Quantum-interference phenomena in point contacts between two superconductors

Physica 41 (1969) 225-254

o North-Holland

QUANTUM-INTERFERENCE BETWEEN

PHENOMENA TWO

A. TH. A. M. DE WAELE Kamerlingh

Publishing

Co., Amsterdam

IN POINT

CONTACTS

SUPERCONDUCTORS and R. DE BRUYN

OUBOTER

Onnes Laboratorium, Leiden, Nederlalzd (Communication No. 370a) Received 19 August 1968

synopsis A point contact, weakly connecting two superconductors, can be represented by an ideal Josephson junction (is = ir sin v) and an effective resistance (R, = V/i,; V = = -(fi/Ze)(ag~/CV)) shunted in parallel to it. By solving the equation i = i, + i, = = il sin v(t) - (1/Rn)(R/~)(6’~(t)/at) = constant, the d.c. current (I) - d.c. voltage (7) characteristic for a single point contact is found to be V(t) = R,,/(is - i:), independent of the magnetic field B,. The relationship between d.c. current (I), d.c. voltage (7) and magnetic field (B,) is also derived for a double junction, taking the self-inductance (L) of the area (0) embraced by the contacts into account. If nLil < h/2e then the self-induced flux is ignored and the current i = 2ii sin p)(t) cos c

B, 0 -

R

1 Ii

-

fi %J = constant 2e at

and V(t) = R,,/(i2 - iE(B,)) for i > i, = 2il Icos(e/h) B, 01. This result explains the observed voltage oscillations in the resistive-superconductive region as a function of the external magnetic field B, when the applied current is constant and exceeds the critical value. If the self-induced flux is taken into account it is impossible to give a simple analytical solution and numerical methods are used. The total magnetic flux enclosed appears to be approximately quantized: B, 0 + Licirc M n h/2e, giving rise to a circulating current (icirc) which adds to the applied current in one of the contacts and subtracts in the other. The modulation in the critical current is approximately given by i. M 2il - (2/L) In h/2e - B, 01, when II Lil > h/2e. The relationship between 6, 7 and BI is in agreement with experiment and can be represented by a corrugated surface periodic in B,.

1. I~trodzlctiolz.

In 1962 Josephson 1) predicted

that :

1) When two superconductors are separated by a very thin oxide layer (< 20 A thick) a d .c. t unnel current of limited magnitude can flow through the junction even when there is no voltage between the superconductors (the d.c. Josephson effect).

226

A. TH.

A. M. DE

WAELE

AND

R. DE

BRUYN

OUBOTER

2) When a voltage V is established across the junction the current oscillates at a frequency 2eV/h (the a.c. Josephson effect). Using evaporated film techniques, the ideal d.c. Josephson effect was observed for the first time by Anderson and Rowells) (1963) and its diffraction properties in applied magnetic fields have been studied by Rowe11 (1963) 3). A quantum interferometer consisting of two parallel Josephson tunnel junctions was constructed by Jaklevic, Lambe, Mercereau and Silvera) (1964). The a.c. Josephson effect was first demonstrated by Shapiro5) (1963) and then experimentally investigated by Fiske6) (1964) ; Yanson, Svistuno and Dmitrenko7) (1965) and, very intensively, by Eck, Scalapino, Parker, Taylor and Langenbergs) (1964-I 967). In 1964’65 Zimmerman and Silver constructed interferometers in which the oxide junctions were replaced by superconducting point contacts. The behaviour of the evaporated tunnel junctions (oxide layers) is fully understood contrary to the case of point contacts which is of quite a different nature. In this type of weak links there is no dielectric barrier between the bulk superconductors. It is even possible to construct a narrow constriction in a superconductor, a technique due to Anderson and Dayernie), and to observe a.c. Josephson effects as a consequence of vortex motion (Anderson’s phase slippage concept). In this article it will be shown that good agreement with experiment is obtained if we approximate a point contact by an ideal Josephson junction with an ideal (i.e. without self-inductance and capacitance) normal resistance shunted in parallel to it. One usually assumes that the voltage between the two superconductors is constant in time. This is not always realized in practice. In an experimental situation it is impossible to avoid a capacitive coupling of order IO-10 F between the two superconductors. A typical frequency of the a.c. voltage is of order 109 Hz, so 1/WC is of order 1Q. If the impedance of the voltage supply is also of this order or larger (L > lo-10 H) than the voltage cannot be assumed to be constant in time. In this article we shall assume both that V is constant and that the current through the junction is constant in time. The experimental situation is described as a gradual transition from i = = constant to V = constant when r is increased. Paying our attention to a double point contact the d.c. voltage (v) versus d.c. current (z’) characteristics at different applied magnetic field values are calculated in several ideal cases: 1. V respectively i constant in time. The self-induced flux (&) in the hole is ignored (nLir < h/2e or Qs = 0). The critical currents of the individual junctions are equal (symmetrical double junction). 2. V respectively i constant in time. The self-induced flux in the hole is ignored. The critical currents of the individual contacts are unequal (asymmetrical double junction).

QUANTUM-INTERFERENCE

PHENOMENA

IN SUPERCONDUCTORS

227

PV 1200

615

WA

49c 36C

250

it 0

60

I v

250

PV

Fig. la Fig. ZOne B,O

la. X-Y recorder plots of two v characteristics. at B,O = n h/2e and one at = (n + 4) h/Ze.

0

V

1 0 0

L-lLhxL 100 PA

300

Fig. 1b. Voltage oscillations at constant applied current (above) and critical-current oscillations (below) as a function of applied magnetic field. The critical-current oscillations can be described with eq. (17) with il = 230 PA and is = 90 ,uA.

ic (00

I

_ c

1

Fig. lb

ga”56

2

228

A. TH. A. M. DE WAELE

AND R. DE BRUYN

OUBOTER

3. V respectively i constant as a function of time. The self-induced flux in the hole is not ignored ($ # 0). The critical currents of the individual contacts are equal. Also the limit (k/2e)/(7&i) -+ 0 is considered. So far only case 1 at I’ = 0 (the i, - B, dependence) has been treated in the literature. By extending the theory to the above mentioned cases a better understanding of the observed phenomena can be obtained. We now shall review the experimental results on double point contact junctions. a. The relationship between d.c. current, d.c. voltage and magnetic field forms a corrugated surfaceii). The critical current and the d.c. voltage (at constant applied current) are both periodic functions of the applied magnetic field with a period dB, = (12/2e)/O(0 being the area enclosed between the two contacts). A maximum in the critical current corresponds to a minimum in the voltage. Double contact junctions were investigated where a small area was enclosed (0 = 60 ,us, L M 10-12-10-11 H). The results are presented in fig. la, b. Experiments were also performed on junctions with large enclosed areas (0 M 4 mm2, L M IO-9H). In this case only the voltage oscillations N b - screws

I.

o-4

insul&ion

b

IO.1 pV

Q

a

invar

Sn

5.1d6Gauss -

v,o

B.L

‘awl

=

1.2mA=

constant

Fig. 2. Cross section of a double point contact with a large area (4 mma) which was used to measure the v - B, dependence at constant applied current. The double shaded areas represent tin blocks separated by an insulation layer (1O-30 pm). In the upper tin block there are two Nb-screws that penetrate the insulation layer. These can be adjusted at room temperature as to give a contact resistance of about 1 Q. At the bottom of the figure the measured r - B, dependence is given. This can be compared with fig. 10~. The sharp peaks in fig. 10~ cannot be seen in this figure because an a.c. magnetic field of the order 5 x 10-7 oersted averages this out.

