1
Fundamentals of Microwave Superconductivity
Short Course TutorialSuperconductors and Cryogenics in Microwave Subsystems
2002 Applied Superconductivity ConferenceHouston, Texas
Steven M. Anlage
Center for Superconductivity ResearchPhysics Department
University of MarylandCollege Park, MD 20742-4111 USA
2
ObjectiveTo give a basic introduction to superconductivity, superconducting electrodynamics, and microwave measurements as background for the Short Course
Tutorial “Superconductors and Cryogenics in Microwave Subsystems”
3
Outline
• Superconductivity• Microwave Electrodynamics of
Superconductors• Experimental High Frequency
Superconductivity• Current Research Topics• Further Reading
4
Superconductivity
• The Three Hallmarks of Superconductivity
• Superconductors in a Magnetic Field
• Where is Superconductivity Found?
• BCS Theory
• High-Tc Superconductors
• Materials Issues for Microwave Applications
5
The Three Hallmarks of Superconductivity
Zero Resistance
I
V
DC
Res
ista
nce
TemperatureTc
0
Complete Diamagnetism
Mag
netic
Indu
ctio
n
TemperatureTc
0
T>Tc T<Tc
Macroscopic Quantum Effects
Flux Φ
Flux quantization Φ = nΦ0Josephson Effects
6
Zero ResistanceR = 0 only at ω = 0 (DC)
R > 0 for ω > 0
EQuasiparticles
Cooper Pairs
2∆
0
The Kamerlingh Onnes resistance measurement of mercury. At 4.15K the resistance suddenly dropped to zero
EnergyGap
7
Perfect DiamagnetismMagnetic Fields and Superconductors are not generally compatible
The Meissner Effect
The Yamanashi MLX01 MagLevtest vehicle achieved a speed of
343 mph (552 kph) on April 14, 1999
Super-conductor
T>Tc T<Tc
H
H
Spontaneous exclusion of magnetic flux
H
H
( ) 00 =+= MHB
µ
T
λ(T)
λ(0)
Tc
LzeHH λ/0
±=
( )zH
z
vacuum superconductor
λ
λ is independent of frequency (ω < 2∆)
magneticpenetration
depth Β=0
surfacescreeningcurrents
λ
8
Macroscopic Quantum EffectsSuperconductor is described by a singleMacroscopic Quantum Wavefunction
φieΨ=Ψ
Consequences:Magnetic flux is quantized in units of Φ0 = h/2e (= 2.07 x 10-15 Tm2)
R = 0 allows persistent currentsCurrent I flows to maintain Φ = n Φ0 in loopn = integer, h = Planck’s const., 2e = Cooper pair charge
Flux Φ
Isuperconductor
Sachdev and Zhang, Science
Magnetic vortices have quantized flux
vortexlattice
B
screeningcurrentsx
B(x)|Ψ(x)|
0 ξ λ
Type IIξ << λ
vortexcore
A vortexLine cut
9
Macroscopic Quantum Effects Continued
Josephson Effects (Tunneling of Cooper Pairs)
111
φieΨ=Ψ 222
φieΨ=Ψ
I
DC Josephson Effect( )21sin φφ −= cII
( )
DCVe∗
=− 21 φφAC Josephson
Effect
Quantum VCO:VDC
1 2
(Tunnel barrier)
+=
∗
0sin φtVeII DCc
μVMHz593420.4831*
0
=Φ
=he
10
AbrikosovVortex Lattice
Superconductors in aMagnetic Field
The Vortex State
T
Η
Hc1(0)
Tc
Meissner State
Hc2(0)
B = 0, R = 0
B ≠ 0, R ≠ 0
NormalState
Type II SC
vortex
LorentzForce
Moving vorticescreate a longitudinal voltage
V>0
I
0Φ×= JFL
J
vortexDrag vF η−=
Vortices also experiencea viscous drag force:
11
What are the Limits of Superconductivity?
Phase Diagram
SuperconductingState
NormalState
Ginzburg-Landau free energy density Applied magnetic fieldCurrentsTemperature
dependence
Tcµ0Hc2
Jc
12
BCS Theory of Superconductivity
http://www.chemsoc.org/exemplarchem/entries/igrant/hightctheory_noflash.html
First electron polarizes the lattice Second electron is attracted to the concentration of positive chargesleft behind by the first electron
Sv+ +
++
Sv + +
++
s-wave ( = 0) pairing
Spin singlet pair
Cooper Pair
NVDebyec eT /1−Ω≅
ΩDebye is the characteristic phonon (lattice vibration) frequencyN is the electronic density of states at the Fermi EnergyV is the attractive electron-electron interaction
An energy 2∆(T) is required to break a Cooper pair into two quasiparticles (roughly speaking)
∆⋅=
FvξCooper pair size:
A many-electron quantum wavefunction Ψ made up of Cooper pairs is constructedwith these properties:
Bardeen-Cooper-Schrieffer (BCS)
13
Where do we find Superconductors?
