Angular dependence of Josephson currents in unconventional superconducting junctions

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Angular dependence of Josephson currents in unconventional superconducting junctions

T. Yokoyama,1Y. Sawa,1Y. Tanaka,1and A. A. Golubov2

1_{Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan} and CREST, Japan Science and Technology Corporation (JST) Nagoya, 464-8603, Japan 2_{Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands}

共Received 26 September 2006; published 3 January 2007兲

Josephson effect in junctions between unconventional superconductors is studied theoretically within the model describing the effects of interface roughness. The particularly important issue of applicability of the frequently used Sigrist-Rice共SR兲 formula for Josephson current in d-wave superconductor/insulator/d-wave superconductor junctions is addressed. We show that although the SR formula is not applicable in the ballistic case, it works well for rough interfaces when the diffusive normal metal regions exist between the d-wave superconductor and the insulator. It is shown that the SR approach only takes into account the component of the d-wave pair potential symmetric with respect to an inversion around the plane perpendicular to the inter-face. Similar formula can be derived for general unconventional superconductors with arbitrary angular mo-mentum l.

DOI:10.1103/PhysRevB.75.020502 PACS number共s兲: 74.20.Rp, 74.50.⫹r, 74.70.Kn

A number of phase-sensitive experiments have convinc-ingly demonstrated the realization of d-wave pairing state in high-TC cuprates.1–4 Because of such unconventional sym-metry, the study of Josephson effect in high-TC supercon-ducting共HTS兲 junctions attracted a lot of interest. A while ago, a simple formula for the Josephson current of d-wave superconductor/insulator/d-wave superconductor 共DID兲 junctions was proposed by Sigrist and Rice共SR兲.5_According

to the SR formula, the Josephson current is proportional to cos 2␣cos 2␤, where the␣共␤兲 denotes the angle between the normal to the interface and the crystal axis of the left共right兲

d-wave superconductor.5

Although the SR formula can explain experiments with the so-called␲ junctions,1,2 _{this formula does not take into}

account the effect of midgap Andreev resonant states 共MARS兲 formed at junction interfaces.6,7_{Actually, as shown}

in Ref. 8, SR formula does not work in ballistic d-wave junctions for ␣⫽0 and ␤⫽0 where MARS influence se-verely the charge transport at low temperatures. It was shown both theoretically9,10 _{and experimentally}11,12 _{that MARS}

in-duce a nonmonotonic temperature dependence of the maxi-mum Josephson current in DID junctions. On the other hand, SR formula has been extensively used to analyze experi-ments with various types of HTS Josephson junctions.13–15

Experiments with HTS junctions are of high importance for basic understanding of high-Tc superconductivity since they may provide information on possible subdominant admix-tures to the d-wave symmetry.16–18_{Therefore it is of}

funda-mental interest to understand the physical mechanisms that determine the angular dependence of Josephson current in HTS junctions. For this reason, the determination of the con-ditions of applicability of the SR formula is an important issue which is addressed in the present paper.

In the following, we study the Josephson current in

D / DN/ I / DN/ D junctions, where DN denotes diffusive

nor-mal metal and could be formed between the insulator and

d-wave superconductors. The calculations are based on the

quasiclassical Green’s function method applicable to uncon-ventional superconductor junctions.19,20 _{We find that the}

re-sulting Josephson current in D / DN/ I / DN/ D junctions is well fitted by the SR formula. Near the transition tempera-ture, it is proven analytically that Josephson current follows the SR formula. We also confirm that this formula does not hold in the ballistic junctions. It is clarified that in the SR formula, the component of the pair potential that is antisym-metric by the inversion operation around the plane perpen-dicular to the interface is neglected. We also study p-wave superconductor/diffusive normal metal/insulator/diffusive normal metal/p-wave superconductor 共P/DN/I/DN/ P兲 junctions. The resulting Josephson current is also well fitted by cos␣cos␤, where ␣ 共␤兲 denotes the angle between the crystal axis of left 共right兲 p-wave superconductor and the normal to the interface. This is a corresponding version of the SR formula in the p-wave superconductor junctions. Fur-thermore, it is possible to extend the theory for an unconven-tional superconductor 共US兲 with arbitrary angular momen-tum l. For US/DN/I/DN/US junctions, the expected Josephson current is proportional to cos l␣cos l␤. The ob-tained results may serve as a guide for the analysis of the experiments in unconventional superconductor junctions.

