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Flux pinning and creep in high-Tc superconductors
Physica C 162-164 (1989) 239-240 North-Holland
FLUX PINNING AND CREEP IN HIGH-T¢ SUPERCONDUCTORS V.GESHKENBEIN, A.LARKIN, M.FEIGEL'MAN, V.VINOKUR ° Landau Institute for Theoretical Physics, Moscow • Institute of Solid State Physics, Chernogolovka, Moscow Region, USSR The theory of collective pinning in the presense of thermal fluctuations of vortex lines is developed. Critical current j, is shown to decrease rapidly with temperature. Flux creep under the action of weak (j << j,) current is considered and j-dependence of the effective energy barrier U(j) is estimated. At the current decrease U(j) grows initially as j-% and then saturates at a finite value Ups, due to plastic deformations coming into play. The relations is discussed between this theory and the existing experiments. 1. INTRODUCTION There are a number of unusual features of the bchaviour of high-T, nmterials in the mixed state: i) a giant flux creep characterized by a very low pinning energy up at low temperatures and in rather veak magnetic field H ~ 0.1T (e.g.Up ~ 200K was observed in 1, and even Up ~ 50K in 2); nonmonotonic temperature behavior of the logariphmic relaxation rate T/Up(T) implying the increase of the effective barrier value U~(T) with temperature 3. ii) rapid fall of the critical current density j, oc ezp(-T/T1) with 7"1 << T, 4.s in the field range 0.2-4 T. iii) rather steep depinning of the flux lines lattice (FLL) at the temperature Td(H) well below T, (Td is measured by mechanic oscillation method and interpreted as melting of FLL 6). iv) the existence of broad temperature region well below T¢ where linear I-V curve with resistivity p(T) oc ezp(-U~(H)/T) was observed in the weak current region r.s , the value of U~(H) being much higher then Up measured in critical state relaxation experiments (item i). It is a great challenge for theorists to account for all these features as unite picture. The approach developed below may provide the step on this way. 2. COLLECTIVE CREEP We assume that pinning in high-T, materials (especially in Bi-based ones) is mainly due to weak shortrange disorder, so that the length scale of random potential that flux lines (FLs) see is rl ~ ~.Then in the low T, low B region the picture of the collective pinning of individual FLs holds, leading to well-known result 9 0921-4534/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
: Jc ~ Ja,p,~(~/L,)2,where the pinning length L, depends upon the strength of disorder 9 . The value of pinning barrier Up at j ~ j, can be estimated as
up = JoL,'~o,'u = ¢ 0 ~ / ~ [ n ~ j ~ / 4 ~ c ] 1/2
(1)
this expression being valid for the case X _1_(a. b) plane. An estimate (1) gives Up ~ 102K in qualitative agreement with 1,2. One mechanism giving rise to the strong temperature and time dependence of the critical current is the flux creep. To describe the creep phenomenon the Anderson formula j(t) = j~(1 - (T/Up)log(t/to)) is commonly used, which is valid at ( j , - j) << j,. In high-T~ superconductors, however, this condition breaks down at T > 10K. The theory of collective creep developed by authors 13 includes the case o f j << j, and,therefore, can be employed in wider temperature region. The main result of ta is that the activation barier, U(j), associated with the currents j << Jc, grows with j decrease, U(j) o~ j - ~ . Since U(j(t)) = Tlog(t/to) 14, then for j << j, one finds j oc (TIogt)-ll% In the region j, > j > j,(L,/ao) r15 we obtain la a = 1/7. An appropriate interpolation between this result and Anderson formula is given (in the intermediate region Up < Tlog(t/to) < 7Up ) by j(t) = j,(l + Tlog(t/to)/TUp) -r ~ j, ezp(-T/T1) with 7"1~ 10K. The time dependence of the magnetization current is given by j(t) oc t-a, with/9 = T/Up ~ 0.05 + 0.15. In the region jl > j > j~ = jl(ao/:~) 2 (where a0 is the F L L spacing) creep is controlled by hopping of the small bundles with transverse size, R, such that ao < R < ,~ and
240
V. Geshkenbein et al. / Flux pinning and creep in high-T~ superconductors
