Localized Ferromagnetic Resonance Using Magnetic Resonance Force Microscopy

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LOCALIZED FERROMAGNETIC RESONANCE USING MAGNETIC RESONANCE FORCE MICROSCOPY

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Jongjoo Kim, B.S.,M.S.

*****

The Ohio State University

2008

Dissertation Committee: Approved by

P.C. Hammel, Adviser D. Stroud Adviser F. Yang Graduate Program in K. Honscheid Physics

ABSTRACT

Magnetic Resonance Force Microscopy (MRFM) is a novel approach to scanned probe imaging, combining the advantages of Magnetic Resonance Imaging (MRI) with Scanning Probe Microscopy (SPM) [1]. It has extremely high sensitivity that has demonstrated detection of individual electron spins [2] and small numbers of nuclear spins [3].

Here we describe our MRFM experiments on Ferromagnetic thin ﬁlm structures.

Unlike ESR and NMR, Ferromagnetic Resonance (FMR) is deﬁned not only by local probe ﬁeld and the sample structures, but also by strong spin-spin dipole and exchange interactions in the sample. Thus, imaging and spatially localized study using FMR requires an entirely new approach.

In MRFM, a probe magnet is used to detect the force response from the sample magnetization and it provides local magnetic field gradient that enables mapping of spatial location into resonance field. The probe field influences on the FMR modes in a sample, thus enabling local measurements of properties of ferromagnets. When sufficiently intense, the inhomogeneous probe field defines the region in which FMR modes are stable, thus producing localized modes. This feature enables FMRFM to be important tool for the local study of continuous ferromagnetic samples and structures.

ii In our experiments, we explore the properties of the FMR signal as the strength of the local probe ﬁeld evolves from the weak to strong perturbation limit. This un- derlies the important new capability of Ferromagnetic resonance imaging, a powerful new approach to imaging ferromagnet. The new developed FMR imaging technique enables FMR imaging and localized FMR spectroscopy to combine spectroscopy and lateral information of ferromagnetic resonance images [4][5]. Our theoretical approach agrees well with spatially localized spectroscopy and imaging results. This approach also allows analysis and reconstruction of FMR modes in a sample.

Finally we consider the effect of strong probe fields on FMR modes. In this regime the probe field significantly modifies the FMR modes. In particular we observe the complete local suppression of the FMR mode under the probe. This provides as a new tool for local study of continuous ferromagnetic thin films and microstructures.

iii To Sohyun and Reeya

iv Acknowledgement

First of all, I would like to express my sincere gratitude to Professor P. Chris

Hammel for his endless interests, encouragements, supports, and guides to Physics.

The first day when I knocked his door and decided to join the best Physics group is not unforgettable in my life. I found the future, the attitude, the energy, and the promise as one of physicists in the first day. In the group, he kindly opened any topics during so-called coffee hours. Everyday I have learned and exchanged valuable ideas and social news. This activity also made me move toward the goal of my life.

In addition, there was another luck with the group because I met solid scientists and mentors, and friends. I am grateful to Dr. Tim Mewes who taught and promoted my interests to the FMRFM system and treats me as a friend, Dr. Denis Pelekhov who supported our results of FMRFM and helped all other things even though they were not related to physics, Prof. Philip E. Wigen who regularly visited me and gave endless interests and discussions, Dr. Palash Banerjee who prepared for many cantilevers and advices for me, and ﬁnally Dr. Yuri Obukhov who could not be forgotten through my research and life in many aspects.

Dr. Yuri Obukhov was not only the physicist who has endless promotions but also the right mentor who keeps suggesting the guideline in my future research. My physics was being developed and becomes matures with him. His attitude was always appreciated because he tried to make me understand hidden physics and problems.

Sometimes we spent many hours or days to solve some problems.

With all scientists and friends whom I mentioned above, I thank Ross, Inhee,

Jay, Vidya, Mike, Claudia, Tom, Bob, and Physics department at the Ohio State

Univerisity.

v Finally I appreciate Sohyun, Reeya, mothers, and Woojae who are always standing with me.

vi TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

List of Figures ...... xi

Chapters:

1. Introduction ...... 1

1.1 Motivation of the study of ferromagnetic nanostructures ...... 1 1.2 Strengths and weaknesses of conventional FMR ...... 1 1.3 New approach to magnetic resonance using Force detection ..... 2 1.4 History of MRFM ...... 3 1.5 History of FMRFM ...... 3 1.6 Chapter summary ...... 5

2. Basic concept of Magnetic Resonance Force Microscopy ...... 8

2.1 Introduction ...... 8 2.2 Brief description of Magnetic Resonance ...... 8 2.3 Geometry of MRFM experiment ...... 10 2.4 Force detection ...... 11 2.5 Measurement approach in MRFM ...... 12 2.6 Comparison of force sensitivity in MRFM and MFM ...... 13 2.7 Force noise ...... 14 2.8 Strengths and weaknesses of mechanical detection ...... 15

vii 3. Theory of Ferromagnetic Resonance ...... 17

3.1 Summary ...... 17 3.2 Definition of effective magnetic field Heff ...... 18 3.2.1 Zeeman Energy ...... 18 3.2.2 Demagnetizing energy ...... 18 3.2.3 Anisotropy energy ...... 19 3.2.4 Exchange energy ...... 19 3.3 Spin dynamics in ferromagnetic materials ...... 20 3.4 Herring-Kittel equation ...... 22 3.5 Spin dynamics in thin ferromagnetic film ...... 24 3.6 Damon-Eshbach approach ...... 25 3.7 Comparison between Kalinikos-Slavin and Damon-van de Vaart dispersion relations ...... 26 3.8 Dispersion relation for our experimental conditions ...... 27 3.9 Dispersion relation in presence of inhomogeneous demagnetizing field 27

4. FMRFM Experimental set-up ...... 29

4.1 Introduction ...... 29 4.2 Design of the experimental set-up ...... 30 4.3 Cantilever characterization ...... 34 4.3.1 The measurement of cantilever spring constant ...... 34 4.3.2 The measurement of quality factor of cantilever ...... 35 4.3.3 Characterization of probe magnet ...... 36 4.3.4 Measurement of the probe ﬁeld gradient ...... 39 4.4 Detection of the cantilever position ...... 40 4.4.1 Fiber optic interferometer ...... 40 4.4.2 Measurement of the probe-sample distance ...... 43 4.5 Microwave Resonator ...... 45 4.6 Conclusions ...... 46

5. FMRFM spectroscopy in circular disk array ...... 47

5.1 Introduction ...... 47 5.2 Experimental conditions ...... 47 5.2.1 Sample properties ...... 47 5.3 Cantilever properties ...... 48 5.3.1 Properties of the probe magnet ...... 49 5.3.2 FMRFM measurement protocol ...... 51 5.4 FMRFM spectra ...... 53

viii 5.5 FMRFM in antiparallel conﬁguration ...... 54 5.6 Quantization of FMR modes in a disk sample ...... 56 5.7 FMR images using local spectroscopic information ...... 58 5.8 Conclusion ...... 60

6. Ferromagnetic resonance with weak ﬁeld perturbation ...... 62

6.1 Motivation ...... 62 6.2 Experimental conditions ...... 63 6.2.1 MW resonator ...... 64 6.2.2 Properties of new probe magnets ...... 65 6.2.3 Force noise of the cantilever ...... 65 6.2.4 New protocol to detect the force signal ...... 65 6.3 Results ...... 68 6.3.1 FMR Spectra for center of permalloy disk ...... 68 6.3.2 New scanning method: spatially resolved FMR spectroscopy 71 6.3.3 Spatially resolved FMR spectroscopy for the second order FMR mode ...... 72 6.3.4 FMR with weakly perturbing probe field ...... 72 6.3.5 Reconstruction of the probe magnetic field ...... 77 6.3.6 The influence of the probe magnetic field into FMR mode . 77 6.3.7 Observation of FMR mode splitting ...... 78 6.3.8 FMR with strongly perturbing probe field ...... 80 6.4 Conclusion ...... 82

7. FMR Mode Suppression at strong ﬁeld perturbation ...... 83

7.1 Motivation ...... 83 7.2 Experimental conditions ...... 84 7.2.1 An isolated disk sample ...... 84 7.3 Probe magnet ...... 85 7.3.1 Signal measurement method ...... 86 7.4 Results ...... 88 7.4.1 Strong probe field behavior of FMRFM spectra ...... 88 7.4.2 Theory of FMR in a strongly perturbing field ...... 89 7.4.3 Numerical simulation results ...... 94 7.4.4 Spatially resolved scanning influenced by the strong probe field 94 7.5 Conclusion ...... 95

8. Conclusion ...... 97

ix Bibliography ...... 99

x LIST OF FIGURES

Figure Page

2.1 Schematic of MRFM. The sample is positioned in the external magnetic

field Hext and Microwave field HMW. The cantilever with a magnetic probe is positioned above the sample. The position of the cantilever is detected by the fiber optic interferometer. The red region corresponds to the magnetic resonance slice where spins meet the resonance condition. The red dotted curves show inhomogeneous magnetic field of the probe magnet...... 10

2.2 Schematic comparison of MRFM and MFM force sensitivity in the case when the force is defined by dipole-dipole interaction. The blue curve shows the force-distance curve. The red curve corresponds to the force-distance curve when the spins of the sample are flipped. The horizontal axis is the distance from a sample to the probe magnet.The oscillation of the probe position is shown by the blue sinusoidal curve. The force response in MFM case is shown by the blue arrows and marked δFMFM. The force response in MRFM case is shown by the red arrows and marked δFMRFM. The figure shows the advantages of MRFM detection in comparison with MFM...... 13

3.1 The illustration of the dynamics of the magnetic moment M in the presence of the effective magnetic field Heff . The precession frequency is defined as ω. The black vector τ represents the torque acting on the magnetic moment M...... 20

3.2 The schematic is illustrating the orientation of the spin wave k vector relative to the effective magnetic field. The z-direction corresponds to the direction of effective magnetic field. The Θ is the angle between the effective magnetic field and k-vector of spin wave...... 23

3.3 The dispersion manyfold derived by C. Herring and C. Kittel (see text). 23

xi 3.4 The dispersion relations for a thin ferromagnetic film saturated in the external field perpendicular to the film plane. The red curve represents the Kalinikos-Slavin dispersion relation (See Eq. 3.18). The blue curve is the dispersion relation obtained by Damon and van de Vaart (See Eq. 3.21). The dispersion relations are shown with the assumption ωH=ωM, following Damon and van de Vaart [6]...... 26

4.1 The schematic of the MRFM set-up. The reddish square object is the XYZ superconducting magnet. The 1 inch diameter FMRFM experimental stages are located inside the XYZ superconducting magnet. . 31

4.2 Schematic of low temperature FMRFM experimental stages (see text). 32

4.3 Schematic of the cantilever motion in the presence of the external magnetic ﬁeld. The red arrow represents the magnetization of the probe magnet...... 36

4.4 The dependence of the probe field gradient on the probe-sample distance. Inset shows the FMRFM spectra for thin film DPPH at different probe-sample separations. The bottom spectrum is taken at 400 nm away from the sample surface and the top spectrum is taken at 880 nm away. The black arrows show the field where the sensitive slice of the probe magnet enters DPPH sample. Resonance field pointed by the black arrows is used for calculation of the probe magnetic field and its gradient (See text)...... 38

4.5 Schematic of ﬁber optic interferometer(see text)...... 41

4.6 The dependence of signal of ﬁber optic interferometer on the probe- sample distance. The red arrow shows the position where the cantilever touches the sample surface. This position is obtained by the primary ﬁber optic interferometer used for the detection of cantilever position, (See text) ...... 43

4.7 Schematic and geometrical sizes of MW resonator used in our experiments (Designed by D. Pelekhov)...... 44

xii 4.8 The S12 and S11 reﬂection parameters for MW resonator. The red and blue curves represents S parameters obtained for our resonator using simulation software (Ansoft HFSS). The black and green curves are the experimental results (See text)...... 46

5.1 A patterned array of 1.5 µm diameter and 50 nm thickness NeFe disc is used. The center to center separation is 1.8 µm. The patterned array is deposited on a low doped Si substrate using E-beam lithography (Provided by S. Batra at the Seagate Technologies in Pittsburgh, USA.). 48

5.2 A SmCo particle was glued at the end of a commercial Si cantilever. The dimension of the cantilever was 450 µm long and 30 µm wide. After gluing the SmCo particle, it was transferred into a Focus Ion Beam milling facility and then its shape was milled at low E-beam energy. The sharp tip was intentionally milled in order to provide strong inhomogeneous magnetic ﬁeld (Ref. [4])...... 49

5.3 The resonance frequency of a cantilever with a SmCo probe is monitored in changing the external magnetic ﬁeld. Initially the external magnetic ﬁeld is set at 9 kG and then it changes from 9 kG to -9 kG and from -9 kG to 9 kG...... 50

5.4 FMRFM spectra were measured at 7.7 GHz with 60 % amplitude modulation depth. Each spectrum was measured at different probe-sample separations shown in Figure. The external magnetic field is applied in the same direction of the probe magnetic field. The solid vertical lines provide to keep tracking the FMR resonances which are independent of the probe-sample separations (Ref. [4])...... 52

5.5 FMRFM spectra were measured at 7.7 GHz with 60% amplitude modulation depth. Each spectrum was measured at different probe-sample separations shown in Figure. The external magnetic field are applied in the opposite direction of the probe magnetic field. The solid vertical line guides the position of each zero field resonance (Ref. [4])...... 55

xiii 5.6 An FMRFM spectrum was measured at 7.7 GHz. The amplitude of MW excitation was modulated at the resonance frequency of the cantilever with a depth of 60%. During the measurement, the probe- sample separation was kept at 0.4 µm. The external magnetic ﬁeld was applied in the same direction of the magnetic moment of the cantilever. For each FMR modes, the transverse components of magnetization were sketched. Up to sixth order FMR modes were observed in the spectrum (Ref. [4])...... 56

5.7 Experiment results compared to theoretical results based on the dispersion relation done by Kakazei et al. [7]. The solid line corresponds to theoretical results while disks are the measurements of FMRFM at a few diﬀerent MW frequencies...... 57

5.8 Images were obtained at 150 nm as tip-sample separation at 7.7 GHz and 60% amplitude modulation. The lower right panel shows the FM- RFM spectrum obtained with the probe located at the center of a permalloy disk. Each image was obtained when the external magnetic ﬁelds was set at 11960, 11980, and 12040 G respectively in the parallel conﬁguration (Ref. [4])...... 59

6.1 Cantilevers magnetometry data for probes, used in our experiments. (a) The SEM picture (inset) and magnetometry data for cantilever used in experiments presented in this Chapter. (b) The SEM picture (inset) and magnetometry data for cantilever used in experiments presented in Fig. 6.6 (The probe in (a) was provided by R. Steward and the probe (b) was provided by P. Banerjee)...... 64

6.2 FMRFM spectra of an array of 2 µm diameter permalloy dots acquired for two orientations of the external magnetic field Hext relative to the probe magnetic moment mtip (tip, shown in Fig. 6.1a): parallel (a) and antiparallel (b). The first order TFR modes of the dots close to the micromagnetic probe are indicated as peaks 2 and 3, and the first and the second order ZFR magnetostatic modes are indicated as peak 1 and 4, respectively; these arise from dots far from the probe tip (tip-sample separation 200 nm). The spectra were acquired with the probe magnet located directly over the center of one of the dots as shown in the inset which also schematically indicates the dipolar gradient pattern of the probe magnet mtip...... 69

xiv 6.3 Lateral-Field Scan for external field parallel to the tip magnetization (tip, shown in Fig. 6.1a): (b) Lateral-Field Scan for 1st order mode. FMR spectra recorded by sweeping the external magnetic field Hext and spatially scanning the probe along the one-dimensional trajectory indicated by the dotted line above the diagram of the dots in (a) (tip- sample separation is 200 nm). The dashed curves in (b) show the analytically calculated dependence of the TFR resonance field on probe position (see Eq. 6.2); these agree excellently with the experimental data. Arrows show the signal from the rows of the dots aside the tip trajectory (adjacent rows). (d) Spectrum extracted from image (b) with the tip located over the center of the dot [see panel (a)]. Numerical labels are the same as in Fig. 6.2a. (c) Lateral-Field Scan for 2nd order mode. Arrows show the splitting of the signal from adjacent rows. (e) Spectrum extracted from image (c) with the tip located over the center of the dot. Blue colored images are experimental fixed-field ((f) Hext = 12.23 kG; (g) Hext = 12.27 kG; (h) Hext = 12.30 kG; (i) Hext = 12.34 kG; (j) Hext = 12.38 kG) 2D images (4.8 µm × 4.8 µm) of the spatial X-Y variation of the FMRFM signal. Dotted line in (f) shows the direction of lateral scan used for (b) and (c) and presented in (a). 73

