# The Shot Noise Thermometer

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```					The Shot Noise Thermometer

Lafe Spietz, K.W. Lehnert,
I. Siddiqi, R.J. Schoelkopf

Department of Applied Physics, Yale University
Thanks to:
Michel Devoret, Daniel E. Prober, and Wes Tew
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=0.02 K to 300 K*

*Lafe Spietz et al, Science 300, 1929 (2003)
Thermometry
Desirable Characteristics for a Thermometer:
•   Wide Range
•   Fast
•   Primary
•   Accurate
•   Easy and simple to use
•   Physically compact

Secondary: Needs to be calibrated from some
outside standard, e.g. resistive thermometers

Primary: Needs no outside calibration—based on
understood physics, e.g. ideal gas thermometer
Cryogenic Thermometry: Overview

300 K
Johnson                 100 K
Noise
10 K     Resistance
Thermometers
1K       RuOx      50 mK
CBT        0.1 K      3He   Melting Curve
0.01 K
Nuclear
Orientation
The Kelvin
Water Triple Point = 273.16 K    By Definition

The Kelvin (K) unit of thermodynamic temperature
is the fraction 1/273.16 of the thermodynamic temperature
of the triple point of water.
ITS-90: Overview
T=1000 K
962 K
Radiation Thermometer
Platinum Thermometer
Not Primary!

25 K
{          13.8 K
Constant Volume
Gas Thermometry
3K
5K

0.65 K
}          Vapor Pressure
Thermometry
Below 0.65 K      Nothing!
PLTS 2000: 0.9 mK-1 K Scale

3He   Melting Curve

Superconducting Fixed Points
Fundamental Noise Sources
Johnson-Nyquist Noise
4k BT  A2 
SI ( f )         Hz 
R  
• Frequency-independent
• Temperature-dependent
• Used for thermometry

Shot Noise
 A2 
S I ( f )  2eI    Hz 
    
• Frequency-independent
• Temperature independent
Conduction in Tunnel Junctions
I

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  I L R  I R  L  GV
Assume: Tunneling amplitudes and
D.O.S. independent of energy Conductance (G)
Fermi distribution of electrons  is constant
Thermal-Shot Noise of a Tunnel
Junction*
Sum gives noise:

S I ( f )  2e( I LR  I R L )
 eV 
S I ( f )  2eI coth         
 2 k BT 
I  GV

*D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974)
Thermal-Shot Noise of a Tunnel
Junction

2eI
Shot Noise

Transition Region              4kBTJohnson Noise
eV~kBT                      R

 eV 
S I ( f )  2eI coth         
 2 k BT 
Self-Calibration Technique
P(V) = Gain( SIAmp+SI(V,T) )
P(V)

2eI
 4k T             
G  B  S IAmplifier 
 R                
                  
{
V
2kBT / e
Experimental Setup: RF + DC
Measurement

P
5m
SEM

Al-Al2O3-Al Junction
High-Bandwidth Measurement

P 1
P  B

8
B ~ 10 Hz ,  = 1 second    P 104
P
Noise Versus Voltage
 eV         eV  
Fit = Gain       Coth         - T
 2k B       2k BT  
Universal Functional Form
Agreement over four decades in temperature
Comparison With Secondary
Thermometers
High Precision Measurement
 e(V - V        )         e(V - V )  
Fit = Gain           off
Coth            -T
off

 2k   B                   2k T  
B
Residuals

 2  1.04
T  502.5mK  .094mK
Gain  1.0001  6.7 105
Offset  18nV  4.2nV
Uncertainty vs. Integration Time
Correlations of Fit Parameters

 e(V - Voff )       e(V - Voff )  
Fit = Gain               Coth                - T
 2k B               2k BT  
Thermodynamic Uncertainties
of Temperature Scales
Thermodynamic
Uncertainty of
PLTS-2000

500 mK

SNT
High Bias Nonidealities
eVmax ~10kBT
High T        High Bias

SI (V )  2eI (V )
Nonlinear Current and Noise
R
~ 800 ppm
R

R
~ 6%
R

SIjunction (V ) R junction (V )
T junction   
4kB
Shot Noise and Inelastic Tunneling
117 mV  Al2O3 Vibrational mode

T=4 K

117 mV
Self-Heating
Thermal Circuit:
V2/Rjunction

Rlead                       Rlead

V~T, P~T2, G~T
T/T ~ constant

Even with all cooling through leads, can have
negligible effect on SNT measurement
Null-Balancing Noise
Measurement for High Precision
Noise Contours in Voltage-Space

Small range of noise keeps
detector in linear range
Modular SNT Package
Copper Tubing for DC lines         Copper Plumbing parts

SMA Connectors for RF                          Tunnel Junction

Built-in Bias Tee
(on-board SMT
Components)

Total cost of package <10\$
Work In Progress: High Accuracy
Comparisons
Water
Triple
Point
(273.16 K)

High Precision
4 K to 300 K
Cryostat
- Calibrated RhFe Comparison
- Hydrogen Triple Point
(13.8033 K)
Future Work
• Determine effect of nonlinearity on shot noise
• Measure heating effects with dirty film
• Improve room temperature results
• Measure hydrogen triple point
• Make SNT more modular and easy to use for
use in other labs and for commercialization
• Push the lower temperature end with lower
system noise temperature and more careful
filtering
Summary
• Demonstrate functional form of junction noise
0.02 - 300 Kelvin*

• Use as fast, accurate thermometer

• As good as 200 ppm precision, 0.1% accuracy

• Relates T to V using only e and kB
Possible kB determination?

*Lafe Spietz et al, Science 300, 1929 (2003)
Tien-Gordon Theory

Tucker and Feldman, 1985
Tien-Gordon for Noise of Junction
Diode Nonlinearity
Vdiode = GP + bG2P2

b= -3.1 V-1   1mV => 3x10-3 fractional error
Conductance

R=31.22Ohms
More Conductance
Fano Factor Has No Effect:

 eV 
2eI coth       
 k BT 

1           eV      8k BT
2eI coth         
3           k BT    3R
Temperature Measurements
Over Time
Tfit
6.0                TRhFe          75.0
Tnoise
Gain

Gain [10 V/K]
5.5
T and Tnoise(K)

74.5

5.0
74.0

-6
4.5
73.5
4.0
73.0
0   2   4     6    8   10
Time [hours]
Experimental Setup:RF + DC
Measurement and Thermometry
RhFe
Thermometer
capacitors

device               RuOx
Thermometer
inductors
Fit With Two Parameters
 eV         eV  
Fit = Gain       Coth         - T
 2k B       2k BT  
Residuals

 2  1.49
T  502.5mK  .094mK
Gain  1.0001  6.7 105
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)
• Frequency dependence*
• No B-dependence
• Compact electronic sensor                • Self-heating

• Possibility to relate T to
frequency!*
*R. J. Schoelkopf et al., Phys Rev. Lett. 80, 2437 (1998)
Tunnel Junction
(AFM image)
R=33 W
Area=10 mm2        Al-Al2O3-Al Junction
V+
I+

I-

V-

```
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