phase-sensitive measurements on superconducting quantum...
TRANSCRIPT
University of Illinois at Urbana-Champaign
2019 NSF/DOE/AFOSR Quantum Science Summer School (QS3) June 3-14, 2019 --- Penn State University
Dale J. Van Harlingen
LECTURE 1: Monday, June 3
Phase-sensitive measurements on superconducting quantum materials and hybrid superconductor devices
LECTURE 2: Tuesday, June 4
S-TI-S Josephson junction networks: a platform for exploring and exploiting topological states and Majorana fermions
Josephson physics and techniques useful for exploring superconductor materials and devices, focusing on probing unconventional superconductors and junctions
A specific device architecture that may support Majorana fermions and shows promise for manipulating them for quantum computation processes
Significant role of SUPERCONDUCTIVITY in Quantum Information Science
Quantum materials
Quantum sensing
Quantum computing and
Quantum simulation
Superconductors were the first and the most-studied many-body system and continue to be critical to materials research. Every time we think the field is fading out, new classes of superconductors emerge
e.g. HTSC, heavy fermion superconductors, pnictide superconductors, topological superconductors, twisted graphene superconductivity , …
Superconductors and devices are legendary for their sensitivity as detectors and probes of exotic phenomena. This in part because of their required low temperature operation that has driven cryogenic physics and technology, and in part because of their intrinsic quantum naturee.g. MRI, SQUIDs, quasiparticle mixers, axion detectors, …
Conventional superconductor-based qubits (transmons, etc.) are one of the leading technologies for quantum simulation and quantum computing, and topological qubits based on Majorana fermions are one of the most explored for next-generation candidates
Quantum communications Photons rule in this space, so one of the grand challenges in quantum information science is “transduction” --- the transfer of quantum state information from qubit platforms into telecommunication photons
Phase-sensitive measurements on superconducting quantum materials and hybrid superconductor devices
Agenda
1. Superconductivity --- very little, but enough
2. Phase-coherence
3. Josephson effect
4. Phase-dynamics ---- the RSJ-model
5. Multiply-connected geometries --- SQUIDs and arrays
6. Josephson interferometry
7. Applications to quantum materials
A. Measuring the order parameter of unconventional superconductors
B. Measuring the current-phase relation of Josephson junctions with unconventional barriers
Cryogenics – low temperature physics 1. Get rid of thermal motion2. Observe onset of new phenomena
Heike Kamerlingh Onnes Leiden, Netherlands
The Discovery of Superconductivity (1911)
First to liquify helium in 1908 Kamerlingh Onnes Lab, Leiden
Superconductivity R=0
Temperature (K)
Resistance ()
+ thousands of metallic compounds and alloys
More illuminating: what materials are NOT superconducting?
Magnetic materials (Mn, Fe, Co, Ni)
Extent of Superconductivity
Good metals (Cu, Ag, Au)
q
k1
k'1 k2
k'2
e e- -
Conventional (“classic”) superconductivity
BCS theory:Bardeen, Cooper, Schrieffer (1957)
MECHANISM = attractive phonon-mediatedelectron-electron interaction Cooper pairing
ky
kx
kz
(k) =
“s-wave”
EXCITATIONS = normal “quasiparticles” with an isotropic energy gap
GROUND STATE = superfluid pair condensate = ns e i macroscopic phase coherence
23.2K
Liquid He
John Bardeen
Leon Cooper
Bob Schrieffer
-3 -2 -1 0 1 2 30
1
2
3
e V /
G/G
(eV
>>
)
quasiparticle tunneling spectroscopy reveals the fully-formed energy gap
Tc increased slowly from 4K to 23K over 75 years from 1911 to 1986
phase
High Temperature Superconductivity (1986)Woodstock of Music (1969)
3 days / 33 acts / 500,000 hippies
Woodstock of Physics (1987)
8 hours / 50 talks / 2500 physicists
Georg BednorzAlex Müller
“Our lives will be changed” … this did not turn out at expected but is rather true for physicistschallenged our understanding of condensed matter physics
