Developing a temperature sensitive tool for studying spin dissipation
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Kurtis Jon Wickey
Graduate Program in Physics
The Ohio State University
2015
Dissertation Committee:
Professor Ezekiel Johnston-Halperin, Advisor
Professor Fengyuan Yang
Professor Mohit Randeria
Professor Harris Kagan
Copyright by
Kurtis Jon Wickey
2015
Abstract
Measuring the thermodynamic properties of nanoscale structures is becoming
increasingly important as heterostructures and devices shrink in size. For example, recent
discoveries of spin thermal effects such as spin Seebeck and spin Peltier show that
thermal gradients can manipulate spin systems and vice versa. However, the relevant
interactions occur within a spin diffusion length of a spin active interface, making study
of these spin thermal effects challenging. In addition, recent ferromagnetic resonance
studies of spatially confined nanomagnets have shown unique magnon modes in arrays
and lines which may give rise to unique magnon-phonon interactions. In this case, the
small volume of magnetic material presents a challenge to measurement and as a result
the bulk of the work is done on arrays with measurements of the magnetization of
individual particles possible through various microscopies but limited access to thermal
properties. As a result, tools capable of measuring the thermal properties of nanoscale
structures are required to fully explore this emerging science. One approach to addressing
this challenge is the use of microscale suspended platforms that maximize their
sensitivity to these spin thermal interactions through thermal isolation from their
surroundings. Combining this thermal decoupling with sensitive thermometry allows for
the measurement of nanojoule heat accumulations, such as those resulting from the small
heat flows associated with spin transport and spin relaxation. As these heat flows may
manifest themselves in a variety of spin-thermal effects, the development of measurement
ii
platforms that can be tailored to optimize their sensitivity to specific thermal
measurements is essential.
To address these needs, I have fabricated thermally isolated platforms using a
unique focused ion beam (FIB) machining that allow for flexible geometries as well as a
wide choice of material systems. The thermal characteristics of these platforms were
rigorously tested by measuring the heat capacity of a 6.2 ng Au sample using a
microscale suspended SiNx platform. The heat capacity measurement was selected for its
ability to provide meaningful metrics in the evaluation and optimization of our platform
design. The results match closely with the values obtained for bulk samples despite a 1012
difference in mass and our platforms perform near the state-of-art for thin film
calorimetry.
Having established the sensitivity of the SiNx platforms, I designed and built a
custom vacuum setup for use in a microwave cavity for the thermal measurement of
resonant spin dynamics. The long term goal is to thermally study the resonant spin
transfer across a FM/N interface utilizing the decay of the spin polarization through
magnon-phonon and electron-phonon interactions. Initial measurements were done and
challenges identified and improvements suggested.
iii
Dedication
This document is dedicated to my parents and my grandparents.
iv
Acknowledgments
I would like to thank my advisor Ezekiel Johnston-Halperin for mentoring me and
working with me to better myself as a scientist and researcher. I have learned how to
think critically and methodically approach solving intractable problems under his
tutelage.
I would like to thank my colleagues in the lab for their advice and technical support that
helped enable my research: Yi-Hsin Chiu, Howard Yu, Justin Young, Yu Sheng Ou, and
Matt Sheffield. I would like to thank my colleague Mike Chilcote for help with sample
fabrication and publication editing. I want to thank Kris Dunlap for helping me navigate
graduate school bureaucracy. I want to thank Pete Gosser for machining parts on time,
and working closely with me to make sure all the tolerances are being met, including the
ones that I forget to include. I want to thank Denis Pelekhov for well-maintained user
facilities enabling consistent processing as well as for allowing me to serve as a super
user for multiple pieces of equipment. I want to thank the late Bob Wells for rescuing me
from multiple equipment failures.
Finally, I want to thank my parents and grandparents for unconditionally supporting me
throughout my lengthy schooling.
v
Vita
June 2004 .......................................................Centreville High School, Michigan
May 2008 .......................................................B.S. Physics, Applied Math, Western
Michigan University
May 2008 .......................................................B.A. Economics, Western Michigan
University
2008-present ..................................................Graduate Teaching Associate, Department
of Physics, The Ohio State University
Publications
1.
“Nanogram calorimetry using microscale suspended SiNx platforms fabricated via
focused ion beam patterning,” K. J. Wickey, M. Chilcote, and E. Johnston-Halperin,
Rev. Sci. Instrum. 86, 014903 (2015)
2. “High Temperature Thermal Stability and Oxidation Resistance of Magnetron-sputtered
Homogeneous CrAlON Coatings on 430 Steel,” A. Kayani, K. J. Wickey, M. I.
Nandasiri, A. Moore, E. Garratt, S. AlFaify, X. Gao, R. J. Smith, T. L. Buchanan, W.
Priyantha, M. Kopczyk, P. E. Gannon, and V. I. Gorokhovsky AIP Conf. Proc. Vol 1099,
pp. 303-307 (2009)
3. “Ion Beam Analysis of the Thermal Stability of Hydrogenated Diamond-Like Carbon
Thin Films on Si Substrate,” M. I. Nandasiri, A. Moore, E. Garratt, K. J. Wickey, S.
vi
AlFaify, X. Gao, A. Kayani, and D. Ingram AIP Conf. Proc. Vol 1099, pp. 318-322
(2009)
4. “Femtosecond coherent control of spins in (Ga,Mn)As ferromagnetic semiconductors
using light,” M. D. Kapetanakis, I. E. Perakis, K. J. Wickey, C. Piermarocchi, and J.
Wang Phys. Rev. Lett. 103, 047404 (2009)
Fields of Study
Major Field: Physics
vii
Table of Contents
Abstract ............................................................................................................................... ii
Dedication .......................................................................................................................... iv
Acknowledgments............................................................................................................... v
Vita..................................................................................................................................... vi
List of Tables .................................................................................................................... xii
List of Figures .................................................................................................................. xiii
Chapter 1 : Introduction ...................................................................................................... 1
1.1 Overview ................................................................................................................... 1
1.1.1 Types of magnetism............................................................................................ 1
1.1.2 New applications for magnetism ........................................................................ 3
1.2 Spintronics................................................................................................................. 5
1.3 Traditional spin injection/detection ........................................................................... 8
1.3.1 Electrical spin injection/detection ...................................................................... 8
1.3.2 Spin Hall spin injection/detection..................................................................... 13
1.4 Emerging techniques for spin injection ................................................................... 15
1.4.1 Resonant spin injection ..................................................................................... 15
viii
1.4.2 Thermal spin injection ...................................................................................... 17
1.5 Opportunities for additional studies ........................................................................ 18
1.6 References ............................................................................................................... 19
Chapter 2 : Dynamically-driven spin physics ................................................................... 24
2.1 Introduction ............................................................................................................. 24
2.2 Thermally driven spin dynamics ............................................................................. 24
2.2.1 Magnons ........................................................................................................... 26
2.2.2 Magnon-phonon interaction ............................................................................. 29
2.2.3 Electron-phonon coupling (spin-phonon coupling) .......................................... 30
2.2.4 Magnon-electron coupling ................................................................................ 31
2.2.5 Magnetocaloric effect ....................................................................................... 34
2.2.6 Spin Seebeck effect .......................................................................................... 36
2.2.7 Other spin thermal effects................................................................................. 40
2.3 Resonant spin dynamics .......................................................................................... 43
2.3.1 Magnetization dynamics ................................................................................... 43
2.3.2 Resonance condition ......................................................................................... 47
2.3.3 Non-equilibrium spin transfer (spin pumping) into adjacent films .................. 48
2.4 Thermal interactions in resonant spin dynamics ..................................................... 49
2.4.1 Resonant phonon driven spin dynamics ........................................................... 49
ix
2.4.2 Thermal detection of magnetization dynamics ................................................. 51
2.4.3 Need for sensitive tools .................................................................................... 53
2.5 References ............................................................................................................... 54
Chapter 3 : Developing customized experimental probes ................................................ 60
3.1 Need for thermally isolated platforms ..................................................................... 60
3.2 Constraints on platform design ............................................................................... 61
3.3 Fabrication of platforms .......................................................................................... 63
3.4 Overview of measuring heat capacity ..................................................................... 68
3.4.1 Adiabatic Calorimetry ...................................................................................... 69
3.4.2 Differential Scanning Calorimetry (DSC) ........................................................ 69
3.4.3 AC Calorimetry ................................................................................................ 70
3.4.4 Thermal Relaxation Calorimetry ...................................................................... 71
3.5 Experimental Methodology ..................................................................................... 75
3.6 Results and Discussion ............................................................................................ 81
3.7 Conclusion............................................................................................................... 86
3.8 References ............................................................................................................... 87
Chapter 4 : Thermal detection of magnetization dynamics .............................................. 90
4.1 Motivation ............................................................................................................... 90
4.1.1 Proposed experiment ........................................................................................ 91
x
4.1.2 Expected signal size.......................................................................................... 92
4.2 Experimental setup .................................................................................................. 93
4.2.1 Vacuum design ................................................................................................. 94
4.2.2 Measurement electronics and wiring ................................................................ 94
4.2.3 Platform design ................................................................................................. 96
4.3 Measurements.......................................................................................................... 97
4.3.1 Expected thermal resonant response................................................................. 99
4.3.2 Possible reductions in thermal signal ............................................................. 100
4.3.3 Future improvements to experimental design ................................................ 101
4.4 Conclusion............................................................................................................. 104
4.5 References ............................................................................................................. 105
Appendix ......................................................................................................................... 106
A1: Photolithography with a maskless aligner ............................................................ 106
A2: Metal deposition parameters on the Lab-18 ......................................................... 109
A3: E-beam patterning of SiNx membranes ................................................................ 112
A4: FIB patterning of SiNx membranes ...................................................................... 117
A5: Measurement wiring ............................................................................................. 122
A6: Miscellaneous ....................................................................................................... 126
Bibliography ................................................................................................................... 128
xi
List of Tables
Table 1: Parameters for every metal film deposition. ..................................................... 110
xii
List of Figures
Fig. 1.1 A schematic of a spin valve ................................................................................... 6
Fig. 1.2 A schematic of a Datta-Das spin transistor. .......................................................... 7
Fig. 1.3 A lateral spin valve with the spin valve and Hanle signal ................................... 12
Fig. 1.4 A cartoon of the spin Hall and inverse spin Hall effect....................................... 14
Fig. 1.5 Cartoon picture of resonant spin injection and detection .................................... 16
Fig. 1.6 A schematic of thermal spin injection into a semiconductor............................... 18
Fig. 2.1 A cartoon of magnons.......................................................................................... 26
Fig. 2.2 Measured magnon thermal conductivity ............................................................. 28
Fig. 2.3 Spin pumping across an interface ........................................................................ 32
Fig 2.4 A study of the spin-mixing conductance as measured by three separate spin
injection methods .............................................................................................................. 33
Fig 2.5 S vs T diagram of the magnetocaloric effect ........................................................ 36
Fig. 2.6 Transverse geometry of the spin Seebeck effect ................................................. 37
Fig. 2.7 Phonon and magnon temperature profiles as a function of sample position ....... 39
Fig. 2.8 Reciprocal effects: spin Peltier and spin Seebeck ............................................... 42
Fig. 2.9 Magnetization precession cone. ........................................................................... 44
Fig. 2.10 Damping of magnetization precession. ............................................................. 46
Fig. 2.11 Resonant phonon spin pumping ........................................................................ 50
xiii
Fig. 2.12 Thermally detected FMR ................................................................................... 52
Fig. 3.1 An electron micrograph of SiNx platforms .......................................................... 64
Fig. 3.2 Failed approaches towards fabrication ................................................................ 67
Fig. 3.3 A schematic showing the 1-D heat flow model ................................................... 72
Fig. 3.4 Systematic checks before applying heat model. .................................................. 74
Fig. 3.5 Schematic of the measurement circuit. ................................................................ 76
Fig. 3.6 Resistance of the Pt resistors vs temperature....................................................... 78
Fig. 3.7 Averaged thermal pulse and noise ....................................................................... 80
Fig. 3.8 Extracted thermal time constant and Keff. ............................................................ 82
Fig. 3.9 Frequency dependence of Keff. ............................................................................ 84
Fig. 3.10 Heat capacity as a function of T. ....................................................................... 85
Fig. 3.11 Specific heat of Au compared to literature. ....................................................... 86
Fig. 4.1 In-plane FMR response of a 60 nm Py thin film. ................................................ 93
Fig. 4.2 FMR vacuum setup .............................................................................................. 95
Fig. 4.3 Modified platforms for FMR measurements ....................................................... 97
Fig. 4.4 Non-resonant heating under microwave application. .......................................... 98
Fig. 4.5 Thermal detection with field scan ....................................................................... 99
Fig. 4.6 High power field scan ........................................................................................ 103
Fig. A.1 SiNx membrane and Al holder .......................................................................... 107
Fig. A.2 A developed pattern using photoresist. ............................................................. 109
Fig. A.3 Incomplete liftoff of Pt pattern. ........................................................................ 112
Fig. A.4 E-beam masking pattern. .................................................................................. 117
xiv
Fig. A.5 FIB master pattern for machining the platforms............................................... 119
Fig. A.6 Electron micrograph during the FIB process. ................................................... 122
Fig. A.7 Wired sample and measurement puck .............................................................. 124
Fig. A.8 Blown Pt leads. ................................................................................................. 125
xv
Chapter 1 : Introduction
1.1 Overview
Magnetism has been studied throughout modern history with discussions
regarding its origins dating back to the ancient Greeks.
Magnetic properties were
exploited through navigating with magnetized compass needles in the middle ages. The
present age sees the use of magnets for power generation (generators), refrigeration
(magnetocaloric effect), information storage (giant magnetoresistance effect), and
beyond.
Investigations into magnetization still continue due to the rich underlying
physics and myriad of interactions that contribute to magnetic effects along with the
many potential applications.
1.1.1 Types of magnetism
Magnetism is the result of a macroscopic ordering of localized electron’s intrinsic
angular momentum (spin).
While paramagnetism or diamagnetism rely on an external
field to align the spins, permanent spin alignment (ferromagnetism, anti-ferromagnetism)
results from interactions between adjacent spins.
This interaction is known as the
exchange interaction and is a result from the Pauli exclusion in overlapping electron
1
orbitals.
To understand this interaction and the resulting spin alignment, we can write
down the internal energy of a 1-D chain of spins using the Heisenberg model:
𝑈 = −2𝐽 ∑𝑁
𝑝=1 𝑺𝑝 ∙ 𝑺𝑝+1 ,
(1.1)
where J is the exchange constant (material dependent), and Sp is the spin at site p. As this
is the an approximation of the internal energy for the spin system alone (only nearest
neighbors interactions with no dipolar coupling), it is not able to account for other energy
costs which can lead to frustrated spin ordering systems such as spin ice1. Minimizing
the spin energy depends on the sign of J. For positive J, the energetically favorable
ordering is for the neighboring spins to align parallel (ferromagnetism). For negative J,
the energetically favorable ordering is for neighboring spins to align anti-parallel (antiferromagnetism). For a non-existent J=0 (no electron interaction), there will be no
ordering without an applied magnetic field (paramagnetism).
Unlike these effects,
diamagnetism is the electron’s orbital motion that couples with field not the spin degree
of freedom, and the resulting induced magnetic moment opposes the applied magnetic
field (anti-parallel). Diamagnetism is present in all materials; however, it is usually
hidden by much stronger paramagnetism from un-paired electrons and therefore only
seen in materials with full electron shells (noble gases, alkali or halide ions).
For many practical purposes, it is desirable to have a magnetic moment absent an
applied magnetic field.
Although anti-ferromagnets have spin ordering without an
applied field, they do not exhibit a net magnetic moment due to canceling contributions
from the anti-aligned electron spins.
Ferromagnets’ spin ordering results in a net
2
magnetic moment and are widely used for practical applications due to the persistent
magnetization.
1.1.2 New applications for magnetism
Recent discoveries2-3 have shown that thermal gradients applied across
ferromagnet can generate spin currents through the spin Seebeck effect which in turn can
generate voltages through spin-dependent scattering in an attached metal4.
While
generating voltages from thermal gradients is nothing new (thermoelectrics),
interdependent material parameters (electrical conductivity and thermal conductivity)
makes materials engineering difficult for designing high efficiency thermoelectric
conversion. While voltages generated by spin Seebeck are small (a few μV compared to
mV for classic thermoelectric effects), the relevant parameters are completely different
from thermoelectrics and may be optimized independently. Recent calculations5 have
predicted a potential efficiency factor of 0.5 which is near state of the art for current
thermoelectrics. Work has already been done to adapt material systems suitable for
commercial use for converting waste heat into power generators using the spin Seebeck
effect6.
Another developing use for ferromagnets uses the spin polarization inherent in
ferromagnets for spin logic. Spin-based electronics, or spintronics, utilizes the intrinsic
angular momentum instead of charge as the state variable. Using a quantum mechanical
description, spin can be oriented in two ways: up and down states. The system has some
probability of being found in either state: the coefficients of each state squared summed
3
together must be one (the system has a probability of 1 to be found in a state). However,
it is often easier to use a semi-classical description when working with spin: spin as a
vector with a fixed magnitude and makes some angle with respect to the eigenbasis of the
system. Reliable creation, control, and detection of this angle is the foundation for
functionality of spin devices.
One of the potential benefits of using spin as the state variable instead of charge
involves the energy cost to perform an operation. Conventional charge devices require
energy to move from ON to OFF state (or vice versa). This energy can be thought of as
lowering a barrier height between two potential wells to transfer an electron between the
two (the state being determined by the location of the electron). Calculations have
estimated7 that the thermodynamic minimum for this operation is ~23 meV. However
real devices operate well above this minimum with the estimated8 gate switching energy
in 2018 for a 10 nm wide gate is 15 eV (103 larger). The energy required to change states
for a spin device is much lower as a spin rotation is needed (which can be achieved by
external magnetic fields) not a barrier height manipulation. Using a gate electrode (via a
Rashba spin-orbit coupling) to achieve this rotation is thought to be much closer to the
fundamental limit9. The end result is that spintronic devices are predicted to require
much less power to operate than electronic devices; a critical parameter as devices
become smaller and power density rises. The following sections are devoted to exploring
spintronics as a developing field and identifying potential areas where thermal
investigations could make significant contributions towards furthering the field.