QUANTUM-INTERFERENCE

PHENOMENA

IN SUPERCONDUCTORS

229

are measured because the amplitude of the critical-current oscillations is very small and difficult to measure. A typical example is given in fig. 2. b. The a.c. Josephson effect was observed by Parker, Taylor and Langenbergis) and the relation V = (h/24 v was confirmed within 6 ppm. In our laboratory the behaviour of a double point contact with small enclosed area (hole) is studied in its own resonant electromagnetic field by putting it in the center of a coaxial cavity is). If the applied flux on the hole is equal to an integral number of magnetic flux quanta (& maximum, d.c. voltage minimum) the current-voltage curve shows pronounced steps at constant voltages corresponding to a resonant frequency of the coaxial cavity. These current steps disappear when the applied flux is equal to a half integral number of flux quanta. A typical d - r curve is given in fig. 3. c. In an asymmetric junction a rectification process takes place due to the fact that the absolute values of the voltage oscillations are shifted with respect to each other for different signs of a d.c. applied currenti4) (see fig. 11). The aim of this article is to try to understand these phenomena and to discuss the original qualitative explanations given in previous articles. The Josephson relations will be taken as the basic equations. A short derivation of these relations given by Feynmanis) will be summarized here. When two superconductors are separated by a weak link the two (base state) wave functions $1 and $2 of these two superconductors are coupled and in the absence of a magnetic field are related in the following way: ifi

in

-

a*1 = at a*2

at=+

2eV -

2

$1

+

K42,

$2

+

WI,

(1)

2eV

-

2

mA

0.5

0 it____0

Fig. 3. The current-voltage the bottom

v

50

pv

100

curve for a double point contact placed in the center of

of a coaxial cavity. The regions of constant voltage are most pronounced

when B,O

= n h/2e and are not observed when BIO

=

(n + *) h/2e.

A. TH.

230

A. M. DE

WAELE

AND

where V is the voltage difference Cooper-pair

R. DE

across the junction,

and K is a coupling constant.

2), equating

the real and imaginary

2KJprps/fi = ii, 0s link) it is found that

BRUYN

81 = v(t) (=

Substituting

OUBOTER

2e is the charge of a IJ~= J& eiej (i =

1,

parts of eqs. (1) and (2) and taking the phase difference

across the weak

i(t) = ii sin q(t)

(3)

and 1

p)(t) =

v(o)

4

-

V(f)

dt’.

(4)

s

0

In the presence of a magnetic field the coupling constants K in eqs. (1) and (2) change by a factor exp(-i(2e/&) j A, ds) and exp(+i(2e/fi) J A, ds) respectively where the integral is to be taken across the contact (Kis = K&). Eqs. (1) and (2) now change to __

=

at

-

2eV

2

&

+

K

e-i(2e/R)Ad

__

t,h2

+

K

e+i(2elR)Ad

,/ 2,

*

(24

1,

where A(r) is the magnetic vector potential in point I, which is defined as B = curl A. The vector A is taken to be constant in the small contact region so JdA, ds = Ad (6 is the thickness of the contact layer). Performing changes to

the calculation

i(t) = ii sin v(t) (

_ V>

as given

A ds

above

it is obtained

= ir sin v*.

that eq. (3)

(34

d

For a double point contact junction (fig. 2) in an applied magnetic field the phase difference between P and Q calculated along the point contact b is equal to Q AyPbQ

=

(02 -

‘%)b

+

A, ds-

+s

(.JunotioPnbexcluded) Q Iqb+$

A,

ds,

s Pb Junction

excluded

QUANTUM-INTERFERENCE

PHENOMENA

where vb is the phase difference

IN SUPERCONDUCTORS

231

across point contact b. Along a:

Q

+P~Q

=

s

A, ds.

~a + f

Pa

Junction excluded

As the wave function i.s single valued the line integral of VT along the closed loop PaQbP (equal to A~P~Q - AcppbQ)must be equal to 2nn. Without loss in generality we take TZ= 0. In this way one obtains that: Q 0

=

AVP~Q

-

&pb&

=

9%

-I-

Q

&ds-vb-;

$

s

s Pb

(Junctions .

A,ds=

Pa

excluded) Q

Q

c

J

A,ds

-

$

i!t,ds-qb-$ s

J

Pb

A,ds

+ 9

5s

A8d.s, b

and from eq. (4) it follows that

9’; + ‘?‘: = 2

‘?‘b + %

2

= V(0) -

(6)

(Here we assumed jB A, ds = - jb A, ds). All these expressions are gauge invariant under the transformations

2. The behaviour of two sqberconductors weakly connected by a single point contact. In this section the relation between d.c. current (i) and dc voltage (7) will be derived for the case where the influence of the self-inductance and the capacitance of the point contact can be ignored. Furthermore it is assumed that the linear dimensions of the contact are so small that the influence of an applied magnetic field can be neglected. Especially the two cases in which respectively the current and the voltage are constants as a function of time will be discussed. According to Josephson the supercurrent through a single

232

A. TH.

A. M. DE

WAELE

AND

weak link between two superconductors

is(t) = il sin q(t) = ii sin { ~(0) -

2”h

R. DE

BRUYN

OUBOTER

(eqs. (3) and (4)) is equal to

JV(t’)

dt’}.

0

In eq. (7) is(t) is th e supercurrent

through the junction;

ir is its maximum

value, v(t) and ~(0) are respectively the phase differences between the superconductors at time t and at t = 0 and V(t) is the voltage difference across the junction. If the junction is brought into the resistive-szl~erconductive region (V(t) # # 0), then a normal current also has to be taken into account. We assume I that this normal current in(t) is equal to in(t) = v(t)/Rn

(8)

in which Rn is an ideal resistance. Hence we suppose that the single point contact can be represented by an ideal Josephson junction (through which the supercurrent i, = il sin 9 flows) and an ideal resistance R, shunted in parallel with this junctionra). The total current is equal to

i(t) = i, + i, = ii sin 0

The cases in which the voltage V(t) = V, independent of time and in which the current i(t) = i, independent of time, are of special interest because they can be realized experimentally to a good approximation. V(t) = V = constant, independent can be evaluated easily and gives i(t) = il sin{q(O) -

of time.

