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Influence of a linear term on the density of states of high-Tc superconductors
PhysicaC 202 (1992) 33-36 North-Holland
Influence of a linear term on the density of states of high-To superconductors Anna Maria Cucolo a, Canio Noce ~'and Alfonso R o m a n o b " Dipartimento di Fisica, Universitlt di Salerno, 1-84081 Baronissi (Salerno), Italy b Dipartimento di Fisica Teorica e Sue Metodologie per le Scienze Applicate, Universitit di Salerno, 1-84081Baronissi (Salerno), Italy
Received 1OJuly 1992
The temperaturedependenceof the zero-biasconductance,the electronicspecificheat and the ultrasonicattenuation in highTcsuperconductorshavebeenanalyzedwithinthe frameworkofa phenomenologicalmodel,basedon a densityof states expressed as a superpositionof a linear term to a BCS standard one. By usingthe same valuesof the fitting parameters,we have obtained a satisfactoryagreementwith the experimentalresults, togetherwith a consistentphysicalinterpretationof the weightof the linear contribution.
1. Introduction
Tunneling experiments performed on high-To superconductors show an unusual behaviour of the background conductance, which up to very high biases ( ~ 200 meV) follows a linear dependence on the absolute value of the applied voltage. This peculiarity is routinely observed in point-contact and planar junctions fabricated on copper oxides LSCO, YBCO, BSCCO and TBCCO as well as in bismuthates, regardless of the junction preparation method. This experimental evidence strongly supports the idea that the linear background conductance is an intrinsic property of high-To superconductors. Several authors have given theoretical explanations of this behavior in terms of density of states effects. Anderson and Zou [ 1 ] have derived a normal state linear conductance from simple assumptions on the spectrum of holon and spinon excitations in a bidimensional RVB state. On a more general ground, it has been suggested that linearity could be a consequence of electronic correlation effects near the metal-insulator transition, leading to a breakdown of the conventional Fermi liquid description. This point of view is explicitly present in the theory of marginal Fermi liquids formulated by Varma [ 2 ], where the hypothesis of a normal den-
sity of states of the form N ( E ) =N0+N~ IEI is found to agree with the basic assumption of the theory, concerning the structure of the charge and spin density excitation spectrum. We also mention the approach by Phillips [3 ] who, within a quantum percolation theory, separates the density of states into an extended part, responsible for the superconducting properties, and a localized part linear in energy, responsible for the normal properties. Motivated by these considerations, we have presented in a previous paper [4 ] a model for the description of tunneling experiments on YBa2Cu307_~ junctions, based on the assumption of a density of states expressed as a superposition of two BCS-like contributions and a linear one. We have made the microscopic hypothesis that two distinct BCS-like gap structures are associated with the CuO2 planes and the CuO chains along the b-axis, respectively, whereas the linear contribution is supposed to take into account the anomalous behavior along the c-axis. By means of this model, satisfactory quantitative fittings have been obtained for tunneling experiments performed both on fully oxygenated (To= 90 K) and oxygen deficient (To=60 K) YBCO junctions, in the superconducting as well as in the normal state. We have found that the increase of the weight of the linear term well describes the reduction of the oxygen
0921-4534/92/$05.00 © 1992ElsevierSciencePublishersB.V. All rightsreserved.
34
A.M. Cucoloet aL ~Influenceof a linear term on DOS
content leading to the decrease of the critical temperature. In this paper we use a simplified one-gap version of this model, previously presented on a purely phenomenological ground [5 ], for the qualitative evaluation of the temperature dependence of other physical quantities of interest in the study of high-To superconductors, namely, the zero-bias conductance, the specific heat and the ultrasonic attenuation. We point out that this model, compared with the one specifically formulated for YBCO in ref. [ 4 ], has the advantage that it is more suited for the study of other high-T~ superconductors.