6.4 Components of convolution problem used for theoretical prediction (see Eq. 6.2) of Lateral-Field Scan presented in Fig. 6.3b by dashed line. z −11 z-component of the tip moment mtip = 1.1 · 10 J/T, lateral com- x −11 ponent of tip moment mtip = 0.55 · 10 J/T. Eﬀective tip dipole to st 2 sample distance z0 = 4 µm. (a) 1 order mode µ (r) for permalloy dot of our sample. (b) Tip ﬁeld. (c) theoretical prediction for signal Fig. 6.3b obtained by convolution of (a) and (b). Dashed line in Fig. 6.3b correspond to the data extracted from (c) along dashed line. 74

xv 6.5 Lateral-Field Scan for external field anti-parallel to the tip magnetization (tip, shown in Fig. 6.1a): (c) Lateral-Field Scan for 1st order mode. FMR spectra recorded by sweeping the external magnetic field Hext and spatially scanning the probe along the one-dimensional trajectory indicated by the dotted line above the diagram of the dots in (a) (tip-sample separation is 200 nm). The dashed curves in (c) show the analytically calculated dependence of the TFR resonance field on probe position (see Eq. 6.2); these agree excellently with the experimental data. (e) Spectrum extracted from image (c) with the tip located over the center of the dot [see panel (a)]. Numerical labels are the same as in Fig. 6.2b. (b) Lateral-Field Scan for 2nd order mode. (d) Spectrum extracted from image (b) with the tip located over the center of the dot. Blue colored images are experimental fixed-field ((f) Hext = -12.77 kG; (g) Hext = -12.75 kG; (h) Hext = -12.70 kG; (i) Hext = -12.67 kG; (j) Hext = -12. 64 kG) 2D images (4.8 µm × 4.8 µm) of the spatial X-Y variation of the FMRFM signal. Dotted line in (f) shows the direction of lateral scan used for (b) and (c) and presented in (a)...... 75

6.6 Spatially resolved FMR scans for tip presented in Fig. 6.1b. (a) and (b) are the Scans made at tip-sample separation z = 1.9 µm and z = 1.4 µm respectively, that shows behavior excellently described by small perturbation approach (see dashed lines). Scan (c) was acquired at z = 1 µm and shows a strong deviation from small perturbation approach (see dashed line) and demonstrate behavior better described by appearance of localized mode (solid line)...... 79

7.1 (a)The sample is prepared using Optical Lithography on low doped Si Substrate whose thickness is 125 µm. Each disk has 5 µm as diameter and 40 nm as thickness. The interdisk separation is 25 µm from center to center. (b) The MFM image is taken using FMRFM aparatus at 4K. Before FMR measurement from this diluted disk sample, we found one of disks using MFM image and the probe is located at the exact the center of the found disk sample...... 84

xvi 7.2 (a) The magnetic moment at various external fields is shown. The red arrows shows the direction of the external field change. Initially it starts from 2T to -2T. After that, the field changes from -2T to 2T. In FMRFM measurement, it performs from 1.3T to 1T to prevent unwanted magnetic moment change. Within the experiment range from 1T to 1.3T, the magnetic moment is approximately 1.0 · 10−12 J/T. (b)The top view of the SEM image of the probe was shown. The probe was sitting on the chopped off tip of the cantilever. (c) The side view of the SEM image of the probe was shown. From (b) and (c), the cubic shape of the probe was prepared after FIB process. The dimension of the cube was 1 × 1 × 1µm. (The probe provided by I. Lee) 85

7.3 FMRFM spectra of an individual 5 µm diameter permalloy disk at various tip-sample separations z. The magnetic probe for all spectra was positioned in the center of Py disk. Spectra are oﬀset in vertical direction for convenience. The distance z change from top spectrum to bottom as z = 1669; 1552; 1434; 1316; 1199; 1081; 963; 846; 728; 610; 493; 375; 258 nm. The magnetic moment of the probe magnet is parallel to the external magnetic ﬁeld...... 87

7.4 Dependance of FMR resonance field for 1st order mode on probe sample separation z. Circus marked line is experimental data taken from Fig. 7.3 . Dashed line is a theoretical calculation in assumption, that FMR mode shape was not changed by probe field, performed using Eq. 7.3 (see text). Squares marked line is numerical simulation. 2D inserts represent the mode shape at different distances z, obtained by numerical simulation...... 89

7.5 Numerically calculated spatial profile of the first order FMR mode m excited in a 5 µm diameter, 40 nm thick Py disk (4πMs = 11 kG) in an external magnetic field Hext = 13 kG: a) uniform external field, b) in the presence of the field from a spherical magnet with magnetic moment 10−12 J/T located 250 nm above the center of the disk with tip magnetic moment of the probe magnet parallel to Hext. The mode is confined out of the region of strong field beneath the probe. The mode amplitude plotted along a line through the center of the dot is shown with the dotted line; the corresponding magnitude of the total magnetic field Heff is shown with solid line; the dashed line indicates the resonant Hres field of the mode. Pluses and minuses illustrate effective magnetic charges picture used in discusion (Ref. [8])...... 91

xvii 7.6 a) Illustration of probe motion for Lateral-Field Scan. Probe is positioned at constant height z above the sample and FMR spectra recorded at different probe lateral position, following the line through the center of the sample disk. b-d) Lateral-Field Scans for different probe-sample separation z: b)z = 3625 nm,; c) z = 1050 nm; d) z = 580 nm. Dashed curves is theoretical calculation of ferromagnetic resonance field evolution with changing of probe lateral position in assumption, that FMR mode shape doesn’t changed by probe field performed using Eq. 7.3 for 1st order Bessel mode (see text). Force for each Lateral-Field Scan is normalized to its maximum value. 2D inserts represent the mode shape at different lateral position, obtained by numerical simulation...... 93

xviii CHAPTER 1

INTRODUCTION

1.1 Motivation of the study of ferromagnetic nanostructures

The study of patterned magnetic nanostructure is growing rapidly due to the demands of high density information storage. High density storage of course requires reduction in the physical size of each bit, and an increase in the density of bits without loosing their stability. The stability and switching behavior of ferromagnetic nanostructure memory element is deﬁned by its coercivity and dynamic properties.

Recent progress in spin torque experiments [9] provides a new mechanism for magnetic reversal of memory bits, where the dynamics of ﬂipping process becomes extremely important. There are various tools to understand details of their dynamics like the

Magnetic Force Microscopy (MFM) and Magnetic Optic Kerr Eﬀect (MOKE) for static properties and Brillouin Light Scattering (BLS) and Ferromagnetic Resonance

(FMR) for the dynamic properties.

1.2 Strengths and weaknesses of conventional FMR

Ferromagnetic Resonance is widely used to evaluate and extract magnetic properties of various sample structures, for example thin magnetic layers or patterned structures. From the FMR we could extract the eﬀective internal magnetic ﬁeld

1 which consists of external magnetic field, demagnetizing field, exchange field and others. In addition, the resonance frequencies define the dynamics of ferromagnetic structures. This can be used to evaluate how fast the magnetization of the sample is remagnetized and rearranged in optimized ways. This is important for understanding the stability of the magnetic storage devices.

However the widely used conventional FMR has limited sensitivity and requires relatively large sample area in the order of mm2. In addition, conventional FMR excites spins in the entire sample volume, and it is literally impossible to extract magnetic properties and dynamics from a localized region of the sample. In order to understand the complimentary physics from the submicron or nano sized patterned magnetic structures, it is required to study individual microstructures and obtain the local information from thin ﬁlm ferromagnetic samples.

1.3 New approach to magnetic resonance using Force detection

In order to overcome the limited sensitivity based on inductive detection method such as MRI or FMR, a different approach is required. Such approach was proposed by J. Sidles in 1991 [1]. The main idea of this approach was the mechanical detection of Magnetic resonance. Briefly the magnetic spins in a sample are manipulated by the Microwave field. The force sensor (a micromechanical cantilever with a magnetic probe) is positioned near the sample and experiences force variations due to spin inversion. This approach allows better coupling of spins in the sample to the sensor that provides extremely high sensitivity of the method. Magnetic Resonance Force

2 Microscopy (MRFM) is the new method for detecting magnetic resonance which com- bines the advantages of Magnetic Resonance imaging and scanning probe microscopy techniques.

1.4 History of MRFM

The ﬁrst experimental Electron Spin Resonance (ESR) using MRFM was demonstrated by D. Rugar et al. [10] in 1992 with a particle of Diphenylpicrylhydrazyl

(DPPH) mounted on a mechanical resonator. In the experiment, a commercial silicon cantilever was used in vacuum at room temperature and they achieved a sensitivity better than 30ng in weight and spatial resolution of 19 µm in one dimension.

After the ﬁrst demonstration of ESR, the Nuclear Magnetic Resonance (NMR) using MRFM was also demonstrated by D. Rugar et al. [11] in 1994. With an Ammo- nium nitrate sample mounted on a silicon cantilever, they demonstrated sensitivity of 1.67 × 1013 protons. With high magnetic ﬁeld gradient, 600 T/m, generated by a millimeter sized Fe particle and a new modulation technique called cyclic adiabatic inversion, they achieved 2.6 µm spatial resolution.

In 2004, D. Rugar et al. [2] conclusively demonstrated the single electron spin signal with ultrashort cantilever. Their result proved that MRFM should be the ultimate tool with at least 10−18 N force sensitivity.

1.5 History of FMRFM

Ferromagnetic Resonance using MRFM (FMRFM) was ﬁrst demonstrated by

Zhang et al. [12] in 1996 with Yttrium Iron Garnet (YIG) thin ﬁlm whose dimension

3 was 20×40 µm and thickness was 3 µm. Their experiment was performed in ambient pressure and temperature conditions.

The FMR was reliably demonstrated, but the question of the spatial resolution was not really addressed. In 2000, Midzor et al. [13] demonstrated one dimensional spatially resolved imaging of FMR mode based on spatial force contrast of MRFM signal with spatial resolution ∼ 2 µm. Later Urban et al. [14] demonstrated influence of probe magnetic field on the FMR modes. In 2006, T. Mewes et al. [4] demonstrated the first two dimensional imaging using FMR signal on 50 nm thick NiFe disk array of 1.5 µm diameter, at liquid Helium temperature.

In our experiments on permalloy disks samples, we consider the probe ﬁeld as small perturbation of FMR [5]. We developed a new FMR imaging technique, that allows to combine spectroscopy and lateral information of ferromagnetic resonance images [4][5]. The proposed theoretical approach allows us to explain these spatially resolved spectroscopic data, and is in excellent agreement with experimental results.

This approach allows an analysis reconstruction of FMR modes in a sample, based on spectroscopy information. This implies that we used spectroscopy contrast for FMR imaging unlike force contrast in earlier experiments [13][14]. This allows us to achieve the spatial resolution deﬁned by the line width of FMR, similar to MRI techniques and unlike the previous experiments where the spatial resolution was deﬁned by the probe size.

Later in our experiments, we consider strong probe induced perturbation of FMR modes in a sample. We found that in this regime the probe ﬁeld could signiﬁcantly modify FMR modes. In particular we demonstrated the complete local suppression

4 of the FMR mode under the probe. Hence this provides a tool for local study of continuous ferromagnetic thin ﬁlms and microstructures.

These experiments demonstrate, for the first time the possibility of FMR imaging with submicron resolution and establish FMRFM imaging as a powerful tool for local study of nanoscale ferromagnetic structure. The high sensitivity of MRFM allows to study individual nanostructures. The strong inhomogeneous probe field could locally perturb the FMR mode in the sample, and thus provides the local information about ferromagnetic material. In the regime of strong probe induced perturbation, the probe field could provide boundary conditions for local FMR mode. These provide the basis for the local study of ferromagnetic thin film materials and structures.

In spite of presented advantages of FMRFM, it brings particular challenges in comparison with the conventional FMR technique. The most important comes from the strongly inhomogeneous probe ﬁeld which modiﬁes the FMR mode, and complicates the analysis of spectroscopy data.

Basically FMRFM experiment requires very strong theoretical support of modeling and simulation for each particular experiment. Only then this enables the extraction of reliable local information about sample structures. Other weaknesses are more technical and diﬃculties also arise from the strong interaction between the probe and the sample, and the interaction between the probe and microwave ﬁeld.

These usually increase the noise in the measurements and reduce the sensitivity of

FMRFM method.

1.6 Chapter summary

Here a brief summary of the following chapters in this thesis is presented.

5 The main purpose of this work is to demonstrate localized FMR in individual permalloy disk. Moreover it focuses on the investigation of an individual permalloy disk instead of the FMR in the whole sample.

In Chapter 2, the basic concept of MRFM is discussed. We describe in detail the

MFRM protocols that include spin manipulation; sensitive slice concept; sensitivity; role of the probe ﬁeld gradient; limitations of sensitivity. In addition, some advantages and disadvantages of mechanical force detection are addressed. Finally the resolution and sensitivity of MRFM in application to FMR is discussed.

In Chapter 3, we discuss the classical theory of FMR, based on Herring-Kittel and Damon-Eshbach approaches. We show the basic theoretical results in application to thin film ferromagnetic structures. We also discuss the theoretical approaches for the case of FMR in the presence of highly inhomogeneous magnetic field. These theoretical results allow us to explain later our experimental data for thin film structures obtained by FMRFM, where inhomogeneous probe field influence FMR in the structures.

Chapter 4 includes the description of the experimental apparatus and key measurement techniques. They include: design of the cryogenic, UHV, high magnetic

ﬁeld, vibration isolated setup; the ﬁber optic interferometer; characterization of mechanical resonator and magnetic probe; scan-probe design; design of the Microwave resonator as the spin excitation device.

From Chapter 5 to Chapter 7 key experimental results are presented in detail.

In Chapter 5, we discuss the results of FMR in 1.5 µm in diameter and 50 nm in thick NiFe disk. We demonstrate that FMRFM allows extraction of FMR data from individual NiFe disks as well as collections of disks, similar to conventional

6 FMR technique. The obtained data is in good agreement with those obtained by conventional FMR technique. In addition FMRFM allows to get information about individual structures (disks) near the probe that allows FMR imaging of ferromagnetic samples.

In Chapter 6, we use 2 µm micron in diameter and 50 nm thick NiFe disk and consider the influence of inhomogeneous probe field on spectroscopy of individual disk. We consider probe induced change of the spectrum as small perturbation problem, where the probe field is treated as a perturbation. Our experimental results show perfect agreement with the proposed approach. Finally it allows us to develop spectroscopy technique for imaging of FMR modes in ferromagnetic structures. The lateral resolution of this technique is defined by probe magnetic field gradient and

FMR linewidth, similar to MRI, not by the probe size like in MFM.

In Chapter 7, we consider the case when the probe field is not small and could not be treated as small perturbation. We show that probe magnetic field could dra- matically affect the shape of FMR mode. In particular we demonstrate the local suppression of Bessel FMR mode by strong probe field. The micromagnetic simulations show a good agreement with our experimental results and confirm our qualitative explanation of experiment.

In conclusion, in Chapter 8, we discuss FMRFM as a powerful tool for local FMR study of ferromagnetic micro and nanostructures.

7 CHAPTER 2

BASIC CONCEPT OF MAGNETIC RESONANCE FORCE MICROSCOPY

2.1 Introduction

The basic concept of MRFM is presented in this chapter. First we brieﬂy discuss

Magnetic Resonance from the Quantum and Classical point of views. Then we discuss mechanical force detection in application to Magnetic Resonance experiments in detail. Later we compare the sensitivity of MRFM to MFM in order to understand why MRFM is the high sensitivity tool. Meantime, we consider some advantages and disadvantages of the MRFM method.

2.2 Brief description of Magnetic Resonance

In order to understand the magnetic resonance phenomenon, we consider it from the quantum point of view. Each electron and most nucleons have internal angular momentum (spin). The relationship between the magnetic moment and spin angular moment could be written as:

µ = γ~S (2.1) where γ is the gyromagnetic ratio and ~ is the Plank constant, and S and µ are respectively the spin and magnetic moment of a particle . In the external magnetic

8 ﬁeld, Hz, the Zeeman energy of the particle is written as

Ez = µzHz = γ~SzHz (2.2)

1 where Sz = ± 2 for electron spin. That transition from spin down to spin up state could be driven by microwave power with the frequency.

∆E ω = z = γH (2.3) ~ z

From the classical point of view, if we assume that a single spin is placed in a uniform external magnetic ﬁeld, the spin starts precessing around the external magnetic ﬁeld.

The frequency of its precession is called as the Larmor frequency.

Its dynamics is ruled by the Landau Liftshitz equation as

dM = −γ[M × H ] (2.4) dt z

Considering magnetic moment is the sum of static magnetic moment Mz and the

iωt dynamic magnetic moment m, M = Mz + me , the Larmor frequency can be driven as following

ω = γHz (2.5) where ω is the Larmor frequency, γ is the gyromagnetic ratio, and Hz is the external magnetic ﬁeld in z direction.

