Magnetoresistance due to scattering on the flux lines in a superconductor-insulator-semiconductor multilayer

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Magnetoresistance due to scattering on the flux lines in a superconductor-insulator-semiconductor multilayer

D. Kuptsov, M. Moiseev

To cite this version:

D. Kuptsov, M. Moiseev. Magnetoresistance due to scattering on the flux lines in a superconductor- insulator-semiconductor multilayer. Journal de Physique I, EDP Sciences, 1991, 1 (8), pp.1165-1171.

�10.1051/jp1:1991198�. �jpa-00246402�

J. Phys. 1Fraucel (1991) l165-l171 AOUTI991, _PAGE l165

Classification Physics Abstracts

74.20 74.70J 74.75

Magnetoresistance due to scattering

_on

the flux fines in

_a

superconductor-insulator-semiconductor multilayer

D. A.

Kuptsov

and M. Yu. Moiseev

Physics Depa~ment,

Moscow State

University,

Moscow 117234, U-S-S-R-

(Received

4 March 1991, accepted 25

April

1991)

Abstract. The solution of the electron scattering

problem

in the field of _an Abrikosov vo~ex is obtained. The flux line lattice, which arises in the

superconductor

film of _a

superconductor-

insulator-semiconductor

multilayer

in extemal

magnetic field,

creates a modulated field pattern in the semiconductor. The

inhomogeneity

of the field _can lead to additional

scattering

of the

charge

carriers in the

semiconducting

material and, as a result, to a correction to

magnetoresistance.

This

correction is calculated for the _case of _a

large separation

between the vortices.

1. Inwoducfion.

During

the

past

^few years the

properties

of the

superconductor,insulator-semiconductor multilayers

have attracted much attention. If the extemal

magnetic

field is

applied perpendicular

to the

multilayer

^{the flux} lattice in the

superconductor produces

a

highly inhomogeneous magnetic

field which _can

change

the transport

properties [1,2]

and the energy spectrum

[3,4]

of the

charge

carriers in the semiconductor

layer.

Under certain

conditions Abrikosov vortices _can

play

the role of scatterers for the

charge

carriers in the semiconductor. Electron

scattering by

the flux lines _can result in _{an extra} contribution to the

resistivity

of

semiconducting

material which has been

recently

observed in the

Pb/GaAlAs/GaAs

structure

[5] (1).

In this _{paper we} present a solution for the

problem

of the electron

scattering by

a

magnetic

field of the Abrikosov vortex and _an estimate for the correction to the

resistivity

of the

semiconducting layer

due to the presence of the flux lines.

We take into account both the interaction of the electron

magnetic

moment

p _~ = 9 _x 10~ ^~'

~@

^{with the vortex}

magnetic

field and the effect of the vector

potential

A upon the

phase

of the electron _wave function. Our

analysis

shows that

usually

the latter

(') Although

bulk Pb is _a type ^I

superconductor,

its thin films pass into the mixed rather than intermediate state in

perpendicular magnetic

field, cf. [7].

1166 JOURN~L DE PHYSIQUE I K 8

mechanism

gives

the main contribution into the

scattering amplitude.

So the

problem

of the electron

scattering by

the vortex

magnetic

field resembles the well-known Aharonov-Bohm effect

[6].

Note in this connection that

recently

the shift of the

phase

of the electron _wave

function caused

by

the vector

potential

has been observed

experimentally using

the method of the electron

holography [7j.

At the _same time the influence of

spin

effects _can be

considerably

enhanced in

superconductor-insulator-semimagnetic

semiconductor structures. The

exchange

field of the

magnetic

ions in the semiconductor material may result in effective

magnetic

field

« enchancement _», the _« enhancement

» factor

being

as

high

as 300

[8].

and

spin

effects will

prevail.

In this _{case one can} expect a much

larger

correction to the

resistivity

than in

nonmagnetic

semiconductor.

As far _as the _vortex lattice is

always

distorted

by pinning

forces and the distance

d between the vortices is much

larger

than the Fermi

wavelength

we need _not take _into

account the interference of the electron _waves scattered

by

different vortices. In _a small

external field the vortices _are well

separated

^{and the} distance d is greater ^{than the London}

penetration depth A~.

Therefore _we may consider the electron

scattering

_on a separate

~ortex. In this _case the correction _to the

resistivity

AR due _to

scattering by

the _{vortices is}

proportional

_to their concentration, I-e- to the external field H~~j. ^Below we shall consider

just

this situation. In [5] ^{the linear}

relationship

AR

H~~j)

_was observed in the fields up to 50 G. In greater ^{fields the flux lattice becomes} more dense and the

dependence AR(H~~j)

de,>fates from

linearity

^{towards smaller} AR.

