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Superconducting long-range order in mixed state
Phystea C 235-240 (1994) 1949-1950 North-Holland
Superconducting
PHYiCA
L o n g - R a n g e O r d e r in M i x e d S t a t e
G.G.Sergeeva NSC Kharkov Insut,,i~ ufPhys, and Technology, 310108, Kharkov, Ukraine. It is shown that the controversies between different defivittons of the gauge-mvariant order parameter :-nase can be removed by taking into account that in the Meissner state transverse component A is gauge-invartant q~ ~.attty and under gaage transformation only longitudmal component A changes As a consequence, gauge-invar~aat order parameter phase and the phase within London gauge satisfy the same equattons and boundary conditmn ,:or three &mensional superconductors m the mixed state it is shown that the destruction of superconducting iong-r:nge order parameter by thermal fluctuatmns does not depend on the defimtton of the order parameter.
VVO(r) =-KVQ, Q = A-lhcVtD(r)
1. INTRODUCTION The problem of the influence o f the thermal fluctuattons on the existence of a long-range order in the lattice o f Abrikosov vortices remains open since the pioneermg works, where it was shown that thermal fluctuatmns destroy the superconducting long-range order (SLRO) in the rmxed state. In spite of the narrowness af the fluctuatton regmn, the broadening of the transitmn to nuxed state was dtscovered m experiments wtth low-temperature superconductors. In HTSC, because of shorter correlatmn lengths the thermal fluetuattons are stronger than in low temperature ones. Thetr account is by far more comphcated and leads to contradictory results dependmg on the defimtlons of the order parameter [1-3]. In this report we compare various definltion~ of the gradlent-mvanant order parameter (GIOP) and show that m the nuxed state SLRO ts deotroyed by thermal fluctuatton irrespective of the defmtton of the GIOP.
(2)
where Q is the superflutd velocay, ,.hlch is measurable and gauge-mvanant quantity, ts the Gmzburg parameter. Within the London choice of gauge one has V A = 0 , and as tt fol~ ws from Eq.(2),the phase q~L(r) tS also the solution of Eq (2). l:rom the expressmns (1) and (2) it folio,,- that the phase tI~*(r) also satisfies Eq (2) Vd>*(r) = - KQ VVcD*(r) = -~:VQ
(3) (a)
The gra&ent ~*(r) depends on both Iongd~._hnal and transverse components of the superflmd velocity Q = Q I +Qt : Qi = k (kQ)/k ~,
V Qt = 0
(5)
One can see from the Eqs. (2) and (5) that VV(D*(r) =-K VQ!
(6)
2. R E S U L T S At present, two defimtmns for the GIOP phase are known The first of them q~*(~ ts ordinary and follows from the gauge transformatsons of the vector potentml A and the phase q~(r) of the order parameter 'F(r) [41" cl~*(r) = q~(r)- ~ I A d|
(!)
Accordmg to the second defimtmn the GIOP phase (-) (r) ts the solution of the equatton I21.
and hence VV(I)'~(r) depends only on Qr~ The contradtctton between Eqs (3) and (6) can be removed by taking mto account that for singly ~_onnected superconductors the transverse component At ~s gaugemvartant quantity and only Iongttudmal component A! changes under gauge trasformattons Thts means that for single-valued phase q)~,(r) of the Metssner state Eqs.(3) ar,,I (6) have no contradtctum, and the extstence of SLRO m Melssner state does no~ depend on the definmon ol the GIOP phase
0921-4534/94/$07 00 © 1994 - I.lscvlcr Scmncc B V All rights reserved SSDI 0921-4534(94)01541-4
1950
G G Sergeeva/Physwa C 235-240 (1994) 1949-1950
In the tmxed state singly connected superconductor becomes multiply connected and we must take rote account the multi-valued character of the phase and a contribution of both components of Q into O*. The multi-valued part of the phase leads to the localized vorttclty of both V O*(r) and transverse component Qt at the vortex core. It should be noted that in Ref.[2] the correlation function G(r,r') was actually calculated with both components of Q taken Jnto account .The comparison of the Eqs.(2.28) and (2.29) for O from Ref.[2] and the Eqs.(3),(4) for ~ * shows that VOo and transverse component Qt satisfy the same equation and boundary condition [5]. Therefore the exponential decay of G(r,r'), calculated in Ref.[2] represents the contribution of the transverse component of superfluid velocity. This means that G(r,r' ) tends to zero ff Ir-r' I ~ o 0 ,i.e. the SLRO vamshes for the dimenstonahty less then four irrespechve of the defimhon of the GIOP phase. 3. DISCUSSION In conclusion, we have demonstrated that smularly to the phases ®(r) and ~L(r) the gauge-mvanant phase tD*(r) sattsfies Eq.(6) wtthout the transverse component Q. Nevertheless, the general solution of Eq.(6) can be written for the boundary condthons which depend on the state of the superconductor. In the Melssner state ~M(r) r, a ,,ingle-valued funchon. In this report tt ~.,, ~hown that ~t,, gradient does not depend on Qt, and G(r,r') tend,, to constant d the dlmenstonahty I~ more then two. The gauge mvanance of At leads to the absence of dfff~rencles between the pha~e within the London gauge and the GIOP phase In the mixed state the phase gradient depends on both components of Q . and QI gwes the dominant contrabuhon to the correlahon funchon. The followmg hypotheses about the consequencc~ of the SLRO destructton by thermal fluctuahons m the
