1

Josephson effect

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: April 27, 2012)

Brian David Josephson, FRS (born 4 January 1940; Cardiff, Wales) is a Welsh physicist. He

became a Nobel Prize laureate in 1973 for the prediction of the eponymous Josephson effect. As

of late 2007, he was a retired professor at the University of Cambridge, where he is the head of

the Mind–Matter Unification Project in the Theory of Condensed Matter (TCM) research group.

He is also a fellow of Trinity College, Cambridge.

http://en.wikipedia.org/wiki/Brian_David_Josephson

1. DC Josephson junction

Fig. Schematic diagram for experiment of DC Josephson effect. Two superconductors

SI and SII (the same metals) are separated by a very thin insulating layer (denoted

2

by green). A DC Josphson supercurrent (up to a maximum value Ic) flows without

dissipation through the insulating layer.

Let 1 be the probability amplitude of electron pairs on one side of a junction. Let 2

be the probability amplitude of electron pairs on the other side. For simplicity, let both

superconductors be identical.

12

21 ,

Tt

i

Tt

i

ℏℏ

ℏℏ

where Tℏ is the effect of the electron-pair coupling or (transfer interaction across the

insulator). T(1/s) is the measure of the leakage of 1 into the region 2, and of 2 into the

region 1. Let

2

1

22

11 ,

i

i

en

en

where

2

2

2

1

2

1

n

n

Then we have

ienniTt

int

n21

11

1

2

1

ienniTt

int

n

212

22

2

1

where

12 .

Now equate the real and imaginary parts of Eqs.(3) and (4),

3

cos

cos

sin2

sin2

2

12

1

21

212

211

n

nT

t

n

nT

t

nnTt

n

nnTt

n

If 21 nn as for identical superconductors 1 and 2, we have

tt

21

or 0)( 12

t

.

t

n

t

n

21

The current flow from the superconductor S1 and to the superconductor S2 is proportional

to t

n

2 . J is the current of superconductor pairs across the junction

)sin( 120 JJ ,

where J0 is proportional to T (transfer interaction).

sin0II . (5)

2. AC Josephson effect

Fig. Schematic diagram for experiment of AC Josephson effect. A finite DC voltage is

applied across both the ends.

4

Let a dc voltage V be applied across the junction. An electron pair experiences a potential energy difference qV on passing across the junction (q = -2e). We can say that a pair on

one side is at –eV and a pair on the other side is at eV.

212

121

eVTt

i

eVTt

i

ℏℏ

ℏℏ

or

ienniTieVn

tin

t

n21

111

1

2

1

ℏ,

ienniTieVn

tin

t

n

2122

22

2

1

ℏ.

This equation breaks up into the real part and imaginary part,

cos

cos

sin2

sin2

2

12

1

21

212

211

n

nT

eV

t

n

nT

eV

t

nnTt

n

nnTt

n

ℏ

ℏ

.

From these two equations with 21 nn ,

ℏ

eV

tt

2)( 12

.

)](sin[0 tJJ .

with

Vdte

tℏ

2)0()( .

5

When V = V0 = constant, we have

tVe

t 0

2)0()(ℏ

.

]2

)0(sin[ 00 tVe

JJℏ

.

The current oscillates with frequency

Ve

ℏ

20 .

A DC voltage of 1 eV produces a frequency of 483.5935 MHz.

Ve

ℏ

ɺ 2 .

((Note))

Suppose that V = V0 = 1 V. The corresponding frequency is estimated from the relation,

00 2

2

ℏ

eV,

or

MHzeV

5935.4831005459.12

10)1060219.1(2

2

227

612

00

ℏ.

3. I-V characteristic of Josephson tunneling junction

We now consider the I-V characteristic of the Josephson tunneling junction where a

insulating layer is sandwiched between two superconducting layers (the same type). A capacitor is formed by these two superconductors. In this type of Josephson junctions, one can see the

quasiparticle I-V curve which is different with increasing voltage and decreasing voltage

(hysteresis). There are two voltage states, 0 V and 2/e, where is an energy gap of each superconductor. The I-V curve is characterized by (i) maximum Josephson tunneling current of

Cooper pairs at V = 0 and (ii) Quasi-particle tunneling current (V>2/e).

6

Fig. Schematic diagram of quasiparticle I-V characteristic (usually observed in a S-I-S

Josephson tunneling-type). Josephson current (up to a maximum value Ic) flows at V = 0.

is an energy gap of the superconductor . The DC Josepson supercurrent flows under V

= 0. For V>2/e the quasiparticle tunneling current is seen.

