Electronic refrigeration using superconducting tunnel junctions - - PowerPoint PPT Presentation

Electronic refrigeration using superconducting tunnel junctions

Sukumar Rajauria

• H. Courtois, F. W. J. Hekking and B. Pannetier

I

Motivation

Quantum nano-electronics:

• New devices with new functionality (SET, qubits, …)
• High performance at (very) low temperature.

On-chip cooling of a nano-device:

• Improved efficiency, more compact,
• N-I-S micro-coolers promising.

Basic knowledge on:

• N-I-S junction with a heat perspective,
• Heat transport at micro- or nano-scale.

Motivation

First S-I-N-I-S cooler – Helsinki Prototype cooler – N. I.S.T.

Quasiparticle tunneling in N-I-S junction

E Empty States Forbidden states 2∆ Principle of N-I-S cooler The superconductor energy gap induces an energy-selective tunneling.

T = 0 K

N I S N

Quasiparticle tunneling in N-I-S junction

E Forbidden states 2∆

~kT

Principle of N-I-S cooler The superconductor energy gap induces an energy-selective tunneling. Empty States S

T > 0 K

N I S N

Quasiparticle tunneling in N-I-S junction

eV

Empty States Quasiparticle tunnel current:

[ ] dE

) E ( f ) eV E ( f ) E ( n eR 1 I

S N S N T

∫

∞ ∞ −

− − =

2

IT S

T > 0 K

N I S N

• 2
• 1

1 2

• 2
• 1

1

T = 0.49Tc ITeRn/∆ eV/∆ T = 0.07Tc

Quasiparticle tunneling in N-I-S junction

eV

Empty States Quasiparticle tunnel current:

[ ] dE

) E ( f ) eV E ( f ) E ( n eR 1 I

S N S N T

∫

∞ ∞ −

− − =

Net Cooling Power:

[ ] dE

) E ( f ) eV E ( f ) E ( n ) eV E ( 1 P

∫

∞

− − − =

PCool S

T > 0 K

N I S N

[ ] dE

) E ( f ) eV E ( f ) E ( n ) eV E ( R e 1 P

S N S N 2 Cool

∫

∞ −

− − − =

PCool ≈ (Ē/e). IT – V.IT Joule heat Cooling

Quasiparticle tunneling in N-I-S junction

0.06

eV

Empty States PCool Net Cooling Power:

[ ]dE

) E ( f ) eV E ( f ) E ( n ) eV E ( R e 1 P

S N S N 2 Cool

∫

∞ ∞ −

− − − =

PCool ≈ (Ē/e). IT – V.IT > 0 PCool is symmetric to bias.

• 1.0
• 0.5

0.0 0.5 1.0 0.00 0.02 0.04

T = 0.49Tc

P

Coole 2R N/∆ 2

eV/∆

T = 0.07Tc

S N I S N

T > 0 K

S-I-N-I-S = 2 N-I-S junction in series Cooling power increases by a factor of 2

S-I-N-I-S junction

E E

2∆ Empty States

eV eV

2∆ IT IT PCool PCool

• F. Giazotto, T. T. Heikkila, A. Luukanen, A. M. Savin and J. P. Pekola, Rev. Mod. Phys. 78

78, 217 (2006).

Better thermal isolation

• f N-island

I S Occupied States

T > 0 K

N S I

Thermometer Junction

2 µm

Cu

Al Al

10

• 1

10

Cooler ON 134 mK

hermometer

Cooler OFF 288 mK

Cooler with External thermometer

) exp(

N BT

k eV I I ∆ − ≈

• E. Favre-Nicollin et. al.

Cooler junctions

• 2,0
• 1,5
• 1,0
• 0,5

0,0 0,5 1,0 1,5 2,0 10

• 2

dI/dVThe VThermometer/∆

Outline

• Electronic temperature without thermometer
• Thermal model
• Andreev current contributions
• Conclusions

Quasiparticle diffusion based heating in S-I-N-I-S cooler heating in S-I-N-I-S cooler

Probe Junction: N electrode is strongly thermalized, litlle cooling effect expected. 1 µm Al

Cooler with NO external thermometer

I Cu Cu Al Cooler junctions: N electrode is weakly coupled to external world, strong cooling effect expected.

