EECS 598 Week 13 Single spin detection by magnetic resonance force microscopy

1
EECS 598 Week 13Single spin detection by
magnetic resonance force microscopy

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

2
Outline

• Magnetic Resonancetheory behind itNMR and MRI
  applications
• MRFM instrumentcantilever fabricationsample
  preparation principle
• Single Spin Detectiondetection method
• MRFM results single spin signal
• Other applicationqubit readout device

3
Nuclear Spin

• Spinning charge on proton generates magnetic
  dipole

Classical representation of a proton precessing
in a magnetic field of magnitude Bo in analogy
with a precessing spinning top
4
Spin in a field

• Spin state of a nucleus is affected by an
  externally applied magnetic field

High energy state (b) versus the low energy state
(a)
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Spin states
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Spin energy difference

• The difference between the two spin states
  depends on the strength of the magnetic field

The absorbed frequency n depends on the
gyromagnetic ratio, g of the particle.
DEhn
n g B
Where n is the Larmor frequency
7
Resonance
z-component of the spin angular momentum (mspin
quantum number)
Iz mh/2p
resultant magnetic moment is connected with its
spin angular momentum.
µz?Iz
The energy of a magnetic moment µ when in a
magnetic field B0
E -µzB0
Therefore resulting
E -mh?B0 / 2p
The energy gap between our a and ß states is
?E h?B0/2p
Resonance if RF applied with DE hn.
? ?B0/2p
g is the gyromagnetic or magnetogyric ratio, a
fundamental nuclear constant, g 2pm/hm
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Population distribution

• Distribution of 2 Million protons at different
  field strengths

Boltzmann Statistics
At room temperature, the number of spins in the
lower energy level, N, slightly outnumbers the
number in the upper level, N-. The signal is
thus proportional to the population difference
between the states.
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Relaxation
At equilibrium the magnetic moment Mz lines up
with applied field. A pulse of resonant frequency
can lead to zero moment Mz. Relaxation is the
time to return to the equilibrium position
Mz Mo ( 1 - e-t/T1 )
Two relaxation times Spin-lattice or
longitudinal relaxation process (T1), involves
transfer of energy from the excited protons to
the surrounding protons tumbling at appropriate
frequency Spin-Spin or transverse relaxation
(T2), involves transfer of energy among the
precessing protons, resulting in dephasing, line
broadening, and signal loss.
10
Magnetic resonance

• When the energy of the RF matches DE absorption
  of energy occurs which can be detected.
• The DE also depends on the surrounding molecules.
• In NMR spectroscopy, n is between 60 and 800 MHz
  for hydrogen nuclei. (or carbon atoms using
  13C-NMR spectroscopy phosphorus atoms using
  31P-NMR spectroscopy)
• In clinical MRI, n is typically between 15 and 80
  MHz for hydrogen imaging.

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Spatial resolution

• If each of the regions of spin was to experience
  a unique magnetic field we would be able to image
  their positions. A gradient in the magnetic field
  is what will allow us to accomplish this.

U.S. Patent 3,789,832 (the '832 patent), filed on
March 17, 1972 by Raymond V. Damadian 2003 Nobel
prize in Medicine to Paul Lauterbur and Sir Peter
Mansfield
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Magnetic resonance

• Magnetic Resonance Imaging (MRI)
• MRI based on NMR principles - an image of the NMR
  signal in a thin slice through the human body.
• The human body is primarily fat and water - human
  body approximately 63 hydrogen atoms.
• Two or more particles with spins having opposite
  signs can pair up to eliminate the observable
  manifestations of spin. An example is helium. In
  nuclear magnetic resonance, it is unpaired
  nuclear spins that are of importance.

13
Comparison of microscopy techniques

• Electron microscopy
• Radiation damage
• Specimen preparation in TEM
• Scanning probe microscopy
• Can only image the atoms at the surface.
• X-ray crystallography and NMR spectroscopy
• Both require homogeneous samples, consisting of
  highly purified solutions or well-ordered
  crystals. Purification is often difficult,
  crystals dont form, etc.
• Traditional MRI
• Inductive technique of magnetic resonance
  detection is not sensitive ? 1012 nuclear spins
  needed to generate a detectable signal.

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MRFM

• Magnetic Resonance Force Microscopy (MRFM)
  Combines the best of MRI and SPM
• MRI characteristics
• 3D, sub-surface imaging
• Chemical-species specific due to local magnetic
  environment
• SPM characteristics
• Scan a probe with a magnet across the sample
• Detection of force from a single nucleus or
  electron

15
Single spin sensitivity demonstrated

• Rugar et al. detected the force from the spin of
  a single electron, thus demonstrating the
  ultimate resolution limit of MRFM.
• Basic idea
• The magnetic moment of the electron exerts a
  force on a magnet mounted on a cantilever
• The cantilevers resonant frequency fc shifts due
  to the change in effective stiffness.
• Challenge detect the tiny frequency shift dfc.

