Last lecture (#4) · 2016-07-22 · 1 Last lecture (#4): We completed the discussion of the B-T...
Transcript of Last lecture (#4) · 2016-07-22 · 1 Last lecture (#4): We completed the discussion of the B-T...
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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
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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 " $2
d % exp(i& ) "1
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Voltage-Biased Voltage-Biased Josephson Josephson Weak LinksWeak LinksThe The Josephson Josephson Current-Phase EquationCurrent-Phase Equation
R
VIIII Jns +=+= ! sin
Consider the resistively shunted junction (RSJ):
The total current with bias voltage V is
Since V = (ћ/2e)∂ϕ /∂t, we can rewrite I in terms of the phase ϕalone
This is a strange circuit equation unlike any known in conventionalcircuit theory and it leads to remarkable I-V characteristics. IJ isknown as the Josephson critical current of the weak link and is aconstant that depends on the microscopic details of the junctions.Typical values of IJ are in the range 10-6 A to 10-2 A.
IJ
VI
In
Is
O
teRII J
!
!+=
""
2 sin
h
R
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The The Josephson Josephson Current-Voltage RelationCurrent-Voltage Relation
From the phase-voltage relation V = (ћ/2e)∂ϕ /∂t, we can write ϕas an integral over V(t)
where ϕ0 is a constant. Thus, the current-voltage form of theJosephson equation becomes
Consider first the case V = 0. Then In = 0 and
Current flows without an applied voltage, i.e., Is is indeed asupercurrent flowing through the weak link. This is theDC-Josephson effect.
!
" = "0
+2e
hV(t )dt
0
t
#
0 sin !
JIII s ==
!
I = IJ sin "
0+
2e
hV(t )dt
0
t
#$
% & &
'
( ) ) +
V
R
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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
!
!"!#!
!
""
++
++$$
%
&
''
(
)= *
+
,+=
)(cos 0 tVVVRFRF !+=
side band frequencies
),sin()()sinsin(
xJxodd
!""!
!#=$
%$=
)cos()()sincos(ven
xJxe
!""!
!#=$
%$=
*
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The AC voltage generates a current response at the sidebandfrequencies ωJ0 +νωRF. The DC part of the current is just V0/Runless the the Josephson frequency matches a multiple of theAC frequency, ωJ0 + νωRF =0. In that case we generate a DCsupercurrent |Is| = IJ Jν(ωJRF /ωRF). This is known as the inverseAC Josephson effect. Even though there is a quantuminterpretation – the RF photons supply the energy needed to lifta pair across the junction – the effect is really more subtle. Inparticular the DC tends to zero as the RF power is increased(since Jν ! 0 for all ν).In the DC I-V characteristicthe supercurrent appearsat the so-calledShapiro spikes as shown right.
Shapiro SpikesShapiro Spikes
In = V0/R
V0
I
!
"V0 =h#
RF
2e= fRF$0
ultra sharp spikes (parts in 109)
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Gauge Invariant PhaseGauge Invariant Phase
So far we have ignored the effect of the coupling of the changeto the vector potential. This coupling requires that we look for agauge invariant form of the phase ϕ. Recall that to obtain agauge invariant current we required (lecture 2)
By integration we arrive at a gauge invariant generalization ofthe phase ϕ
This has major consequences for a wide weak link and for twoweak links in an applied field. Here we consider two weak linksused in the design of a SQUID.
!
" = (#1$ #
2) % (#
1$ #
2) $
2e
h
&
' (
)
* + A , ds
1
2
-
h
eA2+!"! ##
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Macroscopic Quantum Interference Between Two Weak Links:Macroscopic Quantum Interference Between Two Weak Links:Matter Field InterferometerMatter Field Interferometer
! "#=
! "#=
b
b
a
a
dsAe
b
dsAe
a
path12
path12
2) path(
2
) path(
h
h
$$
$$
!
"a# "
b= #
2e
hA $ ds = #
2e
h%& = #2'
%
%0
• Phase change ϕ12 from (1) to (2) is given in two ways
• Since ϕ12 (path a) = ϕ12 (path b), we get
a
Itot (1)
Ib
Ia
flux φ(2)
b
ϕa is phase change across junction a,
& ϕb is phase change across junction b
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!""!""hh
ee
avebavea+=#=
• Define ϕa + ϕb = 2ϕave , so that
• The total current Ia +Ib is then
ave
aveave
J
JJtot
I
eI
eII
!"
"#
"!"!
sin cos 2
)(sin )(sin
0$$
%
&
''
(
)=
++*=hh
a
Itot (1)
Ib
Ia
flux φ(2)
b
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• 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
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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
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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 +=