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Asymptotic behavior of the magnetic field and attractive interaction between flux lines in type-II superconductors
Volume 45A, number 3
PHYSICS LETTERS
24 September 1973
ASYMPTOTIC BEHAVIOR OF THE MAGNETIC FIELD AND ATFRACTIVE INTERACTION BETWEEN FLUX LINES IN TYPE-Il SUPERCONDUCTORS F. MANCINI Istituto di Fisica, Università di Salerno, Via Vernieri 42, 84100 Salerno, Italy Gruppo Nazionale di Struttura della Materia del CNR, Mostra d’Oltremare Pad. 19, 80125 Napoli, Italy Received 26 July 1973 It is shown that type-LI superconductors with a Ginzburg-Landau parameter a first order transition at H~ 1.
In the last two years or so considerable attention, both from theoretical and experimental point of view, has been paid to the study of magnetic of type-I! superconductors. One of the main results is that there is a range of values of K, the Ginzburg-Landau parameter, such that the transition at Hci, between the Mejssner state and the mixed state, is first order [1, 2]. This result has been theoretically interpretated as due to the fact that for values of K less than a critical ~ there is an attractive interaction between flux lines. This hasfield beenofintuitively related to theattractive fact thatinteraction the magnetic a single vortex changes direction at a certain distance d 0 in correspondence the circulating persistent currents will flow in the opposite direction and an attractive interaction will occurr. The first theoretical derivation of the possibility of the existence of an attractive interaction was obtained in ref. [3] by means of the Gor’kov equations and, independently, in ref. [4] by means of a different approach. In the boson formulation of superconductivity [5,6] the magnetic field of a single isolated vortex is given by h(r) = F(k) exp (ik r), (1) (2?T)2 where 4) is the unit quantum flux and
4)J’-~_-~.-
2 + c(k)F1
(2)
F(k) = c(k)[X~ k XL is the London penetration depth and c(k) is the boson characteristic function, defined in refs. [5] and [6] ; both the quantities depend on the temperature.
K
less than a critical value
exhibit
On the other hand, we have also shown [4,6] that the interaction energy between two vortices is given by E(d) = ~h(d)/4ir,
(3)
where d is the distance between centers. Eqs. (1) and (3) immediately relate the presence of an attractive interaction to the phenomenon of field reversal. To compute the magnetic field we shall use in this article the following expression for the function c(k): 2]’, (4) c(k) = [1 + a~k where ~ is the BCS temperature-dependent coherence length, ~0(T) = vF/7re~(T),and a is a function of temperature. Use of expression (4) is justified by the following arguments. The boson characteristic function has been computed from the microscopic formulation 2 + 0(k4), with a = 0.44 at T 0°K.It has alfora~k small values of k; the results is [7]: c(k) = 1 so been shown [4,8] that in order to have a stable vortex lattice c(k) must decrease at least as 1/k2 when k goes to infinity. On the other hand, we have proved [6,9] that the form factor for scattering of —
neutrons by the vortex lattice is given by the function F(k), as defined in eq. (2). Use of expression (4) gives [10] a form factor which is in good agreement with the experimental data obtained in the case of scattering of neutrons by Nb 0 27Ta073 [11]. By putting expression (4) in eq. (1) we can easily compute [8] the distribution of the magnetic field of a single vortex. Let us distinguish two cases: i) x> ii) K
179
Volume 45A, number 3
h(r)
—p-— ~
=
[K0(r/ A1)
27rX~k where ~
A2
=
~
A1 (I
=
PHYSICS LETTERS
—
=
K0(r/ A2
(5)
)1
ALVc~l-I-K/K),
i4)
K
—
—
k/K).
[K0(br) _~Ko(b*r)1
(6)
27rA~K ______
2,
with
k
=
b
1 +ib2, b* =b1 -ib,,
\/~ —K
_______--—f-). 1--v~-(l
__________
bI
=
~ XL \/~-Kc (I
+
b 2 =— XL
-f-),
c
Kc
defined as K = AL/So) which should be compared with the experimental value K~= 1.14 of ref. [11. The agreement is sufficiently good and we expect that the situation could be improved by using a more accurate ex~ pression for the function c(k). According to our conclusions the value K~should not depend on the special material; this point could be experimentally investigated. The function dO/ALthe depends on the particular material only through Ginzburg-Landau parameter
b
=
second order, while for K
In the case ii) we obtain:
h(r)
24 September 1973
~: this result agrees with the experimental data of ref.