The strong nonlinearity in the quasiparticle I-V curve of a tunneling junction is not an

appropriate to the application to the SQUID element. This nonlineraity can be removed by the

use of thin normal film deposited across the electrodes. In this effective resistance is a parallel

combination of the junction. The I-V characteristic has no hysteresis. Such behavior is often

observed in the bridge-type Josephson junction where two superconducting thin films are

bridged by a very narrow superconducting thin film.

Fig. Schematic diagram of I-V characteristic of a Josephson junction (usually observed in

bridge-type junction), which is reversible on increasing and decreasing V. A Josephson

supercurrent flows up to Ic at V = 0. A transition occurs from the V = 0 state to a finite

7

voltage state for I>Ic.. Above this voltage the I-V characteristic exhibits an Ohm’s law

with a finite resistance of the Junction. The current has an oscillatory component of

angular frequency (= 2eV/ħ) (the AC Josephson effect). Note that in order to avoid the

hysteresis of I-V characteristic, the normal metal is used as a shunt resistance R. Two

superconductors are connected in parallel with this resistance (resistively shunted

junction (RSJ), see below).

________________________________________________________________________

_________________________________________________________________________

4. RSJ (Resistively shunted junction) model: Josephson junction circuit application7

Here we discuss the I-V characteristics of a Josephson tunneling junction using an equivalent

circuit shown below. This circuit includes the effect of various dissipative processes and the

distributed capacity with so-called lumped circuit parameters (connection of R and C in parallel).

4.1 Fundamental equation

Fig. Equivalent circuit of a real Josephson junction with a current noise source. (RSJ

model). J.J. stands for the Josephson junction. IN(t) is the noise current source.

We now consider an equivalent circuit for the Josephson junction, which is described above. J.J.

stands for the Josephson junction.

ItIVCR

VI Nc )(sin ɺ ,

where the first term is a Josephson current, the second term is an ohmic current, the third term is

a displacement current, and IN(t) is the noise current source. Here we neglect this term. Since

Ve

ℏ

ɺ 2 ,

we get a second-order differential equation for the phase

8

sin22

cIIeRe

C ɺ

ℏɺɺ

ℏ,

with = I/Ic, where 0 is a magnetic quantum flux. ________________________________________________________________________

4.2 Differential equation for For the sake of simplicity, we use the dimensionless quantities. Here we assume that

,RI

V

c

cI

I , tJ ,

where J is the Josephson plasma frequency and is defined by

2/1

2

C

eIcJ

ℏ ,

Jd

d

dt

d

d

d

dt

d

.

Similarly we have

2

22

2

2

d

d

dt

dJ .

Then we get

sin22 2

22

d

d

eRId

d

eI

CJ

c

J

c

ℏℏ,

or

sin2

2

d

d

d

dJ ,

where

RCC

eI

eRIeRIeRI J

c

JcJc

J

c

JJ

12

222

2

ℏ

ℏℏℏ.

The normalized voltage is described by

9

d

d

d

d

eRIdt

d

RIeRI

VJ

c

J

cc

2

1

2)(

ℏℏ,

since

Ve

ℏ

ɺ 2 .

4.3. I-V characteristic for J >> 1

For the special case J>>1 (small capacitance limit), the above equation is reduced to

sind

dJ ,

TRId

d

d

TRI

d

dRIV J

T

JJc

0

0

0 21

,

where T is a period;

JT

1

22

.

Here

2

0

2

00sin

d

d

d

ddT J

T

,

or

ddddT

J

]sin

1

sin

1[

sinsinsin000

2

0

,

or

1

22

J

T for >1,

0J

T

for <1,

So we get

10

122

RI

TRIV c

Jc ,

or

12

RI

V

c

,

or 2

1

RI

V

c

.

________________________________________________________________________ ((Mathematica))

Clear@"Global`∗"D;

K1@x_D := IntegrateB1

x − Sin@φD+

1

x + Sin@φD, 8φ, 0, π<,

Assumptions → x > 1F

K1@xD2 π

−1 + x2

eq1 = V1 ==2 π Ic R

K1@xD; eq2 = Solve@eq1, xD; I1 = x ê. eq2@@2DD ê. 8R → 1, Ic → 1<;

Plot@8I1, V1<, 8V1, 0, 4<, PlotStyle → 88Red, Thick<, 8Blue, Thick<<,AxesLabel → 8"VêIcR", "IêIc"< D

1 2 3 4VêIcR

1

2

3

4

IêIc

11

Fig. I/Ic vs V/(IcR) curve for J>>1 (red curve). The blue curve shows the Ohm’s law:

I/Ic = V/(IcR)

5. Flux quantization

We start with the current density

ss qc

q

m

qvAJ

2**

2

*

*

)( ℏ

ℏ.

