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Internal friction and smearing of superconducting gap structures in high-Tc superconductors
PHYSICS LETTERS A
Physics LettersA 178 (1993) 310—314 North-Holland
Internal friction and smearing of superconducting gap structures in high-Ta superconductors Y.N. Huang, Y.N. Wang, S.O. Chen, H.L. Zhou, Q.M. Zhang, H.M. Shen National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210008, China
Z.X. Zhao Institute of Physics, Academia Sinica, P.O. Box 603, Beijing 100080, China
and G.H. Chao Beijing Science and Technology University, Beijing, (‘hina Received 16 March 1993; revised manuscript received 12 May 1993; accepted for publication 18 May 1993 Communicatedby J. flouquet
For both BiSrCaCuO and TlBaCaCuO samples, the internal frictions, Q~,in the kHz range reveal a plateau, Q~,above 7, and drop rapidly below T, with the turning points located just at T, for various samples with different T, values. This kind of anomaly cannot be observed for non-superconducting samples. The drop of Q~below T, is found to be closely related to superconducting condensation. The existence of Q;’ for high-T, superconductors can be reasonably explained using the couplingmodel among carriers with local dynamic distortion. Further, taking into account the smearing of the superconducting gap structure resulting from the recombination ofquasi-pai-ticles and by modifying the BCS relative jump rate as S(E, E’, F) = Re [I — (E—iF)(E’ —iF)], the calculated results of the internal friction below 7’, are in fair agreement with the experimental data. The superconducting gap, A, and the damping rate, F, for both BiSrCaCuO and TIBaCaCuO have also been obtained. These values are in accordance with those obtained by tunneling spectrum and NMR methods, etc.
1. Introduction In conventional superconductors the ultrasonic attenuation drops exponentially below T, and generally obeys the formula derived from the BCS theory [1]. This attenuation, called electronic damping, is attributed to electron—phonon scattering, thus ultrasonic measurements can be used to determine the energy gap and verify the mechanism of superconductivity [1]. Sun and Wang [2] discovered that the internal friction in the kHz range for Bi (Pb) SrCaCuO reveals an obvious plateau Q1~’ above T~and drops linearly with temperature below T, with an apparent turning point located at T~,which is similar to the results measured in the MHz range 3 10
by Saint-Paul et al. [3] and Pankert et a!. [4]. Although the turning point is obscured in YBaCuO due to the existence of relaxation peaks around I’,,
Q1~above T, is obviously higher than that below T, [2,5]. What is more important is that Q~ in YBaCuO decreases monotonically with the decrease of the oxygen content [2,5]. It is generally recognized that the concentration of carriers in YBaCuO decreases as the oxygen content is reduced [6]. Therefore, the decrease of Q~with reduced oxygen content may be caused by the decrease of the carrier density [2]. Because the traditional electronic damping should be too small to be detected in the kHz range [1], and besides, Q1~in the kHz range only changes 10% when the frequency varies three
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PHYSICS
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LETTERS A
times, which is different from the conventional electronic damping [1], the mechanism of Q~’needs further study, as well as whether Q~’dropping linearly or exponentially below T~for high- T~superconductors (HTSCs), which may be important to determine the energy gap. One impressive feature of the tunneling data of HTSCs is the broadening of the gap structure [7—9], which is usually observed in tunneling spectrum experiments. Whether this phenomenon will appear in the internal friction measurements is both interesting and useful for having an insight into the superconductivity of HTSCs. In this paper, we have carefully studied the internal friction Q versus temperature in the kHz range for BiSrCaCuO and TlBaCaCuO. The mechanism of Q” is discussed, and the energy gaps, A, and the damping rate, F, have been calculated. ‘
2. Samples and experimental method To measure the internal friction related to carriers, we selected the method of reed-vibration with a very low strain amplitude (<106), which can minimize
12 July 1993
the influence of some kinds of amplitude-dependent peaks induced by phase-like transitions [10]. We have elaborately calibrated the nodes ofthe samples until the exponential decay and the reproducibility were very good. Four kinds of BiSrCaCuO samples have been used in our experiments. The T~ofthe first and the third kind of these samples determined by ac susceptibility are 104.5 and 84 K, respectively. The second shows two superconducting transitions with one T~at 102 K and another at about 88 K; and the fourth is non-superconducting above 50 K. The TIBaCaCuO samples reveal T~at 98 K.