Kc
K 0 is the modified Bessel function of zero order. Let us first dicuss the case i). Since A1 > A2, the
[II.
magnetic field goes exponentially to zero when the distance from the axis center goes to infinity. For K > Kc the interaction energy is always repulsive and the transition at H~ is second order. The situation is different in the case ii). Using the asymptotic expression of the Bessel function we obtam the following behavior for the magnetic field for r ~ XL:
h(r)
4)K =—
[
1
~i1/2
I
~L 2~rlb U
exp(—rb1) sin(-~0
+
rb2), (7)
where 0 = arctan (b2/b1). For K
[n ir
—
~ 0]
n
1, 2,...
=
We thus find that materials
(8)
with K
teraction energy which becomes attractive at a distance d0 given by
d0 XL
2 K~
1/2r
I
K(K~—K)J
Let us note that d0
_____
Iir—-1 [ 1-arctafl -~
00
as K
(9)
___
Kc.
—~
Summarizing, we have found that there is a critical value Kc, such that for K > Kc the transition at is
180
The author wishes to thank Prof. H. Umezawa for many valuable discussions.
References [1] J. Auer and H. Ullmaier, Phys. Rev. B7 (1973) 136. [2] A.E. J. Low Phys.complete 10 (1973) We refer Jacobs, to this paper forTemp. an almost list137. of experimental and theoretical ~ticles on the subject of phase transitions at He1 and H~2. [31 G. Eilenberger and H. Büttner, Z. Physik 224 (1969) 335. 141 L. Leplae, F. Mancini and H. Umezawa, Phys. Rev. B2 (1970) 3594. Leplae, F. Mancini and H. Umezawa, preprint UWM-4867-73-4 (1973). [61 F. Mancini, Ph.D. Thesis, University of WisconsinMilwaukee (1972). [71 F. Mancini and H. Umezawa, Lettere al Nuovo Cim. 7 (1973) 125. [81 Leplae, F. Mancini and H. Umezawa, Rev. B6 [9] L. F. Mancini and H. Umezawa, Phys. Lett.Phys. 42A (1972)
[51 L.
(1972) 4178.
287. [101 F. Mancini, G. Scarpetta, V. Srinivasan and H. Umezawa, Preprint UMW-4867-73-1 (1973). [111 J. Schelten, H. Ullmaier and W. Schmatz, Phys. Stat. Sol. (b) 48 (1971) 619.
PHYSICS LETTERS
24 September 1973
ASYMPTOTIC BEHAVIOR OF THE MAGNETIC FIELD AND ATFRACTIVE INTERACTION BETWEEN FLUX LINES IN TYPE-Il SUPERCONDUCTORS F. MANCINI Istituto di Fisica, Università di Salerno, Via Vernieri 42, 84100 Salerno, Italy Gruppo Nazionale di Struttura della Materia del CNR, Mostra d’Oltremare Pad. 19, 80125 Napoli, Italy Received 26 July 1973 It is shown that type-LI superconductors with a Ginzburg-Landau parameter a first order transition at H~ 1.
In the last two years or so considerable attention, both from theoretical and experimental point of view, has been paid to the study of magnetic of type-I! superconductors. One of the main results is that there is a range of values of K, the Ginzburg-Landau parameter, such that the transition at Hci, between the Mejssner state and the mixed state, is first order [1, 2]. This result has been theoretically interpretated as due to the fact that for values of K less than a critical ~ there is an attractive interaction between flux lines. This hasfield beenofintuitively related to theattractive fact thatinteraction the magnetic a single vortex changes direction at a certain distance d 0 in correspondence the circulating persistent currents will flow in the opposite direction and an attractive interaction will occurr. The first theoretical derivation of the possibility of the existence of an attractive interaction was obtained in ref. [3] by means of the Gor’kov equations and, independently, in ref. [4] by means of a different approach. In the boson formulation of superconductivity [5,6] the magnetic field of a single isolated vortex is given by h(r) = F(k) exp (ik r), (1) (2?T)2 where 4) is the unit quantum flux and
4)J’-~_-~.-
2 + c(k)F1
(2)
F(k) = c(k)[X~ k XL is the London penetration depth and c(k) is the boson characteristic function, defined in refs. [5] and [6] ; both the quantities depend on the temperature.
