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Specific heat evidence for a large energy gap in YBCO
Physica C 153-155 (1988) 1020-1021 North-Holland, Amsterdam
SPECIFIC HEAT EVIDENCE FOR A LARGE ENERGY GAP IN YBCO
J.W. LORAM and K.A. MIRZA School of Mathematical & Physical Sciences, University of Sussex, Brighton BN7 9QH, U.K.
Using a high precision differential calorimeter we have determined the temperature dependence of the electronic specific beat coefficient 7s(T) of superconducting YBa2Cu307 (YBCO) from 1.5K to above the transition temperature Tc~9OK. 7s(T) i s shown to have a constant plus an exponential term, the l a t t e r corresponding to a large energy gap A for excitations with 2&(O)/kTc = 6.0+.5. The normal s t a t e value 7 n = 16.3 mJ/mole.K2 and the thermodynamic c r i t i c a l field Hc(O) = 8.15..05 kOe.
I. INTRODUCTION To determine 7s(T) we have measured directly the difference in specific heat between a superconducting YBCO sample and a quenched n o n - s u p e r c o n d u c t i n g YBCO r e f e r e n c e sample (Q) between 1.5K and 300K using a high precision d i f f e r e n t i a l calorimeter with r e l a t i v e precision better than 1:104 (1). The bulk of the phonon specific heat common to both samples i s eliminated in this way leaving a rather small correction to be made for the residual difference in phonon terms. To test the sensitivity of our results to metallurgical factors we have measured two superconducting samples prepared in different laboratories and find excellent agreement for the curves for 7s(T). Each sample was slntered In air, reground and compressed into 4g discs. The superconducting samples DP and $2 were then slowly cooled from 950C in flowing oxygen and the non-superconducting sample q was quenched from 900C. In the following discussion the subscripts 1 and 2 refer to non-superconducting and superconducting samples respectively. 2. RESULTS A f t e r c o r r e c t i n g t h e low t e m p e r a t u r e r e s u l t s for rather large Schottky anomalies (presumably of magnetic origin) we find electronic terms 72(0) = 5.5 and 5.2.2 mJ/mole K2 for the superconducting samples DP and $2 respectively and 71 = 16.3 mJlmole K 2 for the non-superconducting quenched sample Q. The non-zero values of 72(0) for the superconducting samples could r e s u l t either from regions of t h e Fermi surface with very small or zero energy gap, or to regions of the sample which are not superconducting. The rather large and sample independent value of 72(0) makes t h i s l a t t e r explanation improbable and suggests an intrinsic origin. The measured curves for AC/T up to room temperature (details of which will be published elsewhere) show, in addition to the anomaly at T c, a broad negative peak at around 36K resulting from the difference in phonon specific heats between the
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Pliysics Publishing Division)
superconducting and quenched samples. The magnitude and temperature dependence of t h i s term ~s e n t i r e l y consistent with the differences in phonon spectra revealed by i n e l a s t i c neutron s c a t t e r i n g (2) which shows a downward s h i f t in frequency on quenching from 18 meV to I I meV for around 3% of the phonon modes. Correcting ~ C / T for t h i s phonon term (consistently with the neutron results), and for the low temperature magnetic term, we obtain the curves for the difference in electronic terms ~7 = 72,s-71 shown in Fig. 1. The curves for the two samples correspond very closely, with equa] step heights ~7(Tc) = 33 mJ/mole K2 and peak temperatures 88K and 86K f o r BP and $2 r e s p e c t i v e l y . We conclude from t h e s e r e s u l t s (in c o n j u n c t i o n with t h e e n t r o p y c u r v e s in Fig. 2a) t h a t t h e normal s t a t e v a l u e s 72, n and 71 a r e a p p r o x i m a t e l y equal. If t h i s were not t h e c a s e ~7 would differ significantly in the helium and higher
40
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.
.
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.
.
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T (I'O FIGURE 1 ~7 (in mJ/mole.K2) vs T a f t e r correcting AC/T for magnetic and phonon terms. ~7 = 72,s-72. n is the difference in electronic specific heat coefficients in the superconducting and normal states. The curves show 72,s i f the broken line at -16 mJ/mole.K2(= -Y2,n) i s taken as the zero line.