a=3/2. At smaller currents j < J2 the bundles with R > ), give the main contribution and a = 7/9. Our theory can be relevant for the intermediate temperature/current region between low-T creep and high-T TAFF 15 3. COLLECTIVE PINNING OF FLUCTUATING FLUX LINES As the temperature increases up to Tv ~ 0.3Up, thermal fluctuations of FLs appear to be relevant and reduce very strongly the value of critical current calculated in the picture of independent FLs. Practically it means that at T > Tv rapid crossover takes place to the region of collective pinning of FLL. Note here that an appearence of the strong j~(B) dependence at T > 40K was observed in 4 in the same field range where j~(T < 30K) was B-independent. The mean- squared fluctuation in FL position is given now by u~. = ~2(T/TL(B)) 10, where TL(B) = = Oao/2BX12/(27r~)2, a = A/L To obtain j¢(T, B) dependence one can merely replace r! = ~ by r1(T ) = (~z+u~) t12 in formulas (54),(56) of paper 9, the T,B-independent parameter 7 (W(T) = 7r74(T)B/@o) being fixed; for more detailes see 1°.In particular, in the region ja,p,~,/a 2 << j~ << jd,p,i~(B/H,2) ; T << T~(B), we obtain j,(T,B)/j,(O,B) oc ezp(-eonsf TB) (of.4). Moreover, since the picture of thermal fluctuations in the elastic FLL gives rise to the sharp fall of j¢ at T > TL(B), some doubts, whether the experimental results 6 are really associated with FLL melting, may arise. As the melting temperature T,~(B) is determined by the relation u~ = c~a20(et, is the Lindemann parameter, see e.g.ll), one easily gets that T~(B) < TL(B) provided B > 4c~He2. In the opposite case an intermediate phase of perfect, unpinned FLL should exist in the temperature range TL < T < TM(B). To find out if such a state really exists, the measurements of FLL shear modulus C66 could be very useful (see e.g. discussion of this subject in 12). Another interesting feature of 3d elastic collective pinning is that the growth of rp(T) leads also to the increase of the effective pinning barrier Up(T) (e.g. Up(T) o~ BaT 4 in the region where j: < j~,p,i,/~ 2) so that the decrease of relaxation rate T/Up(T) can be accounted for. 4. PLASTIC TAFF At high temperatures and low currents the activa-
tion energy associated with the elastic collective creep process is very large and another,plastic, mechanism of thermally activated flux motion becomes operating.In fact, similiar idea has already been proposed by Tinkham 16, who considered "the motion of a row of fluxons past neighbour rows" ,however our results differ from that of 16. The plastic deformations of FLL axe due to thermally activated motions of dislocations over the Peierls barriers associated with the periodic structure of FLL. The activation barrier for such a motion is aproximately the energy of double-kink configuration which can be estimated as
Upl ~- ~ao)~-2 c<(T¢ - T)H -I12
(2)
Linear Uv,(T) dependence (2) can also account for an apparent T-independence of the U, observed in resistive measurements 7: p(T) oc exp(-Up~/T) = exp[U°~(1/Tc- l/T)], where U~s ~ H -1/2 and does not depend on temperature. Moreover, the fact that Up, ~ (Tc-T) is in agreement with the observed 5 behaviour of the irreversibility temperature 7~, as the function of measurement's frequency: dTirr/dlogf = const. REFERENCES 1. Y. Yeshurun et al, Phys.Rev. B37 (1988) 11828. 2. A.C. Mota, private communications 3. M. Tuominen et al, Physiea C153-155 (1988) 324. 4. S. Senoussi et al, Phys.Rev. B37 (1988) 9792. 5.
J. van den Berg, C.J. van der Beeket al, Super-
cond.ScLTehn. 1 (1989) 249. 6. P.L. Gammel et al, Phys.Rev.Le~t. 61 (1988) 1666. 7. T.T.M. Palstra et al, Phys.Rev.Lett. 61 (1988) 1662. 8. J.Z. Sun et al, Appl.Phys.Let$. 54 (1989) 663. 9. A.I. Larkin, Yu.N. Ovchinnikov, J.Low Terap.Phys. 34 (1979) 409. 10.M.V. Feigel'man, V.M. Vinokur, Phys.Rev.B, in press. II.A. Houghton et al, preprin~ (1989). 12.D.S. Fisher, Phys.Rev. B22 (1980) I190. 13.M.V.Feigel'man, V.B.Geshkenbein, A.I.Larkin, V.M. Vinokur, submitted to Phys.Rev.Lett (1989). 14.V.B. Geshkenbein, A.I. Larkin, ZhETF 95 (1989) II08. 15.P.M. Kes et al, Supercoad.Sci.Tehn. I (1989) 242. 16.M. Tinkham, Phys.Rev.Lett 61 (1988) 1658.
