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Vortex ring excitations in layered superconductors
PhysicsLettersA 183 (1993 ) 102-106
PHYSICS LETTERS A
North-Holland
Vortex ring excitations in layered superconductors Yu.M. I v a n c h e n k o Department of Physics, Polytechnic University, Six Metro Tech Center, Brooklyn, NY 11201, USA Received 17 August 1993;acceptedfor publication 1 October 1993 Communicated by L.J. Sham
A specialkind of thermal excitationscan be found in the vicinityof Tcin a layeredsuperconductingstructure. The excitations are vortextings positionedin the layerwith weakersuperconductivity.For parameterscorrespondingto high-Tosuperconductors these excitations should consist of two coupled vortex rings with opposite helicities. Fixed in position by pinning forces and frozen, these excitationscontribute to randomization of the superconductingstate at low temperatures. Estimates are made for the densityof states within the gap regionand characteristiclength of the random structurecausedby the doublering excitations.
It is now widely recognized that thermal excitations of 2D vortices play an important role in the magnetic and transport properties of anisotropic layered superconductors (SC). This phenomenon substantially influenced extremely anisotropic systems like Bi- or Tl-based high-To oxides or artificial superconducting superlattices. A conventional model describing such compounds was first advanced by Lawrence and Doniach (LD) [ 1 ]. This model deals with a periodic array of SC layers coupled by Josephson weak links. It admits two kinds of topological excitations. They are vortex points [2] (or pancake vortices [ 3] ) confined to superconducting layers and fluxon loops [ 2 ], which can also be seen as loops formed by Josephson vortices [ 4 ]. The latter are confined between superconducting layers and lead to their decoupling. The thermodynamics of loop excitations in layered structures has been studied [ 5,6,2 ] in the framework of the LD model. Friedel [ 5 ] supposed that in the case of 3D systems formed by weakly coupled layers a phase transition (caused by vortex rings) should occur at a temperature Tf which decreases to zero with the decrease of coupling between the layers. Thus, the temperature Tf, according to Friedel, is always less than the 2D Berezinskii-Kosteditz-Thouless transition temperature Tv [7,8 ]. Later, Korshunov [6 ] showed, that in a simplified Gaussian approximation of the Josephson coupling Tr> Tv. In a strict consideration Horovitz [ 2 ] has confirmed the statement that Tf> T, and proved that the fluxon temperature Tf determines the transition temperature To. In the present work a model similar to that used in ref. [ 9 ] for a squeezed vortex is applied to the description of noninteracting fluxon excitations. I assume that a cell of a layered structure consists of two layers with thicknesses 2dl and 2d2. Each layer is a London superconductor characterized by the bulk field penetration length ,~.x or ~,2 with the same critical temperature and the same bulk correlation radius ~. These layers alternatively form a periodic structure in the zdirection (see fig. 1 ). The magnetic field distribution h ( r ) can be found as a solution of the London equation, which for this case takes the form (see ref. [ 10 ] ) h ( r ) = VX [22(z)Vxh (r) ] = O0 ~ d l J ( r - l ) .
(1)
Here ~o is the magnetic flux quantum and the integration should be carried out along the curve of the ring with the length L = 2~p. Equation ( 1 ) must be completed by boundary conditions reflecting continuity of the magnetic field and the normal component of the current. I consider a ring parallel to layers with the center positioned at rj. = 0 in the middle of the layer (the plane z = 0), with the penetration length 2, (weak superconductor), which is the largest of all characteristic lengths 22, dl, and d2. The solution to eq. (1) can be ob102
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Volume 183, number 1
PHYSICSLETTERSA
29 November 1993
z
i
2dl
/////////,)//////////// / r a = 2(p + 1) / ] / p + 1 / / / / / / / 2 d 2 ~
? / / / / / / /Y/V..,/ / / / / / / / /
~
/ "// / / / / , Y / r " / / / / / / / / / / /
Fig. 1. Geometry of the model. The layers of thicknesses 2dl and 2d2 with different bulk penetration lengths are periodically arranged. The middle of layers are in the planes z,. = m (dl + d2 ). The integer p numerates "lattice ceils" in z-direction.
