Transcript slides from group 3 - EECS @ University of Michigan

Paul Lee
Wayne Fung
George Ioannou
Smitesh Bakrania
Capacitance standard using
an electron pump
Based on work published by
Keller, Martinis, Zimmerman
and Steinbach (1996 to 1999)
Outline
Introduction
• Current Standard for Capacitance
• Counting electrons
• Josephson voltage standard
Approach
• Components:
1. Electrometer
2. Bridge, Calculable capacitor circuit
3. Pump:
• SET theory: Coulomb blockade
• Pumping mechanism
• Pump operation
• Fabrication process
Results and analysis
• Measurements - accuracy, repeatability
• Analysis - comparison
Conclusion
Paul Lee
Wayne Fung
Introduction
Current Standard for Capacitance
Counting electrons
Josephson voltage standard
Current Standard for Capacitance
“Calculable Capacitors”
•
Thompson and Lampard discovered the
cylindrical cross capacitor in 1956
•
Special arrangement of electrodes:
• Capacitance/unit length proportional
to just one length to be measured
• Factor of proportionality depends
only on the magnetic constant µ0 and
the speed of light c.
•
A,B,C, D are four circular cylinders (cross
capacitor) enclosed by a movable screen
E, spaced to limit cross capacitance.
•
F and G tubular electrodes screen out
the internal capacitances.
•
G can move axially relative to F so that
the capacitance varies linearly by (ln
2)/42 e.s.u. per cm displacement
Partially reflective coating on the opposing
surfaces of F and G
W.K. Clothier, Metrologia 1,36 (1965)
Why Consider an Alternative?
• A beam of monochromatic light is
directed along the axis of the guard tubes
• This produces circular fringes, and any
change in spacing can be viewed in the
fringes.
• The change in capacitance can be
determined with a relative uncertainty of
about 1x10-8 F (comes from limited
accuracy of the velocity of light)
Further Considerations
• Need precise alignment of electrodes of
order 1 m in length
• Compensation of end effects needed to make
a system of finite length behave like an infinite
system over a limited range
http://www.ptb.de/en/org/2/26/inhalt26_en/th_la_en.htm
W.K. Clothier, Metrologia 1,36 (1965)
Let’s Go Smaller!
Using SET Devices to Measure C
•
•
•
•
•
•
Allows precise manipulation and detection
of single electrons
Can be used to create a capacitance
standard based on the quantization of
electric charge
Energy required to charge a capacitance
C0 with one electron (Coulomb Blockade
Energy):
E = e2/2C0
C0  0.1 fF and e2/2C0  10 K with today’s
nanolithography
When cooled to 0.1 K, SET effects
completely dominate thermal fluctuations
Individual electrons are manipulated using
an electron pump (more details to follow)
Keller, Mark W. Science Vol 285
www.estd.nrl.navy.mil/code6870/lith/lith.html
Main Idea Behind Electron Counting
Capacitance Defined Based
on Counting Electrons
•
•
Very Simple Concept!
The charge transfer of a single
electron of charge Q from one
conductor to the next creates a
potential difference V
•
C = Q / V = Ne / V
•
Larger sample size gives better
accuracy
Proposed standard requires pumping
~108 electrons onto a 1 pF capacitor
with uncertainty in the number of
electrons of 1
•
Keller, Mark W. Science Vol 285
Uncertainties in new standard
Ne
C
V
•Uncertainty of N is 0.01 ppm.
•Uncertainty of e is 0.6 ppm.
•V measured using Josephson standard, which has uncertainty 0.8
ppm
•However, e is correlated to V via fine structure constant and
Josephson constant.  the nonexperimental uncertainty of C is just
the uncertainty of the fine structure constant, which is 0.09 ppm.
Quantum Metrology Triangle
• According to theory, 3 relations,
2 unknowns e and h  test the
triangle.
•Measure the voltage across a
Josephson junction driven at
frequency f  V = f / KJ
• Measure the current of an SET
pump, pumping at frequency f  I
=Qf
•Measure the quantum Hall
resistance RH
•Compare whether V / I = RH
•If experiments do not agree with
the triangle to at least 0.01 ppm,
then one of the relations is not
valid  new physics needed
George Ioannou
Wayne Fung
Smitesh Bakrania
Approach
1. Electrometer
2. Bridge Circuit and Ref. Capacitor
3. Electron Pump
Schematic of Experiment
Electrometer
Used to measure Vp
across the external
island
Necessary because of
very sensitive
measurements
Electrometer History
•
In 1784, Coulomb developed the torsionbalance electrometer, a sensitive device that
measures electric forces.
•
Measured very small charges, estimated the
attractive and repulsive forces between bodies
of known surface area.
