FEW ELECTRON PARAMAGNETIC RESONANCES DETECTION TECHNIQUES

ON THE RUBY SURFACE

By

Xiying Li

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Dissertation Adviser: Dr. Massood Tabib-Azar

Co-Adviser: Dr. J. Adin Mann, Jr.

Department of Electrical Engineering and Computer Science

CASE WESTERN RESERVE UNIVERSITY

August, 2005

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

______

candidate for the Ph.D. degree *.

(signed)______(chair of the committee)

______

______

______

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______

(date) ______

*We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents

TABLE OF CONTENTS ...... II

LIST OF FIGURES ...... IV

ABSTRACT...... VII

CHAPTER 1 INTRODUCTION ...... 1 1.1 STUDY OBJECTIVES...... 2

1.2 HIGHLY LOCALIZED EVANESCENT MICROWAVE MICROSCOPIC PROBE TECHNIQUES ...... 3

1.3 ELECTRON SPIN...... 5

1.4 THE MAGNETIC SUSCEPTIBILITIES ...... 9

REFERENCES...... 12

CHAPTER 2 ELECTRON PARAMAGNETIC RESONANCE (EPR) ...... 14 2.1 ABSORPTION LINE SHAPES ...... 14

2.1.1 Lorentzian Type Line Shapes...... 15

2.1.2 Gaussian Type Line Shapes ...... 16

2.1.3 EPR Absorption Line Shapes...... 18

2.2 CLASSIC EPR DETECTION TECHNIQUES...... 19

2.2.1 Microwave Cavity ...... 21

2.2.2 Magnetic Field Modulation...... 23

2.2.3 Frequency Modulation...... 31

3+ 2.3 MATHEMATICAL MODEL OF CR ION IN AL2O3 EPR...... 32

2.4 LARMOR FREQUENCY...... 40

2.5 EPR SENSITIVITY...... 40

REFERENCES...... 42

CHAPTER 3 MICROSTRIP RESONATOR ...... 45 3.1 MICROSTRIP TRANSMISSION LINE RESONATOR...... 45

I 3.2 MAGNETIC MICROSTRIP PROBE TUNING...... 49

3.3 SYSTEM OVERALL Q FACTOR...... 51

3.4 MICROWAVE REFLECTIVE COEFFICIENT...... 53

REFERENCES...... 59

CHAPTER 4 EPR SINGLE POINT MEASUREMENT...... 61 4.1 THEORETICAL BACKGROUND...... 61

4.2 MEASUREMENT APPARATUS...... 63

4.2.1 Magnetic Field Modulation Coil...... 64

4.2.2 Concept of the Circulator...... 65

4.2.3 Phase Sensitive Detector (PSD) ...... 66

4.3 EPR REAL TIME DETECTION...... 69

4.4 COMPARISON OF EXPERIMENTAL AND THEORETICAL LINE SHAPES ...... 71

4.5 THE EPR EMM PROBE SENSITIVITY...... 74

4.5.1 Background Noise ...... 74

4.5.2 The Spin-Spin Relaxation Time...... 74

4.5.3 The Total Number of Spins...... 75

4.5.4 The Minimum Number of Spins ...... 76

REFERENCES...... 77

CHAPTER 5 EPR MEASUREMENT ON THE RUBY SURFACE ...... 79 5.1 SINGLE POINT SURFACE EPR DETECTION ...... 79

5.2 EPR SURFACE SIGNAL ANALYSIS...... 82

5.2.1 The Total Number of Spins near the Surface...... 84

5.2.2 The Minimum Number of Detectable Spins near the Surface...... 86

5.3 COMPARISON BETWEEN THE SURFACE EPR AND ENCLOSED EPR ...... 87

5.4 MAGNETIC FIELD DISTRIBUTION ...... 88

REFERENCES...... 90

CHAPTER 6 CONCLUSIONS AND FUTURE APPLICATIONS OF EPR...... 91 6.1 SUMMARY OF THE EPR MEASUREMENT ON THE RUBY SURFACE AND FUTURE WORKS ...... 91

II 6.2 APPLICATIONS OF SURFACE EMM EPR ...... 94

6.2.1 Quantum Computing...... 95

6.2.2 Defects in Material ...... 95

6.3.3 Aging Process Detection...... 96

REFERENCES...... 97

APPENDIX A 3-D SCHRÖDINGER EQUATION...... 99

APPENDIX B MAGNETIC DIPOLE MOMENT ...... 103

APPENDIX C THE ATOMIC FORCE MICROSCOPY (AFM) ...... 106

REFERENCES...... 107

III List of Figures

Figure 1-1 Magnetic Dipole and Electric Dipole EMM Probe 4

Figure 1-2 Stern’s-Gerlach’s Experiment Illustration 5

Figure 1-3 Zeeman Effects for single electron 8

Figure 1-4 Magnetic Susceptibility of Different Material 9

Figure 1-5 LCR measurement of a Solenoid Coil with Ruby Sample 11

Figure 2-1 Normalized Lorentzian type and Gaussian Type Line Shapes 15

Figure 2-2 First Derivatives of Gaussian and Lorentzian Type Line Shapes 17

Figure 2-3 EPR First Derivative Line Shape 18

Figure 2-4 The Classic CW EPR Setup 20

Figure 2-5 Microwave Magnetic and Electric Fields in a Rectangular Cavity 22

Figure 2-6 Magnetic Field Modulation Detections 25

Figure 2-7 Phase Sensitive Detector Input Voltage Waveform 27

Figure 2-8 Demodulator Output Voltage Signals Before and After LP Filter 29

Figure 2-9 Derivation of the Relation between Power Absorption Slop and Magnetic

Field Modulation 30

Figure 2-10 Cr3+ Energy Levels Splitting With External Magnetic Field 33

Figure 2-11 EPR Line Shapes of the Natural Ruby 37

Figure 2-12 First Derivative EPR Signal for Nature Ruby 39

Figure 2-13 the Filling factor of the Cavity 41

Figure 3-1 Cross Section of the Microstrip Line Resonator 46

Figure 3-2 Structure of the Microstrip Line Resonator 47

Figure 3-3 Magnetic Field Distribution around the EMM Probe Tip 48

IV Figure 3-4 Lump Circuit of the Microstrip Resonator and the Ruby Sample 49

Figure 3-5 Reflective Coefficient at the Critical Coupling 50

Figure 3-6 Quality Factor Q Estimate from S11 Measurement 52

Figure 3-7 Network Analyzer Setup for Measurement of the S21 54

Figure 3-8 S11 vs. Frequency Plots with Varying Magnetic Field Strength 55

Figure 3-9 Microwave Reflective Coefficient 56

Figure 4-1 Theoretical Background of EMM EPR 62

Figure 4-2 EPR Single Point Measurement Apparatus 63

Figure 4-3 Port Diagram of a Three-port Circulator 65

Figure 4-4 Phase Lock-in Amplifier Components 67

Figure 4-5 Program Flow Chart of the Real Time EPR 69

Figure 4-6 Real Time EPR Result for Single Point Measurement 70

Figure 4-7 First Derivative of EPR Absorption Signal Comparison between

Experiment Data and Theoretical Lorentzian Line Shape 72

Figure 4-8 First Derivative of EPR Absorption Signal Cross Zero at 1363 Oe with

Enclosed EMM probe 73

Figure 4-9 Integrated EPR Line Shape 76

Figure 5-1 EPR Surface Measurement Apparatus 79

Figure 5-2 Single Point Surface EPR Real Time Detection 81

Figure 5-3 Comparison of the First Derivative of the EPR Absorption Line Shape and

Gaussian Line Shape 82

Figure 5-4 Comparison of the First Derivative of the EPR Absorption Line Shape and

Lorentzian Line Shape 83

V Figure 5-5 The integrated EPR Absorption Line Shape 84

Figure 5-6 The Integrated Line Shape of the Surface EPR Signal 85

Figure 5-7 The Comparison of the First Derivative Line Shapes between Enclosed and Surface Measurement 87

Figure 5-8 The Comparison of the Integrated Line Shapes between Enclosed and

Surface Measurement 88

Figure 5-9 Magnetic Field Distribution near the EMM EPR Probe tip 89

Figure 6-1 Schematic of Future Enhanced EMM EPR Setup 92

Figure 6-2 Future Micro –Fabricated AFM Tip with Carbon Nano Tube 93

Figure 6-3 Future Research Road Map 94

Figure 6-4 Coordinates of an Electron 99

Figure 6-5 Spherical Electron Harmonics for s and p Orbital 101

Figure 6-6 Spherical Electron Harmonics for d Orbital 102

Figure 6-7 Macroscopic and Microscopic Magnetic Moments 103

Figure 6-8 Magnetic Moment of One Single Electron 104

Figure 6-9 AFM Imaging of silicon (Si3N4) 106

VI Few Electron Paramagnetic Resonances Detection Techniques on the Ruby Surface

Abstract

By

XIYING LI

A method based on the highly localized Evanescent Microware Microscopy

(EMM) is developed to spatially resolve small number of Electron Paramagnetic

Resonance (EPR), also called Electron Spin Resonance (ESR) transitions on the surface

3+ of a signal crystal ruby (Al2O3 doped with Cr ). The EMM probe operates at a resonate frequency of 3.7 GHz, corresponding to a classical S-band EPR, in a precisely controlled

biasing electromagnetic field in the range of 100 to 7000 Gauss. To obtain the highest

signal to noise ratio in the overwhelming noise background, a magnetic field modulation

with amplitude of 2.0 Gauss and frequency of 5.2 kHz has been applied along with a

Lock-in Amplifier, which detects weak signals in very narrow band frequency. Three

distinct EPR peaks detected at 1300, 2800 and 5510 Gauss, have demonstrated that three

unpaired Cr3+ electrons have four distinct energy levels in the presence of an external

magnetic field.

Real-time EPR signal measurement software has been developed to control the

biasing magnetic fields, collect and display the EPR signals in real time. To observe the

weak EPR signal, the measurement speed is set at 400 ms per data point with the time

constant of the lock-in amplifier set at 30 ms. The current EMM probe with system

overall Q factor of 4,500-5,500, is capable of resolving 20,000 spin transitions with spin-

VII spin relaxation time of around 3 Nano seconds. While the minimum number of detectable spin centers for the commercially available EPR instruments is more than ten million without any spatial information for the spin centers. With more spatially confined EMM probe, the minimum detectable spin transition is expected to reach about

2,000 spin transitions or lower. The ultimate goal of this research is to achieve the minimum detectable spin transition of one single electron using micro-fabricated Atomic

Force Microscopy (AFM) EMM probes.

VIII Chapter 1 Introduction

One of the most important aspects of the electrical engineering is to modulate electron flows in various conductors and semiconductors at different frequencies. Today, the

electronic devices scales down dramatically in physical sizes, the size of the electronic

devices is expected to reach 10 nm or less by 2010, which reduces the number of

electrons in the device to about 10. With such a small number of electrons determining on/off behavior of the device, the number fluctuation and other noise sources become more important. Thus, an attempt to use other degrees of freedom such as electron spins in the electronic devices to improve the signal to noise ratio has attracted researchers’ attentions. One of the manifestations of the electron spin is its resonant transition

(flipping) in the presence of external magnetic and suitable RF field, which is called the

Electron Spin Resonance, also known as Electron Paramagnetic Resonance (EPR).

The EPR is a spectroscopic method that explores the milli-eV or even micro-eV

energy levels instead of the electron volt energy levels explored by the widely used

optical spectroscopy. The EPR transitions from different energy levels can be detected

by monitoring the power absorbed from the RF magnetic field, just like the atomic

transitions detected by the absorption of the light [1]. Even without the spatial location

information, the existing EPR is a very powerful tool that widely used in many different

applications. When combined with the spatial information, the future applications may

include,

• Reveal the formation and destruction of free radicals in liquid and solid samples.

1 • Map free radicals in biological tissues that cause aging and cancer [2]. EPR signal

is very selective, only a molecule with unpaired electrons (also known as free

radicals) can generate EPR signals. Therefore, EPR has been widely used in

biology applications, especially in aging process, such as brain neuron

degenerations, where thousands of other non-free-radical molecules may be

present.