The integral in eq. (9)

(2e/ti) Vt} + V/Rn.

If V = 0, this expression reduces to i(t) = il sin v(O). Thus between -il and ii can be obtained. If V # 0, is(t) is the sinusoidal a.c. Josephson current 2eV/h and an amplitude ii. The time average of is(t) is equal the normal electrons contribute to the d.c. current. The i can be described with the relations

v=

0,

V # 0,

0



i(t) = V/R,,

and is given in fig. 4a.

any d.c. current at a frequency to zero and only 7 characteristic

(104 (1Ob)

QUANTUM-INTERFERENCE

PHENOMENA

i(t) = i = constant,

independent

IN SUPERCONDUCTORS

of time.

i
233

V(t) = 0 and y

is independent of time. Eq. (9) again reduces to i = ii sin ye In order to solve eq. (9) when i > ir we make use of q)(t) (eq. 6) : t

v,(t)

v(o)

=

s

$

-

V(f) dt’.

0

Now eq. (9) gives -

v(t) = R,il

&ii

sin v(t) = -

fi 2e

a&) at

k

and c(t)

--dp,

t=& n

il sin pl ’

i -

s

P(O) where V(t) is a periodic function periodic in time with a period

of p)(t) with a period 2x

Hence

V(t) is

-2n

1

-dP

_&& Y

n s

i -

il sin v

(11)

*

0

The time average

of the voltage is equal to the time average in one period: T

T V(t) =

+

V(t) dt =

+

Y $

I

0

dt.

0

(12) Hence eq. (11) gives v(t) =

_-2* 2nRn s

i -

-dp, ii sin y

0

which has the solution i > il, Together 0
V(t) = R,t/q.

(w

V(t) = 0

Wb)

with
7 dependence

can be calculated

and is given in fig. 4b.

234

A. TH. A. M. DE WAELE single

, v=constant

AND R. DE BRUYN point

OUBOTER

contact

i = constant

Fig. 4. i - P characteristic of a single point contact (calculated). a. When the voltage is a constant as a function of time. b. When the current is constant as a function of time. c. With a capacitance parallel to the junction. At small (resp. large) 7 the Zr- v dependence is approximately the same as in fig. 4b (fig. 4a). At an intermediate value there is a gradual transition between these two limits. When one tries to measure the z?- v dependence with a circuit having high resistance and self induction it is impossible to measure the part with a negative resistance and the z?- v dependence has an irreversible character.

Eqs. (10) and (13) have also been derived by StewardiG). Voltage biasingcanbe achieved by shunting the junctionwith a parallelideal resistance R (R < R,), see ref. 9, or an ideal capacitance C(l/oC < Rn) 16). Current control can be experimentally realized if one applies the current with a circuit having a high resistance and self-inductance. All the experiments carried out in our laboratory were with such a circuitll~rs~ r4). However, if one tries to measure the 5 - 7 characteristic with this constant-current circuit it is impossible to avoid capacitive coupling between the two superconductors of order 10-10 F. In this way there is a gradual transition from the case of constant current (R, < 1/UC), to voltage biasing (Rn > 1/o.L) as 7 is increased (because of the increasing frequency (12)). Therefore it is possible to have a theoretical curve as given in fig. 4c (solid plus dashed line). With a constant-current circuit it is impossible to measure the part with negative resistance. The measured curve is given in fig. 4c with the solid lines and has an irreversible character. 3. The behaviour of two sufierconductors weakly connected by a double fioint contact, when the self-induced flux in the enclosed ayea is ignored. The critical current of this double jurtction (I’ = 0, as = 0). Although many experiments are performed with only one weak link between two superconductors, situations in which the superconductors are connected by two weak point contacts are of more experimental and theoretical interest. Here not only the a.c. effect occurs, as in the case with one weak link, but in the double contact it is also possible to vary the critical current and the voltage by varying a magnetic field (B,) applied perpendicularly to the area 0 enclosed by the contacts. This is a consequence of the fact that the phases between

QUANTUM-INTERFERENCE

PHENOMENA

the two superconductors we restrict

ourselves

IN SUPERCONDUCTORS

across the two contacts

to a symmetrical

junction

235

are not independent.

If

with V = 0 eq. (5) gives

We now assume that we can ignore the self-induced flux in the area 0 enclosed by the two contacts. The total flux in 0 is than equal to the applied flux, B,O.

From eq. (5) we obtain

(14) The total current through the double junction is equal to the sum of the currents through the individual contacts (ii = is) : i(B,,

cpa) = il sin vi + is sin vz = = ii sin vz + ii sin

= 2ii sin The critical

current

‘p: + $

at a certain

respect to vz, therefore known result 4315 v17) i,(B,)

= 2ii I,,,_B

This relationship

vi + $

o

(15)

(e/h) B,O)

=

of i(B,,

L 2e 810

ya) with

& 1. This gives the well

01 1. *

ti

symmetrical

h

=

B, value is the maximum

(16)

between i, and B,

-zG

>

B,O ) cos ($0).

sin (qt + e

B,O

is given in fig. 5a.

double

211.0 2e

point

-- h 2e v --_)

k=o

contact,

0

L 2e

2h z

B_LO

Fig. 5. a. Critical-current oscillations as a function of applied magnetic flux in an area 0 for a symmetrical double point contact when the self-induced flux in the hole is ignored (ms = 0). b. Two T - r characteristics (i(t) = i = constant). One at an applied flux of n h/2e and one at (n + 4) h/2e. When the applied flux is not equal to one of these two values the I - v characteristic is between these two curves according to eq. (36). c. Voltage oscillations as a function of the applied magnetic flux at the constant applied current indicated by the arrow in fig. 5b.

A. TH. A. M. DE WAELE

236

AND R. DE BRUYN

OUBOTER

One of the significant features of this curve is that the critical current is equal to zero when the embraced magnetic flux is equal to a half integral number of flux quanta, B,O = (n + 4) k/2e. However, this is never observed and we must conclude that the assumptions made for deriving eq. (16) are never realized in practice. If one drops the condition that the critical currents of the individual contacts are equal (ii # is) and again ignores the selfinduced flux one obtains: i(B,,

P)~)= il sin 9’: + is sin

91: + 9

B,O

, >

from which the following result is derived:

W,)

= =

J J

2e i; + i; + 2ilis cos h B,O

=

(il - i~)~ + 4ili2 cos2

(17)

The curve representing this relationship for ii = 5;s is given in fig. 6a. In section 7 we shall compare this result with the results of experiments performed with double point contacts enclosing a small area. The formalism described above for the case that V = 0 is not limited to a double point contact but can be extended to a junction with N point contacts arranged on a straight line at equal distance from each other, thus resulting in an interference grating 17). For simplicity we assume that all contacts and holes are identical and that again we can ignore the flux in the holes due to self-inductance and mutual inductance. The total flux in the asymmetrical

tk

i,+iz,

double

___ --a 0

!&=o

contact,

it nh 2a - ;+--_ -/

ix -- h 20

point

h2hO 20 2P 40

I;J

/’

(n A, 2

L 20

b

h 20 -

v

0

L

20

2h

2a

8~0

Fig. 6. a. Critical-current oscillations as a function of applied magnetic flux in an area 0 for an asymmetrical double junction. The self-induced flux in the hole is ignored (di, = 0). b. Two i - v characteristics (i(t) = i = constant). One at an applied flux n k/2e and one at (n + 4) 12/2e. The B - 7 characteristic at any other applied flux value is between these two according to eq. (37). c. Voltage oscillations at the applied current indicated by the arrow in fig. 6b.

