AC Magnetic Susceptibility of a Thin Film of Permalloy
Isaiah Gray
Submitted under the supervision of Dan Dahlberg to the University
Honors Program at the University of Minnesota-Twin Cities in partial
fulfillment of the requirements for the degree of Bachelor of Science,
summa cum laude, in Physics.
May, 2014
Acknowledgements
I would like to thank my labmates, Dan Endean, Bern Youngblood, and Barry Costanzi
for helpful suggestions throughout the experiment and for wire bonding. I would also
like to thank Professor Dan Dahlberg for the idea for the experiment and invaluable
discussion of the physics involved.
This work was supported by the Undergraduate Research Opportunities Program
at the University of Minnesota - Twin Cities.
i
Abstract
Measurements of noise in magnetic thin films have variously reported 1/f noise, white
noise, and random telegraph noise. The 1/f noise experiments claim magnetic noise by
relating the magnetic noise to the susceptibility with use of the fluctuation-dissipation
theorem. However, neither the linearity of the susceptibility necessary for application
of fluctuation-dissipation nor the frequency dependence of susceptibility was explored.
To investigate more fully the frequency dependence and linearity of the magnetic sus-
ceptibility of a magnetic film, we measured the AC susceptibility of a 100 nm thick
film of permalloy as a function of the magnitude HAC and frequency f of an applied
AC magnetic field over the full hysteresis loop of the films, i.e., the measurements were
performed while slowly varying an applied DC magnetic field, HDC . The AC frequency
range was from 20 Hz to 5 kHz while the AC field range was between 0.2 G and 1 G.
At HDC = 0 the response of the system was measured as a function of HAC at 200
Hz. It was found to be nonlinear but became reasonably close to linear for HAC < 1G
- the coercive field was approximately 15 G. The in-phase and out-of-phase compo-
nents approximately follow power laws with frequency, with exponents 0.69 and -0.2.
The behavior of the out-of-phase component is roughly consistent with previously mea-
sured white magnetic noise and a simple harmonic oscillator model, but the in-phase
component does not follow the prediction of this model.
ii
Contents
Acknowledgements i
Abstract ii
List of Figures ii
1 Introduction 1
2 Background 3
2.1 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Magnetic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . 7
3 Experimental Setup and Methods 11
3.1 Sample Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Anisotropic Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Results 18
4.1 Susceptibility as a function of DC Field: Testing Linear Response . . . . 18
4.2 Susceptibility as a function of DC Field: Mapping out the Hysteresis Loop 20
4.3 Susceptibility as a function of Frequency: Predicting magnetic noise . . 22
5 Analysis and Implications for Magnetic Noise 27
5.1 Comparison to a Simple Harmonic Oscillator Model . . . . . . . . . . . 27
6 Conclusion and Next Steps 30
References 32
i
List of Figures
2.1 Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Example White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Parkin 1/f Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Smith white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Nonlinear and linear responses . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Parkin Noise and Prediction Comparison . . . . . . . . . . . . . . . . . . 9
3.1 Sample schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 AMR data and illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 AMR full field sweeps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Schematic of the experimental setup . . . . . . . . . . . . . . . . . . . . 14
3.5 Amplifier circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.6 Wire bond schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.7 Imaginary background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 χ(HAC) at 0 HDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 χ(HAC) at 0 HDC with analysis . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 AMR curve and slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 AMR curve with cartoons . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Re(χ(f,HDC)) Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Im(χ(f,HDC)) Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.7 Frequency Dependence of Re(χ) at 0 DC Field . . . . . . . . . . . . . . 25
4.8 Frequency Dependence of Im(χ) at 0 DC Field . . . . . . . . . . . . . . 25
4.9 Frequency Dependence of Re(χ) at 2.5 G DC Field . . . . . . . . . . . . 26
4.10 Frequency Dependence of Im(χ) at 2.5 G DC Field . . . . . . . . . . . . 26
5.1 Re(χ(f)) at 0 G DC log-log . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Re(χ(f)) at 0 G DC log-log . . . . . . . . . . . . . . . . . . . . . . . . . 29
ii
Chapter 1
Introduction
Microscopic magnetic systems, especially thin film stacks such as tunnel junctions, are
extensively used for magnetic field sensors and random access memory devices [1]. Al-
though critical to the engineering process, it is arguable that magnetization dynamics on
small length scales remain poorly understood [2]. In addition, magnetic noise - thermally
driven fluctuations of the magnetization - is important from a physics perspective as it
provides information about the dynamics of the system in thermal equilibrium, without
perturbing the system. In principle, magnetic noise could be determined by measuring
magnetization as a function of time, but directly measuring magnetization in a micro-
scopic system is experimentally difficult and often requires specialized equipment [3]. In
practice, noise in microscopic magnetic systems is often probed by measuring the resis-
tance fluctuations associated with giant magnetoresistance, relating the magnetization
to the resistance of the sample [4]. Several groups have performed noise measurements
in multilayer tunnel junctions using giant magnetoresistance and find that the power
spectral density, which measures the magnitude of the Fourier component of the noise
as a function of frequency [5], the noise varies inversely with the probing frequency [6],
[2]. Other groups performing measurements in similar systems report white noise, or a
frequency-independent power spectral density [7].
