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Superconductive state localized on inclusion in semiconductor or semimetal
Solid State Conmunications, Vo1.54,No.3, pp.227-230, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
SUP%CONMJCTIVR STATH LOCALIHEDON INCLUSIONIN SHMICONDUCTOBOR SEHIHETAL VANabutovskii The Instituteof InorgenicChemistry,Siberian Branch of the Academy of Sciences,Novosibirsk,630090,USSR (Received13 Hay 1984 by E.A.Kener) The electron densitynear inclueionin semiconductormay be high so that localizedsuperconductivestate(ISS)may appear at temperatureor magneticfield many orders exceeding their bulk value. We diecuss*ISScriticalproperties,structure and currentpercolation.The media with inclusionsmay be diamagneticat high temperature but no superconductive current exista down to low tempera-Le.
I9T field Ho, sise d, and briefly euperconductlvity currentpercolation and We do restrictour60 "x" the mall to Tc Hc2, but investigatethe where To exceedmanytlmee bulk
ANK IWCIHSIOHinto eemlconductor,eemimetal ormetalbeoomee charged due to the differenceof electronchemicalpotentialabetween Inclusionend bulk malmrlal. lie a result additionalelectron or hole deneity appearsIn region adjacent to the inclusion.The eleatron atatea density at the Per&&surface ~A~~~~t~ectron-electrOn interacti(r)=KO,r)Vimt tarn inhomogeneow anf their values near the lnclu~~lonmayexceedei@flcantlybulk one6 W(0 >, g(w). The qua&its VL~~ is thema trG element of the electron-eleotron interaction.one can suppose that local increaaineof g(r) map lead to ~at~~~ring l&allEdd-&w+rconductive 1 near the inclu8lonattemnerat&&To; thou& .electronlo &as in bulk may be nonauperconductive and even mag be nondegenerate. The I86 cri%icalpropertlee have been investigatedin our p~pers"~ for inhoa0
4i”eati meteile,
ass*
HlectronDensity Interaction.
Restrict to the model. one charge species (electrons) ma68 m bulk density existe. Inclusion the sphere radius B potentialat surface be Here and subscripte00 R are to the and inclueion respectively.Aa was mentioabove we that g(r) the inclusion leas than . Evewbelow ilrkg=l tiand are the end Noitsman The potential cp(r) near inclusionsatisfies equati-
mooth-
no88 o a(r)ind ite maximali-& inside the lnho&kcbeits. In the case-under
coneideratIonwe-assumeg(r) marlmalify at the Inclusionsurface.me critical temmrature TC increases due to the ele&zon dens%* redletrZbutionby the externaleleatrlcfield auolied to the eemlaonductorpl e surfa% have been dlaayyd earlie3 . The papers (e.g. 9 ) have been two-dimensionaleleatronia.gaeat the quantlsedflratlevelinducedby the eleatriofleld.On the eontram in this paper we dfeauee the eituafion, whenmany levels are preeentin the region of IS8 that Is under the quasiclaeeicalsituatlon.The claseicalanalo e of thiq caee hae been lnveetigato8"in paper and for the euperconductivity near the plane eudace(for Tc variation) .the-paper8. In paperwereport the ofourlnvestigation XSS appearing
and Rlect-
Heree the bulkpermittlvity. densityn(r) general ca8e a funcof U(r). ISS arises the domain the electron is dege& one apply the (TP) mo&el TV equation. n{U~~/3w)m3/2[P(r>+u]3/2
(2)