// / / //////y / / / ////,/ // // " ,~ = o/77//7, ~ = o W ~ ~ X / / - ///////////////////// i
tained with the help of the Fourier transformation b(r±, z) = f ( d k / 2 ~ ) 2 h k ( Z ) exp(ik'r.L ). As a result eq. ( 1 ) can be reduced to
.kXn [ 1 +22(z) (k 2 - 02/022) ]Ilk(Z) ------2~pl - - ~ ¢~oJ1(kp)~(z) ,
(2)
where n is the unit vector in the z-direction, J1 (kp) is the Bessel function of the first order. Substituting hk(Z) = - - 2 n i k x n J ~ ( k p ) Z , ~ ( z ) / k with integers m numbering layers (rnk= 2p for the layers with the penetra-
tion length 21 ), one can obtain the boundary conditions for eq. (2) in the form Z~=Z~+~,
Z ~ + I =Z~cv+~ ),
2 + 2 2~Z2v,,=A2Z2o+~,z,
2 + 2 -22Z2v+l,z=,~lZ2(v+l),z,
(3)
where the Z -+ denote upper ( + ) and lower ( - ) boundary of a layer, Zz is a derivative with respect to z. The explicit expressions for Z,,,(z) can be represented as Zm = A + exp [Xm(Z--Zm)] "Ji-A~nexp[ --Xm(z--z,,,) ] ,
(4)
where z , , , = m ( d l + d 2 ) , i.e. Zm is positioned in the middle of the mth layer, X2o-~)el=~r~2+k 2, and X2p+x-X2 = v/~2- 2 + k 2" With the use of eq. (4) one can rewrite the set of boundary conditions (3) in the matrix form L t A2r = L y A2p+ 1 and L ~ A2p+ 1= L i- A2~p+1), where the following matrices are introduced,
o)
_
..-\amy
Yk=a~Zk,
X,,=a,,Z,~.
(5)
The solutions for the columns Am can be written in the form A2o=NUAo, where the matrix N is defined as + and is equal to
N= (LF)-~L~(L~)-~L
exp (2xl) [ cosh ( 2 x 2 ) + a + sinh ( 2 x 2 ) ] N= a_ sinh(2x2)
- a _ sinh ( 2 x 2 )
)
exp( -2Xl ) [cosh(Xxk)- a + sinh(2x2) ]_ '
(6 )
where a+_= ½( y d y 2 + Y 2 / y ~ ) . As is seen from eq. (6) the matrix N has d e t N = 1 with two real and positive eigenvalues (n+) satisfying the equality n+n_ = 1, i.e. n+ > I > n_. The amplitudes A~ should decrease with the increase of m. Therefore, one must nullify terms increasing as n~. Imposing this requirement one arrives at the following solvability condition, ( n_ - n l l ) A ~ =-nl2Aff
,
(7)
where the n~k are the matrix elements defined by eq. (6). An additional condition can be obtained from the solution to eq. (2) for the layer containing vortex singularity. As a result the columns can be found in the form 103
Volume 183, number 1
PHYSICSLETTERSA
n p(n12) Azv= 221z)fi(n-~ntl-n12 ) \ n _ - n l l
29 November 1993 (8)
"
The rotational symmetry of the problem helps to represent the magnetic field distribution as
h(r, z) = ~ 0 ~ ; dk kZ m(k, z)Jl (kr)J1 (kp) , m
(9)
0
where ~ris the azimuthal unit vector, Z,, ( k, z) - Z,, (z). So far we have not made any approximations and hence eq. (9) is exact. A simple analytical expression for h can be obtained after the integration over k in eq. (9) in the limit when 21 >> d2 > 2 2> dl >> q~oP h(r, 0) - 2d,2~ [O(r-p)I, (p/2 ±)K, (r/2±) +O(p-r)K, (p/2 ~)I, (r/2,) ] ,
(10)
where 1, and K, are Bessel functions of imaginary argument. In this equation the value 2 . is the penetration length for magnetic fields perpendicular to the layers. It is equal to 2 . =2 ~x/rf-,/22. One can also obtain in this limit the effective penetration length for magnetic fields parallel to the layers 2, = 22 (d2 + d, )/d2. The solution for the magnetic field distribution helps to define the energy of a vortex ring. The energy E(p) can be expressed as an integral along the curve of the ring
e(p)=~-
dlh(l).