Suppose that 2

sn =constant. Then we have

AJℏℏ c

q

nq

ms

s

*

*

*

,

or

lAlJl dc

qd

nq

md s

sℏℏ

*

**

*

.

The path of integration can be taken inside the penetration depth where sJ =0.

ℏℏℏℏ c

qd

c

qd

c

qd

c

qd

****

)( aBaAlAl ,

where is the magnetic flux. Then we find that

ℏc

qn

*

12 2 ,

where n is an integer. The phase of the wave function must be unique, or differ by a multiple of

2 at each point,

nq

c*

2 ℏ .

The flux is quantized. When |q*| = 2|e|, we have a magnetic quantum fluxoid;

e

ch

e

c

e

c

22

20

ℏℏ = 2.06783372 × 10-7 Gauss cm2

((Note))

12

The current flows along the ring. However, this current flows only on the surface boundary

(region from the surface to the penetration depth ). Inside of the system (region far from the

surface boundary), there is no current since c/4 JH and H = 0.

6. DC SQUID (double junctions): quantum mechanics

DC SQUID consists of two points contacts in parallel, forming a ring. Each contact forms a

Josephson junctions of superconductor 1, insulating layer, and superconductor 2 (S1-I-S2).

Suppose that a magnetic flux passes through the interior of the loop.

Fig. Schematic diagram of superconducting quantum interference device. 1 and 2

refer to two point-contact weak links. The rest of the circuit is strongly

superconducting.

Here we have

bbaad 2112 l .

or

0

2112 2

bbaa

or

0

21 2

where )( 111 ab is the phase difference between the superconductors a and b through the

junction 1 and )( 222 ab are is the phase difference between the superconductors a and b

through the junction 2.

13

When B = 0 (or = 0), we have 021 . In general, we put the form

c

e

ℏ01 ,

c

e

ℏ02 .

The total current is given by

)cos()sin(2

)]sin()[sin(

)]sin()[sin(

0

00

2121

c

eI

c

e

c

eI

IIII

c

c

c

ℏ

ℏℏ

or

)cos()sin(20

0

cII .

since

c

e

ℏ

0

.

The current varies with and has a maximum of 2Ic when sc

e

ℏ (s: integers),

or

sse

hcs

e

c0

2

ℏ.

The simple two point contact device corresponds to a two-slit interference pattern, for which the

physically interesting quantity is the modulus of the amplitude rather than the square modulus, as it is for optical interference patterns.

7. Critical current

The maximum of I (called as the critical current) is given by

)cos(20

max

cII .

Note that

cII 2max

14

when = 0. This means that the critical current is a periodic function of the magnetic flux .

Fig. Ideal case for the IB/Ic vs ext/0 curve in the DC SQUID, where IB is the maximum

supercurrent. IB = 2 Ic when ext/0 = n (integer) and IB = 0 for ext/0 = n +1/2.

_________________________________________________________________________

8. Analogy of the diffraction with double slits and single slit

Fig. Diffraction effect of Josephson junction. A magnetic field B along the z direction, which

is penetrated into the junction (in the normal phase).

We consider a junction (1) of rectangular cross section with magnetic field B applied in the plane

of the junction, normal to an edge of width w,

]2

sin[]sin[2

1

10

2

1

10 lAlA dc

eJd

c

qJJ

ℏℏ ,

0.5 1 1.5 2 2.5 3

ΦextêΦ0

0.5

1

1.5

2

IBêIc

15

with q = -2e. We use the vector potential A given by

)0,2

,2

()(2

1 BxBy rBA ,

)0,0,(' By AA ,

where

2

Bxy .

Then we have

]2

sin[])(2

sin[)( 1010 yWc

eBJdxBy

c

eJyJ

b

a

x

xℏℏ

,

dyyWc

eBLJLdyyJdI ]

2sin[)( 101

ℏ ,

or

2/

0

10

2/

0

10

2/

0

10

2/

2/

101

)2

sin(1

sin

)2

sin(2

1sin2

)2

cos(sin2

]2

sin[

t

t

t

t

t

yWc

eB

Wc

eBLJ

yWc

eB

Wc

eBLJ

dyyWc

eBLJ

dyyWc

eBLJI

ℏ

ℏ

ℏ

ℏ

ℏ

ℏ

.

Then we have

)sin()sin( 101 Wtc

eB

eBW

cLJI

ℏ

ℏ .

Here we introduce the total magnetic flux passing through the area Wt ( BWtW ),

16

c

eBWt

e

c

BWtW

ℏℏ

2

20

, or c

eBWtW

ℏ

0

.

