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.