the chemicalpotential densityn,. The of TP approximation in our case arer d)r,-R,where d is the width of the auperconduotivelayer,r, is the radius of the &round level of an
medium. These inequalitieshold if 227
Vol. 54, No. 3
SUPERCONDUCTIVESTATE LOCALIZED ON INCLUSION
228
u >uo,where Uo=e/eao,m/e is small in o8mparisonwith the mam of a free elec-
tron andnia highenough. Substituting(2) into (1) one obtains T!Requation.The boundaryoonditiona are: U(R)=UR,I&,,,=O. Give asymptotically accuratesolutionin two domains. Let r be thqfiootof eq.U(ru)=u, Er-R$nzdf=U 0) . p =r -R, U(r)>u and the functio;v' I 'd$& on ?? meter Ro, where R&a~of, f&JR/U, F”' . If BtRo then U(r>=U,(r)
Restrictourselvesby the ndirty'* superconductors.In this case the )&y@r,_, equationfor To(H) (or Ho(T)) is ~L+o(r)=[X(~2~)+V(r)]~o(r)=Eo~o(r) (8) ~~z)=~(l/2+~/2)-'9(1/2)~(l+n%/4)(9) V(lr)=l/g(r)=V~[(U(r)/u)+l]-1/2(AC) &-iv+(e/c)(Hr],~&v,$/6zT
(11)
E,=-ln(T,/1.14~,,)
U,(r)+RF(sr/R)/rF(s), s3F(s)= w+(fa/Q14,
(3) 3/2 , P(O)=1 d'/zBl'(~)a(Z)
burg-Le.ndau(GL) equation.Here and below l=r~5~/2 and lR, looare its Values at inclusionsurface end in bulk. ISS CriticalTemperaturein Zero Field
ti
radius r (arbitrary units) Fig. I.
The functionF(z) has been tabulated (sees).PracticallyPRO. In this case F(zpe-3 and "4 U,(r)=UR(R/r)4,ru=Rw, w=(UR/u)
The solutionof the set of equations (8)-(11)dependson "potential"V(r). Consideressentiallimitingoaeee* "Yukawapotential"case arises if ISS nshellnthlckuessd is large, ~RI?, (fig.2a).In this case U/Ml, &$W, so GL eq. is valid and E&y+V, , where WL
(4)
If R> then x&R and the "plane"solution 9 o the TF eq. nay be applied Uq(r)=UR/(1+x/xO)4, xo=4e5ao/f9
(51
xgxo(w-I) In the domain 2, r>ru TF equationmay be linearizedend its solutionis U(r)=U2, Ut=(uru/r)eqk(ru-r),k2=6z&,ee/u (6) Here l/k is the Debye radius of the degenerateelectrongas. If UR(u, the domain 1 does not exist and U(r) takes farm(6) where UR end R have to be substitutedhorlu and ~f~~e~~F~~~~)~n~~~~%ue~~~2~b~~ns g(r)=(4*/z)m3'2[v(r)+u]"~,,,p */2, &=(M@/u)m 3/e ga[l+U(r)/ul u l/%1'.', Basic Equationsfor ISS CriticalLine
Pig. 2.
Vol. 54, No. 3
SUPERCONDUCTIVE STATE LOCALIZED ON INCLUSION
Vo=V~~u=MqJ,
vu
(14)
The accurateboundary conditionat the inclusionsurface in this case is not important.l?orany inclusionmaterial it may be chosen as finitness$0(O). Adding the condition+o(o)=O one obtains the well-Imownproblem of the ground state in the Yukawa potential.Using simple variationalfunction+o=em(-r/d) and mini&sing (+oIHL)+o.)in respect to d, one obtains (in parametricform)Q E+vgQ-W/4(&, &Vo(klJ2
(15)
p=Wkd
=(I+P)'/P(P+~),To=Tu,+'pr,
Ta=l.14~exp(-l/g,,), !L+w It is clear that ISS exists only if p>l or q=K-PO. There are two limits. 0d. In this case I&S arises in the region adjacent to the inclusionsurface(fig.2b)and boundm conditionsdepend essentially on inclusionmaterial.In crude ap roximationthese conditions are: s for metallic inolusionand a+ (Rt%%* for the dielectricone. The s%nplestsolution msy be obtainedif la(Lo, where Lo%/2for R>Ro. In L =R/2 for X % I turns into G.Lequation &is case eq.( -l;+~+(V~o-~)~o=o,
K+,=VR+RL, (18)
vR=l/gR=v,(umR)l/2, $,=Ai(x/d$,>, do=(l$ogR)
“3t
~=C~(VR1R/LoIJ3,
IBS
CriticalBield Ho(T).