2. The model
Our basic hypothesis is that the superconducting density of states can be expressed in the form Ns(E, T) =a+ blEI +cNBcs(E, T ) ,
( 1)
where NBcs(E, T) = R e ~ T ) 2
,
lowing expression for the zero-bias conductance G (0, T): G(O,T)=CNo
a+2bln2~
1 + ~cfl f dENBcs(E, T ) [ I - t g h 2 ( - ? ) ] }
(4)
--oo
where p = 1/kBT. The temperature dependence of G(0, T) determined by this equation is shown in fig. 1, where we report G (0, T), normalized to its value at T/Tc = 1.5, for the following choices of the parameters: (1) a = b = 0 , c= 1 (BCS curve); (2) a=0.5, b=0.5, c= 1; (3) a=0.5, b=2, c = l ; ( 4 ) a = 0 . 5 , b=10, c = l (a and c are measured in meV- 1, b in meV -2). The figure clearly indicates that, as the weight of the linear term tends to become dominant with respect to the BCS contribution, the discontinuity at the transition temperature is gradually smoothed, with a corresponding disappearance of the exponential behavior below To. The experiments in the inset have been re-
(2)
1.5
zJ(T) being the BCS superconducting energy gap. In the following we study how this assumption affects the temperature dependence of the zero-bias conductance, the specific heat and the ultrasonic attenuation coefficient.
I
=
00,,0
0,1.0 ' o
oe °
!
o •
o° "~0.8
o •
*,~
,,*
oo
t.O
I
o* ° o**
,,
to.
,.,"
1 t
l
2. I. Zero-bias conductance The current I flowing in a superconductor-insulator-normal metal (S-I-N) junction at finite temperature is given by
~"
0.5
o
+oo 0.0
I(V, T)=CNo J dENs(E, T)[f(E, T)
O.l]O
0.t0
0.80
1.20
--oo
- f ( E + e V , T) ] ,
(3)
where V is the applied voltage, C is a constant conraining all the tunneling probabilities, No is the density of states of the normal counterelectrode (assumed constant) a n d f ( E , T) is the Fermi function. From eqs. ( 1 ) and (3) one readily obtains the fol-
T/Tc Fig. 1. Temperaturedependenceofthe normalizedzero-biasconductance for a=0, bffi0, c=1 (curve 1) BCS curve, a=0.5, b=0.5, cffil (curve 2); a=0.5, b=2, cffil (curve 3); a=0.5, b=10, c=l (curve 4); (a and c are measured in meV-t, b in meV-2). The inset showsexperimentaldata on 90 K (circles) and 60 K (stars) YBCOphases,takenfrom Cueoloet al. [6].
A.M. Cucoloet al. I Influence of a linear term on DOS ported in refs. [6] normalization.
and
[7]
with a different
35
0.5
0.4
2.2. Specific heat tO
Inserting eq. ( 1 ) in the general definition of the specific heat 4"oo
II 0.3
p,
v
i
r~ 0.2 [-, v iP
it takes the form 0.1
Cr( T) +oo
S
T) [I
0.0
0.0
+ cEf ( E, T) - 0~ NBcs ( E, T ) } .
i
,o / ~..
-:,ol
1.5
'
i
=t .... "
i
1""
,
],0 t,o, 1/I
:--" t'°:/I
"'
"
e. 0.5
0.0
O.O
0.4
0.2
0.8
0.3
0.4
0.5
T/Tc
(6)
The curves obtained from this equation, normalized to Cv at T~ T~= 1.5 and with the same values of the parameters a, b and c used for the zero-bias conductances curves, are reported in fig. 2. An important feature is that increasing values o f the param2.0
0.!
Fig. 3. Low temperature behavior of the electronic specificheat (enlargementof fig. 2).
eter b give rise to a drastic reduction of the specific heat j u m p AC at To. Since the increase of the weight of the linear contribution satisfactorily simulates the reduction of the oxygen content from 7 to 6.6 [4], the curves of fig. 2 are consistent with the experimental observation [ 8,9 ] that oxygen-deficient samples are characterized by AC jumps at T = T~ lower than those found in the fully oxygenated compound. This is shown in the inset of fig. 2, that refers to the results obtained by Wiihl [9]. As far as the behavior at low temperatures is concerned, it seems now widely accepted that the electronic contribution is described as a linear term 7*T, or more generally a T ~-'~ term (ct<< 1 ). This property is qualitatively well reproduced by our model, as one can see in fig. 3, showing C v ( T ) for 0 < T / To< 0.5. The slope of the curves and the range of temperature where the linear behaviour is exhibited are both increasing functions of the parameter b.