(11)
In the following we restrict the consideration to the most interesting case 2, >> 22 >> ~, d,, d2. In this limit the effective penetration lengths are
/22~d, + 2~d2 .~2,x/d,/ (d, +dE)
2±=,4
(when d~ ~ dE) and 2 IL= 2 z (d, + d2)/dx/-~d~. After carrying out the integration in eq. (11 ) we arrive at (compare refs. [ 11,12 ] )
E,(p)=Epln(2~/~±) =~pln(p/~±)
when 2,<
(12)
where e = (q~ ~/ 8rt2,22) d2x/-d~ and ~± = ~ 2,/22 is the characteristic length which has, in the limit ~< ~±, the meaning of the transverse correlation radius. The effective correlation radius in the z-direction is ~1,= 2 (d, + dE) when ¢ < ~,. The energy of a ring excitation (RE) can be made essentially smaller if one considers a pair of vortex rings of opposite helicity separated by a layer with the strong superconductivity. The latter is a consequence resulting from the compensation of the large logarithms in eq. (12). This compensation occurs due to the attractive interaction between these rings. The energy of such an excitation will be dependent on two radii p~, Pz and the separation distance R, and it can be calculated in the same manner. In this case instead of eq. ( 11 ) we have q~o ( ~ d l h , ( l ) + ~dlh2(l)+ (~ E(p,,pE,R)= ~-~ _ dl h 2 ( l ) l
2
1
+
~ dl/h(/)),
(13)
2
where the indices 1, 2 in the integrals define integration paths along the first or the second ring, hi is the field generated by the ith ring. The smallest energy E2 at a fixed vortex path length L=2n(p~ +Pz) is realized when PI =pz=L/4n=p and R = 0 . This energy is equal to E2(p).~ep. Though the interaction between rings is at104
V o l u m e 183, n u m b e r 1
PHYSICS LETTERS A
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29 N o v e m b e r 1993
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Fig. 2. T h e d e p e n d e n c e o f t h e e n e r g y o f a d o u b l e r i n g e x c i t a t i o n o n the r i n g s e p a r a t i o n u=z/2(d~ + d 2 ) (F(u) =E2(p, z)/E(p) ) f o r d~ = d~, d~/2~= 10 -~, a n d 2 d 2 2 = 3.
Fig. 3. Sketch of streamlines of the s u p e r c u r r e n t in the s e c t i o n of the d o u b l e r i n g e x c i t a t i o n .
tractive they cannot annihilate due to the large (in the limit ~<<~j_) energy barrier equal to The dependence E2(p, z) on the ring separation z is shown in fig. 2. In fig. 3 one can see current stream lines in the section of the double ring excitation (DRE). Now we will estimate an entropy contribution resulting from the freedom implied by the possible change of P t = P + ~r, P 2 - - P - ~ r, and R. In this case, according to the general principles of statistical mechanics, we have to define the free energy as
Eb(p) =~jE2(p)/¢.