• Facilitate quantum computing, high density memory device (Spin Net) and other

future applications.

• Discover defects in material research, since most defects trap unpaired electrons.

• Detect explosives and food freshness

1.1 Study Objectives

For most of the EPR experimental setups, the samples are located in the center of a

microwave cavity, which is usually a metal box and only the density of the spin centers

can be measured. The first study objective is to explore the possibility of exacting both

spatial and spin density information at the same time. The EMM probe is an excellent

candidate for the spatial information extraction, because the spatial location of the probe tip is always well defined and the sample is placed close to the tip of the probe. Also the

EPR power absorption is mainly from the magnetic field and the RF magnetic field strength is at the maximum at the probe tip. Therefore, the spatial location of the spin transitions can be determined.

2 For the classic EPR, the typical minimum number of detection is in the range of

107 to 109, which is quite large. The second study objective is to increase the sensitivity of the detecting system, which in turn reduces the number of the minimum detectable

spin centers and ultimately reaches single electron detection combining with AFM [6-8].

1.2 Highly Localized Evanescent Microwave Microscopic Probe Techniques

Two types of EMM probes are widely used in a variety of applications, the Electric

Dipole Probe (EDP) and the Magnetic Dipole Probe (MDP). As the name implies, an

EDP uses the electric field as the main detecting field, while a MDP uses the magnetic

field interacting with the sample [3]. The simplest construction of an EDP and a MDP is

to use the Micro Stripline construction, as shown in Fig. 1-1. Under resonance condition,

an EDP has the coupling gap at λ 2/ , which ensures that the standing Electrical Magnetic

waveform has the highest electrical field at the probe tip at resonance. While the MDP

has the coupling gap at λ 4/ [4] and the probe tip is shorted to the ground plate, which

results the strongest magnetic field at the probe tip.

3 H H E E

Electric Dipole EMM Probe Magnetic Dipole EMM Probe

λ/2 λ/4

εr εr= 2.2

Sample Sample

Figure 1-1 Magnetic Dipole and Electric Dipole EMM Probe

The electric dipole probe experiences a short at typical EPR frequency (above 2

GHz) and the EPR power absorption is from the magnetic field. Therefore, in order to get the maximum sensitivity, the MDP is chosen because the probe tip magnetic field strength is the strongest. Also the electric fields at the MDP tip will be at the minimum to minimize the dielectric losses at the probe tip. At critical coupling, the energy stored in the electric field equals to the energy stored in the magnetic field and the relative phase angle between electric field and magnetic field is exactly 180o. The electromagnetic waveform is a standing wave where the Poynting vector is

avg = × HSP = 0 (1-2-1)

The reflected voltage of the EMM probe is close to zero at the critical coupling, in

another word, the probe perfectly matches the transmission line. When EPR condition is

4 reached, the EPR power absorption changes the load impedance and breaks the perfect

match that causes the reflected voltage increases dramatically.

1.3 Electron Spin

One of the earliest discovered Quantum phenomena is the Stern’s-Gerlach’s experiment

performed in 1921[6]. When the electrical neutral silver vapor beam passes a non-

homogeneous magnetic field, the beam splits itself and lands on two separate spots on the

detection glass plate as shown in Fig. 1-2.

S=-1/2

S=1/2 S Glass Plate

N

Figure 1-2 Stern’s-Gerlach’s Experiment Illustration

Classical physics would predicate a vertical line distribution, which is incorrect.

Quantum physics explains this phenomenon nicely with the spatial quantization of the spin magnetic moment, which is the intrinsic characteristic of an electron. The electron

5 configuration of silver atom is 1s2 2s2p6 3s2p6d10 4s2p6d10 5s1, where the outer fifth shell

has one single unpaired electron whose spin is not balanced by any other electrons. This unpaired electron acts like a small magnetic dipole (or a small magnet bar) when it is

passing through the magnetic field. For the electron spins up, the electromagnet South

Pole pulls it upwards and vise versa for the electron spins down. Similar experiments

were conducted for hydrogen beam and other transition metals, such as copper, good,

sodium and potassium. Although the nucleus also has spins, the magnetic dipole moment

is much smaller than that of the electron, due to the heavy mass of the proton and

neutrons.

To precisely define a single electron in an atom, four quantum numbers are

needed as shown in Table 1. Among these four numbers, only the spin magnetic is the

intrinsic property of an electron, whether it is trapped in an atom, buried in a chemical

bond or free in the space. Unlike the Nucleus Magnetic Resonance (NMR), the electron

spins along its own axis. The following table describes four parameters to define an

electron in an atom precisely.

Table 1 the Energy States of an Electron in an Atom

Quantum Name Description Values Number N Principal Energy Level (Distance from the 1, 2, 3, … nucleus) l Orbital Shape of symmetry 0,1, 2, .., n-1 m Orbital Orbital orientation -l, …, 0, ..+ l, magnetic S Spin angular Spin orientation ± ½ momentum

6 In 1940’s, the EPR absorption in solids was detected by Cummerow and Haliday.

The EPR is a branch of microwave spectroscopy, where the sample with unpaired

electrons is immersed in a homogeneous magnetic field. The microwave radiation

achieves a resonance condition when the microwave energy equals to the electron energy

gap, which can be detected through proper microwave detectors. Among many EPR samples, Ruby is the one of few samples that has been studied extensively and it was treated as the standard EPR reference material, because of the relative strong EPR signal and high observation temperature (room temperature). According to the selection rules, three possible transitions for the Cr3+ ions, at 3.77 GHz, took place at 0.55 Tesla, 0.14

Tesla and 0.28 Tesla.

To describe the interact energy of the unpaired electrons with the nucleus and

other electrons in a constant magnetic field, an energy equations can be derived, which is

called the Spin Hamiltonian. To simplify the study, energy such as the crystal, electronic,

spin-orbit interaction, spin-spin interaction, hyperfine structure and quadruple energy and

nuclear spin energy [7-8] are considered to be negligence when the external magnetic field

is changing. Therefore, the simple form of the Spin Hamiltonian can be written in term

of the Zeeman energy [5].

= Zee = μΗΗ S = gμ B ⋅ HSH (1-3-1)

When an unpaired electron immersed in a magnetic field, two possible electron spin states (spin up and spin down) are degenerate from the ground state. The spin of the electron can be either in the same or in the opposite direction of the external magnetic field. The two possible electron alignments have different energy levels, such that the

7 electron ground state energy level is split into two levels S = -1/2 and S = 1/2. This

phenomenon is called the Zeeman effects, which is shown as following.

Evacuum

E2 Emission photon

−= EEhv 23

= μ B Hg E1 Absorption photon μB Hg −= EEhv 23

= μ B Hg

E3

H = 0 H ≠ 0

Figure 1-3 Zeeman Effects for single electron

1 1 The microwave energy required for spin transition from S −= to S = is 1 2 2 2

v −24 hfE g μ zS, BDC ×=⋅⋅== 1050.2 J (1-3-2)

To write the complete Spin Hamiltonian [3], the Electron Zeeman Interaction

(EZI), Zero-Field Interaction (ZFI), Nuclear Zeeman Interaction (NZI), Nuclear

Quadruple Interaction (NQI) , the Electron-Electron Interaction (EEI) between electron spins and the Hyperfine Interaction (HFI) must be considered.

= ∑[]EZI ZFI )()( ++ ∑[ NZI NQI )()( ]++ ∑[]EEI iHiHiHiHiHH )( i k > ji

+ ∑[ HFI kiH ),( ] (1-3-3) ,ki

8 In this study, only two Hamiltonians, the Electron Zeeman Interaction and Zero-

Field Interaction are significant.

1.4 The Magnetic Susceptibilities

Magnetism arises from the motion of charges on an atomic scale and the magnetic moment is the result of the intimate relation between the charge and mass. If an atom has unpaired electrons in the outer shell and the electrons are not in any ionic or covalent

bonds, the atom will exhibit Paramagnetism. The magnetic moment is closely related to

the magnetic susceptibility, which is the ratio of the volume magnetic moment to the

magnetic field as shown in Fig. 1-4 [12].

T-Ba-Cu-O Au (-2.74E-6) Cr (3.6E-6) Mn (6.6 E-5) Fe (1.0E5) (-0.20) Super Diamagnetic AntiFerro- Paramagnetic Ferromagnetic Conductor Materials magnetic Materials Materials Materials

-106 -104 -102 -1 -10-2 -10-4 0 10-4 10-2 1 102 104 106 Magnetic Susceptibility

Figure 1-4 Magnetic Susceptibility of Different Material

For the diamagnetic material, all electron spins are paired, while for a paramagnetic material, it has no macroscopic magnetic moment in the absence of an

9 external magnetic field. When the paramagnetic material is placed inside a magnetic field, it displays magnetic moment whose orientation is aligning with the external magnetic field. The magnetic susceptibility can be written as,

M χ m +== jχχ "' H (1-4-1)

where χ" is the imaginary part of the susceptibility, which indicates dissipative processes

in the sample and directly related to the EPR power absorption. The transition metal

group of atoms is excellent study objects because of their incomplete 3d, 4d, 5d, 4f or 5f

shells. Certain defects such as vacancies or foreign atoms in a crystal like the Chromium

atom in the Al2O3 crystal loses three atoms to the chimerically bonded host and produces

a very strong magnetic moment.

To study the effect of the magnetic field to the susceptibilities of the ruby sample,

a solenoid coil is made with 24 turns, and 7.85 E-05 m2 area. With the Ruby file inserts, the coil is placed in a time varying DC magnetic field. The inductance of the solenoid coil is

μ 2 AN 2 AN μμμ 2 AN L == r 0 = 0 (1+ χ ) (1-4-2) l l l

By using a HP 4284 Precision LCR meter, the following measurements were

taken to exam the inductance changes under different magnetic field strength and

orientations. The LCR meter sends a test signal with 1 MHz frequency to the solenoid coil and measures the impedance of the coil, then extract the inductance and the Q factor

of the coil. The experiment results are shown in Fig. 1-5. The inductance of the coil

decreases as the magnetic field increases, meanwhile the Q factor increases.

10 Electromagnet Gap: 12 mm Electromagnet Gap: 25 mm Electromagnet Gap: 25 mm Electromagnet Gap: 12 mm (1) Air (1) Air No Power (1) Air (1) Air Ls: 4.98273 uH Q: 18.8 Ls: 5.22500 uH Q: 35.1 Ls: 5.22414 uH Q: 35.2 Ls: 4.98319 uH Q: 17.8 (2) With inside Power 0 A (2) With inside Power 0 A (2) With inside Power 0 A (2) With inside Power 0 A Ls: 4.98271 uH Q: 18.8 Ls: 5.22512 uH Q: 35.1 Ls: 5.22394 uH Q: 35.2 Ls: 4.98431 uH Q: 17.8 (3) Insert Ruby 0A (3) Electromag Current -5 A (3) Electromag Current -5 A (3) Electromag Current -5 A Ls: 4.98043 uH Q: 18.8 Ls: 5.22360 uH Q: 35.0 Ls: 5.22232 uH Q: 35.1 Ls: 4.90568 uH Q: 15.4 (4) Electromag Current -5 A (4) Electromag Current -10A (4) Electromag Current -10A (4) Electromag Current -10A Ls: 4.96882 uH Q: 18.6 Ls: 5.21986 uH Q: 36.0 Ls: 5.21839 uH Q: 36.1 Ls: 4.88119 uH Q: 16.3 (5) Electromag Current -10A (5) Electromag Current -19A (5) Electromag Current -19A (5) Electromag Current -19A Ls: 4.96024 uH Q: 18.9 Ls: 5.21593 uH Q: 37.0 Ls: 5.21402 uH Q: 37.2 Ls: 4.86251 uH Q: 17.3 (6) Electromag Current -19A (6) Electromag Current 0A (6) Electromag Current 0A (6) Electromag Current 0A Ls: 4.95340 uH Q: 19.4 Ls: 5.22576 uH Q: 35.1 Ls: 5.22375 uH Q: 35.2 Ls: 4.91778 uH Q: 17.8 (7) Electromag Current 0A (7) Electromag Current 5 A (7) Electromag Current 5 A (7) Electromag Current 5 A Ls: 4.97758 uH Q: 18.8 Ls: 5.22424 uH Q: 35.0 Ls: 5.22227 uH Q: 35.1 Ls: 4.90429 uH Q: 15.3 (8) Electromag Current 5 A (8) Electromag Current 10A (8) Electromag Current 10A (8) Electromag Current 10A Ls: 4.97028 uH Q: 18.6 Ls: 5.22079 uH Q: 36.0 Ls: 5.21870 uH Q: 36.1 Ls: 4.88211 uH Q: 16.2 (9) Electromag Current 10A (9) Electromag Current 19A (9) Electromag Current 19A (9) Electromag Current 19A Ls: 4.96113 uH Q: 19.0 Ls: 5.21667 uH Q: 37.1 Ls: 5.21407 uH Q: 37.2 Ls: 4.86215 uH Q: 17.3 (10) Electromag Current 19A (10) Electromag Current 0A (10) Electromag Current 0A (10) Electromag Current 0A Ls: 4.95430 uH Q: 19.3 Ls: 5.22640 uH Q: 35.1 Ls: 5.22393 uH Q: 35.2 Ls: 4.91697 uH Q: 15.4