opened new opportunities for superconductor research
IBM Zurich Research Laboratory
Family tree of superconductors
pnictidesconventional
fullerenes
carbon
hydrogen sulphide
q
k1
k'1 k2
k'2
e e- -
Conventional (“classic”) superconductivity
BCS theory:Bardeen, Cooper, Schrieffer (1957)
MECHANISM = attractive phonon-mediatedelectron-electron interaction Cooper pairing
ky
kx
kz
(k) =
“s-wave”
EXCITATIONS = normal “quasiparticles” with an isotropic energy gap
GROUND STATE = superfluid pair condensate = ns e i macroscopic phase coherence
23.2K
Liquid He
John Bardeen
Leon Cooper
Bob Schrieffer
-3 -2 -1 0 1 2 30
1
2
3
e V /
G/G
(eV
>>
)
quasiparticle tunneling spectroscopy reveals the fully-formed energy gap
Tc increased slowly from 4K to 23K over 75 years from 1911 to 1986
phase
Phase constraint:
SC:
JJ:
2 2emn es
J + As
2 n
n h n
2e 0
J 0s
1 2
0
2 2 n
Phase Coherence in multiply-connected superconductorsrequirement of single-valuedness of the condenstate wavefunction
Ring with Josephson Junction (rf SQUID)
gauge-invariant phase magnetic flux
Superconducting Ring
flux quantization
1 2 1
1 20 02 1 2
2 2( - ) - A d A d A d
in center of SC
2 2emn es
J + As
150 2.07 x10 Wb
20 20 G m
1
21
2
2 2A d 2 e e n
2
1 20 1
2( - ) + A d 2 n
Josephson Effect
Josephson junction can be thought of as the “transistor” of superconducting electronics
IsS
S
IIc
Ib
50th Anniversary of the Josephson Effect
,
Josephson Effect
In general : “current-phase relation”
Cooper pair correlations enable a supercurrent in a tunnel junction that depends on the gauge-invariant phase difference across the junction:
I1 2
cI I sin
cI I cpr
2
1 20 1
2= ( - ) - A d
Ic is the maximum current before switching into a finite voltage state
d 2eV=dt
2e 2eV(t) Vdt t c c J
2eVI(t) I sin t I sin 2 f t
Phase dynamics --- phase winds according to the “Josephson relation”
(483.6 THz/V)J2eVf =
hJosephson frequency
483597.84841698 GHz/VJosephson standard volt defined by:
c c 0J
I Iwhere E2e 2
Josephson supercurrent
Josephson coupling energy
JU E cos
At constant V:
RSJ (Resistively-Shunted Junction) model
Josephson dynamics: “phase particle” moving in a tilted washboard potential
U
I
I=0
I=Ic
cV dVI I sin CR dt
2
c 2
d C dI I sin2eR dt 2e dt
dV =2e dt
2
c2
C d d ( I I cos ) 02e dt 2eR dt
I < Ic: static solution = constant V=0
I > Ic: dynamic solution evolves in time V > 0voltage oscillates at the Josephson frequency
“mass” “damping” “potential”
Model junction as a Josephson junction in parallel with a resistor and capacitance
S
S
I
1/42
c cp
cJ
2eI cos 2eI1 I= 1C C IL C
Josephson inductance
small oscillation frequency in washboard potential well
non-linear (depends on the current)0J
c
L ( ) =2 I cos( )
cI I sin
Phase dynamics in the superconducting state
cc
2eIdI dI cos cos Vdt dt
J
dIV Ldt
Important for superconducting qubits by creating anharmonic potential that allows lifting degeneracy of harmonic oscillator states
Equilibrium phase shifts
JU = 2E
Josephson plasma frequency
0c
IarcsinI
Barrier height drops
I
0
Bp U/k TT e
2
p p
U 0.877.2 1Rcp
Q e2
Critical current -- escape of phase particle from the potential well
Thermal activation over the barrier)
Macroscopic Quantum Tunneling (MQT) though the barrier
Thermal activation rate:
MQT rate:
Transition to the finite voltage state always occurs before Ic,the “thermodynamic critical current”, due to fluctuations:
(the numerical factors come from a WKB approximation for tunneling through the washboard potential barrier and damping corrections)
MQT dominates when
p B7.2k T
RSJ model ---current-biased
Shape of the I-V characteristic depends on the damping
I
V
2c
c0
2 I R C
“McCumber parameter”Low damping (large R): c > 1
I-V is hysteretic
I
V
High damping (small R): c < 1
I-V is single-valued
2 2cV R I I
IS tt
IS
IN
IS
For I>Ic, the junction switches abruptly to finite voltage and the supercurrent is sinusoidal and averages to zero