The lack of consensus on the functional form of magnetic noise in thin film sys-
tems is more problematic as recent studies cast doubt on the accuracy of the previous
reports of observations of 1/f noise in many of the studies [8]. The magnetic afteref-
fect is associated with the finite response time of a magnetic system when the field is
adjusted. Plotting this time-dependent response in the frequency domain yields a 1/f
spectrum similar to observed 1/f spectra, so more information is needed to distinguish
time-dependent effects from true magnetic 1/f noise.
A powerful tool to interpret observations of magnetic noise is the fluctuation-dissipation
1
2(FD) theorem [9], which relates the noise spectra of an observable of a system at ther-
mal equilibrium to the susceptibility, or response of the system to small perturbations.
This has been the approach taken by many researchers in their explorations of noise. In
principle, to apply the theorem it is necessary to know the response in the limit of van-
ishingly small perturbations. Experimentally, the perturbations must at least be small
enough to observe linear response, so that the data can be extrapolated to estimate
zero-perturbation response. A ferromagnet is a strongly nonlinear system, however,
exhibiting hysteresis for oscillating fields [10], so it is not clear that FD applies in this
situation. Assuming a linear response, since noise is measured as a function of frequency,
the susceptibility must also be measured as a function of frequency. There are many
studies of AC magnetic susceptibility as a function of DC field amplitude in the litera-
ture [2] [6], but the frequency dependence over a large enough range to effectively apply
the FD theorem has been largely unexplored. The goal of the present experiment was
two-fold: to test the experimental applicability of the fluctuation-dissipation theorem
by measuring the response of a magnetic thin film to small AC applied fields, and to
measure the frequency dependence of AC magnetic susceptibility.
Chapter 2
Background
2.1 Magnetization
A material placed in a magnetic field H will acquire a magnetization M [11], defined to
be the net magnetic dipole moment per unit volume. In a paramagnet or a ferromagnet
above the Curie temperature the response is proportional to the applied field, and the
constant of proportionality is called the susceptibility, χ:
M = χH (2.1)
If M and H are collinear, χ is simply a scalar. The response of a ferromagnet to an
applied field is more complex. As shown in Fig 2.1, M is not linear with H nor even a
single-valued function of H. The magnetization at a given field depends on the previous
history of the field as well as its present value [10].
Figure 2.1: An example of a hysteresis loop in a ferromagnet. The arrows represent the
direction of field sweep, from negative to positive and back to negative.
3
4Since M is not linear with H, the susceptibility is now DC field dependent and we
can define the field-dependent χ as
χ =
dM
dH
. (2.2)
A ferromagnetic material in an AC magnetic field can be modeled as a damped
driven harmonic oscillator for many purposes [12], giving rise to the phenomenon of
ferromagnetic resonance. Just as there is a phase shift between the driving force and
the displacement in a damped harmonic oscillator, there is a phase shift between the
magnetization and the AC applied field. If the driving force is
H = H0 sinωt, (2.3)
the response is
M = M0 sin(ωt+ φ). (2.4)
Expanding out the sine,
M = M0 cosφ sinωt+M0 sinφ cosωt. (2.5)
The response is now seen to have an in-phase component, M0 cosφ, and an out-of-
phase component M0 sinφ. Alternatively, we may define the susceptibility as a complex
number:
χ = χ′ + iχ′′, (2.6)
where the real part χ′ is the in-phase component and the imaginary part χ′′ is the
out-of-phase component. The phase angle can then be written:
φ = tan−1
χ′
χ′′
. (2.7)
52.2 Magnetic Noise
At any non-zero temperature, the direction of spin of the electrons in a ferromagnet,
and therefore the magnetization, will fluctuate slightly around a mean value. These
fluctuations can be experimentally measured in microscopic thin films, where they are
called magnetic noise to distinguish from other sources of noise. An example of generic
white noise measured with arbitrary units in the time domain is shown in Fig 2.2 below.