K
less than a critical value
exhibit
On the other hand, we have also shown [4,6] that the interaction energy between two vortices is given by E(d) = ~h(d)/4ir,
(3)
where d is the distance between centers. Eqs. (1) and (3) immediately relate the presence of an attractive interaction to the phenomenon of field reversal. To compute the magnetic field we shall use in this article the following expression for the function c(k): 2]’, (4) c(k) = [1 + a~k where ~ is the BCS temperature-dependent coherence length, ~0(T) = vF/7re~(T),and a is a function of temperature. Use of expression (4) is justified by the following arguments. The boson characteristic function has been computed from the microscopic formulation 2 + 0(k4), with a = 0.44 at T 0°K.It has alfora~k small values of k; the results is [7]: c(k) = 1 so been shown [4,8] that in order to have a stable vortex lattice c(k) must decrease at least as 1/k2 when k goes to infinity. On the other hand, we have proved [6,9] that the form factor for scattering of —
neutrons by the vortex lattice is given by the function F(k), as defined in eq. (2). Use of expression (4) gives [10] a form factor which is in good agreement with the experimental data obtained in the case of scattering of neutrons by Nb 0 27Ta073 [11]. By putting expression (4) in eq. (1) we can easily compute [8] the distribution of the magnetic field of a single vortex. Let us distinguish two cases: i) x> ii) K
179
Volume 45A, number 3
h(r)
—p-— ~
=
[K0(r/ A1)
27rX~k where ~
A2
=
~
A1 (I
=
PHYSICS LETTERS
—
=
K0(r/ A2
(5)
)1
ALVc~l-I-K/K),
i4)
K
—
—
k/K).
[K0(br) _~Ko(b*r)1
(6)
27rA~K ______
2,
with
k
=
b
1 +ib2, b* =b1 -ib,,
\/~ —K
_______--—f-). 1--v~-(l
__________
bI
=
~ XL \/~-Kc (I
+
b 2 =— XL
-f-),
c
Kc
defined as K = AL/So) which should be compared with the experimental value K~= 1.14 of ref. [11. The agreement is sufficiently good and we expect that the situation could be improved by using a more accurate ex~ pression for the function c(k). According to our conclusions the value K~should not depend on the special material; this point could be experimentally investigated. The function dO/ALthe depends on the particular material only through Ginzburg-Landau parameter
b
=
second order, while for K
In the case ii) we obtain:
h(r)
24 September 1973
~: this result agrees with the experimental data of ref.
Kc
K 0 is the modified Bessel function of zero order. Let us first dicuss the case i). Since A1 > A2, the
[II.
magnetic field goes exponentially to zero when the distance from the axis center goes to infinity. For K > Kc the interaction energy is always repulsive and the transition at H~ is second order. The situation is different in the case ii). Using the asymptotic expression of the Bessel function we obtam the following behavior for the magnetic field for r ~ XL:
h(r)
4)K =—
[
1
~i1/2
I
~L 2~rlb U
exp(—rb1) sin(-~0
+
rb2), (7)
where 0 = arctan (b2/b1). For K
[n ir
—
~ 0]
n
1, 2,...
=
We thus find that materials
(8)
with K
teraction energy which becomes attractive at a distance d0 given by
d0 XL
2 K~
1/2r
I
K(K~—K)J
Let us note that d0
_____
Iir—-1 [ 1-arctafl -~
00
as K
(9)
___
Kc.
—~
Summarizing, we have found that there is a critical value Kc, such that for K > Kc the transition at is
180
The author wishes to thank Prof. H. Umezawa for many valuable discussions.
References [1] J. Auer and H. Ullmaier, Phys. Rev. B7 (1973) 136. [2] A.E. J. Low Phys.complete 10 (1973) We refer Jacobs, to this paper forTemp. an almost list137. of experimental and theoretical ~ticles on the subject of phase transitions at He1 and H~2. [31 G. Eilenberger and H. Büttner, Z. Physik 224 (1969) 335. 141 L. Leplae, F. Mancini and H. Umezawa, Phys. Rev. B2 (1970) 3594. Leplae, F. Mancini and H. Umezawa, preprint UWM-4867-73-4 (1973). [61 F. Mancini, Ph.D. Thesis, University of WisconsinMilwaukee (1972). [71 F. Mancini and H. Umezawa, Lettere al Nuovo Cim. 7 (1973) 125. [81 Leplae, F. Mancini and H. Umezawa, Rev. B6 [9] L. F. Mancini and H. Umezawa, Phys. Lett.Phys. 42A (1972)
[51 L.
(1972) 4178.
287. [101 F. Mancini, G. Scarpetta, V. Srinivasan and H. Umezawa, Preprint UMW-4867-73-1 (1973). [111 J. Schelten, H. Ullmaier and W. Schmatz, Phys. Stat. Sol. (b) 48 (1971) 619.