Proximity and Josephson
effects in superconductors
Content
1. Quasiparticles
in superconductors
2. Tunneling (general)
3. Tunneling in N/S, S/S
junctions
4. Josephson effect
5. Andreev reflection
Nobel price 1973:
Esaki, Giaever, Josephson
A.F. Andreev
The Nobel Prize in Physics 1972
"for their jointly developed theory of superconductivity, usually called the BCS-theory"
The BCS theory (1957)
John Bardeen Leon Neil Cooper John Robert Schrieffer
USA USA USA
University of Illinois Urbana, IL, USA
Brown University Providence, RI, USA
University of Pennsylvania Philadelphia, PA, USA
b. 1908 d. 1991
b. 1930 b. 1931
Fermi liquid and quasiparticles
Quasiparticle concept
(Landau 1956, 1957)
electron-like QP
Fermi sea
hole –like QP
Fermi liquid - a system of interacting Fermi particles
Quasiparticle (QP) – excitation in a Fermi liquid, it
resembles an excitation in an ideal Fermi gas,
but not equivalent
Due to interaction with other electrons and ions,
quasiparticle effective mass m* differs from
the free electron mass me
)(2/2/ *2*2
FFF ppvmpmp Excitation energy
Quasiparticles have finite lifetime due to interaction e.g. with phonons
Electron-phonon interaction
Diagram illustrating e-e interaction
via emission and subsequent
absorption of a phonon
In the BCS picture, only electrons
within narrow layer near the Fermi
surface attract each other via phonons
The BCS ground state
Superconducting ground state
Cooper pairs ),(
kk : opposite momenta k, -k and
opposite spins
The discovery of Leon Cooper:
if there is attractive interaction
between electrons, the standard
Fermi distribution of electrons
becomes unstable !
Total spin S=0, =>
charge carriers are bosons. This justifies the Ginzburg-Landau theory
Why a band gap is needed for superconductivity ?
Kkm
E F
2
Elementary process leading to electrical resistance (energy dissipation)
in a normal conductor:
Scattering of an electron from the state 1 to the state 2
This process involves energy transfer
and thus is not allowed if E is smaller than the gap Δ
Energy gap and critical temperature
1)0(
2)0(/1
VN
e
ep
Dep
MeT DDc
ep1
,14.1/1
=> isotope effect
Typically, Debye temperature is of the order of 300 - 500 K
- energy required to created an
excitation in a superconductor
- coupling constant
Qasiparticles (QP) in a superconductor
Density of states 22
)0(/)()(
E
ENENE
=>
QP energy E(k) in a superconductor
-> energy gap Δ0
Tunneling: NIN and NIS junctions
NIN
NIS
Tunneling: SIS junctions
I-V curve (left) and corresponding tunneling diagram in ‘semiconductor’ model
V=0 branch: Josephson effect
Josephson effect
,02
22
dx
d
Example: two superconductors connected by a short bridge
Linearized GL
equation
0)2/(,1)2/(
)(1)( 21
21
dfdf
xfexfeii
General solution: superposition of two wave functions
Coherence length
m2
Josephson effect
)(2
** m
iejs
Current density
)(1)( 21
21 xfexfeii
substituting
we obtain 21,sin cs jj
Detailed calculations show that e
RI NC2
Josephson effect:
Macroscopic quantum phenomenon
Current Biased JJ U(z )
z
Quantum Tunneling
I ext = 0
I ext > 0
I ext
Macroscopic Variable: Phase Difference
Transition from
Superconducting (0 voltage) State
to
Finite Voltage State ( )
Andreev reflection
Energy diagram
A(E)+B(E)+C(E)+D(E)=1, C(E)=0, D(E)=0 for E<
Probability of Andreev reflection A(E), of normal reflection B(E) and of
transmission with and without branch crossing C(E), D(E)
Potential barrier U(x)=W(x-x0) FvWZ /Barrier parameter
Summary of the Andreev reflection physics
Normalized Conductance
ΔeV-1 1
2
(no barrier) (high barrier)
ΔeV-1 1
Normalized Conductance
Cooper pair Fermi energy
energy gap
electron
hole
SC N
(N: Normal metal, SC: superconductor)
Andreev reflection
Transition from metallic to tunneling regimes in superconducting junctions:
Phys. Rev. B 25, 4515 (1982)
Model of Blonder, Tinkham and Klapwijk (BTK)
dEEBEAEfeVEfSevNI FNS )]()(1)][()([)0(2
Current across NIS junction
Transition from metallic to tunneling regimes in superconducting
microconstrictions: Excess current, charge imbalance, and
supercurrent conversion: Phys. Rev. B 25, 4515–4532 (1982)
Model of Blonder, Tinkham and Klapwijk (BTK)
dEEBEAEfeVEfSevNI FNS )]()(1)][()([)0(2
Current across NIS junction
Multiple Andreev reflections in SIS junctions
Steps on I-V curves appear
at
eV=2/n, n=1,2,3,…
due to MAR
=>
Proximity effect in normal metal -superconductor (NS)
structures
222 2 pmp rpiv
u
v
ur
p
pexp)(
Josephson coupling in SNS junctions
(d.l.)