Electromagnet Electromagnet Electromagnet Electromagnet

12 mm 25 mm 25 mm 12 mm Electromagnet Electromagnet Electromagnet Electromagnet

Figure 1-5 LCR measurement of a Solenoid Coil with Ruby Sample

The Curie-Weiss Law indicates that

2 2 M 0 μμ B SSgN + )1( χ m == (1-4-3) H 3 B ()− TTk C

Where the N is the number of electron moments per unit volume. The Boltzmann

− −− 12223 constant is k B ×= 1038.1 Kkgsm , the T (K) is the absolute temperature. S is the electron spin. The continuous wave EPR signal intensity is proportional to the magnetic susceptibility as following [11],

S = "ηχ PZQV 0 (1-4-4)

Where η is the filling factor of the resonator, Q is the Q factor of the resonator off

resonance condition, P is the microwave power into the resonator and Z 0 is the characteristic impedance of the transmission line. At the EPR resonance condition, there

11 are two things take place, the first one is the dispersion and the second one is absorption.

In this study, only the absorption line is obtained and studied extensively. The energy

absorption of the EPR measurement is related to the imaginary part of the AC

susceptibility.

REPR ∝ χ" (1-4-5)

References

[1] D. Halliday, R. Resnick, J. Walker, “Fundamentals of Physcis”, John Wiley & Sons, Inc. New York, 2003, 6th. ISBN 0-471-22858-3

[2] Oxtoby, Gillis, Nachtrieb, “Principles of Modern Chemistry” Thomson Learning Inc. 2002, 5th. ISBN 0-03-035373-4

[3] OxJ. D. Kraus, “Electromagnetics” McGraw-Hill, Inc. Thomson Learning Inc. New York, 1992, 4th. ISBN 0-07-035621-1

[4] F. Monaco, “Introduction to Microwave Technology” Merrill Publishing Co. Columbus, 1989. ISBN 0-675-21030-5

[5] C. P. Poole, Jr. and H. A. Farach, “Handbook of Electron Spin Resonance Vol. 2” Springer-Verlag New York, Inc., NY, 1999, ISBN 1-56396-044-3

[6] C. Durkan and M. E. Welland, “Electronic Spin Detection in Molecules Using Scanning-Tunneling-Microscopy-Assisted Electron Spin Resonance.” Applied Physics Let., Vol 80, No., 3, Jan, 2002

[7] C. Durkan, “Detection of Single Electronic Spins by Scanning Tunneling Microscopy.” Contemporary Phys., Vol. 45, No. 1, Jan, 2004, pp. 1-10

[8] A. Blank, C. R. Dunnam, P. P. Borbat, and J. H. Freed, “High Resolution Electron Spin Resonance Microscopy.” J. of Magnetic Resonance, July, 2003

[9] P. Challaghan, “Principles of Nuclear Magnetic Resonance Microscopy”, Oxford University Press, Oxford, 1991

12

[10] D. Shachal and Y. Manassen, “Mechanism of Electron-Spin resonance Studied with Use of Scanning Tunneling Microscopy”, Physical Rev. B, Vol. 46 No. 8, Aug, 1992

[11] L.K. Aminov, M.P. Tseitlin, K.M. Salikhov, “1-D EPR imaging of conducting and lossy-dielectric samples”, Appl. Magn. Reson. 16, 1999, pp. 341–362.

[12] M. Petelin, U. Skaleric, P. Cevc, M. Schara, “The permeability of human cementum in vitro measured by electron paramagnetic resonance”, Arch. Oral Biol. 44, 1999, pp. 259–267.

13 Chapter 2 Electron Paramagnetic Resonance (EPR)

EPR is a spectroscopic method, which detects the paramagnetic presences caused by

unpaired electrons in the samples. Instead of frequency scanning, the EPR employs a DC

biasing magnetic field scanning while the source frequency is unchanged. EPR requires

continuous intermediate power microwaves and the detection of a very small change in

this power level. However, the EPR energies are 2000 times as big as NMR energy [1]. It

is an ideal technique to produce meaningful structural and dynamic information in a wide

range of application areas, including Biology, Material Science, Chemistry and Physics.

The typical samples with unpaired samples are transition metal, free radicals, organic ion-

radicals, odd electron molecules, etc.

2.1 Absorption Line Shapes

The typical EPR detection result is a EPR absorption line shape, which is Distribution of

the relative strength of resonance as a function of magnetic field strength. Plenty useful

information, such as spin center density, spin to lattice and spin to spin relaxation time can be derived from this particular spectral line shape. Two most common line shapes to describe the probability distributions are Lorentzian Type and Gaussian Type, which mathematically describe EPR absorption line shapes. Both of the two line types are symmetrical lines and simple mathematical forms.

14 2.1.1 Lorentzian Type Line Shapes

The Lorentzian type line shape is a function obtained from the Fourier transformation of an exponential function, which can be expressed as following,

ym HY )( = 2 (2-1-1) 1 [()0 / Δ−+ HHH ]

Where the ΔH is the line width at the half the peak ym , and the H 0 is the center of the

EPR line shape. ΔH is the full line width at ym / 2 . The first derivative formula is

2 2 ⎡ ⎤ ⎡ ⎛ ⎞ ⎤ ⎢ − HH ⎥ ⎢ ⎜ − HH ⎟ ⎥ = yHY '16)(' 0 ⎢3 + ⎜ 0 ⎟ ⎥ (2-1-2) ⎢ m 1 ⎥ / 1 ⎢ ΔH ⎥ ⎢ ⎜ ΔH ⎟ ⎥ pp ⎢ ⎜ pp ⎟ ⎥ ⎣⎢ 2 ⎦⎥ ⎣ ⎝ 2 ⎠ ⎦

Figure 2-1 Normalized Lorentzian type and Gaussian Type Line Shapes

15 By using MATLAB simulation, Fig. 2-1 shows the comparison between the normalized Lorentzian type and Gaussian Type line shapes with the line width of 12

Gauss and the center field of 1359 Gauss. It is obvious from the comparison that both

Lorentzian and Gaussian type line shapes bear the same line width at the half line width.

Lorentzian type line shape has narrower width at the top and wider width at the bottom than the Gaussian type.

The Lorentzian line shape are related to the spin-spin relaxation time Trelaxation as following [1],

22 Trelaxation = = (2-1-3) γΔH 3γΔH pp

2.1.2 Gaussian Type Line Shapes

The Gaussian line type is a bell shaped form, which is proportional to an exponential function. The mathematical expression for Gaussian type line shape is shown as following,

2 ⎡ ⎛ ⎞ ⎤ ⎢ ⎜ − HH ⎟ ⎥ yHY exp)( ⎢−= ⎜ 0 ⎟ 2ln ⎥ (2-1-4) m 1 ⎢ ⎜ ΔH ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ 2 ⎠ ⎦

2 ⎛ ⎞ ⎡ ⎛ ⎞ ⎤ ⎜ − HH ⎟ ⎢ 1 ⎜ − HH ⎟ ⎥ = ')(' eyHY 2/1 ⎜ 0 ⎟exp⎢− ⎜ 0 ⎟ ⎥ (2-1-5) m 1 2 1 ⎜ ΔH ⎟ ⎢ ⎜ ΔH ⎟ ⎥ ⎜ pp ⎟ ⎢ ⎜ pp ⎟ ⎥ ⎝ 2 ⎠ ⎣ ⎝ 2 ⎠ ⎦

16

Figure 2-2 First Derivatives of Gaussian and Lorentzian Type Line Shapes

Since most EPR line shape was first derivative of the absorption line, it is

important to study the first derivative of the Gaussian and Lorentzian types. The first

derivative comparison between the Gaussian type and Lorentzian type is shown in Fig. 2-

2. The actual EPR line shape is neither a pure Gaussian or Lorentzian type. Some researches use a combination function of both Gaussian and Lorentzian types.

17 2.1.3 EPR Absorption Line Shapes

If the microwave power level is sufficiently low (no saturation occurs) and the magnetic field modulation amplitude is much less than the EPR line width, the area under the line shape is proportional to the number of spins. By integration of finite area of A’, the total absorption power can be calculated, which can be used to derive the number of spins at the resonant condition.

P P'

P'm2 Absorption H Pm Hj-1 Hj H0 Pj 0 H Absorption Derivative

P'm2 Pj-1 Hm1 H Hm2 0 Hj-1 Hj H0 (a) Absorption (b) Absorption Line Shape Derivative Line Shape

Figure 2-3 EPR First Derivative Line Shape

As shown in Fig. 2-3, the area under the line shape can be calculated by m = ∑ j (− HHPA jj −1 ) (2-1-6) j=1

The magnetic field scan uses constant scan rate,

m ⋅Δ= ∑ PHA j (2-1-7) j=1

For the nth moment of the absorption power can be defined as

18 m n ΔH n H = ∑()j − 0 PHH j (2-1-8) A j=1

The criterion used for defining the resonance peak field H0 is,

H 0 1 ∞ )( == )( dHHYAdHHY (2-1-9) ∫ ∞− 2 ∫H 0

In most EPR setup, magnetic field modulation and phase sensitivity detector are

used to improve signal to noise ratio and the output signal is the first derivative of the

absorption line shape, the P' is the first derivative of the power absorption, which can be

written as,

d P'= P (2-1-10) dH

The area A is proportional to the number of total electron spin centers in the sample,

which can be written as

m m 2 ()−= HHA jj −1 ∑∑Pi ' (2-1-11) j==1 ji

2.2 Classic EPR detection techniques

In Continuous Wave (CW) EPR, the resonance frequencies vary over a very broad range,

which can be divided to L-band (1~2 GHz), S-band (3~4 GHz), C-band (5~8), X-band

(9~12 GHz), Ku-band (13 GHz ~17 GHz), K-band (18 GHz -26 GHz) and Ka-band (27

GHz~40 GHz). In theory, one can detect the EPR signal using a DC detection method.

Because of the overwhelming noise background and weak signal, the typical setup for the

classic EPR is shown as Fig. 2-4, which consists of a microwave source, a magnet with adjustable field, a resonant cavity, a magnetic field modulation coils, a crystal detector, a

phase-sensitive detector, Data Acquire board and a PC for data processing. At

19 microwave frequencies, the RLC resonator is replaced by the cavity, which is an excellent device for storing electromagnetic energy.

Circulator Microwave Phase-Sensitive Source Detector Crystal Detector

Waveguide Reference Signal

Field Field Modulation Modulation Coils Source

Cavity Magnetic Field Sweep Control Signal Computer DAQ N Sample S

ElectroMagnet ElectroMagnet

Figure 2-4 The Classic CW EPR Setup

The Microwave frequency is fixed to keep the Q factor of the cavity constant during the magnetic field sweep. At a low magnetic field where the spin system is off resonance, the cavity is tuned to be perfect matched to the waveguide and no power transmitted to the crystal detector and the reflected signal to the phase-sensitive detector is close to zero. The Microwave source generates an orthogonal low amplitude high frequency magnetic field.