4
1.2 Spintronics
Investigations into using spin as a state variable began with the discovery of the
giant magnetoresistance effect10-11 (GMR) where a macroscopic spin polarization
(manifested by a magnetization) influences the properties of a spin polarized charge
current moving through it. This effect is encapsulated in spin valve: a device which
consists of normal metal layer sandwiched between two ferromagnetic layers shown in
Fig. 1.1 (a). Any current passing through the metal stack will be spin-polarized by the
first electrode (source) as there exists a spin-polarization of electrons at the Fermi energy.
The normal metal sandwiched between the ferromagnetic layers has zero spinpolarization at the Fermi energy. As spin polarized electrons pass through the normal
layer, they lose their spin-orientation. Provided the normal spacer layer’s thickness is
chosen accordingly (less than the spin diffusion length in the material) then polarization
will remain at the second ferromagnetic electrode (drain). The relative orientations of the
ferromagnetic layer (parallel or anti-parallel) determine whether the spin valve is in a
high resistance state (anti-parallel orientation) or low resistance state (parallel orientation)
as the density of states at the Fermi energy is polarization dependent shown in Fig. 1.1.
5
Fig. 1.1 (a) A spin valve with magnetizations oriented in a parallel configuration. Density of states
diagram shows a lot of majority spin states available for conduction in the F2 contact indicating a low
resistance state. (b) A spin valve oriented in the anti-parallel configuration. Density of states diagram
shows there are not many states available in F2 for the majority spins in F1 indicating a high
resistance state. Source: [12]
The relative orientation of the electrodes is usually controlled through an applied
magnetic field. The ability to differentially address each magnetic layer is critical for the
device functionality. This can be accomplished in a variety of ways such as different
layer thickness so the shape anisotropy energy is slightly different leading to different
coercive fields13. Another option is through exchange biasing one electrode with an
adjacent anti-ferromagnetic layer to pin the magnetization direction through exchange
energy14-15. Regardless of the specific methods used, GMR and similar effects have
given rise to commercial applications such as magnetic random access memory (MRAM)
6
and other storage based applications.
There has been a broad push for further
functionality in spin-based devices where computation could be combined with storage16.
One of the ways to add functionality in spin-based devices is to separate the
source and drain electrodes and have the spin-polarized electrons flow in a channel where
the spin orientation is allowed to rotate. This idea is encapsulated in a Datta-Das spin
field effect transistor17 (spin-FET) shown in Fig. 1.2. Thus in addition to the parallel and
anti-parallel states like the spin-valve mentioned in the previous paragraph, through
controllable spin precession in the channel additional ON and OFF states (or anything in
between) can be reached. For this device to function there are four key features that need
to be addressed: spin injection from the source into the channel, spin transport through
the channel, controllable spin precession in the channel, and spin detection at the drain.
Fig. 1.2 A Datta-Das transistor. The spin orientation in the channel is controlled by the gate electrode
through spin orbit coupling. Increasing the gate voltage will increase the precessional rate of the
electron spins in the channel.
7
The Datta-Das transistor serves as more of an instructional device rather than a
practical one as a functioning spin transistor has yet to have been achieved (other than
proof-of principle). Other spin transistor device geometries have been proposed such as
magnetic bipolar transistors18, spin functionalized CMOS transistors19, or a magnon
transistor20. Regardless of the operating principle proposed, most require an injection of
non-equilibrium spin population into an adjacent layer; therefore the next few sections
will discuss current progress on spin injection techniques.
1.3 Traditional spin injection/detection
Injecting a spin polarization into a normal material (zero spin-polarization under
ambient conditions) has been achieved using a variety of methods such as optical spin
injection21, spin Hall22, or through use of an applied spin-polarized current (which is the
method that founded the spintronics field10-11,23). This section will detail the use of
electrical spin injection and spin Hall injection, and addressing some of the challenges
faced by both approaches.
1.3.1 Electrical spin injection/detection
Using an electrical current for spin injection is a natural extension of the existing
electronics architecture: addressing the state variable through applied/measured voltages.
Initially, spin injection was done using all-metal systems: injector/channel/detector23.
However, a lot of progress has been made to realize a metal/semiconductor/metal
system24-28 or an all semiconductor system29-31. Semiconductors have comparably long
8
spin lifetimes and spin diffusion lengths than metals making them superior material
systems for the spin transport.
As previously mentioned, ferromagnets have a permanent magnetization without
the application of an external field. For spin injection purposes, this translates into a
spin-polarization at the Fermi energy (shown in Fig. 1.1) which can be transferred to
adjacent layers through an applied bias. This was shown definitively in work done by
Johnson and Silsbee23 where spin polarization of permalloy (a Ni80Fe20 alloy) was passed
to an adjacent Al layer. The efficiency of the spin injection depends on the initial spin
polarization of the injector, spin-flip scattering at the interface (the spin-mixing
conductance g↑↓) and on the ratio of conductivities of the injector and channel. Using
charge continuity conditions and a two channel model, Schmidt et al derived32 the spin
polarization in the channel, α, as:
𝛼=𝛽
𝜎𝑠𝑐 𝜆𝑓𝑚
2
𝜎𝑓𝑚 𝑥0
𝜎 𝜆𝑓𝑚
(2 𝑠𝑐
+1)−𝛽 2
𝜎𝑓𝑚 𝑥0
,
(1.2)
where β is the spin polarization in the injecting ferromagnet far from the interface, λ fm is
the spin diffusion length in the injecting ferromagnet, x0 is the length of the channel
(between the electrodes), and σsc, σfm are the conductivities for the semiconducting
channel and injecting ferromagnet respectively. The maximum that α can be is β (the
injecting ferromagnet’s spin polarization). This expression should be viewed as an upper
bound on the channel’s spin polarization since neither spin-flip scattering at the interface
nor spin relaxation in the channel is considered. However despite these limitations, the
expression is illuminating as a typical metallic ferromagnet/semiconducting channel will
have σfm = 104 σsc making α very small. Indeed for a typical device32 (β = 60%, x0 = 1
9
μm, λfm = 10 nm) this would translate to α < 0.002%. This lack of spin injection into a
resistive channel is known as the conductivity mismatch problem.
One way around this conductivity mismatch is to use semiconducting
ferromagnets so that σfm ≈ σsc. This has been shown in material systems such as Ga130
xMnxAs/GaAs
and Zn1-xMnxSe/GaAs31 with an injected spin polarization higher than
50%. However in keeping with the CMOS architecture, it would be more beneficial to
continue using metal electrodes as typical semiconducting ferromagnets have Curie
temperatures much less than room temperature. It was proposed by Rashba33 that adding
a tunnel barrier between the injector and the channel could solve the conductivity
mismatch problem. This will work only if the interfacial resistance is greater than the
effective resistances of the injector and channel combined. The spin injection coefficient
is controlled by the largest effective resistance in the FM-T-N-junction. The presence of
the tunnel barrier allows for large difference in the spin-dependent chemical potentials of
the injector and channel. This combined with slow spin relaxation allows for efficient
spin injection. In addition to successful spin injection into semiconductors utilizing
tunnel barriers24-26, spin injection has also been achieved using Schottky barriers27-28.
Once a spin polarization has successfully been injected into the channel, the next
challenge is to detect it. This can also done electrically with a ferromagnetic electrode
although in principle a normal electrode can also be used34. A normal detector will detect
the averaged spin chemical potential whereas a ferromagnet detector will detect
whichever spin species chemical potential the majority spin is. Fig. 1.3 (a) shows a
typical device geometry. To reduce potential artifacts, a four contact geometry is used
10
where the injected current path is kept separate from the spin detection. In this geometry,
no current flows through the detection electrode. This is known as the ‘non-local’
geometry. Work has also been done using a three-contact (local) geometry34, but it has
been shown that the detected signal is not guarenteed to be the electron spins in the
channel rather than a superposition between spin-polarized interface states local to the
detector and spins located in the channel35. Observation of a spin-valve signal (such as
Fig. 1.3 (b)) is often not considered sufficient to prove a spin signal as spurious voltages
from anomalous magnetoresistance (AMR) may also give rise to spin valve like signals36.
The more rigorous proof for spin signal is made by applying a perpendicular magnetic
field (relative to the spin orientation of injected spins) allowing the injected spins to
precess. At high enough field values, the injected spins precess enough to dephase by the
time they reach the detection electrode: suppressing the signal. This is known as the
Hanle effect23,37 and can be observed in Fig. 1.3 (c).
11
Fig. 1.3 (a) Schematic of a lateral spin valve. Current path is on the left and detection
electrodes are on the right. (b) Spin valve signal as measured by a voltage change in
the detection electrodes. The peak corresponds to an anti-parallel orientation. The
bottom part of panel b has a linear background subtracted. (c) Measured voltage of the
spin valve as a function of an out-of-plane applied field. Spin signal dephases at high
enough field due to Hanle effect. Bottom pannel has a quadratic background
subtracted. Source: [38]
12
1.3.2 Spin Hall spin injection/detection
Using spin-orbit coupling, it is possible to create a spin polarization in the absence
of ferromagnetism. Dubbed the spin Hall effect, a spin polarization is created transverse
to an applied electrical current through spin-dependent effects (scattering or band
mechanisms). This can be seen schematically in Fig. 1.4 (a). The spin Hall effect was
first observed in GaAs using optical Kerr detection39 in 2004 although its existence was
predicted40 since the early 1970s. For high Z metals where spin-orbit coupling is strong
(such as Pt), the generated spin current can be as high41 as 7% of the applied charge
current. Devices using the spin Hall effect in a high Z metal to transport spin across an
interface are rare: work done so far focuses on generating spin accumulation through an
applied current in the material under questions (such as HgTe42). However, recent work
has shown successful spin Hall spin injection across a tunnel barrier43 into a second
material.
The spin Hall effect has been more widely used for its reciprocal effect. The
recriprocal effect, labeled the inverse spin Hall effect4 (ISHE), converts a spin current
into a transverse charge current as seen in Fig. 1.4 (b). This charge current should look4
like:
𝒋𝑐 = 𝜃𝑆𝐻 𝒋𝑠 × 𝝈,
(1.3)
where js is the spin current, σ is the polarization direction of the injected spins. The
material dependent spin Hall angle θSH is defined as the ratio of the spin Hall and charge
conductivities and is a measure of the material’s spin-orbit coupling. Useful for detecting
spin currents, this generated charge current leads to a potential build up (under open
13
circuit conditions) and is detected by a voltmeter. Because of the cross-product between
the spin polarization and the spin current, the ISHE detection is insensitive to
magnetization out of the plane (where the spin polarization and spin current into the
attached normal layer are collinear). The ISHE is sensitive to a spin current, not spin
density. Thus it is useful with so called ‘spin batteries’ such as spin injection through
ferromagnetic resonance4,44-45 (FMR) or thermal effects2,3.
Fig. 1.4 (a) An applied charge current is converted into a spin current through the spin Hall effect as
different electron spin species deflect in opposite ways due to spin-orbit coupling. (b) An applied spin
current is converted into a charge current through the inverse spin Hall effect using the same
scattering mechanisms as for the spin Hall effect. Source [46]
14
1.4 Emerging techniques for spin injection
There are two recent spin injection techniques that share an underlying
mechanism: spin Seebeck and resonant spin pumping. Spin Seebeck uses a thermal
gradient across a ferromagnet to generate a spin current while resonant spin pumping
uses microwaves to generate a spin current from a ferromagnet. Both of these effects
excite magnetization dynamics in a ferromagnet and use the dynamics to transfer spin
across an interface into a separate material. An introduction to both effects is provided
here; the fundamental interactions behind these effects are discussed in chapter 2.
1.4.1 Resonant spin injection
In 2002, it was shown that under an applied microwave field a ferromagnet can
transfer a spin polarization into an adjacent layer without an accompanying charge
current47-48. This process is known as ‘spin pumping’ and has been studied extensively
since. It has been postulated that this spin transfer doesn’t suffer the same limitations that
electrical spin injection does with conductivity mismatch.
Provided resonance conditions are met (dependent on applied field strength and
frequency of the microwave excitation), a ferromagnet’s magnetization will precess
around the equilibrium magnetization. The system will damp back to equilibrium if a
microwave field is not present. It is this dissipation that causes angular momentum to
flow in/out of the spin system into the surroundings (electrons and phonons). If a normal
layer is attached, electrons in this layer will participate in the damping process and
acquire a polarization, thus spin transfer occurs from the ferromagnet to the normal layer.
15
The spin transfer is normally detected in metal films using the ISHE effect4. This is
shown schematically in Fig. 1.5.
In addition to metals, this method has been demonstrated to inject a spin
polarization into a semiconductor without a need for a schottky or tunnel barrier in both
GaAs49 and p-doped Si50 although these results are controversial. Very recent work with
epitaxial Fe3Si/GaAs also indicates successful spin injection into a semiconductor51.
Fig. 1.5 (a) The ferromagnetic thin film’s magnetization precesses under application of microwaves
making some cone angle θ with the equilibrium magnetization. (b) Injected spin current from the
precessing magnetization is converted to a voltage from ISHE. Source [4]
16
1.4.2 Thermal spin injection
In addition to microwave excited magnetization dynamics leading to spin transfer,
thermal gradients can also excite the magnetic system and transfer spin. In 2008, the
thermal spin Seebeck effect was discovered2 where an applied thermal gradient collinear
with a ferromagnet’s magnetization excited non-equilibrium magnetization dynamics and
caused a spin current to flow out-of-plane and be detected through ISHE in an adjacent Pt
layer. The thermal excitation is not resonant, unlike the microwave excitation, but
studies indicate that the spin Seebeck effect may be more efficient52 at producing a spin
current, although there are no reports of spin Seebeck spin injection into semiconductors
to date.
Similar to the spin Seebeck effect, the spin-dependent Seebeck effect has been
used to achieve thermal spin injection into materials. The thermal gradient doesn’t
interact with the magnetization directly with this spin-dependent effect, rather spinsplitting of the density-of-states leads to spin-dependent thermoelectric voltages. Using
standard electrochemical detection, thermal spin injection has been used to push a spin
polarization into an attached metal layer53 and into a semiconductor—Si—through a
tunnel barrier54 with perpendicular magnetization and thermal gradient shown
schematically in Fig. 1.6.
17
Fig. 1.6 Schematic of geometry used for thermal spin injection into Si. Changing the thermal flow
direction changes the sign of the injected spin polarization. Source [54]
1.5 Opportunities for additional studies
After the discovery of the spin Seebeck effect, there is a lot of interest in
understanding and harnessing spin-thermal interactions.
One promising route for
studying this physics lies in studying the thermal part of the spin-thermal interactions.
Thermal effects are intrinsically linked with ferromagnetism. For example, it is a wellknown fact that ferromagnetism is temperature dependent. Above the Curie temperature,
electron-electron interactions coordinating the spin alignment are weaker than the thermal
energy. As a result, spin coordination dies and ferromagnetism is lost. Thermal studies
on coherent magnetic excitations (magnons) have shown that thermal conduction is
possible through the magnons and measurable55 provided the phonon-magnon
interactions aren’t too strong56.
Investigating these spin-thermal interactions will require thermally sensitive tools
as the relevant interactions occur within a spin diffusion length of a spin active interface,
18
making study of these spin thermal effects challenging. These tools should be able to
thermally address small samples which allow access to unique magnon modes. This will
be the topic of this thesis: designing/building a system to study small thermal flows
interacting with various spin systems. In Chapter 2 I will give more detailed backstory
on thermal-spin interactions as well as delving into the physical mechanisms behind the
interactions. In Chapter 3 I will detail my work on designing and building isolated
thermal platforms for measuring small thermal effects, as well as testing the platform’s
functionality through heat capacity measurements. In Chapter 4 I will attempt to use
these platforms to study magnetization dynamics under the application of microwaves
and thermal dissipation arising from decaying magnons.
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23
Chapter 2 : Dynamically-driven spin physics
2.1 Introduction
Ferromagnetic resonance (FMR) driven spin injection and spin Seebeck share a
common mechanism: magnetic excitations coupling with conduction electrons and
decaying causing an angular momentum transfer. The difference between the two spin
injection methods lies in the manner with which the magnetic excitations are excited:
microwaves and thermal gradients. This chapter will discuss the mechanisms which
drive the magnetic system and how the resulting magnetic excitations interact with
environment. By studying the mechanisms by which the magnetic system interacts with
the environment, parameters can be identified to engineer material systems and device
geometries for enhancing the spin injection for applications.
2.2 Thermally driven spin dynamics
Using thermal gradients to drive a system out of equilibrium is not a new concept.
Thermal gradients have been used to manipulate electrons since the 1800’s with the
discovery of the Seebeck effect.
Under the application of a thermal gradient, an
electromotive force is generated through a temperature dependence1 in the density of
states at the Fermi level. The direction of this electromotive force (parallel or anti24
parallel to the applied temperature gradient) depends on the polarity of the charge carriers
in the material. The charge Seebeck effect has been widely used for temperature sensing
(thermocouples) and in industry for thermoelectric generators.
Through the interaction with electrons, thermal gradients also interact with
applied magnetic fields. As the charge Seebeck causes carriers to move parallel (antiparallel) to the applied temperature gradient, an out-of-plane magnetic field will cause a
charge accumulation transverse to the applied temperature gradient due to deflection of
the charge carriers. The Nernst effect is the thermal analogue to the Hall effect with a
temperature gradient setting up the electric field rather than an applied voltage. Through
spin-dependent scattering, this will also lead to an anomalous Nernst effect (ANE) in the
presence of a permanent magnetization (analogue to the anomalous Hall effect (AHE)).
Thus thermal gradients can interact with magnetization via the interaction with electrons.
