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The fluctuation enhanced conductivity in amorphous superconductors
Solid State Commumcatlons,Vol
17, pp. 799401,
Pergamon Press
1975
Printed m Great Bntam
THE FLUCTUATION ENHANCED CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS B Keck and A Schnud Instltut fur Physlk der Umverntat Dortmund, Germany (Recezved 29 Aprzl 1975 by B Muhlschlegel)
Conndermg the effective phonon density of states obtamed by tunnel spectroscopic measurements, we conclude that m amorphous metals such as Bl, Ga, and Pb, the electron-phonon colhslon rate becomes as large as 1012/sec at the transttlon temperature Thus 1s attnbuted to the colhsion drag effect whch couples low frequency transverse phonons to the electrons It 1s shown that such a large collision rate suppresses the Malu contnbutlon to the fluctuation enhanced conductlvlty
IN THE AMORPHOUS state of the metals Bl, Ga, and Pb, Glover’ has lscovered an enhanced conductlvlty above the superconducting tranntlon temperature, which he mterpreted to anse from evanescent Cooper pnrs created by thermal fluctuations The theory of Aslamasov and I_arkm,2 which was presented nnmedlately afterwards, was m excellent agreement ~th the experunents Almost at the same time, however, Malu3 showed that the mlcroscoplc theory requires the existence of an addItional contnbutlon to the excess conductivlty Though bemg absent m the matenals Glover investigated, this contnbutlon became mamfest afterwards m granular ahunmmm 4 Of the numerous theoretical mvestlgatlons on the Malu contnbutmn, the one of Thompson’ was the first and, perhaps, the most important. It ISwidely recogmzed now,6 that the Malu contnbutlon c& depends strongly on pair breakmg, m contrast to the Aslamasov-Larkm contnbutlon uLL This dependence 1s seen m the well-known relation &=-------
e2
1
8hd EC-e
h,4 E’
where e = (T - T,)/T,, E, IS the par breakmg parameter, and d IS the film thickness When E, 2 1, the vanation of u;W111the relevant temperature remon IS evidently so small that only a contribution to the
residual res&lvlty results However, No p;ur breakmg large enough was found to explam the apparent absence of oh m the expenments of Glover. In particular, thermal phonon pour breakmg was thought to be msuffclent ’ However, it has been overlooked that m amorphous metals, the electronphonon mteractlon may mcrease considerably at low frequencies Tlus can be mferred from the behamour of the Ehashberg function cr’F(o), which 1s the phonon density of states F(w) multlphed by an appropnate average cy2of the electron-phonon couphng strength Comparmg, for mstance, the Ehashberg functions of amorphous and pure crystallme lead obtamed from tunnel spectroscopic measurements as shown 111Fig 1, one recognizes that there 1sa large increase of a’F(w) m the low frequency range where thermally activated phonons scatter the electrons Thus, the melastlc colhslon rate OD 1 QL2 F(w) - = 4n do (2) s shhw/kB T, TM 0 of the electrons, and hence the parameter of par breakmg by phonons,’ nh $h
=
f%
TPph
may become so large that oh 1sneghgble
(3)
CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS
800
Vol. 17, No 7
Table I Expenmental and theoretical values of eBh The superscnpt gwes the reference Amorphous metals have an electromc mean free path I of about 8 a 1 The theoretical values have been obtatned for the mean free paths gwen m the last column
Metal
State
Bl
Amorphous
Ga Pb
eph Theory
IlA
22* 1 5” 1 416
25 23 17
5 10 20
Amorphous
1 513 1615
38
10
Amorphous
Lo* 1 615 16=
13
10
01
00
1o-3
10
a2F(ul
Pb Al
