1

Last lecture (#4):Last lecture (#4):

We completed the discussion of the B-T phase diagram of type-Iand type-II superconductors. In contrast to type-I, the type-IIstate has finite resistance unless vortices are pinned by defects.

B

f

Jvortex

• Homogeneous state → Meissner effect → type-I sc• Inhomogeneities at isolated points → vortices → type-II sc• Inhomogeneities at weak links → Josephson effect, SQUIDs

Jtr

2

Lecture 5:Lecture 5:

• Weak links and the Josephson phase relation• Josephson critical current• DC and AC Josephson effect with voltage source

(current source given in an appendix)• Gauge-invariant phase• Quantum interference for weak links• The DC SQUID• Applications of SQUIDS• Other applications of Josephson phenomena:

frequency mixers and voltage standards

• Literature: Waldram chs 6 & 18

3

Weak LinksWeak Links

• Bulk superconductors 1 & 2 are separated by a very thin region of normal metal or insulator.

• A simple model of a 1D superconducting weak link, d<<ξ, and strong link, d>>ξ, is given in the appendix. Here we

consider a more general phenomenological approach

• Phase difference ϕ = θ1 − θ2 evolves as (lecture 2)

ψ

2

2

|| !

"

"

ie

=

h/2/ eVt =!!"

1

1

|| !

"

"

ie

=V

SC 2SC 1

d

4

• In GL free energy density and in expression for current wereplace

• The current through the link is then of the form

I = IJ sin ϕ

In a weak link (i.e., d<<ξ for 1D sc link) the current is periodic in ϕ with period 2π.

• The current I is consistent with a free energy term of the form ΔF = -F0 cos ϕ, where F0 = ћIJ /(2e). Proof:

!

Power = "#F / "t =F0

sin$ "$ / "t = IV as required.

!

"#$ by$1 " $2

d % exp(i& ) "1

5

Voltage-Biased Voltage-Biased Josephson Josephson Weak LinksWeak LinksThe The Josephson Josephson Current-Phase EquationCurrent-Phase Equation

R

VIIII Jns +=+= ! sin

Consider the resistively shunted junction (RSJ):

The total current with bias voltage V is

Since V = (ћ/2e)∂ϕ /∂t, we can rewrite I in terms of the phase ϕalone

This is a strange circuit equation unlike any known in conventionalcircuit theory and it leads to remarkable I-V characteristics. IJ isknown as the Josephson critical current of the weak link and is aconstant that depends on the microscopic details of the junctions.Typical values of IJ are in the range 10-6 A to 10-2 A.

IJ

VI

In

Is

O

teRII J

!

!+=

""

2 sin

h

R

6

The The Josephson Josephson Current-Voltage RelationCurrent-Voltage Relation

From the phase-voltage relation V = (ћ/2e)∂ϕ /∂t, we can write ϕas an integral over V(t)

where ϕ0 is a constant. Thus, the current-voltage form of theJosephson equation becomes

Consider first the case V = 0. Then In = 0 and

Current flows without an applied voltage, i.e., Is is indeed asupercurrent flowing through the weak link. This is theDC-Josephson effect.

!

" = "0

+2e

hV(t )dt

0

t

#

0 sin !

JIII s ==

!

I = IJ sin "

0+

2e

hV(t )dt

0

t

#$

% & &

'

( ) ) +

V

R

7

AC-Josephson AC-Josephson EffectEffect

teV

h

2

0+= !!

The main surprise comes when we apply a finite voltage. Considerfirst a DC voltage V. This leads to a time-dependent phase

and thus to an oscillatory component in the current

is the Josephson frequency.

!

fJ

="

J

2#=

V

$0

= (4.8359... x 108Hz/1µV) V

0/2

2 where ,)( sin

0!"##$ V

eV

R

VtII

JJJ==++=

h

In = V/R

V

I

|Is| · IJ

Remarkably, a DC applied voltagedrives an oscillating DC super-current at a frequency that is (1/φ0)per unit of voltage applied. This isthe AC-Josephson Effect. The DC I-Vcharacteristic of a RSJ weak link isgiven on the right.

