Capacitance standard using an electron pump

1
Capacitance standard using an electron pump

• Paul Lee
• Wayne Fung
• George Ioannou
• Smitesh Bakrania

Based on work published by Keller, Martinis,
Zimmerman and Steinbach (1996 to 1999)
2
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

3
Introduction

• Paul Lee
• Wayne Fung

Current Standard for Capacitance Counting
electrons Josephson voltage standard
4
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)
5
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)
6
Lets 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 todays
  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
7
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
8
Uncertainties in new standard

• 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.

9
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 Q f
• 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

10
Approach

• George Ioannou
• Wayne Fung
• Smitesh Bakrania

1. Electrometer
2. Bridge Circuit and Ref. Capacitor
3. Electron Pump

11
Schematic of Experiment
12
Electrometer

• Used to measure Vp across the external island
• Necessary because of very sensitive measurements

13
Electrometer History

• In 1784, Coulomb developed the torsion-balance
  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.

14
Modern Electrometers

• Researchers have scaled Coulombs 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

15
Electrometer Fabrication
SOI Cross-section
16
Electrometer Fabrication
Top View
Cross Section
17
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

18
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

19
Comparison of standards to test accuracy

• AC Bridge Circuit
• Measure C using old standard
• Compare it to the value obtained from counting
  electrons

I1
C
Vary V1 and V2 until the null detector reads zero
potential difference.
I2
Cref calibrated with a 10-pF silica-dielectric
capacitor traceable to NISTs calculable
capacitor at 1592 Hz.
20
Electron Pump
Single electron tunneling (SET)
Criteria Charging energy must be larger than
thermal energy Ec gt kBT Electron number on dot
needs to be well-defined junction contact
resistance must be larger than resistance
quantumRcontact gt h/e2
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Electron Pump
Single electron tunneling (SET) Theory
Electrostatic energy for island with N electrons
Q2/2C (Ne)2/2C __________________________ Tota
l energy on islandU(N) ?Ei (Ne)2/2C
__________________________ Electrochemical
potentialmN1 (N0.5)e2/C e/C
?CiVi __________________________ Charging
energy?? ?N1 - ?NEce2/C
22
Electron Pump
Coulomb blockade
Coulomb oscillations
mS and mD are electrochemical potentials of
source and drain, respectively.
23
Electron Pump
Coulomb diamonds with ZERO bias
24
Electron Pump
Coulomb diamonds with NON-ZERO bias
25
Electron Pump
Coulomb diamonds with constant gate Voltage
26
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
27
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)
28
Electron Pump
side effects
Stray capacitanceisland 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 gt0.2 fF - junctions Cg possible to
make less than 0.1CjCstray choosing substrate
with small ?Cself reducing island size
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Electron Pump
Multiple junction electron pump
30 times more accurate
Error/e 500 x 10-9hold time 10 s
Error/e 15 x 10-9hold time 600 s
Sapphire substrate (? 10)
Fused quartz substrate (? 3.75)
Cg 0.03 fFCstray 0.16 fFCself 0.02 fF
Cg 0.02 fFCstray 0.06 fFCself 0.01 fF?
reliable fabrication
30
Electron Pump
side effects
Cross capacitancesmall 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
31
Electron Pump
Multiple junction electron pump
Cryogenic capacitor
Magnetically controlled switches
Electron pump and electrometer
Switches
32
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
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Electron Pump
Pump operation digital logic section

• Allows control of - number of electrons pumped-
  direction of pumping- wait time between pumped
  electrons
• Triangular voltage pulses on 6 output channels by
  charging/discharging capacitors.
• DC bias adjusted to compensate for background
  charges on islands of pump.
• Pulses and dc biases are summed to produce set of
  Vg on gate lines.
• Cross-capacitance cancellation circuit performs
  transformation producing set of Vg

4
1
3
2
34
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 (VgVg) For a
particular geometry, this needs to be done only
once.
35
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.
36
Electron Pump
Fabrication two angle evaporation
Mask patterned through e-beam lithography Oxidatio
n 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.
37
Electron Pump
Fabrication two angle evaporation
How can we make a tiny island
http//www.ptb.de/en/org/2/24/244/winkel.htm
38
Electron Pump
7-junction electron pump
http//www.jcnabity.com/nistpump.htm
Gate Capacitor
Tunnel junctions
Dual image due to two-angle evaporation
1mm
39
Results and Analysis

• George Ioannou
• Paul Lee

40
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.

41
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.

42
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 bywith a 0.021

43
Temperature Effect on Pump Accuracy and Leakage
Rate

• High Temperatures
• Theoretical Expressions for Error due to
  Thermally Activated Processes
• ?thb 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)
• ?Epenergy barrier (pump)
• ?Ehenergy barrier(hold)

• Pumping requires pulse height and shape, cross
  capacitance cancellation and dc biases be
  properly adjusted, while hold mode only requires
  dc biases.

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
Appl. Phys. Lett., Vol.69, No. 12, 16 September
1996
44
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 lt 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/p
rb_accotun.pdf
45
Repeatability of SET Capacitance Standard

• 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!

46
Comparison to Calculable Capacitance Measurement
lt?Vgt?10 V over 24 hours

• Uncertainty bars on the electron counting values
  are ?UtotC, where
• Utot 0.0920.0120.12(2?)21/2
• owing to the uncertainty from ?, 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.

lt?Vgt?10 V over 10 days
Keller, Mark W. Science Vol 285
47
3Rs 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
48
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) 12371242
49
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) 12371242
50
Thank you
51
Slide distribution

• 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
• Components1. 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

GeorgeWayneSmiteshPaul
52
Fine-structure constant

• A dimensionless number (?)
• 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 a to change by 4, carbon
  would no longer be produced in stellar fusion. If
  a were greater than 0.1, fusion would no longer
  occur in stars.