It was only recently found that thermal gradients interact with the magnetization
directly without the need for electron mediation. In 2008, a spin analogue to the Seebeck
effect was discovered2 in the metallic ferromagnet Ni81Fe19 (permalloy). A temperature
gradient was applied to a ferromagnetic thin film sample, and an out-of-plane spin current
was detected in an attached Pt film through the ISHE voltage. This out-of-plane spin
current was also detected in insulators21 and semiconductors22. Its discovery sparked
interest in a variety of spin-thermal effects collectively3 known as ‘spin caloritronics’ and
the role of the interactions between electrons, magnetic excitations (magnons), and
thermal excitations (phonons) in transport. Therefore, efforts at understanding these
interactions lie at the core of this developing field. Before diving into these interactions,
25
however, we need to first understand the elementary magnetization excitations
(magnons).
2.2.1 Magnons
In order understand magnons, we need to first write down the energy of the spin
system. The 1-D Heisenberg model says that the internal energy should look like:
𝑈 = −2𝐽 ∑𝑁
𝑝=1 𝑺𝑝 ∙ 𝑺𝑝+1 ,
(2.1)
where J is the exchange constant (material dependent), and Sp is the spin at site p.
Minimizing this energy depends on the sign of J. For positive J, the energetically
favorable ordering is for the neighboring spins to align parallel (ferromagnetism). For
negative J, the energetically favorable ordering is for neighboring spins to align antiparallel (anti-ferromagnetism).
Fig. 2.1 Energy efficient magnetization excitations, magnons are spins oscillating with a well-defined
phase. Source [70]
26
At finite temperatures there are thermal fluctuations to the spin alignment. The
energy cost of the lowest excited state (one spin anti-parallel to its neighbors) is 8JS2
more than the ground state from (2.1). However, this energy cost can be decreased if the
spins are allowed to tilt with respect to their neighbor and spread the single spin flip
across the tilting of many spins. This can be seen in Figure 2.1. These spin excitations
are wave-like in nature and called magnons.
In a ferromagnet provided the spin
excitations are small enough, a magnon dispersion relation can be derived4:
ℏ𝜔𝑘 = 4𝐽𝑆(1 − cos 𝑘𝑎),
(2.2)
where k is the reciprocal momentum and a is the lattice constant. At long wavelengths
(such that ka << 1)
ℏ𝜔𝑘 ~2𝐽𝑆𝑎2 𝑘 2 .
(2.3)
This is different from the long wavelength acoustic phonon dispersion (ℏ𝜔𝑘 ~𝑘).
Since magnons are excitations and have an associated momentum vector, they
also transport and store energy similar to phonons. It is possible to measure the magnon
portion of the thermal conductivity and heat capacity by applying a large magnetic field.
By applying a field, a Zeeman energy cost is introduced causing the spins to align with
the field direction. This can be represented in the dispersion relations by:
ℏ𝜔𝑘 = 𝜇𝐵 𝑔𝐻 + 2𝐽𝑆𝑎2 𝑘 2 ,
(2.4)
where μB is the Bohr magneton, g is the gyromagnetic ratio and H is the applied field. As
the strength of H is increased, a gap opens up in the magnon spectra. At high enough
fields, the magnons ‘freeze out’ or don’t have enough thermal energy to pay the Zeeman
cost.
However, the complete suppression of magnons is only possible at low
27
temperatures due to the disparity of the energy scales (𝑘𝐵 𝑇~25 𝑚𝑒𝑉 at RT and
𝜇𝐵 𝑔𝐻~100 𝜇𝑒𝑉 at 1 Tesla).
One such measurement5 of the magnon thermal
conductivity of the Heisenberg magnet EuO is shown in Figure 2.2. Around 1 K, the
magnon contribution is around 75% of the total thermal conductivity.
Fig. 2.2 Normalized thermal conductivity as a function of field. Saturation occurs once the magnetic
field is high enough to completely suppress the magnon contribution. Source [5]
28
2.2.2 Magnon-phonon interaction
In the absence of complete suppression of magnons, it becomes difficult to
measure the magnon contribution to the thermal conductivity.
The straightforward
relation of:
𝜅𝑇 = 𝜅𝑝 + 𝜅𝑚 ,
(2.5)
the total thermal conductivity (κT) is the sum of the phonon thermal conductivity (κp) and
the magnon thermal conductivity (κm) does not hold true at intermediate field values due
to magnon-phonon coupling. From the magnon dispersion relations such as (2.4), we see
that the magnon modes are dependent on the spacing between atoms (the lattice constant
a) as well as the exchange coupling constant J (these parameters are not independent as J
is also dependent on the lattice constant). As phonons are lattice excitations which
locally change the lattice constant, magnons and phonons will interfere with each other
and scatter. The simplest of these collisions is where the magnon and phonon dispersion
relations cross as energy and momentum can be conserved with one magnon-one phonon
scattering. Near this crossing, the phonons and magnons will hybridize6 forming so
called ‘magnetoelastic modes’ and a gap will form in the dispersion curve. This has been
observed7 in YMnO3 at 1.5 K by neutron scattering.
Higher order scattering events can also occur such as a one phonon-two magnon
processes8 among others. One way to parametrize all the magnon-phonon scattering
events is the magnon-phonon relaxation time, τmp. It is defined8 by:
𝑑
𝑑𝑡
Δ𝑇 = − 𝜏
29
1
𝑚𝑝
Δ𝑇,
(2.6)
where ΔT = Tp-Tm is the difference in the phonon and magnon temperatures. Since both
magnons and phonons follow Bose-Einstein statistics, the temperature of each system can
be defined by the average occupancy (nk) of each mode, ωk. The average occupancy is
defined through:
⟨𝑛𝑘 ⟩ =
1
,
(2.7)
ℏ𝜔
exp( 𝑘 )−1
𝑘𝐵 𝑇
where kBT is the thermal energy. Intuitively (2.6) makes sense as the time it takes for the
magnon system to thermalize with the phonon system depends on how far they are from
𝑑
equilibrium (ΔT) and the rate at which they approach equilibrium ( 𝑑𝑡 Δ𝑇). This time has
been measured through spin resonance techniques in YIG at low temperatures9 and found
to be 1.5x10-6 s. This is considered to be relatively weak magnon-phonon coupling. In
general, the shorter τmp is the stronger the magnon-phonon coupling. As τmp → ∞, there
is no magnon-phonon coupling.
2.2.3 Electron-phonon coupling (spin-phonon coupling)
In addition to magnons, phonons can interact with conduction electrons carrying a
spin degree of freedom. A phonon will distort the lattice from the ‘official’ position
causing the local wave function of an electron to change which makes the electron liable
to be scattered from its previous course. For this scattering to take place, the total wave
vector must not change (except by a reciprocal lattice vector) and a phonon must be
either emitted or absorbed10. If the electron-phonon coupling is sufficiently large enough,
electrons and phonons can hybridize in a similar fashion to the magnetoelastic modes
30
discussed for magnons and phonons. These modes are called polarons for electrons and
phonons11.
In the case of an applied temperature gradient, strong phonon-electron
interactions are often referred4 to as ‘phonon drag.’ Phonons preferentially move antiparallel to the thermal gradient thus phonon-electron scattering events impart a net
momentum to the electrons in the direction of the thermal gradient. This is seen in
enhanced thermoelectric effects in the temperature regime where phonon-electron
scattering becomes dominant12.
This also applies to spin-polarized electrons where
phonon-electron scattering will directly affect the spin polarization. The giant spinSeebeck effect13 observed in InSb at high magnetic fields and low temperatures in order
to fully spin polarize the electrons is a direct result of this phonon drag effect.
2.2.4 Magnon-electron coupling
Magnons also interact with conduction electrons directly without the need for
phonon mediation.
Conduction electrons (in s character bands) can interact with
stationary electrons (in localized d orbitals) through s-d exchange40. This interaction can
cause an electron to undergo spin-flip scattering, where it scatters from the localized spin
while exchanging its spin orientation with the localized spin. It is through this interaction
that magnons can interact with conduction electrons since magnons are comprised of
these localized spins precessing coherently.
This spin-flip exchange between conduction electrons and localized electrons can
be thought of as an effective torque acting on the respective spin polarizations. In the
31
case of a metal attached to a ferromagnet, this torque has been modeled71 using an
interfacial spin mixing conductance, g↑↓. Although initially g↑↓ was developed72 using a
two channel model for conductance (one for spin up, one for spin down), the spin mixing
conductance exists even when the non-mixing conductances through an interface vanish
(i.e. valid for insulator/metal interfaces).
The main consequence of the magnon-electron coupling is a spin current
perpendicular to an interface occurs when the magnon population becomes excited out of
equilibrium. This can be seen schematically in Fig. 2.3 where an excited magnetization
damps to equilibrium through an exchange with the nearby conduction electrons. The
efficiency of the magnetization damping depends on g↑↓.
Fig. 2.3 As the magnetization damps back to equilibrium, angular momentum is transferred between
the electron and magnon system. A spin diffusion current carries angular momentum across the
interface into the normal metal.
This effect is known as spin pumping.
The interface is not
transparent to the spin current (symbolized by the series resistance shown), and the interfacial spin
resistance is captured in a phenomenological spin mixing conductance parameter g↑↓.
32
Fig 2.4 (a) Spin current injection as measured by ISHE in a Pt film by three different methods: spin
pumping by FMR, spin Seebeck, spin Hall magnetoresistance. (b) Spin mixing conductance as a
function of the Pt detection layer thickness. The spin mixing conductance is independent of the source
used to push spin from the YIG source into the Pt layer. Source [Ref. 57]
33
Interestingly, the spin mixing conductance g↑↓ seems to be independent of the
process driving the magnetization out of equilibrium, as seen in Fig. 2.4 in work done by
Weiler et al57 where three different methods were used in exciting the magnetizations
dynamics that are driving the spin current across an interface: spin pumping from FMR,
spin Seebeck, and spin Hall magnetoresistance.
Having now discussed all of the relevant fundamental interactions between
electrons, magnons, and phonons for understanding dynamic spin-injection (thermal or
resonant), specific examples of magnon-phonon interactions will be discussed in the
following sections.
2.2.5 Magnetocaloric effect
One example of interactions between a spin system and phonons is the
magnetocaloric effect (MCE) first discovered14 in the ferromagnet Fe. As a magnetic
field is applied to a spin system (either a ferromagnet or paramagnet at low enough
temperatures), the system undergoes an adiabatic temperature change (ΔTad) due to the
suppression of the spin entropy (aligning spins with field direction) which leads to an
enhancement of the phonon entropy from the 2nd law of thermodynamics. Although
implied with the term adiabatic, the MCE is an equilibrium effect which relies on the
interactions between the spin system and the phonons (either electron-phonon or
magnon-phonon interactions depending on the spin system) in order to transfer entropy.
The entropy transfer is illustrated in Fig. 2.5 where either the MCE can proceed
adiabatically and result in a temperature change, or isothermally and transfer entropy out
34
of the system. Should the MCE proceed adiabatically (without a thermal link to the
surroundings), the expected temperature change can be derived from Maxwell’s
relations15:
𝐻
𝑇
𝜕𝑀(𝑇,𝐻)
Δ𝑇𝑎𝑑 (𝑇, Δ𝐻) = − ∫𝐻 2 (𝐶(𝑇,𝐻)) (
1
𝐻
𝜕𝑇
) 𝑑𝐻,
𝐻
(2.8)
where C is the specific heat, M is the magnetization. From this relation, it is clear that a
strong temperature dependence of M is desirable to maximize ΔTad. For this reason,
ferromagnets are chosen near the Curie temperature as the M varies strongest with
temperature in this regime such as Gd5(Si2Ge2)16. Paramagnets can also be used at low
temperatures due to low C, and high magnetic susceptibility. With programmable control
over thermal coupling of the system to the surroundings, a refrigeration cycle can be
established by isothermally applying magnetic field followed by adiabatic relaxation back
to zero field. Using such techniques, paramagnetic salts have been cooled below helium
temperatures17-19.
35
Fig 2.5 Entropy vs temperature offset from Tc. The colored curves represent S(T,H) for different
values of applied field. Adiabatically applying field leads to a temperature increase moving to
S(T,H1). Source [20]
2.2.6 Spin Seebeck effect
Magnon-phonon interactions also play a dominant role in the spin Seebeck effect
where a temperature gradient is applied to a ferromagnet and a spin current is observed in
an attached normal metal through the inverse spin Hall effect (ISHE). Though spin
Seebeck was first discovered in a metal2, it has since been observed in a variety of
magnetic material systems such as insulators21 and semiconductors22.
36
There are two main geometries for studying the spin Seebeck effect: longitudinal
where the thermal gradient is perpendicular to the magnetization, and transverse where
the thermal gradient is parallel to the magnetization. The longitudinal geometry is only
usable with magnetic insulators as it becomes impossible to separate out the spindependent ISHE voltage from the anomalous Nernst effect23 (ANE) for conducting
materials. The transverse geometry has to be implemented with care as out-of-plane
thermal gradients will always be present24-27 leading to the aforementioned ANE effect.
One of the defining characteristics of the transverse spin Seebeck effect is the sign
reversal across the center of the sample2.
Through positional dependence, the spin
voltage can be resolved28 from thermal voltages arising from ANE.
Fig. 2.6 Transverse geometry shown here for GaMnAs. Temperature gradient is applied along the
sample and the magnetization is collinear with the thermal gradient. Voltage is measured across the
attached Pt strips. Source [22]
37
For the following discussion, we will focus on the transverse geometry shown in
Fig. 2.6. Applying a temperature gradient is done by directly coupling with the phonon
temperature at the ends of the sample. The magnon temperature, however, doesn’t
observe the same boundary conditions as the phonon temperature. Instead the magnon
temperature establishes a steady-state non-equilibrium with the phonon temperature
through magnon-phonon coupling. Since the magnons also conduct heat, the magnon
temperature is not necessarily the same as the phonon temperature. Sanders and Walton
derived an expression29 for the magnon temperature as a function of position with an
applied thermal gradient across a sample of length L thermally anchored at ends ±L/2:
𝑗
𝑇𝑚 (𝑥) = 𝑇0 − 𝜅𝑄 (𝑥 −
𝑇
𝑠𝑖𝑛ℎ𝐴𝑥
𝐴𝑐𝑜𝑠ℎ
𝐴𝐿
2
),
(2.9)
where jQ is the applied heat current, κT is the total thermal conductivity, and T0 is the
temperature at the middle of the sample where Tm=Tp from symmetry arguments. The
parameter A is defined as:
𝐴=(
𝐶𝑝 𝐶𝑚 𝜅𝑇
1
𝐶𝑇 𝜅𝑝 𝜅𝑚 𝜏𝑚𝑝
)
1⁄
2,
(2.10)
with C being the specific heats of the respective systems (phonon, magnons, and total
specific heat), κ is the thermal conductivity of the respective systems, and τmp is the
magnon-phonon relaxation time constant. The phonon and magnon temperatures are
plotted as a function of sample position in Fig. 2.7 (a). For large magnon-phonon
coupling (small τmp), the magnon temperature will mirror the phonon temperature. For
no coupling (large τmp), the magnon temperature will be flat at T0.
38
Fig. 2.7 (a) Phonon and magnon temperatures are plotted as a function of sample position. (b) The difference
between the two temperature profiles is shown. (c) Magnetization decreases at the hot end and increases at the
cold end shown by the reduction in cone angle. (d) Hysteresis loops show the detected voltage in the Pt strips.
The voltage reverses with the magnetization and changes sign across the sample. Source [30]
39
The change in sign of the spin current across the sample can be readily seen from
Fig. 2.7 (b) which shows the temperature difference between the magnon and phonon
populations. At the hot side of the sample, the magnons are being heated by the phonons
causing the magnetization to decrease (c) as more magnons are present.
Angular
momentum is conserved ensuring that the reduction in the magnetization is accompanied
by a transfer of angular momentum out of the magnon system. The presence of the
attached normal metal allows this transfer of angular momentum to take place as a spin
current carried by conduction electrons. Likewise, at the cold end of the sample the
phonons are cooling the magnons causing an increase in the magnetization.
The
subsequent transfer of angular momentum is opposite in sign leading to the voltage
switch shown in (d). At the center of the sample, the magnon and phonon temperatures
are equal and the spin current is zero.
2.2.7 Other spin thermal effects
In addition to the spin Seebeck effect, there is also the spin-dependent Seebeck
effect31 in which electron-phonon coupling plays a dominant role not magnon-phonon
coupling. Spin-dependent Seebeck is where a thermal gradient is applied to a metal or
semiconductor and spin-polarized charge carriers move in the direction of the thermal
gradient forming a spin accumulation at the ends of the sample. Using this effect, a spin
polarization has been injected into an attached metal32 or semiconductor33.
40
While spin dynamics can be thermally driven through magnon-phonon or spinpolarized electron-phonon coupling, the opposite is also true: spin dynamics can be used
to apply thermal gradients through the reciprocal spin Peltier effect.
As electrons
(magnons) carry thermal energy, a charge (magnon) current will convey heat4 as well as
charge. At a junction of two separate materials with different abilities to convey heat per
charge (Peltier coefficient), a thermal buildup will occur (or reduction depending on the
current direction). Peltier cooling has been observed with a spin-polarized current—spindependent Peltier34—as well as with a magnon current—spin Peltier35. The reciprocity is
illustrated for spin Peltier/spin Seebeck in Fig. 2.8.
41
Fig. 2.8 Top: Spin Peltier effect. An electron flux in the normal metal drives the spin Hall polarization
accumulation at the ferromagnet interface which excites magnons. The magnons then propagate away
from the interface carrying heat away eventually decaying into phonons. Bottom: Spin Seebeck effect.
A phonon flux in the ferromagnet interacts with the magnons which drive a spin accumulation at the
normal interface. Spins diffusing from the interface drive the inverse spin Hall charge accumulation.
