- Email: [email protected]
The dc Josephson current in cylindrical junctions
Volume 85A, number 2
PHYSICS LETTERS
14 September 1981
THE DC JOSEPHSON CURRENT IN CYLINDRICAL JUNCTIONS P.B. BURT~andM.D. SHERRILL Department ofPhysics andAstronomy, Clemson University, Clemson, SC 29631, USA Received 18 May 1981
The dc Josephson current between co-axial cylinders is calculated for arbitrary Josephson penetration depth. The results support; the interpretation of recent experiments which determined the magnetic-field dependence of the Josephson critical currelit density. The interpretation centers on the conclusion that the Josephson current vanishes when the fluxoid quantum nuilnbers differ.
The Josephson supercurrent density is given by J0 S~fl7, where is the Josephson critical current density and ‘y is the difference in phase between the wave functions on the two sides of the junction. Magnetic fields and currents cause 7 to vary spatially, and electric fields or potential differences cause 7 to vary temporally. Untilrecently all the Josephson effects could be explained by these phase changes. Recent experiments on cylindrical Josephson junctions have shown that the spatial variations of 7 with magnetic field may be suppressed, allowing a determination of the intrinsic dependence ofJ0 upon field [1,21. The arguments given ~nsupport of this point of view were, however, valid oilily when thesmall sample circumference 2liro and length I were both compared to the Josephson penetilation depth, Xj. In this letter we cal-
and n240, with n1 and n2 integers. The flux within the junction is (n2 n1)4)~.The dc Josephson current may therefore have only two values; its maximum value when n2 = 0, and zero when n2 ~ 0. This argument may be generalized to include thin-walled cylinders in which the fluxoid is quantizedbut the flux is not. The Ginzburg—Landau wave functions in the two cylinders which, because they must be single valued, take the form —
—
= =
—
I’I’i,., f~(r)exp(in1O) ‘I’21f2(r) exp[i(n20 + 6)]
where n1 and are theof fluxoid numbers rather than then2number fluxonsquantum in the cylinder. This leads to a current density J 0 sin [(n2 n1 )O + 61, which in turn gives a current = 2irrl ~0 sin 6 when n1 = n2 andlj = 0 for n1 n2. It is now clear that if the measured value of I~is not zero we can conclude that n1 = n2, and further that any dependence of the maximum Josephson current (6 = ir/2) upon field may be interpreted as a dependence ofJ~upon field. The arguments just given are deficient in that they do not allow for the self-field of the Josephson current which became important when Xj ~ 2w- or ~ 1. A complete phenomenological description of the Josephson effect (apart from the quasi-particle tunneling current) is given by the equation [4] 2a2’y/ao2+ a2 2 ,r2a2 2 = Sin 7. (1) —
culate 2irr the Josepl~soncurrent for arbitrary X1/l and Xj/ 0 and conclude that the original arguments, though restrictive, lead to an essentially correct interpretation of the experiments. The cylindrical Josephson junction was first discussed by Tffley in 1966 [3]. He considered Josephson tunneling betw$n two bulk co-axial cylinders in an axial magnetic field. He deduced some unique properties of the cyllnc~ricaljunction by analogy with the flat singly connected junction. In a flat junction the dc Josephson current has its principal maximum when there is no flux in the junction and is zero when an integer number of flux quanta ~o are included within the junction. The flux within each cylinder is n140
r~
0031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
7/az
—
7/at
97
Volume 85A, number 2
PHYSICS LETTERS
Here r0 is the radius of the junction and v is the yelocity of electromagnetic waves in the junction. Cheishvili [5] has given solutions of this equation for the special case that both cylinders are much thicker than the superconducting penetration depths (flux in the junction is quantized) but he neglected the dependence of 7 upon z. We 2are interested the dc our 2 0. Weinrestrict Josephson effect so that a 1/at attention to 2a27/ao2 + a2 2 = Xj2 sin y. (2)
r0
7/az
14 September 1981
Since y(2ir, z) 7(0, z) + 2irn, we see that I 0 for k0 * 0. This result and eq. (4) together give: = ~w-0X~J0 [(d7/dz)1 (d’y/dz)0], k0 = 0, (6) = o, k0 # 0. —
Consequently depends 0 the total Josephson currentifis7 zero. Thisupon corresponds to the earlier result for n 2 n1 #0 since in that case k0 we mustn1)/r0. return To to eq. (5) which is now one dimen= (n2 evaluate the current for k0 = 0 —
—
We can now calculate the dc Josephson current since
sional:
jr=.~Osi~~7:
d27/dz2=k
2~rI
I=J 0 X~
27/az2) dO ~
f f(r~1a2y/a02 0
k’
0
(7)
Fork 0 = 0,7, J~and the axial field B5 are indepen. dent of 0. However, 7, J~and the azimuthal field B0 which results from the Josephson current, afl depend
+r08
0
2Xj2siny. 2T
}
~ f [(a7/a0)2~ (a7/ao)0] f [(a7/az), (a~/az)0]do
upon z. The axial is independent of z also. is dependent only field uponB5 r and this dependence is dueIt to the cylindrical superconductors junction and not to the Josephson which current.form We the are in0 terested in solutions to eq. (7) when the applied field Now, 7(z, 0 + 2ir) = 7(z, 0) + 2irn, with n an integer is in the z-direction and the azimuthal field B 0 is due to insure that is single valued. Therefore (~7/~0)2~ solely to the Josephson current. We will require that = (~7/~0)0 so that the first term in the integral is zero, the Josephson current be supplied symmetrically from For the special case that a7/az is independent of 0 both ends. If supplied from one end the Josephson this gives: current will be confined to one end when x~ ~ I. The =
+ r0
—
2ir
—
~
.