Tbath = 304 mK Al I 1 µm Cu Cu

Cooling in S-I-N-I-S junction

Cooler

0.1 1 1 10 dI/dV (norm.) Cooler Probe

Probe follows isothermal prediction at Tbath. High resolution measurement (log scale) ) T k eV exp( I I

e B

∆ − ≈

Probe

0.001 0.01 0.01 0.1

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 V/(2) ∆

10

• 3

10

• 2

10

• 1

10 ITeR

n/∆

Te = 98 mK Isotherm T

e = 304 mK

Cooler

• two refrigerating junction are symmetric;
• N-metal is at quasi-equilibrium;
• Ideal superconductor;

Temperature determination

600

0.0 0.1 0.2 0.3 0.4 0.5 10

• 4

10

• 3

V (mV)

0.0 0.2 0.4 150 300 450 Te (mK) V (mV)

Thermal model

( ) ( ) ( ) ( ) ( ) [ ]

∫

+∞ ∞ −

+ − − = dE eV E f E f E n eV E eR 1 V P

N S S N Cool

N electrons, T

e

S, Tbath S, Tbath Power flow from N electrons to the S electrodes remaining at base temperature

The thermal model

N phonons, Tph Substrate phonons, Tbath Electron - phonon coupling Kapitza thermal coupling

( )

4 ph 4 bath K

T T KA P − =

( )

5 e 5 ph ph e

T T U P

• Σ

=

N electrons, T

e

S, Tbath S, Tbath Power flow from N electrons to the S electrodes remaining at base temperature

The thermal model - Hypothesis

( ) ( ) ( ) ( ) ( ) [ ]

∫

+∞ ∞ −

+ − − = dE eV E f E f E n eV E eR 1 V P

N S S N Cool

N phonons, Tph = T bath Electron - phonon coupling Kapitza thermal coupling Hyp.: N phonons are strongly thermalized

( )

5 e 5 bath ph e

T T U P

• Σ

=

Substrate phonons, Tbath

For Tph = Tbath

( )

5 bath 5 e Cool

T T U P 2 − Σ =

5 Cool 5 e

T P 2 U 1 1 T T Σ − =      

Hypothesis of phonon thermalized to the bath

0.4 0.6 0.8 1.0

Tbath (mK) Σ(*10

9 Wm

• 3K
• 5)
• 292

1.21 489 1.02 586 0.80

• (Te/Tbath)

5

5 bath bath

T U 1 T Σ − =      

20 40 60 80 100 0.0 0.2

2PCool/Tbath

5 (pW/K 5)

Fitted Σ much smaller than expected (2 nW.µm-3.K-5) Impossible to fit data with a given Σ

( )

5 e 5 ph ph e

T T U P − Σ =

−

N electrons, T

e

S, Tbath S, Tbath N phonons, T Electron - phonon coupling N phonons can be cooled

The thermal model

Cool

P

Cool

P

( )

4 ph 4 bath K

T T KA P − = N phonons, Tph Substrate phonons, Tbath Kapitza thermal coupling N phonons can be cooled

Two free fit parameters: Σ = 2 nW.µm-3.K-5 K = 55 W.m-2.K-4 Kapitza coupling smaller by a factor

• f 3 than bulk.

300 400 500 600

Tbath

T (mK)

T

Phonon cooling

• f 3 than bulk.

0.0 0.1 0.2 0.3 0.4 100 200

model experiment Te

V (mV)

Tph Sukumar Rajauria, P. S. Luo, T. Fournier, F. W. J. Hekking, H. Courtois and B. Pannetier, PRL (2007) (2007)

Phonon cooling dominant at high temperature.

• How much can we lower the electronic temperature ?
• Can we reach below 10mK starting with a dilution temperature ?
• What about the other contribution like Andreev Current etc. ?

What now?

• What about the other contribution like Andreev Current etc. ?
• Is a quantitative analysis possible ?

Andreev current-induced dissipation dissipation

Experiment at a very low temperature

10

• 2

10

• 1

10

340

dI/dV norm.

450 mK

Zero-bias anomaly. Not a linear leakage. Cannot be fitted with a smeared D.O.S or a non-

• 0.8
• 0.4

0.0 0.4 0.8 10

• 4

10

• 3

10

340 240

V(mV)

90

smeared D.O.S or a non- equilibrium distribution in N. Likely two electron tunneling process.