Source [36]
42
2.3 Resonant spin dynamics
Resonant spin dynamics refers to coherently driving the spin system with a
narrow range of frequencies as opposed to the incoherent broadband frequency
excitations used with thermal excitation methods. One of the primary ways of driving
resonances in spin systems is through applied microwave fields. Spin systems that can be
driven in this fashion include nuclear magnetic resonance37 (NMR), electron spin
resonance38 (ESR), and ferromagnetic resonance39 (FMR). For the purposes of this
dissertation, we will focus on FMR.
2.3.1 Magnetization dynamics
FMR was first discovered in 1912 by V.K. Arkad’yev41, although it wasn’t until
the mid-1940s when it was re-discovered and explained42-44. Conventional FMR is where
a sample with a permanent magnetization is placed in a static magnetic field. An AC
field component transverse to the static magnetization is applied to torque the
magnetization out of equilibrium as shown in Fig. 2.9. If the frequency of the AC field
matches the natural frequency of the magnetic system, the magnetic moment of the
system will precess around the equilibrium magnetization direction. This precession is
often called the uniform mode, and is equivalent to exciting magnons with a wave vector
k = 0.
The magnitude of the precession, measured by the angle the rotating
magnetization makes with the static position (called the cone angle), will depend on the
magnitude of the applied AC field. In addition to the resonant uniform mode, additional
43
magnetization dynamics can be excited in a ferromagnetic system such as magnetostatic
dipolar-dominated surface modes45 labeled ‘Damon-Eshbach modes’.
Z
𝑴0
𝑴(𝑡)
Heff
θ
Y
X
Fig. 2.9 An effective field is applied along the z-axis parallel to the static magnetization vector, M0.
The time dependent magnetization vector M precesses around M0 with an angle θ.
FMR has been used as a spectroscopic technique to study magnetization dynamics
in various material systems. Information can be gleaned from the linewidth of the
resonance about the damping mechanisms of the material such as magnon-phonon46
interactions, magnon-magnon47 interactions etc. More recently however, it has been used
44
as a tool to push a non-equilibrium spin polarization into adjacent non-magnetic layers4850
(spin pumping).
In order to understand how the magnetic systems evolve under microwave
excitation, the phenomenological Landau-Lifshitz-Gilbert equation is used to predict
magnetization precession and relaxation following a disturbance from equilibrium. At
equilibrium, there will be no torque exerted on the magnetization:
(2.11)
𝑴 × 𝑯𝑒𝑓𝑓 = 0.
Heff is the effective magnetic field51 experienced by the magnetization and contains
several contributions:
𝑯𝑒𝑓𝑓 = 𝑯𝑎 + 𝑯𝑀 + 𝑯𝐴𝑁 + 𝑯𝐸𝑋 ,
(2.12)
where Ha is the applied field, HM is the field due to the magnetization, HAN is the
anisotropy field and HEX is the exchange field.
When the system is not in equilibrium, M × Heff ≠ 0, it will have some time
dependent evolution. According to Landau and Lifshitz52 this time dependence should
look like:
𝜕𝑴
𝜕𝑡
= −𝛾𝐿 𝑴 × 𝑯𝑒𝑓𝑓 −
𝛼𝛾𝐿
𝑀𝑆
𝑴 × (𝑴 × 𝑯𝑒𝑓𝑓 ),
(2.13)
the first term is the magnetization precession where γL is a gyromagnetic ratio-esque
constant which is different from γ and sets the precession rate, and the second term is the
dissipation driving the system back to equilibrium characterized by α (a damping
parameter) which encompasses anything that is dissipative such as magnon-phonon
interactions, magnon-magnon interactions or conduction electron-magnon interactions.
Gilbert53 also wrote an equation for the magnetization dynamics:
45
𝜕𝑴
𝜕𝑡
𝛼
= −𝛾𝐺 𝑴 × 𝑯𝑒𝑓𝑓 + 𝑀 𝑴 ×
𝜕𝑴
𝑆
𝜕𝑡
.
(2.14)
This is called the Landau-Lifshitz-Gilbert (LLG) equation and equivalent to (2.13)
provided that:
𝛾𝐺
𝛾𝐿
= 1 + 𝛼2.
(2.15)
In an applied field, the damping parameter α is often called the Gilbert damping
parameter. With this damping term in place the magnetization will decay to equilibrium
in a spiral fashion as shown in Fig. 2.10.
Fig. 2.10 The time dependent magnetization under a damping torque will relax back to the equilibrium
value M0 in a spiral fashion as each rotation has a smaller cone angle.
46
2.3.2 Resonance condition
When an AC field is applied in such a way as to equal the dampening torque from
whatever dissipative processes are present, the magnetization will continue to precess (as
in Fig 2.9).
The frequency of the applied field must meet certain conditions for
continuous precession which depend upon the Heff that the magnetization experiences.
Since we will work on thin films, the HAN component of Heff will be made explicit in the
following discussion through the introduction of demagnetization factors Nx, Ny, Nz.
These demagnetization factors capture the shape anisotropy of the sample and obey the
relationship Nx+Ny+Nz = 1 (in SI units, 4π in CGS units). Kittel4 shows that for the
applied field, Ha, (in the z direction for derivation purposes, choice of applied field
directions will be discussed shortly) the resonance frequency will be (in SI units):
𝜔02 = 𝛾 2 [𝐻𝑒𝑓𝑓 + (𝑁𝑦 − 𝑁𝑧 )𝜇0 𝑀𝑠 ][𝐻𝑒𝑓𝑓 + (𝑁𝑥 − 𝑁𝑧 )𝜇0 𝑀𝑠 ].
(2.16)
The frequency ω0 is the uniform mode frequency, when all the spins in the ferromagnet
precess in phase and with the same amplitude. In the case of a sphere where N x=Ny=Nz,
(2.16) reduces to ω0=γHeff. For thin films, there are two cases worth considering: applied
field out-of-plane (OOP) and in-plane (IP). For OOP geometry, Nx = Ny = 0 and Nz = 1.
Then (2.16) reduces to:
𝜔0 = 𝛾(𝐻𝑒𝑓𝑓 − 𝜇0 𝑀𝑠 ).
(2.17)
For IP geometry, Nz = Nx = 0 and Ny = 1 (one could also choose Ny = 0, Nx = 1):
𝜔0 = 𝛾√𝐻𝑒𝑓𝑓 (𝐻𝑒𝑓𝑓 + 𝜇0 𝑀𝑠 ).
47
(2.18)
As can be seen in (2.17) and (2.18) that for a given resonance frequency ω0, the applied
field will need to be greater out-of-plane to compensate for the shape anisotropy (the
magnetization has lower energy if oriented in-plane).
2.3.3 Non-equilibrium spin transfer (spin pumping) into adjacent films
It was seen that sandwiching a thin ferromagnetic film between two nonferromagnetic metal layers will increase the Gilbert damping54 due to increased electronmagnon interactions. This was measured by an increase in the linewidth of the resonance
since the linewidth is proportional55 to the Gilbert damping by:
∆𝐻 ∝ 𝛼
𝜔0
𝛾
.
(2.19)
Since angular momentum is conserved, the transfer of energy out of the magnetic system
must also be accompanied by a transfer of angular momentum48 (a spin current). This
transfer of angular momentum to the normal layer does not have an associated charge
current, and is thus an example of a pure spin current being generated across the
interface.
The efficiency of the injected spin current is not well understood and is very
dependent on the interface56 between the ferromagnet and normal layer. This is seen in:
1
𝐽𝑠 = 2 𝑔↑↓ ℎ𝜔𝑃𝑠𝑖𝑛2 𝜃,
(2.20)
where g↑↓ is the spin mixing conductance, h is Plank’s constant, ω is the driving
frequency, P is a correction factor for the ellipsoid precession of the magnetization (due
to anisotropy effects) and θ is the cone angle of the precession. The uncertainty lies in
g↑↓ which is not well understood and mostly used as a phenomenological parameter.
48
2.4 Thermal interactions in resonant spin dynamics
There isn’t a clear dividing line between thermally excited spin dynamics and
resonantly excited spin dynamics. Often there is a crossover: thermal interactions in
resonance or using resonant thermal interactions. For example, the presence of magnonphonon coupling included in the Gilbert damping term ensures that magnon decay results
in increased temperature of the phonon bath. We will examine two of these cases.
2.4.1 Resonant phonon driven spin dynamics
Using a piezoelectric material, it is possible to create phonons at a particular
frequency (a harmonic of the drive frequency). Uchida et al showed58 that using these
frequency selective phonons, a spin current could be driven into a Pt layer from a
ferromagnetic YIG film. This spin current was measured by ISHE and shown in Fig.
2.11 (a) as a function of frequency. Careful monitoring of the temperature of the sample
showed no thermal gradient present at the resonance frequency (Fig. 2.11 (b)).
Further measurements done59 by Weiler et al on a Co/Pt system show that the
ISHE signal can be contaminated by microwave pickup from the piezoelectric acoustic
phonon generator. By doing time resolved measurements, they were able to separate the
electromagnetic wave (EMW) and spin current contributions due to the different
velocities of propagation of the two wave types (EMW vs acoustic phonons). The
acoustic phonons arrived later due to the speed of sound of the substrate being much
smaller than the speed of light.
49
Both of these experiments demonstrated resonant phonon-magnon coupling as the
narrow bandwidth of the generated acoustic phonons drives the magnetization dynamics
which in turn cause a spin current to flow into the detector.
Fig. 2.11 Top: ISHE voltage measured in a Pt strip as a function of the applied acoustic phonon
frequency. Inset: zero test with OOP geometry for Pt and IP geometry for Cu. Both signals should be
zero if originate from spin. Bottom: temperature measured as a function of frequency. Source [58]
50
2.4.2 Thermal detection of magnetization dynamics
On resonance, the spin system dissipates energy via magnon-phonon coupling
which leads to uniform heating.
Therefore, the resonance can be detected through
changes in the sample’s temperature as first demonstrated60 by J. Schmidt et al in 1966.
This detection method has been applied using techniques such as scanning thermal
microscope-detected FMR (SThM-FMR)61-64, bolometry utilizing cantilevers65, and on
free-standing membranes66. It is possible to thermally detect FMR without thermally
engineered devices67 such as on bulk Si as shown in Fig. 2.12 (a), although this inhibits
the sensitivity. Thermal detection of magnetization dynamics beyond the uniform mode
resonance has been demonstrated as thermal detection of magnetostatic modes66 as
shown in Fig 2.12 (b) or thermal detection through the anomalous Nernst effect (ANE) of
spin waves in a Py waveguide68.
Thermal detection of FMR offers some advantages over conventional inductive
detection methods (where the output microwave power is monitored and dips are seen in
absorbed power). It is more sensitive than cavity FMR, with a claimed sensitivity of 109
spins by Ref. 66 (compared to 1012 spins sensitivity of cavity detected FMR). Thermal
detection is not as sensitive as magnetic resonance force microscopy (MRFM) which has
detected a single spin in resonance69.
Electrically detected FMR (ED-FMR) offers
comparable sensitivity to thermal detection, but is prone to anomalous magnetoresistance
(AMR) effects which have to be resolved through angular dependence55. This artifact
scales with resistance, therefore larger samples will have larger AMR voltages. Thermal
detection also has artifacts such as off resonant heating by eddy currents 65, but are easier
51
to separate than AMR artifacts in ED-FMR. Furthermore, thermal detection can address
electrically insulating samples where ED-FMR struggles.
Fig. 2.12 Top: FMR resonance detected thermally through the use of a thermocouple of a Py strip on a
bulk Si substrate. The two peaks seen per frequency is a result of frequency modulation of 5 GHz.
Source [Ref 67] Bottom: uniform and magnetostatic modes detected by a thermocouple on a SiN
membrane. Source [Ref 66]
52
2.4.3 Need for sensitive tools
Moving beyond resonant magnetization dynamics, the presence of magnonphonon coupling ensures the relevance of thermal detection in the field of spin
caloritronics. Using thermal detection, spin-dependent Peltier34 and spin Peltier35 were
discovered.
Phonon temperature profiles were compared with magnon temperature
profiles directly through thermal imaging and Brillion light scattering (BLS) by Agrawal
et al73 and cut off frequencies for magnons responsible for the magnon-phonon
interactions were established for spin Seebeck. Many advances to date in the spin
caloritronics field relied on the ability to reliably and sensitively thermally address the
system under consideration.
For thermometry challenges, we again address work done on resonant thermal
detection to highlight potential improvements. The detected thermal signal at resonance
in Fig. 2.12 is only 2.4 mK. The lowest order magnetostatic mode is near the detection
limit of ~300 μK for the experimental geometry used. To further study these effects,
more sensitive tools are needed. Increasing thermal sensitivity can be approached two
ways: use of a tailored thermocouple that exhibits maximum sensitivity for the
temperature range studied, or decreasing the thermal link to the surroundings to
maximize the system’s thermal response. Maximizing thermocouple composition could
yield about 6x improvement while thermal engineering of the surroundings could get
roughly 10x improvement of the thermal response on resonance from Fig. 2.12.
The plethora of thermal-spin effects mentioned in this chapter suggest that
flexibility is also needed in the tool design. Different effects need different geometries to
53
study them. For example, spin Seebeck requires a temperature gradient—one that could
be made conventionally large (>50 K) by thermal engineering such as work74 done by
Avery et al. This dissertation will focus developing flexible thermally isolated platforms
designed to study a variety of spin thermal effects. Chapter 3 will explore tool building
in detail in addition to benchmarks on the tool’s capabilities. Chapter 4 will be a first
pass at applying the tool towards spin dissipation of a ferromagnet on resonance similar
to the works presented here. Ultimately, we hope to extend thermal detection of not only
resonance, but spin pumping across an interface and begin to disentangle magnonelectron interactions through thermal studies.
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59
Chapter 3 : Developing customized experimental probes
3.1 Need for thermally isolated platforms
Measuring the thermodynamic properties of nanoscale structures is becoming
increasingly important as heterostructures and devices shrink in size. For example, recent
discoveries of spin thermal effects such as spin Seebeck1 and spin Peltier2 show that
thermal gradients can manipulate spin systems and vice versa. However, the relevant
interactions occur within a spin diffusion length of a spin active interface, making study
of these spin thermal effects challenging. In addition, recent ferromagnetic resonance
studies of spatially confined nanomagnets have shown unique magnon modes in arrays3
and lines4 which may give rise to unique magnon-phonon interactions. In this case, the
small volume of magnetic material presents a challenge to measurement and as a result
the bulk of the work is done on arrays3 with measurements of the magnetization of
individual particles possible through various microscopies4,5 but limited access to thermal
properties. As a result, tools capable of measuring the thermal properties of nanoscale
structures are required to fully explore this emerging science. One approach to addressing
this challenge is the use of microscale suspended platforms that maximize their
sensitivity to these spin thermal interactions through thermal isolation from their
surroundings. Combining this thermal decoupling with sensitive thermometry allows for
60
the measurement of nanojoule heat accumulations, such as those resulting from the small
heat flows associated with spin transport and spin relaxation. As these heat flows may
manifest themselves in a variety of spin-thermal effects, the development of measurement
platforms that can be tailored to optimize their sensitivity to specific thermal
measurements is essential. We select heat capacity as the first measurement to optimize
due to a combination of its broad utility (such as yielding information about the density
of states of electrons and phonons, as well as phase transitions, etc6-9.) and its ability to
provide meaningful metrics in the evaluation and optimization of our platform design.
Here we present the measurement of the heat capacity of a 6.2 ng Au sample using a
microscale suspended SiNx platform. Our results match closely with the values obtained
for bulk samples10 despite a 1012 difference in mass.
3.2 Constraints on platform design
A key limitation to measuring the heat capacity of small samples is the residual
heat capacity of the empty calorimeter, sometimes referred to as addenda. Since heat
capacity is an extensive property, the calorimeter background must itself be on the order
of tens of nanojoules per kelvin to measure nanojoule heat accumulations. In an effort to
achieve this goal, recent calorimeters11-17 have moved toward suspended membrane
geometries to reduce the calorimeter background, achieving addenda as low as 55 nJ/K
(Ref. 12) at a temperature of 250 K.
A second constraint arises from the coupling of the calorimeter to the heat bath.
Ideally this coupling should be small relative to the internal equilibration in order to
61
ensure a uniform temperature distribution. Traditionally, membrane calorimeters have
relied on the inclusion of a metallization layer to ensure thermal uniformity; however,
this approach adds mass to the calorimeter, leading to addenda in the tens to hundreds of
nanojoules per kelvin and limiting sample masses to micrograms and above11-14. An
alternate approach relies on perforating the membrane to reduce thermal coupling without
increasing the addenda15-19. This is the same approach used in platforms optimized for
thermal transport20-22 with separate platforms containing heater and thermometer
respectively to allow for thermal transport across a sample spanning the two.
Following this approach, the structures reported here rely on a flexible focused
ion
beam
(FIB)
nanomanufacturing
technique
to
fabricate
two
suspended
100 µm × 100 µm square platforms (one active and one reference calorimeter, Fig. 3.1)
from a commercial SiNx membrane. Without a thermal conduction layer the dominant
background comes from the SiNx platform itself, significantly reducing the addenda. The
addenda can be further reduced by scaling down the size of the platform. This scaling
also reduces the internal thermal equilibration time of the finished calorimeter, improving
the effective isolation from the bath. Conversely, this reduced equilibration time makes
measurements of thermal transients more challenging. The 100 µm × 100 µm square
platform size used here represents a compromise between these factors.
Platforms
optimized for thermal transport do not need to measure thermal transients and therefore
can scale down the platform size without regard for instrument limitations.
62
3.3 Fabrication of platforms
The calorimeters presented here consist of two platforms, one active and one
reference, fabricated simultaneously using FIB to avoid discrepancies in the addenda
(Fig. 3.1). Platinum wiring and thermometry are defined by photolithography followed by
metal deposition and liftoff. A Au film is similarly patterned and deposited on one of the
platforms to define the sample under study. Details of this fabrication process follow
below.