3. Experimental results The results of the internal friction of BiSrCaCuO and TlBaCaCuO are shown in fig. 1. They reveal an obvious plateau of the internal friction, Q;’, above T~,that drops rapidly below T~with an apparent turning point located at T~for superconducting samples with different T~values (curves I, II, III and V), but no such anomaly appears for non-superconducting samples (curve IV). Even more interesting are the two turning points respectively at 102 K and
3.0
2.0
a 2.0
10
•
b
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1 / ~ e~~”PT
2
~
:
s,.. I
a
a
~
v ~r~I”~1J~’
0.4
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~
,~
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~
~
i~o
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TEMPERATURE (K) Fig. 1. Q’ of BiSrCaCuO and TIBaCaCuO versus T measured at0.8 and 1.2 kHz, respectively. Curves 1,11, III and IV refer to the first, second, third and fourth BiSrCaCuO samples, and curve V to the TlBaCaCuO sample, Curves a: expected Q~”inthe normal state; b: fits according to formula (4); and c: derived from the traditional BCS theory in the weak-coupling limit.
311
Volume 178, number 3,4
PHYSICS LETTERS A
around 88 K for the second kind of BiSrCaCuO. In order to determine the reliability of our results, we have measured more than ten samples of BiSrCaCuO and two of TlBaCaCuO, and all the results manifest the same features as the ones shown in fig. 1. The results on heating are almost coincident with those on cooling, and the data for successive cycles are nearly the same. Connecting the results of ref. [2] with our experimental results, it can be concluded that Q~is closely related to the carriers and the rapid decrease of Q —‘ below T~for superconducting samples (fig. I) could be regarded as caused by the superconducting condensation because there is no relaxation peak around T~for both BiSrCaCuO and TlBaCaCuO.
4. Discussions
in accordance with the traditional electronic damping [1]. Therefore, the mechanism of Q~needs further study. One of the striking features of the strong-correlation systems is the coupling property [13], which leads to dissipation proportional to when wr << 1, where r is the relaxation time and n is the correlation factor (0< n < I). n = 0 indicates the Debye relaxation, and n = I expresses the limit ofstrong correlation [1 3]. HTSCs are strong-correlation systems, and otherwise the photoinduced absorption [14] and neutron scattering studies [15] have shown that there exists lattice distortion around the carriers (it may be accompanied by distortions of the magnetic orders). These distortions with various effective directions are expected to be related to the softest shear modulus [16] (WI)!
,~., — ‘~
The ultrasonic attenuation [11] and the internal friction [12] in the kHz range for YBaCuO also decrease below T~,and often manifest exponential variations with temperature. Thus, they have been used to calculate the superconducting energy gaps [11,12]. However, as mentioned above, there always exist relaxation peaks around T~[2,5] for YBaCuO, the decrease of the ultrasonic attenuation or the internal friction below T~cannot be recognized to be related with superconducting condensation only. But fortunately, there is no such a prob1cm for both BiSrCaCuO and TlBaCaCuO samples, the rapid decrease of Q1’ below T~may be used to determine the energy gaps. As mentioned in the introduction, Q~’in the kHz range is different from the conventional electronic damping [1], and experimental results indicate that Q~~’is closely related to the carriers and drops below T~with a turning point located at T~.But according to the conventional electronic damping [I], the energy dissipation due to the carriers Q~’‘—o.i when WI ~z 1, i.e. ql<< 1, where w and q are the angular frequency and the wave vector of the acoustic waves, respectively, and ‘r and / are the relaxation time and the free-path of the electrons, respectively. Therefore, Q~ be toohand, small the to berelation detected in the kHz range. On will the other between dissipation and frequency according to the experimental data of Q~measured in the kHz range [2] is not 312
12 July 1993
ç~
I
—212’.
