contributions of the electronic i and lattice i components: i"i #i . In the normal state the sample o: i "¸ ¹/o (Wiedemann—Franz rela * Corresponding author.
(¹'¹ ) i can be estimated from the resistivity of tion, ¸ "2.4;10\ WX is the Lorentz number). This relation gives the upper limit for i . To explain the behaviour of i (¹) and i (¹) in the normal state the measurements of i and o must be done on the same sample and with the same directions of the heat and electric currents due to strong anisotropy of the layered cuprate crystals even in the ab-plane and poor reproducibility of the values of transport coefficients from sample to sample [1]. As was
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 4 4 1 - 0
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
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shown recently [4,10,11], information on the behaviour of i (¹) and i (¹) in the superconducting state can be obtained from measurements of the ‘thermal Hall effect’ in perpendicular magnetic fields or studies of absorption of electromagnetic waves (the resistivity of the normal electronic excitations in the bulk), in addition to the static measurements (see Ref. [3]).
2. Experimental results and discussions Our measurements of thermal conductivity and resistivity in the ab-plane and in the c-direction were performed on the same samples of the HTSC Y- and Bi-cuprate crystals in the temperature range 250—7 K [1]. The temperatures of superconducting transition in our Y-crystals C1 and C2 were close to ¹ & 93 K with *¹+0.5 and 2 K. The temperature dependencies of thermal conductivity of Y-samples are shown in Fig. 1. The dependencies of K(t)"i (¹)/i (¹ ) and ?@ ?@ K (t)"i (¹)/i (¹ )"(¸ ¹/o)/i (¹ ) on the ?@ ?@
Fig. 1. Thermal conductivity of YBaCuO samples: 1,2-our crystals, in-plane and 2A along c-axis; 3-a ceramic sample; 1 —3 the electronic contribution; dotted lines samples [3,6].
Fig. 2. Thermal conductivity as a function of the reduced temperature in relative units.
reduced temperature t"¹/¹ are shown in Fig. 2. Triangles (1) and open squares (2) show the behaviour of K(t) for our samples C1 and C2 in the ab-plane. The dark squares 2A correspond to the total thermal conductivity K (t) in the c-direction A for the sample C2. The thermal conductivity of the Y-ceramic sample is shown by the curve 3. Curves 1e—3e describe the behaviour of the electronic component K (t) of the same samples. The dotted curves k?, k@ and k?@ correspond to the thermal conductivity of the best of previously studied Ycrystals [3,6]. Thermal conductivity of Bi-crystals with transition temperatures ¹ +89 K (crystal C 1), 86 K (C2) and 82 K (C3) are shown in Figs. 3 and 4 (in reduced units). Open triangles, squares and rhombus (1, 2, 3) and dotted lines 1e, 2e correspond to K(t) and K (t) of the same samples in the ab plane. The dark points 2A, 3A show conductivity K (t) along the c-axis. Dotted lines, curves 4 and A 5 correspond to the data obtained from the literature [7,8]. Solid and dotted lines 6, 6e pertain to the homogeneous ceramic sample [1].
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V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
Fig. 4. Temperature dependence of thermal conductivity in relative units. Fig. 3. Thermal conductivity of Bi-samples: in-plane 1,2,3-our crystals, 4,5 from [7,8] and along c-axis dark points 2A, 3A. Curve 6 is for ceramic sample; dashed curves 1 —6 correspond to electronic contributions.
The main experimental results are as follows: 1. The temperature dependencies of thermal conductivity of Y- and Bi-cuprate crystals in reduced scales (Figs. 2 and 4) look nearly identical. The in-plane thermal conductivity K(t) in the normal state is weakly dependent on temperature. In the superconducting state the value of K(t) below ¹ increases with reducing temper ature and passes through a maximum at ¹ (0.5 ¹ . The higher the relative height of
the maximum the steeper the temperature dependencies of K(t) near to ¹ .
Similar maxima have been observed in ceramic samples. 2. The electron component K (t) in the ab-plane in the normal state can be comparable with that of the lattice K (t) in Y-crystals, but in all our Bi-crystals K (t);K (t). And we have not ob served any correlation between the total thermal
conductivity K(¹) of the Bi crystals and the contribution from the electron system K (¹). C 3. The ratio K(t)"i (¹)/i (¹ ) in more anisot?@ ?@ ropic Bi-crystals can reach 2 as in less anisotropic Y-crystals and with increases in this ratio the position of ¹ shifts to lower temperatures.
