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Current transitions of superconducting whiskers
Volume 42A, number 7
PHYSICS LEUERS
29 January 1973
CURRENT TRANSITIONS OF SUPERCONDUCTING WHISKERS H.J. FINK University of California, Davis, California 95616, USA Received 5 December 1972
In long one-dimensional superconductors, such as whiskers, new, current-carrying states above the critical current were found that are due to spatially periodic solutions of the orderparameter and spatially periodic super and normal transport currents in parallel. These solutions lead to voltage steps in the DC voltage-current characteristic and are in good agreement with experiments.
Recent experiments [1,2] have shown that the voltage across a superconducting whisker increases by a number of finite steps as the current is increased. Because the radius, a, of the whiskers was smaller than X(T) and ~(T) the whiskers are one-dimensional in the sense of the Ginzburg—Landau theory. The proposed explanation is based on the[3] time dependent Ginzburg—Landau (GL) equations which in the usual GL normalization are
a~/at= (1_~2)~+ V2I4~./K2 ci + acI~/at=
4~i
—
.
~
~,2
2Q.
(1) (2)
The time t is normalized by the relaxation time r of the orderparameter, ~2= 2pr/71, and p = e V is the electrochemical potential related to the normal current component i~.The orderparameter is ~I(r,t)exp i’1(r,t), where ~1iand 1 are real functions. Q = —V4~’/ic+ A is the normalized superfluid velocity and the electron charge e < 0 in the above normalization. a~/at= —2p1,ri~i,where p.~,is the pair electrochemical potential related to the supercurrent i5. The total current density iT = in + i~in normalized notation is 2
sin2Ø = (1_/i2/i~)/(1~(i~2/~) sin2~)
(6)
sin2a=2/(~i2+Q2)—1.
(7)
(3)
5Ic=—h,Lil
4ir I Ic = —QI ~,
make the Ansatz Q = (Q0/~’0)~P where Q0 and ~ are values of Q and ~ at some point x = x0 at which d~ifdx= 0. VJQ is defined in such a way that i/i/~I~is always positive. The solution of i~(x)is then from eqs. (1) and (2): 2a), (5) = cd(u Isin where u F(~i\a)is the incomplete elliptic integral of the first kind and the inverse of the incomplete ell~ptic integral, the Jacobian elliptic functioncd(pIsin~) is the spatially periodic solution with the definitions
—
ahV~u/rce
(4)
T
where a is the normal state conductivity. Let us assume that the modulus of the orderparameter i~ti(x)is periodic and time independent and that Q(x) has the same spatial periodicity as I ~i(x)I. We
°
°
Applying boundary conditions to eq. (5), it follows from eq. (6) that ~ = ir/2 when ~~(x 1)= 0. Assuming that the order-parameter is ~11~at the contacts, we may set (x1 —x0) = L/2n where n is the number of zeros of ~i(x)in the whisker and L is the length of the whisker. It follows from eq. (2) when evaluating the potential atx1 that ~re VmIIIfl
=
—3Q0dOP/~i0)/dx~,
(8)
where eVm = 2n(p~ —p~ ) is the total voltage drop across the whisker. ‘the le?t hand side of eq. (8) is in conventional units. Eq. (5) with eq. (8) is: I v I = nDIQ I (9) m
0
where D = 3h/rieI-s/~with a 1x 1 —x0 I >>
-~
ir/2, provided
~.
465
Volume 42A, number 7
PHYSICS LETTERS
10r
‘1
40
‘
I!
29 January 1973
it
t
i
‘14’
‘1
Sn — Whiskers 0.9
\
/
L =0.255mm A=0.18(~m)2
—
08
5, S
13.5 m°K
Normal State
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30-
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/ Normal State
~06 /
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100 18
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Fig. 1. Shown are the experimental results of ref. [1]. The reduced scales are for ~T = 13.5 m°K with Vc ICR. R is defined by the slope of the straight line. The dashed curves are calculated from eq. (10), where n is the number of singularities of p (x) in the whisker. The experimental and theoretical coordinate systems have been shifted for the best fit. Uncertainties in some of the experimental parameter, of r, and recorder drift might be responsible for the shift.
The total current density in terms of the spatially averaged normal and supercomponents 1T = + becomes with eqs. (7) and (9): 1T
Vm
Tc
c
F 3~~fi ‘~‘c
L1
~
3\/~(~‘~c \3
(Vm \2
——~~~J ~y-j c
1 ].