16
Configuration of MFRM
Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
17
MRFM Instrument

• Cantilever fabrication
• Sample preparation

18
Cantilever Fabrication

• Cantilever requirements and responses
• Minimize dissipation
• Uniform thickness
• Made from SCS, very clean surface
• Minimize RF/laser induced self-heating
• Must have low electrical conductivity
• SCS must be undoped or light doped
• Minimize reduce clamping losses
• Overhang at base of cantilever must be minimized
• Base should be thickened and stiffened

19
Fabrication Procedure

• Silicon-on-insulator (SOI) substrate
• Selective undoped silicon epitaxy to form the
  mass.
• a) Low-temperature oxide (LTO) layer deposited
  and patterned to form a mask.
• Forms the hinge
• b) Selective SCS epitaxy is grown
• Does not grow over oxide.
• b) LTO removed with HF
• c) LTO layer deposited and patterned to form a
  second mask.
• Exposes mainly the base of the cantilever
• d) Selective SCS epitaxy is grown
• d) LTO removed with HF
• Thickness of the base is 5 pm, providing the
  structural rigidity required for reducing
  clamping losses in the cantilever.
• e) Cantilever and base lithographically
  patterned, level defined using Si plasma
  etch
• f) Backside lithography followed by DRIE, HF etch
  used to remove the buried oxide and release the
  cantilever
• To improve yield, a temporary nitride-LTO
  protective layer may he deposited on the front
  side of the wafer before the backside etch,
  removed by plasma etching before the HF release.
• Finally, deposit a SmCo magnetic tip on mass

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Alternative Fabrications

• a) Long-hinge design
• b) Short-hinge design
• c) Design using LOCOS
• (LOCal Oxidation of Silicon)

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Cantilever Quality

• Quality factor is very good overall
• At T 270 K, Q 15,000

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Suppressing noise

• High-order mode noise must be suppressed
• ensure reliable sensitivity of the device
• Mass-loaded cantilever effectively filters out
  many noise peaks

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Cantilever Sensitivity

• Minimum detectable force in a bandwidth
• Fmin (SFB)(1/2) (wt2/lQ)(1/2)(E?)(1/4)(kBTB)(1
  /2)
• The ferromagnetic tip of the beam will suffer
  magnetostatic forces on the order 10-16 N (aN)
• For our beam, Fmin 36 aN
• Within desirable limits

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Sample Preparation

• Substrate consists of vitreous Silica (Suprasil
  W2)
• Irradiated with 2-Gy dose of Co60 gamma rays
• Produces a low concentration of Si dangling bonds
  containing unpaired electron spins
• Known as E? centres

25
Proposed E? centre models

• E? modeled as a single electron
• E1? model
• trapped at a Si ion
• located between two oxygen vacancies
• E2? model
• trapped on a defect silicon ion
• next to a Si vacancy
• non-bridging oxygen ion removed during irradiation

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Pinpointing Where to Detect Spin

• Create a resonant slice in the sample using
  both the
• 1) Microwave magnetic field (B10.3 mT)
• 2) Inhomogeneous magnetic tip field
• Key Properties
• Gradient of the microscopic magnetic probe is ? 2
  gauss per nanometer, so that the force generated
  on the cantilever by an individual electron-spin
  can be detected at 2 10-18 N
• Field gradient causes spins at different depths
  to resonate at different frequencies for
  selective excitation of spins (and thus imaging)
• The slice is a bowl shaped surface that extends
  about 250 nm below the tip

http//www.nature.com/nature/journal/v430/n6997/fu
ll/430300a.html
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Magnetic Field Setup

• Condition for Electron Spin Resonance
• B0(x,y,z) ?Btip(x,y,z)z Bext ??rf/??rf
  frequency of the microwave field?
  gyromagnetic ratio
• In the given experiment, ?rf / 2? 2.8 1010
  Hz T-1 and ? / 2? 2.96 GHz, leading to
  B0(x,y,z) 106 mT
• Due to perpendicular cantilever orientation, the
  cantilever can only detect force in the
  x-direction (ie, spin either in front or behind
  the cantilever in the x direction)
• The spin must be located either slightly in front
  of or behind the cantilever for there to be any
  substantial response

