Negative superfluid density: Mesoscopic fluctuations and reverse of
the supercurrent through a disordered Josephson junction
Beenakker, C.W.J.; Titov, M.; Jacquod, Ph.
Citation
Beenakker, C. W. J., Titov, M., & Jacquod, P. (2002). Negative superfluid density:
Mesoscopic fluctuations and reverse of the supercurrent through a disordered Josephson
junction. Retrieved from https://hdl.handle.net/1887/1214
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PHYSICAL REVIEW B, VOLUME 65, 012504
Negative superfluid density: Mesoscopic fluctuations and reverse of the supercurrent through a
disordered Josephson junction
M Titov, Ph Jacquod, and C W J Beenakker
Instituut-Loientz Umveisiteit Leiden, PO Box 9506, 2300 RA Leiden, The Netherlands
(Received l June 2001, published29 November 2001)
We calculate the Josephson couphng energy Uj(<f>) [related to the supercurrent I=(2e/fi)dU}Ιάφ] for a
disordered normal metal between two superconductois with a phase difference φ We demonstrate that the symmetry of the scattenng matrix of nonmteracting quasiparticles m zeio magnetic field imphes that i//(0) has a mimmum at φ=0 A maximum (that would lead to a π junction or negative superfluid density) is excluded for any reahzation of the disorder
DOI 10 1103/PhysRevB 65 012504 PACS number(s) 74 80 Fp, 72 15 Rn, 73 63 Rt, 74 40 +k
The question of whether mesoscopic fluctuation can re-verse the supercorrent through a disorderd Josephson junc-tion was posed ten years ago by Spivak and Kivelson,1 m their search for mechanisms that would lead to a negative local superfluid density m a dnty superconductor A negative mstead of a positive superfluid density means that the Jo-sephson couphng energy U ] ( φ ) has a maximum mstead of a mimmum for zero phase difference φ of the superconductmg order parameter The supercurrent Ι(φ) = (2 e/h)dUj/d(/) then has the opposite sign äs usual for small φ This has a vanety of observable consequences, mcludmg a ground state with a nonzero supercurrent,2 Aharonov-Bohm oscillations with penod h/4e, and negative magnetoresistance 3 A Josephson junction with a negative superfluid density is known äs a π junction,2 because the ground state for large magnetic mductance is close to φ = ττ mstead of äs usual at The known mechamsm for the creation of a rr junction1"4
m equihbnum5 in a nonmagnetic matenal6 requires strong
Coulomb repulsion to create a locahzed spin on a resonant impunty level Spivak and Kivelson asked the question whether purely one-electron conductance fluctuations might be sufficient to produce a locally negative superfluid density near the insulatmg state A suggestive argument that this might be possible comes from the relative magnitude of the mesoscopic sample-to-sample fluctuations of the supercur-rent m a disordered superconductor-normal-metal-superconductor (SNS) junction7 The ratio (SI2)m/(I)
— e2/hG of the root-mean-squared fluctuations over the
mean supercurrent is <^ l if the conductance G of the normal metal is large compared to the conductance quantum e2lh
The ratio becomes of order umty on approachmg the insulat-ing state, suggestmg that the supercurrent might have a nega-tive value m some samples
Of course, the root-mean-square amphtude of the super-current fluctuations does not distmguish between a positive and negative sign, so that this argument is only suggestive We were motivated to settle this issue because of recent ex-penments on locahzation in quasi-one-dimensional superconductors 8 This has renewed the interest in the
funda-mental question whether mesoscopic fluctuations aie suffi-cient or not to create a negative local superfluid density The answer, äs we will show, is that they are not
The two superconductors that form the SNS junction have order parameters Ae'*/2 and Ae~"*/2 The contacts to the
normal metal have W propagatmg modes at the Ferrm energy
EF, so that the elastic scattenng by the normal metal at
energy E = EF+s is charactenzed by a 2NX2N scattenng
