Transcript The Ideal Electron Gas Thermometer

The Ideal Electron Gas
Thermometer
Lafe Spietz, K.W. Lehnert, I. Siddiqi,
R.J. Schoelkopf
Department of Applied Physics, Yale University
Thanks to:
Michel Devoret and Daniel E. Prober
Introduction
• Johnson-Schottky transition of the noise in
tunnel junctions
• Relates T and V using only e and kB
 primary thermometer
• Demonstrate operation from
T=0.26 K to 300K
Fundamental Noise Sources
Thermal(Johnson) Noise
4 k BT  A2 
SI ( f ) 
 Hz 
R  
• Frequency-independent
• Temperature-dependent
• Used for thermometry
Shot(Schottky) Noise
SI ( f )  2eI
 A2 
 Hz 


• Frequency-independent
• Temperature independent
Conduction in Tunnel Junctions
I
M I M
V
G
I l  r   f l (1  f r )dE
e
G
I r l   f r (1  f l )dE
e
Difference gives current:
Fermi functions
I  Il r  I r l  GV
Assume: Tunneling amplitudes and
D.O.S. independent of energy
Fermi distribution of electrons
Conductance (G)
is constant
Thermal-Shot Noise of a Tunnel
Junction*
Sum gives noise:
SI ( f )  2e( Il r  I r l )
 eV 
S I ( f )  2eGV coth 

 2kBT 
I  GV
*D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974)
Thermal-Shot Noise of a Tunnel
Junction
2eGV=2eI
Shot Noise
Johnson-Schottky
Transition Region
eV~kBT
4kBT
Thermal Noise
R
 eV 
S I ( f )  2GV coth 

 2kBT 
Johnson-Schottky Transition:
Direct relationship between T and V
Tunnel Junction
(AFM image)
R=33 W
Area=10 mm2
Al-Al2O3-Al Junction
V+
I+
I-
V-
Experimental Setup:RF + DC
Measurement
Experimental Setup:
Pumped He Cryostat
Noise power vs. bias voltage:
High bandwidth:
hence fast
For  = 1 second,
8
B ~ 10 Hz
 noise
noise

2
B
~ 10
4
Self-Calibration Technique for
Thermometry
P = Gain*B( SIAmp+SI(V,T) )
Noise Versus Voltage

 eV  
eV
Fit = Gain  TNOISE +
Coth 

2k
2k
T
B
 B 

T=4.372 K
Tnoise=5.128 K,
Gain=29.57 mV/K
Universal Functional Form:
Agreement over three decades In temperature
Comparison With Secondary
Thermometers
Temperature Measurements
Over Time
Tfit
TRhFe
Tnoise
Gain
5.5
75.0
74.5
5.0
4.5
73.5
4.0
73.0
0
2
4
6
8
Time [hours]
10
-6
74.0
Gain [10 V/K]
T and Tnoise(K)
6.0
Merits Vs. Systematics
Merits
Systematics
• Fast and self-calibrating
• I-V curve nonlinearities
• Primary
• Amplifier and diode
• Wide T range
nonlinearities
(mK to room temperature)
• No B-dependence
• Compact electronic sensor
• Frequency dependence*
• Self-heating
• Possibility to relate T to
frequency!*
*R. J. Schoelkopf et al., Phys Rev. Lett. 80, 2437 (1998)
Summary
• Ideal Electron Gas Thermometer based on
Johnson-Schottky transition of noise in a
tunnel junction (thermal-shot noise.)
• Fast, accurate, primary thermometer
• Works over a wide temperature range
• Relates T to V using only e and kb
applications for metrology
Diode Nonlinearity
Vdiode = GP + bG2P2
b= -3.1 V-1
1mV => 3x10-3 fractional error
Conductance
R=31.22Ohms
More Conductance
2
3
1
4