Therefore we have

0

011

)sin(

)sin(W

W

cII

.

where

LtJIc 0 .

The total current is given by

0

0

0

0

0

021

21

)sin(

)cos()sin(2

)sin(

)]sin()sin([

W

W

c

W

W

cc

I

II

III

,

. (32)

where

)cos(sin2)]sin()[sin(0

021

cc II

The short period variation is produced by interference from the two Josephson junctions, while

the long period variation is a diffraction effect and arises from the finite dimensions of each junction. The interference pattern of |I|2 is very similar to the intensity of the Young’s double

slits experiment. If the slits have finite width, the intensity must be multiplied by the diffraction pattern of a single slit, and for large angles the oscillations die out.

((Example))

The pattern of |I|2 vs 0/ is very similar to the pattern of the Young's double slit experiments.

17

Fig. Plot of 2

0

])sin(2

[cI

I vs

0

, where A/(Wt) = 10. The envelope arises from

2

00

2 /)(sin

WW

and the oscillation with short period arises from

)(cos0

2

_____________________________________________________________________________

9. Young’s double slit experiment

We consider the Young’s double slits (the slits are separated by d). Each slit has a finite

width a.

AêHWtL=10

-20 -10 10 20FêF0

0.2

0.4

0.6

0.8

1.0

squares of amplitude

18

Fig. Geometric construction for describing the Young’s double-slit experiment (not to scale).

((double slits))

E is the electric field of a light with the wavelength . d is the separation distance between the

centers of the slits.

Fig. A reconstruction of the resultant phasor ER which is the combination of two phasors (E0).

2cos2 0

EER .

The intensity:

)cos1(22

cos42

0

22

0

2

EEEI R .],

where the phase difference is given by

sin2 d

((single slit))

We assume that each slit has a finite width a.

19

Fig. Phaser diagram for a large number of coherent sources. All the ends of phasors lie on the

circular arc of radius R. The resultant electric field magnitude ER equals the length of the

chord.

RE 0 ,

2

2sin

2sin2

2sin2 0

0

EE

RER .

where the phase difference is given by

sin2 a

. Then the resultant intensity I for the

double slits (the distance d) (each slit has a finite width a) is given by

222)

2

2sin

)(cos1(2

)

2

2sin

(2

cos

mm

III .

10. Principle of DC SQUID

For the special case J>>1 (small capacitance limit), we have

sind

dJ ,

for the RSJ model. In the DC SQUID, the two Josephson junctions (junctions 1 and 2) are

connected in parallel.

20

sB

Jd

d

2

sin 11 , for the Josephson junction-1

sB

Jd

d

2

sin 22 , for the Josephson junction

where

0

12 2

ext . (3)

From these three equations, we have

2)sin()cos()(

0

1

00

1Bextextext

Jd

d

.

When we introduce a new parameter

10

1

ext ,

we have the final form

2sin)cos( 1

0

1 BextJ

d

d

.

We are interested in the DC current-voltage characteristic so we need to determine the time

averaged voltage

2

20

111Jc

T

JcJc RId

d

d

T

RI

d

dRI

dt

d

eV

ℏ.

Here

'

)cos(

1

sin')cos(sin)cos(

2 0

2

0 1

1

0

2

01

0

1

2

0 1

1

extext

J

extB

J dd

d

d

d,

)cos(2

'

0

ext

B

,

where

21

dddd

]sin'

1

sin'

1[

sin'sin'sin''

000

2

0

,

or

1'

2'

2

for >1,

0' for <1.

Then we have

)cos(1')cos('

212

0

2

0

extc

extcJc RI

RIRIV

,

or

1')cos( 2

0

ext

cRI

V,

or

c

B

extextc

I

I

RI

V

)cos(2

1

)cos(

1'

0

2

0

,

or

2

0

2 )()(cos2 c

ext

c

B

RI

V

I

I

,

or

)(cos)2

(0

22

ext

c

B

c I

I

RI

V ,

or

22

)(cos42 0

222

extcB II

RV . (45)

When this equation for the voltage is compared with that for one Josephson junction with J»1

22

cIIRV .

We find that the critical current is )cos(20

extcI . This means that the critical current is 2Ic for

next 0/ (integer) and zero for next 0/ +1/2. In other words, the critical current is a

periodic function of with a period 0 . However, the actual critical current does not oscillate

between 0 and 2Ic because of the finite self-inductance L. In the above model, L (or = 0) is

assumed to be zero. The critical current varies between 2Ic and finite value depending on the

value of (see the detail in Sec.7.1).