QUANTUM-INTERFERENCE

PHENOMENA

IN SUPERCONDUCTORS

holes between contact 1 and contact K is than equal to (k analogous to eq. (5) we obtain 4p; = go);+ ;

(k -

(k = 2, 3, . . . . N).

1) B,O

1) B,O

237

and

(18)

The total current through the junction is equal to N

= il sin q~t+ (N -

1) G B,O

In order to derive the critical current we take sin@& + (N - I)(@) B,O) = = f 1 resulting in eq. (19) :

i,(B,)

= il

(19)

From eq. (19) it is possible to derive the diffraction formula for a weak contact of nonzero extensiond). To derive this formula we introduce Ot,t = = (N - 1)0 and itot = Nil, where Ot,,tis the total area of the junction and itot is the maximum critical current of the junction. Substituting these new variables in eq. (19) we obtain . i,(B,)

sin &

(i

B,Ot,t)

= 9 sin &($

B,%t)

’

We take the limit N -+ co keeping all other variables constant sin(t

lim QB,) N+ca

B, Om)

= itot

(20) +

B, Otot

Eq. (20) can be compared with the Fraunhofer diffraction formula in optics and describes the diffraction experiment of Rowells). 4. The behaviour of two sq5ercolzductors weakly connected by a symmetrical dotible point contact, when the self-idzcced

f&x in the embraced area is taken

A. TH.

238

A. M. DE

WAELE

AND

R. DE

BRUYN

OUBOTER

i&o account. The critical current of this dozlble junction (V = 0, Qs # 0). Till now it was assumed that the self-induced flux ($) in the hole between the contacts can be ignored. This is not at all a physical reality. It is unsatisfactory because the magnetic flux quantum h/2e SW2 x lo-15 weber, so that for example, if the self-inductance of the hole is equal to 10-10 H a circulating current of 20 PA is sufficient to generate a flux h/2e in the hole. As the critical current of an individual contact is of the order 10 ~A-10 mA and L = 10-12-10-s H it is in fact not justified to ignore the self-induced flux because different currents in the individual contacts can easily generate a flux in the hole of the order of a flux quantum. Therefore the fact that L # 0 has its influence on the i, - B, dependence. In order to obtain a better description of the experimental observations it is necessary to include the effects of the self-inductance in the calculations. We restrict ourselves to two superconductors connected by two identical point contacts. The flux Gs generated in the hole by the current through the individual contacts is equal to CD,= *Lii(sin 9’: -

sin c&) = Licirc

(21)

in which icrrc = &r (sin vl - sin c&) represents the net circulating current. Twice the circulating current is equal to the difference of the currents flowing through both junctions. From eq. (5) we obtain: y;=cp:++ Substituting

m3 -= Lil

A,ds=pl:+$(B,O+@,).

9

(22)

this in eq. (21) results in a relation for OS sin vi -

@S zzz Lil

sin;

sin vt + $-

(B,O

(B,O + CD*)cos

+ C-IQ = 2 y,

pi + g

(B,O + %)

The total current through the double junction

is equal to

i(B,,

q:) = il sin cpi + 1-rsin vz + $ i

= 2il sin vz + $

(B-0

+ c&) cos$

(B, 0 + @,)

1

(B,O + C&J.

.

(23)

=

(24)

Eqs. (23) and (24) are exact and can only be solved together. From eq. (23) the flux Qs generated by the circulating current can be derived graphically as a function of B, and 9:. This function must be substituted in eq. (24). The critical current is the maximum of i(B,, &) with respect to 9:.

QUANTUM-INTERFERENCE

_--1

h

2 20

0

010

h

PHENOMENA

IN SUPERCONDUCTORS

239

3h 2 20

-Z

a Fig. 7. u. Curves representing resp. the critical current of the double junction (-), the current in point contact a (----) and in b (. . . . .) as a function of the applied flux through the hole. The self-induced flux Ds is taken into account. In this example nLil = 5h/2e. When the value of B,O is increased from zero, the current through point contact b is constant apart from the B,O values near zero. The current through contact a decreases. As the critical current is the sum of these two currents the critical current also decreases. The difference between the current through a and b can be interpreted as twice the circulating current inducing an amount of flux in the hole opposing B,O. When the applied flux BIO surpasses the value ih/Ze a transition takes place. This transition can be interpreted as a change of sign of the circulating current, thus inducing a magnetic flux in the same direction as B,O. Now the current through point contact a is large and almost equal to ii while the current through b increases when B,O is increased, in this way leading to an increase of i,. The currents through the contacts are periodic in B,O with a period h/k Thus when B,O = h/Ze the currents in the junction are the same as when B,O = 0. The only difference is that a flux h/Ze is enclosed by the hole between the contacts. b. Flux induced in the hole by the current through contact a and b, and the total flux in the hole as a function of the applied flux BIO. For this value of nLil the flux quantization is not complete.

The results of these calculations (for nLir = 51z/2e) are shown in figs. 7a and lOa. It is also possible to derive the total embraced magnetic flux B,O + !Ds as a function of B,. The result is given in fig. 7b. Taking the limit nlir/(h/2e) -+ 0, it follows from eqs. (23) and (24) that (2e/h) QS -+ 0 and that the calculated i, - B, dependence is the same as the i, - B, dependence derived in section 3 (eq. (16)), where the self-induced flux was ignored. Therefore the results of section 3 are valuable only when nlir/(k/2e) < 1.

A. TH. A. M. DE WAELE

240

AND

R. DE BRUYN

OUBOTER

In the limit (Iz/2e)/(nLii) --f 0 (or 7&1/(12/2e) -+ co) we obtain from eq. (23) 0 = sin f

(B,O + QS)cos 9): + + (B,O + Q8) , I

I

from which we derive sin + (B,O + Q8) = 0 or B,O

+ cDs = n h/2e.

(25)

This relation shows that the total flux in the hole is quantized in units h/2e. Eq. (24) simplifies to i(B,,

~)a) = 2i1 sin(~; + ZWZ) cos nn.

This is a maximum when p’i = (m + +) YZand therefore the critical current is equal to i,(B,)

= 2ii.

