Phase-Charge Duality in a Josephson Junction coupled to an - - PowerPoint PPT Presentation

Phase-Charge Duality in a Josephson Junction coupled to an electromagnetic environment

Silvia Corlevi, David B. Haviland

Nanostructure Physics, Royal Institute of Technology Stockholm, Sweden

Wiebke Guichard, Frank W. J. Hekking

Universite´ Joseph Fourier – CNRS Grenoble, France

Outline

Small-capacitance Josephson junctions: Phase and Charge dynamics Measurement of Bloch oscillations: Single SQUID in a tunable electromagnetic environment Thermal fluctuations in the overdamped quasicharge regime: Quasicharge diffusion Cooper pair transistor in the high impedance environment

Small-capacitance Josephson Junctions

ϕ − = cos E C 2 Q H

J 2

[ ]

ei 2 Q , = ϕ

Josephson energy

N Q J

R 2 ) ( R E ∆ =

2

C 2 e EC =

Charging energy RQ = h/(2e)2 ~ 6.45 kΩ Electromagnetic environment Z(ω)

) ( Z / V I

b b

ω =

Hz E E

C J p 10 2 / 1

10 ~ / ) 8 ( ~ h = ω ω

Voltage-biased junction Current-biased junction

Phase and Charge Dynamics of a Josephson Junction

ϕ- tunneling

Bloch

• scillations

EJ/EC RQ/R Schmid ´83

∞ ∞

Classical Josephson effect

q- tunneling

Z(ω)=R<<RQ EJ>>EC Phase dynamics Z(ω)=R>>RQ EC>>EJ Quasicharge dynamics

Classical Dynamics of the Josephson Phase

ϕ ϕ ϕ sin

2

+ + = & & & Q I I

C b C J Q

E E R R Q 2

2

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = π

Overdamped phase dynamics

1 << Q

Underdamped phase dynamics

1 >> Q R V I I

b −

=

IC IR I

“Load Line” slope=-1/R

V

C b

I I <

C b

I I >

C b

I I = = ϕ & ≠ ϕ & 1 << Q 1 >> Q

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ϕ ϕ ϕ

C b J

I I E U cos ) (

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + + = e C e R I dt dV C R V I I

C C b

2 2 1 sin sin ϕ ϕ ϕ ϕ & & h & h

N C

eR I 2 / ) ( ∆ = π

Thermal fluctuations in the case of

• verdamped-phase dynamics (theory)

) ( 2 ) ( ) ( τ δ τ δ δ R T k t I t I

B n n

= +

Overdamped phase dynamics: EJ<<EC Z(ω) =R <<RQ Langevin-type eq. for the phase ∞ = α

, 10, 3

dt dV C R V sin + + ϕ IC Ib+In= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ α α =

α − α −

) ( I ) ( I Im I I

v i v i 1 C C B RI

/ V v = T k / E

B J

= α

Ivanchenko and Zil’berman ´68

Supercurrent peak at finite voltage Phase diffusion

Thermal fluctuations in the case of

• verdamped-phase dynamics (experiment)

Steinbach, Joyez et al. 2001

Environment suppresses phase fluctuations at all frequencies

ω + = ω

− − B 1 B 1

iC R ) ( Z

Quantum fluctuations of the phase: ”P(E) theory”

ϕ- tunneling

Bloch

• scillations

RQ/R EJ/EC

∞ ∞

Classical Josephson effect

q- tunneling

Incoherent CP tunneling

Limit of small EJ

R) / R ( E / E

Q C J

<<

Z(ω)=R>>RQ EC>>EJ Z(ω)=R<<RQ EJ>>EC Quasicharge dynamics Phase dynamics

Quantum fluctuations of the phase: ”P(E) theory”

Ingold and Nazarov ´90 Ingold, Grabert and Eberhardt ´94

R/RQ= 2, 20,100 Z(ω)=R LC-harmonic oscillators 2eVb=EC

[ ]

) eV 2 ( P ) eV 2 ( P eE I

b b 2 J S

− − π = h

[ ]