Figure 2.2: An example of white noise plotted in the time domain.
It is more useful, however, to perform a Fourier transform on the time domain signal
and consider noise in the frequency domain, so that
Mˆ(ω) =
∫ t
−∞
M(t′)e−iωt
′
dt′. (2.8)
We define the power spectral density [5] as
S(ω) = ‖Mˆ(ω)‖2. (2.9)
S(ω) is a measure of the amplitude of the various frequencies which together comprise
the noise signal and is the standard measure of noise in a system.
6Several groups beginning with Parkin et. al. [6] have performed noise experiments
in thin film stacks comprised of a layer of non-magnetic Cu sandwiched between layers
of Co. The width of the spacer Cu layer is chosen so that at zero applied field the Co
layers align antiferromagnetically, or antiparallel. In a large applied field the layers ro-
tate with the magnetization into a parallel state, and in a smaller applied field the angle
between the magnetizations fluctuates with time, resulting in magnetic noise. Noise was
measured using the giant magnetoresistance effect, which relates the electrical resistance
measured through the layers of the stack to the angle between the magnetizations of
the two layers [2]. The power spectral density of the noise measured was found to vary
inversely with frequency as shown in Fig 2.3 below.
Figure 2.3: Data taken from [2]. A log-log plot of the power spectral density as a function of
the probing frequency. The curve is linear with a slope of -1, showing a 1/f spectrum.
By contrast, Smith et al [7] have performed similar magnetoresistance measurements
with GMR thin films in which the active elements were a CoFe layer whose magneti-
zation was fixed or pinned by antiferromagnetically coupling to a layer of PtMn, and a
free magnetic layer of CoFe, whose magnetization would rotate with an external field.
Magnetic noise was indirectly measured through the giant magnetoresistance effect as in
7Parkin’s experiments but the DC fields were applied from -200 G to 200 G, as opposed
to Parkin’s range of 0 - 45 G. The power spectral density was found to be constant or
flat with frequency as shown in Fig 2.4 below.
Figure 2.4: From [7]. Another noise measurement in systems similar to those in [6], but the
power spectral density is approximately flat from 0 - 250 MHz, disagreeing with a 1/f spectrum.
Giant magnetoresistive stacks offer the advantage of a large, easily measurable signal
but also the disadvantage of a complex system with potential nonmagnetic sources of
noise, such as defects in the antiparallel state [6]. In addition, Guo’s work [8] demon-
strates that the time lag of the response when a changing field is applied can appear
very similar to noise. Since noise data by itself is insufficient to determine the source of
the noise, these qualitatively different results demonstrate that interpreting noise data
in magnetic thin film stacks is not straightfoward. Additional information is required
to distinguish between magnetic noise and other sources of noise, which in both these
experiments comes from application of the fluctuation-dissipation theorem.
2.3 The Fluctuation-Dissipation Theorem
The fluctuation-dissipation (FD) theorem relates fluctuations in a dynamical variable
of a system at thermal equilibrium to the response of a system to a small perturbing
force [9]. In general the theorem states:
8S(ω) =
2kBT
ω
χˆ′′(ω), (2.10)
where χˆ′′(ω) is the imaginary part of the susceptibility as a function of frequency,
obtained by taking the Fourier transform of χ′′(t).
In a magnetic system, if the susceptibility to applied magnetic fields is known as
a function of frequency, the functional form of magnetic noise at thermal equilibrium
can be predicted. However, one of the conditions necessary to apply the theorem is
that the response of the system be linear with the perturbing force. This condition is
necessary because χˆ′′(ω) is a differential quantity, and although it is not possible exper-
imentally to measure dM with zero dH amplitude, if dM is linear with dH the data
may be extrapolated to obtain χˆ′′(ω). Fig 2.1 shows that a ferromagnet is inherently
a non-linear system under an AC perturbation, so in principle FD cannot be applied
in this case. However, one may expect that for small enough amplitudes, the response
may be approximately linear as shown in Fig 2.5, and FD would be approximately valid.