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
28
Manipulating Spins

• When no electron spins are present, the
  cantilever with the attached ferromagnet acts as
  a harmonic oscillator. Any unpaired electron
  spins behave like magnetic dipoles and exhibit
  perturbing forces on the cantilever.
• iOSCAR (interrupted oscillating cantilever-driven
  adiabatic reversal) is used to manipulate spins,
  allowing the cantilever to detect a readable
  force signal
• The cantilever is part of a gain-controlled
  positive-feedback loop, which adjusts to maintain
  cantilever oscillation at both
• 1) a specifiable set amplitude (ex. 16 nm)
• 2) the fundamental frequency of the cantilever (
  fc 5.5 kHz), which is dependent on spin forces
  and the material
• The cantilever is the frequency-determining
  element in the feedback loop, so the vibration
  frequency will automatically vary in response to
  tip-sample interactions to maintain cantilever
  oscillation

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
29
iOSCAR Setup

• The oscillator control amplifier provides the
  positive feedback to keep the cantilever
  operating at its resonant frequency.
• An analog frequency demodulator is used to detect
  any frequency shift in the cantilever.

T.R. Albrecht, P. Grutter, D. Horne, and D.
Rugar, J. Appl. Phys. 69, 668 (1991)
30
iOSCAR Explained

• The positive feedback forces the cantilever into
  mechanical oscillation.
• Vibration of the cantilever tip causes the
  resonant slice to sweep back and forth rapidly
  through the sample
• If the slice sweeps through the location of a
  electron spin, the spin will be cyclically
  inverted in synchrony with the cantilever motion
  because of adiabatic rapid passage
• The synchronous inversion of the spin creates an
  alternating magnetic force on the cantilever that
  mimics a change in cantilever stiffness
• The cyclic spin causes a slight shift of the
  cantilever frequency

Ting, M., Hero, A.O., Rugar, D., Yip, C.-Y.
Fessler, J.A. Electron spin detection in the
frequency domain under the interrupted
oscillating cantilever-driven adiabatic reversal
(iOSCAR) protocol. Preprint at http//xxx.lanl.gov
/abs/quant-ph/0312139 (2003).
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iOSCAR Explained

• The back-action force on the magnetic tip from
  the spins results in a frequency shift of the
  cantilever.
• The resulting shift in cantilever frequency is
  given by
• k cantilever spring constant
  Xpeak peak vibration amplitude of the
  cantilever G ? ?B0/?x lateral field
  gradient ?B magnetic moment of the
  electron (9.3 x 10-24 J T-1)
• Sign of frequency shift depends on relative
  phase of spin inversions with respect to the
  cantilever motion
• The two polarities correspond to adiabatic rapid
  passages with spin either aligned or anti-aligned
  with respect to the effective field in the
  rotating frame
• In the experiment, (G 2 x 105 T m-1, k 0.11
  mN m-1, xpeak 16 nm), ?fc 3.7? 1.3 mHz

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
32
OSCAR in detail magnetic moment in a magnetic
field
Classically, the motion of the magnetic moment
under the influence of a magnetic field is
described by
Quantum mechanically, the equation remains valid
if µ is replaced with ltµgt, which is what we are
actually dealing with.
If µ is set at an angle with B0 then the equation
implies it precesses around B0 at the Larmor
frequency ?B0.
33
OSCAR in detail
OSCAR in detail oscillating magnetic field
Experimental Condition
B1, which is linearly polarized, may be viewed as
the superposition of two circularly polarized
vectors rotating in opposite directions.
B0 (106 mT, static)
B1 (0.3 mT, 2.96 GHz)
Effectively µ sees
Near resonance, µ will absorb much more energy
from the component rotating in the same direction
as the µs precession around B0. So the other
rotating component may be neglected.
34
Magnetic resonance
OSCAR in detail rotating reference frame
Observed in a rotating reference frame, the
motion of the magnetic moment behaves as if the
magnetic field has been modified.
µ magnetic moment vector µi coordinates in
the rotating frame O angular velocity vector of
the rotating frame.
The velocity of µ as seen in the rotating frame.
35
Magnetic resonance
OSCAR in detail rotating reference frame
then the field in the rotating reference frame is
constant with time.
36
OSCAR adiabatic rapid passage

• Assuming B1 ltlt B0 and µ is initially aligned with
  B0
• The application of B1 ? µ precesses around Beff
  with a angle dictated by its starting direction
  B0.
• At frequencies below the Larmor frequency, the
  precession angle is near zero because Beff is
  nearly parallel to B0
• If the frequency is increased by a small amount,
  the precession angle around Beff remains near
  zero.
• As the frequency is slowly increased, the
  z-component of Beff will change sign when the
  frequency increases above the Larmor frequency,
  with µ following suit. This method of flipping µ
  is called adiabatic rapid passage.

