Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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262 S. L. Prischepa

integral surface effects in multilayers. Theoretical work is in progress in

order to get a deeper insight into the problem.

Conclusion

In conclusion, we have performed a systematic study of the influence of the

finite dimensions of the S/N proximity coupled multilayers on the H-T

phase diagram and angular dependences of the upper critical magnetic field.

Our experiments were performed with two different systems, namely

Nb/Cu and Nb/Pd, with different ratios between d

and d

and different N

values (i.e. symmetry of the samples). The observed H

c2||

(T) dependences

differ with respect to the position of the geometrical symmetry plane. For

samples with the symmetry plane located in the middle of the N layer the

measured H

c2||

(T) curves were typical for S/N multilayers, presenting the

well known 3D-2D crossover at a certain T*<T

. For samples with the

symmetry plane in the middle of S layer the H

c2||

values were higher in the

region T→T

, revealing the characteristic features of 2D behavior up to T

The systematic study of the angular dependences reveal that for samples

with central S layer the H

(Θ) curves always show the 2D-like behavior in

agreement with the H

c2||

(T) results. At the same time for samples with the

symmetry plane in N layer the H

(Θ) behavior was more complicated,

most likely due to the influence of a complex interface effect on the spatial

distribution of the superconducting wave function.

Acknowledgments

This work has been partially supported by the national Program

"Nanoelectronics" under the grant 01-3109. Valuable discussions with

V.N. Kushnir and C. Attanasio are gratefully acknowledged. The

experimental work has been done with assistance of C. Cirillo and

M. Salvato.

References

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Phys 69:39

Andreev Reflection in Ballistic Superconductor-

Ferromagnet Contacts

L. R. Tagirov

1,2

, B. P. Vodopyanov

Kazan State University, 420008, Kazan, Russia

Kazan Physico-Technical Institute of RAS, 420029 Kazan, Russia

Abstract: On the basis of Eilenberger-type equations of superconductivity for

metals with exchange-split conduction band we derive an original

expression for the Andreev conductance of superconductor-

ferromagnet point contacts. We give particular formulas for different

relations between Fermi momenta of contacting metals, and calculate

the Andreev conductance as a function of exchange splitting of the

ferromagnet conduction band. From comparison with experimental

data on point contact Andreev reflection spectroscopy we estimate

the conduction band spin polarization for a series of ferromagnets in

contact with superconductors.

Keywords: Quasi-classical theory of superconductivity, Andreev reflection,

ferromagnet-superconductor point contacts

Introduction

At low temperatures, an electric current from a normal metal enters

through a contact in a superconductor by means of Andreev reflection [1].

In the process of Andreev reflection an electron coming to the normal

metal/superconductor interface is reflected back into the normal metal as a

hole with the opposite spin, and the formed Cooper pair moves through the

superconductor transferring a charge

2e . This results in doubling of the

differential conductance of a pure N/S nanocontact which has been

demonstrated theoretically in Ref. [2] (BTK) based on the solution of the

Bogoliubov equations. In Ref. [3] it was emphasized that the Andreev

265

R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 265–275.

266 L. R. Tagirov, B. P. Vodopyanov

reflection in ferromagnet/superconductor (F/S) contacts is suppressed as the

spin polarization of the ferromagnet conduction band sets up. The general

understanding of this suppression is by a reduction of the efficiency of the

Andreev reflection process associated with the decrease of the number of

conducting channels in the minority spin-subband (the subband with the

lower value of the Fermi momentum). It has been proposed to use the

suppression of Andreev reflection in F/S contacts to determine the spin

polarization of the conduction band of ferromagnets (Andreev spectroscopy

of ferromagnets) [4-9]. Experimental data were interpreted on a

phenomenological basis [4, 5, 7]. The BTK theory was generalized and

applied to F/S point contacts in the theoretical works [8-13]. However, the

general requirement that the Fermi momenta of contacting ferromagnetic

and superconducting metals are not the same has not been taken into

account. The number of experiments on Andreev spectroscopy of diverse

ferromagnets grows [14-20], what demands an adequate theoretical

description. In this paper we derive particular formulas for the Andreev

conductance of a superconductor in a point contact with different types of

ferromagnets. From comparison with experiments on Andreev spectroscopy

we estimate the polarizations of the conduction bands of ferromagnetic

metals.