Amorphous SiNx is chosen as the membrane material for its excellent thermal
properties – low thermal conductivity23 (0.8 W/cmK) and low specific heat23 (0.4 J/gK) –
as well as high mechanical strength24 (Young’s modulus of 280 GPa). Transmission
electron microscopy (TEM) membranes are purchased from Ted Pella, Inc. and are
composed of a 200 nm film of low-stress, silicon-rich SiNx (Si:N ratio varying from
1.04:1 to 1.16:1) deposited on a Si wafer via low-pressure chemical vapor deposition
(LPCVD). The membranes are produced by back-etching a 750 µm × 750 µm window
centered within a circular Si support-frame 3 mm in diameter. In preparation for the
fabrication process, the TEM membranes are attached to a Si carrier wafer with an
indium bond. This step reduces the number of fabrication steps that involve handling the
membranes directly, greatly reducing the likelihood of breaking them during subsequent
processing.
63
Fig. 3.1 (a) An electron micrograph of a membrane with Pt resistors deposited (light gray lines). (b)
Another electron micrograph of the platforms partway through the cutting process. The debris is only
electrostatically attached and is removed during the etch steps. (c) A finished device. The strip in
between the platforms is remnant SiNx and an artifact of the cutting process. (d) An optical
micrograph showing the platforms are residue free.
Platinum resistive thermometry is used to measure the temperature on the
platforms. Platinum is selected based upon its availability and temperature dependent
resistivity.13-14,25 Note that this fabrication paradigm allows for inclusion of other
materials for resistive thermometry in the future such as AuGe11 or NbN19,26. In order to
deposit the Pt resistors and leads, the SiNx membrane is coated with a bilayer of Shipley
1805 resist atop LOR 3A and patterned via photolithography. After development, a 1 nm
64
Ti adhesion layer and 35 nm of Pt is evaporated onto the sample and lifted off. The two
resistor patterns (Fig. 3.1) consist of a 1.5 µm wide wire in a meander pattern that is
11 µm wide and 90 µm long for both the heater and thermometer. These meanders are
each contacted via four 3 µm wide Pt wires in a four probe geometry to subtract the lead
resistance from measurements of the meander. The heater and thermometer are separated
by 60 µm – the active area of the calorimeter.
For the heat capacity measurement, a Ti(1 nm)/Au(200 nm) sample is similarly
patterned and evaporated onto a 40 µm wide by 90 µm long rectangle between the heater
and thermometer resistors one of the platforms as shown in Fig. 3.1. A 10 µm gap
between the sample and resistors is used to avoid any electrical shorting of the
thermometry by the conductive sample. During metal evaporation, changes in the
resonant frequency of a quartz crystal monitor located within the deposition chamber are
recorded. From the frequency change, Saurbrey’s equation27,28 is used to calculate the
density of the deposited Au:
∆𝑓 = (
2
−2𝑓𝑟𝑜
√𝜌𝑞 𝐺𝑞
)
∆𝑚
𝐴
,
(3.1)
Where Δf is the frequency change from the metal deposition, fro is the resonant
frequency of the crystal monitor, and ρq, Gq are the density and shear modulus of the
crystal monitor (2.648 g/cm3 and 2.95E11g/cm s2 respectively). Normalizing by the area
of the crystal monitor, the density of evaporated Au is calculated to be 8.4 g/cm3.
Together with the dimensions of the Au pattern and atomic force microscopy (AFM)
characterization of film thickness, the mass is found to be 6.2 ng ± 0.2 ng. This mass will
be used to normalize the heat capacity and obtain the specific heat for comparison with
65
accepted bulk values. After the Au deposition, the calorimeter is ready to be machined
using the FIB.
Focused ion beam machining of the free standing SiNx membranes is difficult to
implement. Early iterations using this method saw the membranes shatter upon contact
with the FIB beam. One such membrane can be seen in Fig. 3.2 (a). It is not known what
leads to this failure mode, but it is likely a combination of charging effects from the FIB
beam and thermal effects. What is clear and shown in Fig. 3.2 (b) is that the presence of
metal leads on the membrane during the FIB cutting reduces membrane failure (though
does not eliminate it entirely, as seen in Fig. 3.2 (b)).
Following this to the logical conclusion, a 200 nm Au film was deposited on the
membrane and shapes were cut into the membrane without membrane failure (Fig. 3.2
(c)). However, measurements aren’t possible with a metal film coating everything.
Attempting to remedy this, the stabilizing Au film was patterned around the leads and
platform in order to stabilize the cutting without interfering with any measurements. As
can be seen in Fig. 3.2 (d), this approach leads to localized membrane failures in the
regions of the membrane not covered in Au.
66
Fig. 3.2 (a) An electron micrograph of a shattered membrane (light gray areas) after FIB cutting
process. Darker areas are a bulk Si substrate (membrane is lying face down) (b) Another electron
micrograph of a membrane with Pt leads deposited (light gray lines) post FIB session. The membrane
can be seen peeling away from the frame center of the picture (c) A successfully cut platform using a
Au anti-charging layer. (d) Patterned Au on the membrane (light gray areas) surrounding the platform
area and legs. The crack in the membrane can be seen near the lower left leg.
Therefore, an approach had to be adopted that included a metal film present
everywhere on the membrane during the FIB cutting process followed by removal of the
metal film post cutting. To achieve this, the membrane is patterned with the electronbeam resist poly(methyl methacrylate) (PMMA), leaving behind resist masking the legs
and platform of the unfinished device. PMMA is chosen as it resists degradation from
exposure to the focused ion beam. In contrast, the S1805 photoresist used in preceding
fabrication steps is difficult to remove using standard cleaning methods when used as a
67
masking layer. This is done to protect the calorimeter from the metal film soon to be
deposited.
Next, 200 nm of Al is sputtered to stabilize the membrane and dissipate heat
during FIB machining. Al is chosen for the stabilization layer based upon its high thermal
and electrical conductivity and, unlike the exploratory Au film, can be removed using
tetramethylammonium hydroxide (TMAH) based developers. Portions of the membrane
are removed with the FIB over several steps due to the limited field of view of the beam.
First, the 100 µm × 100 µm square platform and the four 10 µm wide by 325 µm long
legs are defined by line cuts. Then, the undesired portion of the membrane is sliced into
sections to facilitate material removal – portions of which can be seen in Fig. 3.1 (b).
Temporary SiNx bridges are left to stabilize the platform and legs (Fig. 3.1 (b)). The SiNx
strip between the platforms seen in Fig. 3.1 (b)–(d) is an artifact of this stabilization.
After the platform and legs are defined, the Al layer is etched away with a 5% TMAH
solution, leaving behind PMMA resist, which is subsequently dissolved in acetone. The
device is dried and then reloaded into the FIB where the temporary stabilization bridges
are detached from the suspended structures (shown in Fig. 3.1 (c)).
3.4 Overview of measuring heat capacity
Heat capacity is the amount of heat required to raise the temperature of the system
by one Kelvin, and is attributed29 to Joseph Black, a Scottish physicist and chemist, as
early as 1760. Specific heat can be formally defined as:
𝑑𝑄
𝐶𝑃 = lim (𝑑𝑇 ) ,
𝑑𝑇→0
68
𝑃
(3.1)
Where the P subscript indicates the system is held at constant pressure. Specific heat is
heat capacity normalized by the sample’s mass. For the rest of the chapter, we will refer
to the un-normalized heat capacity except where explicitly stated. There exist a variety
of ways to measure specific heat, and the following is a brief introduction of the more
common methods.
3.4.1 Adiabatic Calorimetry
One of the oldest methods to measure heat capacity, the adiabatic method uses the
definition of heat capacity in (3.1) by applying a known amount of heat to the sample and
measuring the temperature change once the sample achieves thermal equilibrium. This
method implicitly assumes that there are no heat leaks to the surroundings. In practice,
this constraint is impossible to achieve, limiting this method to large thermally isolated
samples where the effects of heat leaks to the surroundings are small relative to the heat
used to drive the temperature change in the sample. This method is obviously not
suitable for our microscale platforms and small sample sizes.
3.4.2 Differential Scanning Calorimetry (DSC)
This method is similar to the adiabatic method in that the thermal losses to the
surroundings are neglected. Rather than engineering the sample size to be large enough
that the heat leaks are minimal, the DSC method subtracts a reference calorimeter that
should in principle have the same thermal link to the surroundings as the loaded
calorimeter being measured. In addition, the system is not measured in equilibrium, but
69
rather the temperature is scanned at a constant rate and the difference in power supplied
is monitored between the bare and loaded calorimeters. However, one of the primary
reasons for selecting the heat capacity measurement as a first test for the microscale
platforms was to give meaningful metrics on platform design: one such metric is the
thermal link of the platforms to the surrounding heat bath. Therefore, DSC is not suitable
for our purposes.
3.4.3 AC Calorimetry
In 1968, Sullivan and Seidel30 developed a technique capable of measuring small
samples by applying a time varying heating current and measuring the thermal frequency
response. The thermal response to the driving heater will have both an AC and DC part.
In the regime where the system is in thermal equilibrium, the DC response of the
calorimeter is neglected and the AC part can be shown to be:
∆𝑇𝑎𝑐 =
𝑃
2𝐾𝑒𝑓𝑓 √1+4𝑤2 𝜏2
,
(3.2)
Where P is the power delivered by the heater, w is the frequency of the applied heating
current, Keff, is the thermal link to the surroundings, and τ is the thermal time constant.
We will talk about Keff and τ in further detail in the following sections. If the frequency
w is chosen such that w2τ2 >> 1, then (3.2) can be simplified to:
𝑃
∆𝑇𝑎𝑐 = 4𝑤𝐶,
Where C is the heat capacity.
surroundings.
(3.3)
Not seen in (3.3) is Keff, the thermal link to the
Thus if the frequency condition is met, the thermal link to the
70
surroundings can be neglected.
This strength of the AC technique also makes it
unsuitable for our purposes.
3.4.4 Thermal Relaxation Calorimetry
In 1972, Bachmann et al31 pioneered the use of the thermal relaxation technique
for measuring heat capacity. This technique applies thermal pulses to the sample and
measures the thermal response as a function of time as the system moves toward thermal
equilibrium. By solving the 1-D heat flow equation, a solution for the heat capacity is
found as long as several assumptions hold true. This technique has several advantages
for small samples: multiple thermal pulses can be easily averaged which improves the
signal-to-noise, and internal thermal relaxation within the sample can be easily identified
and corrected for. This will be discussed in detail as this is the method we have chosen to
characterize our platforms and measure the heat capacity of the attached Au film. For the
following analysis of thermal relaxation we will follow Bachmann’s original paper31.
Consider a one-dimensional model of heat flow from the sample and platform
down the legs to the Si frame held at some temperature T0 as shown in Fig. 3.3. The heat
balance equation can be written then as:
𝜕𝑇
𝜕𝑇
𝑃 = 𝐴𝜅 𝜕𝑧 + 𝐶(𝑇) 𝜕𝑡 ,
(3.4)
where P is the input heating power, A is the cross-sectional area of the legs, κ is the
thermal conductivity, and C is the heat capacity. In writing the heat balance equation in
this way, an assumption is made that the lateral heat diffusion from the sample and
71
platform to the legs is much faster than the longitudinal heat diffusion down the legs
(equivalent to τint<<τext). We will discuss at the end how to detect if this assumption fails.
Fig. 3.3 A schematic showing the 1-D heat flow model. Input power on the platform diffuses through
the legs with an effective thermal resistance of 1/Keff to the bath (where Keff is the thermal
conductance).
Next we re-arrange (3.4) for C(T), and integrate along the length of the leg:
𝑑𝑇 −1
𝐶(𝑇) = ( 𝑑𝑡 )
𝑇
[𝑃 − ∫𝑇 𝐾(𝑇 ′ )𝑑𝑇 ′ ],
0
(3.5)
where we have replaced κ with the thermal conductance K(T) (which is κ(T)*A/l). Next
we make another assumption that the temperature change is small enough that K(T)
doesn’t change appreciably over the temperature range:
𝑇
∫𝑇 𝐾(𝑇 ′ )𝑑𝑇 ′ = 𝐾(𝑇𝐴𝑉𝐸 )∆𝑇,
0
72
(3.6)
where TAVE is the average temperature between T0 and T. Finally, we set P =0 at t=0
(shutting the thermal pulse off and letting the temperature to decay):
−𝐾(𝑇
)
𝐴𝑉𝐸
𝐶 = 𝑑(𝑙𝑛∆𝑇)
.
(3.7)
⁄𝑑𝑡
The temperature should follow an exponential decay such that d(lnΔT)/dt = -1/τ, leading
us to
𝐶 = 𝐾eff 𝜏ext ,
(3.8)
where K(TAVE) has been replaced by Keff (dropping the average nomenclature), and τext is
the thermal time constant associated with thermal relaxation to the bath temperature, T0.
In order to use (3.8) to find C, we must measure Keff – the effective thermal
conductance—in addition to extracting τext. Keff can be calculated by:
𝐾eff = 𝑃⁄∆𝑇.
(3.9)
This derivation has assumed both Keff, and C do not vary over the temperature range used
during the heat pulse. We have also required that the bath temperature, T0, remain
constant.
In the event that the initial assumption of the model about heat diffusion does not
hold, the temperature profile of the sample and platform should look like:
𝑇 = 𝑇0 + ∆𝑇𝑒
−𝑡⁄
𝜏ext
+ ∆𝑇𝑒
−𝑡⁄
𝜏int ,
(3.10)
where ΔT is the temperature change of the calorimeter induced by the heat pulse, and τext
and τint are the external thermal time constant (mentioned previously) and internal thermal
time constant, respectively. These time constants describe thermal equilibration to the
external surroundings (τext) or equilibration within the calorimeter (τint). In the regime
where τext ≈ τint, (3.8) no longer holds, and the calculation of C becomes more complex.
73
However, we will show that our platforms are in the regime where τext >> τint—where the
calorimeter is essentially isothermal—and can extract a single time constant, τext, from the
thermal decay back to equilibrium.
Fig. 3.4 (a) Power applied vs the ΔT of the heat pulse. Linear data shows Keff remains roughly
constant over the temperature interval. (b) Background voltage of thermometer before the heat pulse.
A series of heat pulses at the given power are made at much higher powers than used for calorimetric
measurements. The background doesn’t change, indicating the thermal link of the Si frame to the heat
bath is sufficient such that the frame T remains constant during the heat pulse.
Our calorimeter satisfies all the requisite conditions for the thermal relaxation
model. The input heating power is carefully selected such that ΔT/T < 2%. This condition
ensures that C and Keff are constant during the heat pulse, and is tested by varying P and
verifying ΔT is linear as seen in Fig. 3.4 (a). Additionally, the temperature of the bath T0
74
is monitored at high input heating power (such that ΔT/T ~ 5%) shown in Fig. 3.5 (b). No
change is observed in T0, indicating sufficient thermal anchoring of the device to the
cryostat. Finally, the calorimeter is engineered in the regime where τext >> τint by
enhancing τext through thermal isolation rather than by minimizing τint through additional
thermally conductive layers. This conditions is further verified in Section 3.5.
3.5 Experimental Methodology
Having identified the general methodology for this experiment, we now consider
the more specific challenge of implementing relaxation calorimetry with our microscale
platforms. Principally, this involves careful consideration of both the measurement
electronics, challenging due to the transient (milliseconds) thermal response of the
calorimeter, and the calibration of the Pt thermometer within the context of the overall
experimental protocol.
75
Fig. 3.5 Schematic of the measurement circuit. RT and RH indicate the thermometer and heater
resistors, respectively. RT is biased with an AC voltage source and both the local voltage drop,
Uac,T, and voltage drop across the load resistor, Uac,L, are measured by lock in amplifiers. RH is
pulsed with a DC voltage source and both the current and voltage drop, Udc,H, across the resistor are
monitored.
The measurement circuit used for relaxation calorimetry is schematically shown
in Fig. 3.5. During operation, a Keithley 2400 Sourcemeter supplies a pulsed current
with a square waveform through the heater while simultaneously measuring the voltage
response, Udc,H, in a four probe configuration to determine the heating power, P.
Concurrently, a Stanford DS45 function generator provides an alternating voltage at
3.5 kHz to the thermometer through a 1 MΩ load resistor, yielding a typical sense current
of 350 nA as measured by a Signal Recovery 7265 lock-in amplifier that monitors the
potential drop, Uac,l, across the load resistor. The response of the thermometer, Uac,T, is
monitored by either a second Signal Recovery 7265 or a Signal Recovery 7270 lock-in
amplifier. A trigger is sent from the Keithley 2400 to the lock-in monitoring Uac,T to
76
coordinate acquisition with the heater current pulse. The results are stored to a local data
buffer and transferred to a control computer between each pulse.
The sample is measured in a Quantum Design Physical Property Measurement
System (PPMS) using a custom break out box to provide access for the instruments
described above. Electrical contact between the sample and the PPMS measurement puck
is made with 0.001″ diameter Cu wire attached with an indium bond to the contact pads
located on the Si frame. The device is mechanically anchored to the measurement puck
with rubber cement. The thermal coupling across this interface is validated by comparing
the temperature of the Si frame to that of the sample puck and the suspended platforms,
yielding a variation of less than 50 mK at heater powers 4× above what was used for the
actual measurement (Fig. 3.4 (b)). Finally, a copper shield is installed around the sample
to ensure an isotropic thermal environment.
The in situ thermometry within the PPMS is used to calibrate the temperature
response of the calorimeter’s Pt thermometers. The local resistance of the platform and
frame thermometers are measured in a four wire configuration at cryostat temperatures
ranging from 310 K down to 20 K in 5 K increments with a wait time of 30 minutes at
every temperature to allow the device to come to thermal equilibrium with the cryostat.
All measurements are made during a single temperature sweep to avoid thermal cycling
effects (Fig. 3.6). Two third order polynomials are fit to the resulting curve, one for
temperatures above 70 K and one for temperatures below, in order to accurately account
for the transition from dominantly phonon scattering to dominantly impurity scattering in
the resistivity.