FIG 1 The Ehashberg function a* F for amorphous (a) and pure crystalhne (b) lead The full hnes represent the expernnental results of reference 16 (Note that the amorphous state of Pb has been stabtied by an addltton of 10% Cu ) The theoretical results for mean free paths I= 10 A and I = * are gven by the broken hnes Quite generally, F(o) as well as a* may undergo changes m the transltlon to the amorphous state. As pomted out by Bergmann, lo there IS a posslbtity that (11’increases when the restnctlons imposed by momentum conservation on electron-phonon colhslons are partially removed by mtermedate nnpurlty scattermg ns effect 1s agmficant only for transverse phonons where It ISalso known as the colhnon drag effect l1 In this rather complex sltuatlon of mtertwmed phonon and unpunty scattermg, it 1sby no means obvrous that the function a’F(o) retams rts umversal meanmg By this we mean that m the pure and m the amorphous mater& a*F(o) appears 111the same form m the expressions (1) for the melastlc hfe-tnne of the electrons, (n) for the ultrasonic attenuation, and (m) for the frequency dependent order parameter. Based on the mvestlgatlons of one of us,12 we have been able to show that tlus umversahty persists A deWled report on this SubJect will be pubhshed later These mvestlgatlons are based on the Debye model where
epph Experunent
Crystalhne Granular
the electron fluid is coupled by its stresses to the strams of the lattice Furthermore, Umklapp-processes and changes m the phonon spectrum have been neglected In this case, we obtamed c: w2
or*F(w) = Xp - 4h
(ci4M4CI)
+ 2Cr4 &(4C,))
(4) Here, $ IS the electron-phonon couplmg constant of the pure metal; cI and ct are the two sound velocities, qD 1sthe Debye wave vector and I the mean free path of of the electrons The functions & and #t have been mtroduced m reference 12 In particular, we have &(m) = 1 and &(m) = 0, whch means that m a pure metal, only lon@tudmal phonons couple to the electrons For values of the parameters whch seem to be reasonable for Pb, and for mean free paths of I = 10 A and 1= 00,a graph of equation (4) is shown m Fig 1 Companng the expenmental and the theoretical results, one should remember that m the former case, there 1sa poor resolution at small frequencies, and that m the latter case, the neglect of Umklapp-processes and of the broademng of the phonon density of states causes rather large errors at large frequencies In Table 1, we gWevalues of eph for various metals calculated from expenmental data and from theory Although there 1s a considerable scatter m the data of lfferent expenments, as well as m the results from expenment and theory, there can be no doubt
Vol 17,No 7
CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS
that epph1slarger than 1 m the amorphous state of Bl, Ga and Pb Thus, we can understand Clover’s result
801
the Malu contnbutlon 1ssuppressed In this case, only fluctuating Cooper pairs contnbute to the excess conductivity above the transition temperature
The theoretical value eph = 0 1 for pure Pb should be compared with the value 0 05 found m reference 7 from an analysts of the Mah contnbutlon The small value of eph for granular Al can be explamed by the large ratio of the Debye temperature to the transition temperature Of course, when Q, ISthat small, other contnbutlons to the pair breakmg, for instance, those discussed m references 17-19 have to be taken mto account
This large colhslon rate results from the large values of the Ehashberg function at small frequencies as found m tunnehng expenments We have explamed this fact by an increase m the effective strength of the electronphonon mteractlon We leave open the posslblhty that other effects may contribute as well to the form of the Ehashberg function and that a thorough mvestlgation *’ of the properties of amorphous metals seems to be a very pronusmg and interesting task