Equations of Superconductivity and General Boundary

Conditions for Superconductor-Ferromagnet Contacts

The equations for the equilibrium thermodynamic Green functions (GF)

read:

l ll l

ˆˆˆˆ

n( ) ( ) [ ] 0

ˆˆ

n( ) ( ) [ ] 0

ˆˆ

xj xj

gggKg

GGGKG









 

 

(1)

The quantities which enter the above equation are defined as follows:

ˆˆ

()()2

ˆˆˆˆ

xj xj x x

xj xj

Kii i

a b ab ba



 '6    

 r

(2)

Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts 267

() ()

ˆˆ

(2 )

ic u



 _ _ 

(3)

icu

WW S

  ! _ _ 

(4)

where the functions

and

G are the quasiclassical GF of incident and

reflected quasiparticles,

are the Pauli matrices,

is the spin index,

(2 1)

 is the Matsubara frequency, ' is the order parameter,

is the Fermi momentum of the ferromagnet spin-subbands,

p is the Fermi

momentum of a superconductor,

u is the potential of interaction of

electrons and impurities,

c is the concentration of impurities,

is the

mean free time of electrons in a superconductor, and brackets mean

averaging over the solid angle:

/4d

! :

. Hereafter we use the unit

system in which

1c = , so that we do not distinguish momentum and

wave number, for example. We assume that the F/S interface coincides with

the plane 0x . Then, ()x r ȡ , and ()

z ȡ is the coordinate within the

contact plane,

v the matrix of the Fermi velocities projections to the x

direction, and the other notations are standard.

The above Eilenberger-type equations for the metal with the exchange-

field-split conduction band had been derived for the first time in [21, 22].

They are valid for arbitrary band splitting and arbitrary spin-dependent

mean free paths within the quasiclassic approximation. The system of

Eqs. (1) must be supplemented with boundary conditions at the F/S

interface (see details in Refs. [22, 23]):

















ˆˆ

 

(5)

12 34 34

12 34 12

ˆˆ ˆˆ ˆˆ

ˆˆˆˆ

ˆˆ ˆˆ ˆˆ

ˆˆˆˆ

aaa a

gggg

bb bb bb

gggg

bb bb bb

 

 

 

 

(6)

The

b matrices are given by

268 L. R. Tagirov, B. P. Vodopyanov

ll ll

ˆˆ

ˆˆ ˆˆ

ˆˆ

ˆˆ ˆˆ

ss ss

gg gg

GG GG

gg gg

GG GG

 

 

 

 

 

(7)

where

()

and

()sa

G are the matrices, which are symmetric (s) and

antisymmetric (a) with respect to the variable

p :

l ll

()

ˆˆˆ

ggg

GGG

!

r r

ªº

¬¼

(8)

Note that only the diagonal part of the functions is continuous at the F/S

interface (Eq. (5)). These BCs take into account explicitly the spin-

dependence of F/S interface.

General Expression for the Andreev Conductance of F/S

Point Contacts

We start with the equation for the current I in the linear approximation

with respect to the electric field

(00)

E E . The current is calculated on

the ferromagnet side of the contact at

0xo :

111 1

lim

24cosh(2)

()() ( )

ITr

mxx T

dG E G

HWH

§·

¨¸

©¹

f





ww 

u 

rrrr rr

(9)

Here,

()RA

G is the retarded (advanced) GF, which is obtained from the

temperature dependent GFs, Eqs. (1), by substituting

r for

. After a

Fourier transformation with respect to 'ȡȡ coordinate, we obtain the

ballistic conductance



of the F/S point contact:



ll ll

16 cosh ( 2 )

ˆˆ ˆˆ

FS z

RA RA

zz z z

ss aa

Ae d

GTr

gg gg

GG GG

WWW W



f





ªº

u  

¬¼

(10)

Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts 269

Now, we have to solve the first of Eqs. (1) with the BCs given by

Eqs. (5)-(7). When

is independent of ȡ the solution to Eq. (1) takes the

form

ˆˆ

sgn( ) sgn( )

ˆˆ

()

jx jx

Kx Kx

Fj j

eCe C



p

(11)

The matrices

C represent the values of GF

at large distances from the

F/S interface, and for the superconductor we have

§·

¨¸





¨¸

©¹

'