(26)

This equation shows that the critical current is independent of the magnetic field. This is a consequence of the fact that we took (h/2e)/(nLil) exactly equal to zero. If we take the less severe restriction 0 < (h/2e)/(nLil) << 1 the flux quantization is almost complete and the &oscillations can be derived by considering the behaviour of the individual contacts. In previous publicationsii) the i, - B, oscillations were explained starting from complete flux quantization (eq. (25)). It was supposed that there is a circulating current around the hole in such a way that B, 0 + Licirc = n h/2e.

(27)

This means that the total flux in the hole is quantized for every applied magnetic field and every applied current (as long as V = 0). When the applied current (i) is increased from zero this current will divide equally between both weak contacts. Through one contact a current ii + icirc flows and through the other ii - icirc. The maximum supercurrent that can flow through one individual contact is equal to ii, so the critical current (&) for the total junction can be derived from the equation:

or with eq. (27)

(28)

QUANTUM-INTERFERENCE

PHENOMENA

I

b:” :

:’

\ Xc’

241

.\ : \ ,:. I ‘4

)__a__, CL.?...

.. . .+_a__ .:

IN SUPERCONDUCTORS

. . .b. . . . .. _a__:

\

b,.'

‘\a \

:

\

‘,:

:’

‘,a

b.:’ :.

\

1

:

1.:

Fig. 8. Critical-current oscillations () and the currents through the individual contacts (when the critical current flows through the junction) as a function of the applied flux on an area 0 when it is assumed that the total flux in the hole is quantized (eq. (24a)). When the applied flux is increased from zero a circulating current is induced in such a way that the total flux in 0 remains zero. In one of the contacts the circulating current adds to half the applied current and in this way the critical current of the total junction is smaller than when B,O = 0. Increasing B,O increases the circulating current and ic is decreased. The current through b is constant and equal to the critical current of the individual contact b (ii). When the B,O value surpasses +h/2e the circulating current changes sign and now the current in contact a has its maximum value (ii). The total flux in the hole is now equal to h/B. When B,O is further increased the circulating current decreases, the current through b increases, and in this way the critical current increases. When B,_O = h/2e the circulating current is zero and the critical current is again equal to 2ii.

The value of n in eq. (28) is determined by the condition that iC is a maximum but smaller than 2ii. The results obtained in this way are given in fig. 8. This figure can be compared with fig. 7~. In both figures we see that in one of the contacts a current flows being (almost) equal to the critical current. 5. The behaviour of two sufiercondtictors weakly connected by a double point contact in the resistive-su$erconductive region. a) A general introduction. In the introduction we mentioned that a double point contact in the resistive-superconducting region shows a periodic behaviour in the d.c. voltage as a function of the applied magnetic field at constant applied current (see figs. 1 and 2). This behaviour will be explained in this section. We assume again that the normal current flows through an ideal resistance R, shunted in parallel with an ideal Josephson junction carrying the supercurrent. When i, represents the externally applied supercurrent then is/2 will flow through each junction. In addition to this there is a circulating supercurrent associated with the self-inductance of the double

242

A. TH.

A. M. DE

WAELE

AND

R. DE

BRUYN

OUBOTER

junction. In one of the junctions both currents have the same direction and in the other they are opposite. Hence the total supercurrent through each junction is respectively equal to: is/2 + Girc,s = ii sin p)I, is/2 - i&c,8 = ii sin &. By addition and subtraction we obtain &/ii = sin vt + sin & = =

2

sin

‘pi + v);Icos d - 6 = 2

2 =

2

sin

%

+

2

9%

cos$

Q

A,ds = 2sin&)

cosi

A, ds

(29)

and

%irc8 =

)

sin p’i - sin 91: =

il

=-

2 sin %

Q

A,dscosv(t)

= 2$.

(30) 1

To simplify the calculations we assume that there is no circulating current due to normal electrons. This assumption is justified in many experiments because the Josephson frequency (Y)is of the order 109 Hz and L/4R, m 10-11s. The normal circulating current (i ci rc, *) is of order (d@/dt) /4R,,, while the circulating current due to the superconducting electrons is of order (h/2e) L. The ratio of icirc,n and iCrrC,B is of the order

With the above mentioned values of v and L/4R,, we obtain icirc, n/&c, 8 m 3.10-s < 1. Hence eq. (30) also gives the total circulating current. The total current through the junction is again equal to the sum of the supercurrent and the

QUANTUM-INTERFERENCE

PHENOMENA

IN SUPERCONDUCTORS

243

normal current : i(i) = i, + in =

When we substitute eqs. (5) and (6) in this equation and in eq. (30), we obtain i(t) = 2ii cos$

(B,O + &) sin 0

@s -= Lil

- sini

(B,O + CD*) cos

(32) 0

or with eq. (6) i(t) = 2ir cos f

VP)

(B,O + !D8)sin p)(t) + R-,

(3’4

n

@s -= Lil

- sin + (B,O + @8)

COS v(t).

Wa)

b. When the self-induced flux in the embraced area is ipored

(@s = 0). In

this case !& = 0 and eq. (31a) is reduced to

VP)

i(t) = 2ii sin q(t) cos $ B,O + R,

(33)

n

hence V(t) is a periodic function of go.As in the case for a single point contact we solve this differential equation for two cases: V(t) = V = constant

i(t) = 2ii sin ~(0) - $

in time. Vt cos & B,O + g_. >

n

(34

When V = 0 this expression reduces to i = 2ii sin ~(0) cos i

B,O

and is identical with eq. (15). As before the critical current is given by eq. (16). When V # 0 the first term in the right-hand side of eq. (34) is sinusoidal instime and has a time average that is equal to zero. The time average of the total current is equal to V/Rn.

A. TH.

244

The a -

v =

A. M. DE

WAELE

v characteristic 0

0,

v # 0,

AND

R. DE

BRUYN

OUBOTER

can be described with the relations

C2il)cosfB,0~:


w4 (3w

i(t) = V/R,,.

This relationship

is the same as shown in fig. 4a if ii is replaced by

2il/cos~B**~. i(t) = i = constant

in time.

R,i - R,2ir cos f

B,O sin v(t) =

Following

the same procedure

0
v(t)

=

k ddt)

-

2e

dt

as for a single point contact


v(t)

(364

= 0;

v(t) =

i > G,

one obtains

R&

--

- i;.

(3W

r dependences for i, = 2ii, (B,O = n h/2e), and for iC = 0, (B,O = (n + 4) h/2e), are given in fig. 5b.

The C -

These calculations can be easily extended to an asymmetric (ir # i2). By straightforward calculations we obtain 2e

i; + i; + 2ili2 cos a

OQi
-

f

0

v(t) =

R&

- i:.

