научная статья по теме JOSEPHSON EFFECT IN SIFS TUNNEL JUNCTIONS WITH DOMAIN WALLS IN WEAK Физика

Текст научной статьи на тему «JOSEPHSON EFFECT IN SIFS TUNNEL JUNCTIONS WITH DOMAIN WALLS IN WEAK»

Pis'ma v ZhETF, vol. 101, iss. 11, pp. 863-868

© 2015 June 10

Josephson effect in SIFS tunnel junctions with domain walls in weak

link region

In memory of V.F.Gantmakher

S. V. Bakurskiy+*, A. A. Goluhov*x, N. V. Klenov+, M. Yu. Kupriyanov*oV1\ 1.1. Soloviev° + Physics Department, Lornonosov MSU, 119991 Moscow, Russia * Moscow Institute of Physics and Technology, 141700 Dolgoprudniy, Russia x Faculty of Science and Technology and MESA+, Institute for Nanotechnology, University of Twente, 7522 NB Enschede, Netherlands ° Skobeltsyn Institute of Nuclear Physics, Lornonosov MSU, 119991 Moscow, Russia vInstitute of Physics, Kazan (Volga region) Federal University, 420008 Kazan, Russia

Submitted 5 May 2015

We study theoretically the properties of SIFS type Josephson junctions composed of two superconducting (S) electrodes separated by an insulating layer (I) and a ferromagnetic (F) film consisting of periodic magnetic domains structure with antiparallel magnetization directions in neighboring domains. The two-dimensional problem in the weak link area is solved analytically in the framework of the linearized quasiclassieal Usadel equations. Based on this solution, the spatial distributions of the critical current density, Jc, in the domains and critical current, Ic, of SIFS structures are calculated as a function of domain wall parameters, as well as the thickness, and the width, W, of the domains. We demonstrate that Ic{cIf,W) dependencies exhibit damped oscillations with the ratio of the decay length, £i, and oscillation period, £2, being a function of the parameters of the domains, and this ratio may take any value from zero to unity. Thus, we propose a new physical mechanism that may explain the essential difference between £1 and £2 observed experimentally in various types of SFS Josephson junctions.

DOI: 10.7868/S0370274X15110090

It is well known that properties of Josephson structures with ferromagnetic (F) material in a weak link region depends on relation between the complex decay length, £ (£_1 = and geometrical parameters

of these junctions [1-3]. If F metal is in the dirty limit and exchange energy, H, sufficiently exceeds the critical temperature of superconducting (S) electrodes, nTc, then from Usadel equations it follows that £1 « £2 • However, it was demonstrated experimentally [4-12] that there could be a noticeable difference between £1 and £2. Previously the difference has been attributed either to the presence of strong paramagnetic scattering in the F layer [7], or to violation of the dirty limit conditions in ferromagnetic material [12, 13]. However, application of the first of the mechanisms for the experimental data interpretation requires the existence of unreasonably strong paramagnetic scattering in the weak link material [7]. The relation between an electron mean free, £, and £1, £2 in typical experimental situation is also closer

e-mail: mkupr@pn.sinp.msu.ru

to the dirty limit conditions, £ < £1, £2 rather than to the clean one.

In this article we prove that the existence of a ferromagnetic domain walls in F layer can also lead to appearance of substantial differences between £1 and £2 even in the absence of strong scattering by paramagnetic impurities, and under the fulfilment of the dirty limit conditions in the F material.

Model. Consider multilayered SIFS structure presented in Fig. 1. It consists of superconductor electrode (S), insulator (I), and FS bilayer as an upper electrode. We assume that the F film has a thickness, dp, and that it subdivides into domain structure with antiparallel direction of magnetization vector in the neighboring domains. The width of the domains is W and they separated by atomically sharp domain walls oriented perpendicular to SF interfaces. Due to periodicity of the structures we, without any loss of generality, can perform our analysis within its half of the period, that is from —W/2 to W/2. This element is enlarged in Fig. 1. It consists of two halves of domains and domain wall separating them.

-W/2

» 1

F!® H

I

S

W/2

ix

Fig. 1. Geometry of the considered SIFS Josephson junction and its enlarged part, which includes two halves of domains and domain wall separating them. The insulating barrier I has a small transparency (shown by a blue line)

We will suppose that the condition of dirty limit is fulfilled for all metals and that effective electron-phonon coupling constant is zero in F material. We will assume further that either temperature T is close to the critical temperature of superconducting electrodes To or the suppression parameters ^bs — RbsAbn / Pf£,f at SF interface is large enough to permit the use of the linearized Usadel equations in F film of the structure. We will characterize the FF interface (domain wall) by the suppression parameter 7 = 1, and the suppression parameter 7bf = Rbf-Abf/Pf^f, which can take any value. Here Rbs,Rbf and Abn,Abf are the resistances and areas of the SF and FF interfaces, and S,f = {Df/2-kTc)1!2 are the decay lengths of S, F materials, while ps and pF are their resistivities, DF is diffusion coefficient in the F metal.

Under the above conditions the proximity problem in the SF part of SIFS junction (0 < x < dF) reduces to solution of the set of linearized Usadel equations [13,14]

d2 d2 1 ~ TU

w + w}FF-Q+FF = 0'0-y-T> (1)

d2 d2 1 ~ TU

^ + W\FF-Ü.FF = 0,-Y<y<0, (2)

where Q = w/ttTg,0± = \Q\ ± itegn(w), h = H/ttTc, H is exchange energy of ferromagnetic material, u> = = :jyT(2ii+ 1) are Matsubara frequencies. The spatial coordinates in (1), (2) are normalized on decay length . To write these equations we have chosen the x and y axis in the directions perpendicular and parallel to the

SF plane and put the origin in the middle of SF interface to the point, which belongs to the domain wall (see Fig-1).