(3)
~,.
I = 2irr~X~J0[(dy/dz)1
—
(dy/dz)0], (4)
boundary conditions on 7 are (~1)0/d)(a7/~z)0 = —B0(0) = p~Ij/4irr0
7(0,Z)7(Z).
=
To obtain solutions to eq. (2) and to evaluate the second term in eq. (3) when ~ is not independent of 0 we will restrict our attention to solutions of the form 7(0, z) = ~y(r0k0O + k5z)= 7(y) with this restriction eq. (2) becomes 27/dy2 = k2X~2 Sifl 7 k2 = k~+ k~. (5) d The solutions to this equation are well known [6,7] but for the moment we need only the symmetry properties of ‘y(r 0 k00 + k5z). For solutions of this form o7/az = (k5/r0 Ic8) 87/SO so that eq. (3) becomes:
=
~ Jo (k5 /k9)
f 0
[(a7/ao)1 (a’1/ao)0] dO
=
X~J0(k5/k0) [y(2ir,1)
—
7(21T, 0) +
—
—
7(0, 1)
where d is the junction thickness (X1 + X2 for bulk but modified for films). Owen and Scalapino [7] have solved eq. (7) for a flat junction lying in the y—z plane with the Josephson netic field in current the —yindirection. the +x direction Therefore andB, the~ andJj magdepend only upon z as for the cylindrical case. Their results for zero applied field are identical to the cylindrical case with k0 = 0. To obtain this exact correspondence it is necessary to supply the current from both ends of the cylindrical junction since in a flat junction the Josephson current density appears at both ends even when A1 41. The solutions for .Jj are in terms of the jacobian effiptic functions. The results of Owen and Scalapino for 1 = 15 X~are shown in
fig. lb. 98
7(0,0)].
—B0(l) = p01J/41rr0
Volume 85A, number 2
_____
PHYSICS LETTERS
compare the measured Josephson current when supplied from both ends to the Josephson current when supplied from one end since the former is twice as large if X~41 and the two are equal when A1 1.
B —i _____
14 September 1981
_____
O~?J4~}~l7~)
~‘
(a)
Jo
J~ ______________________
(b)
Josephson junction in an axial magnetic field. (b) The’ current density distribution for! 15 is reproduced front ref. [7]. If the current is supplied from one end of the inner cylinderIj will be half as large with the Fig. 1. (a) A cylin4rical
In conclusion, we have calculated the dc Josephson current for a cylindrical junction for arbitrary values of the parameter A~/l.The calculations are valid for solutions of the form 7 = 7(r0k00 + k5z), where J~ = .10 sin ‘y. We find that I~=0 when k0 ~ 0, that is when the two cylinders are in different fluxoid quantum states. When k0 = 0, Jj varies with z as in a flat junction. These conclusions are important in interpreting recent experiments [1,2] determining the magnetic-field dependence of the Josephson critical current density.
Xj
References
current density at the supply end for this case. [1] M.D. Sherxill, Pbys. Lett. 82A (1981) 191.
It is possible to apply an azimuthal field B9 to a cylindrical junctlion by sending an auxiliary current down the central cylinder. Again, if the Josephson current is suppli~dfrom both ends the variation of Josephson curreflt with azimuthal field is like, that of a flat junction in a field parallel to the plane of the junction. For diagnostic purposes it may be useful to
[2] M. Jihushan and M.D. Sherrill, submitted for presentation LTTilley, XVI (Los Angeles, CA, USA). [3] at D.R. Pbys. Left. 20(1966)117. [4] B.D. Josephson, Adv. Phys. 14 (1965) 419. [5] O.D. Cheishvili, Soy. Phys. Solid State 11(1969)138. [6] R.A. Feud and R.E. Prange, Phys. Rev. Lett. 10 (1963) 479. [7) C.S. Owen and D.J. Scalapino, Phys. Rev. 164 (1967) 538.
