Giant Shapiro steps in Josephson junction arrays

Physica B 169 (1991) North-Holland

707-708

GIANT

SHAPIRO

Thomas

C. HALSEYt

The James The

Franck

University

5640 South Chicago,

STEPS

IN JOSEPHSON

and Stephen

Institute

JUNCTION

ARRAYS

A. LANGERI

and Department

of Physics

of Chicago

Ellis

Avenue

Illinois

60637

U.S.A.

Recent

experiments

have shown

well-known

single junction

With

assumptions,

some

dependence

Shapiro a formula

of the step width

that

Josephson

steps.

junction

We examine

for the step

upon resistance

width

arrays

exhibit

this phenomenon may be derived.

inhomogeneity

coherent

both

with

large-scale and without

We also present

effect

a range

voltage.

(1-f) field. of currents

The

steps

junction

These

steps

are regions

at which

steps

vn-

II. Harmonic

are

In the limit through rents,

of the n’th

Recently,

various

fiW

(1.1)

2e

of “giant” (1).

Shapiro

These

out a transverse

steps

of Josephson

steps are seen both with and with-

magnetic

array of junctions,

in arrays

the obser-

field.

these steps

For a square

appear

N x N

at voltages

rect

fiW

(1.2)

2e

which

correspond

to an ordinary

of the array.

In this width

cell of

with

p&/q,

quantum,

steps

Shapiro

step

fluxes

across

per unit

p,q integers and as the magnetic

are seen at voltages

!Supported

by the Science

+Supported

by the Materials

Center

Laboratory

change.

limit,

The

a magnetic

field.

of

This is

phases

(We have verified of small

the array

total

free energy;

show sidiary arrays

Shapiro

state

to equi-

this

Josephson

calculated steps

by di-

junction

assumption is always

of this

a local

this assumption

with respect that

this

(and

to array

is equivalent

more

dangerous)

may be modelled are generally

of Chicago

analytically

for general

calculation

to i,,,

(one can

ic).

A sub-

<

voltage

in the current

change

times

assumption

as being

is

of the

to be true in

conditions

relaxation

the

magnetic

minimum

is likely

in which the slow boundary

for Superconductivity. at the University

we have

principal

that

ical arrays

and Technology Research

flux

of an indi-

the dynamics

with the time scale over which

simulation

of the giant

the limit

For magnetic

supercur-

current

to analyze

here the time scale for junction voltages

currents

with

arrays).

slowly each junction

it is possible

numerical

fields.

vN;n = --Nn,

normal

compared

<< io, with is the critical

is fast compared

applied have reported

steps

the typical

are small

an array both with and without because

step.

authors

in which

a junction i,,,

librate for the voltage

and sub-harmonic

vidual junction,

-n,

(1.3)

2e q

with a occur

w of the r-f field by

-

vation

to a

in

exposed

Ai are compatible

voltages

to the frequency

junctions

leading

I’,,,;,, = --.

of the ac

of Shapiro

of a Josephson

to a radio-frequency

related

consequences

is the appearance

the I-V characteristic in which

field.

arguments

hw Nn

One of the most remarkable

single

of the

magnetic

in the arrays.

I. Introduction

Josephson

versions

is that driven.

driven

the

Phys-

regime.

T.C. Halsey, SA. Lmger

708

The final assumption quasi-one

dimensional

as the “staircase

state.”

to be the ground .‘.

fluxes

state

This

we may take a special,

form for the array In this state,

flow along the diagonals rational

is that

I Giant Shapiro steps in Josephson junction arrays

of a square

state,

known

constant

array.

tion

currents

This is known

of the array for certain

low order

f = p/q per unit cell, f = l/2,

l/3,

215,

that

With

these restrictions,

the appearance dition,

we are able to account

of the steps

we find that

there

mentioned

should

above.

appear

parallel locally

for

In ad-

per bond that

can flow through

rf driv-

(2.1) with p a dimensionless

measure

driving

appearance

The

studies

steps

is currently

of the strength

or in numerical

of some controversy

the persistence

of quenched

disorder,

of these

either

or in the critical

here on the first

to derive a stability

criterion

of steps in the presence that state,

of resistance

we assume

that

with

voltage

fixed

a junction

array

differences

flow, and voltage

v = 0 in the direction

excess

to the net

normal

currents

consider acteristic parallel

current

variation

flow.

This

per-

will lead

to

with L,,L,

flow.

in this region

The

Now

the char-

perpendicular total

at

resistance,

in the resistances.

L,L,,

of the region

to the net current

rent generated

is

U, =

of the order of i, - v,6R/R2

a region of volume dimensions

current

as supercurrent

L,Ai

is

+ L,io

we must have I,,,

that

the higher

will be washed

arguments

state

(3.2)

< Isup for domains the criterion

for step

(3.3)

normal

order

out

steps,

in any

can be extended

for which

real

to obtain

of the fluctuations

Ai

experiment. the width

in the normal

resistance: E Ai(0)

Ai(6R)

with a a constant

- a(s)2(g)2,

(3.4)

of order unity.

Acknowledgements

to the net current

the nodes of the array, where R is the average and bR is a typical

current

the total

case.

randomness.

parallel

differences

steps

for the persis-

tLwn/2e a.cross its junctions pendicular

-

and sizes, we obtain

implies

is small,

of the

We will concentrate

Suppose

the maximum Thus

&i-Lug

in the nor-

currents

It is simple

in a locked

moving

In the direction

stability,

This

(4).

junctions. tence

of is.

the boundary

of a step as a function

We have also studied mal sta.te resistances

step.

(or non-appearance)

in experiments

a matter

that

of all shapes

of the rf

of disorder

in the presence

In the

flow, an additional

Ai can flow without

IsUp(LZ,LY)

These Effects

is incorrect.

to the net current,

is of the order

If we realize

field (3).

of order

off the current

boundassump-

will be given approximately

Ai,,,

by

of subharmonic

are locked

to the net current

per bond

the array

the region

or else the original

subharmonic

For small

= (hw/2e)(Nn/mq).

at VN;~,~

ing, the step widths

III.

the voltages

direction current

must flow through

as supercurrent,

perpendicular

(2).

steps

current

ary entirely

We are grateful helpful

to C.L.

Henley

and C.J.

Lobb

for

discussions.

References

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Benz,

Lobb,

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