Most of these materials, and their compounds, display spin-singlet pairing
Nb-Ti, Nb3Sn, A3C60, NbN, MgB2, Organic Salts ((TMTSF)2X, (BEDT-TTF)2X), Oxides (Cu-O, Bi-O, Ru-O,…), Heavy Fermion (UPt3, CeCu2Si2,…), Electric Field-Effect Superconductivity (C60, [CaCu2O3]4, plastic), …
Also:
14
The High-Tc Cuprate SuperconductorsLayered structure – quasi-two-dimensionalAnisotropic physical propertiesCeramic materials (brittle, poor ductility, etc.)Oxygen content is critical for superconductivity
YBa2Cu3O7-δ Tl2Ba2CaCu2O8Two of the most widely-used HTS materials in microwave applications
Spin singlet pairingd-wave ( = 2) pairing
15
HTS Materials Issues Affecting Microwave Applications
High-Tc small Cooper Pair size (ξ – correlation length)
ξ ~ 1 – 2 nm for HTS materials used in microwave applications
Superconducting pairing is easily disrupted by defects:grain boundariescracks
Josephson weak links are created, leading to:nonlinear resistance and reactanceintermodulation of two microwave tonesharmonic generationpower-dependence of insertion loss, resonant frequency, Q
ξ = vF ⋅
∆∝ vF ⋅
1Tc
Most HTS materials made as epitaxialthin films for use in planar microwave devices
film
substrate
grainboundaries
16
Microwave Electrodynamics of Superconductors
• Why are Superconductors so Useful at Microwave Frequencies?
• The Two-Fluid Model
• London Equations
• BCS Electrodynamics
• Nonlinear Surface Impedance
17
Why are Superconductors so Useful at Microwave Frequencies?
Low Losses: Filters have low insertion loss Better S/N, can be made smallNMR/MRI SC RF pickup coils x10 improvement in speedHigh Q Steep skirts, good out-of-band rejection
Low Dispersion: SC transmission lines can carry short pulses with little distortionRSFQ logic pulses – 1 ps long, ~2 mV in amplitude: ( ) psmV07.20 ⋅=Φ=∫ dttV
Superconducting Transmission Lines
E
BJ
microstrip(thickness t)
groundplane
C
KineticInductanceLkin
GeometricalInductanceLgeo
LCvphase
1=propagating TEM wave
tLkin
2
~ λ
L = Lkin + Lgeo is frequency independentattenuation α ~ 0
18
Electrodynamics of SuperconductorsIn the Meissner State
EQuasiparticles(Normal Fluid)
Cooper Pairs(Super Fluid)
2∆
0 Ls
σn
≈Superfluid channel
Normal Fluid channel
EnergyGap
σ = σn – i σ2
J = Js + JnJs
Jn
Current-carrying superconductor
J = σ E σn = nne2τ/mσ2 = nse2/mω
nn = number of QPsns = number of SC electronsτ = QP momentum relaxation timem = carrier massω = frequencyT0
nnn(T)ns(T)
Tc
J
19
Surface Impedance
x
-z
y
J
EH
( ) σωµi
dzzJ
EiXRZ sss ==+=
∫
Surface Resistance Rs: Measure of Ohmic power dissipation
sSurface
sVolume
Dissipated RIdAHRdVEJP 22
21~
21Re
21
∫∫∫∫∫ =
⋅=
Rs
conductor
Surface Reactance Xs: Measure of stored energy per period 222
21~
21Im
21 LIdAHXdVEJHW
Surfaces
VolumeStored ∫∫∫∫∫ =
⋅+=
ωµ Xs
Xs = ωLs = ωµλ
20
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
10-1 100 101 102Su
rface
resi
stan
ce R
s (Ω)
frequency f (GHz)
Cu(77K)
poly
epitaxial
Nb3Sn
T=0.85Tc
T=0.5Tc
YBCO
Two-Fluid Surface Impedance
Because Rs ~ ω2:The advantage of HTS over Cu diminishes with increasing frequency