For I>Ic, the supercurrent is non-sinusoidal has a finite average and averages to zero
dc SQUID
A dc SQUID consists of two junctions embedded in a superconducting loop
c 00
I 2I cos
Ic2
1 20
2 0
2 2c c1 c2 c1 c2I (I I ) 4I I cos
0
c1 1 c2 2I I sin I sin
Symmetric SQUID Ic1 = Ic2 = I0Asymmetric SQUID Ic1 > Ic2
Ic1
Applied flux (0) Applied flux (0)-5 -4 -3 -2 -1 0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
Crit
ical
cur
rent
(I0)
Crit
ical
cur
rent
(I0)
Modulation reduced
Ic1+Ic2
Ic1-Ic2Modulates to zero
(Ic1=1.5, Ic2=0.5)
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
2.5
dc SQUID w/ inductance
A finite inductance modifies the SQUID characteristic because the circulating currents generate a flux in the loop that adds to the applied flux
Inductive SQUID
0
0
2LI 0
loop 1 2L I I LJ
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
2.5
Applied flux (0)
Crit
ical
cur
rent
(I0)
J = I1-I2 = circulating current = applied flux
Modulation reduced
=1
This is an example of “self-field effects” in which the fields from the Josephson currents cannot be ignored
Dependence on
-5 -4 -3 -2 -1 0 1 2 3 4 50
1
2
3
4
0 1 2 3 40
1
2
3
4
Crit
ical
cur
rent
(I0)
Applied flux (0)
Volta
ge (I
0R)
Voltage (I0R)
I/I0 =
4
3.5
3
2.5
2
1.5
1.1
dc SQUID operation
A dc SQUID biased at constant current exhibits a voltage modulation wih applied magnetic flux (period in 0)
For highest field sensitivity, bias just above I0
n0
(n+½)0
Josephson Effect in extended junctions
“local current-phase relation”
The supercurrent depends on the local CPR and the local gauge-invariant phase difference across the junction:
Local relation --- tunneling is highly-directional so the supercurrent depends on the phase difference at each location across the junction
2
1 20 1
2(y) = ( ) - (y)) - A dy
J1 2 Critical current:
w = junction width t = barrier thickness
Phase coherent --- phases at each location are related interference
w/2
cw/2
I (y) = t J(y) dy
cJ( (y)) J (y) cpr( (y))
Small-junction limit --- ignore self-field effects (no screening of field by tunneling currents)
Josephson Interferometry: response to a magnetic field
Phase coherence magnetic field induces a phase variation:
Uniform magnetic field and small junction limit
linear phase variation
y(y)0
Bm
-5 -4 -3 -2 -1 0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
Ic/I
c0
Fraunhofer diffraction
pattern
xmagnetic thickness of barrier
Single-slit optical interference
b
barrierthickness
m = b +2
Uniform junction Fourier transform
Current-phase
relation
Order parameter symmetry
Critical current variation
Magnetic field
variations
Josephson Interferometry: what it can tell you
Gap anisotropyDomainsCharge traps
Flux focusingSelf-field from tunneling currentTrapped vorticesMagnetic particlesUnconventional
superconductivityNon-sinusoidal terms-junctionsExotic excitations e.g. Majorana fermions
Current-phase
relation
Order parameter symmetry
Critical current variation
Magnetic field
variations
Josephson Interferometry: what it can tell you
Gap anisotropyDomainsCharge traps
Flux focusingSelf-field from tunneling currentTrapped vorticesMagnetic particlesUnconventional
superconductivityNon-sinusoidal terms-junctionsExotic excitations e.g. Majorana fermions
Josephson Interferometry: critical current variations
Uniform junction
Narrow junction dc SQUID
Wide junction dc SQUID
Asymmetric junctions (magnitude)
Three-junction SQUID
Josephson Interferometry: more critical current variations
Asymmetric junctions (magnitude and width)
Current-phase
relation
Order parameter symmetry
Critical current variation
Magnetic field
variations
Josephson Interferometry: what it can tell you
Gap anisotropyDomainsCharge traps
Flux focusingSelf-field from tunneling currentTrapped vorticesMagnetic particlesUnconventional
superconductivityNon-sinusoidal terms-junctionsExotic excitations e.g. Majorana fermions
Theorists with new ideas Experimentalists with new materials
What unique phenomena can I probe with Josephson Interferometry
Is it superconducting?