In a real system, the Larmor precession always decays because of the damping eﬀect of the system. In order to maintain its precession, an additional energy should be pumped into the system to maintain its precession. The Microwave (MW) power is used to maintain the precession. When the MW frequency is applied in the perpendicular direction of the external magnetic ﬁeld at the Larmor frequency, the magnetic resonance is obtained.

9 chch

Figure 2.1: Schematic of MRFM. The sample is positioned in the external magnetic field Hext and Microwave field HMW. The cantilever with a magnetic probe is positioned above the sample. The position of the cantilever is detected by the fiber optic interferometer. The red region corresponds to the magnetic resonance slice where spins meet the resonance condition. The red dotted curves show inhomogeneous magnetic field of the probe magnet.

2.3 Geometry of MRFM experiment

Fig. 2.1 shows a basic examples of an MRFM set-up. The sample is positioned in

iωt the external magnetic field, Hext, and the microwave magnetic field, he , is applied perpendicular to Hext. The microcantilever as force detector with a magnetic probe is positioned near the sample. The inhomogeneous probe magnetic field defines the surface where the resonance conditions are met (sensitive slice shown by the reddish color in Fig. 2.1: See Eq. 2.5). The modulation of the MW power changes the magnetization of the spins in the sensitive slice. That changes the force between

10 spins in the sensitive slice and the probe magnet. The force change is detected by the deﬂection of the cantilever or by change of the cantilever eigen frequency.

The sensitive slice can probe diﬀerent regions under the sample surface. This means that MRFM provides 3-dimensional nondestructive image capability for spins in a sample. This capability is similar to the MRI 3-dimensional imaging technique.

The application of this technique on micron or nano scale and the ability to study buried magnetic structures make MRFM extremely an promising and powerful technique.

2.4 Force detection

The force between the probe magnet and spins in the sensitive slice could be written as:

F(r, t) = −[m(r, t) · ∇]Hprob(r) (2.6)

where m is the magnetic moment of the sample and Hprob is the inhomogeneous probe magnetic field. This equation shows that spin sensitivity of the MRFM method strongly depends on the field gradient provided by the probe magnet. On the other side, the probe magnet defines the sensitive slice. The thickness of the sensitive slice is defined by the magnetic resonance linewidth and the gradient of the probe.

Therefore, the high ﬁeld gradient of the probe generated by the probe magnet deﬁnes the sensitivity and spatial resolution in an MRFM experiment.

Usually we make the probe magnet by gluing a high coercivity (SmCo or NdFeBr) magnetic particle to the end of a cantilever and shaping it to micron size using a

Focused Ion Beam (FIB) milling process. In our experiments, the magnetic ﬁeld gradient near such a probe magnet is routinely on the order of 1 G/nm

11 2.5 Measurement approach in MRFM

The cantilever as a mechanical force detector is governed by the equation of a simple harmonic oscillator

mz¨ + Γz ˙ + kz = F (z) (2.7) where m is mass, Γ is damping factor, k is spring constant, and F (z) is the force acting on the cantilever.

The spins in a sample are manipulated by a microwave ﬁeld in phase with the cantilever motion. The cantilever displacement is written as z = Z0 cos(ωt + ∆) where Z0 is the amplitude of the motion of the cantilever and ∆ is the phase, and p ω = k/m is the cantilever eigen frequency. The force acting on the cantilever could be written in terms of in-phase and out of phase components relative to the cantilever oscillation

F = Fi + Fq (2.8)

Thus Eq. 2.8 becomes

Fi0 Fi = Fi0 sin(ωt + ∆) = z (2.9) Z0

Fq0 Fq = Fq0 cos(ωt + ∆) = z˙ (2.10) Z0 Now we can rewrite Eq. 2.7 as · ¸ · ¸ F F mz¨ + Γ − q0 z˙ + k − i0 z = 0 (2.11) Z0 Z0

Now, the in-phase force gives the variation of cantilever eigen frequency of the cantilever oscillation and the out of the phase force aﬀects the variation of the amplitude of the cantilever oscillation. In FMRFM measurements we use both frequency and amplitude detection of the force acting on the cantilever.

12 chch

Figure 2.2: Schematic comparison of MRFM and MFM force sensitivity in the case when the force is defined by dipole-dipole interaction. The blue curve shows the force- distance curve. The red curve corresponds to the force-distance curve when the spins of the sample are flipped. The horizontal axis is the distance from a sample to the probe magnet.The oscillation of the probe position is shown by the blue sinusoidal curve. The force response in MFM case is shown by the blue arrows and marked δFMFM. The force response in MRFM case is shown by the red arrows and marked δFMRFM. The figure shows the advantages of MRFM detection in comparison with MFM.

2.6 Comparison of force sensitivity in MRFM and MFM

The MW manipulation of spins in a sample in MRFM experiments provides sig- niﬁcant advantage for the MRFM sensitivity in comparison with the Magnetic Force

Microscopy. Let us consider some spins of a sample that deﬁne probe-sample force near the sample surface as shown by the blue curve in Fig. 2.2. In the case of MFM detection, we oscillate the cantilever with the amplitude Z0. This oscillation deﬁnes

13 the oscillating force acting on the cantilever with amplitude δFMFM = ∂F/∂z · Z0 where ∂F/∂z is the gradient of the probe-sample force.

In the case of MRFM detection, we oscillate the cantilever with the same amplitude Z0. In the same time, we change the spin magnetization from spin-up to spin-down in phase with the cantilever motion. The oscillating force acting on the cantilever in this case is independent of Z0 and is deﬁned by the probe-sample force

δFMRFM as shown in Fig. 2.2.

1 If the probe-sample force is a dipole-dipole force, F ∝ Z4 , the MRFM force is bigger than the MFM force by factor of 4Z . In a typical experimental case, we have Z0

Z ≈ 1 µm and Z0 ≈ 10 nm. That gives force sensitivity for MRFM signal 400 times better than the sensitivity of MFM.

2.7 Force noise

The fundamental limitation of MRFM sensitivity is deﬁned by the unavoidable thermal ﬂuctuations of the cantilever force sensor. The force noise due to thermal

ﬂuctuations could be written as: s 2kk T ∆f F = B (2.12) noise Qf

, where k is the spring constant of the cantilever, kB is the Boltzmann constant, T is the temperature, ∆f is the bandwidth, Q is the quality factor of the cantilever, and f is the cantilever frequency. The force acting on the cantilever from single electron spin could be written as Fsingle = µB∇H where µB is the Bohr magneton and ∇H is the probe magnetic ﬁeld gradient.

14 Thus the number of electrons that we can detect in the MRFM is: s 1 2kk T ∆f N = B (2.13) µB∇H Qf

To improve the force sensitivity, we need to work at high probe ﬁeld gradient and low temperature. The optimal cantilever sensor should be designed to have small spring constant, high frequency and Quality factor.

2.8 Strengths and weaknesses of mechanical detection

The main advantage in mechanical detection of magnetic resonance in comparison with inductive detection is based on very good coupling of the magnetic probe with spins in the sample. This means that we can position the probe very close to the spins in the sample and obtain better sensitivity. The geometry of a MRFM experiment is scaled down by decreasing the size of the sample. Opposite way, the inductive detection, the sensor (pick-up coil) could not be made too small that deﬁnes limitations of this method with scaling down of the sample.

MRFM has new challenges that could be addressed in comparison with inductive detection. The magnetic probe provides the strong inhomogeneous magnetic ﬁeld that signiﬁcantly limits the spectroscopy ability of MRFM (very strong inhomogeneous broadening).

The microcantilever force detector used in MRFM experiments has resonance frequencies approximately 1 ∼ 10 kHz. This prevents the direct detection of the magnetic resonance, which has typical resonance frequency from 10 MHz to 10 GHz.

In this case, in MRFM detection, we need to manipulate spins at the cantilever frequency. We can do it using adiabatic inversion of spins in the sample in the case of the long spin lifetime, or by cyclic suppression in the case when spin relaxation is fast.

15 This signiﬁcantly limits the measurements protocols in comparison with inductive detection.

Another diﬃculty is the strong interaction between probe and sample, and the interaction between the probe and microwave ﬁeld. These usually increase the noise of measurements and reduce the sensitivity of FMRFM method.

16 CHAPTER 3

THEORY OF FERROMAGNETIC RESONANCE

3.1 Summary

In this chapter, we will briefly discuss classical FMR theory. First we will discuss the main theoretical approaches derived by C. Herring and C. Kittel [15] and by Da- mon and Eshbach [16]. Using the Herring-Kittel equation, we will show the basic dispersion relation for spin waves in an infinite 3-dimensional area. After that, we will discuss the spin wave behavior in thin ferromagnetic films, that was developed by Kalinikos and Slavin [17] using the Herring-Kittel approach. We will compare this result with Damon-van de Vaart [6] theory based on a Damon-Eshbach approach.

The obtained dispersion relations for thin ferromagnetic film allows analysis magnetostatic modes in thin film disk samples. We will also discuss the influence of the inhomogeneous demagnetizing field in a disk on the dispersion relation for thin ferromagnetic film. This approach was developed by Guslienko et al. [18] and Kakazei et al. [7].

17 3.2 Definition of effective magnetic field Heff

For spins in ferromagnetic material, the dynamics of the magnetization includes complicated field contributions not only from the external magnetic field but also various other fields. The effective magnetic fields in a ferromagnet is defined by a number of energy terms that include exchange interaction, Zeeman energy, demagnetizing energy, crystalline anisotropy energy, and exchange bias energy. All the energy terms could be written as effective fields acting on an individual spin by virtual variation of the spin direction.

Heﬀ = −∇E (3.1)

∂ 1 ∂ where ∇ = eθ ∂θ + eφ sin θ ∂φ in polar coordinate if the z axis is deﬁned by the perpendicular direction of the sample plane. More details can be found in Ref. [19].

3.2.1 Zeeman Energy

If a magnetic moment M is positioned in an external magnetic ﬁeld H, the Zeeman energy is deﬁned by

Ezeeman = −M · H = −MzH[cos θH cos θ + sin θH sin θ cos(φ − φH)] (3.2)

where θH and φH are in polar coordinates (See in Fig. 3.2). In order to reach the minimum energy state, it requires that the sample magnetization and the eﬀective magnetic ﬁeld should be parallel.

3.2.2 Demagnetizing energy

In a ferromagnet, each spin is influenced by all other spins. This influence by other spins modifies the internal magnetic field. Its contribution depends on sample

18 geometry and is deﬁned as: Z µ ¶ m(r0) 3[m(r0)(r − r0)](r − r0) H(r) = − d3r0 (3.3) (r − r0)3 (r − r0)5

3.2.3 Anisotropy energy

Spins in crystal lattice have a preferred orientation to minimize energy. When energy involves the spin orbital coupling and the chemical bonding of atomic orbitals in a crystal, the atomic orbital has a preferred orientation due to interactions with the electronic wave functions of surrounding ions. Spin-orbital coupling results in a preferred orientation with respect to the crystal axis.

On the surface and interfaces, due to crystalline stress the anisotropy energy could also appear. Basically, the crystalline energy could be described as uniaxial, four fold and higher order terms depending on crystal symmetry. This anisotropy energy could be written as:

2 4 Ea = Ku2 sin θ + Ku4 sin θ (3.4)

where Ku2 is uniaxial constant and Ku4 is four fold anisotropy constant.

3.2.4 Exchange energy

Unlike the demagnetizing energy which is a long range eﬀect, the exchange interaction acts over a much shorter range. This energy could be written by the Heisenberg exchange hamiltonian:

X X Hex = −2 JijSi · Sj = −2 JijSiSj cos θ (3.5)

where i and j refer to spins at lattice sites, and Jij is the positive value for ferromagnetic materials.

19 chch

Figure 3.1: The illustration of the dynamics of the magnetic moment M in the presence of the effective magnetic field Heff . The precession frequency is defined as ω. The black vector τ represents the torque acting on the magnetic moment M.

3.3 Spin dynamics in ferromagnetic materials

In 1951, C. Herring and C. Kittel [15] proposed a theory that described the dynamics of magnetization in ferromagnetic systems. They started from the Landau-Lifshitz equation, which is the torque equation (the motion of the spin is schematically shown in Fig. 3.1). 1 dM − = τ = [M × H ] (3.6) γ dt eﬀ

where γ is the gyromagnetic ratio, τ is the torque, Heff is the effective magnetic field,

and M is the magnetic moment of the sample.

The effective magnetic field, according for all field contributions, is expected as:

Heﬀ = Hext + Hdem + Hexc + Hani (3.7)

20 where Hext is the external magnetic field, Hdem is the demagnetizing field, Hexc is the exchange magnetic field, and Hani is the anisotropy field.

We will describe free precession of spins in an effective magnetic field, disregarding the damping of spin motion. Basically spins will precess around the local effective

field with frequency defined by the gyromagnetic ratio. We can write the static and dynamic parts of the effective magnetic field and spin magnetic moment as follows: ½ i(ωt+k·r) H = Hz + hde i(ωt+k·r) (3.8) M = Ms + me where Hz is the static magnetic field, hd is a dynamic magnetic field due to the dynamic magnetic moment of m, Ms is the static magnetic moment which equals the saturation magnetization of sample, ω is the Larmor frequency of precession, k is the spin wave vector, and r is a position vector.

We assume that the dynamic terms hd and m are small relative to the static terms

Hz and Ms. ½ h ¿ H d z (3.9) m ¿ Ms

This means that the product of dynamic magnetic ﬁeld (hd)and dynamic magnetic moment (m) can be ignored in Eq. 3.6.

In addition, the static magnetic moment has the same direction of the static eﬀective magnetic ﬁeld, which makes the cross product equal to zero.

Ms × Hz = 0 (3.10)

Finally, the Landau Liftshitz equation Eq. 3.6 is simpliﬁed to the form:

dm = −γ([m × H ] + [M × h ]) (3.11) dt z s d

21 3.4 Herring-Kittel equation

If we consider that the static magnetic ﬁeld and static magnetization of the sample are directed in the z-direction, and that the dynamic magnetization is perpendicular to z, we can rewrite the Landau-Liftshitz equation (Eq. 3.11) and obtain the Herring-

Kittel equation ½ iωm + γ(m H − h M ) = 0 x y ext y s (3.12) iωmy + γ(−mxHext + hxMs) = 0

The relation between hd and m can be determined by the Maxwell equation

2 ∇ hd = −4π∇(∇ · m) (3.13)

With the general solution in Eq. 3.13, the dynamical term of hd is explicitly obtained:

−4π(k2m + k k m ) h = x x x y y (3.14) x k2 2 −4π(kxkymx + k my) h = y (3.15) y k2

2 2 2 2 where k = (kx + ky + kz ). Therefore, the equation could be written as

 ³ ´ ³ 2 ´  −iω + ω kxky mˆ + ω + ω ky mˆ = 0 ³ M k´2 x³ H M k´2 y 2 (3.16)  kx kxky ωH + ωM k2 mˆ x + iω + ωM k2 mˆ y = 0 where two notations ωH = γHz and ωM = 4πγMs .

Eq. 3.16 has a nonzero solution if its determinant is equal to zero. That allows us to obtain the dispersion relation. The dispersion relation obtained for the bulk sampe is

2 2 ω = (ωH + ωexc)(ωH + ωexc + sin Θ · ωM ) (3.17) where Θ is the angle between the eﬀective magnetic ﬁeld and the wave vector k,

2 2 kx+ky 2 A (sin Θ = 2 ) as shown in Fig. 3.2, and ω = αω k where α = and A is k exc M 2πMs exchange stiﬀness.

22 chch

Figure 3.2: The schematic is illustrating the orientation of the spin wave k vector relative to the effective magnetic field. The z-direction corresponds to the direction of effective magnetic field. The Θ is the angle between the effective magnetic field and k-vector of spin wave.

chch

Figure 3.3: The dispersion manyfold derived by C. Herring and C. Kittel (see text).

23 The Herring-Kittel dispersion relation deﬁnes the allowed frequency for dipole exchange plane spin waves in an inﬁnite ferromagnetic sample. This relation is plotted

π for various angles from Θ = 0 to Θ = 2 in Fig. 3.2. If the k-vector is parallel to the effective magnetic field, the dynamic field, h, perpendicular to Ms is zero and the

FMR resonance frequency is defined by the effective magnetic field Heff . On the other hand, if the k-vector is perpendicular to the effective field, the contribution of the dynamic field, h, to the FMR frequency is maximum (Fig. 3.2). The ensemble of dispersion relation for different Θ is called a spin wave manifold.

3.5 Spin dynamics in thin ferromagnetic ﬁlm

The dispersion relation (Eq. 3.17) was derived by C. Herring and C. Kittel for the case of an inﬁnite 3-dimension ferromagnetic sample.