99
PHYSICS LETTERS
14 September 1981
THE DC JOSEPHSON CURRENT IN CYLINDRICAL JUNCTIONS P.B. BURT~andM.D. SHERRILL Department ofPhysics andAstronomy, Clemson University, Clemson, SC 29631, USA Received 18 May 1981
The dc Josephson current between co-axial cylinders is calculated for arbitrary Josephson penetration depth. The results support; the interpretation of recent experiments which determined the magnetic-field dependence of the Josephson critical currelit density. The interpretation centers on the conclusion that the Josephson current vanishes when the fluxoid quantum nuilnbers differ.
The Josephson supercurrent density is given by J0 S~fl7, where is the Josephson critical current density and ‘y is the difference in phase between the wave functions on the two sides of the junction. Magnetic fields and currents cause 7 to vary spatially, and electric fields or potential differences cause 7 to vary temporally. Untilrecently all the Josephson effects could be explained by these phase changes. Recent experiments on cylindrical Josephson junctions have shown that the spatial variations of 7 with magnetic field may be suppressed, allowing a determination of the intrinsic dependence ofJ0 upon field [1,21. The arguments given ~nsupport of this point of view were, however, valid oilily when thesmall sample circumference 2liro and length I were both compared to the Josephson penetilation depth, Xj. In this letter we cal-
and n240, with n1 and n2 integers. The flux within the junction is (n2 n1)4)~.The dc Josephson current may therefore have only two values; its maximum value when n2 = 0, and zero when n2 ~ 0. This argument may be generalized to include thin-walled cylinders in which the fluxoid is quantizedbut the flux is not. The Ginzburg—Landau wave functions in the two cylinders which, because they must be single valued, take the form —
—
= =
—
I’I’i,., f~(r)exp(in1O) ‘I’21f2(r) exp[i(n20 + 6)]
where n1 and are theof fluxoid numbers rather than then2number fluxonsquantum in the cylinder. This leads to a current density J 0 sin [(n2 n1 )O + 61, which in turn gives a current = 2irrl ~0 sin 6 when n1 = n2 andlj = 0 for n1 n2. It is now clear that if the measured value of I~is not zero we can conclude that n1 = n2, and further that any dependence of the maximum Josephson current (6 = ir/2) upon field may be interpreted as a dependence ofJ~upon field. The arguments just given are deficient in that they do not allow for the self-field of the Josephson current which became important when Xj ~ 2w- or ~ 1. A complete phenomenological description of the Josephson effect (apart from the quasi-particle tunneling current) is given by the equation [4] 2a2’y/ao2+ a2 2 ,r2a2 2 = Sin 7. (1) —
culate 2irr the Josepl~soncurrent for arbitrary X1/l and Xj/ 0 and conclude that the original arguments, though restrictive, lead to an essentially correct interpretation of the experiments. The cylindrical Josephson junction was first discussed by Tffley in 1966 [3]. He considered Josephson tunneling betw$n two bulk co-axial cylinders in an axial magnetic field. He deduced some unique properties of the cyllnc~ricaljunction by analogy with the flat singly connected junction. In a flat junction the dc Josephson current has its principal maximum when there is no flux in the junction and is zero when an integer number of flux quanta ~o are included within the junction. The flux within each cylinder is n140
r~
0031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
7/az
—
7/at
97
Volume 85A, number 2
PHYSICS LETTERS
Here r0 is the radius of the junction and v is the yelocity of electromagnetic waves in the junction. Cheishvili [5] has given solutions of this equation for the special case that both cylinders are much thicker than the superconducting penetration depths (flux in the junction is quantized) but he neglected the dependence of 7 upon z. We 2are interested the dc our 2 0. Weinrestrict Josephson effect so that a 1/at attention to 2a27/ao2 + a2 2 = Xj2 sin y. (2)
r0
7/az
14 September 1981
Since y(2ir, z) 7(0, z) + 2irn, we see that I 0 for k0 * 0. This result and eq. (4) together give: = ~w-0X~J0 [(d7/dz)1 (d’y/dz)0], k0 = 0, (6) = o, k0 # 0. —
Consequently depends 0 the total Josephson currentifis7 zero. Thisupon corresponds to the earlier result for n 2 n1 #0 since in that case k0 we mustn1)/r0. return To to eq. (5) which is now one dimen= (n2 evaluate the current for k0 = 0 —
—
We can now calculate the dc Josephson current since
sional:
jr=.~Osi~~7:
d27/dz2=k
2~rI
I=J 0 X~
27/az2) dO ~
f f(r~1a2y/a02 0
k’
0
(7)
Fork 0 = 0,7, J~and the axial field B5 are indepen. dent of 0. However, 7, J~and the azimuthal field B0 which results from the Josephson current, afl depend
+r08
0
2Xj2siny. 2T
}
~ f [(a7/a0)2~ (a7/ao)0] f [(a7/az), (a~/az)0]do
upon z. The axial is independent of z also. is dependent only field uponB5 r and this dependence is dueIt to the cylindrical superconductors junction and not to the Josephson which current.form We the are in0 terested in solutions to eq. (7) when the applied field Now, 7(z, 0 + 2ir) = 7(z, 0) + 2irn, with n an integer is in the z-direction and the azimuthal field B 0 is due to insure that is single valued. Therefore (~7/~0)2~ solely to the Josephson current. We will require that = (~7/~0)0 so that the first term in the integral is zero, the Josephson current be supplied symmetrically from For the special case that a7/az is independent of 0 both ends. If supplied from one end the Josephson this gives: current will be confined to one end when x~ ~ I. The =
+ r0
—
2ir
—
~
.