1

i--ising,-&sin

B,O,

junction

v(t) = 0;

(37a)

-1

dg,

(37b)

The i - V characteristics calculated from eq. (37~2,b) when the B,O values are equal to either an integral or a half integral number of flux quanta are given in fig. 6b. Eq. (37~~)is identical with eq. (17). c. When the self-iladuced fhx in the embraced area is taken into account (CD*# 0). If we drop the condition that the self-induced flux in the hole is equal to zero (QS # 0) it is no longer possible to give a simple analytical

QUANTUM-INTERFERENCE

PHENOMENA

IN SUPERCONDUCTORS

245

solution of the eqs. (31) and (32). However, it is possible to solve these equations with graphical or numerical methods. V(t) = V = constant

in time.

Eqs. (31) and (32) reduce to

@s --= Lil

- sin + (B,O + CD*) cos ~(0) - f (

i(t) = 2ii sin ~(0) -

(

$

Vt , >

Vt cos + (B,O + CDS) + +. II >

(38) (39)

When V = 0 these expressions reduce to

a!

---c

Lil

- sin $ (B,O + G8) cos q(O),

i = 2ii sin ~(0) cos f

(B,O + &).

(40) (41)

These equations must be essentially the same as eqs. (23) and (24). If we replace p1: + (e/ti)(B,O + Q8) in eqs. (23) and (24) by ~(0) we can easily see that they are indeed identical. Therefore the critical current calculated in section 4 is equal to the critical current derived from eqs. (40) and (41). When V # 0 we take ~(0) = 7~without loss in generality. From eq. (38) it is derived that Gp,is a function of cos(2e/ft) Vt and therefore cos(e/Fi)(B,O + !&) is also a function of cos(2e/&) Vt. The supercurrent is(t) = i(t) - V(t)/R, is equal to 2ii sin(2e/ti) Vt times a function of cos(2e/fi) Vi and therefore is(t) = --is(--t). (This result can be verified in fig. 9). Furthermore the supercurrent is a periodic function of time with a frequency 2eV/h and therefore the time average of is(t) is equal to zero: v # 0,

i(t) = V/R,;

(424

v = 0,

0 < i(t) < i, (ic calculated in section 4).

(42b)

In order to calculate the time dependence of i, eq. (38) is solved with graphical methods. We plot both the right-hand side and the left-hand side of eq. (38) as a function of GS at a certain voltage, magnetic field and time in the same diagram (fig. 9). The Qs values of the points of intersection are the solutions of eq. (38). The amplitude of the sin function representing the right-hand side of (38) is equal to lcos(2e/fi) Vtl. By solving eq. (38) for different values oft we obtain as as a function of time. In general there is more than one solution but there is only one solution that is of physical importance : it is the solution that has a continuous development in time. From fig. 9 it can be derived that this solution is situated in the

246

A. TH.

A. M. DE

WAELE

AND

R.

DE

BRUYN

OUBOTER

interval C---B,0 + n Jz/2e, - B,O + (TZ+ 1) h/2e], where n is an integer chosen in such a way that also the origin is in this interval. In principle it is possible that GS is not in this interval, but this is not a stationary solution because every time (2e/lz) Vt = +(K + +) (k integer) the only solution of (38)

____--:-:iwh !!L h

h

2ev

2ev I

I

2i,

I

_I-_

I

is

t

t

L

~--

-2 I,

Fig. 9. Calculation

of the time

supercurrent In the eq.

upper

(40a)

Q5. R,,

dependence

part

of the figure

are given

as a function

S, etc. from

of the total

in the case that the voltage

the points

both

the left-hand

of di, at a certain

of interaction

flux

in the hole

is constant

and of the

in time.

side and the right-hand B,O

and time

Q, R, S etc. of these

2. The

two

side of Ds values

curves

are the

solutions of eq. (40a) and correspond to the flux that can be induced in the hole by currents through the contacts a and b. In order to find the solutions of eq. (40a) at another time the amplitude of the sin function must be changed (amplitude is equal to

]cos(2e/R)

Vtl) and again the & values of the points

of intersection

must

be de-

termined. In this way the time dependence of the solutions of eq. (40a) can be derived. There is only one solution that is continuous in time (when t = (R/2eV)(n/2), then cos(2e/a)

Vt = 0 and S, =

0 is the only

solution).

It is the solution

in the interval

[--B,O + n h/2e, -B,O + (n + 1) h/2e] where n is chosen in such a way that the origin is also in this interval. The solution S, is the only solution that is of physical interest.

From

the time dependence

can be calculated.

The results

of S,, the time dependence

are given

in the middle

respectively.

of DS + B,O

and lower

part

and of i,

of the figure,

QUANTUM-INTERFERENCE

PHENOMENA

IN

SUPERCONDUCTORS

247

is Gs = 0 and the junction has to jump back into this state. Knowing CD&) we can derive the time dependence of the total amount of flux in the hole between the contacts. It has the remarkable property that it oscillates between n h/2e and (n + 1) h/2e (fig. 9). When B,O is exactly equal to an integral number of flux quanta there are several solutions that are continuous in time, but in a practical situation this very special case will never occur. It is a prediction of the theory that the time average of the total flux in the hole is roughly equal to (n + 8) h/2e if the junction is in the resistive region. From B,O + @&) we derive cos(2e/ti)(B,O + c?&)as a function of time. Substituting this function in eq. (39) finally gives the total current as a function of time. A picture of the supercurrent i&) is given in fig. 9. The basic frequency is 2eV/h and there are strong harmonics. Especially the second harmonic seems to have a large amplitude. In the limit nLil/(h/2e) -+co the behaviour of the junction is rather simple. The time dependence of the function B,O + CD&) is a block-shaped function between the values n h/2e and (n + 1) h/2e. Now is(t) is essentially a saw tooth at a frequency 2(2eV/#i). i(t) = i = constant

in time. We now turn our attention to the experimentally interesting limit that the current is independent of time. As before V(t) = 0 when i < i,; i, is the same critical current as derived in section 4 (see fig. 7~2). When i > &, as and therefore also cos(2e/ti)(B,O + Cl&)can be obtained as a function of y(t) from eq. (32~) and in this way V as a function of q(t) can be derived from eq. (31~). V is a periodic function of 9, with period 2n, which means that V is a periodic function of jb V(f) dt’ with a period h/2e. Therefore V is a periodic function of time; v = V(V),

dpl -= dt

-

$

qp), QG) .

dg, J-Rii’

90)

The time period is equal to

The time average of the voltage is equal to the time average in one period so : -2n

T V(f) s 0

dt'

=

-dy

v; s 0

= ;

v.

(44

248

A. TH.

A.

M. DE

WAELE

symmetrical

h -20

o

double

Fig.

10. a. Critical-current 0 for a symmetrical

ignored.

These

point

oscillations double

critical-current

v characteristics

and one at B,O c. Voltage

oscillations

DE

BRUYN

contact,

h -20 v

oscillations

at the applied

211. 24

of applied

magnetic

flux

The

self-induced

flux

in the hole

are

constructed

the

same

way

in an is not as the

in fig. 7~.