II
j
f 22)
12
So, the relaxation of carriers accompanied by distortion clouds may occur and obey the coupling model with a much larger relaxation time t than that of the traditional electronic relaxation. Thus, the energy dissipation Q~’may be expressed as being proportional to (WI)’ when wi ~ 1. After the consideration above, the results of internal friction related to carriers in HTSCs could be explained reasonably. When n ~ 0, the energy loss will decrease slower than that according to the traditional electronic damping when the frequency decreases, i.e. the dissipation will be high enough to be detected in ‘~
the kHz range if n is large enough in addition to the large value oft. When the carriers are condensed below T~,Q,~’will drop. With the above discussion, the energy gap could be calculated by using the internal friction results below T~,in a way similar to that in ref. [11. Curve c as shown in fig. 1 is the result according to the traditional BCS theory in the weak-coupling limit [1], i.e. 2J0/k~T~= 3.52, which is much lower than the experimental data. This may indicate that the conventional BCS theory may not fit to HTSCs, and implies that other mechanisms would be active in HTSCs. In fact, by2 using density of states A2 the /2 N ~ E N~ E E E according to the conventional BCS theory, the tunneling spectrum of HTSCs cannot be well explained ~
—
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PHYSICS LETTERS A
either. One striking feature of the tunneling data of HTSCs, which is usually observed, is the broadening of the gap [7,8]. Hasegawa et a!. [8] have used a broadening BCS density of states in the superconducting state, which was first modeled by Dynes et al. [17], to explain the tunneling results of HTSCs, E—iF N5(E,F)=N~(E)Re([(E.F)242]I,2),
12 July 1993
S(E,E’. r)
I ~o _________________________ 1—0 E—EF
~
,
‘
/
/
where F is the damping rate or the line-width of the quasi-particle energy [18], 2+(l—T/T~) 4/40=l.82(l—T/T~)” x[—O.82+0.09(T/T~)+0.3l75(T/T~)2] N~(E) is the density of states in the normal state. Here, we choose the density of states in the superconducting state as formula (1) for the calculations. According to the traditional BCS theory, the rel-
/
,
—
Fig. 2. Schematic drawing of the relative jump rate S(E, E’, F) versus the energy E.
Table
1
24 0/k5T~,G and F calculated from the experimental results of
ative jump rate S(E, E’) from the energy E to E’ in the superconducting state for conventional 2/EE’ superconductors is expressed as S(E, E’ ) = 1 —4 [1]. Here, we modify the relative jump rate from the energy E toE’ for HTSCs according to the following form when the recombination of quasi-particles is taken into account, i.e. by adding an imaginary part to the energy similar to the method of Dynes et al. [18] for dealing with the density of states, S(E,E’,F)=Re(l_ (E_iF)(E’iF))’
(2)
BiSrCaCuO and TIBaCaCuO, and formula (4) taking into account the energy gap smearing. _______________________________________________ 24 0/k5T, G I’o (meV) BiSrCaCuO (T~=104.5 K) BiSrCaCuO (T0=84 K) TIBaCaCuO (T~=98K)
3.9 3.8 4.0
0.51 0.85 0.35
2.4 0.7 0.8
where W~(E)=9[S(E,E’,F)]S(E,E’,F)