F=-
Tin I dr I
dRexp[-E(p,,p2,R)/T]=E2(p)-Tlnf(p),
(14)
where the function f(p)ocp-1 when E2o/T>> 1. The consideration of ideal gases not in equilibrium given in the book by Landau and Lifshitz [ 13 ] can be generalized for the case when excitations in addition to the configurational has the internal entropy S = l n f . This generalization leads to the following expression for the number of DRE with the radius p, n(p)=exp{- [ E 2 ( p ) - TS(p)I/T}. The DRE created at some temperature in the vicinity of T¢ can be pinned with a decrease of temperature. Unfortunately, at present we cannot estimate the temperature at which this pinning may occur. Nonetheless, one may assume that the pinning is essential not very far below the critical temperature. This mechanism causes a finite density of quasiparticle states in the gap region. Such an effect is always seen in tunneling data. There have been advanced a number of explanations for the phenomenon. Let us estimate the contribution of DRE to this subgap density of states. We assume that, along a vortex line in a layer, some volume V(p) = 2rc2p¢±~ll is roughly in the normal state. Within the layer area S we have
Vn= ~, ~., V(p)n(p)'~S f ~2 ~dp 2~a~_~Jlf(P ) exp [ -
E2 (p) / T] ,
(15)
where we assumed that phase space is characterized by the position vector on the plane and the smallest distinguishable cell has the area ~2. From eq. (15) one can estimate the relative part an = Vn/S~uof the normal volume
an~16~222T/@2d: ,
(16)
where T should be equal to the temperature at which pinning occurs. Quantitative estimates can be easily made for YBa2Cu307 as the most complete set of data is available for this material (see for instance ref. [ 14 ] ) d~ ~ 2d:, 2d2~ 3.8 A, 2_L =7000 A, and 2u,.~ 1400 A. From these data one can obtain 2 ~ 8 . 6 × 104 A, 2 2 ~ 0 . 6 6 × 103 A, 21/22 ~ 13, and ~± ~ 35 A. These data lead to an ~ 0.1 for comparatively low temperature ( T o - T) / Tc = 0.1. 105
Volume 183,number 1
PHYSICSLETTERSA
29 November 1993
Formula (16) can be compared with the available experimental data. In ref. [ 15 ] the density of states within the gap related to the normal density of states is approximately 0.25. This value reasonably agrees with the estimate of a,. One more qualitative fact found in ref. [ 15 ] which agrees with the presence of DRE is that the regions on the surface of the crystal, with lower density of states in the gap, correspond to a better superconductivity which is seen in the increase of the density of states in the vicinity of the gap singularity. In conclusion, we believe that doubled vertex loops at low temperatures, originated from DRE existing in the vicinity of To, manifest themselves in additional randomization (in comparison with the normal state) of superconducting properties in high-To superconductors and layered structures. It is likely that such randomization has already been seen in the experiment of ref. [ 15 ]. Further experiments which could contribute to the clarification of this problem should compare levels of randomization in normal and superconducting states. I am grateful to E.L. Wolf, fruitful discussions with whom stimulated the appearance of this work, My thanks are also to O. Mezrin for enlightening discussions. This work was supported by the Air Force Office of Scientific Research under Grant AFOSR-89-0338 and the US Department of Energy under Grant DE-ED02-87ER45301.
References [ I ] W.E. Lawrence and S. Doniach, in: Proc. 12th Int. Conf. on Low temperature physics, ed. E. Kanda (Academic Press of Japan, Kyoto, 1971) p. 361. [2] B. Horovitz, Phys. Rcv. Lctt. 67 ( 1991 ) 378. [3 ] J.R. Clem, Phys. Rev. B 43 ( 1991 ) 7837. [4] J.R. Clem and M.W. Coffey, Phys. Rev. B 42 (1990) 6209. [5] J. Friedel,J. Phys. (Paris) 49 (1988) 1561. [6] S.E. Korshunov, Europhys. Lett. I 1 (1990) 757. [7 ] V.L. Berezinskii,Soy. Phys. JETP 6 ( 1971 ) I 144. [8 ] I.M. Kosterlitz and D.I. Thouless, J. Phys. C 5 (1972) L124. [9 ] Yu.M. Ivanchenko, V.L. Belevlsov,Yu.A. Oenenko and Yu.V. Medvedev, Soy. J. Low Temp. Phys. (USA ) 17 ( 1991 ) 1239; Physica C 193 (1992) 291. [ l0 ] P.G. de Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1966 ). [ I I ] L.N. Bulaevskii, Soy. Phys. Usp. 18 (1976) 514. [ 12] K.B. Efetov, Soy. Phys. JETP 49 (1979) 905. [ 13 ] L.D. Landau and E.M. Lifshitz,Statisticalphysics (Addison-Wesley, Reading, MA, 1974) p. I 12.
[ 14] B. Batlogg,Phys. Today 44 ( 1991) 44. [ 15] A. Chang, Z.Y. Rong, Yu.M. Ivanchenko,F. Lu and E.L Wolf,Phys.Rev. B 46 (1992) 5692.