(3)
~,.
I = 2irr~X~J0[(dy/dz)1
—
(dy/dz)0], (4)
boundary conditions on 7 are (~1)0/d)(a7/~z)0 = —B0(0) = p~Ij/4irr0
7(0,Z)7(Z).
=
To obtain solutions to eq. (2) and to evaluate the second term in eq. (3) when ~ is not independent of 0 we will restrict our attention to solutions of the form 7(0, z) = ~y(r0k0O + k5z)= 7(y) with this restriction eq. (2) becomes 27/dy2 = k2X~2 Sifl 7 k2 = k~+ k~. (5) d The solutions to this equation are well known [6,7] but for the moment we need only the symmetry properties of ‘y(r 0 k00 + k5z). For solutions of this form o7/az = (k5/r0 Ic8) 87/SO so that eq. (3) becomes:
=
~ Jo (k5 /k9)
f 0
[(a7/ao)1 (a’1/ao)0] dO
=
X~J0(k5/k0) [y(2ir,1)
—
7(21T, 0) +
—
—
7(0, 1)
where d is the junction thickness (X1 + X2 for bulk but modified for films). Owen and Scalapino [7] have solved eq. (7) for a flat junction lying in the y—z plane with the Josephson netic field in current the —yindirection. the +x direction Therefore andB, the~ andJj magdepend only upon z as for the cylindrical case. Their results for zero applied field are identical to the cylindrical case with k0 = 0. To obtain this exact correspondence it is necessary to supply the current from both ends of the cylindrical junction since in a flat junction the Josephson current density appears at both ends even when A1 41. The solutions for .Jj are in terms of the jacobian effiptic functions. The results of Owen and Scalapino for 1 = 15 X~are shown in
fig. lb. 98
7(0,0)].
—B0(l) = p01J/41rr0
Volume 85A, number 2
_____
PHYSICS LETTERS
compare the measured Josephson current when supplied from both ends to the Josephson current when supplied from one end since the former is twice as large if X~41 and the two are equal when A1 1.
B —i _____
14 September 1981
_____
O~?J4~}~l7~)
~‘
(a)
Jo
J~ ______________________
(b)
Josephson junction in an axial magnetic field. (b) The’ current density distribution for! 15 is reproduced front ref. [7]. If the current is supplied from one end of the inner cylinderIj will be half as large with the Fig. 1. (a) A cylin4rical
In conclusion, we have calculated the dc Josephson current for a cylindrical junction for arbitrary values of the parameter A~/l.The calculations are valid for solutions of the form 7 = 7(r0k00 + k5z), where J~ = .10 sin ‘y. We find that I~=0 when k0 ~ 0, that is when the two cylinders are in different fluxoid quantum states. When k0 = 0, Jj varies with z as in a flat junction. These conclusions are important in interpreting recent experiments [1,2] determining the magnetic-field dependence of the Josephson critical current density.
Xj
References
current density at the supply end for this case. [1] M.D. Sherxill, Pbys. Lett. 82A (1981) 191.
It is possible to apply an azimuthal field B9 to a cylindrical junctlion by sending an auxiliary current down the central cylinder. Again, if the Josephson current is suppli~dfrom both ends the variation of Josephson curreflt with azimuthal field is like, that of a flat junction in a field parallel to the plane of the junction. For diagnostic purposes it may be useful to
[2] M. Jihushan and M.D. Sherrill, submitted for presentation LTTilley, XVI (Los Angeles, CA, USA). [3] at D.R. Pbys. Left. 20(1966)117. [4] B.D. Josephson, Adv. Phys. 14 (1965) 419. [5] O.D. Cheishvili, Soy. Phys. Solid State 11(1969)138. [6] R.A. Feud and R.E. Prange, Phys. Rev. Lett. 10 (1963) 479. [7) C.S. Owen and D.J. Scalapino, Phys. Rev. 164 (1967) 538.
99