Rs crossover at f ~ 100 GHz at 77 K
Ls
σn
Superfluid channel
Normal Fluid channel
nsR σλµω 30
2
21
=
ωλµ0=sX
sss iXRZ +=
M. Hein, Wuppertal
Rs ~ ω2
Rn ~ ω1/2
21
The London Equations
τvmEe
dtvdm
−=Newton’s 2nd Law fora charge carrier
Superconductor:1/τ 0
EEmen
dtJd
L
ss
20
2 1λµ
== 1st London Equation
τ = momentum relaxation timeJs = ns e vs
1st London Eq. andyield:
tBE
∂∂
−=×∇
02
=
+×∇ B
menJ
dtd s
s
London
surmise0
2
=+×∇ BmenJ s
s
2nd London Equation
These equations yield the Meissner screening
HHL
2
2 1λ
=∇ LzeHH λ/0
±=
( )zH
z
vacuum superconductor
λL
20 enm
sL µ
λ ≡
λL ~ 20 – 200 nmλL is frequency independent (ω < 2∆)
22
( ) BJBJdtd
EdtJdEJ
sLL
n
L
snn
−=×∇=
+×∇
==
202
0
20
01
1
λµλµ
λµσ
The London Equations continued
Normal metal Superconductor
B is the source of Js,spontaneous fluxexclusion
E is the source of Jn E=0: Js goes on forever
Lenz’s Law
1st London Equation E is required to maintain an ac current in a SCCooper pair has finite inertia QPs are accelerated and dissipation occurs
23
BCS Microwave ElectrodynamicsLow Microwave Dissipation
Full energy gap -> Rs can be made arbitrarily small
( ) Tks
BeR /0∆−≈ for T < Tc/3 in afully-gapped SC
10-5
10-4
10-3
10-2
10-1
0 2 4 6 8 10Su
rface
resi
stan
ce R
s (Ω)
Inverse reduced temperature Tc/T
YBa2Cu
3O
7-x
@ f=87 GHz
sputteredLaAlO
3
coevaporatedMgO
Nb3Sn on
sapphirelogR
s(T) ∝ –∆/kT
c∙T
c/T
90 1040 20 15 T (K)
( ) residual,sBCSs RTRR +=
Rs,residual ~ 10-9 Ω at 1.5 GHz in Nb
M. Hein, Wuppertal
FilledFermiSea
∆skx
ky
HTS materials have nodes inthe energy gap. This leadsto power-law behavior ofλ(T) and Rs(T) and residual losses
( ) ( ) TaT += 0λλTbRR ss += residual,
Rs,residual ~ 10-5 Ω at 10 GHz in YBa2Cu3O7-δ
FilledFermiSea
∆d
node
kx
ky
24
0
0.5
1
1.5
2
0 5 10 15 20 25 30su
rface
resi
stan
ce, R
s (mΩ
)field amplitude, B
s (mT)
4K30K
61K
71K
77K
Nonlinear Surface Impedance of Superconductors
0.1
1
10
100
1000
0 20 40 60 80 100
YBaCuO 1, PLD, 2"YBaCuO 2, sputt, 1"
surfa
ce re
sist
ance
, R s (m
Ω)
temperature, T (K)
YBa2Cu3O7-δ Thin Film Made by Pulsed Laser Depositionand sputtering19 GHz
The surface resistance and reactance values depend on the rf currentlevel flowing in the superconductor
Data from M. Hein, WuppertalSimilar results for Xs(Bs)
25
Why are Superconductors so Nonlinear?
Superconductinggrains
Josephsonweak links
GranularitySmall ξ ~ grain boundary thickness
Intrinsic Nonlinear Meissner Effectrf currents cause de-pairing – convert superfluid into normal fluid
Nonlinearities are generally strongest near Tc and weaken at lower temperatures
( )( ) ( )
22
1,,0
−=
TJ
JTJT
NLλλ JNL(T) calculated by theory (Dahm+Scalapino)
JJs havea strongly nonlinearimpedance
McDonald + ClemPRB 56, 14 723 (1997)
Edge-Current Buildup
0 100 200 300 400 500 600 700
0
20
40
60
80
100
120
LTLS
M R
espo
nse,
a.u
.
Distance, µ
J rf2
(a.u
.)
Microstrip (Longitudinal view)
Scanning Laser Microscope imageYBCO strip at T = 79 Kf = 5.285 GHz, Laser Spot Size = 1 µm
See poster 1EG08
+ Vortex Entry and Flow
Heating
26
How to Model Superconducting Nonlinearity?