∫

+ π = h h / iEt ) t ( J exp dt 2 1 ) E ( P

T k / t i Q t

B

e 1 1 e R )] ( Z Re[ d 2 ) t ( J

ω − ω −

− − ω ω ω = ∫

h

Kuzmin, Nazarov et al. ´91 Grabert and Ingold ´99

Quasicharge description of a Josephson junction

ϕ- tunneling

Bloch

• scillations

EJ/EC RQ/R

Schmid ´83

∞ ∞

Classical Josephson effect

q- tunneling

Z(ω)=R<<RQ EJ>>EC Z(ω)=R>>RQ EC>>EJ Quasicharge dynamics Phase dynamics

Quasicharge description of a Josephson junction

EJ/EC < 1 EJ/EC > 1

ϕ − ϕ ∂ ∂ − = cos E E 4 H

J 2 2 C

) ( e ) 2 (

q , n e 2 / q 2 i q , n

ϕ ψ = π + ϕ ψ

π

e q e ≤ ≤

quasicharge

EJ hωp

∆ >> [EC, kBT] EC ~ EJ Z(ω)=R >> RQ

Likharev, Zorin 1985

dq dE V =

I I dt dq

b

R V − = =

Langevin equation for the quasicharge

+I n

V

Ib-V curve: Coulomb blockade region

EJ/EC<<1

q0

VC C b

I I <

Stationary solution:

R / ] dq / dE max[ I

C=

] dq / dE max[ V

C=

I q = = & R I V

b

=

Coulomb blockade of Cooper pairs

Ib-V curve: Bloch oscillations region

EJ/EC<<1

VC

Ib> IC

Bloch oscillations:

I e 2 f

b B =

q ≠ & q / E V → d = d

Coherent tunneling of Cooper pairs

Ib-V curve: Bloch oscillations region

V(t)

VC

Coulomb blockade t

b b

RI V =

V(t) VC

• VC

<V>

b b

RI V =

t

q / E V → d = d

) ( ) ( t saw V t V

C

→

C b b

V R I V < =

VC

• VC

<V> V(t) t

b b

RI V =

Cooper pair tunneling

C b b

V R I V > =

Ib-V curve: Zener tunneling region

VC

EJ/EC<<1

Ib > IZ

Zener tunneling:

dq / dE V = q & ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π − =

b Z b C 2 J Z

I I exp I e E E 8 exp P h

Transition to higher energy bands

Thermal fluctuations in the case of

• verdamped-quasicharge dynamics (theory)

Overdamped quasicharge dynamics:

I dt dq

b

R V − = +I n ) e 2 / q 2 sin( V V

C

π =

EJ>>EC Z(ω) =R >>RQ

q/2e VC/e Increasing EJ/EC VC/e

Beloborodov, Hekking, Pistolesi 2003 kBT/eVC=0, 0.05, 0.1

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π β π β =

π β − π β −

) / eV ( I ) / eV ( I Im V V

c / R eI i c / R eI i 1 C

b b

<V>/VC I/IC

Thermal fluctuations suppress VC

noise BT

k / 1 = β

Quasicharge diffusion

Phase-Charge Duality

Beloborodov,Hekking and Pistolesi 2003

<V>/VC I/IC

Ivanchenko and Zil’berman 1968

Overdamped phase dynamics EC>>EJ Z(ω) = R <<RQ Overdamped quasicharge dynamics EJ>>EC Z(ω) = R >>RQ

dt dq e 2 2 I π = ) e 2 / q 2 sin( V V

C

π = dt d 2 V ϕ π Φ = ϕ = sin I I

C

High impedance environment

α πε 8 4 8

2

= = c e R Z

Q

h

Fine structure constant

137 / 1 = α

Q

R Z Z << Ω = = 377 / ~ ) ( ε µ ω

Φ= 0 Φ= Φ0 /2

0,00 0,01 0,02 0,03 10

4

10

5

10

6

10

7

10

8

10

9

R0 (Ω) B (T)

5 kΩ < R0 < 50 MΩ

Φ0 /2 Φ

EJ I Φext

The single-junction samples

Single SQUID junction: area ~ 0.04 µm2 SQUID arrays: 60 junctions area ~ 0.06 µm2 ASQUID/Aarray ~ 8-10 Single junction: area ~ 0.02 µm2 SQUID arrays: 60 junctions area ~ 0.06 µm2 non-SQUID arrays: 16 junctions area ~ 0.01 µm2