77
R(k)
H
0.24
0.9
0.20
0.8
0.16
0.7
100
200

T
1.0
300
T(K)
0.6
RT
0.5
RH
0.4
50
100
150
200
250
300
T(K)
Fig. 3.6 Resistance of the Pt resistors used as heater (RH) or thermometer (RT) as a function of
temperature. The slight difference in resistance of the two resistors is due to lithographic tolerances.
Inset: A polynomial fit is made and the local slope is calculated and normalized by the resistance at
every temperature to get the temperature coefficient of resistance, η, for the resistors.
This analytic approach, similar to that employed in Ref. 12, allows for a more
precise determination of the temperature than a simple look up table based on the
calibration data. The uncertainty in this polynomial fit leads to uncertainties in the
determination of the absolute temperature on the order of ± 0.1 K. However, the
fractional uncertainty in the slope of this polynomial is less than 0.2%, smaller than the
fractional uncertainty in the resistance of the Pt thermometer (0.5%). As a result, relative
changes in temperature can be measured to a much higher precision than the absolute
temperature (± 10 mK vs. ± 100 mK) and the uncertainty due to the polynomial fit can be
78
discarded. The temperature coefficient of resistance for our thermometry (a measurement
of resistance thermometry sensitivity) is defined as the slope of the fit curve normalized
by the instantaneous resistance:
1
𝑑𝑅
𝜂(𝑇) = |𝑅(𝑇) 𝑑𝑇 |,
(3.11)
where the local slope, 𝑑𝑅/𝑑𝑇, is calculated analytically from the appropriate fit curve at
every temperature. As can be seen in the inset of Fig. 3.6, the sensitivity reaches a
maximum at a temperature of 60 K before dropping sharply, corresponding to the onset
of saturation in the resistance of Pt below 40 K.
Following the relaxation method discussed in the previous section, a current pulse
is sent through the heater and the transient thermal response monitored in the
thermometer. During this current pulse, the instantaneous voltage across the heater and
the current are monitored in order to directly measure the power delivered to the heater.
In order to reduce noise, at each temperature the data is averaged over 100–200 thermal
pulses. An example of an averaged time scan of the heat pulse is shown in Fig. 3.7(a) for
250 K. The relative temperature change, T, is determined by averaging the temperature
before and during the heat pulse and taking the difference and an exponential is fit to the
temperature decay in order to extract the thermal time constant, ext.
79
Fig. 3.7 (a) A voltage pulse is sent to the heater and the time evolution of the temperature is
monitored. The heat pulse is averaged over 100 scans shown here at 250 K. The thermal decay time
constant, τext, is extracted via an exponential fit of the decay shown as the red solid line. The ΔT is
measured from the dashed blue lines. Inset: a semi-log plot of the thermal decay following the pulse
shows a single linear line. (b) Deviations from the average temperature and exponential fit are shown.
Three regimes are shown: before the heater is turned on, after thermal equilibrium with the heater on,
and the relaxation back to equilibrium when the heater is turned off. The temperature rise regime is
omitted for clarity.
The inset in Fig. 3.7(a) shows a linear dependence of ext on a log scale, affirming
that the calorimeter is isothermal (τext >> τint) under the chosen measurement conditions.
The deviation from these fits is shown in Fig. 3.7(b). The scatter in the data, roughly
100× larger than expected for Johnson noise, is a measure of the sensitivity limit of our
experimental apparatus, and confirms a temperature difference uncertainty of 10 mK
once averaged. Since both Keff and τext are extracted from the data in Fig. 3.7(a) (Eqs.
80
(3.9) and (3.10) respectively), the temperature uncertainty is translated into uncertainty of
τext and Keff through weighted χ2 fitting.
3.6 Results and Discussion
Repeating the transient thermal response measurements described in Fig. 3.7 at
every temperature from 330 K down to 10 K allows us to extract Keff and ext as functions
of temperature. These values are plotted in Fig. 3.8, top and bottom panels, respectively.
Note that the larger time constant for the loaded calorimeter is consistent with the
increase in heat capacity due to the Au film.
In contrast, Keff is comparable for both the bare and loaded calorimeters over the
majority of the temperature range (and roughly 10× smaller than full membrane
calorimeters12,13), with the bare calorimeter exhibiting a relative increase compared to the
loaded calorimeter for temperatures above 250 K. This is consistent with the difference in
emissivity between the bare and loaded membranes and the competition between
radiative and conductive heat loss. Specifically, radiative heat loss plays a greater role at
higher temperatures. The temperature dependence of the radiative power is given by:
𝑃rad = 𝐴eff 𝜀𝜎((𝑇 + ∆𝑇)4 − 𝑇 4 ),
(3.12)
where Aeff is the area radiating heat, ε is the emissivity, and σ is the Stefan-Boltzmann
constant. Following similar analysis to Ref. 24, a Taylor series expansion is made on Prad
keeping only first-order terms in ΔT (valid as long as ΔT << T). This is then normalized
by ΔT to obtain the thermal conductance 𝐾rad ~4𝐴eff 𝜀𝜎𝑇 3 .
81
Fig. 3.8 Top: extracted thermal decay time constant plotted from 310 K down to 20 K. The Au data
(red circles) is consistently larger than for bare SiNx (black squares) as expected. Bottom: Keff is
calculated from the ΔT divided by the input power (not pictured) and plotted as a function of
temperature. There is a discrepancy above 250 K due to differential thermal radiative losses of Au vs.
SiNx. The Pt (blue triangles) contribution is calculated from the Weidmann-Franz law using the
measured electrical conductivity. Inset: the difference is taken between bare and Au Keff above 250 K
and plotted vs T3 and is linear, confirming the discrepancy is from thermal radiation.
This dependence is made evident by subtracting KeffAu from KeffSiN. This
subtraction removes the contribution due to non-radiative sources, as these terms should
be constant between the two calorimeters. The residual variation is plotted versus T3 in
the inset to Fig. 3.8 and shows a clear linear dependence whose slope can be taken as a
measure of the difference in emissivities, confirming that radiative coupling begins to
82
play a major role for temperatures above 250 K. Below 250 K, Keff is dominated by
thermal conductance through the Pt leads. This can be verified using the WeidemannFranz law to calculate the thermal conductance due to the leads based upon the measured
resistivity of Pt (open triangles, Fig. 3.8 bottom panel). The difference in Keff limits the
effectiveness of these calorimeters (as currently constructed) at temperatures above 250
K. However, this limitation can be readily addressed by including a thin Au film on the
bare calorimeter to match the emissivity of the Au sample.
A potential systematic error in the measurement of Keff was reported in Ref. 13:
using an AC voltage source can cause RC attenuation at high frequencies. This
attenuation will cause the measurement of ΔT to be under-reported. To address this
concern, Keff is also measured at 17 Hz (Fig. 3.9), yielding no difference in Keff. The lack
of frequency sensitivity is likely due to the large difference in the impedance of the
thermometry in this work (1 kΩ) as compared to Ref. 13 (greater than 500 kΩ).
Following the extraction of τext and Keff, the heat capacity C can be found using
Eq. (3.8) and is shown as a function of temperature in the top panel of Fig. 3.10. As
expected, the heat capacity of the Au calorimeter is greater than the bare calorimeter. The
background heat capacity (addenda) is 4 nJ/K at 250 K.
83
Fig. 3.9 Keff is measured at two different frequencies to test for attenuation errors. The 17 Hz data,
while noisier than 3500 Hz, shows the same qualitative behavior.
The sensitivity of our calorimeter is limited by the uncertainty in τext. As can be
seen in Eq. (3.8), the uncertainties in Keff and τext contribute equally to the uncertainty in
heat capacity, C. However, the magnitude of the error in τext is significantly greater (Fig.
3.10, bottom) leading to an overall error in an individual measurement of C of 40–80
pJ/K. Since the heat capacity of Au alone is obtained via subtracting the reference
platform, these errors add in quadrature. Thus, the sensitivity of our loaded calorimeters
is 100 pJ/K at 250 K, decreasing to 50 pJ/K below 100 K. This resolution is well below
the 3 nJ/K reported in Ref. 13 and is comparable to the 150 pJ/K reported in Ref. 12 as
well as the 500 pJ/K reported in Ref. 30.
84
Fig. 3.10 Top: the heat capacity of Au and reference SiNx calorimeter measured as a function of
temperature. Bottom: fractional (%) error is plotted as a function of temperature for both Keff and τext.
We measure a much smaller sample than Ref 12, 13 (6.2 ng compared to
microgram samples reported in Ref. 12 and 13) but comparable to the 10 ng sample
reported in Ref. 32. Note that this is the metric of interest for experiments involving
individual nano- or micro-scale magnetic structures, as the total mass of magnetic
material is the limiting parameter. For measurements of thin films and interfaces, a more
useful metric is the sensitivity per square micron. Normalizing by the active area of our
calorimeter yields 17 fJ/K∙µm2 at 250 K in our devices. For comparison, Ref. 12 reports
4 fJ/K∙µm2 at 250 K and Ref. 30 reports 0.5 fJ/K∙µm2 above 300 K.
85
Fig. 3.11 Au data with the reference SiNx heat capacity subtracted and normalized by the molar mass.
It is plotted against bulk data for Au10.
Subtracting the background from the Au calorimeter yields the heat capacity for
Au alone. In Fig. 3.11, this difference is first normalized by the molar mass of the Au
sample and plotted along with bulk data from Ref. 10. The measured Au signal matches
closely with the bulk data over the temperature range measured despite a difference of
1012 in the mass of the two samples.
3.7 Conclusion
We have successfully fabricated suspended platforms from commercial SiNx
membranes using FIB-based nanomanufacturing. We demonstrate the utility of these
platforms by measuring the heat capacity of a 6.2 ng Au sample. The 100 µm × 100 µm
square platform size reduces the background heat capacity of the calorimeter to 4 nJ/K at
86
250 K. The measured Au specific heat matches closely with established bulk data10. The
thermal conductance to the bath is reduced to 120 nW/K at 300 K and is sufficiently
small that these calorimeters are free from τint effects without the inclusion of a metal
thermalization layer. The low thermal conductance of these platforms makes them
suitable for measuring other thermodynamic properties of nanoscale structures such as
those associated with spin relaxation and dissipation arising from spin transport in
magnetic heterostructures and nanostructures.
3.8 References
1
K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa and E.
Saitoh, Nature 455, 778–781 (2008)
2
J. Flipse, F. Dejene, D. Wagenaar, G. Bauer, J. Ben Youssef, and B. van Wees, Phys.
Rev. Let. 113, 027601 (2014)
3
J. Shaw, T. Silva, M. Schneider, and R. McMichael, Phys. Rev. B 79, 184404 (2009)
4
J. Ding, S. Jain, A. Adeyeye, J. Appl. Phys. 109, 07D301 (2011)
5
H. Nembach, J. Shaw, C. Boone, and T. Silva, Phys. Rev. Lett. 110, 117201 (2013)
6
G. Stewart, Rev. Sci. Instrum 54, 1(1983)
7
S. Mabrey and J. Sturtevant, PNAS 73, 3862–3866 (1976)
8
M. McGuire, A. Christianson, A. Sefat, B. Sales, M. Lumsden, R. Jin, E. Payzant, D.
Mandrus, Y. Luan, V. Keppens, V. Varadarajan, J. Brill, P. Hermann, M. Sougrati, F.
Grandjean and G. Long, Phys. Rev. B 78, 094517 (2008)
87
9
S. Li, C. de la Cruz, Q. Huang, Y. Chen, J. Lynn, J. Hu, Y. Huang, F. Hsu, K. Yeh, M.
Wu, and P. Dai, Phys. Rev. B. 79, 054503 (2009)
10
T. Geballe, W. Giauque, J. Am. Chem. Soc. 74, 2368 (1952)
11
B. Zink, B. Revaz, R. Sappey, and F. Hellman, Rev. Sci. Instrum. 73, 1841 (2002)
12
S. Tagliati, V. Krasnov, and A. Rydh, Rev. Sci. Instrum 83, 055107 (2012)
13
D. Queen and F. Hellman, Rev. Sci. Instrum. 80, 063901 (2009)
14
D. Denlinger, E. Abarra, K. Allen, P. Rooney, M. Messer, S. Watson, and F. Hellman,
Rev. Sci. Instrum. 65, 946 (1994)
15
F. Fominaya, T. Fournier, P. Gandit, and J. Chaussy, Rev. Sci. Instrum 68, 11 (1997)
16
W. Chung Fon, K. Schwab, J. Worlock, and M. Roukes, Nano Lett. 5, No. 10 1968
(2005)
17
S. Sadat, Y. Chua, W. Lee, Y. Ganjeh, K. Kurabayashi, E. Meyhofer and P. Reddy,
App. Phys. Lett. 99, 043106 (2011)
18
A. Lopeandia, E. André, J.-L. Garden, D. Givord and O. Bourgeois, Rev. Sci. Instrum.
81, 053901 (2010)
19
E. Dechaumphai and R. Chen, Rev. Sci. Instrum. 85, 094903 (2014)
20
M. Wingert, Z. Chen, S. Kwon, J. Xiang, and R. Chen, Rev. Sci. Instrum. 83, 024901
(2012)
21
P. Ferrando-Villalba, A. Lopeandia, L. Abad, J. Llobet, M. Molina-Ruiz, G. Garcia, M.
Gerbolès, F. Alvarez, A. Goñi, F. Muñoz-Pascual, and J. Rodríguez-Viejo,
Nanotechnology 25, 185402 (2014)
88
22
L. Shi, D. Li, C. Yu, W. Jang, Z. Yao, P. Kim, A. Majumdar, J. Heat Transfer 125, 881-
888 (2003)
23
B. Zink, and F. Hellman, Solid State Comm. 129, 199-204 (2004)
24
A. Khan, J. Philip, and P. Hess, J. App. Phys. 95, 1667 (2004)
25
B. Zink, B. Revaz, J. Cherry, and F. Hellman, Rev. Sci. Instrum. 76, 024901 (2005)
26
O. Bourgeois, E. André, C. Macovei and J. Chaussy, Rev. Sci. Instrum. 77, 126108
(2006)
27
G. Sauerbrey, Phys. Verh. 8, 113 (1957)
28
G. Sauerbrey, Z. Phys. 155, 206 (1959)
29
F. Holmes, “Lavoisier and the Chemistry of Life: an Exploration of Scientific
Creativity” (1987)
30
P. Sullivan, and G. Seidel, Phys. Rev. 173, 3 (1968)
31
R. Bachmann, F. DiSalvo, T. Geballe, R. Greene, R. Howard, C. King, H. Kirsch, K.
Lee, R. Schwall, H.‐U. Thomas, and R. Zubeck, Rev. Sci. Instrum. 43, 205 (1972)
32
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89
Chapter 4 : Thermal detection of magnetization dynamics
4.1 Motivation
Having successfully demonstrated the utility of the thermally isolated platforms,
we now turn to identifying a suitable system/geometry for studying spin-thermal
interactions.
Spin dependent non-equilibrium transport phenomena such as spin
Seebeck1 or spin Peltier2 require a bridge geometry with two platforms. Utilizing such a
geometry, it is possible to leverage the thermal isolation of the platforms and engineer
large thermal gradients3 (up to 50 K) to look for spin Seebeck or look for spin dependent
Peltier heating and cooling of the platforms with applied spin polarized currents. With
the one platform design (or including an uncoupled reference platform), it is possible to
study equilibrium spin-thermal effects such as magnetocaloric or magnon specific heat
through heat capacity measurements.
It is also possible to use the one platform design to study magnon-phonon
couplings through dissipation of a magnetization dynamics detected by bolometric
measurements4-11. These measurements are performed by monitoring the temperature on
the platform which has a known thermal coupling to the surroundings. Any thermal flux
incident on the platform will result in temperature response. Such thermal fluxes can be
driven by exciting a spin system out of equilibrium, enabling dissipation as a way to
90
study spin-thermal interactions. There are multiple ways of driving a spin system out of
equilibrium, such as optical spin injection or electrical spin injection. Another such
option is using microwaves to excite select magnon modes in a ferromagnet. Here, we
propose operating the platforms as bolometers to study thermal dissipation of a nonequilibrium spin system through the resonant excitation of magnetization dynamics in a
ferromagnet by microwaves.
4.1.1 Proposed experiment
Combining thermal detection with spin pumping offers a potentially exciting way
of studying spin dynamics in an attached normal metal without the parasitic AMR
voltage, as well as the ability to measure the field out-of-plane geometry (where the ISHE
voltage goes to zero). In addition, the sensitivity of thermal detection lends itself to small
sample volumes and enables access to spin dynamics in nanostructures difficult to
address using conventional cavity detection techniques.
Here, we present steps towards realizing thermal detection of spin pumping by
utilizing the thermal platforms developed in Chapter 3 to measure the resonant
dissipation of a Py thin film under the application of microwaves. We want to eventually
attach a non-magnetic layer to the Py thin film, ideally both a spin preserving material,
such as Cu, and an efficient spin sink like Pt. Through comparison of both the spinpreserving and non-spin-preserving layers, it may be possible to tease out the spin
dissipation of the conduction electrons in the normal layer. In addition, a variation from
91
in-plane to out-of-plane (IP to OOP) geometries would be studied and the angular
dependence of the spin pumping signal resolved.
4.1.2 Expected signal size
In order to derive an expression for the dissipation of the precessing
magnetization, we will follow Bakker et al11 and start with the Zeeman energy of a
magnet with magnetization M in a magnetic field Heff:
(4.1)
𝐸 = − ∫ 𝑴 ∙ 𝑯𝑒𝑓𝑓 𝑑𝑉,
The dissipation can be calculated from the time derivative of E assuming a uniform Heff
and M:
𝑑𝐸
𝑑𝑡
𝑑𝑴
= −𝑉 ( 𝑑𝑡 ∙ 𝑯𝑒𝑓𝑓 + 𝑴 ∙
𝑑𝑯𝑒𝑓𝑓
𝑑𝑡
),
(4.2)
where V is the volume of the magnet. The first term in (4.2) represents the dissipation of
the magnetization due to motion, and the second term represents the absorbed microwave
energy. In equilibrium, the averaged energy change must be zero, thus the two terms
must equal each other. In order to get an expression for the dissipated power, a solution
must be found for dM/dt. Using a linearized set of the LLG equations, assuming a small
precessional cone angle and averaging over high frequency variations, Bakker et al 11
found:
𝑑𝐸
1
2
⟨ ⟩ = 𝜒𝑧′′ 𝛾ℎ𝑟𝑓
𝜔𝑉𝑀𝑠 ,
𝑑𝑡
2
(4.3)
where hrf is the strength of the AC field, ω is the applied frequency, and χz” is a
complicated combination of the in-phase and out-of-phase magnetic susceptibilities
92
containing α, the Gilbert damping parameter. Thus we see that the dissipation scales with
volume, saturation magnetization, applied AC field, and frequency.