where E’ =E+hw, /1 is the Planck constant. Differ2/EE’ ent from the jump rateS(E, E’ )= 1 [1], —4 there according torelative the conventional BCS theory is a non-zero part of S(E, E’, F) inside the gap near the Fermi level, and the effective gap 4eff, which is equal to A for F= 0, becomessmaller after the broadening ofthe energy gap edge (F~0) is taken into account (see fig. 2). According to Coffey [19] and Wolfet a!. [20], for HTSCs F can also be written in the following form,
X [f(E) f(E’=0)]N~(E, F)N~(E’, 9[S(E, E’,—F)] when S(E, E’, F)F), <0 and ø[S(E, E’, F)] = 1 when S(E, E’, F) >0, f(E) is the Fermi function, W~(E)= [f(E)—f(E’ )]N~(E)N~(E’), ~ Q~’are the internal frictions related to the carriers in the superconducting and normal states, respectively, and Qt~is the background ofthe internal friction unrelated to the carriers. From the data of the superconducting BiSrCaCuO and T1BaCaCuO (fig. 1) and by using formula (4),
F=Fo+GkBTC(T/TC)3,
it is obtained that 2A
(3)
0/k5T~,G, and F0 (the fitted where F0, G are constants independent of temperature. From (1), (2) and ref. [1], we obtain
Q” —Q~ J~”W~(E)dE Q~ ,f~W~(E)dE’ —
—
—
(4)
curves in fig. 1) are in fair agreement with the ex-
perimental results, as shown in table 1. As the samples used in our experiments are polycrystals, the energy gap calculated from the experimental data is the weight average of the energy gaps along the c direction and in the basal plane. The re313
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PHYSICS LETTERS A
sults show that 2A0/kBTC is 4—8 in the basal plane and approximately 3.5 along the c direction in tunneling measurements [7,8], which indicates that our results are reasonable. The above discussion confirms further that the mechanism of HTSCs is characterized by a weak coupling along the c direction and a strong coupling in the basal plane and the quasiparticles have a broadening lifetime due to the strong coupling in the basal plane [7,8]. The values of F measured by tunneling and NMR methods are dispersive [8,19,20], our results for F are also reasonable when compared with other experiments. Because the variation of Q1i~ in the related carTiers above and below 1’, is about I O~,it can only be observed with careful adjustment of the nodes of the samples which can minimize the external energy dissipation as mentioned above, and with good sampies in which the amount of superconducting phase must be high enough. On the other hand, F is sensitive to the defects in the samples used, and from formula (4) we can see that there will be no turning point for the energy dissipation at T~when F is large enough. These may be the reason why some authors observed no obvious drops or slower drops of the internal frictions below T~[2,21].
Acknowledgement This work was supported by the National Center for Research and Development on Superconductivity of China.
References [1] J. Bardeen and JR. Schrieffer, in: Progress in low temperature physics, Vol. III, ed. C.J. Gorter (1961); HE. Bommel, Phys. Rev. 96 (1954) 220;
314
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A.B. Pippard, Philos. Mag. 46 (1955) 1104; C. Kittel. Acta Metall. 3 (1955) 295. [2] L. Sun and Y. Wang, Phys. Lett. A 154 (1991) 59; Y.N. Wang, invited talk on BICHTSC, Beijing, 1992, to be published in J. Mod. Phys. 131 M. Saint-Paul et al., Physica C 166 (1990) 405.
