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doi:10.1016/j.m

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Microelectronics Journal 39 (2008) 1231–1232

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Crossed Andreev reflection in superconducting junctions

Shirley Gomeza,�, William J. Herreraa, Jesus V. Ninob, Diego A. Manjarresa

aDepartamento de Fısica, Universidad Nacional de Colombia, Bogota, ColombiabDivision de Ciencias Basicas, Universidad El Bosque, Bogota, Colombia

Available online 17 March 2008

Abstract

In this work crossed Andreev reflection (CAR) and elastic cotunneling (EC) are studied for junctions ðN1ISIN2Þ, where N1 and N2 are

normal metals, S is a high T c superconductor and I is an insulator. This study is carried out based on the analytical solutions of

Bogoliubov de Gennes equations for anisotropic superconductors. The influence of different pair potential symmetries on the CAR and

crossed conductance is analyzed. We show that CAR and EC are higher in dx2�y2 symmetry than in s symmetry. In the case of normal

electrodes without magnetization, EC is the predominantly process for dx2�y2 symmetry, while in s symmetry, both processes decay with

the same amplitude.

r 2008 Elsevier Ltd. All rights reserved.

PACS: 74.20.Rp; 74.45.þc; 74.50.þr; 74.81.�g

Keywords: Superconductivity; Crossed Andreev reflection; Elastic cotunneling; Crossed conductance

1. Introduction

When an electron is injected from a normal regionto an interface where the pair potential is inhomo-geneous, the electron can be reflected as a hole (Andreevreflection—AR) and a Cooper Pair is induced in thesuperconducting region. Now let us consider an N1ISIN2

junction where N1ð2Þ are normal metals and I is aninsulating barrier; an electron that comes from N1 tothe superconductor can be reflected as a hole in the N2

(crossed Andreev reflection—CAR) or as an electron in N2

(elastic cotunneling—EC), see inset (a) of Fig. 1. Thepossibility to observe CAR dominating the electric trans-port in multi-terminals has called attention in the last yearsdue to the possibility of using it as a source of entangledelectron pairs [1–5]. This system has been studiedexperimentally using s wave superconductors [3], wherethe EC or CAR prevails depending of the voltage appliedon a normal electrode. On the other hand, the AR is a verysensitive tool for the study of electrical transport in highcritical temperature (HTc) superconductors, since it isaffected by the symmetry of the pair potential [6]. Recently,

e front matter r 2008 Elsevier Ltd. All rights reserved.

ejo.2008.01.074

ing author.

ess: sgomezp@unal.edu.co (S. Gomez).

indirect evidences of CAR in HTc superconductors havebeen found [7] but there are no previous works about theinfluence of the pair potential symmetry on the CAR andEC in these systems. In this work we make a first approachto the study of this process in HTc superconductors and weanalyze the variation of the crossed differential conduc-tance dI2=dV1 with the width of the superconductorregion.

2. CAR and differential conductance

Taking into account that an electron comes from N1

to the N1IS interface and solving analytically the Bogoliu-bov de Gennes equations for an N1ISIN2 system, we findthe CAR—PCAR—and EC—PEC—probabilities. Theseprobabilities are functions of the energy E and the yangle with respect to the interface of the incident electron,the width a of the superconducting region, and thesymmetry of the pair potential DðyÞ. The crossed differ-ential conductance is proportional to the difference ofthese probabilities [5], s21 / PCAR � PEC, and can be posi-tive or negative depending of the process that exhibits thehigher probability. When abx0 and EojDðyÞj, PCARðECÞðyÞdecays in an exponential manner, expð�2a=px0Þ, where

ARTICLE IN PRESS

7 91 50

PEC

PCAR

2.5X10-4

7.5X10-4

3

PEC

PCAR

1X10-5

a/ξo

1 3 5 7 9

(b)

3X10-5

NI NIIS

a0 x

I I

e e

|Δ|

E

NI NIIS

a0 x

I I

e h

|Δ|

E(a)

a/ξo

Fig. 1. EC and CAR probabilities as a function of the width of the

superconducting region for dx2�y2 symmetry. Both probabilities are

normalized to the tunneling probability when the width of the super-

conducting region tends to zero. Inset (a) NISIN system illustrating the

two types of reflection processes CAR (right) and EC (left). Inset (b) EC

and CAR probabilities as functions of the width of the superconducting

region for s symmetry.

1

-0.04

-0.02

0

3 5 7 9

-0.02

-0.01

0

0.01

0.02

3 5 7 9a/ξo

1

σ 21

σ 21

a/ξo

Fig. 2. Total crossed differential conductance s21 as a function of the

width of the superconducting region for dx2�y2

symmetry. In the inset is

plotted the differential conductance when the normal terminals have

antiparallel magnetizations. Both conductances are normalized to the

conductance when the width of the superconducting region tends to zero.

S. Gomez et al. / Microelectronics Journal 39 (2008) 1231–12321232

xðyÞ ¼ x0D0=ðjDðyÞj2 � E2Þ1=2 is an effective coherence length,

larger than the BCS coherence length x0, for y40.In order to compute s21 we must calculate the integral in

y, since PCAR and PEC have contributions of quasiparticlesthat undergo a pair potential DðyÞðxðyÞ4x0Þ, the decreaseof these probabilities with a is slower for d

x2�y2symmetry

than for s symmetry, as is shown in Fig. 1. In this figure wealso see that PEC is in average higher than PCAR for dx2�y2

symmetry and therefore s21 is negative, see Fig. 2.For s symmetry both processes decay with the same

amplitude and PCAR and PEC are out of phase whereas fordx2�y2 symmetry these probabilities are in phase. In orderto obtain the predominant behavior of CAR on EC wemay consider the normal metals with antiparallel magne-tizations [5], which leads to PCAR4PEC. In the inset ofFig. 2 s12 is plotted as a function of a for zero voltage, anexponential decay is observed, but the average value ispositive, being CAR predominant.

In conclusion we have found that CAR and EC arehigher in dx2�y2 symmetry than in s symmetry. We showthat in the case of normal electrodes without magnetizationEC is the predominant process for d

x2�y2symmetry. When

we use electrodes with antiparallel magnetizations the CARdominate over EC and s21 is positive. These results can berelevant in similar experiments to the type carried out in [3]for the study of CAR in High Tc superconductors.

Acknowledgments

The authors thank the Vicerrectorıa de investigacion ofthe Universidad Nacional de Colombia for the financialsupport.

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