•
Consisted of a horizontal insulating needle
(missing in the Museum's instrument) with a
small ball of conducting material at one end
and counterpoise at the other, suspended in a
glass receiver at the end of a thin thread
•
Bodies were introduced into the receiver next
to the ball and their charge measured by the
degree of deflection of the indicator needle or,
to be more precise, by the torsion under which
the thread must be placed in order to bring the
needle back to its original position.
Modern Electrometers
• Researchers have scaled Coulomb’s
invention down to just a few micrometers
in size.
• Andrew L. Cleland of the University of
California, Santa Barbara and Michael L.
Roukes of the California Institute of
Technology in Pasadena fashioned the
miniature electrometer out of silicon
Electrometer Fabrication
SOI Cross-section
Electrometer Fabrication
Top View
Cross Section
Electrometer Usage
• Movable electrode rests on a paddle
attached to a thin, flexible beam that twists
and vibrates in response to electric
attraction
• Motion can be detected by applying a
magnetic field
• Vibrating beam cuts through the magnetic
field, generating a voltage that is sensed
by another electrode in the device
Electrometer Specs
Operates under high input resistance / lower
input incurrent modes
Current sensitivity in the range of 10-15 A
Voltage measurements can be made from
10 MV to 200 mV
Comparison of standards to test
accuracy
AC Bridge Circuit
I1
C
•Measure C using old standard
•Compare it to the value obtained
from counting electrons
Vary V1 and V2 until the
null detector reads zero
potential difference.
C
V2

Cref V1
I2
Cref : calibrated with a 10-pF
silica-dielectric capacitor
traceable to NIST’s calculable
capacitor at 1592 Hz.
Electron Pump
Single electron tunneling (SET)
Criteria:
Charging energy must be larger
than thermal energy
Ec > kBT
Electron number on dot needs to
be well-defined; junction contact
resistance must be larger than
resistance quantum
Rcontact > h/e2
Electron Pump
Single electron tunneling (SET) Theory
Electrostatic energy for island with
N electrons:
Q2/2C = (Ne)2/2C
__________________________
Total energy on island:
U(N) = Ei + (Ne)2/2C
__________________________
Electrochemical potential:
mN+1 = (N+0.5)e2/C + e/C [CiVi]
__________________________
Charging energy:
m = mN+1 - mN=Ec=e2/C
Electron Pump
Coulomb blockade
mS and mD are electrochemical
potentials of source and drain,
respectively.
Coulomb oscillations
Electron Pump
Coulomb diamonds: with ZERO bias
Electron
Pump
Coulomb diamonds: with NON-ZERO bias
Electron Pump
Coulomb diamonds: with constant gate Voltage
Electron Pump
The idea behind single electron pump: a counter
Electron counting accuracy is critical
Even with a coulomb blockade an electron may
virtually tunnel into the island
This is avoided by having multiple junctions
(multiple blockades)
Chain of metal islands separated by tunnel
junctions, with gate electrodes coupled capacitively
to each island
Trapping electrons
Electron Pump
Goals
Proposed standard requires pumping of
~108 electrons onto a 1 pF capacitor with
an uncertainty in the number of electrons of
±1
Small leakage rate when the gate pulses
are turned off (the hold mode) so that the
charge on the capacitor remains fixed while
the voltage is measured
To maximize the coulomb blockade and
thus minimize unwanted tunneling events,
the total capacitance of each island in
the pump must be small
 Charging needs to be large, total
capacitance must be small (Ec = e2/Ctotal)
Electron Pump
“side effects”
Stray capacitance:
island capacitance determined by the
junction capacitance Cj, the gate
capacitance Cg, the stray capacitance to
all nearby conductors Cstray, and the
self capacitance of the island Cself.
Cj – fabrication leads to >0.2 fF - junctions
Cg – possible to make less than 0.1Cj
Cstray – choosing substrate with small 
Cself – reducing island size
Electron Pump
Multiple junction electron pump
Error/e: 500 x 10-9
hold time: 10 s
Sapphire
substrate
( = 10)
Cg ~ 0.03 fF
Cstray ~ 0.16 fF
Cself ~ 0.02 fF
Error/e: 15 x 10-9
hold time: 600 s
30 times more
accurate
Fused quartz
substrate ( = 3.75)
Cg ~ 0.02 fF
Cstray ~ 0.06 fF
Cself ~ 0.01 fF
 reliable fabrication
Electron Pump
“side effects”
Cross capacitance:
small distance between all the islands
and gates
Voltage applied to one gate, island
nearest may polarize with charge
- +
- +
+
+
+
+
-
Solution: electronically adding a fraction
of the applied voltage, with opposite
polarity, to neighboring gates
This geometry had cross-capacitance
value of 20% of Cg
Electron Pump
Multiple junction electron pump
Magnetically
controlled
switches
Cryogenic
capacitor
Electron pump
and electrometer
Switches
Electron Pump
Pump operation
Close needle switch to measure
voltage-current curve of the pump
Open needle switch to detect
intentionally pumped electrons or
errors
Plot: shows ±e pumping mode, single
electron pumped on and off the
external island.