As Ho(T )=O, in adjacenttemperaturedomain H ?T> is small and satisfiesthe inequalityaR>R,d, aec/2eH. In this "weak field" case eqr(8)-(II) msy be solved by the perturbationtheory using as a first approximationsolutionfor H=O. It gives for the coexistenceline (To-T)/To=CdZ(eH/c>2(d+R>2, WI
(20)
Let 1, be the value of 1 at RTo. At zero temperaturea "1,, and if ature domain 1 ,(d,R then in the tempe!! a8jacent to T=O the magnetic field is For the dielectricinand IS8 represents a tube (or rin ) along the inclusion equator(fig.2b f. The tube wall thickness is a and length is b, beWaHR. Here is t#e V(r) averagedover a distsn$f$ -*a. For be metallic inclusionH=AH(T)+ l&b(Veff).The LSS representsa tube of ra ius R, wall thicknessa ba"d2+2dR (fig.2a).The coexistenceH' ine in this domain satisfiesthe set of eqs3 Ro(T)=Hc2(Veff,T>+AH(T), Y&/R&?-Z&R
, w=c;/a& AV=V(r)-Veff
In particularfor Yukawa potential
Here Ai is the Airy function,subscripts 1 and 2 correspondto the metallic and dielectricinclusion. In more realisticcase xu>;LNLopqs (4),(5)and (10) yields V(r)=VR(dq, L,=R if IKRo and L,=x,if IbRo. It is impossibleto expend x(&$P2) in eq.(8) to obtain GL ea. As Ladecreasesand tends the t%nsitidn temperaturefalls to to far less rapi ly from high value 9 ect. T2due to the proximityef
3
Lc=vR
metallic inclusionand for dielectric onee/(kl$, +G.
4,(x)f(&
TR=l.l‘Wpe~p(-VR),I+,CVR
.5’pR(L,-Lc>fic,
If -9, $ and klop'lthen soluti;, given y ormulas(8), where T I2 and I/k have to be substit&d or
L24,~+~V(x)40=~R/H,~0,L2m~'(~)6~, (21) T), IX, =c/2eL2,+c(?)= ai=c/2eHc2(VeffS
C,=2.3, Czpl, To=TR(l-EL),d=Cido
To4,+
229
3/2 1R
(--@)
dul.6VRlR,T2=1.1LCWDexp(-V,,&@) "Small corrections"domain, l$xu. Here GL eq. is valid, V(r)=V,+AV(r), TO&Q&AT, l=&,,,&,=V,+AE.In this case solutiondepends on relationbetween quantitiesR, Ro, &,, k. If N&then solution is given by formulas (12)-(17).
-V00,AV=V(r)-V,,5T%T oD 'effAhH,(Vo/~*(w)k50)21n2kaw,b-d
(22)
For ISS near the inclusionboundary Veff=VR,&$TR,AH=-H,EL, b-m
(23)
If H=O then ISS may not exists as for Yukawa potentialif K2(l5). Due to the one-dimensional localizationby magnetic field IS8 has to appear in this case also. The sketch of the possiblecoexistencediagramsis representedin fig.&. The I&S domain is between the curve 1 (Hc (T) for bulk) and one of the curves 3, 3 or 4. The curve 2 correspondsto l-d localizationby magnetic field. The dashed line 6 corresponds to the dielectricinclusion.May be similar type has been observedexperimentally13.