20 hν hν H == (2-2-1) gμ eh B g 4πme

When the external magnetic field is brought into the EPR resonance condition, the sample starts absorbing the microwave power, which changes the load impedance and the

Q factor of the cavity changes dramatically. The reflected power to the crystal detector also increases significantly, which can be picked up the phase-sensitive detector. The crystal detector converts the microwave frequency power to a low frequency voltage signal, which is linearly proportional to the microwave power. The typical EPR microwave power absorption is in the order of Nano Watts. When the resonance condition occurs, only a very small part of the total microwave power is absorbed, therefore, the EPR signal is very weak. Because of the background noise and weak EPR signal, the lower frequency (1 kHz ~ 200 kHz) magnetic field modulation is used to suppress the noise. The output signal is actually a power modulation signal at low frequency and has the same phase as the field modulation. The phase sensitive detector is typically receiving a reference signal that is synchronized with the magnetic field modulation signal, such that the PSD output measures the amplitude of the EPR signal only.

2.2.1 Microwave Cavity

At microwave frequency, the equivalent Parallel LC resonator is a completely enclosed rectangular or cylindrical metal box, which is also called the microwave cavity resonator.

The cavity is a great energy storage device with Q factor in the range of 3,000 to 5,000,

21 which makes it a very good EPR resonator to amplify the weak EPR signal. The Q factor of a microwave cavity can be written as,

ω() time average- energy stored in the system Q = (2-2-2) Energy dissipated secondper

Most energy loss in a cavity is due to the electrical eddy currents on the cavity walls generated by the microwaves. The energy loss is tuned into heat radiated inside or outside of the cavity.

Magnetic Field

Electric Sample Field

Figure 2-5 Microwave Magnetic and Electric Fields in a Rectangular Cavity

When tuned into resonance condition, a standing wave is generated inside of the cavity, which means the electric field and magnetic field are exactly out of phase. As shown in Fig. 2-5, the sample in the center of the rectangular cavity and it sees the maximum magnetic field strength and the minimum electric field generated by the

22 microwave frequency. Since most of samples can absorb microwave energy through the electric field even at non-resonance condition, the sample is always put in the center of the cavity to minimize the electric field effect and maximize the magnetic field strength, which in turn increases the sensitivity of the cavity. Although the EPR information can be obtained, the spatial location of the EPR signal is not readily available.

The size of the cavity directly relates to the EPR measurement sensitivity, which is proportional to the Filling Factor of the cavity. The size of the sample also affects the filling factor, which can be written as,

2 1 dVB ∫V η = Sample (2-2-3) 2 1t dVB ∫V Cavity

B1t is the total magnetic flux density in the cavity and the and B1 is the magnetic flux density inside the sample. In most cases, the flux density inside of the cavity is uniform. Therefore, the filling factor can be approximately written as the ratio of the volume of the sample to the cavity.

2.2.2 Magnetic Field Modulation

When the main magnetic field scans through the EPR resonance region, the unpaired electron absorbs a small amount of the microwave energy and also produces a slight change in the resonance frequency of the microwave cavity. If the EPR signal is strong enough, the direct detection through a crystal detector and a multi-meter is theoretically possible. However, the EPR signal is usually very weak and can not be distinguished

23 from the overwhelming background noise. There are two different approaches for obtaining the EPR signal with high signal to noise ratio, one is magnetic field modulation and the other is the frequency modulation. For the magnetic field modulation, the amplitude of the modulation should be less than the EPR line width. It is practical to use the biasing magnetic field sweeping step as the modulation amplitude, for an instance, if the linear magnetic field is sweeping at one Gauss per step, the modulation amplitude should be at one Gauss or more. Therefore the instantaneous magnetic field at any moment can be written as:

)( = + δ sc + m cos()( ωmtHtHHtH ) (2-2-4)

Where δ s )( 1 ⋅= tCtH and C1 is the magnetic field scanning rate (Gauss/sec.), which is

much slower than the sinusoid component. The term 0 + δ s (tHH ) can be treated as a constant to the sample at any moment of the scanning process.

24 BB'

Vin(t) = α cos(ωmt)

Crystal C C' Detector Absorption

D D' H

Phase-Sensitive Detector t Vref(t) = 1V cos(ωmt)

Magnetic Field H0 H Modulation

Hm cos(ωmt)

H Absorption Derivative

Figure 2-6 Magnetic Field Modulation Detections

The magnetic field H(t) scans the absorption curve slowly as shown in Fig. 2-6.

As the magnetic modulation signal scans a portion of the absorption line, which can be approximated as a straight line. The absorption power modulates at the same frequency as the magnetic field modulation. Therefore, the field modulation can also be treated as absorption power modulation. At the left half of the absorption curve, when the modulation filed is positive, the amount of power absorption increases, while the power absorption decreases at the right half of the absorption curve when the modulated filed is positive. This can be observed as 180 degree phase shift when the constant field swept

− + from the H 0 to H 0 . At the resonance condition, the reflected power from the

25 resonator is equal to the absorption power. The reflected power passes through the crystal detector whose output is proportional to the reflected power or the absorption power with

a sensitivity of kCrystal = /5.0 μWmV , which converts the absorption microwave power to

voltage signalVin .

∂ HP )( )( ktV ⋅= m cosω t (2-2-5) in Crystal ∂H m

Where α is a constant, which is related to the slope of the absorption line shape and its plot is shown in the lower right half of Fig. 2-6. The following equations describes the actual voltage output of the crystal detector at different points on the absorption line shape, the plot of the waveform of each point is shown as Fig. 2-7.

Vm ⎡ 1 1 1 ⎤ Point A: V +−= ()2sin ωmt + ()4cos ωnt π ⎣⎢ 2 3 15 ⎦⎥

Vm ⎡ 1 3π 1 1 ⎤ Point B: V +−= cos()ωmt + sin()ωmt + ()4sin ωnt π ⎣⎢ 2 4 3 15 ⎦⎥

Vm ⎡ 1 3π 1 1 ⎤ Point B: V −−= cos()ωmt + sin()ωmt + ()4sin ωnt π ⎣⎢ 2 4 3 15 ⎦⎥

Point C: = m cos()ωmtVV

Point C’: −= m cos()ωmtVV

Vm ⎡1 3π 1 1 ⎤ Point D: V += cos()ωmt + sin()ωmt + ()4sin ωnt π ⎣⎢2 4 3 15 ⎦⎥

Vm ⎡1 3π 1 1 ⎤ Point D’: V += cos()ωmt + sin()ωmt + ()4sin ωnt π ⎣⎢2 4 3 15 ⎦⎥

26

Figure 2-7 Phase Sensitive Detector Input Voltage Waveform

The magnetic field modulation signal generator also generates a synchronized voltage signal, which has the same frequency as that of the field modulation. It can be written as,

ref = ⋅ ωmtVV )cos(1' (2-2-6)

The PSD has an automatic phase shifter that adds a phase angle to this reference voltage

signal to make the reference signal and the input voltage in (tV ) have exactly same phase, which will get the highest sensitivity as shown as following,

27 ref = ⋅ cos(1 ωmtVV +θ ) (2-2-7)

The PSD multiply these two sinusoid signals,

inref )( α ⋅= m ()ωm ⋅ cos(1cos ωmtVtHtVV +θ ) = α ⋅ m cos(ωm )⋅ cos(ωmttH + θ ) (2-2-8)

By combining and rearranging the terms, Eq. (2-2-1) can be rewritten as,

1 1 )( α HtVV )cos( αθ H (2cos t +⋅⋅+⋅⋅= θω ) (2-2-9) inref 2 m 2 m m

Where, the second AC component is eliminated by a low pass filter and only the DC component remains. The final PSD output signal can be written as,

1 α HV ⋅⋅= θ )cos( (2-2-10) o 2 m

28

Figure 2-8 Demodulator Output Voltage Signals Before and After LP Filter

The PSD output voltage signal waveforms are shown in Fig. 2-8. The phase angle

θ is usually set at 0o to get the maximum DC output, i.e. to get the maximum sensitivity.

29

Figure 2-9 Derivation of the Relation between Power Absorption Slop and Magnetic Field Modulation

The magnetic field modulation is actually a power modulation as shown in Fig. 2-

9, near the EPR resonance, whenever the magnetic field changes, the EPR absorption power changes accordingly. At the left of the EPR peak field, when the magnetic modulation field increases the EPR absorption power increases. On the contrary, at the right of the EPR peak field, when the modulation field increases the EPR absorption power decreases. Therefore, there is an 180o phase shift when the main magnetic field passing the EPR peak field. The crystal detector is a square law device, which converts

the power to voltage with a linear factor crystal ( / μWmVk ). The magnetic field modulation amplitude is fixed and the output of the lock-in amplifier can be written as,

∂ HP ∂ HP )()( ∂V = = (2-2-11) ∂H H m crystal ⋅ Hk m

30 Therefore, the first derivative of the absorption line shape can be plotted with the output of the crystal detector.

2.2.3 Frequency Modulation

Similar to the magnetic field modulation the frequency modulation modulates the frequency around the center resonance frequency, where the instantaneous frequency is shown as:

)( = + mFC sin(ωmtfkftf ) (2-2-12)

where k F is the frequency sensitivity constant related to frequency modulation source.

For a typical EPR signal with a g factor equal to 2, the peak signal occurs around

1356 (Gauss), and a field modulation amplitude is about 1 Gauss. If the 100 kHz frequency modulation is used, the frequency sensitivity constant should be

10 H m f c 1 10 k F ≈⋅= 5 = 74.28 (2-2-13) H c f m 3480 10

With the frequency modulation, the disadvantages of field modulation, such as the passage effect, micro-phonic noise and the side-band effects [4], can be effectively eliminated. However, in order to achieve proper frequency modulation amplitude to scan the EPR absorption line shape, for a equivalent of 2 Gauss field modulation amplitude, the frequency modulation amplitude will be,

31 H m fk mF f 0 =⋅= 56.5 MHz (2-2-14) H 0

Where the H m is the modulation amplitude (2 Oe) and the H 0 is the EPR peak field and

f 0 is the resonance frequency. For very high Q resonator, such high frequency modulation amplitude alters the system Q factor constantly and this can be interference to the real EPR signals.

3+ 2.3 Mathematical Model of Cr ion in Al2O3 EPR

The electron configuration of Chromium is 1s2 2s22p6 3s23p63d5 4s1. In single crystal

Ruby, the Cr atom replaces the Al atom in Al2O3 crystal and gives up 3 electrons, which results the Cr3+ ion electron configuration of 1s2 2s22p6 3s23p63d3. The Cr3+ ions in ruby are surrounded by distorted octahedral of oxygen ions. The distortion of the crystal lattice (octahedral symmetry) gives the energy level degeneracy of three electrons from the 3d shell of the Cr3+ ion, when external magnetic field presents. To determine if the electron state transition is forbidden or allowed in quantum mechanics, the following

Selection Rules determine whether the transition is allowed or forbidden.

n (principal): no rule; all is possible

l (orbital): change in l = ± 1

m(orbital magnetic) : change in m = 0 (for linearly polarized light), = +1

(for right-handed circularly polarized light), = -1 (for left-handed circularly

polarized light).

S(spin angular momentum): change in S = ±1

32 For Cr3+ ion, three allowed transitions or transitions with high probabilities are allowed, as shown in Fig. 2-10.

3d10 3d3

Figure 2-10 Cr3+ Energy Levels Splitting With External Magnetic Field

When the Chromium atom replaces the Aluminum atom in the ruby crystal, it loses three electrons and become Cr3+ ion, which is subject to interacts with the surrounding electrons and nuclei that will perturb its energy levels. There are four different energy levels for the three 3d orbit electrons of Cr3+ ion, due to the hyperfine coupling between the electron spin and the nuclear spin I. According to selection rules, only three transitions are allowed. It is clearly shown that three possible magnetic fields may result this kind of transition, which is shown as three strong signal spikes in the ESR

33 spectrum. When one of these transitions occurs, a tiny amount of energy is absorbed from the microwave energy source.