4.2 Experimental setup
FMR is done in a Bruker ELEXSYS E500 electron spin resonance (EPR)
microwave cavity optimized for the X band (8-12 GHz) with a microwave synthesizer
that outputs 9.5-9.8 GHz. The power output can reach 200 mW provided the cavity is
properly tuned. A typical magnetic response is seen in Fig. 4.1 for a 60 nm Py film
grown under similar conditions as the Py on the platforms.
Fig. 4.1 FMR response of a 60 nm Py thin film (IP geometry) in the Bruker EPR cavity at 9.65 GHz
and 2 mW applied microwave power. The measured response is absorbed microwave power and is a
derivative of a Lorentzian with the center occurring at the zero crossing which occurs at 1068 G and a
peak-to-peak linewidth of 34 G.
93
4.2.1 Vacuum design
The thermally isolated platforms are measured in vacuum ~10-5 torr to avoid
convective thermal losses.
In order to achieve this, a custom vacuum system was
designed and built. An EPR grade 10 mm OD quartz tube was purchased from WilmadLabglass. A T-junction (also of quartz, though not EPR grade as it is located outside of
the cavity) was welded onto the quartz tube by a local glass shop. The top of the tube
was connected to an electrical feedthrough wielded through a blank flange via an UltraTorr fitting wielded to a quick connect flange. The second part of the T-junction was
connected to a pump line via another Ultra-Torr/quick connect flange as seen in Fig. 4.2.
Structural supports were needed in order support the weight of the ensemble, and a clamp
was used with the pumping line in order to damp out vibrations from the vacuum pump.
The completed system can be seen in Fig. 4.2.
4.2.2 Measurement electronics and wiring
In order to propagate the electrical contacts down to the sample in the microwave
cavity, a length of Roger and Holland’s TMM4 Cu clad ceramic was used. It was first
machined to the correct dimensions, then the Cu layer was etched down to 50 microns
(down from ~100 microns) in an ammonium perchlorate solution. Next, the ceramic was
patterned with photoresist with contact pads and leads which masked the Cu for a second
ammonium perchlorate etch. Finally, the ceramic was mechanically attached to the
electrical feedthrough with set screws.
94
Fig. 4.2 Left: An image of the quartz vacuum tube. An O-ring UltraTorr to KF 25 flange is used to
connect the pumping line (upper T junction) and the electrical feed through (bottom T junction).
Right: The completed experimental setup in the Bruker EPR cavity.
Similarly to calorimetry measurement, a Stanford DS45 function generator
supplies an AC current (with a ~ 1 MΩ load resistor in series) and a Signal Recovery
7265 lock-in is used to measure the local resistance (in a four probe geometry) of a Pt
resistor on the platform (for reference, see Fig. 3.5). Due to cavity tuning concerns, only
one Pt resistor on the platform is wired during the measurements.
The presence of conducting material inside the microwave cavity reduces the
quality factor Q, which is a measurement of how much energy the system loses (damping
time constant). If Q is reduced too much, it becomes difficult to operate the cavity at
high microwave power (greater than a few milliwatts) necessary to produce a
measureable thermal signal resulting from magnetization dynamics (the thermal signal
95
scales as Hrf2). To improve Q, wire bonding is used instead of In bonding to reduce the
amount of conducting material in the cavity. The height of the wire bonds is also very
important: everything is kept as flat as possible to reduce the inductance of the wiring to
the microwaves.
Sample placement inside the microwave cavity is also a critical parameter for the
quality factor. Small variations of the sample placement in the cavity arise from the
horizontal position of the electrical feedthrough outside the cavity.
This has been
observed to affect Q by 200-300 where a typical Q of this setup is on the order of 900.
To account for this, the cavity is tuned prior to pumping down in order to make
horizontal adjustments which are no longer feasible once the pumping line is in place.
4.2.3 Platform design
The platform geometry used for thermal detection of FMR is modified from the
platforms designed for calorimetry by the inclusion of support legs on either side of the
platform to suppress platform mechanical deformation by strain introduced by thin film
deposition as seen in Fig. 4.3 (a) and (b). It is important that the Py thin film remain in
the applied field plane to reduce the complexity of the resonance conditions by
simplifying the demagnification factors. The addition of the support legs will increase
the thermal link to the surroundings, but it is shown in the previous chapter’s Fig. 3.8 that
the dominant thermal conduction occurs in the Pt legs. Thus the increase in thermal
conductance is not expected to be more than a few nW/K.
96
Fig. 4.3 Left: An electron micrograph of the dual platforms for the calorimetry measurement from
chapter 3. Notice the curve in the loaded platform from the induced strain from the Au thin film.
Right: An electron micrograph of the adapted platform design for thermal detection of magnetization
dynamics. The side support legs keep the platform from bending.
4.3 Measurements
At a specified microwave power, the system is left to equilibrate over a period of
10 minutes before measurements are begun. If not, thermal drifts will be present as the Si
frame and ceramic holder heat up as a result of eddy currents as shown in Fig. 4.4 (a).
The thermal change of the frame and surroundings is roughly 4 K over the course of a
few minutes. As for the platform, the non-resonant heating from the microwaves is much
more pronounced, as shown in Fig. 4.4 (b). This occurs much quicker than the frame and
surroundings and is on the order of the thermal time constant of the platforms (around 50
ms). After equilibration, the magnetic field is scanned at a fixed frequency, usually
around 9.6 GHz (depending on the tuning of the cavity) while the thermal response of the
97
platform is monitored with the lock-in. The scan is usually around 10 minutes, long
enough that each field point is roughly in thermal equilibrium provided a small enough
field range is chosen (1000-2000 G).
Fig. 4.4 (a) non-resonant heating of the Si frame and surrounds once the microwaves are turned on.
The voltage change corresponds to ~4K and is monotonic except for the small downturn seen near 40
s. (b) non-resonant heating of the platform after turning on the microwaves.
A representative field scan at 54 mW is shown in Fig. 4.5. No resonant features
are seen (expected at 1068 G from the thin film data in Fig. 4.1). From the scatter in the
data points, the resolution is estimated at 30 mK. This is higher than the estimated
resolution of the calorimetry measurement of 10 mK. The discrepancy is partially due to
98
the calorimetry data averaging over more data points than the current scan as well as the
fact that the presence of microwaves also adds some noise into the measurement circuit.
Fig. 4.5 Field scan at 54 mW microwave power. Each data point displayed is an average over
multiple points. The error associated with this averaging (shown as red bars) is displayed at the first
and last data points. No signal seen near magnetic resonance (1068 G)
4.3.1 Expected thermal resonant response
It is possible to calculate the expected signal from a similar experiment 10 on a Py
thin film by I. Rod et al from simple scaling arguments. Under the application of 200
mW microwaves in a cavity, they found their sample absorbed/produced 2.7 nW at
resonance.
Since the power absorbed/produced at resonance is linear with applied
microwave power and sample volume (4.3), we normalize their signal with our sample
99
size and microwave power used (54 mW) and find that our sample should absorb/produce
1.1 μW. A key assumption in the scaling argument is that the coupling of the sample to
the microwaves is the same in our microwave cavity as theirs.
In order to turn this power into an expected ΔT, we need to know the thermal
coupling, Keff, to the surroundings. Unfortunately, because the restrictions of the cavity
tuning only allowed one Pt resistor to be wired at the present time, this was not possible
to measure in situ. However, given the calorimetric platforms had a K eff of 120 nW/K at
room temperature and extrapolating Keff at 435.5 K under the 54 mW of applied
microwave power(from Fig. 4.9 (b)), we estimate a Keff to be 170 nW/K. Combining this
with the estimated 1.1 μW of power, the expected signal size should be 6.9 K, or 2.3 μV
in Fig. 4.5. This estimation takes under consideration the increase in thermal radiation
from the platform (as the surroundings are still at 300 K), but assumes that the
contributions from the extra two support legs are negligible and the thermal conductivity
of the Pt has saturated (not an unreasonable assumption, see the Pt contribution in Fig.
3.8).
4.3.2 Possible reductions in thermal signal
From Fig. 4.5, it is obvious that there isn’t a resonant peak near this magnitude.
The reason for lack of signal is still unknown, but there are several possibilities. A key
assumption in calculating the expected thermal signal based upon previous work was that
our sample coupled to the cavity microwaves in the same fashion as Ref [10]. As
mentioned previously, the sample placement inside the cavity is crucial to the cavity
100
tuning factor, Q. It is very likely that some sample misalignment exists, where the
sample is not in the center of the cavity (which is a node for the electric field component
of the microwaves). The intensity of an EM wave defined as the time average of the
Poynting vector, S, can be written as:
𝐸𝐵
𝐼 = 2𝜇 ,
0
(4.16)
where E, B are the magnitudes of the electric field and magnetic field respectively. The
resonant thermal signal in equation (4.3) is proportional to the square of the microwave
magnetic field, thus any sample misalignment will result in reduced microwave magnetic
field and reduced thermal signal. From a comparison of a large Py thin film sample
between various FMR setups, this reduction in field strength could be as large as a factor
of 5x (data not shown). This would take the 2.3 μV signal at 54 mW down to 460 nV. In
principle this should still be visible as the detection sensitivity is still around 100 nV, but
Keff was calculated under the assumptions the Pt and SiNx contributions are constant
above 300 K—providing a lower bound on Keff. If Keff is larger than estimated, the signal
size will be further reduced.
4.3.3 Future improvements to experimental design
There are two paths to see the resonant thermal response: increasing the size of
the thermal response, and increasing the thermal sensitivity of the platforms. We will
first focus on increasing the signal size. There exist multiple ways in which to achieve a
larger thermal response. We could lower the Keff to increase the ΔT, or we could increase
the power delivered to the platforms. Lowering Keff is possible by using a metal, such as
101
Ti, with a lower thermal conductivity than Pt as leads on the legs. This would result in
more fabrication steps, and would result in roughly a factor of 2x reduction in Keff. As
the resonant thermal signal scales linearly with volume, the Py film could simply be
made thicker. However, the ultimate goal of this study is to thermally detect spin
pumping at interfaces. Increasing the thickness of the sample will reduce the surface to
volume ratio, and the sensitivity of any interfaces. Another way to increase the size of
the thermal response is by increasing the microwave power.
This is a more
straightforward method with potential for 4x increase (maximum 200 mW in the current
system, up from the 54 mW used in Fig. 4.5).
However, increasing the microwave power in the current setup has two technical
issues that need to be addressed before implementing. First, increasing the power past 54
mW needs to be done very slowly as large changes in applied microwave power have
been shown to blow out the Pt leads connecting to the Pt resistors on the platforms. This
is caused by eddy currents from the microwaves, and can be reduced by de-coupling the
platform resistors from the outside (the unconnected resistor is not affected by high
power). A large impedance resistor at the breakout box can be used to reduce the
magnitude of these eddy currents.
Second, the system becomes unstable at higher
powers, and the drift is increased as can be seen in Fig. 4.6 (a). The noise on the detector
circuit at higher power is also increased compared to 54 mW as shown in Fig. 4.6 (b).
102
Fig. 4.6 Left: field scan at 86 mW microwave power. The drift is tens of microvolts. Right: Error on
each data point for two different microwave powers. The system becomes unstable at higher powers
and is seen by much higher spread in the error.
The second approach to seeing resonant thermal response involves increasing
detection sensitivity. Platinum is not the most sensitive of resistance thermometers.
Material systems with activated transport are much more sensitive to thermal variations,
such as NbN12-13. It is estimated from preliminary growths of NbN by sputtering Nb in a
nitrogen atmosphere, that the increased sensitivity could be as much as 5x at room
temperature. This would take the 30 mK sensitivity of the current setup with Pt resistors
down to 6 mK. Although resistance thermometry allows for the measurement of absolute
temperature, a relative temperature measurement could be done with thermocouples. The
Au-Pd thermocouple used in Ref [10] has a claimed sensitivity of 300 μK at room
temperature. This would take our platforms current power sensitivity of 5.1 nW (170
103
nW/K Keff times 30 mK sensitivity) down to 50 pW (170 nW/K Keff times 300 μK
sensitivity).
4.4 Conclusion
We have adapted the thermally isolated SiNx platforms for thermally detected
magnetization dynamics measurements and built a custom vacuum system to be used in
conjunction with a commercial Bruker ELEXSYS E500 EPR cavity.
FMR
measurements were performed on a 60 nm Py film and no resonant thermal effects were
seen. Proposed changes for future attempts include sustainably increasing microwave
power (in a way that does not also increase noise and protects measurement electronics),
as well as different thermometry paradigms such as Au-Pd thermocouples with a capacity
to increase sensitivity by two orders of magnitude.
Once resonant magnetization
dynamics of single Py films are detected, additional layers will be added (both spinpreserving and non-spin-preserving) unlocking new spin dynamics in the form of spin
transfer into the attached film. Thermal detection of this spin pumping offers a new way
to study spin dynamics that is not subject to extraneous resonant voltage artifacts and the
ability to study small detection volumes.
104
4.5 References
1
K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa and E.
Saitoh, Nature 455, 778–781 (2008)
2
J. Flipse, F. Dejene, D. Wagenaar, G. Bauer, J. Ben Youssef, and B. van Wees, Phys.
Rev. Let. 113, 027601 (2014)
3
A. Avery, M. Pufall, and B. Zink Phys. Rev. Lett. 109, 196602 (2012)
4
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105
Appendix
A1: Photolithography with a maskless aligner
Prior to spinning down resist, the SiNx membranes shown in Fig. A.1 (a) are
attached to custom Al holders with Crystalbond shown in Fig. A.1 (b). The purpose of
this is two-fold: first, it allows for easier handling of the membranes during processing
(the portion of the SiNx frame safe for tweezer handling is only 1 mm wide). Second, the
indentation of the Al holder, as shown in the inset of Fig. A.1 (b), is machined to match
the thickness of the Si frame (200 μm) with a central post to mechanically support the
membrane.
The patterning is performed using a Heidelberg Instruments tabletop
maskless aligner system µPG501. This system blows pressurized nitrogen onto the
sample and uses the pressure from the backdraft to determine the focus on the sample.
The SiNx membranes bow under the pressure, and the central Al post limits the amount of
deformation preserving the focus of the laser. The best results occur when Crystalbond is
applied away from the central column to avoid attaching the membrane directly to the Al.
The level surface of the Al holder also helps with focusing as the maskless aligner has
difficulty focusing samples with small cross-sectional areas.
Once secured, the membranes are placed in UV ozone for 5 minutes. The SiNx
membrane samples often have photoresist adhesion difficulties. This UV ozone strips off
106
any organics on the surface and significantly improves resist adhesion.
Next, the
membranes are spin-coated with LOR3A, and S1805 resist. The parameters are: 6000
rpm for 60 seconds for both layers. LOR3A is spun down first, and is cured for 5 min at
170 °C on a hotplate. S1805 is spun down second and cured for 2 min at 115 °C. Prior
to spinning, it is important to place the membrane as close to center of the spinner chuck
as possible. This minimizes the pile up of resist around the edge and maximizes the
usable area on the Si frame for contacts. The centering can be achieved by rotating the
sample by hand and looking for nutation in the orbit.
Fig. A.1 (a) A purchased SiNx membrane from Ted Pella. The membrane is the square in the middle
with a side length of 750 μm. The overall sample is 3 mm in diameter. (b) Aluminum sample holder.
The circle is 200 μm deep with a center post. The side profile of the circle is shown in white.
While the Al sample holders limit the deformation of the membrane in the
maskless aligner, bowing of the membrane still occurs and needs to be corrected for
during the patterning. This is achieved by splitting the first patterning into two steps.
107
First, alignment markers are written onto the membrane (focus the maskless aligner as
well as possible on the membrane). The sample is then developed in MF-319 (a TMAH
based developer) for 60 seconds. After rinsing with DI water and blowing dry with
compressed air, the sample is patterned again for the Pt resistor and contact pads. The
alignment markers are used to focus properly on the membrane, and the freeze autofocus
option is selected during patterning. This approach works as long as the patterns written
on the Si frame have larger tolerances than those on the membrane (maskless aligner
feature size limit ~ 1 μm). If fine features (< 5 μm) are required on the Si frame as well,
then a separate patterning step will need to be undertaken with the Si frame as the focus.
The parameters for both patterning steps are 18 ms exposure time and -10 defocus. A
completed pattern can be seen in Fig. A.2. As a side note, the alignment markers should
not be imaged using the optical microscope prior to the second patterning step unless the
light is properly filtered to prevent exposure to the entire sample.
108
200 μm
Fig. A.2 A developed contact pattern using a LOR3A/S1805 bilayer on a SiN x membrane. The
aluminum holder post can be seen underneath the membrane.
A2: Metal deposition parameters on the Lab-18
All the films deposited in this dissertation were deposited (evaporated or
sputtered) in a Kurt J. Lesker Co. lab-18 thin film deposition system. Prior to metal
deposition, the membrane samples are cleaned to avoid any photoresist residue causing
problems during the metal liftoff step. This cleaning can be done in two ways: a 2 minute
UV ozone clean or an in-situ ion mill (for 10-20 seconds at 21 mA current at a base
pressure less than 5x10-6 Torr). Ion milling gives better results than UV ozone and is the
standard cleaning procedure for most of the samples presented in this work.