µ
37
Magnetic resonance
OSCAR Oscillating Cantilever-driven Adiabatic
Reversals

• In the experiment, the cantilever oscillates in
  the x-direction at a fixed low frequency ? 5.5
  kHz.
• The resonant slice oscillates around a magnetic
  moment, which passes in and out of resonance.
• This is similar to applying a 5.5 kHz frequency
  modulation around the larmor frequency 2.96 GHz.
• Beff, and hence µ, oscillates up and down
  synchronously with the cantilever.

?rf
38
iOSCAR Animation

• This animated movie illustrates the
  cantilever-driven spin inversions that occur
  during the iOSCAR spin manipulation protocol (see
  Fig. 2 in the paper). The "Lock" and "Anti-lock"
  states correspond to the spin being either
  aligned or anti-aligned with respect to the
  effective field in the rotating frame, resulting
  in either positive or negative cantilever
  frequency shifts, respectively. Each time the
  microwave field is interrupted, the spin switches
  between the locked and anti-locked states and the
  phase of the spin inversions with respect to the
  cantilever motion is reversed.

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
39
Why OSCAR

• A single spin results in a small shift in
  cantilever resonant frequency dfc 3.7E-3 Hz
• Long integration times are needed to detect this
  change.
• Integration times are limited by relaxation
  processes.
• In conventional MRI, the signal comes from the
  precession of the transverse (xy plane) component
  of µ.
• Spin-spin relaxation time T2 is difficult to
  control because it is due to fields of nearby
  spins, among other causes.
• T2 lt T1 in general
• In the OSCAR method, oscillations of the
  longitudinal component of µ give rise to the
  signal.
• Spin-lattice relaxation time T1 is mainly caused
  by thermal perturbations from nearby atoms.
• T1 can be lengthened by operating at cryogenic
  temps
• According to the experiment, T1 760 ms ?
  coherent through thousands of spin flip cycles
  (5.5kHz).

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iOSCAR Explained

• The microwave field B1 is turned off
  (interrupted) for one-half of a cantilever
  cycle every 64 cycles
• The interrupted frequency is given by fint
  fc/64 ? 86 Hz
• When B1 is turned off, the cantilever continues
  to oscillate. When the microwaves are turned
  back on after the half-cycle gap, B0 will have
  reversed orientation and the magnetization will
  have changed from locked to antilocked.
• Each interruption leads to a reversal in the
  relative phase of the spin and cantilever,
  causing the frequency to shift to reverse
  polarity.

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
41
iOSCAR Animation

• This animated movie illustrates the
  cantilever-driven spin inversions that occur
  during the iOSCAR spin manipulation protocol (see
  Fig. 2 in the paper). The "Lock" and "Anti-lock"
  states correspond to the spin being either
  aligned or anti-aligned with respect to the
  effective field in the rotating frame, resulting
  in either positive or negative cantilever
  frequency shifts, respectively. Each time the
  microwave field is interrupted, the spin switches
  between the locked and anti-locked states and the
  phase of the spin inversions with respect to the
  cantilever motion is reversed.

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
42
iOSCAR Explained

• The interruptions cause the frequency shift to
  alternate between positive and negative values in
  a square-wave-like fashion with a frequency given
  by fsig fint/2, or 43 Hz.
• The fact that the signal is at a subharmonic of
  fint gives it a very distinctive signature that
  is free of spurious feedthrough artifacts.
• Thus iOSCAR allows one to simply look for a peak
  at fint/2 in the power spectrum of the frequency
  demodulated signal to detect single spins

Mamin, H. J., Budakian, R., Chui, B. W. Rugar,
D. Detection and manipulation of
statistical polarization in small spin ensembles.
Phys. Rev. Lett. 91, 207604 (2003).
43
Frequency Shift

• The frequency shift signal is given by
• A(t) is a random telegraph function that has a
  value of 1 or -1, which accounts for extra
  random spin flips induced by the environment
• A(t) has a lorentzian power spectrum and the
  properties ltA(t)gt 0 and ltA(t)2gt 1
• Only the first harmonic of the signal is
  detected, so the spin signal amplitude will be
  given by
• Where 4/? is the first harmonic Fourier amplitude
  of a square wave