106
PHYSICS LETTERS A
North-Holland
Vortex ring excitations in layered superconductors Yu.M. I v a n c h e n k o Department of Physics, Polytechnic University, Six Metro Tech Center, Brooklyn, NY 11201, USA Received 17 August 1993;acceptedfor publication 1 October 1993 Communicated by L.J. Sham
A specialkind of thermal excitationscan be found in the vicinityof Tcin a layeredsuperconductingstructure. The excitations are vortextings positionedin the layerwith weakersuperconductivity.For parameterscorrespondingto high-Tosuperconductors these excitations should consist of two coupled vortex rings with opposite helicities. Fixed in position by pinning forces and frozen, these excitationscontribute to randomization of the superconductingstate at low temperatures. Estimates are made for the densityof states within the gap regionand characteristiclength of the random structurecausedby the doublering excitations.
It is now widely recognized that thermal excitations of 2D vortices play an important role in the magnetic and transport properties of anisotropic layered superconductors (SC). This phenomenon substantially influenced extremely anisotropic systems like Bi- or Tl-based high-To oxides or artificial superconducting superlattices. A conventional model describing such compounds was first advanced by Lawrence and Doniach (LD) [ 1 ]. This model deals with a periodic array of SC layers coupled by Josephson weak links. It admits two kinds of topological excitations. They are vortex points [2] (or pancake vortices [ 3] ) confined to superconducting layers and fluxon loops [ 2 ], which can also be seen as loops formed by Josephson vortices [ 4 ]. The latter are confined between superconducting layers and lead to their decoupling. The thermodynamics of loop excitations in layered structures has been studied [ 5,6,2 ] in the framework of the LD model. Friedel [ 5 ] supposed that in the case of 3D systems formed by weakly coupled layers a phase transition (caused by vortex rings) should occur at a temperature Tf which decreases to zero with the decrease of coupling between the layers. Thus, the temperature Tf, according to Friedel, is always less than the 2D Berezinskii-Kosteditz-Thouless transition temperature Tv [7,8 ]. Later, Korshunov [6 ] showed, that in a simplified Gaussian approximation of the Josephson coupling Tr> Tv. In a strict consideration Horovitz [ 2 ] has confirmed the statement that Tf> T, and proved that the fluxon temperature Tf determines the transition temperature To. In the present work a model similar to that used in ref. [ 9 ] for a squeezed vortex is applied to the description of noninteracting fluxon excitations. I assume that a cell of a layered structure consists of two layers with thicknesses 2dl and 2d2. Each layer is a London superconductor characterized by the bulk field penetration length ,~.x or ~,2 with the same critical temperature and the same bulk correlation radius ~. These layers alternatively form a periodic structure in the zdirection (see fig. 1 ). The magnetic field distribution h ( r ) can be found as a solution of the London equation, which for this case takes the form (see ref. [ 10 ] ) h ( r ) = VX [22(z)Vxh (r) ] = O0 ~ d l J ( r - l ) .
(1)
Here ~o is the magnetic flux quantum and the integration should be carried out along the curve of the ring with the length L = 2~p. Equation ( 1 ) must be completed by boundary conditions reflecting continuity of the magnetic field and the normal component of the current. I consider a ring parallel to layers with the center positioned at rj. = 0 in the middle of the layer (the plane z = 0), with the penetration length 2, (weak superconductor), which is the largest of all characteristic lengths 22, dl, and d2. The solution to eq. (1) can be ob102
o375-9601/93/$06.00© 1993ElsevierSciencePublishersB.V. All rightsreserved.
Volume 183, number 1
PHYSICSLETTERSA
29 November 1993
z
i
2dl
/////////,)//////////// / r a = 2(p + 1) / ] / p + 1 / / / / / / / 2 d 2 ~
? / / / / / / /Y/V..,/ / / / / / / / /
~
/ "// / / / / , Y / r " / / / / / / / / / / /
Fig. 1. Geometry of the model. The layers of thicknesses 2dl and 2d2 with different bulk penetration lengths are periodically arranged. The middle of layers are in the planes z,. = m (dl + d2 ). The integer p numerates "lattice ceils" in z-direction.