Andreev reflection

E

eV

Transmission probability proportional: t2 For tunnel barrier: t is very small E < ∆ : No quasiparticle tunneling I S

T > 0 K

N

eV

Andreev reflection probability vanishes for a tunnel barrier

jA

• A. F. Andreev, Zh. Eksp. Teor. Fiz. ’64 , D. Saint-James, J. Phys. (Paris) ‘64

S N

Confinement-enhanced of the Andreev current

Nb-I-InGaAs junction

Confinement of electron by disorder + Quantum coherence Enables coherent addition of 2e tunneling amplitudes = Enhances sub gap conductivity

Kastalsky et al PRL 91 van Wees-Klapwijk et al PRL 92

diff 2 N A

R . G G =

Andreev current in disordered N-I-S junction

{ }dE

) eV 2 / E ( f ) eV 2 / E ( f ) E ( I ) V ( I

N N A

∫

∞ ∞ −

+ − − =

where I(E) is the spectral current Hekking and Nazarov model : Tunnel barrier in between N and S. Sub-gap conductivity is more sensitive to disorder.

{ }

r d ) r ( P ) r ( P Se 16 hG ) E ( I

2 barrier E E 3 2 n

∫

−

+ ν π =

where PE(r) is the cooperon. Length scale: Phase coherence length, bias or temperature cut off.

Hekking et al PRL 93 and PRB 94, Pothier et al PRL 94

Isotherm of Andreev and Quasiparticle current

Total current = Andreev current + Quasiparticle current IProbe = IA + IT Fit parameters :

• 2

10

• 1

10 10

1

I (nA) Quasiparticle Current Andreev Current

Tbath = 90 mK

Fit parameters : Lϕ = 1.5 µm Scaling factor 1.4 Good fit for the probe.

0.0 0.2 0.4 0.6 0.8 1.0 10

• 4

10

• 3

10

• 2

Probe eV/∆ Andreev Current

Total current = Andreev current + Quasiparticle current ICooler = IA + IT Fit parameters : Lϕ = 1.5 µm

Quasiparticle cooling fit

10

• 1

10 10

1

I (nA)

Cooler

Extra dissipation missing in the thermal model?

Tbath = 90 mK

Lϕ = 1.5 µm Scaling factor 0.5 K = 120 W.m-2.K-4 Quasiparticle cooling does not fit experiment.

0.0 0.2 0.4 0.6 0.8 1.0 10

• 3

10

• 2

eV/∆

Andreev current added in cooling model

eV/(2∆)

Cool

P

N electrons, T

e

S, Tbath S, Tbath

Cool

P

Thermal model with Andreev heat

I=IA+IT The current source work results in a Joule power in the normal metal

V I P

A A

× =

( )

5 e 5 ph ph e

T T U P − Σ =

−

( )

4 ph 4 bath K

T T KA P − =

N phonons, Tph Substrate phonons, Tbath Electron - phonon coupling Kapitza thermal coupling

Andreev heat subjugate Quasiparticle cooling

Low bias: Andreev heating dominates Near gap bias: Quasiparticle cooling again

10

• 3

10

• 2

10

• 1

(pW)

Heating PA

PA

Quasiparticle cooling again prevails. Net cooling power = PCool - PA

• 1,0
• 0,5

0,0 0,5 1,0 10

• 6

10

• 5

10

• 4

Pcool

V/∆ T = 100 mK P(p

PCool

Experiment Vs Model

Fit parameters : Lϕ = 1.5 µm Scaling factor 0.5 K = 120 W.m-2.K-4 Fits experiment from 430 mK to 90 mK.

10 10

1

430

I(nA)

330 to 90 mK. Andreev reflection contributes both to charge and heat current.

0.0 0.2 0.4 0.6 0.8 1.0 10

• 3

10

• 2

10

• 1

230

eV/(2∆)

Andreev Heat added to the Cooling Model Te = 90 mK

Tbath = 90 mK

Heating at low temperature

150 200 250

T (mK)

Andreev reflection gives:

• a small charge current
• a significant heat current

Andreev reflection-

T

e (mK)

0,0 0,1 0,2 0,3 0,4 50 100

V (mV)

Andreev reflection- induced heating is fully efficient.