For the Spin Hamiltonian of Cr3+, the 3d electron spin to orbit interaction is very weak, therefore, only the Electron Zeeman Interaction and the Zero-Field Interaction

(ZFI) are significant, while the other Hamiltonian terms can be ignored. To describe the interaction between an electron spin and the external magnetic field, the EZI can be written in general terms as,

⎡ ggg xzxyxx ⎤⎡S x ⎤ ⎢ ⎥ H == μμ HHHHgS ggg ⎢S ⎥ (2-3-1) EZI []zyx ⎢ yzyyyx ⎥⎢ y ⎥ ⎢ ⎥⎢ ⎥ ⎣ ggg zzzyzx ⎦⎣S z ⎦

The matrix g is usually symmetric and it can be written as a diagonal

⎡g x 00 ⎤ matrix ⎢ g 00 ⎥ . The above equation can be simplified as ⎢ y ⎥ ⎣⎢ 00 g z ⎦⎥

EZI = μ( + + SHgSHgSHgH zzzyyyxxx ) (2-3-2) in term of either parallel or perpendicular to the electron spin, the g tensor is usually

expressed as g || and g⊥ . Therefore, eq. (2-3-2) can be written as,

EZI = μ[ || zz + ⊥ ( + HSHSgHSgH yyxx )] (2-3-3)

The Zero-Field Interaction is another important Hamiltonian term, which describes the interaction the electron spins interaction at zero magnetic fields.

T 2 22 ZFI ( z ( −+== SSEDSDShSH yx )) (2-3-4)

34 ⎡ 1 ⎤ +− ED 00 D 00 ⎢ 3 ⎥ ⎡ 1 ⎤ ⎢ ⎥ ⎢ ⎥ 1 3 Where D = D2 00 = ⎢ 0 −− ED 0 ⎥ and = DD 3 , ⎢ ⎥ ⎢ 3 ⎥ 2 ⎣⎢ 00 D3 ⎦⎥ 2 ⎢ 0 0 D⎥ ⎣⎢ 3 ⎦⎥

− DD E = 21 . D is the zero-field splitting constant. Therefore, the Spin Hamiltonian can 2 be written as,

2 22 =Η μ[]|| zz + ⊥ ( yyxx ) [ z ( −+++ yx )]hSSEDSHSHSgHSg (2-3-5)

From classical Ruby measurement, g|| = ± 0006.09840.1 , g ⊥ = ± 0007.09867.1 ,

hD ±−= 003.0747.5/ GHz . [2]

For ruby crystal, the three electrons of Cr3+ are very close to free electrons in space, and the electronic g-factor can be approximately by two and the hyperfine interactions between the electron and nuclear spins are ignored. When the external magnetic field is parallel the c-axis, the above two Hamiltonian terms can be simplified as,

2 22 =Η μ || zz [ z ( −++ yx )]hSSEDSHSg (2-3-6)

When the applied microwave energy satisfy the following equation,

ν = Δ = + = μ 0||m1m + +1)D(2mHgE-EEh (2-3-7)

The electron spin resonance absorption occurs. Cr3+ has three electrons on the 3d orbital. The electrons can be in any of the four electron spin states: 3/2, 1/2, -1/2, -3/2.

Based on the Selection Rules, three energy state transitions are allowed, which can be expressed as transition pairs: (3/2, 1/2), (1/2, -1/2) and (-1/2, -3/2).

35 For a resonator, the resonance frequency is usually fixed. Therefore, the electron resonance external magnetic field can be found as

ν +1)D(2m-h H 0 = (2-3-8) g|| μ

At 3.77 GHz, the Microwave energy is

−24 −5 MW hE ν J ×=×== 1056.11050.2 eV (2-3-9)

e The magnetic moment of the Cr3+ atom is given by g μμ S 2 h ⋅⋅≈⋅⋅= S , and the ,zS B 2m energy required for spin transition from S=-3/2 to S = -1/2 is,

v −23 −24 g μ zS, BDC ×⋅=⋅⋅ 10854.12 J/Tesla ⋅ ×= 1077.7a0.2096Tesl J (2-3-10)

The power absorption for the spin depends on the spin-spin relaxation time that will be discussed in the later chapter.

Table 2 Electron Transitions for Cr3+ at 3.77 GHz

ES+1 ES f (Hz) hυ (2S+1)D g||μBBH0 H0 (Oersted) 3/2 1/2 3.77E+09 2.50E-24 -7.62E-24 1.01E-23 5504.5 ½ -1/2 3.77E+09 2.50E-24 0.00E+00 2.50E-24 1359.0 -1/2 -3/2 3.77E+09 2.50E-24 7.62E-24 5.12E-24 2786.5

Based on the g tensor equals to 1.9817 and 2D/h equals to -1.15E+10 Hz, Table 2 shows the theoretical resonance peak magnetic fields are at 1359.0 Oe, 2786.5 Oe and

5504.5 Oe1.

To better understand the EPR absorption of the Cr3+ ions, a visual representation of the line shape can be mathematically calculated. EasySpin [3] is a Matlab toolbox for

1 Note: in the vacuum, the magnetic flux density B has a numerical value equal to the value of the magnetic field strength H, i.e. the magnitude of magnetic field strength H (Oersted) and magnetic flux density B (Gauss) are numerically equal. Therefore, in this paper these two are interchangeable.

36 solving problems in Electron Paramagnetic Resonance (EPR) spectroscopy. The

“pepper” function is the calculation of single-crystal and powder continuous wave EPR spectra for solid state, which gives the EPR resonance magnetic fields, line intensities and line widths. It also takes account of the other Spin Hamiltonians, such as nuclear

Zeeman Interaction, Nuclear Quadruple Interaction, Zero-Field Interaction, Electron-

Electron Interaction and Hyperfine Interactions.

Figure 2-11 EPR Line Shapes of the Natural Ruby

37 EasySpin [3] simulation was done based on the parameters from the following table,

Table 3 EasySpin Simulation Parameters

MW Freq Harmonic Magnetic Temp Orientation No. of (GHz) Field (mT) (oC) Points Fig. 2-11 3.77 0 (Line [125 600] 27 Perpendicular 7200 Shape) Fig. 2-12 3.77 1 [125 600] 27 Perpendicular 7200 (Derivative)

For most Continuous-Wave EPR [5-10], the final detected signals are the derivative of their line shapes. Fig. 2-12 shows the first derivative of the EPR signal for Natural isotope Ruby mixture of 89.5% Cr52 and 9.5% Cr53.

38

Figure 2-12 First Derivative EPR Signal for Nature Ruby

From the simulation, the line width of the EPR absorption line width is very narrow in the range of 10 Gauss to 14 Gauss. The mathematical line width predication predetermined the magnetic field modulation amplitude and also the DC magnetic field sweep step.

39 2.4 Larmor frequency

The Larmor frequency is at which the nuclear spin precess about the external static magnetic field. For a continuous wave magnetic resonance experiment, when the EPR resonance occurs, the following equation must be satisfied [11],

ω0 = γ ⋅ B0 (2-4-1)

Where ω0 is called the Larmor Frequency, which is the angular frequency of the energy level transition in a quantum mechanical description, and the γ is called gyromagnetic ratio with unit of Hz/Tesla [12].

2.5 EPR Sensitivity

When the microwave frequency satisfies the resonance condition, = μHghv 0 , the sample will absorb microwave energy, which changes the overall system Q factor of the resonator, either a microwave cavity or a Stripline resonator [1].

111 += (2-5-1) r 0 QQQ χ

40 Sample Volume

Cavity Volume

Figure 2-13 the Filling factor of the Cavity

The Qχ is due to electron resonance absorption, Qr is the combined Q factor at resonance condition and Q0 is the system overall Q factor off resonance condition. Q0 can be easily measured or calculated as following,

2 ω0 dVB ∫V cavity Q0 = (2-5-2) μ PTotal0

Assume χ >> QQ 0 ,

2 1 dVH ∫∫∫Cavity 1 Q = = (2-5-3) χ 2 χ 1 dVH χη ∫∫∫Sample

Because of the power absorption at the EPR resonance condition, the Q factor decreases,

(ω / μ ) Q = 00 (2-5-4) EPR 2 PT + ()00 χμω "/ dVB ∫V Cavity where η is the filling factor, which measures the fraction of the total microwave energy interacting with the sample. The amount of changes in Q is

2 =Δ χηQQ 0 (2-5-5)

When the microwave resonator is perfectly matched the transmission line and the microwave source, the maximum amount of microwave power incident into the cavity is

41 1 2 ()RnE c 2 Pw Pc = = (2-5-6) 2 2 2 ()+ gc nRR

Where the Pw is the microwave source power. EPR studies the interaction between electron magnetic moments and magnetic fields and detects the energy required to reorient electron moment in a magnetic field. The minimum number of detectable spins for the square law reflective detection [1] is,

KTV ⎛ ΔH ⎞ ⎡ ΔfT ⎛ 1 ⎞⎤ N = sample sample ⎜ pp ⎟ × d ⎜ +109 P ⎟ (2-5-7) min 2 ⎜ ⎟ ⎢ ⎜ d ⎟⎥ 0η SSgQ + )1( ω0 ⎝ H 0 ⎠ ⎣ w ⎝ 500PP d ⎠⎦

Where the Pd is the microwave power that incident on the sample. To increase the sensitivity of the EPR measurement, the minimum detectable number of spins should be made as small as possible, which means that the Q factor, frequency and EPR magnetic resonance field should be made as larger as possible.

References

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42 [5] S. Stoll, A. Schweiger, “Rapid construction of solid-state magnetic resonance powder spectra from frequencies and amplitudes as applied to ESEEM”, Journal of Magnetic Resonance. 163(2), 2003, pp. 248-256

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[8] A. Riedel, S. Fetzner, M. Rampp, F. Lingens, U. Liebl, J. Zimmermann, and W. Nitschke, “EPR, Electron Spin Echo Envelope Modulation, and Electron Nuclear Double Resonance Studies of the 2Fe2S centers of the 2-Halobenzoate 1,2-Dioxygenase from Burkholderia (Pseudomona) cepacia 2CBS*”, the J. of Biological Chemestry, Vol. 270, No. 52 1995, pp. 30869-30873

[9] R. B. Hemphill, P. L. Donoho, and E. D. McDonald, “Spin-Lattice Interaction in Ruby Measured by Electron Spin Resonance in Uniaxially Stressed Crystals”, Phys. Rev. Vol. 146, No. 1, June 1966

[10] S. Geschwind, G. E., Devlin, R. L., Cohen, and S. R. Chinn “Orbach Relaxation 3+ and Hyperfine Structure in the Excited E state of Cr in Al2O3”, Phys. Rev. Vol. 137, No. 4A, Feb, 1965

[11] H. G. Andresen, “Electron Paramagnetic Resonance of Manganese in TiO2”, Phys. Rev. Vol. 120, No 5, December, 1960

[12] G. A. Rinard, R. W. Quine, J. R. Harbridge, R. Song, G. R. Eaton, and S. S. Eaton, “Frequency Dependence of EPR Signal-to-Noise”, J. of Magnetic Resonance, 140, 1999, pp. 218-227

[13] G. K. Fraenkel, Paramagnetic resonance absorption, in “Technique of Organic Chemistry,” Vol. I, Part IV: “Physical Methods of Organic Chemistry” A. Weissberger, Ed. 3rd ed., Chap. XLII, Interscience, New York, 1960.

[14] G. Feher, “Sensitivity considerations in microwave paramagnetic resonance absorption techniques, Bell System Tech. J. 36, 1957, pp. 449–484.

[15] D. I. Hoult, C.-N. Chen, and V. J. Sank, “The field dependence of NMR imaging. II. Arguments concerning an optimal field strength”, Magn. Reson. Med. 3, 1986, pp 730–746.

[16] D. I. Hoult and P. C. Lauterbur, “The sensitivity of the zeugmatographic experiment involving human samples”, J. Magn. Reson. 34, 1979, pp. 425–433 .

