heat conduction by photons through superconducting leads w.guichard université joseph fourier and...
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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
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