Heat conduction by photons through superconducting leads

G R E N O B L E 1

UNIVERSITEJOSEPH F OURIERSCIENCES. TECHNOLOGIE. MEDECINE

W.Guichard Université Joseph Fourier and Institut Neel, Grenoble, France

M. Meschke, and J.P. Pekola Low Temperature Laboratory, Helsinki University of Technology, Espoo, Finland

Thermal conductance

TK

nncqQ

ph

kcoldkhotsoundk k

kkph

)(

TKQ

TK

ffL

Q

el

kcoldkhotkk

kel

1

T1

Heat flow (T1 > T2)

T2

Heat flow Thermal conductance

What conducts heat in a solid ?

Phonons (lattice vibrations)

Quantum of thermal conductanceT T +T

Q

Th

kK B

Q 3

22

and what about photons ?

Electrons (important for metals)

Measurement of quantized thermal conductance

2DEG in a GaAs-AlGaAs heterostructure

Molenkamp et al. Phys. Rev. Lett 68 (1992)

Quantized electronic thermal conductance

Quantized phonon thermal conductance

K. Schwab et al. , Nature 404 (2000)

Silicon nitride membrane

Th

kK B

Q 3

22

Energy relaxation in a submicron metal island

1

1),(

/)(

eBTkµEe

eTEf

0 100 200 300 400 500

100

80

60

40

20

0

RS

INIS

[M]

TBATH

[mK]

0

100

200

300

400

500

Te[m

K]

M.Meschke et al.

In thermal equilibrium:

Electron-electron collissions

Electron-phonon collisions T0

TenvTe

Ge

Gep

fWPmKT

mmm

mKWT

PT

TTP

exe

phex

e

pheex

1100

025.06.04

102

3595 5

55

Pex

Pep

Energy relaxation in a submicron metal island

1

1),(

/)(

eBTkµEe

eTEf

0 100 200 300 400 500

100

80

60

40

20

0

RS

INIS

[M]

TBATH

[mK]

0

100

200

300

400

500

Te[m

K]

M.Meschke et al.

In thermal equilibrium:

Electron-electron collisions

Electron-phonon collisions T0

TenvTe

Ge

Gep

fWPmKT

mmm

mKWT

PT

TTP

exe

phex

e

pheex

1100

025.06.04

102

3595 5

55

+Electron-photon „radiative“ relaxation ?

Pex

Pep

Pe

Heat transported between two resistors

2

1

1e

1**4)(

/

th

hRvS iiV

21

22

22

21

02121

32

)]()([

TTh

krP

dhnhnhrP

B

net

1,

)(

42

21

21

rRR

RRr

Voltage noise emitted by resistor Ri:Ge= ?

1D Black body radiation

R2,T2R1,T1

Th

kKrK

dT

dPK B

QQ 3 ,

22 Quantum of thermal

Conductance:

Net heat flow from hot to cold resistor:

Schmidt et al.,Phys. Rev. Lett., 93 (2004)

Competition between ep- and e- coupling

3/122

15

Vh

krT B

cr

0.05 0.1 0.15 0.2 0.250.310-15

10-14

10-13

10-12

Gep

, = 2.0 109, = 6.0 10-20

Ge, r = 1

Ge, r = 0.2

G (

WK

-1)

T (K)

TCO

Cross-over temperature:

Th

krVTK

TTKr

PTTVP

Bep

enveQeeep

3K 5

)(2

22

e4

221

50

51

Typical experimental set-up

Island size:6.6 m x 0.8 m x 20 nm

SINIS junction size:3 m x 0.1 m

SQUID junction size:3 m x 0.1 m

Iheat

V

Ib

Electrical circuit

Actual experimental configuration: tunable impedance between the resistors

)(

* 21

totZ

RRr

QSQUIDceffe GRRCIrG ,...),,,( 21h

Tkvvx B

thth with /

dxe

exxr

G

G

x

x

thQ

e2

2

0 1

Electrical Model I

0.0 0.5 1.0 1.5 2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

G/G

Q

Ic=1nA Ic=10nA Ic=200nA

cmc

thth 1

cJ I

L2

0

Transmission line:C0 C0 C0

L0 L0 L0

C0 C0 C0

L0 L0 L0

R1 R2

R1 R2

Tunable inductance:

Here:

cJ I

L2

0

L~30 μm

Electrical Model II

0.0 0.5 1.0 1.5 2.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

G/G

Q

Ic=500nA Ic=100nA Ic=20nA Ic=0.1nA

LSQ

CSQ R2R1LSQ

CSQ

CSQUID=30fF

Thermal model

Typical parameter values:P1 = 1 fWP2 = 0

50

522

222

211

2

2

50

511

222

211

2

1

12

12

TTTrTrh

kP

TTTrTrh

kP

eeeB

eeeB

SINIS thermometer

0 100 200 300 400 500 600 700 8000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

V (

mV

)

T (mK)

Measured at I = 9 pA

Probes electron temperature of N island (and not of S!) in the case of T/Tc<0.4

Low leakage of junctions

-0.6 -0.4 -0.2 0.0 0.2 0.4-6

-4

-2

0

2

4

6

215mK 250mK 285mK 320mK 360mK 395mK 430mK

I (n

A)

V (mV)

38mK 48mK 78mK 110mK 145mK 180mK

Measured variation of island temperature:

T0

Te2Te1

Ge

Gep1 Gep2

P2P1

-0.162

-0.161

-0.160

-0.159

-0.158

-0.157

-0.156

-0.155

V S

INIS

[mV

]

Flux [a.u.]

Measured variation of island temperature:variation of bath temperature

Flux Φ0

T0

Te2Te1

Ge

Gep1 Gep2

P2P1

-2 -1 0 1 2

90

100

110

120

130

140

150

160

170 TBATH

= 157mK 147mK 114mK 102mK 75mK 60mK

T[m

K]

Ic=20nACSQUID=15fFR1=R2=200P1=1fWP2=0

Increase island temperature Te1

-2 -1 0 1 2

160

170

180

190

200

210

220

17800 fW2700 fW180 fW70 fW30 fW0 fW

T[m

K]

-2 -1 0 1 290

100

110

120

130

140

150

160

170

180

190

200

210 17800 fW7100fW2700 fW1100 fW450 fW180 fW70 fW30 fW5 fW2 fW0 fW

T[m

K]

Flux Φ0 Flux Φ0

T0<40mK T0=150mKT0

Te2Te1

Ge

Gep1 Gep2

P2P1

Measured variation of island temperature:amplitude of modulation

<40mK 75mK 102mK 114mK 147mK 157mK

T0

Conclusion

-First observation of the crossover from phonon relaxation to radiative photon relaxation at temperatures of about 100 mK

-Thermal and electrical model explain quite well the measured data

-Implications on:performance of bolometers (sensitivity): coupling to the heat bath

removing excessive heat from devices at milli-kelvin range