SUPERCONDUCTIVESTATE LOCALIZED ON INCLUSION
230
One more new
is
the
OS-
. The period of osciloillationof T lation is&$&/S, where0, Is the fluxoid, S=zR is the inclusioncross-section.ft is clear that oscillationmay be observedonly if the inclusionsize distributionis narrow enough14,15. IS3 Structure, Superconductive Current Percolationand Magnetization
To determinethe structureof W below the coexistenceline B (T) one has to solve the nonlinearse8 of eqs j=f{+2,}, E=-ln(T/l.14eD) l?;;?ze,dirty,homogeneous supercon-(24) aTand R had been obtainedin papers16, . i{4,3]=QI(r,123,, f~Q:)=I(~'&+~)/2 (25)
Vol. 54, No. 3
For large inclusionnear its surface (30)
(31)
The magnetizationof a media with inclusionsbefore superconductive percolation(SP)is similarto the magnetiration of the dielectricmatrix, containing superconductive shells,tubes or
+w~{+~~E+,
over ISS volume
differentinclusionsoverlap each other. The SP condition is N~XVcr,Nc -3/[v(T) +4zR3/3$ E.g. for Yukawa p&en&al
Q~"(1/2y)/8z2T2, I=Yf(1/2+rl)etn/4zmT Ncr".l@c31~(a), Ncr+7k3/ln3(Vo/z,$(b) Under SP conditionthe media turns into here I, end In are t$igammaand tetrathe bulk superconductorcontainingthe gamma functions,tl-TeR/c, ConsiderZ%nonsuperconductive voids of radius R ro field case. Let-%e 2=(To-T)/To
& B.Ya.Shapiro,Pi.Z. I. V.Y.Nabutovskii Nizk.Temp., 855(1981). iL'i, Sol.StateCommun.f 2. V.M.Nabutovs
v%
4 , 405(1982). .Nabutovskii & B.Ya.Shapiro, J. Low Timp.Phys., 42, 461(1982). 4. V.B.Sendomirskli, Pis,ma Zh.Eksp. Teor.Fiz., 2, 396(1960).
3.
5. Y.Takada J>s.Soc.Japan, %, ?86(19783.
6. R.J.‘Keil$-&W.Hanke, Phys.Rev., B, 2 , 112, y24(1981). 7. Q% .Nabutovskii & N.A.Nemov, J.Phys. c: 1 3849(1984). 8. B.? a.hhapiro, Sol.State Commun.. (in print) _ 9. L.D.Landau & E.Y.Lifshitz, Quantum Mechsnica(in Russian), Ch.X, Nauka,
Moscow(l9~74). lO.W.R.Werthammer,Phys.Rev., 'Q2, 2440(1963).
ll.P.G.De Gennes, Rev.Mod.Phye., s, 225Q.964). 8, Practical Quantum Mecha12&;zu ? Ij.S.J.+oo& J.Non-Cryst.Sol., 61-2, ll(1984). 14.7 & B.Ya.Shapiro, JETP , 310(1979). 15.V.M.g 2 o & V.I.Su akov, Fiz.Nizk. Temp., &4$X19837, sits, 1, 21(1964). 16.K.Maki, 17.C.Caroli. Y.Csrot. P.G.De Gennes. SolStat& Ck.; fl,17(1966). ’ 18.We do not discuss here granular superconductors (see e.g.19,20 >. lY.O.Ectin-Wolman, A.Kapitulnic & ;;$;yira, Phys.Rev., z, 6464 20.Y.Shapira & G.Deutcher, Phys.Rev., B27, +@+(1983)
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
SUP%CONMJCTIVR STATH LOCALIHEDON INCLUSIONIN SHMICONDUCTOBOR SEHIHETAL VANabutovskii The Instituteof InorgenicChemistry,Siberian Branch of the Academy of Sciences,Novosibirsk,630090,USSR (Received13 Hay 1984 by E.A.Kener) The electron densitynear inclueionin semiconductormay be high so that localizedsuperconductivestate(ISS)may appear at temperatureor magneticfield many orders exceeding their bulk value. We diecuss*ISScriticalproperties,structure and currentpercolation.The media with inclusionsmay be diamagneticat high temperature but no superconductive current exista down to low tempera-Le.