[4]J. Pankert,
G. Marbach, A. Comberg, P.L. Lemmens, P. Froning and S. Ewert, Phys. Rev. Lett. 65 (1990) 3052. [5] G. Cannelli, R. Cantelli and F. Cordero, Phys. Rev. B 38 (1988) 7200; G. Cannelli, R. Cantelli, F. Cordero, M. Ferretti and L. Verdini, Phys.~Rev.B 42 (1990) 7925. 161 R.J. Cava et al., Phys. Rev. B 36 (1988) 5719. [7] B. Batlogg, in: High temperature superconductivity, Los Alamos Symposium 1989, eds. K.S. Bedell, D. Coffey, D.E. Meltzer, D. Pines and J.R. Schrieffer, pp. 67—75. l8]T. Hasegawa, H. Ikuta and K. Kitazawa, in: Physical properties of high T, superconductors III, ed. D.M. Ginsberg (World Scientific, Singapore, 1992), tobe published. [9]J.R. Kirtly, lot. J. Mod. Phys. B 4 (1990) 201. [10] Y.N. Wang, H.M. Shen, M. Zhu and J. Wu, Solid State Commun.76 (1990) 1273. [ll]HeYushengetal.,J.Phys.C21 (1988) 1783. [12]G. Cannelli, R. Cantelli, F. Cordero, G.A. Costa, M. Ferretti and G.L. Olcese, Phys. Rev. B 36 (1987) 8907. [13] K.L. Ngai, Phys. Rev. B 22 (1980) 20066; K.L. Ngai, A.K. Jonscher and C.T. White, Nature 277 (1982) 185. [14] Y.H. Kim et al., Phys. Rev. B 38 (1988)6478. [15] B.H. Toby et al., Phys. Rev. Lett. 64 (1990) 2414; W. Dmowshi et al., Phys. Rev. Lett. 61(1988) 2608. 1161 J. Wu, Y.N. Wang, P.S. Guo and H.M. Shen, Phys. Rev. B (1993), to be published. [17] R.C. Dynes, V. Narayanamurli and J.P. Garno, Phys. Rev. Lett.41 (1978) 1509; R.C. Dynes, J.P. Garno, G.B. Hertel andT.P. Orlando, Phys. Rev. Lett. 53 (1984) 2473. [181 W. Arnold, P. Doussineau, Ch. Frenois and A. Levelui. J. Phys. (Paris) Lett. 42 (1981) L289. [l9]L. Coffey, Phys. Rev. Lett. 64(1990) 1071. [20] E.F. Wolf, H.J. Tao and B. Busla, Solid State Commun. 77 (1991) 519. [21] B. Kusz et al., Solid State Commun. 74 (1990) 595, 375.
Physics LettersA 178 (1993) 310—314 North-Holland
Internal friction and smearing of superconducting gap structures in high-Ta superconductors Y.N. Huang, Y.N. Wang, S.O. Chen, H.L. Zhou, Q.M. Zhang, H.M. Shen National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210008, China
Z.X. Zhao Institute of Physics, Academia Sinica, P.O. Box 603, Beijing 100080, China
and G.H. Chao Beijing Science and Technology University, Beijing, (‘hina Received 16 March 1993; revised manuscript received 12 May 1993; accepted for publication 18 May 1993 Communicatedby J. flouquet
For both BiSrCaCuO and TlBaCaCuO samples, the internal frictions, Q~,in the kHz range reveal a plateau, Q~,above 7, and drop rapidly below T, with the turning points located just at T, for various samples with different T, values. This kind of anomaly cannot be observed for non-superconducting samples. The drop of Q~below T, is found to be closely related to superconducting condensation. The existence of Q;’ for high-T, superconductors can be reasonably explained using the couplingmodel among carriers with local dynamic distortion. Further, taking into account the smearing of the superconducting gap structure resulting from the recombination ofquasi-pai-ticles and by modifying the BCS relative jump rate as S(E, E’, F) = Re [I — (E—iF)(E’ —iF)], the calculated results of the internal friction below 7’, are in fair agreement with the experimental data. The superconducting gap, A, and the damping rate, F, for both BiSrCaCuO and TIBaCaCuO have also been obtained. These values are in accordance with those obtained by tunneling spectrum and NMR methods, etc.