This result is in contradiction with the predictions of the theory of pure electronic heat transport in HTSC cuprate crystals [2,9]. 4. In the c-direction the contribution of the electronic component to the total thermal conductivity i (t) is less than 1% in all the crystals. The A value of i decreases monotonically upon coolA ing the samples, the superconducting transition does not influence the temperature dependence of i (t). A 5. The behaviour of the thermal conductivity of homogenous ceramic samples at ¹&¹ can be described in a first approximation by an expression like: i"(i iHi ) . This means that well ? @ A prepared HTSC ceramic samples consist of a number of crystals of small dimensions, and that the small relative height of the maxima on
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
the i (¹) curves and its shift towards high temA peratures (curves 3 and 6 in Figs. 2 and 4) can be explained by the scattering of phonons and carriers on the crystal boundaries that restricts the maximal values of l and l . This observation might serve as a key to understanding the reasons of wide scatter in values of the transport coefficients from sample to sample of the same composition: different content of structure defects restricts the phonons paths l in bulk in addition to different oxygen content that mainly affects the carrier system l (¹).
677
rier l mean free paths below ¹ (due to a decrease in the probability of normal carrier-thermal phonon scattering). To explain the behaviour of i (¹), one ought to take into account the strong A anisotropy of cuprate crystals and also scattering of phonons by interlayer defects (see, for example, Ref. [5]). The work was supported under the program of RSTP “Superconductivity” Grant “Relok” and by the RFBR grant No. 97-02-17772.
References 3. Conclusions The most interesting questions are concerned with the nature of the maxima on the K(t) curves. Keeping in mind our results, the estimations of the conductivity of the normal electron excitations [3], and recent observations [4,10,11] of the thermal Hall conductivity in superconducting Y-crystals, we can state that while finding the origin of the peaks on the K(t) curves in perfect HTSC cuprate crystals one must take into account the possibility of quick growth of both the phonon l (due to decreasing content of normal carriers) and the car-
[1] V.B. Efimov, L.P. Mezhov-Deglin, Low Temp. Phys. 23 (1997) 204. [2] M. Houssa, M. Ausloos, Phys. Rev. B 51 (1995) 9372. [3] R.C. Yu, M.B. Salamon, J.P. Lu, W.C. Lee, Phys. Rev. Lett. 69 (1992) 1431. [4] K. Krishna, J.M. Harris, N.P. Ong, Phys. Rev. Lett. 75 (1995) 3529. [5] E.I. Salmatov, Phys. Rev. B 56 (1997) 1. [6] T. Saki, M. Sawamura et al., Physica B 194—196 (1994) 2135. [7] M.F. Crommie, A. Zelt, Phys. Rev. B 41 (1990) 10978. [8] P.B. Allen et al, Phys. Rev. B 49 (1994) 9073. [9] M. Houssa, M. Ausloos, Physica C 257 (1996) 321. [10] K. Krishna, J.M. Harris, N.P. Ong, Science 277 (1997) 83. [11] A.N. Taldenkov, A.V. Inyushkin, private communication.