1.2
T/Tc Fig. 2. Normalized temperature dependence of the electronic specific heat. The same choice of the parameters and the same notation as in fig. I have been used. The inset shows the doping dependence of the critical temperatureand the specificheat jump at Tc (taken from W0hl et aL [9]).
2. 3. Ultrasonic attenuation We have evaluated the ultrasonic coefficient as, normalized to its normal state value aN, by using the BCS formula
36
A.M. Cucolo et aL /Influence o f a linear term on DOS
a s (T) otr~(T)
Of(E,T)~]
(f a+t~)dE~ 1- zJ~2)2] -Ns(E'T)20E
,OOdENN(E)2 0f(E,T))
gk-/
with Ns(E, T) givenby eq. (1).Consideringthatthe normal densityof statesin our model reducesto NN(E)=a+bIE[
+c ,
aN is explicitly given by
aN(T)=2I(c+a)2+~4b(c+a)+l,82~ - ] The previous equations give a temperature dependence of the ratio a S / a N shown in fig. 4 for the same choice of the parameters adopted in figs. 1 and 2. Also in this case the linear term in Ns has the effect of making the transition at T c smoother. The experiments concerning the measurement of the ultrasonic attenuation coefficient are rather controversial. Sun et al. [ 10 ] find a change in the slope of the attenuation followed by a decrease less pro1.5
"4
1.O
E-I
}
t
E-'
~"
,
i
,
1 95.
TI~,I )~0,
10
,~"
0.5
""':'"
M
..
...i ~ 6O
:
,
,
,
0.0
0.0
0.4
0.8
Acknowledgement We thank Prof. E. Bonetti for useful comments and discussions.
References
i
o.~.
nounced than that predicted by BCS theory, whereas the results by Saint-Paul et al. [ 11 ] indicate a continuous behavior around Tc with no anomalies up to very low temperatures. These two measurements have been reported for completeness in fig. 4, in the insets (a) and (b), respectively. In the framework of our model, these results could be ascribed to slight differences in the oxygen content. In conclusion, we have introduced a model for the superconducting density of states where a linear term is superimposed to a BCS usual one. With this assumption we have calculated the zero-bias conductance, the specific heat and the ultrasonic attenuation as functions of the temperature for different choices of the parameter of the model. We have found that the increase of the linear term correctly describes the reduction of the oxygen content and a good qualitative agreement with the experimental results has been obtained.
[.2
T/Tc Fig. 4. Normalized temperature dependenceof the ultrasonic attenuation. The same choice of the parameters and the same notation as in fig. 1 have been used. The inset (a) is taken from Sun et al. [ 10], the inset (b) is taken from Saint-Paul et al. [ 11].
[ 1 ] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [ 2 ] C.M. Varma, P.B. Litflewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 1996. [3] J.C. Phillips, Phys. Rev. Lett. 59 (1987) 1856. [4] A.M. Cucolo, C. Noce andA. Romano, Phys. Rev. B: Rapid Comm., Scptember-I (1992), to be published. [ 5 ] A.M. Cucolo, C. Noce and A. Romano, Phys. Lett. A 161 (1991) 176. [ 6 ] A.M. Cucolo, R.C. Dynes, J.M. Valles and L.S. Schneemeyer, Physica C 179 ( 1991 ) 69. [7 ] A.M. Cucolo, R. Di Leo, P. Romano, L.S. Schncemeyer and J.V. Waszczak, Phys. Rev. B 44 ( 1991 ) 2857. [8] A. Junod, D. Eckert, T. Graf, G. Triscone and J. Miiller, Physica C 162-164 (1989) 1401. [9] H. Wtihl, R. Benischke, M. Braun, B. Frank, O. Kraut, R. Ahrens, G. Brauchle, H. Claus, A. Erb, W.H. Fietz, C. Meingast, G. Mfiller-Vogt and T. Wolf, Physica C 185-189 (1991) 755. [ 10] ICJ. Sun, M. Levy, B.IC Sarma, P.H. Hor, R.L. Meng, Y.Q. Wang and C.W. Chu, Phys. Lett. A 131 ( 1988 ) 541. [ 11 ] M. Saint-Paul, J.L. Tholence, P. Monceau, H. No~l, J.C. Levet, M. Potel, P. Gougeon and J.J. Capponi, Solid State Commun. 66 (1988) 641.