1. Introduction In conventional superconductors the ultrasonic attenuation drops exponentially below T, and generally obeys the formula derived from the BCS theory [1]. This attenuation, called electronic damping, is attributed to electron—phonon scattering, thus ultrasonic measurements can be used to determine the energy gap and verify the mechanism of superconductivity [1]. Sun and Wang [2] discovered that the internal friction in the kHz range for Bi (Pb) SrCaCuO reveals an obvious plateau Q1~’ above T~and drops linearly with temperature below T, with an apparent turning point located at T~,which is similar to the results measured in the MHz range 3 10
by Saint-Paul et al. [3] and Pankert et a!. [4]. Although the turning point is obscured in YBaCuO due to the existence of relaxation peaks around I’,,
Q1~above T, is obviously higher than that below T, [2,5]. What is more important is that Q~ in YBaCuO decreases monotonically with the decrease of the oxygen content [2,5]. It is generally recognized that the concentration of carriers in YBaCuO decreases as the oxygen content is reduced [6]. Therefore, the decrease of Q~with reduced oxygen content may be caused by the decrease of the carrier density [2]. Because the traditional electronic damping should be too small to be detected in the kHz range [1], and besides, Q1~in the kHz range only changes 10% when the frequency varies three
0375-9601 /93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
PHYSICS
Volume 178, number 3,4
LETTERS A
times, which is different from the conventional electronic damping [1], the mechanism of Q~’needs further study, as well as whether Q~’dropping linearly or exponentially below T~for high- T~superconductors (HTSCs), which may be important to determine the energy gap. One impressive feature of the tunneling data of HTSCs is the broadening of the gap structure [7—9], which is usually observed in tunneling spectrum experiments. Whether this phenomenon will appear in the internal friction measurements is both interesting and useful for having an insight into the superconductivity of HTSCs. In this paper, we have carefully studied the internal friction Q versus temperature in the kHz range for BiSrCaCuO and TlBaCaCuO. The mechanism of Q” is discussed, and the energy gaps, A, and the damping rate, F, have been calculated. ‘
2. Samples and experimental method To measure the internal friction related to carriers, we selected the method of reed-vibration with a very low strain amplitude (<106), which can minimize
12 July 1993
the influence of some kinds of amplitude-dependent peaks induced by phase-like transitions [10]. We have elaborately calibrated the nodes ofthe samples until the exponential decay and the reproducibility were very good. Four kinds of BiSrCaCuO samples have been used in our experiments. The T~ofthe first and the third kind of these samples determined by ac susceptibility are 104.5 and 84 K, respectively. The second shows two superconducting transitions with one T~at 102 K and another at about 88 K; and the fourth is non-superconducting above 50 K. The TIBaCaCuO samples reveal T~at 98 K.
3. Experimental results The results of the internal friction of BiSrCaCuO and TlBaCaCuO are shown in fig. 1. They reveal an obvious plateau of the internal friction, Q;’, above T~,that drops rapidly below T~with an apparent turning point located at T~for superconducting samples with different T~values (curves I, II, III and V), but no such anomaly appears for non-superconducting samples (curve IV). Even more interesting are the two turning points respectively at 102 K and
3.0
2.0
a 2.0
10
•
b
1~J’
1.0
1 / ~ e~~”PT
2
~
:
s,.. I
a
a
~
v ~r~I”~1J~’
0.4
0.2w
)jc
~
,~
g~
0.25
~
~
i~o
70
D.35
a,
i~o ho
TEMPERATURE (K) Fig. 1. Q’ of BiSrCaCuO and TIBaCaCuO versus T measured at0.8 and 1.2 kHz, respectively. Curves 1,11, III and IV refer to the first, second, third and fourth BiSrCaCuO samples, and curve V to the TlBaCaCuO sample, Curves a: expected Q~”inthe normal state; b: fits according to formula (4); and c: derived from the traditional BCS theory in the weak-coupling limit.
311
Volume 178, number 3,4
PHYSICS LETTERS A
around 88 K for the second kind of BiSrCaCuO. In order to determine the reliability of our results, we have measured more than ten samples of BiSrCaCuO and two of TlBaCaCuO, and all the results manifest the same features as the ones shown in fig. 1. The results on heating are almost coincident with those on cooling, and the data for successive cycles are nearly the same. Connecting the results of ref. [2] with our experimental results, it can be concluded that Q~is closely related to the carriers and the rapid decrease of Q —‘ below T~for superconducting samples (fig. I) could be regarded as caused by the superconducting condensation because there is no relaxation peak around T~for both BiSrCaCuO and TlBaCaCuO.
