Title: EECS 598 Week 13 Single spin detection by magnetic resonance force microscopy
1EECS 598 Week 13Single spin detection by
magnetic resonance force microscopy
- Paul Lee
- Wayne Fung
- George Ioannou
- Smitesh Bakrania
2Outline
- 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
3Nuclear 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
4Spin 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)
5Spin states
6Spin 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
7Resonance
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 Ăź 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
8Population 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.
9Relaxation
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.
10Magnetic 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.
11Spatial 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
12Magnetic 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.
13Comparison 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.
14MRFM
- 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
15Single 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.
16Configuration 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)
17MRFM Instrument
- Cantilever fabrication
- Sample preparation
18Cantilever 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
19Fabrication 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
20Alternative Fabrications
- a) Long-hinge design
- b) Short-hinge design
- c) Design using LOCOS
- (LOCal Oxidation of Silicon)
21Cantilever Quality
- Quality factor is very good overall
- At T 270 K, Q 15,000
22Suppressing noise
- High-order mode noise must be suppressed
- ensure reliable sensitivity of the device
- Mass-loaded cantilever effectively filters out
many noise peaks
23Cantilever 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
24Sample 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
25Proposed 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
26 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
27Magnetic 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)
28Manipulating 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)
29iOSCAR 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)
30iOSCAR 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).
31iOSCAR 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)
32OSCAR 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.
33OSCAR 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.
34Magnetic 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.
35Magnetic resonance
OSCAR in detail rotating reference frame
then the field in the rotating reference frame is
constant with time.
36OSCAR 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
37Magnetic 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
38iOSCAR 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)
39Why 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).
40iOSCAR 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)
41iOSCAR 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)
42iOSCAR 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).
43Frequency 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)
44Overcoming 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)
45Experimental 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)
46Experimental 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.
47Magnetic 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.
48Quantum 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
49Quantum 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!
50Qubit 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
51Qubit readout device
- Procedure
- Initialize qubits (polarize spins)
- Apply unitary transformation to selected set of
qubits - Measure qubits to get final result
52Qubit 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
53Qubit 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
54Qubit readout device
- Measure qubits to get final result
- Use MRFM to measure the result (spins)
55Conclusion
- 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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