Influence of a linear term on the density of states of high-To superconductors Anna Maria Cucolo a, Canio Noce ~'and Alfonso R o m a n o b " Dipartimento di Fisica, Universitlt di Salerno, 1-84081 Baronissi (Salerno), Italy b Dipartimento di Fisica Teorica e Sue Metodologie per le Scienze Applicate, Universitit di Salerno, 1-84081Baronissi (Salerno), Italy
Received 1OJuly 1992
The temperaturedependenceof the zero-biasconductance,the electronicspecificheat and the ultrasonicattenuation in highTcsuperconductorshavebeenanalyzedwithinthe frameworkofa phenomenologicalmodel,basedon a densityof states expressed as a superpositionof a linear term to a BCS standard one. By usingthe same valuesof the fitting parameters,we have obtained a satisfactoryagreementwith the experimentalresults, togetherwith a consistentphysicalinterpretationof the weightof the linear contribution.
1. Introduction
Tunneling experiments performed on high-To superconductors show an unusual behaviour of the background conductance, which up to very high biases ( ~ 200 meV) follows a linear dependence on the absolute value of the applied voltage. This peculiarity is routinely observed in point-contact and planar junctions fabricated on copper oxides LSCO, YBCO, BSCCO and TBCCO as well as in bismuthates, regardless of the junction preparation method. This experimental evidence strongly supports the idea that the linear background conductance is an intrinsic property of high-To superconductors. Several authors have given theoretical explanations of this behavior in terms of density of states effects. Anderson and Zou [ 1 ] have derived a normal state linear conductance from simple assumptions on the spectrum of holon and spinon excitations in a bidimensional RVB state. On a more general ground, it has been suggested that linearity could be a consequence of electronic correlation effects near the metal-insulator transition, leading to a breakdown of the conventional Fermi liquid description. This point of view is explicitly present in the theory of marginal Fermi liquids formulated by Varma [ 2 ], where the hypothesis of a normal den-
sity of states of the form N ( E ) =N0+N~ IEI is found to agree with the basic assumption of the theory, concerning the structure of the charge and spin density excitation spectrum. We also mention the approach by Phillips [3 ] who, within a quantum percolation theory, separates the density of states into an extended part, responsible for the superconducting properties, and a localized part linear in energy, responsible for the normal properties. Motivated by these considerations, we have presented in a previous paper [4 ] a model for the description of tunneling experiments on YBa2Cu307_~ junctions, based on the assumption of a density of states expressed as a superposition of two BCS-like contributions and a linear one. We have made the microscopic hypothesis that two distinct BCS-like gap structures are associated with the CuO2 planes and the CuO chains along the b-axis, respectively, whereas the linear contribution is supposed to take into account the anomalous behavior along the c-axis. By means of this model, satisfactory quantitative fittings have been obtained for tunneling experiments performed both on fully oxygenated (To= 90 K) and oxygen deficient (To=60 K) YBCO junctions, in the superconducting as well as in the normal state. We have found that the increase of the weight of the linear term well describes the reduction of the oxygen
0921-4534/92/$05.00 © 1992ElsevierSciencePublishersB.V. All rightsreserved.
34
A.M. Cucoloet aL ~Influenceof a linear term on DOS
content leading to the decrease of the critical temperature. In this paper we use a simplified one-gap version of this model, previously presented on a purely phenomenological ground [5 ], for the qualitative evaluation of the temperature dependence of other physical quantities of interest in the study of high-To superconductors, namely, the zero-bias conductance, the specific heat and the ultrasonic attenuation. We point out that this model, compared with the one specifically formulated for YBCO in ref. [ 4 ], has the advantage that it is more suited for the study of other high-T~ superconductors.