Figure 2.5: Possible responses of a ferromagnet to a small AC applied field. The left plot shows
a minor hysteresis loop, exhibiting nonlinear behavior, while the right plot schematically shows
a linear response.
To apply FD to understand noise in a ferromagnetic thin film, χˆ′′(HAC) must first
be measured to make sure that the response is indeed linear at small amplitudes so that
9FD applies. Assuming this is done, χˆ′′(ω) must be measured over the same range of
frequencies as S(ω) to adequately compare the two. Both the 1/f and the flat noise
experiments use FD to interpret noise data, but neither measure χˆ′′(ω) over the same
range of frequencies as S(ω). Fig 2.3 from the 1/f experiments shows that S(ω) is mea-
sured over six orders of magnitude from 10−1 Hz to 105 Hz, but χˆ′′(ω) is measured to be
approximately constant over a factor of 2 in frequency around 50 Hz, and then assumed
to be constant over the rest of the frequency range. The observed noise spectrum is then
compared with the spectrum predicted by susceptibility measurements in Fig 2.6 below.
Figure 2.6: From [6]. α on the y-axis is fS(f) and is called the noise parameter. The two
plots compare measured noise and the noise predicted by extrapolating a constant χ(f) and
applying the fluctuation-dissipation theorem. Divergence of the two is strongest at zero applied
field, where χ should be most different from zero.
Measured noise data quantitatively matches a constant imaginary susceptibility at
large DC fields, but the two sets diverge at zero applied field. One might expect that
susceptibility would be near-constant with frequency at large DC fields, since the mag-
netization is trapped in a low-energy state aligned with the DC field, but it is not
obvious that χˆ′′(ω) would be constant at zero DC field where the magnetization is free
to rotate. Indeed, the simple harmonic oscillator model described earlier predicts a
non-constant χˆ′′(ω). Smith et al do not measure magnetic susceptibility but estimate it
10
by approximating the free magnetic layer as a single-domain particle and applying the
Landau-Lifshitz-Gilbert equation [7].
The present experiment attempts to solidify magnetic noise analyses by measuring
χˆ′′(ω) over a frequency range of 20 Hz to 5 kHz. To avoid the complexities of a multi-
layer magnetic system, measurements were performed on the simplest system possible -
a single magnetic thin film. Although susceptibility measurements were comparatively
straightforward to interpret, the different types of magnetization measurement neces-
sary meant that noise measurements were impractical.
Chapter 3
Experimental Setup and Methods
3.1 Sample Design
The sample was a thin film of Permalloy, an 80% Ni and 20% Fe alloy chosen for its
relatively high AMR and low coercivity [13], and was about 25 µm long, 1 µm wide, and
100 nm thick. The sample was patterned with optical photolithography and deposited
by sputtering onto a plain Si substrate. A schematic is shown in Fig 3.1 below.
Figure 3.1: A schematic of the sample with dimensions. The purple region represents the
permalloy sample, and the blue the Si substrate.
3.2 Anisotropic Magnetoresistance
Susceptibility measurements require knowledge of magnetization, a difficult quantity to
directly measure in microscopic systems. However, when current is run through a mag-
netic sample the resistance is dependent on the relative orientation of the magnetization
and the current, a phenomenon called magnetoresistance (MR) [14]. Since resistance
can be measured quickly and accurately, MR is currently one of the preferred method
to measure magnetization in microscopic thin films [4]. There are several types of MR
- the present experiment utilized anisotropic magnetoresistance (AMR), in which the
resistance of the sample is a maximum when the magnetization is collinear with the
11
12
current and a minimum when the magnetization is perpendicular to the current [15], as
shown in Fig 3.2 below.
Figure 3.2: The anisotropic magnetoresistance effect in ferromagnets. The plot shows the
resistance of the permalloy sample as a function of DC magnetic field applied perpendicular to
the current. Since there is no field applied parallel to the current, the full magnitude of AMR
is not observed.
In the present experiment, current was applied through the length of the Permal-
loy bar, and the magnetic field was applied perpendicular to the current. Resistance
changes for full field sweeps in both directions are shown in Fig 3.2. The resistance is
lowest at high fields in either direction and greatest at zero field. Since the field is not
applied parallel to the current, the magnitude of the observed MR, about 0.5%, is as
great as the literature value of 1.5% - 2% in permalloy [15].