Before we proceed with a formal discussion, let us provide qualitative arguments on the physical meaning of the SR formula and explain why it holds in the diffusive junctions. First, we consider d-wave supercon-ductor junctions. The pair potentials of left and right

d-wave superconductors are, respectively, expressed by ⌬L =⌬关fSL共␾兲+ fASL共␾兲兴exp共−i⌿兲, and ⌬R=⌬关fSR共␾兲+ fASR共␾兲兴, with fSL共␾兲=cos 2␾cos 2␣, fASL共␾兲=sin 2␾sin 2␣, fSR共␾兲 = cos 2␾cos 2␤, fASR共␾兲=sin 2␾sin 2␤, where ␾ is the in-jection angle measured from the interface normal,⌬ denotes the maximum value of the pair potential and⌿ is the phase difference across the junction. The terms proportional to cos 2␾, i.e., fSL共␾兲 and fSR共␾兲, correspond to the dx2_−y2-wave pair potential and the terms proportional to sin共2␾兲, i.e.,

fASL共␾兲 and fASR共␾兲, correspond to the dxy-wave pair poten-tial, respectively. Here, fSL共␾兲= fSL共−␾兲, fSR共␾兲= fSR共−␾兲,

fASL共␾兲=−fASL共−␾兲, and fASR共␾兲=−fASR共−␾兲 are satisfied. In the actual calculation of Josephson current in

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D / DN/ I / DN/ D junctions, we have to take an average over

the various ␾. Due to the impurity scattering in DN, the average is taken for the left and right D/DN interface inde-pendently. Then we can drop fASL共␾兲 and fASR共␾兲 and arrive at the SR formula, where only the terms fSL共␾兲 and fSR共␾兲 remain which do not change sign by exchanging␾ for −␾. This fact is in accordance with the recent result that the prox-imity effect is absent in the case of dxy-wave pair potential.19 Consequently, the resulting Josephson current is proportional to cos 2␣cos 2␤.

Similar arguments apply to p-wave junctions. In this case, fSL共␾兲=cos␾cos␣, fASL共␾兲=sin␾sin␣, fSR共␾兲 = cos␾cos␤, and fASR共␾兲=sin␾sin␤ are satisfied. The terms proportional to cos␾, i.e., fSL共␾兲 and fSR共␾兲, corre-spond to the px-wave pair potential and the terms propor-tional to sin␾, i.e., fASL共␾兲 and fASR共␾兲, correspond to the

py-wave pair potential. In the actual calculation for

P / DN/ I / DN/ P junctions, functions fASL共␾兲 and fASR共␾兲 vanish after averaging over angle␾. This is consistent with our previous results that the pair potential with py-wave sym-metry does not contribute to the proximity effect.21,22

Next we formulate the junction model and basic equations starting from the d-wave case. We consider ballistic DID and

D / DN/ I / DN/ D junctions. The DN has a resistance Rdand a length L much larger than the mean free path. The DN/D interfaces located at x = ± L have the resistance Rb

⬘

, while the DN/I interface at x = 0 has the resistance Rb. We model infi-nitely narrow insulating barriers by the delta function U共x兲 = H

⬘

␦共x+L兲+H␦共x兲+H

⬘

␦共x−L兲. The resulting transparen-cies of the junctions Tm and Tm

⬘

are given by Tm = 4 cos2_␾_/_{共4 cos}2_␾_{+ Z}2_{兲 and T}

m

⬘

= 4 cos2_␾_/_{共4 cos}2_␾_{+ Z}

_⬘

2_兲, where Z = 2H /vF and Z

⬘

= 2H

⬘

/vF are dimensionless con-stants共we take ប=kB= 1 in this paper兲 and vF is Fermi ve-locity. Below we assume Z1. The schematic illustration of the models is shown in Fig. 1. Here, ␣ and ␤ denote the angles between the normal to the interface and the crystal axis of the left and right d-wave共or p-wave兲

superconduct-ors, respectively. The lobe direction of the pair potential and the direction of the crystal axis are chosen to be the same. The pair potential along the quasiparticle trajectory with the injection angle␾ is given by ⌬L=⌬ cos关2共␾−␣兲兴exp共−i⌿兲 and⌬R=⌬ cos关2共␾−␤兲兴 for the left and the right supercon-ductor, respectively. For ballistic junctions, we use a similar model without DN and calculate the Josephson current fol-lowing Ref.9.