matrix S (ε) The two properties of S that we use are that it is analyüc m the upper half of the complex ε plane and that it is a Symmetrie matrix, S ( e ) = S(s)T, when time-reversal
symmetry is preserved
The startmg pomt of our calculation is the relationship denved m Ref 9 between the Josephson couphng energy f/y(</>) in equihbnum at temperature T and the scattenng matrix,
n = 0 (1)
The summation runs over the Matsubara frequencies ωη
= (2n + l)irkBT The 4NX4N scattenng matrix SN(s)
de-scnbes the elastic scattenng by disorder m the normal metal of nonmteracting electron and hole quasiparticles with exci-tation energy ε,
5(ε) Ο
Ο S(-e)* (2)
The analytical continuation of SN from real to imagmary
energy (ε—>;ω) follows from S(e)—*5(ιω) and S( — ε)* —>£((&>)* Similarly, the matrix SA(e) describes the
Andreev reflection from the superconductors,
0
,-ιλφ/2
,ιΑ.φ/2
(3a)
; V l - 82/ A2 (3b) Here Λ is a 2NX2N diagonal matrix with elements AJ; = l for ls£js£W and A7 J = - l for N+ l^j^2N
Eq (1) differs from the usual representation of the Josephson energy äs a sum over the discrete spectrum (ε <Δ) plus an Integration over the contmuous spectrum (ε >Δ) The derivation of Eq (1) is based on the analyticity of SA and SN in the upper half of the complex ε plane that
BRIEF REPORTS PHYSICAL REVIEW B 65 012504 discrete and the contmuous spectrum We will now show that
each of these combmations is minimal for 0 = 0, although the contnbutions from the discrete and contmuous spectrum separately are not
Let us abbreviate
') (4)
Usmg the identity lnDet=Trln in Eq (1) one can calculate the first and second denvative with respect to φ of the energy U}(φ) The first derivative is given by
(5)
where the NX N matnces hil and h22 are blocks of the
matnx
hu h 12
h-n h22
(6)
The first derivative is equal to the supercurrent and vamshes at φ = 0 äs dictated by time-reversal symmetry
For the second derivative we obtam
άφι
(7)
= Ζ(1+ΖίΖΓ =
/l l /l 2
/2l /22 (8)
At = 0 the symmetry of S implies that F=FT and H
= Η\ hence /2i =/i2 ) and ^ai = ^i2 Therefore, every term
in the sum (7) is positive We conclude that the Josephson energy U j(4>) has a mmimum at φ=0,
= 0, 4 = 0
d2Uj
αφ2 >0 (9)
This concludes the proof that mesoscopic fluctuations cannot mvert the stabihty of the SNS junction at zero phase, exclud-ing a mechamsm for the creation of a 77 junction proposed ten years ago ' The proof holds for nonmteractmg quasipar-ticles m zero magnetic field at arbitrary temperature, for any disorder potential and any dimensionahty of the junction As a final remark, we conjecture (and have a proof for N=l) that Eq (1) implies dUj/αφΧ) in the entire mterval 0 < φ< π in the presence of time-reversal symmetry
We thank Boris Spivak for urgmg us to solve this problem and for valuable discussions This work was supported by the Dutch Science Foundation NWO/FOM and by the Swiss Na-tional Science Foundation
' B Z Spivak and S A Kivelson, Phys Rev B 43, 3740 (1991)
2L N Bulaevskn, V V Kuzn, a n d A A Sobyanm, Pis'maZh Eksp
Teor Fiz 25, 314 (1997) [JETP Lett 25, 290 (1977)]
3S A Kivelson and B Z Spivak, Phys Rev B 45, 10490 (1992) 4L I Glazman and K A Matveev, JETP Lett 49, 659 (1989) 5 The sign of the supercurrent can be switched qmte easily by
bnngmg the Josephson junction out of equihbnum, see J J A Baselmans, A F Morpurgo, B J van Wees, and T M Klapwyk, Nature (London) 397, 43 (1999) This is not a ττ junction m the original sense, which refers to an equihbnum property of the junction
6 The sign of the supercurrent becomes entirely sample specific if
time-reversal symmetry is broken by a magnetic field or mag-netic order, see F Zhou, Int J Mod Phys B 13, 2229 (1999)
7 B L Altshuler and B Z Spivak, Zh Eksp Teor Fiz 92, 609
(1987) [Sov Phys JETP 65, 343 (1987)]
8 A Bezryadm, C N Lan, and M Tmkham, Nature (London) 404,
971 (2000)
9 P W Brouwer and C WJ Beenakker, Chaos, Solitons Fractals 8,
1249 (1997), A factor of 2 is missmg m Eq (8) of this publica-tion PW Brouwer and CWJ Beenakker, cond-mat/9611162 (unpubhshed) does not contain this misprmt