I9T field Ho, sise d, and briefly euperconductlvity currentpercolation and We do restrictour60 "x" the mall to Tc Hc2, but investigatethe where To exceedmanytlmee bulk
ANK IWCIHSIOHinto eemlconductor,eemimetal ormetalbeoomee charged due to the differenceof electronchemicalpotentialabetween Inclusionend bulk malmrlal. lie a result additionalelectron or hole deneity appearsIn region adjacent to the inclusion.The eleatron atatea density at the Per&&surface ~A~~~~t~ectron-electrOn interacti(r)=KO,r)Vimt tarn inhomogeneow anf their values near the lnclu~~lonmayexceedei@flcantlybulk one6 W(0 >, g(w). The qua&its VL~~ is thema trG element of the electron-eleotron interaction.one can suppose that local increaaineof g(r) map lead to ~at~~~ring l&allEdd-&w+rconductive 1 near the inclu8lonattemnerat&&To; thou& .electronlo &as in bulk may be nonauperconductive and even mag be nondegenerate. The I86 cri%icalpropertlee have been investigatedin our p~pers"~ for inhoa0
4i”eati meteile,
ass*
HlectronDensity Interaction.
Restrict to the model. one charge species (electrons) ma68 m bulk density existe. Inclusion the sphere radius B potentialat surface be Here and subscripte00 R are to the and inclueion respectively.Aa was mentioabove we that g(r) the inclusion leas than . Evewbelow ilrkg=l tiand are the end Noitsman The potential cp(r) near inclusionsatisfies equati-
mooth-
no88 o a(r)ind ite maximali-& inside the lnho&kcbeits. In the case-under
coneideratIonwe-assumeg(r) marlmalify at the Inclusionsurface.me critical temmrature TC increases due to the ele&zon dens%* redletrZbutionby the externaleleatrlcfield auolied to the eemlaonductorpl e surfa% have been dlaayyd earlie3 . The papers (e.g. 9 ) have been two-dimensionaleleatronia.gaeat the quantlsedflratlevelinducedby the eleatriofleld.On the eontram in this paper we dfeauee the eituafion, whenmany levels are preeentin the region of IS8 that Is under the quasiclaeeicalsituatlon.The claseicalanalo e of thiq caee hae been lnveetigato8"in paper and for the euperconductivity near the plane eudace(for Tc variation) .the-paper8. In paperwereport the ofourlnvestigation XSS appearing
and Rlect-
Heree the bulkpermittlvity. densityn(r) general ca8e a funcof U(r). ISS arises the domain the electron is dege& one apply the (TP) mo&el TV equation. n{U~~/3w)m3/2[P(r>+u]3/2
(2)
the chemicalpotential densityn,. The of TP approximation in our case arer d)r,-R,where d is the width of the auperconduotivelayer,r, is the radius of the &round level of an
medium. These inequalitieshold if 227
Vol. 54, No. 3
SUPERCONDUCTIVESTATE LOCALIZED ON INCLUSION
228
u >uo,where Uo=e/eao,m/e is small in o8mparisonwith the mam of a free elec-
tron andnia highenough. Substituting(2) into (1) one obtains T!Requation.The boundaryoonditiona are: U(R)=UR,I&,,,=O. Give asymptotically accuratesolutionin two domains. Let r be thqfiootof eq.U(ru)=u, Er-R$nzdf=U 0) . p =r -R, U(r)>u and the functio;v' I 'd$& on ?? meter Ro, where R&a~of, f&JR/U, F”' . If BtRo then U(r>=U,(r)
Restrictourselvesby the ndirty'* superconductors.In this case the )&y@r,_, equationfor To(H) (or Ho(T)) is ~L+o(r)=[X(~2~)+V(r)]~o(r)=Eo~o(r) (8) ~~z)=~(l/2+~/2)-'9(1/2)~(l+n%/4)(9) V(lr)=l/g(r)=V~[(U(r)/u)+l]-1/2(AC) &-iv+(e/c)(Hr],~&v,$/6zT
(11)
E,=-ln(T,/1.14~,,)
U,(r)+RF(sr/R)/rF(s), s3F(s)= w+(fa/Q14,
(3) 3/2 , P(O)=1 d'/zBl'(~)a(Z)
burg-Le.ndau(GL) equation.Here and below l=r~5~/2 and lR, looare its Values at inclusionsurface end in bulk. ISS CriticalTemperaturein Zero Field
ti
radius r (arbitrary units) Fig. I.