Using the Herring-Kittel approach, Kalinikos and Slavin [17] considered the spin wave behavior in an infinite ferromagnetic film at arbitrary directions of external magnetic field. They derived dispersion relations for dipole exchange spin waves in thin ferromagnetic films. In the simplified case when the external magnetic field is perpendicular to the film plane, the Kalinikos-Slavin dispersion relation is:

2 ω = (ωH + ωexc)(ωH + ωexc + f(kL) · ωM ) (3.18)

A 2 where ωH = γHz, ωexc = 2 ωM k is the exchange contribution, f(kL) = 1 − 2πMs (1−e−kL) kL , ωM = 4πγMs, Ms is the saturation magnetization, and L is the thickness of the film sample. Equation Eq. 3.18 is very similar to the Herring-Kittel dispersion relation, Eq. 3.17. Nevertheless the factor f(kL) shows that the dynamic demagnetizing field in thin film is much less than the same field for bulk material in the case

24 when the k vector of spin waves is smaller than the basic k vector deﬁned by the thin

kL ﬁlm thickness L (f(kL) ≈ 2 for kL ¿ 1).

3.6 Damon-Eshbach approach

An alternate theoretical approach for analysis of spin wave behavior was proposed by Damon and Eshbach [16]. Damon and Eshbach started solving the Landau-

Liftshitz equation using the magnetic potential ψ where hd = ∇ψ.

Similar to the Herring-Kittel method, the magnetic ﬁeld and magnetization in a ferromagnetic sample can be written as ½ iωt Heﬀ = Hz + hde iωt (3.19) M = Ms + me where hd ¿ H0 and m ¿ Ms. Now we plug the above equations (Eq. 3.19) into the Landau Liftshitz equation (Eq. 3.11) with hd = 5ψ and ∇ · B = 0 as Maxwell equations. This gives us the Damon-Eshbach equation in the form: ( 2 2 2 (1 + κ)( ∂ ψ + ∂ ψ ) + ∂ ψ = 0 : inside − ferromagnet ∂x2 ∂y2 ∂z2 (3.20) ∇2ψ = 0 : outside − ferromagnet

ΩH Hz ω where κ = 2 2 ,ΩH = , and Ω = . ΩH −Ω 4πMs 4πγMs Damon and van de Vaart [6] considered a thin ferromagnetic slab in an external magnetic ﬁeld perpendicular to the slab plane using the Damon-Eshbach equation

Eq. 3.20. Finally, Damon and van de Vaart derived the dispersion relation that could be written as follows: ³ ´ 1 1 p 2 tan p 2 ks/2 + nπ/2 = 1 (3.21) where p = −(1 + κ) ΩH κ = 2 2 (3.22) ΩH −Ω Ω = Hi Ω = ω H 4πMs 4πγMs 25 chch

Figure 3.4: The dispersion relations for a thin ferromagnetic film saturated in the external field perpendicular to the film plane. The red curve represents the Kalinikos- Slavin dispersion relation (See Eq. 3.18). The blue curve is the dispersion relation obtained by Damon and van de Vaart (See Eq. 3.21). The dispersion relations are shown with the assumption ωH=ωM, following Damon and van de Vaart [6].

k is the in-plane wave vector, s is the slab thickness. The integer n characterized the quantization of spin waves in thickness of the slab.

3.7 Comparison between Kalinikos-Slavin and Damon-van de Vaart dispersion relations

In order to compare the two different approaches done by Kalinikos and Slavin, and by Damon and van de Vaart [20][6], we plotted the dispersion relations defined by equations Eq. 3.18 and Eq. 3.21 in Fig. 3.4, assuming ωH = ωM = 1, similar to the classical Damon-van de Vaart paper [6]. The two different approaches give very similar results. At small k, the FMR frequency is approximately linear with k, and p π for k À L , ω is independent of k and saturates out at ω = ωH (ωH + ωM ) for both approaches.

26 3.8 Dispersion relation for our experimental conditions

In our experiments, we study permalloy thin film ferromagnetic disk arrays whose thickness is 50 nm and the diameter is around 1 µm. The diameter of disks defines the k-vector for spin waves in our experiments. Basically we work in the long wavelength approximation kL ¿ 1 that allows us to use the simplified Kalinikos-Slavin equation

(Eq. 3.18) that can be written as:

kL ω = ω + ω + · ω (3.23) H exc 4 M

For the case of ferromagnetic disk samples, the k-vector of magnetostatic spin

αn waves will be deﬁned by the radius of the disk, kn = R , where n is the order of the magnetostatic modes and αn is the nth zero of the zeroth order Bessel function

(J0(αn) = 0) and R is the radius of the disk. We now obtain the quantization of resonance frequencies for magnetostatic FMR modes in the disk:

k L ω = ω + ω + n · ω (3.24) n H exc 4 M

According to the Damon-van de Vaart model, the spin wave solution for ferromagnetic disk could be described as the Bessel functions.

mn(r) = µ0J0(αnr/R) (3.25)

where mn is dynamic magnetization for magnetostatic mode, µ0 is the amplitude of the dynamic magnetization and r is the distance from the disk center.

3.9 Dispersion relation in presence of inhomogeneous demagnetizing ﬁeld

Up to now, we have considered the case of homogeneous internal magnetic ﬁeld in samples. However, for the thin ferromagnetic dot sample, the demagnetizing ﬁeld

27 should be treated more precisely because the internal magnetic ﬁeld rises near the edges of the samples.

For the case of inhomogeneous effective magnetic fields through samples, Guslanko et al. [18] and Kakazei et al. [7] proposed to modify the dispersion relation (Eq. 3.18) by considering the inhomogeneous demagnetizing field as small perturbation for unperturbed Bessel solutions (Eq. 3.25).

The perturbation due to the inhomogeneous internal magnetic ﬁeld in the disk sample, Hi = Hext − 4πMsN(r), is R H m2 (r)dr hH i = R i n (3.26) i 2 mn(r)dr and the dispersion relation becomes

k L ω = γhH i + n · γ4πM (3.27) n i 4 s where N(r) is the inhomogeneous demagnetizing factor for the disk sample.

Later in our experiments described in Chapter 5 through Chapter 7, we will use this approach for analysis of our data and for the eﬀect of the inhomogeneous probe magnetic ﬁeld on FMR modes in the samples.

28 CHAPTER 4

FMRFM EXPERIMENTAL SET-UP

4.1 Introduction

In here, we present the experimental set-up for the Ferromagnetic Resonance

Force Microscopy (FMRFM) experiment. Our FMRFM system is very similar to an

ESR-MRFM setup, but there are a few diﬀerences due to speciﬁc physics of FMR experiments. In addition, this system could perform MFM and AFM studies of a sample.

First, we discuss the cryogenic magnetic system that provides the external magnetic ﬁeld in our experiments. Then we examine the details of our cryogenic, ultra high vacuum system including design of 3-dimensional scanners, geometrical solutions for the microcantilever sensor, ﬁber optic interferometer, and MW resonator.

Second, we describe in detail our microcantilever force sensor. We analyze the

ﬁber optic interferometer for reading the cantilever position and discuss the noise of the system. Then we show details of cantilever characterization including precise measurement of the spring constant and quality factor of a cantilever. After that we discuss the characterization of the probe magnet on a cantilever. This information includes coercivity and magnetic moment of the probe magnet. These parameters are very important for quantitative evaluation of the force acting on the cantilever

29 and are crucial for comparison of experimental results and theoretical modeling and simulation.

Third, we discuss a microwave system that includes the MW resonator optimized at ∼ 7.5 GHz whose bandwidth is around 500 MHz to excite spins in a sample. High

MW frequency allows to work at a high external magnetic field that saturates the ferromagnetic sample. This is an important condition for analysis of FMR data. In lower external magnetic field, where the sample is not saturated, the magnetic structure of the sample will be extremely complicated. In this case, theoretical analysis of the FMR signal becomes extremely difficult.

In conclusion, we summarize the technical parameters of our system and its sensitivity for FMR experiments.

4.2 Design of the experimental set-up

The schematic overview of the FMRFM setup is sketched in Fig. 4.1. The cryostat for FMRFM experiment is positioned on a vibration-isolation optical table to prevent unwanted noise through the ﬂoor. The cryogenic dewar including XYZ superconducting magnet (see Fig. 4.1) is attached on the same optical table through a counterweight lifting mechanical system. This way the cryogenic dewar could be lifted up and down to the cryostat.

On the top of the cryostat, the high vacuum pumping system, including the turbo pump, is positioned. It generates high vacuum conditions in our experimental stage.

The microcantilever force sensor is mounted on a positioning system shown in Fig.

4.2. The positioning system consists of a coarse motion part which is implemented by a 3 axis Attocube scanner and ﬁne positioning 3D scanner based on a piezo tube.

30 chch

Figure 4.1: The schematic of the MRFM set-up. The reddish square object is the XYZ superconducting magnet. The 1 inch diameter FMRFM experimental stages are located inside the XYZ superconducting magnet.

31 chch

Figure 4.2: Schematic of low temperature FMRFM experimental stages (see text).

32 The coarse motion stage allows the motion range less than 4 mm and step size on the level 10-100 nm. The ﬁne motion scanner provides the X-Y scan range around 10 µm and Z scan range around 2 µm.

For reading of the cantilever position, we use a fiber optic interferometer that will be discussed in details later in this Chapter. Optical fiber goes through vacuum feedthrough on the top of the cryostat down to the experimental stage and the cantilever. Basically we use two optical fibers, one of which is used for detection of the cantilever motion, the other is assigned to read the Z coarse motion stage that helps to control the distance between the cantilever and sample surface.

In our system, the microcantilever probe scans the surface of a sample that is attached to the stationary MW resonator. The MW resonator is designed in a ”bow tie” geometry that allows to obtain high MW ﬁeld and will be discussed in more detail later in this Chapter. The MW power is transmitted to the resonator by coaxial MW cable. The geometry of the MW resonator is symmetrical, therefore we run two MW coaxial cables from the vacuum feedthrough on the top of the cryostat to the experimental stages. It allows us to check the reﬂection and transmission characteristics of our MW resonator during an experimental run and have a backup extra MW cable for excitation of the MW resonator.

The sample and microcantilever sensor are positioned in the geometrical center of a 3-axis superconducting magnet that allows the maximum ﬁeld in the Z-direction up to ±2 T and in the X-Y direction up to ±1 T. The 3 axis superconducting magnet allows us to perform angle dependent FMR measurements.

33 4.3 Cantilever characterization

The Microcantilever force detector is the most important part of our experimental set-up. The details of force detection were discussed in Chapter 2. For calculation of the FMRFM force acting on the cantilever using our experimental data (cantilever frequency at frequency detection or cantilever amplitude in amplitude detection), it is very important to characterize our cantilever. That means that we should know exactly its spring constant and quality factor. We currently use commercially available

N N low doped Si cantilever, whose spring constant is from 0.1 m to 0.3 m [21].

4.3.1 The measurement of cantilever spring constant

The spring constant measurement of a cantilever is based on the method done by

Sader et al. [22]. They considered the cantilever dissipations due to the motion in viscous media (the air) and in hydrodynamic approximation they derived an equation for the spring constant of a cantilever using its geometrical sizes, quality factor, and eigenfrequency of the cantilever.

2 2 k = 0.1906ρf b LQf Γi(ωf )ωf (4.1)

where ρf is the density of an air, Γi is the hdyrodynamic function deﬁned by Sader et al. [22], b, L, and Qf are respectively the width, length, the quality factor of a cantilever, and ωf is the cantilever resonance frequency. The geometrical size of the cantilever we measure from a SEM image with accuracy 10%. The eigenfrequency of the cantilever is measured using the position of thermal peak of cantilever spectral noise. The density of the air is deﬁned by local atmosphere pressure. The determination of the quality factor Qf is presented in the next section.

34 4.3.2 The measurement of quality factor of cantilever

The quality factor is deﬁned by the energy loss rate during the oscillation at the resonance frequency. There are two conventional methods to determine the quality factor of a cantilever which are the Ring Down Method, and the Power Spectral

Density Method.

In the Ring Down Method, we measure free decay of cantilever oscillation amplitude. The free motion of a harmonic oscillator in absence of driving force is written as:

A(t) = A(0)e−t/τ (4.2) where A(t) is the amplitude of the cantilever oscillation, A(0) is the initial amplitude of the cantilever oscillation, and τ is the time constant. The time constant of exponential decay could be written as follows:

2Q τ = (4.3) ω0 where Q is the quality factor of the cantilever and ω0 is the cantilever eigenfrequency.

We measured the quality factor of our Si cantilever far from the sample surface in order to eliminate the probe-sample interaction. The result was always in the range of 50000 to 100000.

In the Power Spectral Density method, we measure the cantilever amplitude response at constant driving force for diﬀerent driving frequencies. The classical resonator response is the Lorentzian:

F A(ω) = m (4.4) 2 2 ωω0 ω − ω0 + i Q

35 chch

Figure 4.3: Schematic of the cantilever motion in the presence of the external magnetic ﬁeld. The red arrow represents the magnetization of the probe magnet.

where F is the driving force, ω is the driving frequency, ω0 is the eigenfrequency of the cantilever, m is the mass of the cantilever, and Q is the quality factor. The ﬁt of the cantilever response gives us the quality factor Q.

We prefer to use the Ring Down Method, because at high quality factor least above the power spectral density method is more time consuming.

4.3.3 Characterization of probe magnet

For our FMRFM experiments, we need a micron-size magnetic probe with high coercivity better than 1 T. The probe was made from a SmCo particle that was glued to the end of the cantilever by Stycast 1266 and then micromachined by Focused Ion

Beam (FIB) milling down to the size ∼ 1 µm.

The magnetic moment of the probe magnet is measured using the cantilever magnetometery [23][24]. If the cantilever with mounted probe magnet is oscillating in a homogeneous external magnetic ﬁeld, the eigenfrequency of the cantilever is dependent on the external magnetic ﬁeld.

36 In Fig. 4.3, we assume that the cantilever is moving in the external magnetic field and the probe coercivity is infinite. The cantilever motion changes the orientation of the probe magnet relative to the external magnetic field. The energy of the probe magnet in external magnetic field is written as: ( 1 2 Ek(x) = 2 kx 1 x2 (4.5) Em(x) = −m · H ' −mH(1 − 2 ) 2 Leff where the first equation represents the energy of the harmonic oscillator for comparison. Ek is the energy of the harmonic oscillator, k is the spring constant, x is the displacement of the harmonic oscillator. In the second equation, Em is the energy of the probe magnet, H is the external magnetic field, m is the magnetic moment of the probe, and Leff is the effective length of the cantilever.

We could write the eﬀective spring constant due to the magnetic energy Em similar to that for energy of harmonic oscillator Ek

∂2E δk = m (4.6) ∂x2 where δk is the eﬀective spring constant due to the interaction of the probe magnet and external magnetic ﬁeld.

With the assumption of mathematical approximation for small angle variation, the cantilever frequency response in terms of the external magnetic field could be written as: ∆ω δk δk mH = √ = = 2 (4.7) ω k 2k 2kLeff where Leff is the effective cantilever length which is L/1.38. And magnetic moment of the probe will be 2kL2 ∂ω m = eff · (4.8) ω ∂H

37 chch

Figure 4.4: The dependence of the probe field gradient on the probe-sample distance. Inset shows the FMRFM spectra for thin film DPPH at different probe-sample separations. The bottom spectrum is taken at 400 nm away from the sample surface and the top spectrum is taken at 880 nm away. The black arrows show the field where the sensitive slice of the probe magnet enters DPPH sample. Resonance field pointed by the black arrows is used for calculation of the probe magnetic field and its gradient (See text).

This cantilever magnetometery allows to obtain the magnetic moment of the probe from linear dependence of the cantilever frequency on the external magnetic ﬁeld.

These measurements also show the coercivity of the probe magnet because the change of the probe magnetic moment violates linear dependence of cantilever frequency on the external magnetic ﬁeld.

38 4.3.4 Measurement of the probe ﬁeld gradient

If we know the magnetic moment of the probe magnet, we can calculate the magnetic ﬁeld and its ﬁeld gradient near the probe in point dipole approximation.

These parameters are very important for analysis of ferromagnetic modes in a sample and for estimation of the force acting on the magnetic probe. Nevertheless, the point dipole approximation is valid if the probe is close to the spherical shape. For real magnetic probe having the shape which we obtain by FIB is much more complicated.

Moreover the distribution of magnetic moment in the probe material could be very complicated. Therefore, it is very important for quantitative measurement to make accurate characterization of the probe ﬁeld and probe ﬁeld gradient.

We characterize the probe field by performing MRFM experiment on di-phenyl- picryl-hydraz (DPPH) thin film sample. The force signal from ESR DPPH sample could be described analytically that allows reliably extract the map of the probe field.

Fig. 4.3.4 shows the ESR resonance for DPPH at diﬀerent probe-sample distances.

The probe magnetic moment was positive and was oriented opposite to the external magnetic ﬁeld which is negative. The bottom spectrum is measured at the closest distance from the sample surface and the top spectrum is measured at the furthest distance from the sample.