Measurement scheme

• Vb

V

I=(Vout-Vb)/Rf

Rf

+Vb

T = 15 mK

Φ= 0 Φ= Φ0 /2

R0 ~ 10 MΩ 50 kΩ < R0 < 50 MΩ

I-V curve of a tunable single junction in the high impedance limit

A) EJ/EC = 4.5 B) EJ/EC 0.2

≤

B A

VC=max[dE0/dq]

C = 1.8 fF , RN ~ 2.8 kΩ R0 ~ 10 MΩ

Thermal fluctuations in the case of

• verdamped-quasicharge dynamics

Tcryo= 50 mK Tnoise = 160 mK Tcryo= 250 mK Tnoise = 260 mK Tcryo = 300 mK Tnoise = 400 mK

VC = 30 µV Rfit = 150 kΩ

C ~0.9 fF RN ~ 2 kΩ R0 ~ 10 MΩ

EJ/EC~3

Comparison between Tnoise and Tcryostat

Saturation of VC inadequate filtering VC drop quasiparticles tunneling Tcryo=Tnoise

The Cooper Pair Transistor

2 / ) (

1 2

ϕ − ϕ = ϕ

g 2 1

Q Q Q Q + − = ) (

1 2

ϕ + ϕ = φ 2 / ) Q Q ( Q

2 1 +

=

φ

ie 2 ] Q , [ ] Q , [ = ϕ = φ

φ

C ~ 2C

∑

) cos( ) 2 / cos( E 2 C 2 ) Q Q ( C Q H

J 2 g 2

ϕ φ − + + =

∑ φ

High impedance environment modulation of the threshold voltage Low impedance environment modulation of the supercurrent

Zorin et al. 1999

Cooper Pair Transistor: 2D energy band diagram

( ) ( )

2 J 1 J 2 g g 2 1 2 2 2 1

cos E cos E C Q C 2 Q ) Q Q ( C 2 Q C 2 Q H ϕ − ϕ − + − + + = EJ<<ECΣ Qg = 2ne Qg = (2n+1)e 2D Bloch-band picture

2 2 n 1 1 n

dq ) q ( dE dq ) q ( dE V + = VC =max [V(q1,q2,Qg)]

I-V curve of the CPT for different environments

• 0.2
• 0.1

0.0 0.1 0.2

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

I (nA) V (mV)

• 10
• 5

5 10

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

I (nA) V (mV)

SQUID arrays CPT

R0 ~ 55 kΩ R0 ~ 1 MΩ R0 ~ 20 MΩ

EJ/ECΣ ~ 0.6

I-V curves of CPTs with different EJ/EC values

EJ/ECΣ ~ 1.9 EJ/ECΣ ~ 0.75 EJ/ECΣ ~ 8.7

R0 ~ 20 MΩ

Gate-induced modulation of the Coulomb blockade

EJ/ECΣ ~ 0.75 R0 ~ 20 MΩ

Qg=2ne Qg=(2n+1)e

Parity effects dependent on the environment ?

R0~0.3 MΩ

• 200
• 150
• 100
• 50

50 100 150 200

• 4
• 2

2 4

I (nA) V (µV)

• 20
• 15
• 10
• 5

5 10 15 20

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8

I (nA) V (µV)

• 0.10
• 0.05

0.00 0.05 0.10 0.012 0.013 0.014 0.015 0.016

V (µV)

• 0.10
• 0.05

0.00 0.05 0.10 10 12 14 16

V (µV) Vg (V)

Ibias= 2pA

2e-periodicity e-periodicity

EJ/ECΣ= 1.9

R0~ 40 MΩ

Conclusions

Measurement of Bloch oscillations in a single SQUID coulped to a tunable electromagnetic environment (SQUID arrays) Measurement of thermal fluctuations in a junction with overdamped quasicharge dynamics (EJ>EC, R>>RQ).

(S. Corlevi, W. Guichard, F. Hekking, and D. Haviland cond-mat/0510504)

Measurement of Cooper pair transistor in the high impedance limit: 2e-periodic gate induced modulations for R>>RQ