J. IV.. Loram and K.A. Mirza / Specific heat euidence for large gap in Y B C O
temperature regions contrary to observation. Thus we have 72,n = 16,4 mJ/mole K 2 and a7 = 72,s-m2on. An important check on the reliability of am(T) is provided by the entropy d i f f e r e n c e aS el =
;
T am(T) dT and this is shown in Fig. Tc
2a.
The
fact that the extrapolated values asel(0) are close to zero shows that the mean v a l u e of am i n the temperature r e g i o n up t o 60K i s r e a s o n a b l y accurate. In Fig. 2b we show the thermodynamic crit~ca] field Hc(T)
calculated
from Hc2(T)
Tc asel T
= 8~/V
dT
w i t h m o l a r v o l u m e V = 104.6 c c / m o l e . At T = 0 f i n d Hc(O) = 8 . 1 9 a n d 8 . ] 2 kOe f o r DP a n d $2.
we
3. DISCUSSION We c o n s i d e r the results first in terms of a BCS model. In F i g . 3 we c o m p a r e am(T) f o r s a m p l e DP with ys-Yn calculated on the phenomenological "=" m o d e l o f P a d a m s e e e t a l (3) f o r s e v e r a l v a l u e s o f t h e gap parameter ~ = 2A(O)/kT c. The curves are normalised to an idealised s t e p h e i g h t &m(Tc) = 39 mJ/mole K 2 with T c = 91K. It is clear that the anomaly is much too sharp for weak coupl~ng BCS (= = 3.53) and a best fit is obtained with ¢ = 6.0*.5. The corresponding value of Yn is 9.45 mJ/mole K 2 and the ratio r = Am(Tc)/mn = 4.1. These large values for and r could suggest a very strong coupling BCS mechanism (the values for weak coupling BCS being 3.53 and 1.43 respectively) but do in fact exceed the maximum theoretical values for phonon mediated superconductivity. Ignoring this and using the strong coupling expression for r (4) we obtain Tc/~ = .23 and thus ~ = 33 meV (where ~ is the prefactor in the BCS expression for Tc). This value for w H e s well within the range of phonon frequencies (up to 80 meV)
1021
in YBCO. Although this might be taken as support for Cooper pairing (via phonon or low frequency electronic excitations) we f i n d no e v i d e n c e f o r the large enhancement of the normal state Yn e x p e c t e d f o r t h i s l a r g e v a l u e o f ¢. The sJmp]e power law dependence for ms(T) predicted for uncharged local bosons is not consistent with the observed exponential dependence, and although an energy gap is expected for charged bose pairs it is not obvious that it would be as large as that w h i c h we observe. Qualitatively at ]east the results are consistent with the Anderson RVB model (5) which predicts a constant term in ms(T) associated with chargeless Fermions and an exponential term due to an energy gap for the excitation of spinless charged Bose holes. The sample independent magnitude that we observe for the constant term suggests that this may indeed be an intrinsic effect, but it remains to be seen whether the RVB model can provide quantitative agreement with the present results. I
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80 T (K)
n-
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FIGURE 3 Comparison of Am(T) for sample DP with ms-y n calculated on the g model, where q = 2a(0)/kTc. (1)
4
= 3.53
(BCS) (2) c¢ = 4 . 5 (3) oc = 6 . 0 (4) cc = 6.8.
0 4. ACKNOWLEDGEMENTS We wish to express our thanks to Dr D.M. Paul for supplying samples DP and Q and to Mr J. Osborne for the preparation of sample $2.
'~ -200 tO
-400 I
I
o
I
I
,
~
l
40 T (K)
,
=
thermodynamic
critical
"
eo
FIGURE 2 (a) aS el (in mJ/mole.K) vs T where AS el difference in electronic entropy superconducting and normal states. (b) The T.
,
field H c
is the between
(in kOe)
vs
5. REFERENCES (1) J.W. Loram, J. Phys. E16 (1981) 367. (2) D.M. Paul, private communication. (3) H. Padamsee, J.E. Nelghbour, C. Shiffman, J. Low Temp. Phys. 12 (1973) 387. (4) V.Z. Kresin and V.P. Parkhomenko, Soy. Phys. Solid S t a t e 16 (1975) 2 1 8 0 . (5) P.W. A n d e r s o n , O. B a s k a r a n , Z. Zou, T. Hsu, P.R.L. 5 8 (1987) 2 7 9 0 .