2. The model
Our basic hypothesis is that the superconducting density of states can be expressed in the form Ns(E, T) =a+ blEI +cNBcs(E, T ) ,
( 1)
where NBcs(E, T) = R e ~ T ) 2
,
lowing expression for the zero-bias conductance G (0, T): G(O,T)=CNo
a+2bln2~
1 + ~cfl f dENBcs(E, T ) [ I - t g h 2 ( - ? ) ] }
(4)
--oo
where p = 1/kBT. The temperature dependence of G(0, T) determined by this equation is shown in fig. 1, where we report G (0, T), normalized to its value at T/Tc = 1.5, for the following choices of the parameters: (1) a = b = 0 , c= 1 (BCS curve); (2) a=0.5, b=0.5, c= 1; (3) a=0.5, b=2, c = l ; ( 4 ) a = 0 . 5 , b=10, c = l (a and c are measured in meV- 1, b in meV -2). The figure clearly indicates that, as the weight of the linear term tends to become dominant with respect to the BCS contribution, the discontinuity at the transition temperature is gradually smoothed, with a corresponding disappearance of the exponential behavior below To. The experiments in the inset have been re-
(2)
1.5
zJ(T) being the BCS superconducting energy gap. In the following we study how this assumption affects the temperature dependence of the zero-bias conductance, the specific heat and the ultrasonic attenuation coefficient.
I
=
00,,0
0,1.0 ' o
oe °
!
o •
o° "~0.8
o •
*,~
,,*
oo
t.O
I
o* ° o**
,,
to.
,.,"
1 t
l
2. I. Zero-bias conductance The current I flowing in a superconductor-insulator-normal metal (S-I-N) junction at finite temperature is given by
~"
0.5
o
+oo 0.0
I(V, T)=CNo J dENs(E, T)[f(E, T)
O.l]O
0.t0
0.80
1.20
--oo
- f ( E + e V , T) ] ,
(3)
where V is the applied voltage, C is a constant conraining all the tunneling probabilities, No is the density of states of the normal counterelectrode (assumed constant) a n d f ( E , T) is the Fermi function. From eqs. ( 1 ) and (3) one readily obtains the fol-
T/Tc Fig. 1. Temperaturedependenceofthe normalizedzero-biasconductance for a=0, bffi0, c=1 (curve 1) BCS curve, a=0.5, b=0.5, cffil (curve 2); a=0.5, b=2, cffil (curve 3); a=0.5, b=10, c=l (curve 4); (a and c are measured in meV-t, b in meV-2). The inset showsexperimentaldata on 90 K (circles) and 60 K (stars) YBCOphases,takenfrom Cueoloet al. [6].
A.M. Cucoloet al. I Influence of a linear term on DOS ported in refs. [6] normalization.
and
[7]
with a different
35
0.5
0.4
2.2. Specific heat tO
Inserting eq. ( 1 ) in the general definition of the specific heat 4"oo
II 0.3
p,
v
i
r~ 0.2 [-, v iP
it takes the form 0.1
Cr( T) +oo
S
T) [I
0.0
0.0
+ cEf ( E, T) - 0~ NBcs ( E, T ) } .
i
,o / ~..
-:,ol
1.5
'
i
=t .... "
i
1""
,
],0 t,o, 1/I
:--" t'°:/I
"'
"
e. 0.5
0.0
O.O
0.4
0.2
0.8
0.3
0.4
0.5
T/Tc
(6)
The curves obtained from this equation, normalized to Cv at T~ T~= 1.5 and with the same values of the parameters a, b and c used for the zero-bias conductances curves, are reported in fig. 2. An important feature is that increasing values o f the param2.0
0.!
Fig. 3. Low temperature behavior of the electronic specificheat (enlargementof fig. 2).
eter b give rise to a drastic reduction of the specific heat j u m p AC at To. Since the increase of the weight of the linear contribution satisfactorily simulates the reduction of the oxygen content from 7 to 6.6 [4], the curves of fig. 2 are consistent with the experimental observation [ 8,9 ] that oxygen-deficient samples are characterized by AC jumps at T = T~ lower than those found in the fully oxygenated compound. This is shown in the inset of fig. 2, that refers to the results obtained by Wiihl [9]. As far as the behavior at low temperatures is concerned, it seems now widely accepted that the electronic contribution is described as a linear term 7*T, or more generally a T ~-'~ term (ct<< 1 ). This property is qualitatively well reproduced by our model, as one can see in fig. 3, showing C v ( T ) for 0 < T / To< 0.5. The slope of the curves and the range of temperature where the linear behaviour is exhibited are both increasing functions of the parameter b.
1.2
T/Tc Fig. 2. Normalized temperature dependence of the electronic specific heat. The same choice of the parameters and the same notation as in fig. I have been used. The inset shows the doping dependence of the critical temperatureand the specificheat jump at Tc (taken from W0hl et aL [9]).