In conclusion, it can be md that m amorphous metals where the Debye temperature 1snot excessively larger than the transltlon temperature, the melastlc colhslon rate of electrons Hrlth phonons 1s so large that
Acknowledgement
- It IS a pleasure to acknowledge stlmulatmg dlscusslons with Dr Bergmann We are also indebted to the Deutsche Forschungsgememschaft for their financial support
REFERENCES 1 2
CLOVER R.E ,Phys Let? ZSA, 542 (1967),Physzca 55,3 (1971) ASLAMASOV L G & LARKIN A I , Phys Left. 26A, 238 (1968), Fzz. 7berd Tela lo,1104 (1968) [Sov. Phys. (1968)]
SoZzdState 10,875
MAKI K , Progr meotet
Phys (Kyofo) 40,193 (1968)
CROW J E., THOMPSON R S , KLENIN M.A & BHATNAGAR A.K ,Phys Rev Lett 24,371 (1970) THOMPSON R S , Phys Rev Bl, 327 (1970) CRAVEN R A., THOMAS G A & PARKS R D , Phys Rev B7,157 (1973) CROW J E., BHATNAGAR A K & MIHALISIN T ,Phys Rev Lett 28,25 (1972) Evidently, eph mcreases when the electron-phonon mteraction 1smcreased However, it has been shown by Bergmann and Ramer,g that T, increases also m such a case Hence, there is an mconsistency when eph is called a pour breakmg parameter However, no confusion should anse when usmg this expression m the present context 9
BERGMANN G & RAINER D , 2. Phys 263,59 (1973)
10
BERGMANN G , Phys Rev B3,3797 (1971)
11
In contrast to Bergmann’s conclusion, we cannot find that longtudmal phonons contnbute to the mcrease m mteractlon strength
12
SCHMID A, Z Phys 259,421 (1973), 271,251 (1974)
13
LESLIE J.D , CHEN J.T & CHEN T T , Can J Phys. 48,2783
14.
ZAVARITSKII N V , Zh Eksp Teor Fzz 57,752
15
WtfHL H , JACKSON J E & BRISCOE C V ,Phys Rev Left 20,1496
16
KNORR K & BARTH N , J Low Temp Phys 4,469 (1971)
17
PATTON B R ,Phys Rev Left 27,1272 (1971)
18
KELLER J & KORENMAN V., Phys Rev B5,4367 (1972)
19
LARKIN A 1 & OVCHINNIKOV Yu N , J Low Temp Phys 10,407 (1973)
20
A recent investigation on tis SubJeCtcan be found m reference 21
21
KRAUSS G & BUCKEL W , Z Phys B20, 147 (1975).
(1970)
(1969) [Sov Phys -JE7P 30,412 (1970)] (1968)
17, pp. 799401,
Pergamon Press
1975
Printed m Great Bntam
THE FLUCTUATION ENHANCED CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS B Keck and A Schnud Instltut fur Physlk der Umverntat Dortmund, Germany (Recezved 29 Aprzl 1975 by B Muhlschlegel)
Conndermg the effective phonon density of states obtamed by tunnel spectroscopic measurements, we conclude that m amorphous metals such as Bl, Ga, and Pb, the electron-phonon colhslon rate becomes as large as 1012/sec at the transttlon temperature Thus 1s attnbuted to the colhsion drag effect whch couples low frequency transverse phonons to the electrons It 1s shown that such a large collision rate suppresses the Malu contnbutlon to the fluctuation enhanced conductlvlty
IN THE AMORPHOUS state of the metals Bl, Ga, and Pb, Glover’ has lscovered an enhanced conductlvlty above the superconducting tranntlon temperature, which he mterpreted to anse from evanescent Cooper pnrs created by thermal fluctuations The theory of Aslamasov and I_arkm,2 which was presented nnmedlately afterwards, was m excellent agreement ~th the experunents Almost at the same time, however, Malu3 showed that the mlcroscoplc theory requires the existence of an addItional contnbutlon to the excess conductivlty Though bemg absent m the matenals Glover investigated, this contnbutlon became mamfest afterwards m granular ahunmmm 4 Of the numerous theoretical mvestlgatlons on the Malu contnbutmn, the one of Thompson’ was the first and, perhaps, the most important. It ISwidely recogmzed now,6 that the Malu contnbutlon c& depends strongly on pair breakmg, m contrast to the Aslamasov-Larkm contnbutlon uLL This dependence 1s seen m the well-known relation &=-------
e2
1
8hd EC-e
h,4 E’