Sukumar Rajauria, P. Gandit, T. Fournier, F. W. J. Hekking, B. Pannetier and H. Courtois, PRL (2008) (2008)

Tbath

Conclusions

• Direct determination of the electronic

temperature in the N-metal

0.0 0.1 0.2 0.3 0.4 0.5 10

• 4

10

• 3

10

• 2

10

• 1

10 IeRn/∆ V (mV) T = 98 mK Isotherm T = 304 mK Cooler

• Electron and Phonon cooling in N-I-S

nano-junctions

• Andreev current heating at very low

temperature

0,0 0,1 0,2 0,3 0,4 50 100 150 200 250

T (mK) V (mV)

Thanks to

Thesis advisor :

• B. Pannetier and H. Courtois.

Experiment

0.0 0.1 0.2 0.3 0.4 0.5 10

• 4

10

• 3

10

• 2

10

• 1

10 IeRn/∆ V (mV) T = 98 mK Isotherm T = 304 mK Cooler

• T. Fournier, T. Crozes, B. Fernandez and C. Lemonias.
• P. Brosse for Cryogenics.
• Electronics team.
• Ph. Gandit for measurement in dilution refrigerator.

Theory

• F. Hekking, A. Vasenko and M. Houzet.

0,0 0,1 0,2 0,3 0,4 50 100 150 200 250

T (mK) V (mV)

Substrate N S S IT PCool PCool

100 200 300

Te T

e (mK)

Tph

Perspective (2) – Efficient Traps

0.0 0.1 0.2 0.3 0.4 100

model experiment V (mV)

Group velocity of qp (at ∆)

dk dE h 1 = v g

100 200

Te (mK)

Heat returns in N due to backtunneling of quasiparticles ~ Nqp - Nqp0

Perspective (2) – Efficient Traps

Cold N Hot S PCool

0.3 0.4

V (mV)

Net Cooling power: PCool - f.IT Quasiparticles accumulation contribute to return power.

Sukumar Rajauria, H. Courtois and B. Pannetier, submitted (2009)

Fit parameters: K = 55 W.m-2.K-4 f = 15 pW/µA

Perspective (3) – Reduce Andreev heat

100 150

0 µm 0.01 µm Te (mK) Lϕ = 1µm

• N-metal with small Lφ:

AuPd

0.0 0.2 0.4 0.6 0.8 1.0 50

eV/2∆

• Ferromagnetic material
• NSQUIDS

Quasiparticle tunneling in N-I-S junction

Charge current : j1 – j2 – j3 + j4

2 eV j1 j3

[ ] dE

) eV E ( f ) eV E ( f ) E ( n eR 1 I

N N S N T

∫

∞ ∆

+ − − =

• 2
• 1

1 2

• 2
• 1

1 T = 0.49T

c

IeR

n/∆

eV/∆ T = 0.07T

c

S

T > 0 K

N

j2 j4

I S N

Quasiparticle tunneling in N-I-S junction

Charge current : j1 – j2 – j3 + j4 Quasiparticle current : j + j – j – j Joule heat Cooling

eV j1 j3

[ ] dE

) eV E ( f ) eV E ( f ) E ( n eR 1 I

N N S N T

∫

∞ ∆

+ − − =

Quasiparticle current : j1 + j2 – j3 – j4

[ ]dE

) E ( f ) eV E ( f ) eV E ( f ) E ( n R e 1 J

S N N S N 2 q

∫

∞ ∆

− − + − =

Net Cooling Power : – IT.V + E.Jq

[ ]dE

) E ( f ) eV E ( f ) E ( n ) eV E ( R e 1 P

S N S N 2 Cool

∫

∞ ∞ −

− − − =

S

T > 0 K

N

j2 j4

I S N

Andreev and Quasiparticle current

Charge current: j1 – j2 – j3 + j4 Quasiparticle current: j1 + j2 – j3 – j4

+ jA

Joule heat Cool

eV j1 j3 jA

Net charge transfer Net Current : I = IT Net cooling power : P = PCool I S

T > 0 K

N

j2 j4 jA

+ IA

• IA.V