YES NO
Is it unconventional or exotic? Can I make it into the barrier of a Josephson junction?
YES NO
Go away –I don’t care
Measure the pairing symmetry
Measure thecurrent‐phase relation
Go away –I don’t care
YES NO
When all you have is a hammer, everything looks like a nail
(For me, everything looks like a SQUID)
My Friends in Urbana
1. Ceramics --- oxide materials2. Tc --- higher than could be explained by conventional BCS3. Layered materials --- low dimensional effects and strong correlations4. Unusual properties --- thermodynamics, transport, electrodynamics, … 5. Unusual doping dependence --- complicated phase diagram6. Unusual vortex states and dynamics7. UNCONVENTIONAL SUPERCONDUCTIVITY
HTSC --- many exciting phenomena and mysteries
SCIENTISTS: new physics ENGINEERS: new applications of superconductivity
super-conductor
pseudo-gap
strange metal
normal metal
T
hole dopingelectron doping
Mott insulator
superconductor
Unconventional superconductivity
What does “unconventional” mean? Not BCS• Mechanism other than phonon-mediated pairing• Symmetry not s-wave … exhibits anisotropy in phase and/or magnitude
1st indication: UPt3 (heavy fermion) two peaks in specific heat
1st confirmation: YBa2Cu3O7-x (high-Tc superconductor) d-wave
Cuprate superconductors:
YBa2Cu3O7-x Tc = 92K
+_ _+
“d-wave”
0
0 90 180 270 3600 90 180 270 3600
0.5
1
px + i py
0
2
0 90 180 270 3600 90 180 270 3600
0.5
1
py.
0
2
0 90 180 270 3600 90 180 270 3600
0.5
1
px
RELATIVE PHASEMAGNITUDEPAIRING STATE
ODD PARITY STATES
Complex order parameter broken time-reversal symmetry
phase shift 0,
+-
+-
Determining the Pairing Symmetry --- A Roadmap for Experimentalists
Magnitude measurementsprobe quasiparticles ---but can be masked or mimicked by impurities
Phase measurementsgive a distinct signature ---less susceptible to microscopic details
Cuprate candidates
s vs. dx2-y
2
Evolution of the Corner SQUID Idea
DVH Group meetingJanuary 1992
B9 MRLDecember 1991
++
__
Unconventional SCsingle crystal
Conventional SCthin film loop
Josephson interferometry: measuring the phase anisotropy
dc SQUID(Superconducting QUantum Interference Device)
measures the phase shift inside the crystal between orthogonal directions
Wollman, Ginsberg, Leggett, Van Harlingen (1993)
Josephson tunnel junctions
tunneling selects direction in k-space
the corner SQUID
+_ _+
Josephson interferometry ---measuring the phase shift between different directions
SCcrystal
s-wave SC thin film
loopdc SQUID single junction
Josephson junctions
The corner SQUID experiment
-3 -2 -1 0 1 2 3
M a g n e tic f lu x ( 0)-3 -2 -1 0 1 2 3
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
M a g n e tic f lu x ( 0)
Crit
ical
cur
rent
s-wave d-wave+ +
-+
-
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0
1
2
3
4
5
6
7
8
edge SQUIDs
corner SQUIDs
Obs
erva
tions
Phase shift
D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett, Phys. Rev. Lett. 71, 2134 (1993)
100m
6 4 2 0 -2 -4 -6
Magnetic flux-6 -4 -2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
Magnetic flux
Crit
ical
cur
rent
+
The corner junction experiment
s-wave-
++-
d-wave
-1000 -500 0 500 10000
20
40
60
Applied magnetic field (mG)
Crit
ical
cur
rent
( A
)