4. Discussions
in accordance with the traditional electronic damping [1]. Therefore, the mechanism of Q~needs further study. One of the striking features of the strong-correlation systems is the coupling property [13], which leads to dissipation proportional to when wr << 1, where r is the relaxation time and n is the correlation factor (0< n < I). n = 0 indicates the Debye relaxation, and n = I expresses the limit ofstrong correlation [1 3]. HTSCs are strong-correlation systems, and otherwise the photoinduced absorption [14] and neutron scattering studies [15] have shown that there exists lattice distortion around the carriers (it may be accompanied by distortions of the magnetic orders). These distortions with various effective directions are expected to be related to the softest shear modulus [16] (WI)!
,~., — ‘~
The ultrasonic attenuation [11] and the internal friction [12] in the kHz range for YBaCuO also decrease below T~,and often manifest exponential variations with temperature. Thus, they have been used to calculate the superconducting energy gaps [11,12]. However, as mentioned above, there always exist relaxation peaks around T~[2,5] for YBaCuO, the decrease of the ultrasonic attenuation or the internal friction below T~cannot be recognized to be related with superconducting condensation only. But fortunately, there is no such a prob1cm for both BiSrCaCuO and TlBaCaCuO samples, the rapid decrease of Q1’ below T~may be used to determine the energy gaps. As mentioned in the introduction, Q~’in the kHz range is different from the conventional electronic damping [1], and experimental results indicate that Q~~’is closely related to the carriers and drops below T~with a turning point located at T~.But according to the conventional electronic damping [I], the energy dissipation due to the carriers Q~’‘—o.i when WI ~z 1, i.e. ql<< 1, where w and q are the angular frequency and the wave vector of the acoustic waves, respectively, and ‘r and / are the relaxation time and the free-path of the electrons, respectively. Therefore, Q~ be toohand, small the to berelation detected in the kHz range. On will the other between dissipation and frequency according to the experimental data of Q~measured in the kHz range [2] is not 312
12 July 1993
ç~
I
—212’.
II
j
f 22)
12
So, the relaxation of carriers accompanied by distortion clouds may occur and obey the coupling model with a much larger relaxation time t than that of the traditional electronic relaxation. Thus, the energy dissipation Q~’may be expressed as being proportional to (WI)’ when wi ~ 1. After the consideration above, the results of internal friction related to carriers in HTSCs could be explained reasonably. When n ~ 0, the energy loss will decrease slower than that according to the traditional electronic damping when the frequency decreases, i.e. the dissipation will be high enough to be detected in ‘~
the kHz range if n is large enough in addition to the large value oft. When the carriers are condensed below T~,Q,~’will drop. With the above discussion, the energy gap could be calculated by using the internal friction results below T~,in a way similar to that in ref. [11. Curve c as shown in fig. 1 is the result according to the traditional BCS theory in the weak-coupling limit [1], i.e. 2J0/k~T~= 3.52, which is much lower than the experimental data. This may indicate that the conventional BCS theory may not fit to HTSCs, and implies that other mechanisms would be active in HTSCs. In fact, by2 using density of states A2 the /2 N ~ E N~ E E E according to the conventional BCS theory, the tunneling spectrum of HTSCs cannot be well explained ~
—
Volume 178, number 3,4
PHYSICS LETTERS A
either. One striking feature of the tunneling data of HTSCs, which is usually observed, is the broadening of the gap [7,8]. Hasegawa et a!. [8] have used a broadening BCS density of states in the superconducting state, which was first modeled by Dynes et al. [17], to explain the tunneling results of HTSCs, E—iF N5(E,F)=N~(E)Re([(E.F)242]I,2),
12 July 1993
S(E,E’. r)
I ~o _________________________ 1—0 E—EF
~
,
‘
/
/
where F is the damping rate or the line-width of the quasi-particle energy [18], 2+(l—T/T~) 4/40=l.82(l—T/T~)” x[—O.82+0.09(T/T~)+0.3l75(T/T~)2] N~(E) is the density of states in the normal state. Here, we choose the density of states in the superconducting state as formula (1) for the calculations. According to the traditional BCS theory, the rel-
/
,
—
Fig. 2. Schematic drawing of the relative jump rate S(E, E’, F) versus the energy E.