Each spectrum includes sharp negative peak which position does not depend on the probe-sample distance. That means that this signal is defined by spins of the sample that are far from the magnetic probe and the probe field does not affect the resonance field for these spins. The other part of resonance is more broad and it has positive sign. This part of resonance strongly depends on the probe-sample distance.

39 That means that this signal came from spins of the sample which are close to the magnetic probe and probe ﬁeld strongly aﬀects the resonance of these spins.

If we use the concept of the sensitive slice (See Fig. 2.1), the resonance signal will appear when the sensitive slice touches the surface of the sample. It corresponds the position of the lowest field for the probe induced resonance which are shown by black arrows in Fig. 4.3.4. At this point, the spins of the sample directly under the probe are resonating. These spins experienced the magnetic field defined by the external

ﬁeld and probe ﬁeld on the sample surface directly under the probe.

ω = H + H (4.9) γ ext probe

That allows us to extract the probe field on the sample surface directly under the probe. In Fig. 4.3.4, we showed the dependence of the probe magnetic field gradient on the probe-sample distance defined by this method. Normally we have the probe magnetic field gradient on the order of G/nm at the probe-sample distance ∼ 1 µm.

4.4 Detection of the cantilever position

In MRFM, the force signal is detected through the motion of a cantilever. We detect the cantilever position using ﬁber optic interferometer. Interferometer is working using infrared solid state laser whose wavelength is 1550 nm. In this section, we will describe the details of our interferometer and will discuss its sensitivity.

4.4.1 Fiber optic interferometer

In Fig. 4.5, the general schematic of the traveling light beams from a 1550 nm solid laser is shown. From the laser diode, the output beam of intensity I0 is splitted

40 chch

Figure 4.5: Schematic of ﬁber optic interferometer(see text).

on two diﬀerent intensity beams, I90 and I10, by 90/10 coupler (90% and 10% of intensity respectively).

We use the I10 beam and I90 beam is blocked using laser attenuator. The I10 beam is traveling through the optical ﬁber which is aligned with the top surface of a cantilever. Normally in here, we align the end of the optical ﬁber at distance

30 ∼ 50 µm above the top surface of the cantilever.

Part of intensity of the beam I10 is reﬂected by the edge of the optical ﬁber (3-

5% intensity). It is shown as Iend in Fig. 4.5. The transmitted part of the beam is partially reflected by the surface of the cantilever and is also directed back to the optical fiber. This beam is shown as Icanti in Fig. 4.5. These two beams create the interferometry picture which depends on the distance from the cantilever and edge of the optical fiber.

41 The output intensity of the interferometry signal Iinter could be written as

p 4πD I = I + I + 2 I I cos (4.10) inter end canti end canti λ where D is the ﬁber-cantilever distance shown in Fig. 4.5 and λ is the wavelength of the laser. Consequently the interference laser output Iinter directly depends on the distance D and it sinusoidally oscillates as the distance. The maximum and minimum intensity could be obtained at: ½ √ max Iinter = Iend + Icanti + 2√IendIcanti min (4.11) Iinter = Iend + Icanti − 2 IendIcanti

Now, the visibility could be deﬁned as:

max min Iinter − Iinter Visibility = max min (4.12) Iinter + Iinter

If the visibility is close to 1, it means that the interferometry signal is comparably with the maximum signal measured by the photodetector. It allows to get the optimal signal to noise ratio (SNR) at cantilever position measurement. Normally we routinely achieve the visibility of 0.8 in our experiments.

In order to reach the maximum sensitivity of the cantilever displacement, we need to operate the ﬁber optic interferometer in optimal conditions. According to Eq. 4.10, the optimal sensitivity of the interferometer is deﬁned by the maximum derivative of the interferometer intensity by the distance D

dI 4πD inter ∝ sin (4.13) dD λ

Therefore, the optimal response of the interferometer is reached at the distance between the cantilever and the end of optical ﬁber deﬁned as:

(2n + 1)λ D = (4.14) 8 42 chch

Figure 4.6: The dependence of signal of ﬁber optic interferometer on the probe-sample distance. The red arrow shows the position where the cantilever touches the sample surface. This position is obtained by the primary ﬁber optic interferometer used for the detection of cantilever position, (See text)

We could adjust the optimal interferometer working point by changing of the wavelength of the laser diode λ.

Our interferometer system provides accuracy of the cantilever detection Zm = √ −2 3 · 10 A˚/ Hz where Zm is the detection noise of the interferometer in the unit of

cantilever displacement.

4.4.2 Measurement of the probe-sample distance

In our FMRFM experimental set-up, we use two optical ﬁbers. One of them is

assigned for monitoring the cantilever displacement. The another one is assigned for

precise measurement of the distance between the cantilever and the sample surface.

43 chch

Figure 4.7: Schematic and geometrical sizes of MW resonator used in our experiments (Designed by D. Pelekhov).

The second optical ﬁber is pointing a gold coated mirror mounted at the bottom of the piezo shaker shown in Fig. 4.2. The position where the cantilever touches the surface could be detected by monitoring DC voltage of cantilever interferometer signal (interferometer 1). This gives the reference point for the second interferometer.

Therefore, we could measure the probe-sample distance with sub-nanometer accuracy.

The typical signal from the second interferometer is presented in Fig. 4.6. The red arrow shows the touch point of the cantilever to sample surface. This point is detected by the cantilever interferometer (interferometer 1).

44 4.5 Microwave Resonator

Our FMR measurements are performed at X-band MW range (7 GHz ∼ 12.5 GHz).

Microwave power is generated by commercial MW generator (Gigatronics Model

12000A) that provides the output power in the order of 20 dBm. In our FMR experi-

−2 ments, we need to create the MW magnetic ﬁeld on the sample of order H1 ∼ 10 −1

G. To achieve this intensity of MW ﬁeld, we need to amplify the generator signal.

For this purpose, we designed and implemented Microwave strip line resonator.

Fig. 4.7 represented the MW resonator which we used in our experiments. The

MW resonator was optimized for the frequency 7.58 GHz. The geometry of this resonator was simulated by Ansoft Microwave software, HFSS, and the results of simulation are shown in Fig. 4.8 by the solid red and blue curves that represent as

S11 transmission and S12 reﬂection respectively. The bow-tie structure was chosen to increase the MW ﬁeld at the localized bow-tie neck region of the size around 1×1 mm.

The experimental results for our MW resonator at 4 K is presented in Fig. 4.8, by the black S11 and green S12 curves. The experimental results demonstrated a poor performance of our MW resonator which defined by number of reasons. MW power transmitted to the resonator is considerably attenuated by the MW cables running from the room temperature to 4 K stage (10-15 dB). The black curve in Fig. 4.8 demonstrates number of peaks that defined multiple reflections of MW power on the joints of MW cables. The eigen frequency of MW resonator is considerably shifted relative to the simulated results due to resonator-sample coupling. All these degrade the performance of our MW system. Nevertheless this resonator allows to obtain MW magnetic field of order of 1 Gauss at applied power 200 mW. The bandwidth of the resonator is 500 MHz. This data was obtained by the experiments on DPPH sample.

45 chch

Figure 4.8: The S12 and S11 reﬂection parameters for MW resonator. The red and blue curves represents S parameters obtained for our resonator using simulation software (Ansoft HFSS). The black and green curves are the experimental results (See text).

4.6 Conclusions

Summarizing technical characteristics of our experimental set-up, we have FM-

RFM system that allows FMR experiments in temperature 4 K; external magnetic

field ±2 T in Z-direction, external field ±1 T in X and Y-directions; maximum microwave field is ∼ 1 G; the coarse motion range in XYZ directions ∼ 1 mm; fine motion scanning range 10 × 10 × 2 µm for X, Y, and Z directions respectively; force √ sensitivity of microcantilever force sensor is ∼ 20 · 10−18 N/ Hz measured without

probe-sample interaction.

46 CHAPTER 5

FMRFM SPECTROSCOPY IN CIRCULAR DISK ARRAY

5.1 Introduction

The first FMRFM using circular YIG dot array was demonstrated by Zhang et al. [12]. The sample which they had used was typically several tens of micron. In this chapter, we discussed our first experiment on micron size permalloy disk array. We demonstrated that MRFM set-up had adequate sensitivity for study submicron ferromagnetic structures. We analyze FMRFM spectroscopy of our sample that contains global and local sample information. The obtained spectra are in good agreements with data obtained by conventional inductive FMR technique. Moreover FMRFM provides valuable local information about the sample. We demonstrated the first

MRFM 2D images on our patterned structure.

5.2 Experimental conditions

5.2.1 Sample properties

The sample in this experiment was 1.5 µm diameter and 50 nm thick permalloy(Ni90Fe10) disk arrays patterned as square lattice by Electron Beam lithography by Seagate Tech- nologies in Pittsburgh, USA. The disk center to center distance was 1.8 µm (Fig. 5.1).

47 chch

Figure 5.1: A patterned array of 1.5 µm diameter and 50 nm thickness NeFe disc is used. The center to center separation is 1.8 µm. The patterned array is deposited on a low doped Si substrate using E-beam lithography (Provided by S. Batra at the Seagate Technologies in Pittsburgh, USA.).

5.3 Cantilever properties

A commercially available silicon cantilever was used in this experiment (See Fig. 5.2).

For our FMRFM experiment, we glue a probe magnet at the very end of the cantilever tip. For that, we chopped oﬀ the end of the tip pyramid by using Focus Ion Beam milling method to get a ﬂat area whose dimension is around ∼ 2 µm in diameter.

For the probe magnet, we used a SmCo particle. The coercivity of the SmCo is greater than a few Tesla at 4 K. Consequently the direction of the probe magnetic

ﬁeld is permanently pointing in one direction. This behavior guarantees that we could perform the FMR in parallel and antiparallel orientation to the external magnetic

field. In Fig. 5.2 a & b, an irregular shaped SmCo particle was glued on the flat surface of the chopped off ultrasharp tip. After gluing the SmCo particle using Stycast 1266 at room temperature, it was transferred into FIB facility and it was shaped by Focus

48 chch

Figure 5.2: A SmCo particle was glued at the end of a commercial Si cantilever. The dimension of the cantilever was 450 µm long and 30 µm wide. After gluing the SmCo particle, it was transferred into a Focus Ion Beam milling facility and then its shape was milled at low E-beam energy. The sharp tip was intentionally milled in order to provide strong inhomogeneous magnetic ﬁeld (Ref. [4]).

Ion Beam milling process (shown in Fig. 5.2 c). Finally the dimension of the probe was around 2 × 2 × 7 µm .

After the FIB process, the SmCo probe magnet was magnetized in the direction

(red arrow shown in Fig. 5.2) perpendicular to the sample surface in 8 T magnetic

ﬁeld at room temperature.

5.3.1 Properties of the probe magnet

In order to measure probe magnetic moment and its properties, we performed cantilever magnetometery measurement (see Section 4.3.3) shown in Fig. 5.3 at 4 K.

In this measurement, we observed negligible deviation from a linear ﬁeld dependence of the frequency shift in the range of the external magnetic ﬁeld (±1 T). As the

49 chch

Figure 5.3: The resonance frequency of a cantilever with a SmCo probe is monitored in changing the external magnetic ﬁeld. Initially the external magnetic ﬁeld is set at 9 kG and then it changes from 9 kG to -9 kG and from -9 kG to 9 kG.

shown results Fig. 5.3, the orientation of the probe magnetization was not reversed even though we changed the external magnetic ﬁeld within ±1 T. Hence the coercivity of our probe magnet is better than 1 T, and the Eq. 4.7 could be applied to calculate

−11 the magnetic moment. The probe magnetic moment was µprobe = 1.4 × 10 Nm/T.

All other physical properties of the cantilever were also measured. The spring constant was estimated as k = 0.3 N/m by the method of Sader et al. [25] using

Eq. 4.1 (see section 4.3.1) and the resonance frequency at the remote distance from the sample surface was around f0 ≈ 16.5 kHz which is within the speciﬁcation of the manufacture.

50 The quality factor of the cantilever was 200000 as measured by the Ring down method discussed in section 4.3.2. After characterizing the cantilever properties, the cantilever has enough force sensitivity to measure 3000 polarized electron spins.

5.3.2 FMRFM measurement protocol

The measurement of the FMR signal in this experiment was based on a phase locked loop to keep the driving frequency precisely at the cantilever resonance frequency. For this we were modulating the amplitude of the MW power at the cantilever resonance frequency thereby driving the cantilever into oscillations through the spu- rious coupling of MW power and cantilever. The modulating frequency was set using a function generator and a software PID loop.

The synchronization signal from the function generator and the signal from the cantilever were both fed into a lock-in ampliﬁer, which detects both the amplitude of the cantilever signal and the relative phase of the two signals. After determining the resonance frequency initially by sweeping the frequency of the function generator through the cantilever resonance, the relative phase of the driving signal and the cantilever oscillation was kept constant by continuously adjusting the frequency of the function generator.

A change of the cantilever resonance frequency caused by changes of the probe- sample interaction or changing the external magnetic ﬁeld was thereby accounted for in a changed driving frequency of the function generator. The amplitude of the cantilever motion serves as the MRFM signal.

51 chch

Figure 5.4: FMRFM spectra were measured at 7.7 GHz with 60 % amplitude modulation depth. Each spectrum was measured at different probe-sample separations shown in Figure. The external magnetic field is applied in the same direction of the probe magnetic field. The solid vertical lines provide to keep tracking the FMR resonances which are independent of the probe-sample separations (Ref. [4]).

52 5.4 FMRFM spectra

In Fig. 5.4, it showed FMR spectra done by FMRFM. The applied MW frequency was 7.7 GHz. The experiment was performed at 4 K and in the external magnetic

ﬁeld applied the parallel to the magnetization of the probe. The depth of Amplitude modulation of the MW frequency was set by 60%.

All seven spectra in series were measured as we changed the prob-sample separation using a piezo tube from 0.19 µm to 1.05 µm. There are alternative series of pronounced maxima and minima. The minima are located at the low magnetic ﬁelds than the magnetic ﬁelds of the maximum peaks.

The positions of all positive peaks indicated by vertical black lines do not change for the different probe-sample distances. However the positions of all negative peaks are strongly affected by the probe-sample separations. This behavior means that the probe magnetic field cannot influence the FMR resonance for the positive peaks.

We starts to call them as the Zero Field Resonance (ZFR) because the peaks are independent of the probe magnetic field. The Zero Field Resonance (ZFR) peaks are originating from remote disks far away from the probe, where the probe field is negligible and does not affect the position of the resonances.

All negative peaks shown in Fig. 5.4 are called as the Tip Field inﬂuenced Res- onance (TFR). As we changed the probe -sample separation, the TFR was strongly inﬂuenced especially in the proximity region from the probe.

The TFR signal is coming from a few localized disks just underneath the probe magnet and the probe magnetic field significantly affects the position of resonance.

If the probe-sample separation is reduced, the average of the probe magnetic ﬁeld into the sample is increased. The TFR shifts away to the lower external ﬁeld from

53 the ZFR as the probe approaches toward the sample surface. This behavior indicates that TFR is influenced by the probe magnetic field. In addition, the magnetic field difference between the TFR and ZFR is measure of the average magnetic field of the probe through a disk sample. Consequently in FMRFM, we could evaluate the probe magnetic field and its gradient dependence on the probe sample separation shown in

Fig. 5.4 and Fig. 5.5. This allows the probe ﬁeld characterization during the run of experiments.

The ZFR and TFR has diﬀerent sign of force acting on the cantilever in FMRFM spectra (in Fig. 5.4). Experimentally we measure the change of amplitude of the cantilever oscillation and the force of FMR is calculated by the Eq. 2.6. Due to the dipolar nature of the probe ﬁeld, it has positive sign under the probe and changes the sign when we move far from the probe on the sample surface. Therefore, the

ZFR signal which originates from part of the sample far from the probe magnet has diﬀerent sign in comparison to the TFR signal which originates from the part of sample directly under the probe.

5.5 FMRFM in antiparallel conﬁguration

Fig.5.5 were taken at the direction of the external field opposite to the probe magnet (antiparallel configuration). For this case, the FMRFM force signal changes the sign due to change of the sample magnetization. Consequently the negative peaks correspond to the ZFR while the positive peaks do correspond to the TFR. With the same manner which is shown in Fig. 5.4, we demonstrate the probe-sample distance dependent FMR in antiparallel configuration (Fig. 5.5) and its behavior could be explained by what it is demonstrated in the parallel configuration.

54 chch

Figure 5.5: FMRFM spectra were measured at 7.7 GHz with 60% amplitude modulation depth. Each spectrum was measured at different probe-sample separations shown in Figure. The external magnetic field are applied in the opposite direction of the probe magnetic field. The solid vertical line guides the position of each zero field resonance (Ref. [4]).