where e = (T - T,)/T,, E, IS the par breakmg parameter, and d IS the film thickness When E, 2 1, the vanation of u;W111the relevant temperature remon IS evidently so small that only a contribution to the
residual res&lvlty results However, No p;ur breakmg large enough was found to explam the apparent absence of oh m the expenments of Glover. In particular, thermal phonon pour breakmg was thought to be msuffclent ’ However, it has been overlooked that m amorphous metals, the electronphonon mteractlon may mcrease considerably at low frequencies Tlus can be mferred from the behamour of the Ehashberg function cr’F(o), which 1s the phonon density of states F(w) multlphed by an appropnate average cy2of the electron-phonon couphng strength Comparmg, for mstance, the Ehashberg functions of amorphous and pure crystallme lead obtamed from tunnel spectroscopic measurements as shown 111Fig 1, one recognizes that there 1sa large increase of a’F(w) m the low frequency range where thermally activated phonons scatter the electrons Thus, the melastlc colhslon rate OD 1 QL2 F(w) - = 4n do (2) s shhw/kB T, TM 0 of the electrons, and hence the parameter of par breakmg by phonons,’ nh $h
=
f%
TPph
may become so large that oh 1sneghgble
(3)
CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS
800
Vol. 17, No 7
Table I Expenmental and theoretical values of eBh The superscnpt gwes the reference Amorphous metals have an electromc mean free path I of about 8 a 1 The theoretical values have been obtatned for the mean free paths gwen m the last column
Metal
State
Bl
Amorphous
Ga Pb
eph Theory
IlA
22* 1 5” 1 416
25 23 17
5 10 20
Amorphous
1 513 1615
38
10
Amorphous
Lo* 1 615 16=
13
10
01
00
1o-3
10
a2F(ul
Pb Al
FIG 1 The Ehashberg function a* F for amorphous (a) and pure crystalhne (b) lead The full hnes represent the expernnental results of reference 16 (Note that the amorphous state of Pb has been stabtied by an addltton of 10% Cu ) The theoretical results for mean free paths I= 10 A and I = * are gven by the broken hnes Quite generally, F(o) as well as a* may undergo changes m the transltlon to the amorphous state. As pomted out by Bergmann, lo there IS a posslbtity that (11’increases when the restnctlons imposed by momentum conservation on electron-phonon colhslons are partially removed by mtermedate nnpurlty scattermg ns effect 1s agmficant only for transverse phonons where It ISalso known as the colhnon drag effect l1 In this rather complex sltuatlon of mtertwmed phonon and unpunty scattermg, it 1sby no means obvrous that the function a’F(o) retams rts umversal meanmg By this we mean that m the pure and m the amorphous mater& a*F(o) appears 111the same form m the expressions (1) for the melastlc hfe-tnne of the electrons, (n) for the ultrasonic attenuation, and (m) for the frequency dependent order parameter. Based on the mvestlgatlons of one of us,12 we have been able to show that tlus umversahty persists A deWled report on this SubJect will be pubhshed later These mvestlgatlons are based on the Debye model where
epph Experunent
Crystalhne Granular
the electron fluid is coupled by its stresses to the strams of the lattice Furthermore, Umklapp-processes and changes m the phonon spectrum have been neglected In this case, we obtamed c: w2
or*F(w) = Xp - 4h
(ci4M4CI)
+ 2Cr4 &(4C,))