We don't consider here the weak localization effect in

inhomogeneous magnetic

field

Ii

since in the

samples

^with a

large phase

coherence

length L,

it is

important only

ⁱⁿ _;ery small

magnetic

field.

According

to the authors' estimate in the

experiment [5]

^weak localization

becomes

significant

_in the field

H~~jw ~~~

_{l G (Am=?}

x10~~G.cm~ _being

_{the flux}

Lj quantum).

2. Solution for the electron

scattering problem.

In this section _we solve the

problem

of the

scattering

of _{a zone} electron

by

_a

single

vortex. In the _case when the

Ginzburg-Landau

parameter ^{of the}

superconductor

satisfies the condition

aew the _vortex field distribution _is

governed by

the London equation [9]

/

curl curl B ₊ B

=

lfi

o id

z

(P1

where B

=

curl

A,

is the _{unit vector}

along

the _vortex axis, _p is two-dimensional vector in the

plane

of the

layers.

Under the

assumption

that

Ds

W A

~ W D

N where

D~

^{is the} thickness of the

semiconducting layer

and

Ds

is the thickness of the

superconducting layer~

the magnetic ^field distribution has the form

4l

B=

°~K~(

^~

)l,

2rAj

AL

(2)

~

~~~~~~~' ^~~~~ 2~~~ _~~~ ~'~~~~~'

where

Ko(I)

and

Kj (.~j

are the McDonald functions

[10].

As usual _in the London

approach

the structure of the vorte~ core is

neglected.

li° 8 ELECTRON SCATTERING ON THE FLUX LINES 1167

We shall restrict ourselves _to the _case of two-dimensional

(2D)

electron motion in the

semiconducting layer

which is realized in the

experiment [5].

It is convenient to describe the motion of the _zone electrons in terms of the effective _mass

approximation

~

(-ihv~-iA)~$r~-ap~Bd~$r~ =E$r~, (3)

2M _c

where _«~

=

°

,

$r~

^is

two-component spinor («

= ± I and M* is the effective _mass.

0

The coefficient _a in

(3)

takes into account the

exchange

interaction between the localized

magnetic

moments and the conduction electrons in _a

semimagnetic

semiconductor. In _a

nonmagnetic

semiconductor

(such

_as

GaAs)

_a

=

I. In the

experiment [5]

the concentration

of 2D electrons in the GaAs

layer

_was

n~=6x10~~cm~~,

the Fermi _wave vector

k~

=

2

/~

₃

x 10~ _cm~

',

_{the London}

penetration depth

~ 10~ ^~ _cm and _so

k~

A

~ 30. As

k~

A

~ » l the term

proportional

to A~ _may be

dropped

in

equation (3)

and this

equation

_may be rewritten in the form _:

-[vl~«-j~jP~Lz~«-«p~B«~~=E~«, ₍₄₎

where

L-

=

(r

_{x p)~}

=

ih ^~ and _q~ is the

polar angle

in the

plane

of the

layer (2).

%q~

The z axis is

parallel

to the _vortex axis. The

general

solution of the above

equation

is

[I1]

w

#i«

₌

£ ^e'~~

~

Qll(P ^e'~'

,

(5)

m -w

where

x$

_are the

phase

shifts

corresponding

to the solutions of the radial

equation

-( ~~+))-$QQ) +UQ(P)QQ"EQII,

~

~~ (6)

uz(p)=-~ ^~ m-apBB(p)«.

As the Fermi

wavelength ("

_{is much smaller than the} characteristic

length

scale of the

~

magnetic

field modulation

(which

is

~

AD

the electron motion may be treated at the

quasiclassical

level.

Introducing

the

impact

_{parameter po}

=

~'~ the

phase

shift in the WKB

k~

approximation

can be written as

X

~(Po)

"

~

)(~ _j~ ^Uj~k~(fit)

dx.

(7)

(~) To avoid

possible misunderstanding,

note that below A is

multiplied by

a factor _exp

(-

_ep where

e - + 0.

1168 JOURNAL DE PHYSIQUE bf 8

Using

the relations

[12]

j~

~'~~~'~~

wA~ jpoj

-°l

,fi

^~~

^~0(~~~~ ~L

~

~

j~Ko( fi~)d.~=