The detailed mechanism of AMR, although well-understood, is not important to
the present experiment and AMR should be thought of as a reliable way to relate
magnetization to resistance. Quantitatively, the fractional change in resistance when
current is applied perpendicular to the magnetization is given by [14]
13
Figure 3.3: The resistance of the permalloy sample as a function of applied DC field. Since the
resistance is a maximum when M = 0, which occurs at distinct DC fields depending on if the
field were swept from positive to negative or negative to positive, two AMR peaks are observed.
R(H)−R(0)
R(H)
= β
(
M
R(H)
)2
, (3.1)
where R(H) and R(0) are the measured resistances at applied fields of H and 0,
respectively. Taking the derivative with respect to R,
dM
dR
=
R−R0
(R2 −RR0)3/2
(3.2)
Since the magnetic susceptibility χM =
dM
dH and AMR resistance susceptibility χR =
dR
dH are related by
χM =
dM
dR
χR, (3.3)
14
the magnetic susceptibility can be written entirely in terms of measurable quantities.
With the values of R(H) and R(0) in this experiment, dMdR varies by a few percent. Since
χR varies by close to an order of magnitude, to a good approximation
χM ∝ χR. (3.4)
Measurements of χR will therefore be taken throughout this thesis as representative
of χM .
3.3 Experimental Setup
A simplified diagram of the experimental setup is shown in Fig 3.4 below. A constant
current of 0.93 mA was run through the length of the sample and the resistance was
measured through a standard four-terminal configuration.
Figure 3.4: A diagram of the experimental setup. DC and AC fields were applied perpendicular
to the direction of current in the sample.
The magnetization of the sample, M(H), was determined by applying a DC field
and measuring resistance. Similarly, the susceptibility χ = dMdH was determined by ap-
plying a small AC field dH on top of a bias field HDC and measuring the amplitude
15
dR of the AC resistance change produced. The amplitude of the AC field, its frequency
and the amplitude of the DC field were independently varied. A Stanford Research
Systems DS345 function generator was used to generate the AC signal, which was then
amplified by a Yamaha audio amplifier before driving the inner coils. A Kepco bipo-
lar operational amplifier generated DC current to drive the outer coils. The magnetic
field was measured with an FW Bell 5080 gauss meter inside the inner coil. The real
and imaginary parts of the susceptibility were measured with a two-channel Stanford
Research Systems SR530 lock-in amplifier. The collection of data was automated with
a custom LabVIEW program.
The lock-in measured the magnitude of the input AC voltage signal as well as the
phase of the signal relative to a reference signal, in this experiment the applied mag-
netic field. Although the amplitude of the field could be measured accurately with the
Hall probe gaussmeter, at high frequencies the processing circuitry caused a phase shift
between the actual field and the signal, so the phase difference between the resistance
and the field could not be measured with the gaussmeter. However, the magnetic field
was in phase with the current through the coils, so a 7.5Ω resistor was placed in series
with the coils to convert the current signal into a voltage signal. The voltage across
the power resistor was then amplified with a simple 741 op-amp inverting amplifier to
provide a large, clean signal with no phase shifts, and then fed into the lock-in reference
input as shown in Fig 3.5.
The op-amp inverting amplifier was used both to provide a clean reference signal of
about 1 V at low AC fields and to avoid overloading the lock-in reference input at higher
AC fields. Using an amplifier was preferable to simply using a larger resistor, since a
larger resistor meant more current was necessary from the power supply. A carbon-film
resistor decade box was used as the variable resistor to minimize any phase shift that
might result from the finite inductance of a potentiometer. The op-amp was powered
by two 9V batteries wired together.
The phase of the resistance signal relative to the applied field could be roughly mea-
sured by turning off the current through the sample and varying the magnitude of the
16
Figure 3.5: A schematic of the amplifier circuit used to accurately determine the phase of AC
magnetic fields. A 741 op-amp inverting amplifier was used to amplify the voltage across the
power resistor to 1 V AC signal preferred by the lock-in reference input.
AC field. With no current, there was no magnetoresistance, so one would expect zero
response. However, as shown in Fig 3.6 below, the sample was attached by wire bonds
to contact wires.