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Combined DC and AC Applied VoltagesCombined DC and AC Applied Voltages

h/2 00eV

J=!

Now include both a DC and an AC voltageso that

where and .

Substituting V and ϕ in I = IJ sinϕ + V/R, we find after somemanipulations, using well known harmonic expansions*

/2 hRFeVJRF

=!

) sin( 00

ttRF

RF

JRF

J!

!

!!"" ++=

tR

V

R

V

tJII

RF

RFJ

RF

JRF

RF

J

cos

] ) ([ sin

0

00

!

!"!#!

!

""

++

++$$

%

&

''

(

)= *

+

,+=

)(cos 0 tVVVRFRF !+=

side band frequencies

),sin()()sinsin(

xJxodd

!""!

!#=$

%$=

)cos()()sincos(ven

xJxe

!""!

!#=$

%$=

*

9

The AC voltage generates a current response at the sidebandfrequencies ωJ0 +νωRF. The DC part of the current is just V0/Runless the the Josephson frequency matches a multiple of theAC frequency, ωJ0 + νωRF =0. In that case we generate a DCsupercurrent |Is| = IJ Jν(ωJRF /ωRF). This is known as the inverseAC Josephson effect. Even though there is a quantuminterpretation – the RF photons supply the energy needed to lifta pair across the junction – the effect is really more subtle. Inparticular the DC tends to zero as the RF power is increased(since Jν ! 0 for all ν).In the DC I-V characteristicthe supercurrent appearsat the so-calledShapiro spikes as shown right.

Shapiro SpikesShapiro Spikes

In = V0/R

V0

I

!

"V0 =h#

RF

2e= fRF$0

ultra sharp spikes (parts in 109)

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Gauge Invariant PhaseGauge Invariant Phase

So far we have ignored the effect of the coupling of the changeto the vector potential. This coupling requires that we look for agauge invariant form of the phase ϕ. Recall that to obtain agauge invariant current we required (lecture 2)

By integration we arrive at a gauge invariant generalization ofthe phase ϕ

This has major consequences for a wide weak link and for twoweak links in an applied field. Here we consider two weak linksused in the design of a SQUID.

!

" = (#1$ #

2) % (#

1$ #

2) $

2e

h

&

' (

)

* + A , ds

1

2

-

h

eA2+!"! ##

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Macroscopic Quantum Interference Between Two Weak Links:Macroscopic Quantum Interference Between Two Weak Links:Matter Field InterferometerMatter Field Interferometer

! "#=

! "#=

b

b

a

a

dsAe

b

dsAe

a

path12

path12

2) path(

2

) path(

h

h

$$

$$

!

"a# "

b= #

2e

hA $ ds = #

2e

h%& = #2'

%

%0

• Phase change ϕ12 from (1) to (2) is given in two ways

• Since ϕ12 (path a) = ϕ12 (path b), we get

a

Itot (1)

Ib

Ia

flux φ(2)

b

ϕa is phase change across junction a,

& ϕb is phase change across junction b

12

!""!""hh

ee

avebavea+=#=

• Define ϕa + ϕb = 2ϕave , so that

• The total current Ia +Ib is then

ave

aveave

J

JJtot

I

eI

eII

!"

"#

"!"!

sin cos 2

)(sin )(sin

0$$

%

&

''

(

)=

++*=hh

a

Itot (1)

Ib

Ia

flux φ(2)

b

13

• The critical Josephson current for the pair of links is

• Ic oscillates with φ with period equal to the flux quantum φ0.

• Analogy to interference from a pair of Young slits, but now formatter waves instead of light waves.

• Superconducting Quantum Interference Device or SQUID:high-sensitivity measurements of magnetic fields, voltagesand currents in the fT, fV and fA ranges, respectively.

φ

Ic

φ0

|)(cos|2 0!