// / / //////y / / / ////,/ // // " ,~ = o/77//7, ~ = o W ~ ~ X / / - ///////////////////// i
tained with the help of the Fourier transformation b(r±, z) = f ( d k / 2 ~ ) 2 h k ( Z ) exp(ik'r.L ). As a result eq. ( 1 ) can be reduced to
.kXn [ 1 +22(z) (k 2 - 02/022) ]Ilk(Z) ------2~pl - - ~ ¢~oJ1(kp)~(z) ,
(2)
where n is the unit vector in the z-direction, J1 (kp) is the Bessel function of the first order. Substituting hk(Z) = - - 2 n i k x n J ~ ( k p ) Z , ~ ( z ) / k with integers m numbering layers (rnk= 2p for the layers with the penetra-
tion length 21 ), one can obtain the boundary conditions for eq. (2) in the form Z~=Z~+~,
Z ~ + I =Z~cv+~ ),
2 + 2 2~Z2v,,=A2Z2o+~,z,
2 + 2 -22Z2v+l,z=,~lZ2(v+l),z,
(3)
where the Z -+ denote upper ( + ) and lower ( - ) boundary of a layer, Zz is a derivative with respect to z. The explicit expressions for Z,,,(z) can be represented as Zm = A + exp [Xm(Z--Zm)] "Ji-A~nexp[ --Xm(z--z,,,) ] ,
(4)
where z , , , = m ( d l + d 2 ) , i.e. Zm is positioned in the middle of the mth layer, X2o-~)el=~r~2+k 2, and X2p+x-X2 = v/~2- 2 + k 2" With the use of eq. (4) one can rewrite the set of boundary conditions (3) in the matrix form L t A2r = L y A2p+ 1 and L ~ A2p+ 1= L i- A2~p+1), where the following matrices are introduced,
o)
_
..-\amy
Yk=a~Zk,
X,,=a,,Z,~.
(5)
The solutions for the columns Am can be written in the form A2o=NUAo, where the matrix N is defined as + and is equal to
N= (LF)-~L~(L~)-~L
exp (2xl) [ cosh ( 2 x 2 ) + a + sinh ( 2 x 2 ) ] N= a_ sinh(2x2)
- a _ sinh ( 2 x 2 )
)
exp( -2Xl ) [cosh(Xxk)- a + sinh(2x2) ]_ '
(6 )
where a+_= ½( y d y 2 + Y 2 / y ~ ) . As is seen from eq. (6) the matrix N has d e t N = 1 with two real and positive eigenvalues (n+) satisfying the equality n+n_ = 1, i.e. n+ > I > n_. The amplitudes A~ should decrease with the increase of m. Therefore, one must nullify terms increasing as n~. Imposing this requirement one arrives at the following solvability condition, ( n_ - n l l ) A ~ =-nl2Aff
,
(7)
where the n~k are the matrix elements defined by eq. (6). An additional condition can be obtained from the solution to eq. (2) for the layer containing vortex singularity. As a result the columns can be found in the form 103
Volume 183, number 1
PHYSICSLETTERSA
n p(n12) Azv= 221z)fi(n-~ntl-n12 ) \ n _ - n l l
29 November 1993 (8)
"
The rotational symmetry of the problem helps to represent the magnetic field distribution as
h(r, z) = ~ 0 ~ ; dk kZ m(k, z)Jl (kr)J1 (kp) , m
(9)
0
where ~ris the azimuthal unit vector, Z,, ( k, z) - Z,, (z). So far we have not made any approximations and hence eq. (9) is exact. A simple analytical expression for h can be obtained after the integration over k in eq. (9) in the limit when 21 >> d2 > 2 2> dl >> q~oP h(r, 0) - 2d,2~ [O(r-p)I, (p/2 ±)K, (r/2±) +O(p-r)K, (p/2 ~)I, (r/2,) ] ,
(10)
where 1, and K, are Bessel functions of imaginary argument. In this equation the value 2 . is the penetration length for magnetic fields perpendicular to the layers. It is equal to 2 . =2 ~x/rf-,/22. One can also obtain in this limit the effective penetration length for magnetic fields parallel to the layers 2, = 22 (d2 + d, )/d2. The solution for the magnetic field distribution helps to define the energy of a vortex ring. The energy E(p) can be expressed as an integral along the curve of the ring
e(p)=~-
dlh(l).