Spatial oscillations of induced superconducting order parameter
in a ferromagnet in close proximity to a superconductor
Ψ(x) = Ψ0 cos(2Qx)
Q ~ Eex/vF is center of pair mass
momentum
Demler, Arnold & Beasley (1997)
Buzdin & Kupriyanov (1991):
analog of the FFLO state
Proximity effect in ferromagnetic -
superconductor (FS) structures
In the regime Eex>>kT the decay length is given by ex
FF
E
D
Proximity effect in N(F) -S structures
‘0’ and ‘’ Josephson junctions
FS
(x) /
x
SFS
(x) /
x
— spatial oscillations of the
order parameter in the F layer
— thickness of F layer equals
half wave-length of oscillations
)sin(sin :junction
0 ,sin :junction 0
CC
CC
III
III
Possible applications of π-junctions: novel types of logic elements
-0.4 -0.2 0.0 0.2 0.4-1.0
-0.5
0.0
0.5
1.0
dF/2
F2=
3/2
1
1/2
<<1
(x
)/
0
x/dF
0.0 0.5 1.0-10
-5
0
5
10
15
20
The inset shows calculations of Ginzburg-
Landau (GL) free-energy in the F-layer
for the 0- and -phase states.
Ginzburg-Landau free-energy consists of
negative condensation energy (~2)
and positive gradient energy ({ grad }2).
The spatial distribution of the order parameter
in the F-layer of the SFS junction calculated for
various dF/2
F2 ratios
0
GL f
ree-e
nerg
y, a
.u.
dF/2
F2
LOFF state in SFS Josephson junction
)exp()exp(21
0
xi
xGL
red lines – state is favorable
blue lines – 0 state is favorable
Spatial dependence of the
order parameter in the F-layer
0 - transition in SFS junctions: theory and experiment
V.Ryazanov, V.Oboznov, A.Rusanov,A.Veretennikov, A.Golubov, and J.Aarts, Phys.Rev.Lett. 86, 2427
(2001)
S/F/S -junctions
Ryazanov et al., Phys. Rev. Lett. 86, 2427 (2001)
Phys. Rev. Lett. 96, 197003 (2006)
S F S
Types of superconducting memory
I: Spin-valve devices on control
of the critical temperature
A.A. Jara et al.,
Phys. Rev. B
89, 184502 (2014)
S-F-F type: F-S-F type:
N. Banerjee et al.,
Nat. Comm.
5 3048 (2014)
Y. Zhu, et al,
Nature Materials,
16, 195–199 (2017)
P.V. Leksin et al,
Phys. Rev. Let.
109, 057005 (2012)
Change mutual magnetization -> Change TC of superconductive layer S
-> Switch S from superconductive state to normal one
General Drawbacks: Require additional circuit for remagnitization: contour to apply
strong field and source of spin polarized electrons for spin-torque
Difficulties with integration of these elements into circuit
S-layer can be superconductive if dS>ξS
Spin valve works effectively if dS>ξS
Proposed to use ferromagnetic insulators
Benefit: F-layer can be fixed with antiferromagnetic layer (true spin-valve)
Types of superconducting memory
II: Josephson spin-valve devices
S-F-N-F-S type:
B. Baek et al., Nature Comm.,
5, 3888 (2014)
B. M. Niedzielski et al.,
IEEE Tran. on Appl.
Supercond. , 24, 4 (2014)
Change mutual magnetization -> Critical current of Josephson junction
Drawbacks: Remagnitization required
Small ICRN due to multiple F-layers (in comparison with SIS
junctions)– small performance
S-F-s-F-S type:
K. Halterman and M. Alidoust,
SUST 29, 5, 055007 (2016)
Long-range triplet
S-F-F-F-Stype:
Benefit: Josephson junctions (integration)
B. M. Niedzielski et al.,
arXiv:1709.04815 (2017) M. Yu. Kupriyanov et al,
Patent RU 2554612 (2013) M. Houzet and A. I. Buzdin, Phys.
Rev. B, 76, 060504(R) (2007)
Summary
1. Superconducting spintronics is a new and rapidly developing field of
investigations.
2. Josephson spintronics has several lines of development.
SFF’S and SF’FF’’S devices.
Josephson spintronics provides the possibility to realize magnetic memory
compatible with energy-efficient SFQ digital circuits with high switching speed.