55 chch

Figure 5.6: An FMRFM spectrum was measured at 7.7 GHz. The amplitude of MW excitation was modulated at the resonance frequency of the cantilever with a depth of 60%. During the measurement, the probe-sample separation was kept at 0.4 µm. The external magnetic ﬁeld was applied in the same direction of the magnetic moment of the cantilever. For each FMR modes, the transverse components of magnetization were sketched. Up to sixth order FMR modes were observed in the spectrum (Ref. [4]).

5.6 Quantization of FMR modes in a disk sample

The ZFR part of signal is originated from the part of sample far from the probe and is very similar to the FMR information obtained in conventional FMR experiment.

This means that we could analyze the information of ZFR using the dispersion relation derived for FMR in thin ﬁlm disk samples (see Eq. 3.18 done by Kalinikos and Slavin in Chapter 3). The sample diameter was 1.5 µm and its thickness was 50 nm.

All following steps are the similar ways done by Kakazei et al. [7].

56 chch

Figure 5.7: Experiment results compared to theoretical results based on the dispersion relation done by Kakazei et al. [7]. The solid line corresponds to theoretical results while disks are the measurements of FMRFM at a few diﬀerent MW frequencies.

For the case of the ﬁnite diameter of a circular disk sample, the inplane wave vector kk is quantized and its quantization induces the multiresonance spectrum [7].

The quantization of the transverse wave vector could be deﬁned by

βm kk,m = 2 (5.1) ddis

where βm is the m-th root of the zeroth order of Bessel function and ddis is disk

diameter (see Eq. 3.24). The mode proﬁles for diﬀerent m in a circular disk are

sketched in Fig. 5.6 at each ZFR peak.

In order to compare precisely our experimental results to the theoretical calcu-

lation, we also changed the MW frequency within range of 500 MHz around 7.7

GHz. Assuming the demagnetizing factor N=0.905, the saturation magnetization

57 Ms=890 G, the gyromagnetic ratio γ/2π = 3.05 MHz/G, and the exchange stiﬀness

A=1.4·10−6 erg/cm, we draw the theoretical prediction for FMR frequency depending on external magnetic field shown in Fig. 5.7. The experimental results for different modes are in excellent agreement with the dispersion relation derived by Kalinikos and Slavin [26] with modification done by Kakazei et al. [7].

5.7 FMR images using local spectroscopic information

From the probe-sample distance dependent spectra shown in Fig. 5.4 and Fig. 5.5, the information of ZFR is useful for determination of the FMR which could measured by the conventional FMR. However the TFR from the spectra includes the local information about the sample. We set the external magnetic ﬁeld at the position of the TFR of the ﬁrst order FMR mode and made the 2D scan of the sample at constant probe-sample separation. The results of 2D FMR images are shown in Fig. 5.8.

All images shown in Fig. 5.8 were taken as the probe magnet was laterally scanning at diﬀerent magnetic ﬁelds but the probe-sample separation remains the same

(150 nm). The color scale shown in the left side corresponds the force signal. The red is deﬁned as the positive force signal and the blue is deﬁned as the negative force signal in FMRFM.

Fig. 5.8 b) was scanned at 11960 G. The external magnetic field, 11960 G, corresponds to the first order FMR mode at 7.7 GHz when the probe was located at the center of a disk. The lateral dependent FMR image has the maximum force signal at the center of a disk and the force decreases as the probe is moving away from the center of a disk in all lateral directions. The average probe magnetic field on the disk decreases as the probe magnet moves away from the disk center.

58 chch

Figure 5.8: Images were obtained at 150 nm as tip-sample separation at 7.7 GHz and 60% amplitude modulation. The lower right panel shows the FMRFM spectrum obtained with the probe located at the center of a permalloy disk. Each image was obtained when the external magnetic ﬁelds was set at 11960, 11980, and 12040 G respectively in the parallel conﬁguration (Ref. [4]).

59 When the probe was located at the center of a disk at the probe-sample separation as 150 nm, the FMRFM spectrum is shown in Fig. 5.8 d) and the TFR resonance meets at 11960 G. If the probe is moving away from the center of a disk, the average probe magnetic ﬁeld through a disk simultaneously decreases and then the TFR cannot meet the resonance at 11960 G. Hence the force signal monotonically decreases as the probe moves away from the center of a disk at 11960 G.

In Fig. 5.8 b), the external magnetic field tuned at higher magnetic fields 11980 G than 11960 G without changing the probe-sample distance (150 nm). For this case, the TFR cannot meet the FMR condition when the probe is located at the center of a disk because the effective magnetic field exceeds the resonance condition. However as the probe moves away from the center of a disk, the average of the probe magnetic

field becomes smaller and then the effective magnetic field decreases simultaneously.

Consequently the TFR condition could meet when the probe moves away from the center of a disk. The force signal grows and decreases again in Fig. 5.8 c). This non-monotonical behavior shows the resonance-ring images and the radius of the resonance ring increase as the external magnetic ﬁeld does simultaneously increase.

5.8 Conclusion

By using the SmCo hard probe magnet, we could demonstrate the FMR of the circular disk array using MRFM in the parallel and antiparallel orientations of the direction of the probe and external magnetic ﬁelds.

FMRFM enables to extract two diﬀerent categories of FMR resonances, the Zero tip Field Resonance (ZFR) and Tip Field inﬂuenced Resonance (TFR) from a single

60 FMR spectrum. From the probe-sample separation dependent FMRFM measurement, we veriﬁed that the ZFR is coming from the remote disks where the probe magnetic ﬁeld is neglected and the TFR includes local FMR information from a few disks in the localized proximate region of the probe.

The ZFR includes the global dynamic properties of the sample. Therefore they could demonstrated by the conventional FMR experiments. The experimental results of the ZFR were in excellent agreement with the theoretical calculation if we assume that the magnetostatic modes are quantized due to the conﬁned geometrical structure of the sample.

From the TFR, it provides two dimensional FMR images of the sample in which individual disks are resolved as the lateral probe position were varying at the fixed probe-sample separation. Consequently we demonstrate that the probe magnetic field is so sensitive that the TFR is affected by the lateral position of the probe magnet.

61 CHAPTER 6

FERROMAGNETIC RESONANCE WITH WEAK FIELD PERTURBATION

6.1 Motivation

In order to reach a higher sensitivity in FMRFM experiments, it is desirable to make the measurements at small probe-sample separation, where probe magnetic ﬁeld gradient became higher, and an inhomogeneous probe ﬁeld could reach kiloG range.

In this case, we may observe that the probe magnetic ﬁeld strongly inﬂuences FMR conditions in a sample.

Moreover, if the probe magnetic field could totally define the boundary conditions for FMR mode instead of the geometrically confined structure, it may allow to excite a localized FMR mode. Such mode would be excellent tool for imaging of extended magnetic structures like thin films. The evidence of FMR excitations, localized near magnetic probe, was shown experimentally in Ref. [27] in FMRFM experiment on continuous permalloy thin film and considered in more details in Ref. [5]. Unfortu- nately the problem of localized FMR mode in inhomogeneous magnetic probe field is complicated for analytical consideration and definitely need more theoretical efforts.

62 The small perturbation of FMR in thin ﬁlm structures was considered in Ref. [14]

[5], that allowed the analytical description of experimental spectroscopy and imaging data [4][5].

In here we will discuss in more details the spectroscopy FMR imaging. We will show that small perturbation approach could be very useful for analysis of our experimental data. In the frame of this approach we developed the procedure for imaging of spatial FMR mode variations using spectroscopy information. That allows to obtain a spatial resolution defined by ratio of FMR linewidth to the field gradient, generated by the probe (similar to Magnetic Resonance Imaging (MRI)), unlike the earlier experiments [13], where spatial resolution was defined by probe size.

The new way to investigate the spectroscopy FMR imaging method gives us lots of advantages to understand FMR in a disk sample. The FMR images over a disk shown in Chapter 5 spends longer time to finish each image and they cannot include all FMR information at continuous external magnetic field changes. The new way reduces the scanning time but includes complemental information of FMR. They also gives easier understanding the convolution of the probe magnetic field with the FMR mode profile presented by Yu. Obukhov et al. [5].

6.2 Experimental conditions

FMRFM measurements were performed at T = 4.2 K and high vacuum (better then 10−6 torr) on a square array of 50 nm thick, 2 µm diameter permalloy disks with 2.2 µm center to center spacing. The sample were prepared on low doped Si substrate of 125 µm thick that reduces possible attenuation of microwave (MW)

ﬁeld on the sample. The experiment was performed in an external magnetic ﬁeld

63 Figure 6.1: Cantilevers magnetometry data for probes, used in our experiments. (a) The SEM picture (inset) and magnetometry data for cantilever used in experiments presented in this Chapter. (b) The SEM picture (inset) and magnetometry data for cantilever used in experiments presented in Fig. 6.6 (The probe in (a) was provided by R. Steward and the probe (b) was provided by P. Banerjee).

Hext ≈ 1.2 T , large enough to saturate the sample magnetization perpendicular to the sample plane. Normally the saturation magnetization of permalloy sample is less than 1 T even though the ratio of Ni and Fe could aﬀect the saturation magnetization.

6.2.1 MW resonator

In here we used the same MW resonator used in Chapter 5. However the resonance frequency was shifted in ∼ 8 GHz after mounting the new sample. The working area of a resonator is ≈ 1 mm × 1 mm. The bandwidth of resonator remains around 500

MHz

64 6.2.2 Properties of new probe magnets

In here, we used two magnetic probes shown in insets of Fig. 6.1. The ﬁrst probe (Fig. 6.1a), that has a bigger size and was sharpened at the end, was used in experiments shown by Fig. 6.2, Fig.6.3, and Fig.6.5. These experimental data are well described by small perturbation approach.

The second probe (Fig. 6.1b) has smaller size and close to cubic shape. It was used in experiments (Fig. 6.6), where we demonstrated a violation of a small perturbation approach. The component of the probe magnetic moment mtip orthogonal to the plane of the cantilever was measured by vibrating cantilever magnetometry [23][28] discussed in Chapter 4. The calculated magnetic moments for our cantilevers are

1.4 · 10−11 J/T and 2.2·10−12 J/T respectively. Fig. 6.1 shows the ﬁeld dependence of cantilever frequency, that allows to extract magnetic moments of the probes, using

Eq. 4.7, and gives information about their coercivity. The magnetization switching

ﬁeld for both tips was higher then 1.5 T.

6.2.3 Force noise of the cantilever

The cantilever displacement was measured using a ﬁber-optical interferometer.

Far from the sample surface the cantilever has a high Quality factor Q ' 2 · 105, measured by the ring down experiment. The thermomechanical force noise for this √ −17 cantilever was Fnoise ≈ 3·10 N/ Hz.

6.2.4 New protocol to detect the force signal

For FMRFM measurements we manipulate a magnetization of the sample by MW amplitude modulation of ∼ 60% at cantilever frequency and measure the force acting on cantilever. In this experiments the parameters of cantilever (eigenfrequency and

65 Q-factor) are strongly inﬂuenced by the sample and could signiﬁcantly change when we change the lateral position of the probe or probe-sample separation. In the same time we should keep the MW modulation frequency exactly at eigenfrequency of a cantilever to have an optimal sensitivity.

For the experiment reported in Chapter 5, we monitored the cantilever frequency based on the Phase Locked Loop with a Lock-in amplifier. When the cantilever properties is strongly influenced during the scanning over a dot or variation of the external magnetic, its performance failed to find the right cantilever frequency. Hence it requires different ways to overcome the persistent problems.

To track the cantilever frequency changes we implemented a Digital Signal Pro- cessor (DSP) based system [29], which includes self oscillation circuit, unit for measurement of cantilever amplitude and frequency, and unit for generation of MW modulation signal phase locked to cantilever signal. We implemented three basic measurement procedures. In all procedures the cantilever is continuously oscillating at its eigenfrequency.

First, we change the sample magnetization by MW modulation in phase and out- of phase (lock-antilock) with cantilever motion and measure the cantilever frequency change. This additional lock-antilock modulation allows to eliminate from MRFM signal a possible contribution of cantilever frequency variation due to another factors

(cantilever position change, external magnetic ﬁeld change or cantilever frequency drift). These contributions could be described as low frequency noises and additional lock-antilock modulation moves the measurements to higher frequencies. Typically we applied lock-antilock modulation at frequency ∼ 1–10 Hz. This procedure is a typical frequency detection method in MRFM [30] and demonstrated to have a

66 sensitivity limited only by thermomechanical noise in those experiments. However, the optimal bandwidth of these measurements is limited by the frequency region, where thermomechanical displacement noise is higher then the noise of interferometer

(detector noise) [29]. At higher measurement bandwidth the noise of the system is deﬁned by detector noise that in frequency domain rise as Fns ∝ Ω, where Fns is the force noise of the system and Ω is frequency oﬀset from cantilever resonance frequency

ω0 [29]. In FMRFM measurements presented here, especially in imaging, that includes lateral scanning, the eigenfrequency of a cantilever could change a few hundred of hertz during the line scan. That deﬁnes a wide bandwidth for self-oscillation and measurement units. Therefore the noise level in this measurement procedure is usually much higher then the thermomechanical limit.

In second method we applied the MW modulation with π/2 phase shift relative to cantilever motion. In this case the MRFM force acting on cantilever aﬀects the cantilever amplitude, not a frequency. Applying similar additional ”lock-antolock” modulation we measured MRFM force by monitoring of cantilever amplitude change.

If lock-antilock frequency is less then that deﬁned by cantilever time constant τ =

2Q/ω0 (Q is cantilever Q-factor), the cantilever works as natural integrator of force with time constant deﬁned by frequency of lock-antilock modulation. In this case, in spite of self oscillation and measurement units have a broad band, the total bandwidth of the measurements is deﬁned by lock-antilock time constant. That allows to reach the sensitivity higher then in frequency detection method described above. In this regime, if the time constant of ”lock-antolock” is shorter then time constant of a cantilever, the sensitivity of cantilever is decreased, because the cantilever does not

67 reach the maximum amplitude response. Nevertheless it allows to cut the inﬂuence of mentioned above low frequency noises and make the measurement more stable.

In third method we do the same amplitude detection like in second, but exclude a ”lock-antilock” modulation. That provides the highest sensitivity for amplitude detection, but make the measurement susceptible to low frequency noises. We found experimentally that the second method provides optimal combination of stability and sensitivity, and the most experimental data presented in this paper were measured in this regime.

6.3 Results

6.3.1 FMR Spectra for center of permalloy disk

A typical FMRFM spectrum recorded with the probe magnet placed over the center of a disk is shown in Fig. 6.2. These results partially repeat the measurements represented by Chapter 5. Nevertheless these explanations are important for discussion of imaging techniques represented in this Chapter. As the probe magnet is highly coercive, we record FMR spectra for both orientations of the external ﬁeld

Hext relative to the probe magnetic moment m: parallel (Fig. 6.2a) and antiparallel

(Fig. 6.2b).

The spectra exhibit both positive and negative peaks; while the the resonance

ﬁelds of the negative-going peaks (labeled 1 and 4) are independent of the relative orientation of Hext and m, the resonance ﬁeld of positive going peaks (2 and 3) is not.

The negative peaks originate from dots far to the side of the tip where its localized magnetic field Htip is negligible and so does not affect the mode; peak 1 is the first

(n = 1) magnetostatic mode and 4 is the n = 2 mode. Here and further we will refer

68 chch

Figure 6.2: FMRFM spectra of an array of 2 µm diameter permalloy dots acquired for two orientations of the external magnetic field Hext relative to the probe magnetic moment mtip (tip, shown in Fig. 6.1a): parallel (a) and antiparallel (b). The first order TFR modes of the dots close to the micromagnetic probe are indicated as peaks 2 and 3, and the first and the second order ZFR magnetostatic modes are indicated as peak 1 and 4, respectively; these arise from dots far from the probe tip (tip-sample separation 200 nm). The spectra were acquired with the probe magnet located directly over the center of one of the dots as shown in the inset which also schematically indicates the dipolar gradient pattern of the probe magnet mtip.

69 to the signal of this kind as Zero probe Field Resonance (ZFR). The ≈ 400 G spacing between the ﬁrst and the second ZFR modes agrees well with theoretical predictions

[7].

The positive peaks 2 and 3 originate from the disks immediately underneath the probe and their shifts are sensitive to the relative orientation of Htip. If parallel, Htip and Hext will complement each other creating a region of the stronger ﬁeld beneath the probe magnet so these disks will resonate at lower values of Hext than for the

ZFR as shown in Fig. 6.2a. If Htip and Hext are antiparallel the region of weaker magnetic ﬁeld directly under the probe means higher values of Hext will be required as seen in Fig. 6.2b. Hence we will refer to a signal strongly aﬀected by the probe

field as a Tip Field Resonance (TFR). The relative signs of ZFR and TFR signals reflect the dependence of the MRFM force signal F (t) on the gradient of the probe magnetic field ∇Htip: Z

F(t) = (δM(r, t) · ∇) Htip(r)dr, (6.1) V where δM(r) is that component of the static sample magnetization varying at the cantilever frequency in response to the amplitude modulated microwave; the integration is over the entire sample. The dipolar nature of the probe magnetic ﬁeld means the sign of the gradient ∇Htip and hence the force due to magnetization directly below the tip will be opposite to that arising from magnetization oﬀ to the side (Fig. 6.2 inset).