43

[17] E. R. Andrew, “Magnetic Resonance and Related Phenomena” 24th Ampere Congress, Poznan, Elsevier, Amsterdam, 1988, pp. 45-51

[18] H. S. Soedjak, R. E. Cano, L. Tran, B. L. Bales, J. Hajdu, “Preparation and ESR Spectroscopic Characterization of the Zinc (II) and Cadmium (II) Complexes of Streptonigrin Semiquinone”, Biochimica et Biophysica Acta 1335, 1997, pp. 305-314

[19] Y. Manassen, I. Mukhopadhyay, and N. R. Rao, “Electron-Spin Resonance STM on Iron Atoms in Silicon”, Phys. Rev. B, Vol. 61, No. 23, June, 2000

[20] K. Wago, D. Botkin, C. S. Yannoni, and D. Rugar, “Paramagnetic and Ferromagnetic Resonance Imaging with a Tip-on-cantilever Magnetic Resonance Force Microscope,” App. Phs. Lett., Vol. 72, No. 21, May 1998

[21] J. H. Freed, “New Technologies in Electron Spin Resonance,” Annu. Rev. Phys. Chem, 2000, pp. 655-689

44

Chapter 3 Microstrip Resonator

In this study, the critical component is the Microstrip resonator, which serves the same function as the classical EPR cavity and also extracts the spatial information. The

Microstrip resonator has been expensive studied in the past decade [2] – [13]. Two types of near-field Microstrip resonator probes, the magnetic-dipole probe and the electric-dipole probe, have been used for different applications, including non-contact and non- destructive imaging of metals, biology samples, etc. Since the EPR absorption relates to the magnetic field, this study uses a near-field magnetic-dipole probe, which has the strongest magnetic field near the probe tip to maximize the EPR absorption.

3.1 Microstrip Transmission Line Resonator

To transfer energy in the microwave frequency, the two-conductor transmission line is no longer sufficient; instead the Microstrip line or other type of wave guides is used. The

EMM EPR probes are made by a lossless feed line and a Microstrip resonator with the

Microstrip Line grounded at the tip. The cross section of a Microstrip line resonator is shown as following,

45 Microstrip Line

Dielectric

Ground Plate

Figure 3-1 Cross Section of the Microstrip Line Resonator

The dielectric strength of Duroid be 2.2, the width of the top metal strip be 3.37 mm, the thickness of the metal strip is 35 µm, and the height of the dielectric is 0.8 mm. By using two additional field cells to represent the fringing field, the characteristic impedance of this line can be simply given by

1 μ Z = (3-1-1) 0 hw + 2/ ε

The more accurate formula for calculating the impedance of the Microstrip line is,

120π Z = (3-1-2) ⎡W 2 ⎛W ⎞⎤ ε eff ⎢ 393.1 ⎜ +++ 444.1ln ⎟⎥ ⎣ H 3 ⎝ H ⎠⎦

⎡ ⎛ 2ht ⎞ ⎤ Where the effective width and height are eff wW += ⎢ln⎜ ⎟ +1⎥ and eff = − 2thH π ⎣ ⎝ t ⎠ ⎦

c 2π The wavelength is λeff = and the phase is θ = f ε eff λeff

ε r +1 ε r −1 ε eff = + (3-1-3) 2 ⎛ H ⎞ +1212 ⎜ ⎟ ⎝W ⎠

The measured characteristic impedance is close to 50 Ω.

46 Air Gape λ/4 Resonator Dielectric

Micro Strip Line & Resonator x

Figure 3-2 Structure of the Microstrip Line Resonator

For electric-dipole resonator, λ 2/ transform was used to have the maximum voltage difference between the resonator tip and the ground plate. For a shorted circuited transmission line, it started behaving as a parallel resonator circuit where the length is closed to an odd multiple of a quarter wavelengths long at certain frequency range.

The magnetic dipole Microstrip Resonator has been used to successfully mapping the conductivity on metal surfaces [1]. For the magnetic-dipole resonator, the λ 4/ transform is used and the length of the resonator is 3.5 mm as shown in Fig. 3-2, which results the resonance frequency of around 3.7 GHz. Unlike the ESR study done by Johnson’s in

1977[1], the Microstrip line resonator in this study has the sample very close to the

Microstrip line tip. The magnetic field distribution of the probe tip is shown as Fig. 3-3, the RF magnetic field is the strongest in the sample where is closest to the tip of the probe.

47 Magnetic Field Air Gape Strength Copper Strip

Probe Tip

Sample

Figure 3-3 Magnetic Field Distribution around the EMM Probe Tip

Assume off resonance condition, the power incident to the Microwave Stripline

2ω0Wm resonator is P1 = , and at the resonance condition the total dissipated power is Q1

2ω0Wm 1 PP EPR =Δ+ . The amount of EPR power absorption is Q2

⎛ 11 ⎞ ⎛ ΔR ⎞ ⎛ ΔR ⎞ ⎜ ⎟ ⎜ EPR ⎟ ⎜ EPR ⎟ PEPR =Δ 2ω0Wm ⎜ − ⎟ = 2ω0Wm ⎜ ⎟ = 2ω0Wm ⎜ ⎟ (3-1-4) ⎝ QQ 12 ⎠ ⎝ ω0 L ⎠ ⎝ ω0 L ⎠

48 ΔP ⎛ ΔR ⎞ ΔR EPR ⎜ EPR ⎟ The ratio of the EPR power absorption is = ⎜ ⎟Q1 = . When the amount P1 ⎝ ω0 L ⎠ 2R of power absorption is detected, the equivalent of the resistance can be calculated. The micros trip resonator can be described by a Lumped circuit model as following,

V1i R0 L0 C0

REPR Port 1 Cd Cd

V1O Microstrip Resonator Sample

Figure 3-4 Lump Circuit of the Microstrip Resonator and the Ruby Sample

At the tip of the Microstrip resonator, the magnetic field is at the maximum and the electrical field is at the minimum. Therefore, with magnetic field modulation the dielectric loss in the sample can be ignored and the sample can be simply simulated by a

resistance RERP .

3.2 Magnetic Microstrip Probe Tuning

The EMM EPR probe uses a Microstrip Line resonator instead of the classic ESR cavity.

It is crucial to tune the Microstrip line resonator carefully to obtain the maximum Q factor as possible. The higher the Q the higher sensitivity the EMM EPR probe has.

49

Figure 3-5 Reflective Coefficient at the Critical Coupling

The EMM EPR probe uses a coupling screw to tune the air gap at the quarter wavelength point. In theory when the transmission line matches exactly the load, the reflected power is to be reduced to zero. As shown in Fig. 3-5, the transmission line is tuned to have a reflected coefficient of -45 dB at 3.77 GHz.

The Smith chart is a very useful tool to graphically represent the impedance- transformation property, which is crucial in tuning the EMM EPR probe. At the critical coupling, the resonance frequency is at the zero or the center point on the Smith Chart.

For any electromagnetic resonator, the average energies stored in the electric and magnetic fields are equal at resonance condition and the input impedance becomes pure real and equal to a resistance.

50 3.3 System Overall Q factor

Q is the quality factor, or a figure of merit, for a resonant circuit. The quality factor of a

Microstrip resonator is the same as the one of the microwave cavity,

energy total energy stored Q = 2π (3-3-1) cycle 1in energy in decrease in energy 1in cycle

For a loss free shore-circuit line, the current on the line a pure standing wave,

ωtj which can be written as = 0 cos βxeII , where the x is the distance measured from the shored tip to the source input. The total time-average stored energy in the magnetic field

[5] is

λ 2/ 1 2 0 2 λ0 2 m = 0 LIW cos β = 0 LIdxx (3-3-2) 2 ∫0 16

At resonance, the total time average stored energy in magnetic field is the same the

energy stored in the electric field. Therefore the total energy stored is 2Wm . The power loss due to the resistance can be approximately written as,

λ 2/ λ 2/ 1 0 R 2 0 2 λ0 2 P = *dxRII = I 0 cos β = RIdxx 0 (3-3-3) 2 ∫0 2 ∫0 8

Therefore, at resonance condition, the Q for a short-circuited line can also be defined in term of the equivalent system inductance and resistance,

2 W ωω L Q 0 m == 00 (3-3-4) P R 0

For microwave frequency, the resistance is typically very small and the inductance is fairly large, therefore the Q factor for the microwave frequency is typically in the range of 1,000 to 10,000. The Quality factor can be easily measured from the S11

51 f 0 plots, Q = , where the f is the minimum S11 frequency and Δf 3dB is the half power Δf3dB width, which occurs at 1 2/ (-3 dB) voltage width.

Figure 3-6 Quality Factor Q Estimate from S11 Measurement

f GHz791.3 Q = 0 ≈ = 5415 (3-3-5) − ff 12 − GHz7907.3GHz7914.3

Since the ESR probe sensitivity is directly related to the load Q at off resonance condition, the higher the Q the higher the sensitivity. In this study, the probe was fine tuned to -42 dB as shown in Fig. 3-6. When the EPR resonance occurs, there are two things changed, one is frequency change, which is also called dispersion. Another is the

Q change which is resulted by the power absorption only. In this study, only the Q change is detected. Alternatively, the loaded Q off EPR resonance frequency is

52 2 0 Lf 2ππ 0 Lf QL == (3-3-6) 2RL 2Z 0

At EPR resonance condition, the loaded Q is smaller due to the microwave energy absorption, which can be written as

2π 0 Lf Q'L = (3-3-7) 2 0 Δ+ RZ EPR and the change of the loaded Q is

2π Lf 2π Lf ΔR ' QQQ =−=Δ 0 0 −=− π Lf EPR LLL 2 Δ+ RZ 2Z 0 ()2 Δ+ ZRZ 0 EPR 0 0 0EPR (3-3-8) ΔREPR −≈ π 0 Lf 2 2Z 0

Therefore, the amount of Q change over the off resonance Q is

ΔQ 1 ΔR L = EPR (3-3-9) QL Z2 0

3.4 Microwave reflective coefficient

This study utilize the reflective mode operation, which means the microwave source sends microwave power to the load and detection system detects the reflected power to determine where the EPR resonance occurs.

53 Network HP Cable Analyzer

Computer DAQ

Coaxial Circulator Cable

Coaxial DAQ Cable Card

Micro Strip Line Tuning Resonator Screw Gauss Meter

N S

Hall Probe ElectroMagnet Sample ElectroMagnet

Figure 3-7 Network Analyzer Setup for Measurement of the S21

Although it is very easy to measure the S11 using a Network analyzer, the setup is to measure the S21, which matches the later experiment setup. In order to see the effects of the magnetic field to the Microstrip line resonator, a network analyzer was used to record the S21 vs. Frequency plots under different magnetic fields. The setup is shown in

Fig. 3-7, where the microwave signal passes through the port 1 of the circulator and transported through port 2 into the EMM probe. The reflected microwave signal is isolated from the incoming microwave signal and transported through port 3 of the circulator into the connector #2 on the network analyzer. The measurement of the

54 scattering matrix parameter S21 can be used to tune the EMM probe to critical coupling with the transmission line.

Figure 3-8 S11 vs. Frequency Plots with Varying Magnetic Field Strength

As shown in Fig. 3-8, the S11 plots changes with the applied magnetic field strength. The resonate frequency and the Q factor are all affected. The resonance frequency increases while the Q factor of the system decreases. The fundamental concept of EPR is to detect the transitions between the splitting energy level by a magnetic field. The most common way to detect the absorption energy is to set up a microwave frequency resonant circuit and monitoring the reflected energy constantly.

The reflective type detection shows a large amount of reflected power during the EPR resonance condition, due to the system Q factor changes.

55 + Z0 ZS ΔREPR

Vo ZL VS

Figure 3-9 Microwave Reflective Coefficient

As shown in Fig. 3-9, the microwave circuit components of an EPR system are the RF source, transmission line and Resonance load. In EPR measurement, the load can be a microwave cavity or Microstrip resonator. The Voltage Standing Wave Ratio

(VSWR) is tuned to be 1 at the off resonance condition, which means the maximum and minimum voltage ratio in the circuit is one and the energy is oscillating back and forth in the circuit.