We parametrize the quasiclassical Green functions G and

F with a function⌽_␻:23,24 G_␻=

_冑

␻ ␻2₊_⌽ ␻⌽−␻* , F_␻=

_冑

⌽␻ ␻2₊_⌽ ␻⌽−␻* , 共1兲

where␻is the Matsubara frequency. Then the Usadel equa-tion reads25 ␰2␲TC ␻G_␻ ⳵ ⳵x

冉

G␻ 2 ⳵ ⳵x⌽␻

冊

−⌽␻= 0 共2兲

with the coherence length ␰=

冑

D / 2␲TC, the diffusion con-stant D and the transition temperature TC. We solve the Us-adel equation with the boundary conditions in Ref.20 at x = ± L and those in Ref.26at x = 0.

The Josephson current is given by

eIR ␲TC = i RTL 2RdTC

兺

_␻ G_␻2 ␻2

冉

⌽␻ ⳵ ⳵x⌽−␻ * ₋_⌽ −␻ * ⳵ ⳵x⌽␻

冊

, 共3兲

⬘

and

replacing 2␣, 2␤, and 2␾ with ␣, ␤, and ␾ in the above equations.

Next we consider the D / DN/ I / DN/ D junctions. Near TC, we can linearize the Usadel equation as follows:

␰2⳵ 2 ⳵x2⌽j␻− ␻ ␲TC ⌽j␻= 0, 共7兲

where j共=1,2兲 denotes the left or right DN. Similarly the boundary conditions at x = −L, x = 0, and x = L are reduced to

SR formula is applicable as well.

In order to understand the above results qualitatively, let us discuss the symmetry of the pair potential by the inversion operation around the plane perpendicular to the interface. As shown in Fig. 4, dx2_−y2 wave and p_x wave are symmetric while dxywave and pywave are antisymmetric by this op-eration. Only the symmetric pair wave function is taken into

FIG. 2.共Color online兲 Maximum Josephson current for d-wave junctions.共a兲 and 共b兲 DID junctions. 共c兲 and 共d兲 D/DN/I/DN/D junctions. Solid lines in共c兲, 共d兲 are proportional to cos 2␣ cos 2␤.

FIG. 3.共Color online兲 Maximum Josephson current for p-wave junctions.共a兲 and 共b兲 PIP junctions. 共c兲 and 共d兲 P/DN/I/DN/ P junctions with solid lines that are proportional to cos␣ cos ␤.

ANGULAR DEPENDENCE OF JOSEPHSON CURRENTS IN… PHYSICAL REVIEW B 75, 020502共R兲共2007兲 RAPID COMMUNICATIONS

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account in the SR formula. Applying this idea to an arbitrary unconventional superconductor with angular momentum l, one can argue that the Josephson current is proportional to cos l␣cos l␤. It is straightforward to get this result just by

replacing cos 2␾, cos 2␣, and cos 2␤ with cos l␾, cos l␣, and cos l␤ in Eq.共11兲, respectively.

In summary, we have studied the validity and the physical meaning of the Sigrist-Rice formula in d-wave supercon-ductor junctions. According to the SR formula, the amplitude of the maximum Josephson current is proportional to cos 2␣cos 2␤. Although this formula is not applicable to the ballistic junctions, it works well for D/DN/I/DN/D junctions where the DN regions are located between the d-wave super-conductor and the insulator. We have also shown that in

P/DN/I/DN/P junctions, the Josephson current is

propor-tional to cos␣cos␤. The obtained results may help to obtain information about pairing symmetry in experiments with un-conventional superconducting junctions.

T. Y. acknowledges support by the JSPS. This work is supported by Grant-in-Aid for Scientific Research on Priority Area “Novel Quantum Phenomena Specific to Anisotropic Superconductivity”共Grant No. 17071007兲 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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The␾ dependences are given by cos 2␾, cos ␾, sin 2␾, and sin ␾.

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