7.6 mV step is due to change in island
charge of e
Electron Pump
Pump operation: digital logic section
1
3
2
4
Allows control of
- number of electrons pumped
- direction of pumping
- wait time between pumped
electrons
1. Triangular voltage pulses on 6
output channels by
charging/discharging capacitors.
2. DC bias adjusted to compensate
for background charges on islands
of pump.
3. Pulses and dc biases are summed
to produce set of {Vg} on gate lines.
4. Cross-capacitance cancellation
circuit performs transformation
producing set of {Vg’}
Electron Pump
Pump operation: Cross-capacitance cancellation
Vp across the pump at constant bias
current is fundamentally periodic
(period e)
If Vgi polarizes only island i, then Vp is
a periodic function of any Vgi.
If not periodic, more than one island is
polarized ({Vg}={Vg’})
For a particular geometry, this needs to
be done only once.
Electron Pump
Pump operation: error detection
Background charges in the junction oxide or
substrate produce random island
polarization of order e over time.
DC biases tuned so each island charge in the
absence of gate pulse is much smaller than e.
In ±e mode, operate faster than electrometer –
hence constant electrometer signal as long
as no errors are present.
An error causes sudden jump in the signal.
Electron
Pump
Fabrication – two angle evaporation
Mask patterned through e-beam
lithography
Oxidation step for junctions
Aluminum forms conductors while
the junctions are Al oxides
Angle determines overlap
For fixed oxide thickness: Cj aRj-1
(easily measured at room temp.)
Four chips fabricated sequentially
for desired overlap to achieve Cj.
Electron
Pump
Fabrication – two angle evaporation
How can we make a tiny island
http://www.ptb.de/en/org/2/24/244/winkel.htm
Electron Pump
7-junction electron pump
http://www.jcnabity.com/nistpump.htm
Gate
Capacitor
Tunnel
junctions
Dual image due to twoangle evaporation
1mm
George Ioannou
Paul Lee
Results and Analysis
Pump Oscillations
Voltage applied to
capacitor by the
feedback circuit while
pumping electrons on
and off.
For these data:
N = 117,440,513
‹∆V› = 10.048 703 31 V
C = 1.872 484 77 pF
From eq.
Electron Counting Errors
• Pump voltage vs time
showing individual
error events.
(a) Pumping ±e at 5.05
MHz, average error
per electron = 15 ppb.
(b) Hold mode, average
hold time ≈S. T_mc =
35 mK for both plots.
Pump accuracy vs. Time
Under V_p = 0,
T_mc = 35 mK
• Constant error of 16
ppb per electron
• Electron speed used
to create gate pulses
limited experiment to
t_p ≥ 100 ns
• Error given by
with a = 0.021
Temperature Effect on Pump
Accuracy and Leakage Rate
High Temperatures
Theoretical Expressions for Error due to
Thermally Activated Processes:
th=b exp (-Ep/kbT)
(pumping)
th=(d/RC) exp(-Eh/kBT) (hold mode)
T = electron temperature in the pump
b,d= pre-factors (b  0.7, d  0.05)
Ep=energy barrier (pump)
Eh=energy barrier(hold)
Predicted
Measured
Ep
2.0 K  0.2 K
1.7 K  0.1 K
Eh
3.4 K  0.3 K
3.3 K
• Pumping requires pulse height
and shape, cross capacitance
cancellation and dc biases be
properly adjusted, while hold
mode only requires dc biases.
Appl. Phys. Lett., Vol.69, No. 12, 16 September 1996
Temperature Effect on Pump
Accuracy and Leakage Rate (2)
Low Temperatures
•
At low temperatures, error and leakage are both independent of
temperature Tmc (T is not equal to Tmc for Tmc < 100 mK)
•
At low temperatures, the power dissipation due to the electrometer and
to the electrons passing through the pump are so small that any tiny
deviances in temperature are negligible.
•
At low T, the error mechanism comes from photon-assisted cotunneling.
•
An environment containing a time-varying voltage source with spectral
components at frequencies corresponding to the charging energy
(typically 10 GHz) will significantly increase tunneling rates, because it
will generate photons of sufficient energy to overcome the charging
energy barrier.