When the total current IB is constant, the voltage across the DC SQUID periodically changes

with the external magnetic flux. This is the phenomenon one exploit to create the most sensitive

magnetic field detection.

Fig. The critical current of the I-V characteristic changes periodically with the

magnetic flux , with the period of the quantum fluxoid 0. Note that here we

neglect the effect of the penetration of magnetic field into the insulation layers.

Suppose that the current in the circuit is fixed. Then the voltage across the DC

SQUID periodically changes with increasing with the period of the quantum

fluxoid 0.

11. Experimental procedure for Mr. SQUID

V

FêF00 1 2 3

0.5 1.0 1.5 2.0IêH2IcL

0.2

0.4

0.6

0.8

1.0

VêHRê2L

23

This figure is obtained from the Instruction manual of Mr. SQUID.9

Fig. Detected voltage vs the magnetic flux . The current IB is kept at fixed value which is

a little larger than 2Ic. The detected voltage shows a maximum for = (n+1/2)0, and a

minimum for = n0. The detected voltage is a periodic function of with a period of

0. (This figure is copied from the User Guide of Mr SQUID9).

12. Experimental result from Mr. SQUID (Advanced Lab. Binghamton

University)

We use Mr SQUID for the measurement on the properties of the Josephson effect.

24

REFERENCES

1. P.G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, 1966).

2. M. Tinkam, Introduction to Superconductivity (Robert E. Krieger Publishing Co.

Malabar, Florida, 1980).

3. C. Kittel, Introduction to Solid State Physics, seventh edition (John Wiley & Sons, Inc.,

New York, 1996).

4. J.B. Ketterson and S.V. Song, Superconductivity, Cambridge University Press, New

York, 1999).

5. L. Solymar, Superconductive Tunneling and Applications (Chapman and Hall Ltd.

London, 1972).

6. C.M. Falco, Am. J. Phys. 44, 733 (1976).

7. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (John Wiley

& Sons, New York 1982).

8. Josephson effects-Achievements and Trends, edited by A. Barone (World Scientific 1986

Singapore.

9 Mr. SQUID Use’s Guide. Version 6.2.1 (2002).

http://hep.stanford.edu/~cabrera/images/pdf/mrsquid.pdf

25

10. W.G. Jenks, S.S.H. Sadeghi, and J.P. Wikswo, Jr., J. Phys. D: Applied Phys. 30, 293

(1997).

11. Y. Ohtsuka, Chapter 1 (p.1 – 86), SQUID, in Progress in Measuremenrs of Solid State Physics II, edited by S. Kobayashi (in Japanese) (Maruzen, Tokyo, 1993). This book was

very useful for us in writing this lecture note. However,unfortunately this book was

written in Japanese.

12. M. Suzuki; Lecture note on Josephson junction and DC SQUID.

http://www2.binghamton.edu/physics/docs/note-josephson-junction.pdf

APPENDIX-1

Current density for the superconductors

We consider the current density for the superconductor. is the order parameter of the

superconductor and m* and q* are the mass and charge of the Cooper pairs. The current density

is invariant under the gauge transformation.

]ˆRe[*

*

*

ApJc

q

m

qs ,

This can be rewritten as

A

AA

AJ

cm

q

im

q

c

q

ic

q

im

q

c

q

im

qs

*

22*

**

*

*

**

***

*

*

*

**

*

*

)(2

)]()[(2

)](Re[

ℏ

ℏℏ

ℏ

The density is also gauge independent.

2

rs .

Now we assume that

)(

)()(r

rr i

e .

We note that

26

)()(])([)(

)(])()()([)(

)(])([)()(

*2

2*

)()()(

2*

)(**

*

rrArr

rArrrr

rArrAp

rrr

r

ℏℏ

ℏ

ℏ

ℏ

ic

q

c

qeei

ie

c

qe

ic

q

iii

i

.

The last term is pure imaginary. Then the current density is obtained as

ss qc

q

m

qvAJ

2**

2

*

*

)( ℏ

ℏ

or

smc

qvA *

*

ℏ .

Since

Apvπc

qm s

**

we have

ℏsmc

qvAp *

*

Note that Js (or vs) is gauge-invariant. Under the gauge transformation, the wave function is

transformed as

)()exp()('*

rr

c

iq

ℏ .

This implies that

c

q

ℏ

*

' ,

Since AA' , we have

27

)(

)]()([

)''('

*

**

*

A

A

AJ

ℏℏ

ℏℏℏ

ℏℏ

c

q

c

q

c

q

c

qs

.

So the current density is invariant under the gauge transformation.