D. A. Wollman, D. J. Van Harlingen, J. Giapintzakis, and D. M. Ginsberg. Phys. Rev. Lett. 74, 797 (1995)
Tricrystal ring experiment Kirtley, Tsuei (IBM)
-SQUID Hilgenkamp, Mannhart (Augsburg)
Scanning SQUID Microscope
image
Further phase-sensitive measurements
60
75
Angles of the grain boundary junctions determines if there is a -phase shift in the SQUID modulation
Angles of the grain boundary junctions determines if there is a spontaneous circulating current
Manfred Sigrist and T. M. Rice, Rev. Mod. Phys. 67, 503 (1995)
Paramagnetic Meissner Effect
W. Braunisch, N. Knauf, G. Bauer, A. Kock, A. Becker, B. Freitag, A. Grütz, V. Kataev, S. Neuhausen, B. Roden, D. Khomskii, D. Colleen, J. Bock, and E. Preisler. Phys. Rev. B 48, 4030 - 4042 (1993)
Model: spontaneous supercurrents in multiply-connected d-wave grains
Experiment: enhanced magnetic flux in granular BSCCO composites
Grain boundary junctions
Geometry for testing symmetry: 45°-asymmetric junction facets sample different signs of the d-wave order parameter
“Multi-corner junction”
+
Maximum Ic not at B=O
Symmetric with respect to field polarity
S-wave would give Fraunhofer pattern
H. Hilgenkamp, J. Mannhart, and B. Mayer, Phys. Rev. B 53, 14586 (1996)
Cuprates--- all d-wave, all the time?+_ _+
The compilation of experiments --- spectroscopic, thermodynamic, transport, and phase-sensitive --- indicate that all of the cuprates have dx
2-y
2 symmetry
Tested by phase-sensitive Josephson inferometry
Many proposals for inducing alternative pairing symmetries but so far not seen --tested vs. material, temperature, carrier doping, magnetic impurities, …
formation of zero-energy bound quasiparticle states suppression of d-wave at interfaces and defects onset of SC phases with complex order parameters
at low temperature broken time-reversal symmetry
Example: Fragility of unconventional superconductors
s-wave superconductor: scattering does not affect superconductivity “Anderson theorem”+
unconventional superconductor: scattering changes magnitude and phase of the order parameter
dx2-y
2
dx2-y
2 +idxy
Onset of secondary order parameter:* increases B=0 critical current node lifted* breaks polarity symmetry broken TRS
Effect of onset of complex order parameter
45°-asymmetric grain boundary junction highly- sensitive to the addition of a complex component in the order parameter
-30 -20 -10 0 10 20 300
5
10
15
20
25
30
35
40
Crit
ical
Cur
rent
(A)
Applied Magnetic Field (G)
15.4K 9.0K 6.4K 1.5K
PREDICTION
EXPERIMENTNo evidence for a change in the order parameter symmetry over a wide temperature range and with doping of non-magnetic and magnetic impurities:
W. K. Neils and D. J. Van Harlingen, Phys. Rev. Lett. 88, 047001 (2002)
• d-wave superconductivity nucleates in spatially inhomogeneous “puddles” which couple together via the Josephson effect
• Produces a globally “s-wave”-like pairing symmetry despite local d-wave in grains (no preferred anisotropy)
• Overdoped LSCO crystals are a candidate for such a system
Where haven’t we looked? --- the strongly overdoped cuprates
David Hamilton(Illinois)
Masaki Fujita (Tohoku)
Steve Kivelson(Stanford)
Optimal-doping Overdoping
• LSCO (x=0.25, Tc=15K) crystal, grown by the Fujita group, is oriented via XRD, cut/polished, then mounted onto a sapphire substrate.