!"Jc II =

14

The device is highly sensitive: under ideal conditions one canmeasure a change of 10-6 φ0/√Hz. SQUIDs are used as precisionmagnetometers in the examples below:

SQUID ApplicationsSQUID Applications

Scanning SQUIDmicroscopy

for exotic experimentson high-Tcs

(more later…)

Magneto-encephalography

measures tiny magneticfields (fT range) createdby active areas in the

brain

Magnetic propertiesmeasurement system

for susceptibilitymeasurements etc. – candetect moments down to

~10-13 Am2

Pict

ure

cre

dits:

J.

R. Kirtley

; 4-D

Neu

roim

agin

g; Q

uan

tum

Des

ign

15

The exactness of the Josephson frequency-voltage relationωJ = 2eV/ has led to the adoption of Josephson junction arraysas the primary voltage standard: an incident RF field is tuned tomatch the Shapiro steps of the array. Frequency can bemeasured highly accurately, and the Shapiro steps are extremelysharp, giving a relative voltage uncertainty of 1 part in 109.This is one out of several quantum standards that haverevolutionized metrology, the quantum Hall effect resistancestandard being another prominent example.

Some other applications include microwave detectors andfrequency mixers – exploiting the strong nonlinearity and thesideband generation of the weak link, respectively. A lot ofresearch effort is presently going into developing super-conducting transistors and quantum computers usingsuperconducting qubits based on circulating currents andenclosed flux in weak link circuits or non-analytic anyonsin exotic pairing states (more later …).

Other Applications of the Other Applications of the Josephson Josephson EffectEffect

!

h

16

Appendix 1: Short Quasi-1D SuperconductingAppendix 1: Short Quasi-1D SuperconductingWeak LinkWeak Link

• for a weak link, 0<x<d<<ξ , the first GL equation

reduces to ψ” ≅ 0 → ψ = a + bx so that from boundary

conditions

The current is proportional to

• For a strong link, 0 < x < d >> ξ , the first GL equation gives

ψ (1 - |ψ|2) ≅ 0 → ψ = exp(-iϕx/d)

so that (instead of )

d

!""

sin)*(Im =#$

d

! sin

)1(1 !" ie

d

x ###=

0 d

ψ(0) = 1 !" ied#

=)(

x

d

!"" =#$ )*(Im

0)1(''22

=!+ """#

Assume β =−α :

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Appendix 2: Simplified Treatment of Combined DCAppendix 2: Simplified Treatment of Combined DCand AC Applied Voltagesand AC Applied Voltages

RFeV !

2

h=

• IS =

• The term (cos A) B is proportional to

• Thus, we will get a DC Josephson effect whenh

eV

JRF

2

0== !!

] )sin[(2

1 ] )sin[(

2

1

) sin( ) cos(

00

0

tt

tt

JRFJRF

RFJ

!!!!

!!

"++=

)] ( sin [ sin 0

ttIRF

RF

JRF

JJ !!

!! +

] sin cos cos sin[ BABAIJ +=

B

B + …

becomes DC if

A

if ϕ0 = 0

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Appendix 3: Current Biased Weak LinksAppendix 3: Current Biased Weak Links

In practice, it is usually the current rather than the voltagethat is controlled in a weak link. We need to invert theJosephson phase equation. Suppose first that we apply aDC current I0 to the RSJ. Then from p. 5 the phase ϕ isgiven by

If |I0| ≤ IJ, the phase reaches an equilibrium value given by∂ ϕ /∂ t = 0, i.e., I0 = IJ sin ϕequil . Note that ∂ ϕ /∂ t = 0means V = 0. If |I0| > IJ such an equilibrium is not possibleand ϕ keeps changing with time.

ϕϕ

|I0| · IJ:equilibrium

|I0| > IJ:rolling, rolling,rolling…

!!

sin 2

0 JIIteR

"=#

#h

19

It can be shown that for a given I0 the phase ϕ satisfies

where V0 is the mean voltage across the linkdefined by

This gives the current-bias I-Vcharacteristic shown on the right.

2

tan

2 tan 0

0

t

R

VII JJ

!"+=

The I-V characteristic under current bias and under voltage biasare therefore quite different.

Generally, the weak link equations have to be integratednumerically.

V0

I0IJ

– IJ

220

20 /RVII J +=