Figure 3.6: A more detailed schematic of the sample than in Fig 3.1. Indium wire-bond contacts
connected the surface of the sample to a lead box, which led into BNC cables.
An alternating magnetic field through the wire bond loops produced a Faraday EMF,
which produced a voltage 90 degrees out of phase with the applied field. If the reference
phase was set properly, the real part was zero and the imaginary part was linear with
applied amplitude as shown in Fig 3.7.
For every measurement involving an AC magnetic field, data runs were taken with
and without current applied and the imaginary background was subtracted off. This
background is not just an annoyance, since it could be used to check that the reference
17
Figure 3.7: The response of the sample at zero current as a function of AC applied field, showing
the linear background effect from Faraday EMF through the wire bonds. The background was
used to adjust the phase of the lock-in relative to the applied field.
phase of the lock-in was set correctly. If the lock-in reference signal were out of phase
with the applied field, the real component would have a non-zero slope. Using this
method, a frequency-dependent shift between −20◦ and 20◦ was found between the
reference input and the applied field, despite efforts taken to ensure no phase shifts
signal processing. The origin of this shift is unknown - at each measurement, the phase
of the lock-in was manually adjusted to compensate.
Chapter 4
Results
4.1 Susceptibility as a function of DC Field: Testing Lin-
ear Response
To test for a linear response, the real part of the AC response of the permalloy thin film
was measured as a function of AC field amplitude. Data was taken at zero DC applied
field, where the magnitude of the AMR signal was a maximum. Results are shown in
Fig 4.1below.
Figure 4.1: The susceptibility at 0 DC field plotted as a function of HAC . At small amplitudes
the response is approximately linear.
The linearity of χ(HAC) was estimated by performing linear fits on different regions
of the curve as shown in Fig 4.2. The red curve represents a linear fit over the bottom
10 values of applied field, the green curve uses the lowest 25 values, and the blue curve
18
19
uses the lowest 40 values.
Figure 4.2: The same data as Fig 4.1 with linear fits using three different regions of χ. The red
curve extrapolates from the bottom 10 data points, the green curve uses the first 25, and the
blue curve uses the first 40. The slopes differ from each other by about 4%, but a least-squares
weighted mean reveals that they differ from the mean slope by, respectively, 3.2σ, 2.1σ, and
2.3σ.
The slopes for the red, green, and blue curves were found to be (3.91±0.07)×10−3,
(4.07±0.03×10−3, and (4.18±0.02)×10−3, respectively. The weighted mean, obtained
by least-squares analysis, is (4.13 ± 0.02) × 10−3, so the slopes obtained are, respec-
tively, 3.2σ, 2.1σ, and 2.3σ from the mean value. Uncertainties on the data points
were obtained by taking the standard deviation over several runs, but may have been
underestimated. The slopes all differed from the mean by less than 5%, so it was con-
cluded that χ(HAC) was approximately linear in the 0.2 G - 1 G HAC range, so that
the fluctuation-dissipation theorem could be applied. Measurements of χ(f) were taken
within this range.
20
4.2 Susceptibility as a function of DC Field: Mapping out
the Hysteresis Loop
In general χ = χ(HAC , HDC , f), although most attention is focused where χ is greatest,
around HDC = 0. In principle HAC should be as small as possible to best approximate
linear response, but as HAC decreased the AC signal became increasingly noisy. A bal-
ance between signal-to-noise ratio and linear response was found between HAC = 0.2G
and HAC = 1G. To account for the variation of χ with HDC , instead of fixing HAC and
HDC and directly varying f , HAC and f were fixed, χ was measured as a function of
HDC , and the measurement was repeated for different frequencies. As shown in Fig 4.3,
this susceptibility measurement amounts to taking the derivative of the AMR curve.
Figure 4.3: The left plot shows the AMR curve with both sweeps as shown in Fig 3.2, while
the right graph plots χ as a function of HDC at fixed HAC , measuring the slope of the left curve
at every point.
The right curve in Fig 4.3, plotting the real and imaginary parts of the susceptibility
as a function of DC field using an AC field of amplitude 1 G at 1 kHz, may be interpreted
as follows. At high magnitude of applied field, the susceptibility approached zero,
since the thermal energy kT was small compared to the energy of alignment with the
field µB and the spins could not fluctuate very much. The susceptibility was greatest
when M = 0, which occurred at two distinct values of H depending on the previous
21
history of the applied field. A schematic of these low and high-energy states is shown
in Fig 4.4 below. χ is negative for certain fields, reflecting the fact that in performing
the measurement HDC was swept from a negative field (defined by the Hall probe) to
a positive field and back down to negative. Tracing out the hysteresis loop, moving
along a region in which the resistance decreased with applied field resulted in a negative
susceptibility.