(i(t) = i = constant).

= (n + +) h/2e. These

A_ 2e B_LO

as a function

junction.

OUBOTER

k#O

0

C

oscillations b. Two i -

R.

_!L *ho 20 2e 810

area

AND

One at the applied

curves

current

are not perfect

indicated

flux B,O

= n h/.2

hyperbolas.

by the arrow

in fig. lob.

This relationship between voltage and frequency is different from the relation V = (lz/2e) v derived for the case of a constant voltage. In the case of constant current, V is time dependent and has strong harmonics especially when i is only slightly larger than i,. If i> iC the d.c. component is larger than the amplitude dependence are :

of the a.c. component.

The relations giving the a -

v

where i, is derived in section 4 and V(v) is derived from eqs. (3 1a) and (32~). In general the shape of the a - v curve is not exactly a hyperbola but will not be very different from one. The ~7- v characteristic of a double junction with identical contacts and as # 0 is given in fig. 10. The behaviour of the double junction in the limit zLii/(k/2e) -+ 00 is 8) as a function of v,(t) is block shaped again rather simple. cos(e/h)(B,O + CD and therefore V as a function of time is essentially a saw tooth. 6. The behaviour of two supercondwtors weakly connected by an asymmetrical doz161e+oint contact, when the self-inductance of the embraced area is taken into account. The critical current of this double junction (V = 0, Qs # 0, il # iz). This is the most general double junction that will be discussed in this article. Only the critical-current oscillations will be considered. To get an idea of what happens in the junction eq. (25) will be taken as the basic equation.

QUANTUM-INTERFERENCE

PHENOMENA

This is a good approximation

IN SUPERCONDUCTORS

249

when TEL&2 > h/2e where ii and is are the

critical currents of point contacts

a and b respectively.

field B, is applied on the area 0 of the junction induced that is equal to

>

When a magnetic

a circulating

.

current is

(46)

This circulating current adds to the applied current in a and subtracts in b. The current through point contact a equals +i + icirc and that through b equals ii - icirc. When the applied current increases from zero, flux quanta are expelled or trapped in the hole every time the current through one of the contacts is equal to the critical current of that contact. The critical current of the total junction is reached when the current through one of the contacts is equal to the critical current of that contact and when in addition the critical current through one of the contacts would be exceeded if an extra flux quantum were expelled or trapped in the hole. This happens when ii + is - (1 /L) (h/2e) < i < il + iz and i/2 is equal to ii - icirc or is + + icirc. Hence, substituting eq. (46) we obtain il + iz -

-

i, = 2il -

1 L

z

h < i, < il + ia 2e

nl

$

-

B,O >

or i, = 2iz + g

(47)

n2 -& -

B,O >

(rti and n2 are integers). In eqs. (48~) and (48b) there is only one nr or ns value that satisfies eq. (47). Suppose for example that nr is such that eq. (47) is satisfied: il + i2 -

-

Straightforward 2i2 + g

calculation ni -& -

( --

1

h _ < 2ilL 2e 1

h < L 2e

B,O

-

2 L

nr -

h 2e

-

B,O

< il + ip. >

gives >

il + i2 < 2iz + g

nr $-

-

B,O

. >

This inequality means that the value of 2i2 + (2/L)(n2 h/2e - B,O) is either larger than ii + is or smaller than ir + is - (1 /L) (h/2e) for any 122

250

A. TH.

A. M. DE

WAELE

value. Hence eq. (47) is a condition

AND

R. DE

BRUYN

OUBOTER

for the value of %i or ns. When ii = is

the expressions (47) and (48a, b) reduce to the expressions for a symmetrical junction given above. Writing eq. (48a, b) in a different way we obtain i, = il +

iz - +

nl

z”e - B,O - $

(il - i2)

(49a)

or

These equations are the same as for a symmetrical junction in which the critical currents of the individual contacts are equal to (ii + is)/2 and on which a magnetic field of the value B, + L(il - iz)/20 is applied, so in order to find the critical-current oscillations for this asymmetrical junction we may calculate the critical-current oscillations for a symmetrical junction (critical current of the contacts equal to (ii + is)/2) and shift the zero of the B, axes by the value L(il - is)/20. By changing the direction of the applied current without changing the direction of B, and taking absolute values, the following relations for the critical current are obtained:

Ii,1 = il + i2 +

;{kl $ -

B,O

+ +

(iI -

ia))

W)

or

where kr or Ks must be such that condition (47) is satisfied. These equations are the same as eqs. (49a, b) apart from the fact that the zero from the B, axis is shifted in the other direction. By equating the various expressions for these two critical currents the obvious result is found that the two iC curves are the same when the total shift of the two curves (L(ir - is)/O) is equal to an integral number of magnetic-field periods. The other result is that the two i, curves intersect when B,O = (m/2)(bz/2e) (m integer). If one is experimentally interested in finding the B, values which correspond to an integral or half integral number of applied flux quanta one only has to reverse the current through the junction and determine the points of intersection of the two +,I - B, curves. When an a.c. current is applied on an asymmetrical junction with an amplitude larger than the critical current of the junction than a d.c. voltage is observed even when the applied d.c. current is zeroid) (see fig. 1 1). 7. Comfiarison with ex#erimerctal results. In order to compare theory with experiment it should be remarked that the theory only describes a finite

QUANTUM-INTERFERENCE

PHENOMENA

no 0.c. currents constant d.c-curront,i,

60 KV b

IN SUPERCONDUCTORS

251

l”d.cl +

+- + IhIm /7hn! -1

-l/2

no d.c current applied a.c current i = lo cos Wt

0

l/2

1

3/2

BIo (h/2e)

“dc t

Fig. 11. Demonstration of the rectification process for an asymmetric double junction (experiment). a. The critical current in the absence of a.c. currents (no r.f. signal). + = critical current in the positive direction - = critical current in the negative direction The total shift of the two curves is equal to (L jii - isl/O)(2e/h). The points of intersection are situated at B,O/(h/2e) = &rn (m integer). b. Measured structure in the d.c. voltage at constant d.c. current = is, applied in the positive (+) and negative (-) directions, in the absence of a.c. currents. c. The observed rectified d.c. voltage when an a.c. current with amplitude is is applied in the absence of a d.c. current.

number of possibilities while in an experimental situation a contact that seems to be one single point contact is in reality a very complicated system of weak contacts in series and parallel each with self-inductances and mutual inductances and capacitances. Therefore the shape of the i - 7 - B, dependence can be very different for two point contacts that are the same from a macroscopic point of view. It is not a contradiction to the theory that a hyperbola-shaped i - 7 curve is seldom observed. Nevertheless lv characteristics were observed that are hyperbolas within the measuring accuracy. Apart from this observation the agreement between theory and experiment is good. The theoretically deduced C - 7 - B, relationship is a corrugated surface with a period in B, equal to (h/2e)/O. In the case of constant current the current is an increasing function of F and the E - 7 characteristic at a certain magnetic field has no points of intersection with an a - 7 characteristic at an other magnetic field value.