mixed state can be formulated 1. In a magnetic field H the transttton to a supereonductmg state with SLRO occurs only at T* where T* is the boundary line of the Meissner state HeI(T*)=H. At T > T * the mixed state has nearly coherent phases. The boundary temperatures T~(H) and TII(H) between the state with nearly coherent phases and the state with incoherent phases are inversely proportlonai to the sample sizes LII and L, along and across the field, respectively. As the sample size increases, the TI(H) and TII(H) shift towards lower temperatures. 2. By analogy with the topological transihon in twodimensional systems the transltton to a mixed state in three-a~mensinal superconductors in a magnetic field can be also topological. This is a transition of topological vortices and anhvortices mto the state with bound pairs. It leads to their nondlsmpative motion at low temperature m the presence of Abnkosov vort~ces without SLRO. The observation of a contnbuhon to the current dependent d~ssipatlon m the lowtemperature part of the resmtlve curve [6,7] may be considered as experimental evidence that substantiates this hypothesls. REFERENCES 1. A.Houghton, R.A.Pelcovlt% and A Sudbo. Phys. Rev., B42 (1990) 906. 2. M.A.Moore,Ph),s. Rev , B45 (1992) 7336. 3. R.Ikeda, T Ohml,and T T,,uneto, J Phys Soc. Jpn , 61 (1992) 254. 4. P.de Gennes, Superconduchvlty of Metals and Alloys. Benjamin, NY, 1966 5. G.Sergeeva, Low Temp Phys , 20(1994) 3. 6. l.G.Gorlova, and Yu l.Latyshev. Abstracts III, USSR Conf HTSC, p2. p.65, Kharkov,(1991). 7. W.K.Kwok, U.Welp, G.W.Grabtree et al., Phys. Rev Lett ,64 (1990) 969.
Superconducting
PHYiCA
L o n g - R a n g e O r d e r in M i x e d S t a t e
G.G.Sergeeva NSC Kharkov Insut,,i~ ufPhys, and Technology, 310108, Kharkov, Ukraine. It is shown that the controversies between different defivittons of the gauge-mvariant order parameter :-nase can be removed by taking into account that in the Meissner state transverse component A is gauge-invartant q~ ~.attty and under gaage transformation only longitudmal component A changes As a consequence, gauge-invar~aat order parameter phase and the phase within London gauge satisfy the same equattons and boundary conditmn ,:or three &mensional superconductors m the mixed state it is shown that the destruction of superconducting iong-r:nge order parameter by thermal fluctuatmns does not depend on the defimtton of the order parameter.
VVO(r) =-KVQ, Q = A-lhcVtD(r)
1. INTRODUCTION The problem of the influence o f the thermal fluctuattons on the existence of a long-range order in the lattice o f Abrikosov vortices remains open since the pioneermg works, where it was shown that thermal fluctuatmns destroy the superconducting long-range order (SLRO) in the rmxed state. In spite of the narrowness af the fluctuatton regmn, the broadening of the transitmn to nuxed state was dtscovered m experiments wtth low-temperature superconductors. In HTSC, because of shorter correlatmn lengths the thermal fluetuattons are stronger than in low temperature ones. Thetr account is by far more comphcated and leads to contradictory results dependmg on the defimtlons of the order parameter [1-3]. In this report we compare various definltion~ of the gradlent-mvanant order parameter (GIOP) and show that m the nuxed state SLRO ts deotroyed by thermal fluctuatton irrespective of the defmtton of the GIOP.
(2)
where Q is the superflutd velocay, ,.hlch is measurable and gauge-mvanant quantity, ts the Gmzburg parameter. Within the London choice of gauge one has V A = 0 , and as tt fol~ ws from Eq.(2),the phase q~L(r) tS also the solution of Eq (2). l:rom the expressmns (1) and (2) it folio,,- that the phase tI~*(r) also satisfies Eq (2) Vd>*(r) = - KQ VVcD*(r) = -~:VQ
(3) (a)
The gra&ent ~*(r) depends on both Iongd~._hnal and transverse components of the superflmd velocity Q = Q I +Qt : Qi = k (kQ)/k ~,
V Qt = 0
(5)
One can see from the Eqs. (2) and (5) that VV(D*(r) =-K VQ!