~~~exp~-

^~~~~

~

A~

²

A~

integration over x in

(7)

is

readily

done,

yielding

where I

= c1

~~ and M is the free-electron _mass. The _term ^" ~"

exp

~~~

in

j9)

M 4 2

k~

A

~ ~

represents ^{the Zeeman} interaction.

Usually

it _is small due _to the condition

k~A~

WI.

However~ its contribution to the

phase

shift _may be

large

in _a

semimagnetic

wiiiiconductor where

c1 may be _as

large

as 10~

[8].

The other _term

"sgnpo(I-e '~"'~~)=

4

e4lo

~~ ~~

~sgn

poll

_e takes into account the influence of the _vector

potential

on the

4 hc

phase

of the electron _u>ave function _as in the Aharonov-Bohm

problem

For

large

pjj W A

~ this term is

exactly equal

to the

phase

shift in the Aharonov-Bohm

problem

where

e4lo

its value is _{sgn p}

o. However, for small _{po « A}

~ the _vector

potential

contribution _to the

4 hc

phase

shift vanishes because in this _case the electron _passes

through

the

region

with _nonzero

magnetic

field. Note that the

phase

shift _{is not} small in

comparison

with _umtv

([or arbitrary

po)

^{and therefore the} Born

approximation

cannot be used here.

The

scattering

_cross section

(per

unit

length along

the _vortex

axisj

is _given b,

da

=

f(q)

_[~

d~

(In)

where the

scattering amplitude

is

[I I]

~ ~,

f"(q~)=

_

jj (e~~"~-I)e""'

_(llj

1,,/2rkFn<=-~

The condition

k~

A

~ » not

only

makes it

possible

to use WKB approximation but also

restricts the

problem

to small

angle scattering.

Since the electron _{motion is}

quasiclassical

the

sum in jl

I)

_can be transformed _{into an}

integral

over pu.

where _q

=

k~

_~ is the transferred momentum for

a

scattering angle

_~ « l. The formula Ii

corresponds

_to the well-known eikonal

approximation (see~

^for

example. [13]).

It has

a

simple interpretation

the factor

exp(2 ix "(po)) gives

the

change

of the electron

wa,,e function

caused

by

the vortex field and the

integration

_{over po} in (l~)

gives

the Fourier component

corresponding

to the

scattering angle

_~

=

~ « l

[I I].

k~

li° ₈ ELECTRON SCATTERING ON THE FLUX LINES 1169

The

scattering amplitude

_can be written in the

following

form

f«(~p)

= A

~

~~ (rj- iqA ~)

_y ^*

~- ^iqA

~,

T 1

~

("

~2

« 2

F A L

r(iqA ~) y* iqA

~,

T

_i

+

~"

,

(i3)

2 2

kF AL

where

y*(a,x)

is _a

single-valued analytical

function related to the

incomplete

_gamma-

function _:

y*(a, x)

~-a

=

y(a, x) [12].

It _can be

easily proved

that the

scattering

_cross

r(a)

section does not

depend

_upon the

spin value,

I-e-

f~? )(q )

^~

=

f~"

+

)(q )

~.

As

expected,

^for

qA~

Ml the

expression (13)

coincides with the small

angle scattering amplitude

in the Aharonov-Bohrn

problem [6]

:

f(q)

=

fAB(q)

₌

^L

^~

^~~

sin

(

, q~ «

, 4~o =

I (i4)

q « C e

In the

opposite limiting

_case,

qA~

» I

(the angle

_q~ still

being small,

_q~ «

I)

_we obtain

neglecting spin

effects _:

f(q)

= I

(j.

(15) A~

_q

The difference between the

problem

of electron

scattering

_{on a} vortex and the Aharonov- Bohm

problem

becomes evident if _we compare the formulas

(14)

and

(15).

The

specific

if(q)i

/fo

1

3

q/kF

Fig.

I. The

dependence

of the absolute value of the

scattering amplitude

_{on q} for various

AL

A~

= 0

(curve I), kF

_AL =10

(curve 2), k~A~

=

30

(curve 3).

The

scattering amplitude

is _nor-

~~~~~~~ ~~

~° /~

1170 JOURNAL DE PHYSIQUE I M 8

character of _our

problem originates

from the fact that the

magnetic

field is not confined _{to a} tube of _very small radius (as it is

supposed

in the _paper

[6]j

but is _nonzero in _a

region

whose

radius is about

~. Therefore _our result stands in marked _contrast with that of Aharonov and Bohm for

scattering angles larger

than For

qA

~ W the scattering

amplitude (15)

_is

k~

~

smaller than

(14) by

a factor

(qA ~)~,

The

dependence

of

(f(q)(

_on the transferred _{momentum q} is shown in

figure

for different values of

k~ A~ (the spin

effect is

neglected).

The _curve

corresponds

to the _case

k~

A

~ =

0 and represents ^the Aharonov-Bohm result for small

angle scattering.

3. The correction to the

resistivity

due to the electron

scattering by

the vortices.

We shall make _an estimate for the correction to the

resistivity

due _to the carrier scattering

by

the vortices

using

the well-known relaxation time

approximation.