2. 3. Ultrasonic attenuation We have evaluated the ultrasonic coefficient as, normalized to its normal state value aN, by using the BCS formula
36
A.M. Cucolo et aL /Influence o f a linear term on DOS
a s (T) otr~(T)
Of(E,T)~]
(f a+t~)dE~ 1- zJ~2)2] -Ns(E'T)20E
,OOdENN(E)2 0f(E,T))
gk-/
with Ns(E, T) givenby eq. (1).Consideringthatthe normal densityof statesin our model reducesto NN(E)=a+bIE[
+c ,
aN is explicitly given by
aN(T)=2I(c+a)2+~4b(c+a)+l,82~ - ] The previous equations give a temperature dependence of the ratio a S / a N shown in fig. 4 for the same choice of the parameters adopted in figs. 1 and 2. Also in this case the linear term in Ns has the effect of making the transition at T c smoother. The experiments concerning the measurement of the ultrasonic attenuation coefficient are rather controversial. Sun et al. [ 10 ] find a change in the slope of the attenuation followed by a decrease less pro1.5
"4
1.O
E-I
}
t
E-'
~"
,
i
,
1 95.
TI~,I )~0,
10
,~"
0.5
""':'"
M
..
...i ~ 6O
:
,
,
,
0.0
0.0
0.4
0.8
Acknowledgement We thank Prof. E. Bonetti for useful comments and discussions.
References
i
o.~.
nounced than that predicted by BCS theory, whereas the results by Saint-Paul et al. [ 11 ] indicate a continuous behavior around Tc with no anomalies up to very low temperatures. These two measurements have been reported for completeness in fig. 4, in the insets (a) and (b), respectively. In the framework of our model, these results could be ascribed to slight differences in the oxygen content. In conclusion, we have introduced a model for the superconducting density of states where a linear term is superimposed to a BCS usual one. With this assumption we have calculated the zero-bias conductance, the specific heat and the ultrasonic attenuation as functions of the temperature for different choices of the parameter of the model. We have found that the increase of the linear term correctly describes the reduction of the oxygen content and a good qualitative agreement with the experimental results has been obtained.
[.2
T/Tc Fig. 4. Normalized temperature dependenceof the ultrasonic attenuation. The same choice of the parameters and the same notation as in fig. 1 have been used. The inset (a) is taken from Sun et al. [ 10], the inset (b) is taken from Saint-Paul et al. [ 11].
[ 1 ] P.W. Anderson and Z. Zou, Phys. Rev. Lett. 60 (1988) 132. [ 2 ] C.M. Varma, P.B. Litflewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 1996. [3] J.C. Phillips, Phys. Rev. Lett. 59 (1987) 1856. [4] A.M. Cucolo, C. Noce andA. Romano, Phys. Rev. B: Rapid Comm., Scptember-I (1992), to be published. [ 5 ] A.M. Cucolo, C. Noce and A. Romano, Phys. Lett. A 161 (1991) 176. [ 6 ] A.M. Cucolo, R.C. Dynes, J.M. Valles and L.S. Schneemeyer, Physica C 179 ( 1991 ) 69. [7 ] A.M. Cucolo, R. Di Leo, P. Romano, L.S. Schncemeyer and J.V. Waszczak, Phys. Rev. B 44 ( 1991 ) 2857. [8] A. Junod, D. Eckert, T. Graf, G. Triscone and J. Miiller, Physica C 162-164 (1989) 1401. [9] H. Wtihl, R. Benischke, M. Braun, B. Frank, O. Kraut, R. Ahrens, G. Brauchle, H. Claus, A. Erb, W.H. Fietz, C. Meingast, G. Mfiller-Vogt and T. Wolf, Physica C 185-189 (1991) 755. [ 10] ICJ. Sun, M. Levy, B.IC Sarma, P.H. Hor, R.L. Meng, Y.Q. Wang and C.W. Chu, Phys. Lett. A 131 ( 1988 ) 541. [ 11 ] M. Saint-Paul, J.L. Tholence, P. Monceau, H. No~l, J.C. Levet, M. Potel, P. Gougeon and J.J. Capponi, Solid State Commun. 66 (1988) 641.