(11)
In the following we restrict the consideration to the most interesting case 2, >> 22 >> ~, d,, d2. In this limit the effective penetration lengths are
/22~d, + 2~d2 .~2,x/d,/ (d, +dE)
2±=,4
(when d~ ~ dE) and 2 IL= 2 z (d, + d2)/dx/-~d~. After carrying out the integration in eq. (11 ) we arrive at (compare refs. [ 11,12 ] )
E,(p)=Epln(2~/~±) =~pln(p/~±)
when 2,<
(12)
where e = (q~ ~/ 8rt2,22) d2x/-d~ and ~± = ~ 2,/22 is the characteristic length which has, in the limit ~< ~±, the meaning of the transverse correlation radius. The effective correlation radius in the z-direction is ~1,= 2 (d, + dE) when ¢ < ~,. The energy of a ring excitation (RE) can be made essentially smaller if one considers a pair of vortex rings of opposite helicity separated by a layer with the strong superconductivity. The latter is a consequence resulting from the compensation of the large logarithms in eq. (12). This compensation occurs due to the attractive interaction between these rings. The energy of such an excitation will be dependent on two radii p~, Pz and the separation distance R, and it can be calculated in the same manner. In this case instead of eq. ( 11 ) we have q~o ( ~ d l h , ( l ) + ~dlh2(l)+ (~ E(p,,pE,R)= ~-~ _ dl h 2 ( l ) l
2
1
+
~ dl/h(/)),
(13)
2
where the indices 1, 2 in the integrals define integration paths along the first or the second ring, hi is the field generated by the ith ring. The smallest energy E2 at a fixed vortex path length L=2n(p~ +Pz) is realized when PI =pz=L/4n=p and R = 0 . This energy is equal to E2(p).~ep. Though the interaction between rings is at104
V o l u m e 183, n u m b e r 1
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29 N o v e m b e r 1993
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Fig. 2. T h e d e p e n d e n c e o f t h e e n e r g y o f a d o u b l e r i n g e x c i t a t i o n o n the r i n g s e p a r a t i o n u=z/2(d~ + d 2 ) (F(u) =E2(p, z)/E(p) ) f o r d~ = d~, d~/2~= 10 -~, a n d 2 d 2 2 = 3.
Fig. 3. Sketch of streamlines of the s u p e r c u r r e n t in the s e c t i o n of the d o u b l e r i n g e x c i t a t i o n .
tractive they cannot annihilate due to the large (in the limit ~<<~j_) energy barrier equal to The dependence E2(p, z) on the ring separation z is shown in fig. 2. In fig. 3 one can see current stream lines in the section of the double ring excitation (DRE). Now we will estimate an entropy contribution resulting from the freedom implied by the possible change of P t = P + ~r, P 2 - - P - ~ r, and R. In this case, according to the general principles of statistical mechanics, we have to define the free energy as
Eb(p) =~jE2(p)/¢.