The transport ^time

Tj~ is related to the

scattering amplitude f(q) by

~

2hkF~

^~

_ji

-CDs

~D)(f(q)(~d~D, _l~~~

(~( k

u

where _Tu

=

~ is the transport ^{time in the} zero

field,

L is the _mean free

path

ⁱⁿ the absence of vp

the field and _~ is the

scattering angle.

The concentration of the vortices is determined

by

the

external field _n

=

~~~

The correction to the

resistivity

_can be written _as

to

I ~~~ilT~~~~

~~/A

~

~~ ^~~' /~~~) ^~~~~~~

^~~~~

where

l'~~(q)

is the

scattering amplitude

ⁱⁿ the Aharonov-Bohm

problem

_{(see expression}

(14)j

and

Ro

^{is the}

resisti~ity

in the absence of the field. The value of the

integral

_in (17) _is

equal

to 0.3. One _{can see} from (17) that the correction ^~~ is

inversely proportional

to [he

Ro product k~

i which is much smaller for _a semiconductor than for _a

perfect

^{metal. That} is

why

a semiconductor material _is

preferable.

The material parameters ^from [5] are

~ ⁼ 10~ _cm~

L

=

2 _x

10~~

_cm and

k~

=

2

,/2

_rn~

= 3 x 10~ _cm~ ^'

Using

these parameters we get ^from

(17)

that ^~~

=

0.6 x

10~~

in the field H~~j = 25 G. The

experimental

result in the _same field

Ro

is

~~

=

10~~ Taking

the model character of _our calculation into _account the agreement Ro

between theoretical and

experimental

values _seems to be

quite satisfactory. Finally

_we note that it is

just

the

expression (13)

which Trust be used in (17) to make _an estimate of

~~ If _we used the Aharonov-Bohm

amplitude

here instead of

f(q

_)~ we would obtain too

Ro

high

result which differs

by

two orders of

magnitude

from the correct ;alue.

4. Discussion.

In the present _paper ^the

problem

of the electron

scattering

in the field of _a

single

flux line _is

investigated.

For the _ca~e of

a

superconductor-insulator-semiconductor multilayer

_we

li° 8 ELECTRON SCATTERING ON THE FLUX LINES 1171

calculate the correction to the

resistivity

of the normal

layer

due to the carder

scattering by

the vortices. Such _a correction has been

recently

observed in the

experiment [5]

_on

high- quality Pb/GaAlAs/GaAs

structures with _a

large

mean free

path

in the

semiconducting layer.

In _case when _a

nonmagnetic

semiconductor is used _as the normal

layer (as

it is in

[5])

the influence of the vector

potential

of the vortex upon the electron wave function

brings

the main contribution to the

scattering amplitude.

So the value of the flux carried

by

_a vortex determines the character of

scattering.

As this flux is

equal

to

4lo

^due to

singlet pairing

the

change

of the

phase

of the electron _wave function is

equal

to w for _a circle

trajectory

of _a

large

radius

enclosing

the vortex.

Hence,

the interference between two

quasiclassical trajectories corresponding

to the electron motion _on the

opposite

sides of the vortex axis is

especially

strong

leading

to the

divergence

of the

scattering amplitude

_as the

scattering angle

tends to

zero.

In _a number of papers

(see,

for

instance, [14])

the

problem

of

two-quanta

^{vortices in}

unconventional

superconductors

is addressed. In this _case the

phase

difference between the two

trajectories

mentioned above is

equal

to 2w

resulting

in _a

qualitatively

different

~

dependence

of the

scattering amplitude and, consequently,

in _a different value of AR

~

Thus,

to _some extent, the

problem

of the electron

scattering

in the field of _a vortex is

analogous

to the Aharonov-Bohrn

problem. Note, however,

that in _{our case} the size of the

region

where the

magnetic

field is _nonzero is

larger

than the electron

wavelength.

Consequently,

the

scattering amplitude considerably

differs from the Aharonov-Bohm result for the

scattering angles

_{~ m}

~F hL

In conclusion _we

point

out that _{one can}

expect

a much _more

pronounced

effect of the carrier

scattering by

vortices _on the

resistivity

in the

superconductor-insulator-semimagnetic

semiconductor

multilayers

due to the

exchange

_« enhancement _» of the vortex

magnetic

field.

Acknowledgements.

We would like to thank Dr. A. I. Buzdin for _many

helpful

discussions. We also thank Dr. V. V. Moshchalkov for valuable advice. We _express

gratitude

to B.

Vuj16ib

for the

help

in the

preparation

of the

manuscript.

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JOURNAL DE PHYSIQUE i T i, w g, AooT iwi «