(10)
1CR, where R is the normal state resistance and is the critical current of the homogeneous solution [4] Assuming that the deviation of ~i, namely 6 ~ti from its equilibrium value ~ vanishes proportional to exp(—t) cos k~xwe find from eqs. (1) and (2):
Vc
‘c
.
466
r = 3h148k ~T(2+k2)
(11)
B
where k2 > 1 for a stable solution. In fig. lwe have plotted eq. (10) withR = 0.905~, D given by eq. (9), r by eq. (11) with k = ~ L~T = 13.5m°K,a=ir/2,~ = 2300A,XL(O)= 310A, T~= 3.72°K,and have compared these solutions with the experimental results of ref. [1]. For ‘T > 1~the solution with n 1 has the lowest energy and a voltage appears. When ‘T is increased with ~ = const. there exists a maximum current where a voltage step must occur and n must increase.
Volume 42A, number 7
PHYSICS LETTERS
Thus we can account for the experiments [1, 2] by a new, current-cariying, superconducting state with super and normal currents in parallel. These currents and the order parameter are spatially periodic, exist above the critical current I~of the homogeneous solution and lead to DC voltage steps across the superconductor The author thanks W. Buckel. A. Baratoff, L. Kramer, D. Scalapino and H. Voigt for stimulating discussions.
29 January 1973
References [1] J. Meyer and G. Minnigerode, Phys. Lett. 38A (1972) 529. [2] W.W. Webb and R.T. Warburton, Phys. Rev. Lett. 20 (1968) 461; GJ. Rochlin, Conf. on Fluctuations in superconductors, Asiomar, Calif. 1968, p. 259; W. Mueller and F. Baumann, University of Karlsruhe, Germany, 1972, unpublished. [3] A. Schmid, Phys. Condens. Matter 5 (1966) 302; T.J. Rieger, D.J. Scalapino and J.E. Mercereau, Phys. Rev. Lett. 27(1971)1787. [4] P.G. de Gennes, Superconductivity of metals and alloys (W.A. Benjamin, Inc., New York, 1966).
467
PHYSICS LEUERS
29 January 1973
CURRENT TRANSITIONS OF SUPERCONDUCTING WHISKERS H.J. FINK University of California, Davis, California 95616, USA Received 5 December 1972
In long one-dimensional superconductors, such as whiskers, new, current-carrying states above the critical current were found that are due to spatially periodic solutions of the orderparameter and spatially periodic super and normal transport currents in parallel. These solutions lead to voltage steps in the DC voltage-current characteristic and are in good agreement with experiments.
Recent experiments [1,2] have shown that the voltage across a superconducting whisker increases by a number of finite steps as the current is increased. Because the radius, a, of the whiskers was smaller than X(T) and ~(T) the whiskers are one-dimensional in the sense of the Ginzburg—Landau theory. The proposed explanation is based on the[3] time dependent Ginzburg—Landau (GL) equations which in the usual GL normalization are
a~/at= (1_~2)~+ V2I4~./K2 ci + acI~/at=
4~i
—
.
~
~,2
2Q.
(1) (2)
The time t is normalized by the relaxation time r of the orderparameter, ~2= 2pr/71, and p = e V is the electrochemical potential related to the normal current component i~.The orderparameter is ~I(r,t)exp i’1(r,t), where ~1iand 1 are real functions. Q = —V4~’/ic+ A is the normalized superfluid velocity and the electron charge e < 0 in the above normalization. a~/at= —2p1,ri~i,where p.~,is the pair electrochemical potential related to the supercurrent i5. The total current density iT = in + i~in normalized notation is 2
sin2Ø = (1_/i2/i~)/(1~(i~2/~) sin2~)
(6)
sin2a=2/(~i2+Q2)—1.
(7)
(3)
5Ic=—h,Lil
4ir I Ic = —QI ~,
make the Ansatz Q = (Q0/~’0)~P where Q0 and ~ are values of Q and ~ at some point x = x0 at which d~ifdx= 0. VJQ is defined in such a way that i/i/~I~is always positive. The solution of i~(x)is then from eqs. (1) and (2): 2a), (5) = cd(u Isin where u F(~i\a)is the incomplete elliptic integral of the first kind and the inverse of the incomplete ell~ptic integral, the Jacobian elliptic functioncd(pIsin~) is the spatially periodic solution with the definitions
—
ahV~u/rce
(4)
T
where a is the normal state conductivity. Let us assume that the modulus of the orderparameter i~ti(x)is periodic and time independent and that Q(x) has the same spatial periodicity as I ~i(x)I. We
°
°
Applying boundary conditions to eq. (5), it follows from eq. (6) that ~ = ir/2 when ~~(x 1)= 0. Assuming that the order-parameter is ~11~at the contacts, we may set (x1 —x0) = L/2n where n is the number of zeros of ~i(x)in the whisker and L is the length of the whisker. It follows from eq. (2) when evaluating the potential atx1 that ~re VmIIIfl
=
—3Q0dOP/~i0)/dx~,
(8)
where eVm = 2n(p~ —p~ ) is the total voltage drop across the whisker. ‘the le?t hand side of eq. (8) is in conventional units. Eq. (5) with eq. (8) is: I v I = nDIQ I (9) m
0
where D = 3h/rieI-s/~with a 1x 1 —x0 I >>
-~
ir/2, provided
~.