Table
1
24 0/k5T~,G and F calculated from the experimental results of
ative jump rate S(E, E’) from the energy E to E’ in the superconducting state for conventional 2/EE’ superconductors is expressed as S(E, E’ ) = 1 —4 [1]. Here, we modify the relative jump rate from the energy E toE’ for HTSCs according to the following form when the recombination of quasi-particles is taken into account, i.e. by adding an imaginary part to the energy similar to the method of Dynes et al. [18] for dealing with the density of states, S(E,E’,F)=Re(l_ (E_iF)(E’iF))’
(2)
BiSrCaCuO and TIBaCaCuO, and formula (4) taking into account the energy gap smearing. _______________________________________________ 24 0/k5T, G I’o (meV) BiSrCaCuO (T~=104.5 K) BiSrCaCuO (T0=84 K) TIBaCaCuO (T~=98K)
3.9 3.8 4.0
0.51 0.85 0.35
2.4 0.7 0.8
where W~(E)=9[S(E,E’,F)]S(E,E’,F)
where E’ =E+hw, /1 is the Planck constant. Differ2/EE’ ent from the jump rateS(E, E’ )= 1 [1], —4 there according torelative the conventional BCS theory is a non-zero part of S(E, E’, F) inside the gap near the Fermi level, and the effective gap 4eff, which is equal to A for F= 0, becomessmaller after the broadening ofthe energy gap edge (F~0) is taken into account (see fig. 2). According to Coffey [19] and Wolfet a!. [20], for HTSCs F can also be written in the following form,
X [f(E) f(E’=0)]N~(E, F)N~(E’, 9[S(E, E’,—F)] when S(E, E’, F)F), <0 and ø[S(E, E’, F)] = 1 when S(E, E’, F) >0, f(E) is the Fermi function, W~(E)= [f(E)—f(E’ )]N~(E)N~(E’), ~ Q~’are the internal frictions related to the carriers in the superconducting and normal states, respectively, and Qt~is the background ofthe internal friction unrelated to the carriers. From the data of the superconducting BiSrCaCuO and T1BaCaCuO (fig. 1) and by using formula (4),
F=Fo+GkBTC(T/TC)3,
it is obtained that 2A
(3)
0/k5T~,G, and F0 (the fitted where F0, G are constants independent of temperature. From (1), (2) and ref. [1], we obtain
Q” —Q~ J~”W~(E)dE Q~ ,f~W~(E)dE’ —
—
—
(4)
curves in fig. 1) are in fair agreement with the ex-
perimental results, as shown in table 1. As the samples used in our experiments are polycrystals, the energy gap calculated from the experimental data is the weight average of the energy gaps along the c direction and in the basal plane. The re313
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PHYSICS LETTERS A
sults show that 2A0/kBTC is 4—8 in the basal plane and approximately 3.5 along the c direction in tunneling measurements [7,8], which indicates that our results are reasonable. The above discussion confirms further that the mechanism of HTSCs is characterized by a weak coupling along the c direction and a strong coupling in the basal plane and the quasiparticles have a broadening lifetime due to the strong coupling in the basal plane [7,8]. The values of F measured by tunneling and NMR methods are dispersive [8,19,20], our results for F are also reasonable when compared with other experiments. Because the variation of Q1i~ in the related carTiers above and below 1’, is about I O~,it can only be observed with careful adjustment of the nodes of the samples which can minimize the external energy dissipation as mentioned above, and with good sampies in which the amount of superconducting phase must be high enough. On the other hand, F is sensitive to the defects in the samples used, and from formula (4) we can see that there will be no turning point for the energy dissipation at T~when F is large enough. These may be the reason why some authors observed no obvious drops or slower drops of the internal frictions below T~[2,21].
Acknowledgement This work was supported by the National Center for Research and Development on Superconductivity of China.
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