Supplement to Josephson equations for an electron in the presence of a barrier

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Supplement to Josephson equations for an electron in the presence of a barrier

J. Riess, Y. Grandati

To cite this version:

J. Riess, Y. Grandati. Supplement to Josephson equations for an electron in the presence of a bar-

rier. Journal de Physique I, EDP Sciences, 1991, 1 (11), pp.1545-1547. �10.1051/jp1:1991218�. �jpa-

00246434�

L

Phys,

I France I

(1991)

1545-1547 NOVEMBRE199i, PAGE 1545

Cl assificati _On

Physics

Abstracts

03.65G 73.40G 74. 50

Sho~ Communication

Supplement to Josephson equations for

_an

electron in the presence of

_a

barrier

J.

Riess(I)

and Y.

Grandati(2) (hereafter

referred _{to as}

R.G.)

J

Phys.

^{I France} 1

(1991)

261-268

(~)

Centre de Recherches _sur les ltds Basses

lbmp6ratures, CNRS,

B,P

166Yj

38042 Grenoble Cedex, France

(2)

Centre de Recherches

Nud6aires, Physique Th6Orique,

B-P 20

CR,

67037

Strasbourg

Cedex, France

(Received

8

July

I ml,

accepted

12

September

I ml

)

There is _a close

relationship

between our results and those obtained in the context of

persistent

currents in normal metal

loops (see

Ref.

[fl

of

R-G.).

We here wish _to make more

explicit

refer-

ence to

previous

work in this field. Furthermore it has been

argued

that the current oscillations described in R.G. _are

just simple

Bloch oscillations. As a result _we make the

following

comments.

In R.G. _we considered a

single

electron in _a finite one-dimensional

system

ⁱⁿ the presence ^of

a

potential

barrier and of _a

homogeneous

electric field. We used

periodic boundary conditions,

which made the mathematical _treatment

equivalent

to that of _an holated electron on a closed

loop

threaded

by

a

magnetic

flux.

Doubly

connected finite

systems

^threaded

by

a flux

# _appear

in various

contexts,

^either as real

systems (loops, cylinders,...)

or as mathematical models due _to

the _use of

periodic

or

quasiperiodic boundary

conditions

(e.g. [Ii ).

For _a

systematic

discussion

including

the relation to

quasiclassical dynamics

and Bloch oscillations see reference

iii

of R.G.

The

eigen energies E;

and currents

1;

=

-c0E; /0#

of such

systems

are

periodic

in

#

with

period (hc/q(,

whenever the

symmetry

^{in the}

system

^is

sufficiently

reduced.

Therefore,

if

# depends

on

time _as

#

₌ -cELt

(E

= electric field

along

a closed

path

of

length

L

surronding

the flux

ii,

an adhbatic current

I;

oscillates with _a time

period (h/qEL(.

Such _a current may therefore be said to be of

Josephson _type.

The existence of such

Josephson _type

oscillations in normal-metal

loops

has been

pointed

out

by Buttiker, Imry

and Landauer in their well known paper [2] on

persistent

_currents, which has stimulated _a

great

amount of further work

[3,

4]

(for

earlier related work _see

[5]).

Connections with

Josephson

^behaviour have also been made in the

quantum

^{Hall effect}

[(.

In the field of

persistent

currents the central

question is,

how disorder and

temperature

^influence the currents in the

loops,

and

also,

how the summation over many

single-electron

currents is

performed ([2-4],

1546 JOURNAL DE PHYSIQUE I N° II

Ref. [~j ^of

R.G.).

On the other hand in _our

paper (R.G.)

we were interested in the

consistency

between the be- haviour of

microscopic (electron)

and

macroscopic quantum systems (superconductors, superflu- ids),

both described

by complex

wave functions. We showed that

complete analogy

with

Josephson

behaviour is obtained in the presence ^of a

high

barrier.

Here,

with

increasing

barrier

heigh~

the

phase

difference _across the barrier becomes linear in time in the _same way as the

phase

difference of the order

parameter

^of a

superconductor

or of _a

superfluid

in the

presence

^of a weak link. This

linearity

then

implies

a monochromatic current of the electron with

frequency (qEL /h( (in

addi- don to the

general

^time

periodic

^behaviour with time

period _(h /qEL(,

which holds for any ^static

potential

^with

sufficiently

low

symmetry).

We _were not aware of

previous

work

[4],

^{from which} one can conclude that the _current in _a

loop

becomes

monochromatic,

if the transmission

probability

_across obstacles in the

loop

is suf-

ficiently

low. We thank Y.

Imry

for

bringing

reference [4] ^{to our} attention. We would like to add the

foflowing

remark: The results of [4] have been obtained

by

a transfer matrix

method,

which is based _on

#-dependent boundary conditions,

whereas in R.G,

fixed boundary

conditions had to

be used

(corresponding

to the behaviour of the order

parameter

^{of the}

macroscopic quantum

sys- tem in the bulk _on both sides of the weak

link).

These

boundary

conditions _are

important

in the

present

context.

They

lead to a

phase behaviour,

which

corresponds

to the

periodic

_passage of

phase singularities (quantized vortices)

across the centre of the barrier

(we

have treated

explicitly

the

ground

state, since it

corresponds

to the solution of the linear

Ginzburg

Landau

equation

with the

highest surperconducting

critical

temperature,

^{which is the}

physically

relevant

solution).

This was our second

key point (besides

the

monochromaticity

of the

current)

in the

analogy

between the

Josephson

behaviour of _an electron and of

macroscopic _{quantum systems,}

since in supercon- ductors and

superfluids

the

quantized

vortices have a real

physical meaning

and their

periodic

motion _{across a} weak link has been

experimentally

observed.

For

completeness

we remark that the

question

of

boundary

^conditions

(more precisely,

of the choice of the

self-adjoint representation

^of the

Hamfitonian)

ⁱⁿ

systems enclosing

_a

magnetic

flux has ben discussed in

y.

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