Eqs. (1), (2) must be supplemented by the boundary conditions [15]. They have the form

c> A TU W

Ibs-z-ïf = -G0 —, x = 0, —— < y < —-, ox uj 2 2

c> TU TU

-FF = 0,x = dF, -— <y<—.

(3)

At FF interface (y = 0, 0 < x < dF) and in the middle of the domains (y = ±TU/2, 0 < x < dF) we also have

7BF-j^FF(x, +0) = Ff(x, +0) - Ff(X, -0), (4) ±Ff(x,+0) = ^-Ff(X,-0),

o o

— Ff(x, W/2) = -Q-yFF{x, —TU/2) = 0. (5)

Here TU is the width of the domains, Go = uj/\/uj2 + A2, A is the modulus of the order parameter of superconducting electrodes. The critical current density, Jc, of SIFS Josephson junction is determined by s-wave superconducting correlations at IF interface, which is even function of the Matsubara frequencies

T

cJqRN _ 2-kTc ~ WTC

(0

w>0

where $(y) = [FF+u(dF, y)+FF-u{dF, y)}/2, while the full critical current, Ic, is the result of integration of Jc(y) over width of the junction.

T

e!cRN __

2TtTc ~ WTC

E

w>0

GnA

rW/ 2

-W/2

$(y)dy. (7)

Here, Rn is the normal junction resistance.

Solution of Usadel equations in FS electrode.

Solution of two-dimensional boundary value problem (l)-(5) in the F layer (0 < x < dF) is convenient to find in the form of the Fourier series expansion

Ff(x, y) = An(y) cos

n=—oo oo

Ff= Bn(y)

mix W

-J^, 0 < y < y,

mix TU

COS —;—,--< y < 0,

where

4 , X Z

A„(y) = — + «

cosh

Bn(y)

n z

b„ cosh

dF

TU TU

Q— I ?/ H TT

(8) (9)

(10)

(H)

and coefficients an and br,

9 9

lq+ q- J

bn =

l

l

9 9

Lq+ q-J

Zq-S-S '

Zq+S+ 5

q± = \jn±+[ g

Z =

AGn

(12) (13)

7 BsdFu

are determined from boundary conditions (4). Here the coefficients S, C±, and S± are defined by expressions

S = q-q+~{BFS+S- + q-C+S_ + q+S+C-, (14)

(15)

C± = cosh (^r^j , s± = sinh (^T^j ■

Taking into account the symmetry relation q~(—u>) = = q+(oj) for s-wave superconducting component in the F layer at x = dp it is easy to get

$(y>0)=- ]T (-1)

l

l

o n^ o

L q+ q-

z

*(y<0)=T E (-1)

n= — oo

" 1 1

(5 ?

(16)

' 1 1 "

"ë. T_ ?

(17)

r Q ,, 2 yTW\ ( 2 yTW S±=q-S- cosh q+--- -çr+S+cosh q_---

Finally for the critical current from (7), (16), and (17) we have

S(u>)

e!cRN

2tvTc

E (-Dr

T

EZGoA

2WTC ^• uj

(18)

W W 2SS+(q2_ -q\)2

9 1 9

L q+ q~

It is seen that the critical current can be represented as the sum of two terms. The first is the contributions from individual domains separated by fully opaque FF wall

e!clRN T 2 ttTc

Gl A2

Tc — 7 bs^

Re-

1

'f2+ sinh ( dp\/Q

(19)

while the second eIC2RN Ah?T

2 ttTc

Z^ „„„„.a

WdpTc ^bsu

w>(J n= — oo

3 JJj

q+q

(20)

gives the contribution from the domain wall. Here Re(a) denotes the real part of a.

Expression (19) reproduces the well-known result previously obtained for single-domain SIFS structures

[16-18] thereby demonstrating the independence of the critical current on the orientation of the domains magnetization vectors, if they are collinear oriented and the FF interface is fully opaque for electrons.

Limit of large 7bf- For large values of suppression parameter 7bf ^ max {l, (Wq^)^1} expression (20) transforms to

eIC2RN _ 4h2T 2tvTc

g2a2

E

(-ir

„4

WdpTc 1BF1BSU* n^oo 1+q-

(21)

The sum over n in Eq. (21) can be calculated analytically using the theory of residues

eIC2RN _ 2hT 2 nTc

GgA2

¿>1,

WTC ^ ^BFlBS^

(22)

S\= Re

Q

3/2

dpyCl-i

cosh dp\/Q_|

sinh dp\/Q_|

It is seen that Ic2 is vanished as (7bfW)^1 with increase of 7bfW product and scales on the same characteristic lengths £1, £2 as the critical current for singledomain SIFS structures (19).

Limit of small 7bf- In the opposite limit, 7bf <C max {l, (Wq^)^1}, we have

cIciRN 2 ttTc

E

8h2T _ G2A2

WdFTc ^ 7bsw

(23)

^ <?+£ (q-C+S- +q+S+C-y

It is seen that in full agreement with the result obtained in [19] in the considered limit of large domain width, W > Re(q±),

eIC2RN _ 4h2T 2tvTc

j^G2A2

WdpTc ^ 7bswz

E

c-ir

q+q- (q- +q+) (24)

contribution to the critical current from domain wall region falls as Wand decays in the scale of £1.

Limit of small domain width. In the opposite case, W <C Re(</±), presentation of the critical current as a sum of Ici and Ic2 is not physically reasonable and

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