The parameters used for metal films mentioned in this thesis are given in Table 1:
109
Ti
Pt
Au
Py
Al
Rate
(Å/s)
0.2-0.5
0.5
1
0.5
1
Power
8-11 %
19-23%
4-6%
6-8%
350-400 W
Pressure
<5E-6
<5E-6
<5E-6
<5E-6
7 E-3
Tooling factor
247.9
224.2
230.2
271.1
355.9 (Gun 1)
Table 1: Parameters for every metal film deposited in this dissertation. Al shows power in Watts as it
is sputtered (DC), e-beam deposition is used for the rest of the metals.
The powers used for depositions will change over time as the system ages, and material is
replaced in the crucibles. In general, lower power will be required for newly filled
crucibles. It is a good idea to measure thickness of deposited films with AFM or
profilometer from time to time and adjust the tooling factor as needed to make sure the
deposited film thickness matches the desired thickness.
There are two metals that deserve special mention: platinum and permalloy. For
platinum deposition, the samples are placed on one side of the platen so that they are
directly over the e-beam crucible during deposition. The platen is not rotated during this
deposition, and metal buildup on resist sidewalls is minimized. The samples can also be
rotated during deposition, but the liftoff after deposition is worse (measured by Pt
remnants left behind). Py deposition also deserves a special mention as extra mounting
steps will need to be undertaken prior to deposition. For FMR measurements, it is
important to ensure that the only magnetic material in the microwave cavity is the film
under study. Therefore, it is necessary to mask off the edges of the sample with Kapton
110
tape prior to deposition to prevent Py deposition on the side of the sample. Any material
deposited on the sides will make the FMR spectra more complicated as the resonance
conditions for the films deposited on the sample’s side will be different from the film of
interest. This will lead to an artificially broadened resonance peak. Also important in the
Py deposition is the deposition rate at 0.5 Å/s (the narrowest linewidth Py deposited in the
system has used this deposition rate). The deposition rate affects the linewidth more than
the power used during the deposition therefore whatever power is necessary to reach the
0.5 Å/s rate should be used.
Metal liftoff with SiNx membranes is one of the highest failure steps for the
membrane. As a LOR3A layer is used, liftoff should be performed in Remover PG, a
proprietary blend from MicroChem with an active component of N-methyl-2-pyrrolidone
(NMP). LOR3A is not compatible with acetone, and will leave a residue that is difficult
to remove. After liftoff is completed (this takes 1-2 hours), IPA is used to rinse the
sample to reduce any residue from the NMP. Better liftoff results are achieved if the
samples remain submerged in IPA and placed under a stereoscope for inspection and
agitation with a pipette.
111
50 μm
Fig. A.3 A Pt contact pattern after liftoff in NMP. Incomplete liftoff is seen as the dark lines just off
the membrane (rolled Pt), as well as Pt in the gaps of the closely spaced leads on the membrane.
An example of poor liftoff is seen in Fig. A.3. Processing is continued for samples
where the Pt fails to completely liftoff. It is observed that continued processing on
sample with imperfect liftoff can oftentimes improve the liftoff. If it does not, FIB
milling is employed after fabrication is complete to locally remove any metal that shorts
the leads. However, in the case of breaks or gaps in the pattern the sample is discarded.
A3: E-beam patterning of SiNx membranes
Electron beam (e-beam) patterning is done in a FEI Helios Nanolab 600 dual
beam focused ion beam/scanning electron microscope. Nabity NPGS software is used to
translate the designCAD files into a rastering electron beam. For all e-beam patterning,
the magnification factor should be set at 145,000. This is a geometrical factor that
translates the feature sizes in the pattern file to feature sizes on the sample.
112
This
parameter can be found in options → system files → PG.system. It is important to check
prior to patterning as this magnification factor is different for FIB patterning.
Historically prior to developing the photolithography process mentioned above, ebeam patterning alone was used for the different metal layers. After the photolithography
process was developed, only one e-beam patterning step was maintained: masking the
legs and platform with resist prior to Al deposition. E-beam patterning is described here
for the general situation, in case smaller feature sizes than 1 μm are needed in the future.
For an easier liftoff, a bilayer of poly (methyl methacrylate) (PMMA 950 in A4) and
methyl methacrylate (MMA EL 11) resists are used. The spin and bake parameters are
6000 rpm for 60 seconds (both layers), and a bake at 160 °C for 30 seconds for MMA
(first layer) and at 180 °C for 5 minutes for PMMA (second layer). Centering of the
samples is just as important as for the photoresist described earlier. For liftoff using these
resists, acetone is sufficient to dissolve the resist (liftoff time 1-2 hours).
For any e-beam patterning, it is important to correctly focus the e-beam onto the
sample at the same current that the pattern is written at (focused at the same or better
resolution than the highest magnification used during patterning). In some cases this is
enough, but for fine control over a spatially varying focus (for example if the sample is
tilted) a focal plane can be defined and the focus controlled to correct for any variance.
In order to do this, four scratches are made in the resist (in the PMMA, or MMA/PMMA
bilayer) around the edges of the sample to make features to focus on. Four spots are used
instead of the minimum three spots to account for any error in focusing. The steps to
implement this in NPGS are as follows: go to commands → direct stage control → and
113
press enter. Next, maneuver the stage to the first spot of interest. Adjust the focus and
record the position and apparent height (z-position) of the stage in the NPGS program
and proceed to the next stop. A linear fit is made once all the points have been entered.
The option in the run file is selected to allow for automated focus control (Disable X-Y
focus control set to no).
SiNx is an insulating material. If the samples aren’t grounded properly during
patterning, charging effects can occur causing the patterns to buckle and distort.
Sometimes it is necessary (especially if there are no existing platinum leads on the
membrane), to deposit a 5 nm Al charging layer on top of the resist to avoid these effects.
This layer is e-beam deposited at very low powers (< 3%) for 0.2 Å/s to avoid exposing
the resist (from the backscattered electrons). A new crucible may be needed to achieve a
deposition rate at these the low powers
The purpose of the e-beam patterning step discussed in chapter 3 is to protect the
Pt legs and platform from the deposited Al layer. It is meant as a negative pattern,
therefore large areas on the membrane are exposed leaving just the resist on the platforms
as can be seen in the pattern file shown in Fig. A.4. This step is included in case MF-319
etches or modifies the material system of interest. For the Py films mentioned in chapter
4, this step is not included as the FMR spectrum of Py exposed to MF-319 is the same as
pristine Py (indicating no chemical sensitivity to MF-319). The patterning is done at 10
kV and the relevant parameters are:
114
Layer 1, 2, 3: Window
Origin offset: 17.89,-18.27
Magnification: 100
Center to center distance and line spacing: 154.88
Configuration parameter: -5
Measured beam current: 86 pA
Area dose: 5
Layer 4:
Origin offset: 17.89,-18.27
Magnification: 100
Center to center distance and line spacing: 22.13
Configuration parameter: -5
Measured beam current: 86 pA
Area dose: 220
Layer 5:
Origin offset: 22.68, -23.46
Magnification: 80
Center to center distance and line spacing: 110.63
Configuration parameter: -4
Measured beam current: 170 pA
115
Area dose: 240
Layer 6:
Origin offset: 22.68, -23.46
Magnification: 80
Center to center distance and line spacing: 110.63
Configuration parameter: 0
Measured beam current: 2700 pA
Area dose: 240
Total write time: 28:34
The first three layers are windows for aligning this pattern to the sample. Properly
defined layers in the DesignCad file with line style set as ‘hidden’ show up as overlays
during the alignment process. These overlays are matched to the features on the sample
for alignment. Layers 4, 5 are included to expose the resist near the legs and platform at
lower currents and dosage to avoid electron scatter exposing the resist intended to remain
behind. Layer 6 is high current and intended to expose the resist far from the legs where
electron scatter is not as large of concern. The offsets given should be close to the actual
offsets as these offsets do not change substantially over time.
After exposure, the
samples are developed in 1:3 MIBK:IPA for 1 minute followed by a DI water rinse.
Vigorous stirring is recommended as large swatches of resist are being etched away and
the agitation helps to remove the resist quicker.
116
Fig. A.4 E-beam pattern for the masking platforms and legs processing step prior to aluminum
deposition. Different colors represent different layers. The green crosses are the fiducial markers for
aligning to the existing Pt contacts. Red and blue layers are at lower doses and currents than most of
the pattern (yellow) in order to avoid exposing resist intended to remain behind (on the legs and
platform).
A4: FIB patterning of SiNx membranes
Prior to the FIB patterning of the SiNx membranes, the samples are removed from
the aluminum holders introduced in section A.1 via an acetone soak (1-2 hours). They
are re-attached to a small Si substrate (< 5 mm on a side) with Crystalbond. Care is
maintained to ensure no Crystalbond is present directly underneath the membrane as it
carbonizes under contact with the ion beam. The SiNx samples can also be attached with
117
an indium bond, as long as the material under study isn’t air sensitive at 270 °C (the
melting point of In). Equal care must be taken to ensure no In is directly underneath the
membrane as re-deposition of In will occur during FIB patterning. One advantage of
using In to attach the SiNx membranes is the survival of sample adhesion during the Al
etch/solvent chain after the patterning. Sample handling without a carrier wafer is much
more likely to result in a broken sample.
The FIB patterning is performed using a FEI Helios Nanolab 600 dual beam
focused ion beam/scanning electron microscope. The Nabity NPGS software is used
(similar to the e-beam patterning) to translate a designCAD file into a rastering Ga beam.
For all FIB patterning, the magnification factor should be set at 179,000 (different from
the e-beam factor of 145,000). FIB patterning is done with the sample stage at a 52
degree tilt (where the sample is level for the ion gun/detector). There are additional steps
which must be taken prior to FIB patterning compared to e-beam patterning to prevent an
ion beam drilling a hole in the sample being patterned. There is an external switch box
which controls which beam blanker (FIB or e-beam) the NPGS software has access to.
To enable FIB patterning, the button on the switch box labeled enable I Blink ON must
be pressed first. The button labeled Ext I Scan ON must be pressed next. Finally on the
desktop of the support computer, the icon Ion blanker enable must be double clicked.
The pattern run file may now be started and patterning commenced.
The field of view for the ion beam (minimum magnification is 150x) is such that
it requires multiple cutting stages in order to complete the platform pattern. This can be
seen in Fig. A.5 which shows the entire FIB pattern. The dashed boxes show each
118
cutting step with the order of operations being: middle patterns are cut first, followed by
the bottom patterns and finally the top patterns. For the dual platform design, it takes a
total of six cutting steps.
Fig. A.5 FIB master pattern for machining the platform and legs in the SiNx membrane. The white
dashed lines are not part of the pattern, but rather indicate the field of view for each section. The
horizontal blue lines are to cut the SiNx remnants into smaller pieces in order to facilitate removal.
119
The relevant parameters for a single layer are:
Layers: 1,2,3 Window
OriginOffset: 0,0
Magnification: 151x
Center to center distance and line spacing: 144.71
Config parameter: -9
Measured beam current: 9.7 pA
Layer 5: (vertical)
OriginOffset: 0.4, 0.6
Magnification: 151x
Center to center distance and line spacing: 36.18
Config parameter: 4
Measured beam current: 21000pA
Area dose: 430,000
Layer 7: (horizontal)
OriginOffset: 0.4, 0.6
Magnification: 151x
Center to center distance and line spacing: 36.18
Config parameter: 4
Measured beam current: 21000pA
Area dose: 430,000
120
Write time: 12:33
All patterning is done at 30 kV. The first three layers are for alignment to the existing Pt
pattern.
The origin offsets given are the most recently used.
This offset changes
regularly when the Ga source is replaced (which occurs roughly twice a year). It is
recommended a test pattern is written prior to cutting the SiNx membrane as the tolerance
of misalignment of the current patterns is very low (1-2 μm).
The vertical lines are written first to ensure clean separation of the SiNx debris
from the platform and legs.
This debris can be seen in Fig. A.6 (a) and is only
electrostatically attached. It is removed upon the MF-319 aluminum etch and solvent
clean. The etch process is as follows: immersion in MF-319 for twenty minutes followed
by a dip in de-ionized (DI) water then a dip in IPA and acetone. Acetone is used to clean
off the PMMA if the platforms and legs were masked. This step usually requires 10-15
minutes of soak time before immersing back in IPA for an additional 5 minutes. This last
immersion step is to ensure that IPA diffuses all around the platforms as IPA has a lower
surface tension than acetone. The sample is then taken out and left to air dry (around 5
minutes).
121
(a)
(b)
(c)
200 µm
100 µm
Fig. A.6 (a) An electron micrograph of the platforms after the FIB cutting step. (b) The platforms after
a post-FIB session cutting the support legs (after the aluminum etch and solvent clean/dry).
Another feature which can be seen in Fig. A.6 (a) is the addition of support legs
connecting the legs and platforms to the rest of the membrane. These supports were
added to increase the yield from the etch step. They significantly reduce the chance of
platforms breaking during the air dry step (after IPA immersion). These supports are
removed in a second FIB session where local cuts are made resulting in Fig. A.7 (b).
A5: Measurement wiring
For SiNx membrane samples, the pads designed for wire bonding are located
within a few hundred microns of the membrane due to photoresist edge bead limitations.
This makes wiring these membrane samples challenging as the feature sizes (150 μm) are
near the size limit for making indium bonds by hand. Wire bonding is used for the
experimental setups that allow it (thermally detected FMR), but due to space limitations it
122
is not feasible for the calorimetry measurements. These space limitations are set by the
spatial dimensions of the Cu thermal radiation shield shown in Fig. A.7 (a). Two steps
are needed to wire the calorimetry samples in the thermal shield. First, indium bonds are
made on the Si frame and 0.001” diameter Cu wire is pressed into the In using two high
precision tweezers (one to hold the wire in place and the other to press into the In). For
stabilization, it is recommended that both elbows and wrists are braced on the table.
Because the wires will need to be manipulated into place, I recommend burying the wire
in the In not merely pressing it. This can be accomplished by rolling the existing In
around the wire or cutting and placing a small amount of In on top of the wire. The
second step is to bend the wires so that they are vertical and place the sample inside the
thermal shield—the sample is attached to the sample puck with rubber cement. Gently
bend the wires near the desired contacts and press into place. The reason for wiring the
sample before placing it in the thermal shield is due to stabilization concerns. It is
difficult to brace the wrists working with the elevated thermal shield and reaching down
to work with the sample.
123
(a)
(b)
Fig. A.7 a) Wired membrane sample to a PPMS puck with custom Cu thermal shield. b) The
completed sample puck with the thermal shield screwed into the bottom copper piece
When wiring a sample, proper grounding must be used through an anti-static wrist
band. Due to the small lateral dimensions of the Pt resistors used for thermometry (~2
μm wide), static discharges frequently result in blown Pt leads. One such example can be
seen in Fig. A.8 (a). To avoid this, no special precaution (other than anti-static wrist
strap) is needed for the wiring in the calorimetry measurement in the PPMS setup.
However for the FMR setup, a grounding box was made and included to ground the
contacts together while making and breaking cable connections. The procedure for this
is: all switches start grounded 1) flip the instrument ground first 2) flip the sense high (for
a four terminal measurement) 3) flip the sense low 4) flip the instrument high. This order
maximizes the series resistance by making sure the discharge current goes through the 1
kΩ resistor on the platform.
124
In case precautions fail and the Pt leads experience a blow-out, Pt patches can be
made using the dual beam Helios. By flowing a pre-cursor gas and using a heated Pt
needle, local Pt deposition can be achieved by passing a large enough electron current to
ionize the gas.
Pt deposited in this manner contains carbon, and exhibits inferior
conduction properties; yet it is sufficiently conductive to locally repair a break in the Pt
leads. Two such patches can clearly be seen in Fig. A.8 (b). Pt patching is done at 5 kV
and at eucentric height, 0° tilt. The e-beam currents used are 2-5 μA. Because of
charging effects, the best results are obtained by wiring the Pt leads directly to the carbon
tape/system ground of the SEM sample puck. Drift will still be seen at high currents and
magnifications. While Pt patching is useful, it is unreliable with breaks located far away
from the Si frame being much more difficult to patch due to Pt failing to deposit.
(a)
(b)
5 μm
5 μm
Fig. A.8 a) Pt leads after a static discharge overheated them. The rounded Pt droplets indicate
melting. b) Pt leads with Pt patches deposited using the dual beam Helios system.
125
A6: Miscellaneous
The size of the thermal platforms (100 μm per side) is motivated by a compromise
between low heat capacity, and large thermal time constant. The thermal time constant
constraint arises from the model of lock-in (7265 Signal Recovery) used to measure the
thermal decay. The lowest time step for the internal buffer is 5 ms which limits the range
of thermal time constants that can be fitted by an exponential decay (15 ms or higher).
The width of the SiNx legs (10 μm) is selected due to a combination of constraints
between the FIB processing and photolithography of the Pt leads. Any reduction of this
width will increase the thermal isolation of the platforms. However, the 3 μm wide Pt
leads (near the resolution limit of the maskless aligner) require a 2 μm gap to avoid
electrical shorting the leads (based upon the pattern roughness and ease of liftoff); and the
FIB alignment is only tolerant to within 1-2 μm. Thus while the stated leg width is 10
μm, the actual width varies between 7-10 microns between FIB misalignment and
rounding of the edges (from FIB Gaussian spot profile). The 10 μm width of the legs
represents the lowest feasible leg width using the current processing techniques.
The thickness of the Pt leads (35 nm) was selected based upon similar thickness
used in the literature (ref 3.22). No variation of this thickness was attempted. Based
upon the meander pattern on the platforms, the resistance with this thickness was ~1 kΩ.
The heating current and detection schemes were optimized for this resistance. Any
variation of this resistance will require a re-optimization of the heating current, AC sense
current, AC sense frequency, and load resistor. All temperature measurements using Pt
126
resistivity were calibrated during the measurement by the in situ PPMS thermometry.
Thus each resistor has its own resistance vs T and polynomial fitting to extract T.
127
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