The easiest way to understand the EPR reflective energy is through reflection coefficient S11, which is given by

Vreflected VSWR −1 − ZZ 0L S11 == = (3-4-1) Vincident VSWR +1 + ZZ 0L

where Z 0 is the characteristic impedance of the transmission line, Z L is the load

impedance at the off resonance condition and REPR is the simulated resistance at the resonance condition. Therefore, if the load impedance matches the transmission line characteristic impedance, the reflection coefficient is zero and the absorbed power by the

56 load Pabsorbed equals to the total power Pincident incident on the input of the microwave network.

At the resonance condition, the power absorption of the electrons can be expressed as an effective resistance. Therefore, the load impedance at the resonance is,

'= 0L + ΔRZZ EPR (3-4-2)

since Z 0 50 >>Ω≈ REPR , equation 3-4-1 can be approximated by

L '−ZZ 0 REPR S11 = ≈ (3-4-3) L +ZZ 0 2' Z 0

When the system is off EPR resonance, the load is matching into the transmission and the power delivered to the load is the maximum deliverable power,

2 ⎛ R ⎞ V 2 ⎜ L ⎟ (3-4-4) PL = ⎜ 0 ⎟ / RV L = ⎝ 0 + RZ L ⎠ 4Z 0

Assuming the equivalent resistor for the EPR power absorption is ΔREPR ,

2 ⎛ ΔR ⎞ ΔR P =Δ ⎜ ⎟ / RV =Δ V 2 (3-4-5) EPR ⎜ ⎟ 2 ⎝ 0 L Δ++ RRZ ⎠ ()2 0 Δ+ RZ

The power delivered to the load at the EPR resonance is

2 ⎛ R ⎞ Z P '= ⎜ L ⎟ / RV = 0 V 2 (3-4-6) L ⎜ ⎟ L 2 ⎝ 0 L Δ++ RRZ ⎠ ()2 0 Δ+ RZ

Therefore, the change in the power delivered to the load, i.e. the power reflected back to the microwave source due to the impedance mismatching is,

2 V Z 0 2 ΔR 2 ' PPP LLL −=−=Δ 2 V ≈ 2 V (3-4-7) 4Z 0 ()2 0 Δ+ RZ ()2 0 Δ+ RZ

From Eq. (3-4-3) and (3-4-7), the reflected power change is equal to the EPR power absorption. Therefore, the recorded reflected power is the EPR absorption power. In this

57 study the microwave signal generator output power is fixed at 3 dBm, which is about

1.413 mW. Therefore, the incident power is a constant The amount changes in of load power absorption due to the EPR is the power available form the microwave source less power delivered (i.e. the power reflected from the input port to port 2).

When the microwave frequency is the resonate frequency of the Microstrip resonator, there exists a standing wave, where the electromagnetic field generated at the tip of the micro Stripline is at the maximum strength. The splitting energy gap varies with the applied time varying large homogeneous magnetic field. When the energy gap equals the microwave energy, a transition from lower energy level to high energy level occurs. The reflection coefficient of the Microstrip resonator off the EPR resonance condition is

VReflected − ZR 0 1 ==Γ (3-4-8) VIncident + ZR 0

When the EPR resonance condition is reached, the microwave power absorbed can be treated as an added resistance ΔR to the load, therefore, the new reflection coefficient is

V 'Reflected ( + Δ )− ZRR 0 2 =Γ = (3-4-9) VIncident ()+Δ+ ZRR 0

Since the absorption power is extremely small and at critical coupling = ZR 0 , the change of the reflected

−Δ+ ZRR 0 − ZR 0 ΔR ΔR 12 ≈Γ−Γ=ΔΓ − = = (3-4-10) + ZR 0 + ZR 0 + ZR 0 2R

The inductance is related to the permeability μ of the material. The permeability of the material is related to the average magnetic dipole moment. Assume the permeability of

the probe is close to μ0 and the inductance at the resonance condition is

58 μr LL ()(1' χ m L +=+== χ + χ '''1 )Lj , where χ m is the magnetic susceptibility, a dimensionless parameter, which is a measure of the ability of the material to become magnetized. Because of the imaginary part of the susceptibility, there is an additional resistance in the probe during the EPR resonance.

With a frequency that is proportional to the external magnetic field, the permeability of the substance increases dramatically, this also implies that the susceptibility of the substance increases tremendously, which implies the electron spins are aligning with the external magnetic field. The relation between the ESR and the magnetic susceptibility can be summarized as

β 22 SSN + )1( χ = (3-4-11) m 3 ()Tk Δ+

whereS is the spin angular momentum operator, β is the Bohr magneton, Δ is the Weiss constant, and N is the number of spins in the sample.

References

[1] B. Johansson, S. Haralson, L. Peterson, O. Beckman, “A Stripline resonator for ESR”, Rev. Sci. Instrument., Vol. 45, No. 11, pp. 1445-1447, November 1974

[2] M. Tabib-Azar, N. Shoemaker, S. Harris, “Non-destructive Characterization of Materials by Evanescent Microwaves,” Measurement Science and Technology, pp. 583- 590, 1993.

[3] M. Tabib-Azar, S. R. LeClair, “Applications of Evanescent Microwave Probes in Gas and Chemical Sensors,” Sensors and Actuators B, vol. 67, pp. 112-121, 2000.

59 [4] M. Tabib-Azar, D. Akinwande, G. Ponchak, S. R. LeClair, “Novel Physcial Sensors using Evanescent Microwave Probes,” Rev. Sci. Instrum., vol. 70, pp. 3381-3386, 1999

[5] H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev., vol. 66, pp.163- 182, 1944

[6] A. F. Lann, M. Golosovsky, D. DAvidov, and A. Frankel, “Mapping the Thickness of Conducting Layers by a mm-wave near-field Microscope,” 1998 IEEE MTT-S Int. Microwave Symp. Dig., Baltimore, Marland, pp. 1337-1340.

[7] M. Tabib-Azar, P. S. Pathak, G. Ponchak, and S. LeClair, “Nondestructive Super- resolution Imaging of Defects and Non-uniformities in Metals, Semiconductors, Dielectric, Composites, and Plants using Evanescent Microwave,” Rev. Sci. Instrument, vol. 70, pp. 2783-2792, 1999.

[8] M. Tabib-Azar, D. P. Su, and A. Pohar, S. R. LeClair, G. Ponchak, “0.4 μm Spatial Resolution with 1 GHz (λ = 30 cm) Evanescent Microwave Probe”, Rev. Sci. Instrument, vol. 70, pp. 1725-1729, March 1999.

[9] M. Tabib-Azar, B. Sutapun, “Novel Hydrogen Sensors Using Evanescent Microwave Probes,” Rev. Sci. Instrument, vol. 70, pp. 3707-3713, September, 1999

[10] M. Tabib-Azar, T. Zhang, and S. R. LeClair, “Self-Oscillating Evanescent Microwave Probes for Nondestructive Evaluations of Materials,” IEEE Transactions on Instrumentation and Measurement, Vol. 51, No. 5, October, 2002

[11] M. Tabib-Azar, D. Akinwande, G. E. Ponchak, and S. R. LeClair, “Evanescent Microwave Probes on High-Resistivity Silicon and its Application in Characterization of the Semiconductors,” Rev. of Scientific Inst. Vol. 70, No. 7, July, 1999

[12] D. A. Rudman, F. J. B. Stork, J. C. Booth, J. Y. Juang, L. R. Vale, G. J. Beatty, C. I. Williams, J. A. Beall and R. H. Ono, “Role of Oxygen Pressure During Deposition on the Microwave Properties of YBCO Films,” Applied Superconductivity Conf. 1998

[13] S. K. Dutta, C. P. Vlahacos, D. E. Steinhauer, A. S. Thanawalla, B. J. Feenstra, F. C. Wellstood, and S. M. Anlage, “Imaging Microwave Electric Fields Using a Near-field Scanning Microwave Microscope,” Applied Physics Lett., Vol. 74, No. 1, January, 1999

60

Chapter 4 EPR single point measurement

The EMM EPR measurement is similar to the classical EPR measurement, which belongs to the AC Magnetometry [1]. A small low frequency AC magnetic modulation field is superimposed on a slowly changed biasing DC magnetic field and the sample absorbs the microwave energy at the resonance condition. The EPR power absorption changes the impedance of the load and breaks the perfect match between the transmission line and the load, which increases the reflected power. The detection system detects the reflected power from the resonator, which is equal to the EPR absorption power. In this chapter, the Ruby sample is placed inside of the copper wire loop of the Stripline Resonator tip to have the large volume of sample under the microwave radiation, which results a higher filling factor, thus higher ESR signal strength.

4.1 Theoretical Background

Instead of using microwave cavity, a micro strip line resonator is used as the EPR resonator. At the critical coupling, the Microstrip Resonator has excellent sensitivity to even detect the weak EPR signals. As shown in Fig. 4-1, the EMM EPR measurement is using reflective mode of detection and the probe is tuned to have minimum of reflected power, which means that the transmission line and the load are perfectly matched before or after the EPR.

61

Figure 4-1 Theoretical Background of EMM EPR

The reflected power is converted to the voltage by the crystal detector and input to of the Lock-in amplifier, whose output is recorded through a DAQ card to a computer.

Again, the EPR detection technique is to detect the reflected power from the EMM probe instead of detecting the EPR power absorption directly. When the EPR resonance condition is reached, the perfect matching condition is no longer exist and the reflected power increases dramatically, which can be recorded and the results are the different EPR line shapes.

62 4.2 Measurement Apparatus

Synchronized Reference Signal

Circulator Synthesized Lock-in Sweeper Preamplifier Amplifier Crystal Detector

Coaxial DAQ PC/ Real Signal Cable Card Time ESR Generator

Micro Strip Line Tuning Resonator Screw Gauss Meter

N S

Hall Probe Sample ElectroMagnet Field Modulation ElectroMagnet Coil

Figure 4-2 EPR Single Point Measurement Apparatus

The experiment setup is shown as Fig. 4-2, the EMM probe is a microwave strip line resonator with an overall Q factor of more than 5,000. The EMM probe is fine tuned between the two poles of the electromagnet, therefore, when the system starts there is no motion in the overall system. The probe tip was shorted to the ground plate by a 0.15 mm copper wire to generate a strong alternating magnetic field at the microwave frequency. The Synthesized Sweeper generates a Microwave signal at 3 dBm power with the resonance frequency of the micro strip line resonator, which passes through the circulator and incident on the Micro Stripline resonator. After tuning the resonator to

63 critical coupling condition by using the tuning screw, a standing electrical magnetic waveform was generated and sustained at the probe tip and minimum power is reflected to the crystal detector. A 3 feet long coaxial cable was used to make certain that the electromagnet field will not affect the circulator operation.

A large current regulated electromagnet with high permeability iron yoke generates uniform magnetic field, with the capability of sweeping .01 mT to 1.6 Tesla.

The electromagnet is water cooled, which also helps to minimize the temperature effects on the detection signal. A 24 turns copper wire coil wrapped around the main pole of the electromagnet, which generates the magnetic modulation field with a magnitude of 2

Gauss at 5.3 kHz. The probe tip is aligned to be parallel to the electromagnet poles, which means the main magnetic field will be perpendicular to the RF magnetic field.

4.2.1 Magnetic Field Modulation Coil

The magnetic field modulation is critical important for this experiment measurement.

For classic ESR measurement, a typical modulation frequency is from 1 kHz to 100 kHz.

The lock-in amplifier has a frequency range of 1 Hz to 100 kHz, the ideal frequency would be 10 kHz. The 24-turn solenoid coil was wound on the one of the two poles of electro-magnet to generate the modulated magnetic field at a much lower frequency than the microwave frequency, which results the main magnetic field through the sample is still can be considered as DC magnetic field. In order to achieve a magnetic modulation field with amplitude of 2 Gauss, a small capacitor was paralleled with the modulation coil to have an parallel RLC circuit. The signal generator output frequency was swept

64 from 1 kHz and 10 kHz to find the resonance frequency of the parallel resonator. By using the Walker MG-4D AC Gauss meter, the AC magnetic field strength was measured to be 2.0 Gauss at the resonance frequency 5.2 kHz.