Phonon scattering in HTSC cuprate crystals V.B. Efimov, L.P. Mezhov-Deglin* Institute of Solid State Physics RAS, Chernogolovka, Moscow distr. 142432, Russia
Abstract From results of measurements of thermal conductivity (i and i ) and electric resistivity (o and o ) of the HTSC ?@ A ?@ A YBa Cu O and Bi Sr CaCu O single crystals, made on the same samples with transition temperatures ¹ close \V >W to 90 K, we have estimated that above ¹ the phonon contribution i to the total thermal conductivity in the ab-plane i can be comparable or higher than the electronic contribution i in Y-crystals, though in all studied Bi-crystals ?@ i
in contrast to predictions of the theories of electronic heat transport in anisotropic HTSC cuprates. Taking into account the estimations of the electronic contributions i (¹) from the known experimental behaviour of conductivity of the normal carriers below ¹ and recent observations of the thermal Hall conductivity in Y-crystals in the mixed state we can conclude that: (a) the nature of the observed maxima should be attributed to the increase in both the phonon i and electron i contributions below ¹ , and (b) in perfect HTSC crystals the mean free paths of phonons l and normal electron excitations l , restricted at ¹&¹ by the mutual phonon-carrier scattering, should grow quickly the temper ature is lowered below ¹ to overcome the decrease of the appropriate heat capacity terms C and C . In the c-direction i ;i in all the samples, and the total thermal conductivity i (¹) decreases monotonously at temperatures 200—7 K. In A addition to the strong anisotropy of the layered cuprate crystals one should take into account the scattering of phonons by interlayer defects and the sample boundaries when discussing the i (¹) dependence. 1999 Elsevier Science B.V. All rights reserved. Keywords: Thermal conductivity; Superconductor; Electric resistivity; Cuprate crystals
1. Introduction The behaviour of the thermal conductivity j of a superconductor is determined by the sum of the contributions of the electronic i and lattice i components: i"i #i . In the normal state the sample o: i "¸ ¹/o (Wiedemann—Franz rela * Corresponding author.
(¹'¹ ) i can be estimated from the resistivity of tion, ¸ "2.4;10\ WX is the Lorentz number). This relation gives the upper limit for i . To explain the behaviour of i (¹) and i (¹) in the normal state the measurements of i and o must be done on the same sample and with the same directions of the heat and electric currents due to strong anisotropy of the layered cuprate crystals even in the ab-plane and poor reproducibility of the values of transport coefficients from sample to sample [1]. As was
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 4 4 1 - 0
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
675
shown recently [4,10,11], information on the behaviour of i (¹) and i (¹) in the superconducting state can be obtained from measurements of the ‘thermal Hall effect’ in perpendicular magnetic fields or studies of absorption of electromagnetic waves (the resistivity of the normal electronic excitations in the bulk), in addition to the static measurements (see Ref. [3]).
2. Experimental results and discussions Our measurements of thermal conductivity and resistivity in the ab-plane and in the c-direction were performed on the same samples of the HTSC Y- and Bi-cuprate crystals in the temperature range 250—7 K [1]. The temperatures of superconducting transition in our Y-crystals C1 and C2 were close to ¹ & 93 K with *¹+0.5 and 2 K. The temperature dependencies of thermal conductivity of Y-samples are shown in Fig. 1. The dependencies of K(t)"i (¹)/i (¹ ) and ?@ ?@ K (t)"i (¹)/i (¹ )"(¸ ¹/o)/i (¹ ) on the ?@ ?@
Fig. 1. Thermal conductivity of YBaCuO samples: 1,2-our crystals, in-plane and 2A along c-axis; 3-a ceramic sample; 1 —3 the electronic contribution; dotted lines samples [3,6].
Fig. 2. Thermal conductivity as a function of the reduced temperature in relative units.
reduced temperature t"¹/¹ are shown in Fig. 2. Triangles (1) and open squares (2) show the behaviour of K(t) for our samples C1 and C2 in the ab-plane. The dark squares 2A correspond to the total thermal conductivity K (t) in the c-direction A for the sample C2. The thermal conductivity of the Y-ceramic sample is shown by the curve 3. Curves 1e—3e describe the behaviour of the electronic component K (t) of the same samples. The dotted curves k?, k@ and k?@ correspond to the thermal conductivity of the best of previously studied Ycrystals [3,6]. Thermal conductivity of Bi-crystals with transition temperatures ¹ +89 K (crystal C 1), 86 K (C2) and 82 K (C3) are shown in Figs. 3 and 4 (in reduced units). Open triangles, squares and rhombus (1, 2, 3) and dotted lines 1e, 2e correspond to K(t) and K (t) of the same samples in the ab plane. The dark points 2A, 3A show conductivity K (t) along the c-axis. Dotted lines, curves 4 and A 5 correspond to the data obtained from the literature [7,8]. Solid and dotted lines 6, 6e pertain to the homogeneous ceramic sample [1].
676
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
Fig. 4. Temperature dependence of thermal conductivity in relative units. Fig. 3. Thermal conductivity of Bi-samples: in-plane 1,2,3-our crystals, 4,5 from [7,8] and along c-axis dark points 2A, 3A. Curve 6 is for ceramic sample; dashed curves 1 —6 correspond to electronic contributions.