§·



¨¸

'



©¹

'

(12)

In the ferromagnet

must tend to

n( )

at x of, and in the

superconductor

must tend to

at x of. Performing the matrix

multiplication in Eq. (11), we find that these conditions are fulfilled if the

following relationships are satisfied:

ˆˆ ˆ

ˆˆ

() () s

n( )( 1) ( )

j j Fj j jx j Fj

CC p C

  pp p

(13)

From these relationships, it follows that

ˆˆˆ ˆ

(1)

jja ja

CCC C

  

(14)

Passing in Eq. (14) to the functions

, substituting them into the system

of BCs given by Eqs. (6), and solving the resulting equations in the linear

approximation with respect to

ˆˆˆ

12[ ]

aaa

CCC

 r , we find,

(1 )

2[1 (1 ) ]

fRRDD

RR RR g

DD D D

DD DD



r

 



(15)

Now, from Eq. (10) and making use of Eqs. (5) and (15), we find the

Andreev conductance

G of the F/S point contact



(0)

4cosh(2)

(1 ) 4

AFS

GGV Ae

dDD

RR RR



pn pn



u

'

(16)

270 L. R. Tagirov, B. P. Vodopyanov

The experimental data [4-8] are given in the normalized form. The

normalization is, in fact, the conductance at high voltage, V>>ǻ, which is

equal to the conductance between a ferromagnet and a normal metal. This

conductance can be calculated as follows:

(0) sin(2)

sin(2 )

Ae p

GV d D

§·

¨¸

©¹



ppp

§·

¨¸

©¹

nnn



(17)

The equations (16), (17) are valid for an arbitrary transmission

coefficient

. They depend on the relationship between the Fermi

momenta of the ferromagnet spin-subbands

, and the Fermi

momentum of the superconductor,

p . Upon integration on the angles the

conservation of the component of the momentum parallel to the interface

plane must be obeyed:

sin sin sin ,

FFSS

pppp

TTT

ppnn

(18)

which determines the critical angle ș

in equation (17).

Andreev Conductance for Various Relations Between

Fermi Momenta of Contacting Metals

We analyze three possible relations between the Fermi momenta of

contacting metals: 1)

SFF

ppp

, 2)

ppp

, and finally, 3)

ppp

.

Case 1),

SFF

ppp

:

The expression for

G takes the form:

Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts 271



(1)

cosh / 2

sin(2 )

(1 ) 4

Ae p

RRRx

pn pn

u



(19)

The corresponding normal state conductance is given by





(1)

(0) sin(2)( ).

FN S S

Ae p

GV d DD





(20)

For particular calculations we use model expressions for the transmission

coefficients corresponding to the direct contact between metals:

() ()

pp pp





(21)

With these transmission coefficients

(1)



can be calculated analytically:

(1)

222

(2 ) (2 )

6(1)(1)

FFFF

Ae p

GGGG

SG G

§·

¨¸

©¹

nnpp

¨¸



¨¸

©¹







(22)

where,

 and

.

Case 2):

ppp

.

The expression for

G takes the form:



(2)

cosh / 2

sin(2 )

(1 ) 4

Ae p

RRRx

§·

¨¸

©¹

pn pn

u



(23)

272 L. R. Tagirov, B. P. Vodopyanov

The corresponding normal state conductance is given by

(2)

(0) sin(2)

sin(2 ) .

Ae p

GV d D

§·



¨¸

©¹



ppp

§·



¨¸

©¹



(24)

It can be evaluated with the model transmission coefficients as follows:

(2)

22 2

(2 ) ( ) (2 )

6(1)(1)

FFF F

Ae p

GGG G

SG G

§·

¨¸

©¹

nnp p

¨¸



¨¸

©¹







(25)

where

 and

.

Case 3),

ppp

.

The expression for

G takes the form:



(3)

cosh / 2

sin(2 )

(1 ) 4

Ae p

RRRx

§·

¨¸

©¹

pn pn

u



(26)

The corresponding normal state conductance is given by

(3)

(0) sin(2)

sin(2 ) .

Ae p

GV d D

§·



¨¸

©¹



ppp

§·



¨¸

©¹



(27)

It can be evaluated analytically as follows:

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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