• Using dry-film photoresist masking, Josephson junctions are patterned onto a- and b- faces of the crystal
• Au (annealed) is used as the normal barrier. • Superconducting contacts are made with sputtered Nb.
Nb
Au
Exploring the region by Josephson interferometry
Josephson interferometry diffraction pattens --- edge or corner
• Also see broken field symmetry which could arise from complex order parameters or local magnetic field inhomogeneity
• Observe short-range field modulations features suggestive of grain boundary junctions --- indicates domains
Edge JJ Corner JJ
• We do not observe a s-wave state, but we cannot tell corner from edge diffraction patterns --- this may be regarded as definition of “s-wave –like”
Model system with d-wave domains and Wohlleben effects
Circulating currents from Wohlleben effect generates an inhomogeneous field that threads the junction
Field (flux quanta)
Model simulates both the rapid oscillations and the magnetic field asymmetry
Field (flux quanta)
Growing Family of Unconventional Superconductors
-(BEDT-TTF)2Cu[N(CN)2]Br
Organic superconductors
“anisotropic d-wave”
+_ _+
115 superconductors
CeCoIn5 Tc = 2.3K
+_ _+
“d-wave”
Heavy Fermion superconductors
UPt3 TcA = 0.50 TcB = 0.45KSr2RuO4 Tc = 1.5K
Ruthenate superconductors
px+ipy(kx
2-ky2) kz (kx+ iky)2 kz
Tc = 11.6K+_ _+
“d-wave”
Cuprate superconductors
YBa2Cu3O7-x Tc = 95K
The Quest for Complex Superconductors
Sr2RuO4 Tc = 1.5K
Ruthenate superconductors:
px+ipy
Josephson interferometry of complex order parameters
-3 -2 -1 0 1 2 30.0
0.2
0.4
0.6
0.8
1.00
/4
/2
3/4
Magnetic flux (/0)
Crit
ical
cur
rent
-1.0 -0.5 0.0 0.5 1.00.0
0.5
1.00
/2
3/4
/4
Magnetic flux (/0)
Crit
ical
cur
rent
Angle SQUID
Angle junction
= phase shift
Phase shifts from complex order parameters
-3 -2 -1 0 1 2 30.0
0.2
Magnetic flux (/0)
Cr
-1.0 -0.5 0.0 0.5 1.00.0
0.5
1.00
/2
3/4
/4
Magnetic flux (/0)
Crit
ical
cur
rent
Angle SQUID
Angle junction
= phase shift
Polarity asymmetry indicates broken time‐reversal symmetry
Current focus is on topological systems --- interplay of topology and superconductivityNew questions of pairing symmetry --- opportunities for Josephson interferometry
Superconductor/Topological Insulator bilayer Topological Superconductors e.g. CuxBi2Se3, NbxBi2Se3, …
Materials exhibit superconductivity, ferromagnetism, Andreev reflection, gate-dependent surface supercurrents, hints of p-wave superconductivity
S-TI bilayerS electrode
N barrier
Corner Junction experiment to test for proximity-induced p-wave symmetry
s + ps + (px+ipy)
(C. Kurter, A.Finck, E. Huemiller, and Y.-S.Hor)
Testing order parameter symmetry via corner SQUID/junction experiments
What’s next? the growing list of systems in which to determine the pairing symmetry
S/TI bilayersDoped-TIs --- topological superconductor candidatesFeTe1-xSex
PrOs4Sb12
YPtBiLAO/STO interface superconductorsTwisted bilayer graphene superconductorsCeCoIn5
CeCu2Si2UBe13
Sr2RuO4
???