The functionF(z) has been tabulated (sees).PracticallyPRO. In this case F(zpe-3 and "4 U,(r)=UR(R/r)4,ru=Rw, w=(UR/u)
The solutionof the set of equations (8)-(11)dependson "potential"V(r). Consideressentiallimitingoaeee* "Yukawapotential"case arises if ISS nshellnthlckuessd is large, ~RI?, (fig.2a).In this case U/Ml, &$W, so GL eq. is valid and E&y+V, , where WL
(4)
If R> then x&R and the "plane"solution 9 o the TF eq. nay be applied Uq(r)=UR/(1+x/xO)4, xo=4e5ao/f9
(51
xgxo(w-I) In the domain 2, r>ru TF equationmay be linearizedend its solutionis U(r)=U2, Ut=(uru/r)eqk(ru-r),k2=6z&,ee/u (6) Here l/k is the Debye radius of the degenerateelectrongas. If UR(u, the domain 1 does not exist and U(r) takes farm(6) where UR end R have to be substitutedhorlu and ~f~~e~~F~~~~)~n~~~~%ue~~~2~b~~ns g(r)=(4*/z)m3'2[v(r)+u]"~,,,p */2, &=(M@/u)m 3/e ga[l+U(r)/ul u l/%1'.', Basic Equationsfor ISS CriticalLine
Pig. 2.
Vol. 54, No. 3
SUPERCONDUCTIVE STATE LOCALIZED ON INCLUSION
Vo=V~~u=MqJ,
vu
(14)
The accurateboundary conditionat the inclusionsurface in this case is not important.l?orany inclusionmaterial it may be chosen as finitness$0(O). Adding the condition+o(o)=O one obtains the well-Imownproblem of the ground state in the Yukawa potential.Using simple variationalfunction+o=em(-r/d) and mini&sing (+oIHL)+o.)in respect to d, one obtains (in parametricform)Q E+vgQ-W/4(&, &Vo(klJ2
(15)
p=Wkd
=(I+P)'/P(P+~),To=Tu,+'pr,
Ta=l.14~exp(-l/g,,), !L+w It is clear that ISS exists only if p>l or q=K-PO. There are two limits. 0
K+,=VR+RL, (18)
vR=l/gR=v,(umR)l/2, $,=Ai(x/d$,>, do=(l$ogR)
“3t
~=C~(VR1R/LoIJ3,
IBS
CriticalBield Ho(T).
As Ho(T )=O, in adjacenttemperaturedomain H ?T> is small and satisfiesthe inequalityaR>R,d, aec/2eH. In this "weak field" case eqr(8)-(II) msy be solved by the perturbationtheory using as a first approximationsolutionfor H=O. It gives for the coexistenceline (To-T)/To=CdZ(eH/c>2(d+R>2, WI
(20)
Let 1, be the value of 1 at RTo. At zero temperaturea "1,, and if ature domain 1 ,(d,R then in the tempe!! a8jacent to T=O the magnetic field is For the dielectricinand IS8 represents a tube (or rin ) along the inclusion equator(fig.2b f. The tube wall thickness is a and length is b, beWaHR. Here is t#e V(r) averagedover a distsn$f$ -*a. For be metallic inclusionH=AH(T)+ l&b(Veff).The LSS representsa tube of ra ius R, wall thicknessa ba"d2+2dR (fig.2a).The coexistenceH' ine in this domain satisfiesthe set of eqs3 Ro(T)=Hc2(Veff,T>+AH(T), Y&/R&?-Z&R
, w=c;/a& AV=V(r)-Veff
In particularfor Yukawa potential
Here Ai is the Airy function,subscripts 1 and 2 correspondto the metallic and dielectricinclusion. In more realisticcase xu>;LNLopqs (4),(5)and (10) yields V(r)=VR(dq, L,=R if IKRo and L,=x,if IbRo. It is impossibleto expend x(&$P2) in eq.(8) to obtain GL ea. As Ladecreasesand tends the t%nsitidn temperaturefalls to to far less rapi ly from high value 9 ect. T2due to the proximityef
3
Lc=vR
metallic inclusionand for dielectric onee/(kl$, +G.