70 6.3.2 New scanning method: spatially resolved FMR spectroscopy

The influence of the localized probe field on the FMRFM is evident in images of the spatial variation of these resonance fields with the position of the probe as it is scanned over the disks. We will refer these measurements as Lateral-Field Scan.

Fig. 6.3b shows an example of such an image acquired by scanning the probe as indicated in Fig. 6.3a with the probe moment m parallel to Hext.

The plot shows the behaviors mentioned earlier: (a wide black line) is the the ﬁrst order ZFR which is independent of probe position as expected. The series of light colored arcs in Fig. 6.3b show the behavior of the probe-position dependent FMR modes or TFR modes.

The maximum shift of the TFR signal occurs when the probe magnet is positioned directly over the center of a disk (as shown in Fig. 6.3a). The oﬀset decreases as the probe moves away from the center of the disk, and the TFR mode eventually merges with the ZFR signal. The less shifted TFR signals, showed by arrows in Fig. 6.3b, arise from the row of disks adjacent to that immediately under the tip, where probe magnetic ﬁeld is weaker.

We can extract the FMR spectrum for any lateral position of the probe which is one of the main advantages using the Lateral-ﬁeld scanning. The spectrum for probe position in the center of the dot is shown in Fig. 6.3d and is similar to spectrum

Fig. 6.2a.

The 2D map of signal intensity shown in Fig. 6.3f-j was obtained by scanning the tip in both lateral directions X-Y at constant applied ﬁeld Hext = 12.25 kG; 12.28 kG;

12.30 kG; 12.35 kG; 12.40 kG respectively. Each magnetic disk at 2D MRFM image

71 shows the resonance ring with radius defined by probe magnetic field and external magnetic field offset from the ZFR. The small deviation from circular symmetry is due to a lateral component of probe magnetization.

6.3.3 Spatially resolved FMR spectroscopy for the second order FMR mode

Fig. 6.3c shows the Lateral Field Scan for second order mode, that corresponds to lower external ﬁeld. It has the same structure as the scan for ﬁrst order mode

Fig. 6.3b, but much more smaller amplitude of a force. The ZFR does not depends on the probe position. The TFR for second order mode practically repeats the TFR for first order mode. There is some difference for TFR from adjacent rows that are split in two different modes, and are shown by arrows in Fig. 6.3c. We attribute it to probe lateral position drift during the time of experiment (∼ 10 hours). In this case the Lateral Scan possibly no longer goes through the centers of nearest neighbor dots, but in the line slightly shifted. That makes the adjacent row of dots on one side to be more close to the probe then the row on the other side, and the degeneracy of their spectra will no longer exist.

6.3.4 FMR with weakly perturbing probe ﬁeld

The behavior of TFR for Lateral Field Scans could be explained in more details by small perturbation approach. As was shown in the case of magnetic disk [31,

?],[6],[7], the unperturbed magnetostatic modes could be approximated by Bessel functions µn = µn0J0(knr), where µn is a normalized dynamic component of dot qR 2 magnetization for n-th order mode µn(r) = mn(r)/ V mn(r)dr (mn(r) is dynamic magnetization deﬁned by applied MW power), kn = αn/a is the wave vector for

72 Figure 6.3: Lateral-Field Scan for external field parallel to the tip magnetization (tip, shown in Fig. 6.1a): (b) Lateral-Field Scan for 1st order mode. FMR spectra recorded by sweeping the external magnetic field Hext and spatially scanning the probe along the one-dimensional trajectory indicated by the dotted line above the diagram of the dots in (a) (tip-sample separation is 200 nm). The dashed curves in (b) show the analytically calculated dependence of the TFR resonance field on probe position (see Eq. 6.2); these agree excellently with the experimental data. Arrows show the signal from the rows of the dots aside the tip trajectory (adjacent rows). (d) Spectrum extracted from image (b) with the tip located over the center of the dot [see panel (a)]. Numerical labels are the same as in Fig. 6.2a. (c) Lateral-Field Scan for 2nd order mode. Arrows show the splitting of the signal from adjacent rows. (e) Spectrum extracted from image (c) with the tip located over the center of the dot. Blue colored images are experimental fixed-field ((f) Hext = 12.23 kG; (g) Hext = 12.27 kG; (h) Hext = 12.30 kG; (i) Hext = 12.34 kG; (j) Hext = 12.38 kG) 2D images (4.8 µm × 4.8 µm) of the spatial X-Y variation of the FMRFM signal. Dotted line in (f) shows the direction of lateral scan used for (b) and (c) and presented in (a).

73 Figure 6.4: Components of convolution problem used for theoretical prediction (see Eq. 6.2) of Lateral-Field Scan presented in Fig. 6.3b by dashed line. z-component z −11 x of the tip moment mtip = 1.1 · 10 J/T, lateral component of tip moment mtip = −11 st 0.55 · 10 J/T. Eﬀective tip dipole to sample distance z0 = 4 µm. (a) 1 order mode µ2(r) for permalloy dot of our sample. (b) Tip ﬁeld. (c) theoretical prediction for signal Fig. 6.3b obtained by convolution of (a) and (b). Dashed line in Fig. 6.3b correspond to the data extracted from (c) along dashed line.

cylindrical symmetry, αn is n-th zeroth of Bessel function J0(αn) = 0, and a is the radius of the magnetic disk. In the case of perturbation of SW mode by probe magnetic ﬁeld, due to orthogonality of Bessel functions, we could use perturbation theory. The ﬁrst order perturbation gives Z 0 2 0 0 Hn(r) − Hn0 = − Htip(r − r )µn(r )dr (6.2)

0 where r, r are lateral positions, Hn0 is a resonance field for n-th order mode not perturbed by tip field (ZFR), Hn is a resonance field of n-th order mode in a presence of tip field (TFR). Similar approach was used by Kakazei et al. [7] and Guslienko et al. [18] for FMR frequency correction due to inhomogeneous demagnetization

field. Eq. 6.2 means that a new resonance frequency in the presence of the tip field could be found by averaging of tip field through the dot with the squared weight of

74 Figure 6.5: Lateral-Field Scan for external field anti-parallel to the tip magnetization (tip, shown in Fig. 6.1a): (c) Lateral-Field Scan for 1st order mode. FMR spectra recorded by sweeping the external magnetic field Hext and spatially scanning the probe along the one-dimensional trajectory indicated by the dotted line above the diagram of the dots in (a) (tip-sample separation is 200 nm). The dashed curves in (c) show the analytically calculated dependence of the TFR resonance field on probe position (see Eq. 6.2); these agree excellently with the experimental data. (e) Spectrum extracted from image (c) with the tip located over the center of the dot [see panel (a)]. Numerical labels are the same as in Fig. 6.2b. (b) Lateral-Field Scan for 2nd order mode. (d) Spectrum extracted from image (b) with the tip located over the center of the dot. Blue colored images are experimental fixed-field ((f) Hext = -12.77 kG; (g) Hext = -12.75 kG; (h) Hext = -12.70 kG; (i) Hext = -12.67 kG; (j) Hext = -12. 64 kG) 2D images (4.8 µm × 4.8 µm) of the spatial X-Y variation of the FMRFM signal. Dotted line in (f) shows the direction of lateral scan used for (b) and (c) and presented in (a).

75 unperturbed FMR mode. Eq. 6.2 is valid for any eigenmode of hermitian operator, i.e. for Herring-Kittel type integral equation with arbitrary boundary conditions, where

µn is not necessary Bessel function. After Fourier transform of Eq. 6.2 we have:

˜ ˜ ˜2 ∆Hn(k) = −Htip(k) · µ n(k) (6.3)

Here f˜(k) means Fourier image of function f(r), and k is Fourier wave-vector for vari- able r, ∆Hn(r) = Hn(r) − Hn0. Eq. 6.3 represents a standard solution of convolution problem. According to Eq. 6.3 we could reconstruct the distribution of squared mode

2 µm(r), if we know the tip field Htip (or reconstruct tip field knowing the mode shape), which means direct imaging of FMR modes in the sample. This imaging option is different from the mode imaging using force acting on cantilever Z 1 0 2 0 0 Fn(r) = − · ∇Htip(r − r )mn(r )dr (6.4) 2Ms

(∇Htip is gradient of a tip ﬁeld), demonstrated by Midzor et al. [13]. Instead, we use the spectroscopy information and proposed method more close to frequency encoding, which used in Magnetic Resonance Imaging. Moreover the spectroscopy signal from close positioned but independent modes (like from two close positioned disks) does not overlap each other, as could be seen on Fig. 6.3b, and could be resolved with the accuracy of FMR line width. Fig. 6.4 shows the components of convolution problem described by Eq. 6.2. Fig 6.4a represents a 1st order mode (Bessel function) for 2 µm diameter dot, Fig 6.4b shows the Htip and Fig 6.4c shows the convolution of these two functions. Dashed line in Fig 6.3b, obtained by extracting the resonance ﬁeld shift

(Hn(x) − Hn0) along the dotted line in Fig. 6.4c, compares this convolution with our experimental results. We have a good agreement of experiment and theory deﬁned by Eq. 6.2.

76 6.3.5 Reconstruction of the probe magnetic ﬁeld

For direct mode shape study the magnetic probe comparable or smaller, then characteristic size of FMR mode features, should be used. The magnetic moment of the probe should be small to keep regime of small perturbation valid. For our experimental conditions it is easier to reconstruct the probe magnetic ﬁeld. The probe magnetic ﬁeld, obtained from above reconstruction, could be approximated

z −11 as a ﬁeld of point dipole with z-component of moment mtip = 1.1 · 10 J/T, and

x −11 lateral component mtip = 0.55 · 10 J/T positioned at the distance z0 = 4 µm from dot. According to x − y images Fig. 6.3 (blue scale pictures) we can evaluate the

y −11 y component of tip magnetic moment as mtip = 0.3 · 10 J/T. This values are in agreement with tip magnetometry data. The physical size of magnetic tip is about

6 µm and, in spite of sharp edge of a tip (see Fig. 6.1a) and tip-sample separation of 0.2 µm, the center of observed eﬀective tip dipole is positioned more far above the sample. That means the magnetic material at the end of the tip (Fig. 6.1a) is damaged possibly due to Ion Beam milling or/and oxidation.

6.3.6 The inﬂuence of the probe magnetic ﬁeld into FMR mode

According Eq. 6.2 the behavior of TFR for the probe anti-parallel to external

ﬁeld should be very similar to the case, when the probe is parallel to the external

field. The only difference is the relative sign of TFR field shift. The Lateral Field

Scans for an external ﬁeld opposite to the probe magnetic moment are presented in

Fig. 6.5b,c, that corresponds to second and ﬁrst FMR modes respectively. These data are very similar to the case, when probe moment is parallel to external ﬁeld.

77 The dashed curves in Fig. 6.5c shows a theoretical estimation of TFR shift made by Eq. 6.2, that demonstrates a very good agreement with experiment. This proves that the small perturbation approach is valid for this case either. Nevertheless the

z −11 x estimated probe magnet has magnetic moment mtip = 0.2 · 10 J/T and mtip = 0.05 · 10−11 J/T. This moment, according to a ﬁt is positioned in the distance of z0 = 2.5 µm from the sample. This demonstrates that magnetic tip signiﬁcantly changes its magnetization pattern, in spite of the tip magnetic moment doesn’t change the sign. It also demonstrates that these measurements could be us

6.3.7 Observation of FMR mode splitting

Some unusual behavior could be seen in Lateral-Field Scan for second order mode

Fig 6.5b. The ZFR and TFR resonances are split in spite of the the ﬁrst order mode

(Fig. 6.5c) doesn’t demonstrate any splitting. The experimental spectrum for tip positioned in the center of the dot presented in Fig. 6.3d by solid line. We made an FMR micromagnetic simulation (see Ref. [8]) study of this case and found that this splitting is induced by a small sample tilt in external field. In these conditions the second order mode looses cylindric symmetryand degeneracy for directions of k parallel to the plane of the tilt and perpendicular to this plane. When the k is parallel to the plane of the tilt the dynamic fields due to tilt-induced demagnetization factor is parallel to dynamic field due to non zero k vector. In this case the resulting frequency of the mode will be:

2 ω = (ωH + ωex + ωM Nα + ωM f(kL)) · (ωH + ωex) (6.5)

Here we use a dispersion relation and a notation of Kalinikos-Slavin paper [20]: ω is

FMR frequency; k is a wave vector; L - ﬁlm thickness; ωH = γHtot, and Htot is a total

78 chch

Figure 6.6: Spatially resolved FMR scans for tip presented in Fig. 6.1b. (a) and (b) are the Scans made at tip-sample separation z = 1.9 µm and z = 1.4 µm respectively, that shows behavior excellently described by small perturbation approach (see dashed lines). Scan (c) was acquired at z = 1 µm and shows a strong deviation from small perturbation approach (see dashed line) and demonstrate behavior better described by appearance of localized mode (solid line).

79 static ﬁeld; γ is a gyromagnetic ratio for electrons; ωM = γ4πMs, Ms is saturation

A 2 magnetization, ωex = 2 ωM k - exchange contribution, A - exchange stiffness, and 2πMs f(kL) = 1−(1−exp(−kL))/kL; Nα is a demagnetizing factor of the film in the plane of the tilt and in a direction perpendicular to external field. In the case when the tilt plane is perpendicular to k the FMR frequency will be

2 ω = (ωH + ωex + ωM Nα) · (ωH + ωex + ωM f(kL)) (6.6)

The mode with the k-vector oriented in the plane of tilt has slightly lower frequency then the mode perpendicular to this plane. The results of numerical simulations for described process is shown in Fig. 6.5 by dashed line and is in a good agreement with experimental data. The angular dependent mode splitting affects not only second order mode, but also a higher order modes. Fig. 6.5a demonstrates the higher order mode splitting and their resonance field for different angles of sample tilt relative to external field.

6.3.8 FMR with strongly perturbing probe ﬁeld

Experimental conditions with external field opposite to the tip magnetization was discussed in a number of publication [4, 14] as a way to create a localized mode, where the mode boundary defined by the tip field other then by the boundary of magnetic structures. In Ref. [5] the results of numerical simulation, that specify the conditions for localized mode, were presented. Basically for this localized mode we need to create the localized region of lower magnetic field (Field Well [FW]), that could be achieved, when the probe magnetic moment is opposite to external magnetic field.

The depth of the Field Well should be bigger then the dynamic ﬁeld contribution to

80 the mode resonance ﬁeld, deﬁned by this mode localization. If we write the Kalonikos-

Slavin dispersion relation in long wavelength approximation (kL ¿ 2π), and disregard exchange contribution, we have:

ω = ωH + ωM kL/4 (6.7)

The condition for localized mode could be written as:

ω α |H | ≥ M L/4 · 1 (6.8) tip γ a where α1/a is eﬀective k-vector for localized mode, α1 is a ﬁrst zero of Bessel function

J0(α1) = 0 and a is a radius of localized mode. For instance, to generate the localized

SW mode with the localization radius a = 0.5 µm, the tip ﬁeld should be |Htip| ≥

600 G within the area of radius a.

Fig. 6.6 shows our experimental data, when we tried to create a localized mode inside the permalloy disk of radius 1 µm (used in all above experiments), using the probe with effective tip radius ∼ 0.5 µm (see Fig. 6.1b). When the tip field is smaller then 600 G (Fig. 6.6b,c) the Lateral-Field spectra are very good described by small perturbation approximation (see dashed lines). For higher tip field (see Fig. 6.6c) the experimental data strongly deviate from small perturbation approximation, shown by dashed line, meaning that the Bessel mode is strongly perturbed. The expected behavior for localized mode, obtained by numerical simulation is shown by solid line.

We believe, that in Fig. 6.6c we observe the transition from unperturbed Bessel mode to localized mode.

81 6.4 Conclusion

We demonstrated our experimental data for FMRFM imaging. We showed, that small perturbation approach could be very useful for analysis of our experiments. In the frame of this approach we proposed the procedure for imaging of spatial FMR mode variations using spectroscopy information. That allows to obtain a spatial resolution defined by ratio of FMR linewidth to the field gradient, generated by the probe (similar to Magnetic Resonance Imaging (MRI)), unlike the earlier experiments, where spatial resolution was defined by probe size.

We studied experimentally and theoretically the inﬂuence of the sample tilt in external magnetic ﬁeld on a FMR modes splitting. We also showed experimentally the limitations of small perturbation approach, and demonstrated the experimental conditions that could results in generation of localized FMR mode.