F=-
Tin I dr I
dRexp[-E(p,,p2,R)/T]=E2(p)-Tlnf(p),
(14)
where the function f(p)ocp-1 when E2o/T>> 1. The consideration of ideal gases not in equilibrium given in the book by Landau and Lifshitz [ 13 ] can be generalized for the case when excitations in addition to the configurational has the internal entropy S = l n f . This generalization leads to the following expression for the number of DRE with the radius p, n(p)=exp{- [ E 2 ( p ) - TS(p)I/T}. The DRE created at some temperature in the vicinity of T¢ can be pinned with a decrease of temperature. Unfortunately, at present we cannot estimate the temperature at which this pinning may occur. Nonetheless, one may assume that the pinning is essential not very far below the critical temperature. This mechanism causes a finite density of quasiparticle states in the gap region. Such an effect is always seen in tunneling data. There have been advanced a number of explanations for the phenomenon. Let us estimate the contribution of DRE to this subgap density of states. We assume that, along a vortex line in a layer, some volume V(p) = 2rc2p¢±~ll is roughly in the normal state. Within the layer area S we have
Vn= ~, ~., V(p)n(p)'~S f ~2 ~dp 2~a~_~Jlf(P ) exp [ -
E2 (p) / T] ,
(15)
where we assumed that phase space is characterized by the position vector on the plane and the smallest distinguishable cell has the area ~2. From eq. (15) one can estimate the relative part an = Vn/S~uof the normal volume
an~16~222T/@2d: ,
(16)
where T should be equal to the temperature at which pinning occurs. Quantitative estimates can be easily made for YBa2Cu307 as the most complete set of data is available for this material (see for instance ref. [ 14 ] ) d~ ~ 2d:, 2d2~ 3.8 A, 2_L =7000 A, and 2u,.~ 1400 A. From these data one can obtain 2 ~ 8 . 6 × 104 A, 2 2 ~ 0 . 6 6 × 103 A, 21/22 ~ 13, and ~± ~ 35 A. These data lead to an ~ 0.1 for comparatively low temperature ( T o - T) / Tc = 0.1. 105
Volume 183,number 1
PHYSICSLETTERSA
29 November 1993
Formula (16) can be compared with the available experimental data. In ref. [ 15 ] the density of states within the gap related to the normal density of states is approximately 0.25. This value reasonably agrees with the estimate of a,. One more qualitative fact found in ref. [ 15 ] which agrees with the presence of DRE is that the regions on the surface of the crystal, with lower density of states in the gap, correspond to a better superconductivity which is seen in the increase of the density of states in the vicinity of the gap singularity. In conclusion, we believe that doubled vertex loops at low temperatures, originated from DRE existing in the vicinity of To, manifest themselves in additional randomization (in comparison with the normal state) of superconducting properties in high-To superconductors and layered structures. It is likely that such randomization has already been seen in the experiment of ref. [ 15 ]. Further experiments which could contribute to the clarification of this problem should compare levels of randomization in normal and superconducting states. I am grateful to E.L. Wolf, fruitful discussions with whom stimulated the appearance of this work, My thanks are also to O. Mezrin for enlightening discussions. This work was supported by the Air Force Office of Scientific Research under Grant AFOSR-89-0338 and the US Department of Energy under Grant DE-ED02-87ER45301.
References [ I ] W.E. Lawrence and S. Doniach, in: Proc. 12th Int. Conf. on Low temperature physics, ed. E. Kanda (Academic Press of Japan, Kyoto, 1971) p. 361. [2] B. Horovitz, Phys. Rcv. Lctt. 67 ( 1991 ) 378. [3 ] J.R. Clem, Phys. Rev. B 43 ( 1991 ) 7837. [4] J.R. Clem and M.W. Coffey, Phys. Rev. B 42 (1990) 6209. [5] J. Friedel,J. Phys. (Paris) 49 (1988) 1561. [6] S.E. Korshunov, Europhys. Lett. I 1 (1990) 757. [7 ] V.L. Berezinskii,Soy. Phys. JETP 6 ( 1971 ) I 144. [8 ] I.M. Kosterlitz and D.I. Thouless, J. Phys. C 5 (1972) L124. [9 ] Yu.M. Ivanchenko, V.L. Belevlsov,Yu.A. Oenenko and Yu.V. Medvedev, Soy. J. Low Temp. Phys. (USA ) 17 ( 1991 ) 1239; Physica C 193 (1992) 291. [ l0 ] P.G. de Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1966 ). [ I I ] L.N. Bulaevskii, Soy. Phys. Usp. 18 (1976) 514. [ 12] K.B. Efetov, Soy. Phys. JETP 49 (1979) 905. [ 13 ] L.D. Landau and E.M. Lifshitz,Statisticalphysics (Addison-Wesley, Reading, MA, 1974) p. I 12.
[ 14] B. Batlogg,Phys. Today 44 ( 1991) 44. [ 15] A. Chang, Z.Y. Rong, Yu.M. Ivanchenko,F. Lu and E.L Wolf,Phys.Rev. B 46 (1992) 5692.
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