The main experimental results are as follows: 1. The temperature dependencies of thermal conductivity of Y- and Bi-cuprate crystals in reduced scales (Figs. 2 and 4) look nearly identical. The in-plane thermal conductivity K(t) in the normal state is weakly dependent on temperature. In the superconducting state the value of K(t) below ¹ increases with reducing temper ature and passes through a maximum at ¹ (0.5 ¹ . The higher the relative height of
the maximum the steeper the temperature dependencies of K(t) near to ¹ .
Similar maxima have been observed in ceramic samples. 2. The electron component K (t) in the ab-plane in the normal state can be comparable with that of the lattice K (t) in Y-crystals, but in all our Bi-crystals K (t);K (t). And we have not ob served any correlation between the total thermal
conductivity K(¹) of the Bi crystals and the contribution from the electron system K (¹). C 3. The ratio K(t)"i (¹)/i (¹ ) in more anisot?@ ?@ ropic Bi-crystals can reach 2 as in less anisotropic Y-crystals and with increases in this ratio the position of ¹ shifts to lower temperatures.
This result is in contradiction with the predictions of the theory of pure electronic heat transport in HTSC cuprate crystals [2,9]. 4. In the c-direction the contribution of the electronic component to the total thermal conductivity i (t) is less than 1% in all the crystals. The A value of i decreases monotonically upon coolA ing the samples, the superconducting transition does not influence the temperature dependence of i (t). A 5. The behaviour of the thermal conductivity of homogenous ceramic samples at ¹&¹ can be described in a first approximation by an expression like: i"(i iHi ) . This means that well ? @ A prepared HTSC ceramic samples consist of a number of crystals of small dimensions, and that the small relative height of the maxima on
V.B. Efimov, L.P. Mezhov-Deglin / Physica B 263—264 (1999) 674—677
the i (¹) curves and its shift towards high temA peratures (curves 3 and 6 in Figs. 2 and 4) can be explained by the scattering of phonons and carriers on the crystal boundaries that restricts the maximal values of l and l . This observation might serve as a key to understanding the reasons of wide scatter in values of the transport coefficients from sample to sample of the same composition: different content of structure defects restricts the phonons paths l in bulk in addition to different oxygen content that mainly affects the carrier system l (¹).
677
rier l mean free paths below ¹ (due to a decrease in the probability of normal carrier-thermal phonon scattering). To explain the behaviour of i (¹), one ought to take into account the strong A anisotropy of cuprate crystals and also scattering of phonons by interlayer defects (see, for example, Ref. [5]). The work was supported under the program of RSTP “Superconductivity” Grant “Relok” and by the RFBR grant No. 97-02-17772.
References 3. Conclusions The most interesting questions are concerned with the nature of the maxima on the K(t) curves. Keeping in mind our results, the estimations of the conductivity of the normal electron excitations [3], and recent observations [4,10,11] of the thermal Hall conductivity in superconducting Y-crystals, we can state that while finding the origin of the peaks on the K(t) curves in perfect HTSC cuprate crystals one must take into account the possibility of quick growth of both the phonon l (due to decreasing content of normal carriers) and the car-
[1] V.B. Efimov, L.P. Mezhov-Deglin, Low Temp. Phys. 23 (1997) 204. [2] M. Houssa, M. Ausloos, Phys. Rev. B 51 (1995) 9372. [3] R.C. Yu, M.B. Salamon, J.P. Lu, W.C. Lee, Phys. Rev. Lett. 69 (1992) 1431. [4] K. Krishna, J.M. Harris, N.P. Ong, Phys. Rev. Lett. 75 (1995) 3529. [5] E.I. Salmatov, Phys. Rev. B 56 (1997) 1. [6] T. Saki, M. Sawamura et al., Physica B 194—196 (1994) 2135. [7] M.F. Crommie, A. Zelt, Phys. Rev. B 41 (1990) 10978. [8] P.B. Allen et al, Phys. Rev. B 49 (1994) 9073. [9] M. Houssa, M. Ausloos, Physica C 257 (1996) 321. [10] K. Krishna, J.M. Harris, N.P. Ong, Science 277 (1997) 83. [11] A.N. Taldenkov, A.V. Inyushkin, private communication.