SPECIFIC HEAT EVIDENCE FOR A LARGE ENERGY GAP IN YBCO
J.W. LORAM and K.A. MIRZA School of Mathematical & Physical Sciences, University of Sussex, Brighton BN7 9QH, U.K.
Using a high precision differential calorimeter we have determined the temperature dependence of the electronic specific beat coefficient 7s(T) of superconducting YBa2Cu307 (YBCO) from 1.5K to above the transition temperature Tc~9OK. 7s(T) i s shown to have a constant plus an exponential term, the l a t t e r corresponding to a large energy gap A for excitations with 2&(O)/kTc = 6.0+.5. The normal s t a t e value 7 n = 16.3 mJ/mole.K2 and the thermodynamic c r i t i c a l field Hc(O) = 8.15..05 kOe.
I. INTRODUCTION To determine 7s(T) we have measured directly the difference in specific heat between a superconducting YBCO sample and a quenched n o n - s u p e r c o n d u c t i n g YBCO r e f e r e n c e sample (Q) between 1.5K and 300K using a high precision d i f f e r e n t i a l calorimeter with r e l a t i v e precision better than 1:104 (1). The bulk of the phonon specific heat common to both samples i s eliminated in this way leaving a rather small correction to be made for the residual difference in phonon terms. To test the sensitivity of our results to metallurgical factors we have measured two superconducting samples prepared in different laboratories and find excellent agreement for the curves for 7s(T). Each sample was slntered In air, reground and compressed into 4g discs. The superconducting samples DP and $2 were then slowly cooled from 950C in flowing oxygen and the non-superconducting sample q was quenched from 900C. In the following discussion the subscripts 1 and 2 refer to non-superconducting and superconducting samples respectively. 2. RESULTS A f t e r c o r r e c t i n g t h e low t e m p e r a t u r e r e s u l t s for rather large Schottky anomalies (presumably of magnetic origin) we find electronic terms 72(0) = 5.5 and 5.2.2 mJ/mole K2 for the superconducting samples DP and $2 respectively and 71 = 16.3 mJlmole K 2 for the non-superconducting quenched sample Q. The non-zero values of 72(0) for the superconducting samples could r e s u l t either from regions of t h e Fermi surface with very small or zero energy gap, or to regions of the sample which are not superconducting. The rather large and sample independent value of 72(0) makes t h i s l a t t e r explanation improbable and suggests an intrinsic origin. The measured curves for AC/T up to room temperature (details of which will be published elsewhere) show, in addition to the anomaly at T c, a broad negative peak at around 36K resulting from the difference in phonon specific heats between the
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Pliysics Publishing Division)
superconducting and quenched samples. The magnitude and temperature dependence of t h i s term ~s e n t i r e l y consistent with the differences in phonon spectra revealed by i n e l a s t i c neutron s c a t t e r i n g (2) which shows a downward s h i f t in frequency on quenching from 18 meV to I I meV for around 3% of the phonon modes. Correcting ~ C / T for t h i s phonon term (consistently with the neutron results), and for the low temperature magnetic term, we obtain the curves for the difference in electronic terms ~7 = 72,s-71 shown in Fig. 1. The curves for the two samples correspond very closely, with equa] step heights ~7(Tc) = 33 mJ/mole K2 and peak temperatures 88K and 86K f o r BP and $2 r e s p e c t i v e l y . We conclude from t h e s e r e s u l t s (in c o n j u n c t i o n with t h e e n t r o p y c u r v e s in Fig. 2a) t h a t t h e normal s t a t e v a l u e s 72, n and 71 a r e a p p r o x i m a t e l y equal. If t h i s were not t h e c a s e ~7 would differ significantly in the helium and higher
40
l.--
s2.j
2O
v
-20
~
0
.
.
. . 40
.
.
.
. . 812
.
.
120
T (I'O FIGURE 1 ~7 (in mJ/mole.K2) vs T a f t e r correcting AC/T for magnetic and phonon terms. ~7 = 72,s-72. n is the difference in electronic specific heat coefficients in the superconducting and normal states. The curves show 72,s i f the broken line at -16 mJ/mole.K2(= -Y2,n) i s taken as the zero line.
J. IV.. Loram and K.A. Mirza / Specific heat euidence for large gap in Y B C O
temperature regions contrary to observation. Thus we have 72,n = 16,4 mJ/mole K 2 and a7 = 72,s-m2on. An important check on the reliability of am(T) is provided by the entropy d i f f e r e n c e aS el =
;
T am(T) dT and this is shown in Fig. Tc
2a.