Figure 4.4: χ(HDC) at fixed f and HAC as before. In this case, f = 1 kHz HAC = 1 G
peak-to-peak. At high fields, the magnetic domains align strongly with the field and do not
fluctuate much, resulting in a low susceptibility. χ is a maximum when the magnetization is
zero, which occurs near H = 0.
22
4.3 Susceptibility as a function of Frequency: Predicting
magnetic noise
Data runs were taken at AC amplitudes of 0.2 G, 0.5 G, and 1 G AC, searching for
the optimum balance between most linear response at low HDC and largest measured
signal at high HDC . At each AC amplitude, dR was measured as a function of HDC
over the whole hysteresis loop at 20 Hz, 50 Hz, 100 Hz, 200 Hz, 500 Hz, 1 kHz, 2 kHz,
and 5 kHz for a total of 24 data runs. Nonlinear background effects were observed at
frequencies above 5 kHz. Plots of Re(χ(HDC)) and Im(χ(HDC)) taken at HAC = 0.5G
for different frequencies are shown in Fig 4.5 and Fig 4.6 below.
The two values of HDC at which the frequency dependence of χ was most interest-
ing: HDC = 0, where the susceptibility was greatest, and HDC = 2.5G, where the AMR
resistance was a maximum. For every HDC two values of χ were measured depending
on the direction of field sweep as shown in Fig 4.3, so the difference between the two
was taken to be the value of χ. The reason for the large value of χ at 20 Hz is not un-
derstood, but it is common to all data sets taken. Final plots of the real and imaginary
parts of χ as a function of frequency at 0 DC field are shown in Fig 4.7 and Fig 4.8 and
at 2.5 G DC field in Fig 4.9 and Fig 4.10.
χ(f) at 2.5 G DC exhibits the same general trend as χ(f) at 0 G DC, except that the
data appears to be noisier, especially at lower frequencies, and trends are less evident.
This result is not surprising since the dR measured is about a factor of two reduced at
HDC = 2.5G from its value at HDC = 0G.
Returning to the fluctuation-dissipation theorem,
S(ω) =
2kBT
ω
χˆ′′(ω), (4.1)
if χˆ′′(ω) ∝ ω, S(ω) should be constant, not 1/f, consistent with the results of [7] and
the deviation at H = 0 found in [6].
23
Figure 4.5: The real part of χ(HDC) for different frequencies. Runs at 50 Hz, 200 Hz, and 1
kHz were taken but are not shown to avoid cluttering the figure. The two peaks at positive and
negative H are evident at low frequencies but move to H = 0 at high frequencies as magnetic
hysteresis effects become less significant. Since the imaginary part dominates the real part of
the susceptibility at high frequencies, the data at 5 kHz does not follow the shape of the data
at lower f .
24
Figure 4.6: The imaginary part of χ(HDC) for different frequencies. The difference between
the positive and negative values of Im(χ(HDC)) was determined and then plotted as a function
of frequency to form Fig 4.8
25
Figure 4.7: The difference between the two values of Re(χ(HDC = 0)) as a function of frequency
from 20 Hz to 5 kHz, for HAC = 0.2 G, 0.5 G and 1 G. Different AC amplitudes were chosen to
find the optimum balance between large signal size for large HAC and best linear response for
small HAC .
Figure 4.8: The difference between the two values of Im(χ(HDC = 0)) as a function of frequency
from 20 Hz to 5 kHz, for HAC = 0.2 G, 0.5 G and 1 G.
26
Figure 4.9: Taken from the same data set as Fig 4.7 except that each point now represents the
difference between the two values of Re(χ(f)) at HDC = 2.5G instead of Re(χ(f) at HDC = 0G.
Qualitatively the results are similar to Fig 4.7, except that the data appears more noisy.
Figure 4.10: Im(χ(f)) at HDC = 2.5G from 20 Hz to 5 kHz.