465
Volume 42A, number 7
PHYSICS LETTERS
10r
‘1
40
‘
I!
29 January 1973
it
t
i
‘14’
‘1
Sn — Whiskers 0.9
\
/
L =0.255mm A=0.18(~m)2
—
08
5, S
13.5 m°K
Normal State
/
30-
/ / 0.7
/ Normal State
~06 /
E
0.5
-
/
,/
/
/ /
/
,/ -
I’
/
I
-
‘N /1
,._-“
/
/
/
/
—
3 / / I /
/
/
V
“ /
/
4/I /
/
/
/
/
,‘
/
/
/ /
LU
\
/
/
/
—
/ /
,..----
,
/
0.4
-
/ /
,-
/ / 10-
0.3
—
/
.—
,fl-
,~.‘n=2
/
/
-
/ 0.2
/
-
/ / 0.1
o
~
I
/
—--
I__-4
—--
I
40
~
0 0
0.2
0.4
06
I
60 0.8
1.0
1.2
I
I
80 1.4
1.6
I
100 18
2,0
Fig. 1. Shown are the experimental results of ref. [1]. The reduced scales are for ~T = 13.5 m°K with Vc ICR. R is defined by the slope of the straight line. The dashed curves are calculated from eq. (10), where n is the number of singularities of p (x) in the whisker. The experimental and theoretical coordinate systems have been shifted for the best fit. Uncertainties in some of the experimental parameter, of r, and recorder drift might be responsible for the shift.
The total current density in terms of the spatially averaged normal and supercomponents 1T = + becomes with eqs. (7) and (9): 1T
Vm
Tc
c
F 3~~fi ‘~‘c
L1
~
3\/~(~‘~c \3
(Vm \2
——~~~J ~y-j c
1 ].
(10)
1CR, where R is the normal state resistance and is the critical current of the homogeneous solution [4] Assuming that the deviation of ~i, namely 6 ~ti from its equilibrium value ~ vanishes proportional to exp(—t) cos k~xwe find from eqs. (1) and (2):
Vc
‘c
.
466
r = 3h148k ~T(2+k2)
(11)
B
where k2 > 1 for a stable solution. In fig. lwe have plotted eq. (10) withR = 0.905~, D given by eq. (9), r by eq. (11) with k = ~ L~T = 13.5m°K,a=ir/2,~ = 2300A,XL(O)= 310A, T~= 3.72°K,and have compared these solutions with the experimental results of ref. [1]. For ‘T > 1~the solution with n 1 has the lowest energy and a voltage appears. When ‘T is increased with ~ = const. there exists a maximum current where a voltage step must occur and n must increase.
Volume 42A, number 7
PHYSICS LETTERS
Thus we can account for the experiments [1, 2] by a new, current-cariying, superconducting state with super and normal currents in parallel. These currents and the order parameter are spatially periodic, exist above the critical current I~of the homogeneous solution and lead to DC voltage steps across the superconductor The author thanks W. Buckel. A. Baratoff, L. Kramer, D. Scalapino and H. Voigt for stimulating discussions.
29 January 1973
References [1] J. Meyer and G. Minnigerode, Phys. Lett. 38A (1972) 529. [2] W.W. Webb and R.T. Warburton, Phys. Rev. Lett. 20 (1968) 461; GJ. Rochlin, Conf. on Fluctuations in superconductors, Asiomar, Calif. 1968, p. 259; W. Mueller and F. Baumann, University of Karlsruhe, Germany, 1972, unpublished. [3] A. Schmid, Phys. Condens. Matter 5 (1966) 302; T.J. Rieger, D.J. Scalapino and J.E. Mercereau, Phys. Rev. Lett. 27(1971)1787. [4] P.G. de Gennes, Superconductivity of metals and alloys (W.A. Benjamin, Inc., New York, 1966).
467