The
fact that the extrapolated values asel(0) are close to zero shows that the mean v a l u e of am i n the temperature r e g i o n up t o 60K i s r e a s o n a b l y accurate. In Fig. 2b we show the thermodynamic crit~ca] field Hc(T)
calculated
from Hc2(T)
Tc asel T
= 8~/V
dT
w i t h m o l a r v o l u m e V = 104.6 c c / m o l e . At T = 0 f i n d Hc(O) = 8 . 1 9 a n d 8 . ] 2 kOe f o r DP a n d $2.
we
3. DISCUSSION We c o n s i d e r the results first in terms of a BCS model. In F i g . 3 we c o m p a r e am(T) f o r s a m p l e DP with ys-Yn calculated on the phenomenological "=" m o d e l o f P a d a m s e e e t a l (3) f o r s e v e r a l v a l u e s o f t h e gap parameter ~ = 2A(O)/kT c. The curves are normalised to an idealised s t e p h e i g h t &m(Tc) = 39 mJ/mole K 2 with T c = 91K. It is clear that the anomaly is much too sharp for weak coupl~ng BCS (= = 3.53) and a best fit is obtained with ¢ = 6.0*.5. The corresponding value of Yn is 9.45 mJ/mole K 2 and the ratio r = Am(Tc)/mn = 4.1. These large values for and r could suggest a very strong coupling BCS mechanism (the values for weak coupling BCS being 3.53 and 1.43 respectively) but do in fact exceed the maximum theoretical values for phonon mediated superconductivity. Ignoring this and using the strong coupling expression for r (4) we obtain Tc/~ = .23 and thus ~ = 33 meV (where ~ is the prefactor in the BCS expression for Tc). This value for w H e s well within the range of phonon frequencies (up to 80 meV)
1021
in YBCO. Although this might be taken as support for Cooper pairing (via phonon or low frequency electronic excitations) we f i n d no e v i d e n c e f o r the large enhancement of the normal state Yn e x p e c t e d f o r t h i s l a r g e v a l u e o f ¢. The sJmp]e power law dependence for ms(T) predicted for uncharged local bosons is not consistent with the observed exponential dependence, and although an energy gap is expected for charged bose pairs it is not obvious that it would be as large as that w h i c h we observe. Qualitatively at ]east the results are consistent with the Anderson RVB model (5) which predicts a constant term in ms(T) associated with chargeless Fermions and an exponential term due to an energy gap for the excitation of spinless charged Bose holes. The sample independent magnitude that we observe for the constant term suggests that this may indeed be an intrinsic effect, but it remains to be seen whether the RVB model can provide quantitative agreement with the present results. I
I
I
I
w
I
i
i
I
40;
o
E
~ V
2O
E
.
.
.
." , L
.
<1 -20 -
2
.x"
_ _ _ _ ~ ' J ]
'
-40
0
'
'
'
'
40
'
'
'
'
80 T (K)
n-
O
FIGURE 3 Comparison of Am(T) for sample DP with ms-y n calculated on the g model, where q = 2a(0)/kTc. (1)
4
= 3.53
(BCS) (2) c¢ = 4 . 5 (3) oc = 6 . 0 (4) cc = 6.8.
0 4. ACKNOWLEDGEMENTS We wish to express our thanks to Dr D.M. Paul for supplying samples DP and Q and to Mr J. Osborne for the preparation of sample $2.
'~ -200 tO
-400 I
I
o
I
I
,
~
l
40 T (K)
,
=
thermodynamic
critical
"
eo
FIGURE 2 (a) aS el (in mJ/mole.K) vs T where AS el difference in electronic entropy superconducting and normal states. (b) The T.
,
field H c
is the between
(in kOe)
vs
5. REFERENCES (1) J.W. Loram, J. Phys. E16 (1981) 367. (2) D.M. Paul, private communication. (3) H. Padamsee, J.E. Nelghbour, C. Shiffman, J. Low Temp. Phys. 12 (1973) 387. (4) V.Z. Kresin and V.P. Parkhomenko, Soy. Phys. Solid S t a t e 16 (1975) 2 1 8 0 . (5) P.W. A n d e r s o n , O. B a s k a r a n , Z. Zou, T. Hsu, P.R.L. 5 8 (1987) 2 7 9 0 .