FLUX PINNING AND CREEP IN HIGH-T¢ SUPERCONDUCTORS V.GESHKENBEIN, A.LARKIN, M.FEIGEL'MAN, V.VINOKUR ° Landau Institute for Theoretical Physics, Moscow • Institute of Solid State Physics, Chernogolovka, Moscow Region, USSR The theory of collective pinning in the presense of thermal fluctuations of vortex lines is developed. Critical current j, is shown to decrease rapidly with temperature. Flux creep under the action of weak (j << j,) current is considered and j-dependence of the effective energy barrier U(j) is estimated. At the current decrease U(j) grows initially as j-% and then saturates at a finite value Ups, due to plastic deformations coming into play. The relations is discussed between this theory and the existing experiments. 1. INTRODUCTION There are a number of unusual features of the bchaviour of high-T, nmterials in the mixed state: i) a giant flux creep characterized by a very low pinning energy up at low temperatures and in rather veak magnetic field H ~ 0.1T (e.g.Up ~ 200K was observed in 1, and even Up ~ 50K in 2); nonmonotonic temperature behavior of the logariphmic relaxation rate T/Up(T) implying the increase of the effective barrier value U~(T) with temperature 3. ii) rapid fall of the critical current density j, oc ezp(-T/T1) with 7"1 << T, 4.s in the field range 0.2-4 T. iii) rather steep depinning of the flux lines lattice (FLL) at the temperature Td(H) well below T, (Td is measured by mechanic oscillation method and interpreted as melting of FLL 6). iv) the existence of broad temperature region well below T¢ where linear I-V curve with resistivity p(T) oc ezp(-U~(H)/T) was observed in the weak current region r.s , the value of U~(H) being much higher then Up measured in critical state relaxation experiments (item i). It is a great challenge for theorists to account for all these features as unite picture. The approach developed below may provide the step on this way. 2. COLLECTIVE CREEP We assume that pinning in high-T, materials (especially in Bi-based ones) is mainly due to weak shortrange disorder, so that the length scale of random potential that flux lines (FLs) see is rl ~ ~.Then in the low T, low B region the picture of the collective pinning of individual FLs holds, leading to well-known result 9 0921-4534/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
: Jc ~ Ja,p,~(~/L,)2,where the pinning length L, depends upon the strength of disorder 9 . The value of pinning barrier Up at j ~ j, can be estimated as
up = JoL,'~o,'u = ¢ 0 ~ / ~ [ n ~ j ~ / 4 ~ c ] 1/2
(1)
this expression being valid for the case X _1_(a. b) plane. An estimate (1) gives Up ~ 102K in qualitative agreement with 1,2. One mechanism giving rise to the strong temperature and time dependence of the critical current is the flux creep. To describe the creep phenomenon the Anderson formula j(t) = j~(1 - (T/Up)log(t/to)) is commonly used, which is valid at ( j , - j) << j,. In high-T~ superconductors, however, this condition breaks down at T > 10K. The theory of collective creep developed by authors 13 includes the case o f j << j, and,therefore, can be employed in wider temperature region. The main result of ta is that the activation barier, U(j), associated with the currents j << Jc, grows with j decrease, U(j) o~ j - ~ . Since U(j(t)) = Tlog(t/to) 14, then for j << j, one finds j oc (TIogt)-ll% In the region j, > j > j,(L,/ao) r15 we obtain la a = 1/7. An appropriate interpolation between this result and Anderson formula is given (in the intermediate region Up < Tlog(t/to) < 7Up ) by j(t) = j,(l + Tlog(t/to)/TUp) -r ~ j, ezp(-T/T1) with 7"1~ 10K. The time dependence of the magnetization current is given by j(t) oc t-a, with/9 = T/Up ~ 0.05 + 0.15. In the region jl > j > j~ = jl(ao/:~) 2 (where a0 is the F L L spacing) creep is controlled by hopping of the small bundles with transverse size, R, such that ao < R < ,~ and
240
V. Geshkenbein et al. / Flux pinning and creep in high-T~ superconductors