(4) Here, $ IS the electron-phonon couplmg constant of the pure metal; cI and ct are the two sound velocities, qD 1sthe Debye wave vector and I the mean free path of of the electrons The functions & and #t have been mtroduced m reference 12 In particular, we have &(m) = 1 and &(m) = 0, whch means that m a pure metal, only lon@tudmal phonons couple to the electrons For values of the parameters whch seem to be reasonable for Pb, and for mean free paths of I = 10 A and 1= 00,a graph of equation (4) is shown m Fig 1 Companng the expenmental and the theoretical results, one should remember that m the former case, there 1sa poor resolution at small frequencies, and that m the latter case, the neglect of Umklapp-processes and of the broademng of the phonon density of states causes rather large errors at large frequencies In Table 1, we gWevalues of eph for various metals calculated from expenmental data and from theory Although there 1s a considerable scatter m the data of lfferent expenments, as well as m the results from expenment and theory, there can be no doubt
Vol 17,No 7
CONDUCTIVITY IN AMORPHOUS SUPERCONDUCTORS
that epph1slarger than 1 m the amorphous state of Bl, Ga and Pb Thus, we can understand Clover’s result
801
the Malu contnbutlon 1ssuppressed In this case, only fluctuating Cooper pairs contnbute to the excess conductivity above the transition temperature
The theoretical value eph = 0 1 for pure Pb should be compared with the value 0 05 found m reference 7 from an analysts of the Mah contnbutlon The small value of eph for granular Al can be explamed by the large ratio of the Debye temperature to the transition temperature Of course, when Q, ISthat small, other contnbutlons to the pair breakmg, for instance, those discussed m references 17-19 have to be taken mto account
This large colhslon rate results from the large values of the Ehashberg function at small frequencies as found m tunnehng expenments We have explamed this fact by an increase m the effective strength of the electronphonon mteractlon We leave open the posslblhty that other effects may contribute as well to the form of the Ehashberg function and that a thorough mvestlgation *’ of the properties of amorphous metals seems to be a very pronusmg and interesting task
In conclusion, it can be md that m amorphous metals where the Debye temperature 1snot excessively larger than the transltlon temperature, the melastlc colhslon rate of electrons Hrlth phonons 1s so large that
Acknowledgement
- It IS a pleasure to acknowledge stlmulatmg dlscusslons with Dr Bergmann We are also indebted to the Deutsche Forschungsgememschaft for their financial support
REFERENCES 1 2
CLOVER R.E ,Phys Let? ZSA, 542 (1967),Physzca 55,3 (1971) ASLAMASOV L G & LARKIN A I , Phys Left. 26A, 238 (1968), Fzz. 7berd Tela lo,1104 (1968) [Sov. Phys. (1968)]
SoZzdState 10,875
MAKI K , Progr meotet
Phys (Kyofo) 40,193 (1968)
CROW J E., THOMPSON R S , KLENIN M.A & BHATNAGAR A.K ,Phys Rev Lett 24,371 (1970) THOMPSON R S , Phys Rev Bl, 327 (1970) CRAVEN R A., THOMAS G A & PARKS R D , Phys Rev B7,157 (1973) CROW J E., BHATNAGAR A K & MIHALISIN T ,Phys Rev Lett 28,25 (1972) Evidently, eph mcreases when the electron-phonon mteraction 1smcreased However, it has been shown by Bergmann and Ramer,g that T, increases also m such a case Hence, there is an mconsistency when eph is called a pour breakmg parameter However, no confusion should anse when usmg this expression m the present context 9
BERGMANN G & RAINER D , 2. Phys 263,59 (1973)
10
BERGMANN G , Phys Rev B3,3797 (1971)
11
In contrast to Bergmann’s conclusion, we cannot find that longtudmal phonons contnbute to the mcrease m mteractlon strength
12
SCHMID A, Z Phys 259,421 (1973), 271,251 (1974)
13
LESLIE J.D , CHEN J.T & CHEN T T , Can J Phys. 48,2783
14.
ZAVARITSKII N V , Zh Eksp Teor Fzz 57,752
15
WtfHL H , JACKSON J E & BRISCOE C V ,Phys Rev Left 20,1496
16
KNORR K & BARTH N , J Low Temp Phys 4,469 (1971)
17
PATTON B R ,Phys Rev Left 27,1272 (1971)
18
KELLER J & KORENMAN V., Phys Rev B5,4367 (1972)
19
LARKIN A 1 & OVCHINNIKOV Yu N , J Low Temp Phys 10,407 (1973)
20
A recent investigation on tis SubJeCtcan be found m reference 21
21
KRAUSS G & BUCKEL W , Z Phys B20, 147 (1975).
(1970)
(1969) [Sov Phys -JE7P 30,412 (1970)] (1968)