Chapter 5
Analysis and Implications for Mag-
netic Noise
5.1 Comparison to a Simple Harmonic Oscillator Model
Starting with the damped simple harmonic oscillator with a sinusoidal driving force
mx¨+ γx˙+ kx = F0 sinωt, (5.1)
taking the Fourier transform and solving for the susceptibility yields
χ =
x
F
=
(ω20 − ω2)− iβω
m((ω20 − ω2)2 + β2ω2)
. (5.2)
Using the Kittel formula [16], the resonant frequency for this system is on the order
of 1 GHz, so a low-frequency approximation applies. In this limit,
Re(χ) ∝ 1
ω2/ω20
, (5.3)
Im(χ) ∝ ω
ω0
(5.4)
The data at 0 G DC are replotted as log-log plots for clearer comparison to the SHO
prediction in Fig 5.1 and Fig 5.2. In logarithmic form, the SHO model predicts that
Re(χ(f)) and Im(χ(f)) should be linear with slopes of -2 and -1 respectively.
Performing least-squares fitting, the slopes of the imaginary log-log plots at 0.2 G,
0.5 G, and 1 G are 0.69, 0.68, and 0.69, and the slopes of the real curves are -0.18, -0.22,
and -0.18, respectively. Neither component quantitatively matches the predictions of
the SHO model, but the discrepancy in the real part is significantly greater than in
27
28
Figure 5.1: The data in Fig 4.7, redrawn as a log-log plot for clearer analysis. Modeling the
magnetic domains as a driven damped simple harmonic oscillator predicts that Re(χ(f)) ∝ 1/f2,
so the curve would be linear with a slope of -2. The average slope of the curve is about -0.2, so
the real component does not fit the SHO model.
the imaginary part. This discrepancy may reflect additional resistance effects in the
sample contributing to the real response or also an inadequacy of the simple harmonic
oscillator model. The SHO model involves a dissipative force giving rise to an imaginary
susceptibility, but also a restoring force, and there may be no restoring force in this
system.
29
Figure 5.2: A log-log plot of Fig 4.8. The anomalously large value of susceptibility at 20 Hz is
consistent across all measurements. Its origin is unknown, and when fitting to a line the data
points at 20 Hz were ignored. The SHO predicts that Im(χ(f)) ∝ f , in which case the curve
would be linear with slope 1. A linear fit yields an average slope of 0.69.
Chapter 6
Conclusion and Next Steps
In this experiment the AC magnetic susceptibility χ of a permalloy thin film was mea-
sured as a function of applied DC field, applied AC field magnitude and AC frequency.
Anisotropic magnetoresistance was used to relate the magnetization to resistance, which
provided the benefit of a simple physical system and the disadvantage of a small signal
compared to previous measurements using giant magnetoresistance in magnetic tun-
nel junctions. The results were used to test applicability of the fluctuation-dissipation
theorem in microscopic ferromagnetic systems and to predict the functional form of
magnetic noise.
χ was measured first at HDC = 0 as a function of HAC to experimentally test the
linear response necessary for application of FD. At AC amplitudes from 0.2 G to 1 G
the response was found to be linear within 5%. χ was then measured as a function of
f over the entire hysteresis loop at three different AC amplitudes. The imaginary part,
χ′′(f), was found to be very roughly linear with frequency, inconsistent with 1/f noise
measurements but consistent with previous white noise measurements as well as a sim-
ple model of the magnetic domains in an AC field as a damped driven simple harmonic
oscillator. The real part χ′(f) approximately varied as f−0.2, inconsistent with the SHO
prediction that χ′(f) should vary as f−2.
Further work on this experiment would investigate in more detail the frequency
dependence of χ′, looking for extraneous contributions to the signal as well as extensions
to the SHO model. In particular, the SHO model involves a restoring force or spring
constant, and it is possible that there is no analogous restoring force on domains in
an AC magnetic field and that the domains simply dissipate energy. This hypothesis
could be tested by measuring the AC susceptibility over higher frequencies. The SHO
model predicts a maximum real part at a resonant frequency, in which case the real
susceptibility would be zero. However, the resonant frequency of 1 GHz may not be
30
31
practical in this experiment due to the large amount of current required to overcome the
inductive impedance of the coils at high frequencies. AC susceptibility measurements
should be performed on more permalloy samples to control for defects in the permalloy
layer, which may pin the magnetization and lead to anomalies in the susceptibility. The
experiment could also be repeated on GMR layers as in [6].
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