Appl. Phys. Lett., Vol.69, No. 12, 16 September 1996
http://www.fys.ku.dk/flensberg/publications/prb_accotun.pdf
Repeatability
of SET Capacitance Standard
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Operating at 40 mK, the relative
variations in the measured C are of
order 1x10-6 = 1 ppm
Standard deviation is 0.3 ppm over
24 hour period, and 0.7 ppm over
ten day period  Accuracy
decreases over longer measuring
periods
Longer measurement periods
contribute to fluctuations in the
dimensions of the vacuum-gap
capacitor
***Pumping different numbers of
electrons still shows the same
value of C, so there is no voltage
dependence!
Comparison to Calculable
Capacitance Measurement
<V>10 V over 24 hours
•
Uncertainty bars on the electron counting
values are UtotC, where
Utot = [0.092+0.012+0.12+(2)2]1/2
owing to the uncertainty from a, N, the
voltmeter, and statistical variations in
each set of data
•
Uncertainty bars in the commercial
calculable capacitance bridge are 2.4
pm owing to the uncertainty in the 10-pF
capacitor at 1000 Hz
•
The measurement of electron counting
agrees with the measurement of the
calculable capacitance.
<V>10 V over 10 days
Keller, Mark W. Science Vol 285
3R’s for Future Improvements
To Realize the SET Capacitance Standard:
•
1) Reduce the frequency dependence of the cryogenic capacitor because the
measurement of C by counting electrons occurs at much lower effective frequency
than that used for bridge comparisons
•
2) Reduce the input noise of the electrometer to allow the feedback circuit to
maintain virtual ground between the pump and capacitor
•
3) Reduce the magnitude of all uncertainties in both pump and hold modes in the
circuit
Keller, Mark W. Science Vol 285
Conclusion
•
Experiments have demonstrated that using the 7-junction electron pump as an
electron counter is an effective means to placing capacitance metrology on a
quantum scale.
•
For this SET capacitance standard to be adopted, it must be developed into a
robust and easy process to use, with total relative uncertainty of order 0.1 ppm.
•
“A long-term metrology goal is to combine the new capacitance standard with the
calculable capacitor and the Josephson voltage standard to achieve a new
measurement of the fine structure constant”
Zimmerman, Meas. Sci. Technol. 14 (2003) 1237–1242
References
M. W. Keller, J. M. Martinis, A. H. Steinbach and N. M.
Zimmerman, Accuracy of electron counting using a
7-junction electron pump, APL, 69 (1996) 1804-1806
M. W. Keller, J. M. Martinis, A. H. Steinbach and N. M.
Zimmerman, A Seven-Junction Electron Pump:
Design, Fabrication and Operation, IEEE Trans. Inst.
Meas., 46 (1997) 307-310
M. W. Keller, A. L. Eichenberger, J. M. Martinis and N.
M. Zimmerman, A Capacitance Standard Based on
Counting Electrons, Science, 285 (1999), 1706-1709
N M Zimmerman and M W Keller, Electrical
metrology with single electrons, Meas. Sci. Technol.
14 (2003) 1237–1242
Thank you
Slide distribution
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Introduction
Current Standard for Capacitance (calculable)
Counting electrons – capacitance, Figure 1(b) Science
Using Josephson (Ne/V) accuracy
Approach
How to count – circuit diagram, Figure 2(a) Science
Components:
1. Electrometer workings,
2. bridge, calculable capacitor circuit (c-ref comparison), Feedback loop (?)
3. Pump:
Pumping mechanism  Coulomb blockade theory, coulomb diamonds
Requirement for multiple pumps (one to many) – co-tunneling
Issues with multiple pumps: cross-capacitance and stray capacitance.
Pump operation: circuit schematic, cross-c canceling
Fabrication process: 2 angle evaporation of Al (video)
Results and analysis
1. Measurements:
Pump oscillations (Fig. 2, APL, Fig. 3 Science)
electron counting errors (Fig. 3, APL)
pump accuracy vs time (Fig. 4, APL) – equation 1 in APL
Temperature dependence (Fig. 5, APL) – include the equations 2 and 3 in APL
2. Analysis:
comparison to calc. cap. Standard (Fig. 5, Science),
repeatability (Fig. 4, Science)
Conclusion
A summary of what was achieved
George
Wayne
Smitesh
Paul
Fine-structure constant
•
A dimensionless number (a)
•
Ratio of energy required to bring two
electrons from infinity to some distance S
and the photon energy of wavelength 2S
•
It was first used to explain the size of
splitting of the hydrogen lines
•
“For instance, were α to change by 4%,
carbon would no longer be produced in
stellar fusion. If α were greater than 0.1,
fusion would no longer occur in stars.”
e2
a
hc