a=3/2. At smaller currents j < J2 the bundles with R > ), give the main contribution and a = 7/9. Our theory can be relevant for the intermediate temperature/current region between low-T creep and high-T TAFF 15 3. COLLECTIVE PINNING OF FLUCTUATING FLUX LINES As the temperature increases up to Tv ~ 0.3Up, thermal fluctuations of FLs appear to be relevant and reduce very strongly the value of critical current calculated in the picture of independent FLs. Practically it means that at T > Tv rapid crossover takes place to the region of collective pinning of FLL. Note here that an appearence of the strong j~(B) dependence at T > 40K was observed in 4 in the same field range where j~(T < 30K) was B-independent. The mean- squared fluctuation in FL position is given now by u~. = ~2(T/TL(B)) 10, where TL(B) = = Oao/2BX12/(27r~)2, a = A/L To obtain j¢(T, B) dependence one can merely replace r! = ~ by r1(T ) = (~z+u~) t12 in formulas (54),(56) of paper 9, the T,B-independent parameter 7 (W(T) = 7r74(T)B/@o) being fixed; for more detailes see 1°.In particular, in the region ja,p,~,/a 2 << j~ << jd,p,i~(B/H,2) ; T << T~(B), we obtain j,(T,B)/j,(O,B) oc ezp(-eonsf TB) (of.4). Moreover, since the picture of thermal fluctuations in the elastic FLL gives rise to the sharp fall of j¢ at T > TL(B), some doubts, whether the experimental results 6 are really associated with FLL melting, may arise. As the melting temperature T,~(B) is determined by the relation u~ = c~a20(et, is the Lindemann parameter, see e.g.ll), one easily gets that T~(B) < TL(B) provided B > 4c~He2. In the opposite case an intermediate phase of perfect, unpinned FLL should exist in the temperature range TL < T < TM(B). To find out if such a state really exists, the measurements of FLL shear modulus C66 could be very useful (see e.g. discussion of this subject in 12). Another interesting feature of 3d elastic collective pinning is that the growth of rp(T) leads also to the increase of the effective pinning barrier Up(T) (e.g. Up(T) o~ BaT 4 in the region where j: < j~,p,i,/~ 2) so that the decrease of relaxation rate T/Up(T) can be accounted for. 4. PLASTIC TAFF At high temperatures and low currents the activa-
tion energy associated with the elastic collective creep process is very large and another,plastic, mechanism of thermally activated flux motion becomes operating.In fact, similiar idea has already been proposed by Tinkham 16, who considered "the motion of a row of fluxons past neighbour rows" ,however our results differ from that of 16. The plastic deformations of FLL axe due to thermally activated motions of dislocations over the Peierls barriers associated with the periodic structure of FLL. The activation barrier for such a motion is aproximately the energy of double-kink configuration which can be estimated as
Upl ~- ~ao)~-2 c<(T¢ - T)H -I12
(2)
Linear Uv,(T) dependence (2) can also account for an apparent T-independence of the U, observed in resistive measurements 7: p(T) oc exp(-Up~/T) = exp[U°~(1/Tc- l/T)], where U~s ~ H -1/2 and does not depend on temperature. Moreover, the fact that Up, ~ (Tc-T) is in agreement with the observed 5 behaviour of the irreversibility temperature 7~, as the function of measurement's frequency: dTirr/dlogf = const. REFERENCES 1. Y. Yeshurun et al, Phys.Rev. B37 (1988) 11828. 2. A.C. Mota, private communications 3. M. Tuominen et al, Physiea C153-155 (1988) 324. 4. S. Senoussi et al, Phys.Rev. B37 (1988) 9792. 5.
J. van den Berg, C.J. van der Beeket al, Super-
cond.ScLTehn. 1 (1989) 249. 6. P.L. Gammel et al, Phys.Rev.Le~t. 61 (1988) 1666. 7. T.T.M. Palstra et al, Phys.Rev.Lett. 61 (1988) 1662. 8. J.Z. Sun et al, Appl.Phys.Let$. 54 (1989) 663. 9. A.I. Larkin, Yu.N. Ovchinnikov, J.Low Terap.Phys. 34 (1979) 409. 10.M.V. Feigel'man, V.M. Vinokur, Phys.Rev.B, in press. II.A. Houghton et al, preprin~ (1989). 12.D.S. Fisher, Phys.Rev. B22 (1980) I190. 13.M.V.Feigel'man, V.B.Geshkenbein, A.I.Larkin, V.M. Vinokur, submitted to Phys.Rev.Lett (1989). 14.V.B. Geshkenbein, A.I. Larkin, ZhETF 95 (1989) II08. 15.P.M. Kes et al, Supercoad.Sci.Tehn. I (1989) 242. 16.M. Tinkham, Phys.Rev.Lett 61 (1988) 1658.