4,(x)f(&
TR=l.l‘Wpe~p(-VR),I+,CVR
.5’pR(L,-Lc>fic,
If -9, $ and klop'lthen soluti;, given y ormulas(8), where T I2 and I/k have to be substit&d or
L24,~+~V(x)40=~R/H,~0,L2m~'(~)6~, (21) T), IX, =c/2eL2,+c(?)= ai=c/2eHc2(VeffS
C,=2.3, Czpl, To=TR(l-EL),d=Cido
To4,+
229
3/2 1R
(--@)
dul.6VRlR,T2=1.1LCWDexp(-V,,&@) "Small corrections"domain, l$xu. Here GL eq. is valid, V(r)=V,+AV(r), TO&Q&AT, l=&,,,&,=V,+AE.In this case solutiondepends on relationbetween quantitiesR, Ro, &,, k. If N&then solution is given by formulas (12)-(17).
-V00,AV=V(r)-V,,5T%T oD 'effAhH,(Vo/~*(w)k50)21n2kaw,b-d
(22)
For ISS near the inclusionboundary Veff=VR,&$TR,AH=-H,EL, b-m
(23)
If H=O then ISS may not exists as for Yukawa potentialif K2(l5). Due to the one-dimensional localizationby magnetic field IS8 has to appear in this case also. The sketch of the possiblecoexistencediagramsis representedin fig.&. The I&S domain is between the curve 1 (Hc (T) for bulk) and one of the curves 3, 3 or 4. The curve 2 correspondsto l-d localizationby magnetic field. The dashed line 6 corresponds to the dielectricinclusion.May be similar type has been observedexperimentally13.
SUPERCONDUCTIVESTATE LOCALIZED ON INCLUSION
230
One more new
is
the
OS-
. The period of osciloillationof T lation is&$&/S, where0, Is the fluxoid, S=zR is the inclusioncross-section.ft is clear that oscillationmay be observedonly if the inclusionsize distributionis narrow enough14,15. IS3 Structure, Superconductive Current Percolationand Magnetization
To determinethe structureof W below the coexistenceline B (T) one has to solve the nonlinearse8 of eqs j=f{+2,}, E=-ln(T/l.14eD) l?;;?ze,dirty,homogeneous supercon-(24) aTand R had been obtainedin papers16, . i{4,3]=QI(r,123,, f~Q:)=I(~'&+~)/2 (25)
Vol. 54, No. 3
For large inclusionnear its surface (30)
(31)
The magnetizationof a media with inclusionsbefore superconductive percolation(SP)is similarto the magnetiration of the dielectricmatrix, containing superconductive shells,tubes or
+w~{+~~E+,
over ISS volume
differentinclusionsoverlap each other. The SP condition is N~XVcr,Nc -3/[v(T) +4zR3/3$ E.g. for Yukawa p&en&al
Q~"(1/2y)/8z2T2, I=Yf(1/2+rl)etn/4zmT Ncr".l@c31~(a), Ncr+7k3/ln3(Vo/z,$(b) Under SP conditionthe media turns into here I, end In are t$igammaand tetrathe bulk superconductorcontainingthe gamma functions,tl-TeR/c, ConsiderZ%nonsuperconductive voids of radius R ro field case. Let-%e 2=(To-T)/To
& B.Ya.Shapiro,Pi.Z. I. V.Y.Nabutovskii Nizk.Temp., 855(1981). iL'i, Sol.StateCommun.f 2. V.M.Nabutovs
v%
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