(6)
2. R E S U L T S At present, two defimtmns for the GIOP phase are known The first of them q~*(~ ts ordinary and follows from the gauge transformatsons of the vector potentml A and the phase q~(r) of the order parameter 'F(r) [41" cl~*(r) = q~(r)- ~ I A d|
(!)
Accordmg to the second defimtmn the GIOP phase (-) (r) ts the solution of the equatton I21.
and hence VV(I)'~(r) depends only on Qr~ The contradtctton between Eqs (3) and (6) can be removed by taking mto account that for singly ~_onnected superconductors the transverse component At ~s gaugemvartant quantity and only Iongttudmal component A! changes under gauge trasformattons Thts means that for single-valued phase q)~,(r) of the Metssner state Eqs.(3) ar,,I (6) have no contradtctum, and the extstence of SLRO m Melssner state does no~ depend on the definmon ol the GIOP phase
0921-4534/94/$07 00 © 1994 - I.lscvlcr Scmncc B V All rights reserved SSDI 0921-4534(94)01541-4
1950
G G Sergeeva/Physwa C 235-240 (1994) 1949-1950
In the tmxed state singly connected superconductor becomes multiply connected and we must take rote account the multi-valued character of the phase and a contribution of both components of Q into O*. The multi-valued part of the phase leads to the localized vorttclty of both V O*(r) and transverse component Qt at the vortex core. It should be noted that in Ref.[2] the correlation function G(r,r') was actually calculated with both components of Q taken Jnto account .The comparison of the Eqs.(2.28) and (2.29) for O from Ref.[2] and the Eqs.(3),(4) for ~ * shows that VOo and transverse component Qt satisfy the same equation and boundary condition [5]. Therefore the exponential decay of G(r,r'), calculated in Ref.[2] represents the contribution of the transverse component of superfluid velocity. This means that G(r,r' ) tends to zero ff Ir-r' I ~ o 0 ,i.e. the SLRO vamshes for the dimenstonahty less then four irrespechve of the defimhon of the GIOP phase. 3. DISCUSSION In conclusion, we have demonstrated that smularly to the phases ®(r) and ~L(r) the gauge-mvanant phase tD*(r) sattsfies Eq.(6) wtthout the transverse component Q. Nevertheless, the general solution of Eq.(6) can be written for the boundary condthons which depend on the state of the superconductor. In the Melssner state ~M(r) r, a ,,ingle-valued funchon. In this report tt ~.,, ~hown that ~t,, gradient does not depend on Qt, and G(r,r') tend,, to constant d the dlmenstonahty I~ more then two. The gauge mvanance of At leads to the absence of dfff~rencles between the pha~e within the London gauge and the GIOP phase In the mixed state the phase gradient depends on both components of Q . and QI gwes the dominant contrabuhon to the correlahon funchon. The followmg hypotheses about the consequencc~ of the SLRO destructton by thermal fluctuahons m the
mixed state can be formulated 1. In a magnetic field H the transttton to a supereonductmg state with SLRO occurs only at T* where T* is the boundary line of the Meissner state HeI(T*)=H. At T > T * the mixed state has nearly coherent phases. The boundary temperatures T~(H) and TII(H) between the state with nearly coherent phases and the state with incoherent phases are inversely proportlonai to the sample sizes LII and L, along and across the field, respectively. As the sample size increases, the TI(H) and TII(H) shift towards lower temperatures. 2. By analogy with the topological transihon in twodimensional systems the transltton to a mixed state in three-a~mensinal superconductors in a magnetic field can be also topological. This is a transition of topological vortices and anhvortices mto the state with bound pairs. It leads to their nondlsmpative motion at low temperature m the presence of Abnkosov vort~ces without SLRO. The observation of a contnbuhon to the current dependent d~ssipatlon m the lowtemperature part of the resmtlve curve [6,7] may be considered as experimental evidence that substantiates this hypothesls. REFERENCES 1. A.Houghton, R.A.Pelcovlt% and A Sudbo. Phys. Rev., B42 (1990) 906. 2. M.A.Moore,Ph),s. Rev , B45 (1992) 7336. 3. R.Ikeda, T Ohml,and T T,,uneto, J Phys Soc. Jpn , 61 (1992) 254. 4. P.de Gennes, Superconduchvlty of Metals and Alloys. Benjamin, NY, 1966 5. G.Sergeeva, Low Temp Phys , 20(1994) 3. 6. l.G.Gorlova, and Yu l.Latyshev. Abstracts III, USSR Conf HTSC, p2. p.65, Kharkov,(1991). 7. W.K.Kwok, U.Welp, G.W.Grabtree et al., Phys. Rev Lett ,64 (1990) 969.