(1) Taylor series expansion of nonlinear I-V curve (Z. Y. Shen)
( ) ( ) ( )43
03
32
02
2
0 !31
!210 VOV
dVIdV
dVIdV
dVdIIVI
VVV
δδδδ +
+
+
+=
===
= 0 if I(-V) = -I(V)
3rd order term dominates
V = V0 sin(ωt) input yields ~ V03 sin(3ωt) + … output
1/R linear term
I
V
(2) Nonlinear transmission line model (Dahm and Scalapino)I
CR L
VtVC
zI
∂∂
−=∂∂
RItIL
zV
−∂∂
−=∂∂
L and R are nonlinear:
ItRC
tI
tLC
tIRC
tILC
zI
∂∂
+∂∂
∂∂
+∂∂
+∂∂
=∂∂
2
2
2
2
additional terms2
0
∆+=
NLIILLL
2
0
∆+=
NLIIRRR
3rd harmonics and 3rd order IMD result
27
Experimental Microwave Superconductivity
• Cavity Perturbation
• Measurements of Nonlinearity
• Topics of Current Interest
• Microwave Microscopy
28
Cavity PerturbationObjective: determine Rs, Xs (or σ1, σ2) from f0 and Q measurements
of a resonant cavity containing the sample of interest
Sample atTemperature T
MicrowaveResonator
Input Output
frequency
transmission
f0
δf
f0’
δf’
∆f = f0’ – f0 ∝ ∆(Stored Energy)∆(1/2Q) ∝ ∆(Dissipated Energy)
Quality Factor
~ microwavewavelength
λ
T1 T2
B
ff
EEQ
δ0
Dissipated
Stored ==Cavity perturbation means ∆f << f0
QRs
Γ= ω
ω∆
Γ=∆
2sX Γ is the sample/cavity geometry factor
29
Measurement of NonlinearitiesIntermodulation is a practical problem
sign
al
frequency2f1-f2 f1 f2 2f2-f1 2f1
Pout (dB)
Pin (dB)
linearω1
20
15
10
5
–5
–10
–15
–20
–25
–15 –10 –5 155 10
3rd-orderinterceptPoint (TOI)
2ω1 - ω2, 2ω2 - ω1 Bandwidth ofpassband
Intermodulation harmonic generation
Nonlinear (i. e., signal strength dependent) microwave responseinduces undesirable signals within the passband by intermodulation.
M. Hein, Wuppertal
SCDevice
Pin Pout
30
Topics of Current InterestIn Microwave Superconductivity Research
Identifying and eliminating the microscopic sources of extrinsic nonlinearityIncrease device yieldAllows further miniaturization of devicesWill permit more transmit applications
Identify the additional Drude term now seen in σ(ω,T)under-doped cuprates show σ2>0 above Tcpseudo-gap electrodynamics
Nonlinear and Tunable DielectricsMgO substrates have a nonlinear dielectric loss at low temperaturesFerroelectric and incipient ferroelectric materials as tunable
microwave dielectrics/capacitors
31
Microwave Microscopy of SuperconductorsUse near-field optics techniques to obtain super-resolution images of:
1) Materials Properties: Nonlinear response2) RF fields in operating devices
200µm loop probe500Å
YBCO film
SrTiO3
J
30° misorientation Bi-crystal grain boundary
0 1 2 3 4 5-105-100-95-90-85-80-75-70-65-60-55
P 2f &
P3f (d
Bm) a
t 60K
Position (mm)
P2f P3f
Noise Level of P2f
Noise Level of P3f
60 KSee poster 2MC10
Scanning Laser Microscopy
Image Jrf2(x,y) in an operating
superconducting microwave deviceImage JIMD
See poster 1EG08re
flect
ance
J rf2
20 µm
20 µm
32
References and Further ReadingZ. Y. Shen, “High-Temperature Superconducting Microwave Circuits,”
Artech House, Boston, 1994.
M. J. Lancaster, “Passive Microwave Device Applications,”Cambridge University Press, Cambridge, 1997.
M. A. Hein, “HTS Thin Films at Microwave Frequencies,”Springer Tracts of Modern Physics 155, Springer, Berlin, 1999.
“Microwave Superconductivity,”NATO- ASI series, ed. by H. Weinstock and M. Nisenoff, Kluwer, 2001.
T. VanDuzer and C. W. Turner, “Principles of Superconductive Devices and Circuits,”Elsevier, 1981.
T. P. Orlando and K. A. Delin, “Fundamentals of Applied Superconductivity,”Addison-Wesley, 1991.
R. E. Matick, “Transmission Lines for Digital and Communication Networks,”IEEE Press, 1995; Chapter 6.