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
44
Overcoming Noise

• Frequency modulation due to the spin is only a
  few mHz, which is small in comparison to
  frequency noise of the cantilever from thermal
  motion and tip-sample interactions (25 mHz)
• Signal averaging is needed to detect the spin
  signal, so we average the square of the signal
  energy (rather than the signal amplitude)
• Frequency modulation of the cantilever is
  detected using an analogue frequency
  discriminator followed by a digital lock-in
  amplifier that has been implemented in software
• Lock-in amplifier consists of a bank of low-pass
  filters that determines the energy (variance) of
  the in-phase ( ?I) and quadrature components (?Q)
  of the frequency-shift signal ?f(t) as a function
  of detection bandwidth
• Signal energy from the spin can be isolated by
  taking ?spin2 ?I2 - ?Q2

?Q2 contains only measurement noise

• I2 ?spin2 ?noise2 contains both spin signal
  and measurement noise

Rugar, D., Budakian, R., Mamin, H.J., and Chul,
B.W. Single spin detection by magnetic resonance
force microscopy. Nature 430, 329-332 (2004)
45
Experimental results
s2spin is non-zero only at a localized position
in the sample. By design, the mean spacing
between spins is 200 to 500 nm. Therefore the
signal likely comes from a single spin. Each data
point is the result of averaging over 13 hours,
owing to the low signal-to-noise ratio (S/N
0.06)
46
Experimental results
Upper graph frequency spectrum of s2spin (power
spectral density of spin signal amplitude) at two
locations in the sample. Lorentzian lineshape is
consistent with a random telegraph model of the
spin signal amplitude. Narrow spectral width
corresponds to a long relaxation time of 760
ms. Bottom graph power spectral densities as a
function of position. Signal is highly localized
spatially and spectrally.
47
Magnetic resonance

• But the ability to detect individual spins is
  about more than imaging it implies the power to
  manipulate individual spins as well. Present-day
  information processing relies on the electron's
  charge, through manipulating and detecting
  voltages in electronic circuits. Exploiting the
  electron's magnetic moment, or spin, could lead
  to significant enhancements in electronic
  information processing, including nonvolatile
  memory, increased integration densities and
  reduced power consumption. Furthermore, the spin
  of the electron is a natural two-state quantum
  system ('qubit') for quantum computing the spin
  can also be isolated from its physical
  environment to achieve the long decoherence times
  needed for successful computation.

48
Quantum Computing

• Classical Computer
• Data stored as bits
• Either 0 or 1
• Quantum Computer
• Data stored as qubits
• Either 0 or 1
• Or both!
• Qubit can exist as both a 0 or 1, with a
  probability for each state
• Allows computations at unimaginable speeds

49
Quantum Computing

• Imagine a system of 500 qubits
• 2500 possible quantum states
• Apply a quantum operation with a particular pulse
  of radio waves (ie. controlled-NOT)
• Would compute not just one machine state, but all
  2500 machine states at once
• Equivalent to performing same operation on 10150
  separate processors!

50
Qubit readout device

• How can we use MRFM to build a quantum computer?
• Use electron spins as qubits
• Apply pulses to the electron spins to perform
  unitary operations
• Unitary operations act like rotations or
  reflections
• product of two unitary operations is a unitary
  operation

51
Qubit readout device

• Procedure
• Initialize qubits (polarize spins)
• Apply unitary transformation to selected set of
  qubits
• Measure qubits to get final result

52
Qubit readout device

• Initialize qubits
• Use magnetic field to create 100 polarization
• With B 10 T, T 1 K, 99.99986 of a given spin
  pointing the right way
• Note during measurement, use an even number of
  pulses to return electron spin to ground state

53
Qubit readout device

• Apply unitary transformation to selected set of
  qubits
• Apply electron or nuclear p pulses
• Interacting through weak Ising interactions
• Example CN Gate
• a) electron p pulse drives the electron
  magnetic moment of the control qubit
• b) a nuclear p pulse cause a transition in
  target qubit if control qubit is in ground state
• c) electron p pulse drives the electron
  magnetic moment back to the ground state

54
Qubit readout device

• Measure qubits to get final result
• Use MRFM to measure the result (spins)

55
Conclusion

• MRFM is capable of detecting individual electron
  spins
• MRFM can image spins below the surface with
  nanometre spatial resolution
• Even a small increase in field gradient can
  dramatically speed up the acquisition time for 2D
  and 3D imaging
• Reducing the measurement time below correlation
  time ?m can enable real-time imaging of the spin
  quantum state!
• The present experiment using iOSCAR presents a
  sensitivity improvement of 107 times over the
  original MRFM experiment, but a further 1000
  fold improvement in magnetic moment sensitivity
  is still needed for molecular imaging
• There is still room (at the bottom) to increase
  the field gradient and lower the operating
  temperature to make this improvement possible!

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