82 CHAPTER 7

FMR MODE SUPPRESSION AT STRONG FIELD PERTURBATION

7.1 Motivation

The purpose of this experiment is to study the influence of the strong inhomogeneous probe (magnetic tip) magnetic field on FMR mode shape. For the regime of the small perturbation discussed in Chapter 6, the probe magnetic field was not strong enough to observe the localized FMR confined by the probe magnetic field [5].

Therefore we prepared a new probe magnet shown in Fig. 7.2 and used a permalloy disk array with disk diameter of 5 µm. That allows to implement the condition when the magnetic probe is much smaller than the sample disk.

As the results of this experiments, we demonstrated FMR signal from an isolated single disk and we observed that probe field allows not only modify the mode frequency (see small perturbation approach discussed in Chapter 6 and Ref. [5]), but also strongly influences the shape of the mode. In extreme the probe field could locally suppress or destroy the FMR mode under the tip. That allows to create localized dead spot or anti mode, that could be employed for local sample characterization similar to localized mode (see Ref. [5]), especially in the case of extended sample like continuous ferromagnetic film [27].

83 chch

Figure 7.1: (a)The sample is prepared using Optical Lithography on low doped Si Substrate whose thickness is 125 µm. Each disk has 5 µm as diameter and 40 nm as thickness. The interdisk separation is 25 µm from center to center. (b) The MFM image is taken using FMRFM aparatus at 4K. Before FMR measurement from this diluted disk sample, we found one of disks using MFM image and the probe is located at the exact the center of the found disk sample.

7.2 Experimental conditions

7.2.1 An isolated disk sample

FMRFM measurements were performed at T = 4.2 K on a square array of 40 nm thick, 5 µm diameter permalloy (Py) disks with 25 µm center to center spacing in a magnetic ﬁeld Hext adequate to saturate the magnetization (excitation frequency ∼

7 GHz) applied perpendicular to the plane of the disks.

The distribution of disks on Si substrate is widely separated. Hence each dot is completely isolated from others. From the previous FMRFM experiments in Chapter

5 and 6, the separation of each dot is less than 0.3 µm and then we simultaneously observed both Zero tip Field Resonance (ZFR) and Tip Field Resonance(TFR).

84 chch

Figure 7.2: (a) The magnetic moment at various external fields is shown. The red arrows shows the direction of the external field change. Initially it starts from 2T to -2T. After that, the field changes from -2T to 2T. In FMRFM measurement, it performs from 1.3T to 1T to prevent unwanted magnetic moment change. Within the experiment range from 1T to 1.3T, the magnetic moment is approximately 1.0 · 10−12 J/T. (b)The top view of the SEM image of the probe was shown. The probe was sitting on the chopped off tip of the cantilever. (c) The side view of the SEM image of the probe was shown. From (b) and (c), the cubic shape of the probe was prepared after FIB process. The dimension of the cube was 1 × 1 × 1µm. (The probe provided by I. Lee)

In the experiments described here, only TFR mode will exits, due to the large interdisk separation.

7.3 Probe magnet

The FMR signal is detected by a microcantilever force detector (0.3 N/m) equipped with a high coercivity SmCo micromagnetic probe tip fabricated by means of Focused

Ion Beam micromachining down to the size 1 × 1 × 1 µm.

85 The component of the probe magnetic moment is orthogonal to the plane of the cantilever, measured by vibrating cantilever magnetometry [23][28], is 1.0·10−12 J/T, and the coercivity exceeds 1.5 T.

However we observed that the magnetic moment depends on the external magnetic

ﬁeld. This behavior means that the probe magnet is possibly composed of multiple domains. Some of them may have less coercivity and they may reduce the total magnetic moment which depends on the external magnetic ﬁeld. In order to understand this observation, more detailed analysis is being followed in our group.

Although the probe magnet has the external dependent magnetic moment, it is stable within the range of FMR experiment which is from 1.3 T to 1 T. During the measurement, the external ﬁeld was sweeping from 1.3 T to 1 T in order to prevent the minor change of the magnetic moment.

7.3.1 Signal measurement method

The FMR force signal was generated by 80% amplitude modulation of the microwave field at the cantilever resonance frequency of ∼ 16 kHz; for more details see Refs. [4][29] and Chapter 6. FMR spectra are recorded by sweeping the external magnetic field Hext and keeping the tip at constant height and lateral position above the sample; such spectra are then recorded for different probe vertical position z

(Fig. 7.3) and lateral position relative to the center of magnetic disk (Fig. 7.6). Here and further we presume that z is tip-sample separation and does not include eﬀective tip radius (∼ 500 nm).

86 chch

Figure 7.3: FMRFM spectra of an individual 5 µm diameter permalloy disk at various tip-sample separations z. The magnetic probe for all spectra was positioned in the center of Py disk. Spectra are oﬀset in vertical direction for convenience. The distance z change from top spectrum to bottom as z = 1669; 1552; 1434; 1316; 1199; 1081; 963; 846; 728; 610; 493; 375; 258 nm. The magnetic moment of the probe magnet is parallel to the external magnetic ﬁeld.

87 7.4 Results

7.4.1 Strong probe ﬁeld behavior of FMRFM spectra

A FMRFM spectra recorded over the center of the disk are shown in Fig. 7.3.

Each spectrum measured at different probe-sample distance z, that was varied from z = 1669 nm (top) to z = 258 nm (bottom). Spectra are offset in vertical direction for convenience. Probe magnetic moment is parallel to external magnetic field.

The spectra exhibit a few negative peaks. All these peaks are strongly aﬀected by the tip-sample separation. Therefore, we eliminated all possible FMR signal coming from remote dots from the probe. For each peaks, we attribute to 1st, 2nd and higher order modes, because at large tip-sample separation (top spectrum) the tip

ﬁeld inhomogeneity is too small to perturb the FMR Bessel modes, deﬁned by disk geometry [6][31][7].

The position of the resonance peaks evolves, when we decrease tip-sample separation z. The evolution of resonance ﬁeld for 1st order mode is shown in Fig. 7.4.

First, when z decreases, the peak position go to lower field, due to tip field, which complements the external field and make resonance happened in lower external field.

This behavior was observed in earlier works (see Tip Field Induced Resonance (TFR) in [4] and [5]) and is well understood. At further z decrease the resonance peak position reaches minimum, at z ∼ 500 nm, and goes to higher external ﬁeld, as if the

ﬁeld on the Py disk decreases closer to the tip. This novel behavior and underlaying physical picture is a main subject of this Chapter.

88 chch

Figure 7.4: Dependance of FMR resonance field for 1st order mode on probe sample separation z. Circus marked line is experimental data taken from Fig. 7.3 . Dashed line is a theoretical calculation in assumption, that FMR mode shape was not changed by probe field, performed using Eq. 7.3 (see text). Squares marked line is numerical simulation. 2D inserts represent the mode shape at different distances z, obtained by numerical simulation.

7.4.2 Theory of FMR in a strongly perturbing ﬁeld

Typically FMR excitation is analyzed using the the Landau-Lifshitz equation discussed in Chapter 3. Assuming that both sample magnetization and the magnetic

ﬁeld undergo small circularly polarized oscillations about the equilibrium value given

iωt iωt by H = Heﬀ + he and M ≈ Ms + me respectively, the dynamics of the magnetization precession can be expressed as:

1 dM ≈ − ([m × H ] + [M × h]) eiωt (7.1) |γ| dt eﬀ s

Here Heﬀ = Hext + Hd(r) + Htip(r) is a sum of external, demagnetizing and probe

ﬁelds respectively (we neglected exchange term, because for our sample size this con-

tribution is small); Ms is saturation magnetization of sample, h and m are dynamic

89 field and magnetization respectively; γ is a gyromagnetic ratio. Demagnetizing field for our disk Hd(r), generally speaking, depends on r. Considering that Heff and Ms is perpendicular to Py disk plane and m and h are in plane we can rewrite Eq. 7.1 as: ω ¡ ¢ m(r) = H + H (r) + H⊥ (r) m(r) − M h(r) (7.2) |γ| ext d tip s

Multiplying both sides of equation by m(r) and making integration through the disk volume we have: R ω h(r)m(r) H = − hH (r)i − hH⊥ (r)i + M VR (7.3) ext d tip s 2 |γ| V m (r) R R 2 2 where hH(r)i = V H(r)m (r)/ V m (r) is average ﬁeld on the sample weighted

2 by the squared FMR mode m (r). Demagnetizing ﬁeld for thin disk is hHd(r)i ≈

−4πMs with small correction due to demagnetizing ﬁeld inhomogeneity near the edges of the disk (see solid line in Fig. 7.5a) , that discussed in details in Ref. [7]. R Integral − V h(r)m(r) could be think about as demagnetizing ”energy” of dynamic component and its contribution is always negative (see discussion below).

In the case when tip ﬁeld doesn’t perturb the mode m(r), the z-dependance of

⊥ resonance ﬁeld is deﬁned only by term hHtip(r)i of Eq. 7.3. For unperturbed trans-

(n) verse magnetization we used Bessel function m (r) = µ0J0(knr), that according to

Ref. [6][31][7], is a good approximation for a disk geometry. Here r is the distance from the centers of the disk, a is the disk radius, αn is the n-th zero of the equation

th J0(αn) = 0 and kn = αn/a is the eﬀective wave vector of the n mode. This approximation is valid even in the case when the inhomogeneity of demagnetizing ﬁeld

Hd(r) is not disregarded [7]. Dependence of resonance ﬁeld on tip-sample separation in this case for mode number n = 1 (1st order mode) is shown by dashed curve in

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Figure 7.5: Numerically calculated spatial profile of the first order FMR mode m excited in a 5 µm diameter, 40 nm thick Py disk (4πMs = 11 kG) in an external magnetic field Hext = 13 kG: a) uniform external field, b) in the presence of the field from a spherical magnet with magnetic moment 10−12 J/T located 250 nm above the center of the disk with tip magnetic moment of the probe magnet parallel to Hext. The mode is confined out of the region of strong field beneath the probe. The mode amplitude plotted along a line through the center of the dot is shown with the dotted line; the corresponding magnitude of the total magnetic field Heff is shown with solid line; the dashed line indicates the resonant Hres field of the mode. Pluses and minuses illustrate effective magnetic charges picture used in discusion (Ref. [8]).

91 Fig. 7.4. According to figure, tip field doesn’t significantly perturb FMR mode at distance grater then ∼ 1.2 µm, bellow which perturbation become obvious.

We could qualitatively explain experimental data in Fig. 7.4 using Eq. 7.3. When magnetic probe approach the disk, the strong tip field suppress the Bessel mode in the center of the disk and we obtain ”doughnut” like shape of the mode (see dotted lines in Fig. 7.6a, b). In this case the weight of the mode shifts from center of the disk to the edges. The term hHdi will increase, because demagnetizing field near edges is higher (see Fig. 7.5a solid line) and it will decrease resonance field Hext according

⊥ Eq. 7.3. The term hHtipi should generally increase due to increasing of probe field, but when mode shape goes from Bessel to ”doughnut”, the weight of the mode shifts to the region of small or negative probe field (Fig. 7.5b) , that decreases this term and respectively increases the resonance field Hext. For analysis of last term it is helpful to consider it as ”energy” and use effective ”charges” picture as shown in

Fig. 7.5. When charges not exist like in case of inﬁnitely long cylinder magnetized along it axis, the ”energy” of system is zero (demagnetizing factor of long cylinder is close to zero). When number of ”charge” pairs increases the ”energy” increases, that proves that contribution of this term is always negative (demagnetizing factor always negative). When mode shape changes from Bessel to ”doughnut” (Fig. 7.4a, b), the ”energy” increases, because the number of charges increases and dynamic magnetization m experiences higher demagnetizing ﬁeld h, that according Eq. 7.3

⊥ decreases the resonance ﬁeld. Basically the term hHtipi and reshaping of the mode from Bessel to ”doughnut” are responsible for resonance ﬁeld increase for experimental data in Fig. 7.4 at tip-sample separation less then 1.2 µm.

92 chch

Figure 7.6: a) Illustration of probe motion for Lateral-Field Scan. Probe is positioned at constant height z above the sample and FMR spectra recorded at different probe lateral position, following the line through the center of the sample disk. b-d) Lateral- Field Scans for different probe-sample separation z: b)z = 3625 nm,; c) z = 1050 nm; d) z = 580 nm. Dashed curves is theoretical calculation of ferromagnetic resonance field evolution with changing of probe lateral position in assumption, that FMR mode shape doesn’t changed by probe field performed using Eq. 7.3 for 1st order Bessel mode (see text). Force for each Lateral-Field Scan is normalized to its maximum value. 2D inserts represent the mode shape at different lateral position, obtained by numerical simulation.

93 7.4.3 Numerical simulation results

To prove the above picture we have solved the problem numerically by lineariz- ing the Landau-Lifshitz equation and ﬁnding the eigenvalues and the eigenstates of the resulting matrix equation. More details are available in Ref. [8]. The result of simulation, presented in Fig. 7.4 by squares marked line, is in a good agreement with experiment. The simulated mode shapes for unperturbed (z = 3.8 µm) and perturbed

(z = 0.7 µm) cases are presented in Fig. 7.4a, b respectively by doted lines. The result of simulation conﬁrms our qualitative picture presented above.

7.4.4 Spatially resolved scanning inﬂuenced by the strong probe ﬁeld

To understand the influence of probe field on mode shape in more details we studied dependence of FMR spectrum on lateral position of the magnetic probe relative to sample. During the measurements the height of the probe z was kept constant and we measured the FMR spectra in different lateral position of the probe. Change of the probe position follows the line above the Py disk running trough the disk center as shown in Fig. 7.6a. Here and later we will call these measurements as Lateral-Field

Scan.

Results of Lateral-Field Scans for diﬀerent probe height z are presented in Fig. 7.6b, c, d. For comparison we calculated expected behavior for unperturbed Bessel mode

⊥ (the term hHtipi of Eq. 7.3) and showed it as dashed line on each picture, similar to dashed line in Fig. 7.4. Far from the sample (z = 3625 nm, see Fig. 7.6b) the Bessel mode perturbation is negligible and approximation showed by dashed curve describes the data pretty good.

94 At distance z = 1050 nm (see Fig. 7.6c) the probe ﬁeld start to perturb the mode and experimental resonance ﬁeld is slightly higher, then suppose to be for unperturbed case. The perturbation is most pronounced when the probe is near the center of the disk.

At z = 580 nm (see Fig. 7.6d) the Lateral-Field behavior of resonance is strongly diﬀerent from expected for unperturbed case. First, the resonance happens at considerably higher ﬁeld, then it suppose to be for unperturbed case. Second, we could recognize separate 1st, 2nd, and 3d order modes when the probe aside of the disk center (numerical simulation for thirst order mode gives in this case a ”skewed” mode shape, see Fig. 7.6d), but when probe close to the center of the disk these modes merge and create single broad resonance peak.

This support our picture that the probe field locally suppresses the mode under the probe. In this case modified 1st and 2nd order Bessel modes (and may be higher order modes) will not be too different. Generally the modes evolution in this case is quite complicated and need more detail studies.

7.5 Conclusion

We demonstrated that localized probe field allows not only modify the FMR mode frequency, but also strongly influences the shape of the mode in the case of a stronger probe magnetic field.

In addition, we propose qualitative explanation of our experimental data and conﬁrmed our picture by numerical simulation. Demonstrated FMR mode shaping by the probe allows to create localized dead spot or anti mode, that could be employed for

95 local sample characterization, especially in the case of extended sample like continuous ferromagnetic ﬁlm.

96 CHAPTER 8

CONCLUSION

In this theses we present our eﬀorts in development of Ferromagnetic Resonance

Force microscopy method. We developed low temperature Scanning Probe experimental setup including, 3D scanner, ﬁber-optic interferometer, Superconducting magnet, microwave resonator, scanning measurement and control system etc. For quantitative analysis of our experimental data, we developed and implemented procedures for cantilever spring constant measurements, magnetic probe evaluation, calibration of a tip-sample distance and cantilever displacement.

This experimental setup was used for Ferromagnetic Resonance experiments on individual magnetic microstructures. Magnetic resonance force microscopy has been used to detect the ferromagnetic resonance of micron-sized permalloy disks. Local spectroscopy at various tip-sample distances enables the separation of spectral features associated with the global and local dynamic properties of the sample. The global properties was used are in good agreement with theoretical predictions and previous experimental investigations using conventional FMR .

The local properties was used to understand and demonstrate the localized FMR conﬁned by the probe magnetic ﬁeld. By using a hard magnetic tip, we are able

97 to image with the tip magnetization aligned either parallel or antiparallel to the external ﬁeld. We also demonstrated that local FMR images could be quantitatively described by small perturbation approach in the case of weak probe magnetic ﬁeld.

The proposed theory is in excellent agreement with our experimental data. We also demonstrated that localized probe field allows not only modify the FMR mode frequency, but also strongly influences the shape of the mode in the presence of strong probe magnetic field.

All mentioned above allows consider FMRFM as a new and very powerfull scanning probe tool for study ferromagnetic micro and nanostructures.

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