252

A. TH.

A. M. DE

WAELE

AND

R. DE

BRUYN

OUBOTER

Therefore a maximum in the voltage oscillations corresponds to a minimum in the i, oscillations. When Z: 7 + 00 the amplitude of the voltage oscillations at constant applied current goes to zero and the i - r curve of the junction approaches the Z - 7 characteristic of the junction in the normal state. The theoretical result that the amplitude of the i, and voscillations goes to zero when L --f 03 is confirmed by experiments with small (L M lo-l2 - lo-11 H) and large (L m 10-a H) enclosed areas. Large amplitudes of the i, and 7 oscillations can only be obtained with small enclosed areas. The B - 7 - B, relationship of double contacts with a small enclosed area was reported previouslyil) and was presented in fig. 1. The agreement between the shape of the i, oscillations as a function of B, and the shape calculated from eq. (17) and ii = 2is is good. With L M 1O-is-IO-11 H and ii M 10-d A, then (2e/lz)/( TL L il ) is of the order 0.7-7 so that the approximation made in section 5b is good for this double contact. The amplitude of the i, oscillations when large areas are enclosed by the contacts is small and difficult to measure. The predicted shape of the voltage oscillations at constant applied current is given in fig. 10~ and is in agreement with experiment (fig. 2). Turning our attention to the a.c. effects rather than to the time-averaged effects, we derived the relation Y = (2e/h) V(t) which has been verified to within 6 ppmis). In the limit (2e/!~)/( z L il ) -+ 0 the current (in the case of a constant voltage) or voltage (in the case of a constant current) is essentially a saw tooth at a frequency of twice the Josephson frequency. When (2e/lz)/(nLii) --f 00 the signal (current or voltage) is sinusoidal and only the first harmonic will be observed when i >>ic. In our laboratoryls) experiments on a.c. effects were carried out with double junctions with (2e/h)/(zLil) M 0.2-2. They were mounted in the centre of the bottom of a coaxial cavity. When the cavity is in a resonant mode the double contact is placed in its own resonant field and is used as a generator and detector at the same time. The l7 characteristic of the double contact mounted in this way was given in fig. 3. The constant-voltage steps correspond to a resonant frequency of the cavity. The values of the step voltage can only be changed by changing the dimensions of the cavity. Since L is only known with an accuracy of a factor 10 and (2e/h)/(nLil) m M 0.2-2 it is reasonable to try to explain the observed phenomena with the model ignoring the self-induced flux ~8~(eq. 33). If the applied flux B,O is equal to an integral number of flux quanta the supercurrents through the individual contacts are in phase and radiation is emitted. When B,O = = (n + 3) h/2e th e supercurrents through the individual contacts are in opposite phase and since the distance between the two point contacts is much smaller than the wavelength of the radiation the net emitted radiation is now much smaller and cannot be detected. Therefore steps of constant voltage are most pronounced when B,O = n h/2e and absent when B,O = = (a + t) h/2e.

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For an asymmetric double junction with Qp,# 0 we derived that the V - B, oscillations with applied currents of i and --i are shifted towards each other with a value of Ii1 - is/ L/O.When an a.c. current (e.g. radiofrequency region) is applied with an amplitude larger than the minimum critical current a d.c. voltage is observed. Experiments of this kind have been reported previouslyl4). Acknowledgments. We take this opportunity to thank Professor K. W. Taconis for his stimulating interest in this work. We express our gratitude to Dr. H. van Beelen for many useful discussions. The experiments discussed in this article were performed in cooperation with Mr. W. H. Kraan 1191%1%17).

REFERENCES 1) Josephson, B. D., Phys. Letters 1 (1962) 251; Rev. mod. Phys. 36 (1964) 216; Advan. Phys. 14 (1965) 419. 2) Anderson, P. W. and Rowe& J. M., Phys. Rev. Letters 10 (1963) 230. 3) Rowell, J. M., Phys. Rev. Letters 11 (1963) 200. 4) Jaklevic, R. C., Lambe, J., Mercereau, J. E. and Silver, A. H., Phys. Rev. 140 (1965) A 1628. 5) Shapiro, S., Phys. Rev. Letters 11 (1963) 80. 6) Fiske, M. D., Rev. mod. Phys. 36 (1964) 221. 7) Yanson, I. K., Svistuno, V. M. and Dmitrenko, I. M., Soviet Physics-JETP 48 (1965) 976 (21 (1965) 650). 8) Eck, R. E., Scalapino, D. J., Taylor, B. N., Phys. Rev. Letters 13 (1964) 15; Langenberg, D. N., Parker, W. H. and Taylor, B. N., Phys. Rev. 150 (1966) 186; Phys. Rev. Letters 18 (1967) 287. 9) Zimmerman, J. E. and Silver, A. H., Phys. Letters 10 (1964) 47; Phys. Rev. 141 (1966) 367. Zimmerman, J. E., Cowen, J. A. and Silver, A. H., Appl. Phys. Letters 9 (1966) 353. 10) Anderson, P. W. and Dayem, A. H., Phys. Rev. Letters 13 (1964) 195. 11) De Bruyn Ouboter, R., Kraan, W. H., De Waele, A. Th. A. M. and Omar, M. H., Commun. Kamerlingh Onnes Lab., Leiden No. 3558; Physica 34 (1967) 525. (See also De Bruyn Ouboter, R., Omar, M. H., Arnold, Miss A. J. P., Guinau, T. and Taconis, K. W., Commun. Leiden No. 348~; Physica 32 (1966) 1448. Omar, M. H. and De Bruyn Ouboter, R., Commun. Leiden No. 352b; Physica 32 ( 1966) 2044). 12) Parker, W. H., Taylor, B. N. and Langenberg, D. N., Phys. Rev. Letters 18 (1967) 287. 13) De Bruyn Ouboter, R., Kraan, W. H., De Waele, A. Th. A. M. and Omar, M. H., Commun. Leiden No. 357~; Physica 35 (1967) 335. 14) De Waele, A. Th. A. M., Kraan, W. H., De Bruyn Ouboter, R. and Taconis, K. W., Commun. Leiden No. 360b; Physica 37 (1967) 114.

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15) Feynman, R. P., Leighton, R. B. and Sands, M., The Feynman Lectures on Physics Vol. III, chap. 21, sec. 21-9 (Addison-Wesley Publ. Comp., New York, 1965). 16) Steward, W. C., Appl. Phys. Letters 12 (1968) 277. 17) De Waele, A. Th. A. M., Kraan, W. H. and De Bruyn Ouboter, R., Commun. Leiden 3683: Physica 40 (1968) 302.