4.2.2 Concept of the Circulator

Figure 4-3 Port Diagram of a Three-port Circulator

The circulator used in this study is Diatom D3C2060, whose frequency range is from 2.0

GHz to 6.0 GHz. The circulator which is a perfectly matched three port circulator, whose scattering matrix can be written as

⎡ SSS 131211 ⎤ ⎡ 00 S13 ⎤ S = ⎢ SSS ⎥ = ⎢S 00 ⎥ [] ⎢ 232221 ⎥ ⎢ 21 ⎥ ⎣⎢ SSS 333231 ⎦⎥ ⎣⎢ S32 00 ⎦⎥

Where 21 32 SSS 13 === 1

65 The microwave signal generated by the RF generator incidents at port 1 as shown in Fig. 4-3. The power transmitted into port 2 without loss. This is also true from the port 2 to port 3 and port 3 to port 1. The power reflected from the resonator due to the

EPR resonance condition transmitted into port 3 and detected by the crystal detector. The circulator makes it feasible to separate the EPR signal from the source microwave.

The circulator is three feet away from the electromagnet to minimize the effect of the strong magnetic field generated by the electromagnet. Although a good circulator has very small power loss and impedance mismatch, it is critical to consider all the effects during the tuning of the Microstrip resonator. Therefore, in this study the tuning includes the circulator in the microwave circuit.

4.2.3 Phase Sensitive Detector (PSD)

Although theoretical the EPR can be detected directly, the EPR signal is too weak to be detected in a noisy background. The phase-lock-in amplifier has two major functionalities; the first one is recovering and amplifying weak signals from an overwhelming noise background through a Band pass filter, which filters out all signals that does not match the external reference frequency. The second one is. Lock-in amplifiers are used to measure the amplitude and phase of signals buried in noise. They achieve this by acting as a narrow Band Pass filter which removes much of the unwanted noise while allowing through the signal which is to be measured.

66 The frequency of the signal to be measured and hence the Pass Band region of the filter is set by a reference signal, which has to be supplied to the lock-in amplifier along with the unknown signal. The reference signal must be at the same frequency as the modulation of the signal to be measured.

Band Low Pass Filter Demodulator Pass Filter DC output Signal Detected A AB Signal

Pre-Amplifier B Amplifier

Reference Signal Phase Shifter

Figure 4-4 Phase Lock-in Amplifier Components

The demodulator is a multiplier as shown in the diagram of Fig. 4-4. It takes the input signal and the reference and multiplies them together. When you multiply two waveforms together you get the sum and difference frequencies as the result. As the input signal to be measured and the reference signal are of the same frequency, the difference frequency is zero and you get a DC output which is proportional to the amplitude of the input signal and the cosine of the phase difference between the signals. By adjusting the phase of the reference signal using the reference circuit, the phase difference between the input signal and the reference can be brought to zero and hence the DC output level from the multiplier is proportional to the input signal. The noise signals will still be present at the output of the demodulator and may have amplitudes 1000x larger than the DC offset, therefore another low pass filter is used to eliminate the high frequency noise.

= ino × ref fVfVV × cos()()( φ) (4-2-1)

67 To obtain the highest sensitivity, the phase between the reference signal and the detected signal is tuned to be zero, which results the highest output DC voltage. In reality, this DC voltage usually contains noise voltage components. By the nature of the noise signal, there is no consistent phase relationship between the noise signal and the reference signal, therefore, the output of the multiplier due to the noise signal is not steady and can easily removed by the PSD.

68 4.3 EPR real time detection

The control flow chart of the EPR real time detection program is shown in Fig. 4-5

Start

Initialize the Field Sweep Range, Field Modulation, Time Constant, etc.

IF Current Field Yes > Main Field

Read Lock-in Amplifier Output, Gauss Meter Reset Electro-Magnet Readings, update display

Current Field = Current Field + Field Step Send Command to Electro- Magnet Controller

END

Figure 4-5 Program Flow Chart of the Real Time EPR

The real time EPR measurement sets the conversion time, the time that is spent on each data point, at 400 ms and the Lock-in Amplifier time constant is set to be 30 msec. When

69 sweeping the electro-magnetic field at one Gauss per conversion period with a magnetic field modulation amplitude of 2.0 Gauss. The energy absorption derivative is obtained through computer DAQ. The sampling frequency of the DAQ is 1 ms and the resolution of the magnetic field sweep is around 1 Gauss. A hall probe is placed near the Microstrip

Resonator tip between the two poles of the electromagnet and real time magnetic field strength is recorded. There are three spikes in the electron resonance spectrum as shown in Fig. 4-6.

Figure 4-6 Real Time EPR Result for Single Point Measurement

70 With the magnetic field modulation, the ESR signal was extracted through a

Lock-in Amplifier, which suppresses the tremendous background noise. As shown in Fig.

4-6, three distinct ESR signals were detected, and the magnetic fields matched the theoretical predications based on the Standard Reference Materials: Electron

Paramagnetic Resonance Intensity Standard: SRM 2601. Based on the line shape of the

ESR signal, the number of spins was in the range of 20 -100 thousand. When the sample was removed from the probe or without the magnetic field modulation, the ESR signal disappeared totally.

4.4 Comparison of experimental and theoretical line shapes

It is important to study each individual EPR peak, which contains rich information, such as number of spins, spin relaxation time, etc. EasySpin [3], a Matlab toolbox is used for comparison of the theoretical and experimental data.

71

Figure 4-7 First Derivative of EPR Absorption Signal Comparison between Experiment Data and Theoretical Lorentzian Line Shape

As shown in Fig. 4-7 and 4-8, the experimental line shape is wider than the theoretical line shape. The EPR power absorption line shape is neither a pure Lorentzian nor Gaussian Type line shape. It is a linear combination of both line shapes.

72

Figure 4-8 First Derivative of EPR Absorption Signal Cross Zero at 1363 Oe with Enclosed EMM probe

From the first derivative line shape the line width ΔH pp is 24.14 Gauss and the g factor is 1.9756, which is wider than the theoretical line width. The EPR absorption line broadening is related the magnetic field modulation amplitude, the time constant of the lock-in amplifier, data conversion time, etc.

73 4.5 The EPR EMM Probe Sensitivity

4.5.1 Background Noise

The background noise is a random signal, which can not be predicated at any moment. The main noise source is the thermal noise or the Johnson noise, detector noise, amplifier noise and low frequency noise. The thermal noise arises in any medium that dissipate energy. When the noise is passing through the preamplifier the amplification applies to both the EPR signal and the noise signal. Only the noise that is not inside the preamplifier bandwidth is filtered out. The same result applies to the Lock-in amplifier also.

Each data point displayed by the Real time EPR measurement program, has 0.4 second conversion time and 40 msec time constant was used for the lock-in amplifier.

Also 201 loops were used to sample to output of the lock-in output to suppress the background white noise. The scan time for one EPR magnetic sweep is about 40 minutes.

4.5.2 The Spin-Spin Relaxation Time

At 3.77 GHz, the gyromagnetic ratio [2] is gμ −24 ⋅×⋅ /102779.91.9756 WbJ γ B == = MHz /4.17 Gauss (4-5-1) ×1063.6 −34 h J − sec 2π

For a single electron, the spin-spin relaxation time (also called T2) is

74 2 −9 Trelaxation = ×== sec10753.2 (4-5-2) 3γΔH pp

Where the line shape width ΔH pp measured from the positive peak and the negative, is about 24.14 Gauss. The EPR line width from other literatures is approximately 12 Gauss

[3]-[5].

4.5.3 The Total Number of Spins

By integrating the first derivative of the EPR absorption signal, the EPR absorption line shape can be obtained to reveal the total number of electron spin centers in the sample.

N = PdH (4-5-3) Total ∫ ΔH

75

Figure 4-9 Integrated EPR Line Shape

The above plot shows three sets of EPR data for the first EPR peak. The error bar is at 60% for the integrated line shape series (1).

4.5.4 The Minimum Number of Spins

The peak of the EPR line shape is the maximum power absorption point, which is 744 above the noise floor, as shown in Fig. 4-9. Since the measurement sensitivity range of the lock-in amplifier is 120 at 10 μV . Therefore the maximum power reflected from the resonator or the maximum power absorption is

76 V Measurement ⋅10μV 120 −4 PAbsorption = ×= 1024.1 μW (4-5-4) SCrystal_Detector ⋅G erPreamplifi

Where the crystal detector sensitivity S Crystal_Detector = /5.0 μWmV and the preamplifier

gain GPreamplifier = 1000 has a 1000 X gain.

1 1 For S −= to S = transition, the microwave energy absorption 2 2 is hν ×= 104995.2 −24 J . Since the microwave power is sufficient low, such that no saturation occurs during the EPR scanning process. Assume the phone emits when the electron jumps from high energy state to low energy state were all absorbed by the crystal lattices and the sample temperature is nearly constant. The power required at resonance

[6] condition for one single electron is approximately by / thv Relaxation , which is

×10079.9 −10 μW . Therefore, the maximum number of spin centers is

−4 PAbsorption ×1024.1 μW 5 N max = = −10 ×= 1037.1 (4-5-5) PElectron ×10079.9 μW

The signal to noise ratio (SNR) is around 6.8:1, therefore, the minimum number of

N detectable spin centers is N max ×== 1001.2 4 min SNR

References

[1] R. Wang, F. Li, M. Tabib-Azar, “Calibration Methods of a 2 GHz Evanescent Microwave Magnetic Probes”, Rev. Sci. Instrument., Vol. 76, May, 2005

[2] C. Gao and X. D. Xiang, “Quantitative Microwave Near-field Microscopy of Dielectric Properties” Rev. of Sci. Inst. Vol. 69, No. 11, November, 1998

77

[3] R. F. Wenzel, Y. W. Kim, “Line width of the Electron Paramagnetic Resonance of (Al2O3)1-x(Cr2O3)x” Physical Rev. Vol. 140, 5A, November 1965

[4] W. J. C. Grant and M. W. P. Strandberg, “Line Shapes of Paramagnetic Resonances of Chromium in Ruby” Physical Rev. Vol. 135, 3A, August 1964

[5] A. Murali and J. Lakshmana Rao, “Electron Paramagnetic Resonance and Optical Spectra of Cr3+ ions in Fluorophosphates Glasses” J. Phys.: Condens Matter 11, 1999, pp. 1321-1331

[6] R. B. Griffiths, “Theory of Magnetic Exchange-Lattice Relaxation in Two Organic Free Radicals” Phys. Rev. Vol 124, No. 4, November, 1961

78 Chapter 5 EPR Measurement on the Ruby Surface

Although many EPR experiment has been carried to study the number of spins, spin to spin, spin to nuclear interactions, etc. The spatial location of the spins has never been clearly defined, especially at the surface with high resolutions. The near-field microwave microscopy is famous localized probe with high spatial location resolutions [2-6]. With the successful measurement of the EPR signals when the sample is inside of the EMM probe tip wires, the following sections describe the EPR measurement [1] when the sample is outside of the EMM probe tip and the EMM probe tip is touching the sample surface.

5.1 Single Point Surface EPR Detection

Micro Strip Line Tuning Resonator Screw

Position Support Control

Sample

Figure 5-1 EPR Surface Measurement Apparatus

79 The experiment setup for the EPR surface measurement is almost identical to the EPR single point measurement described in Chapter 4, except the sample is outside of the

Micro Stripline resonator loop and the tip is touching the sample surface as shown in Fig.

5-1. The EMM probe tip can be setup to move around on the tip to image the number of spin centers [1].

When the sample was moved outside of the EMM EPR probe wire loop, the transmission line characteristic impedance changed slightly. By adjusting the tuning screw, the EMM probe has a new resonant frequency of the 3.73 GHz in critical coupling condition. The EPR magnetic fields where the EPR peak occurs also altered as shown in

Table 4.

Table 4 the EPR peak Magnetic Field at 3.73 GHz

ES+1 ES f (Hz) hυ (2S+1)D g||μBHo Ho (Oersted) 3/2 1/2 3.73E+09 2.47E-24 -7.62E-24 1.01E-23 5490.1 1/2 -1/2 3.73E+09 2.47E-24 0.00E+00 2.47E-24 1344.6 -1/2 -3/2 3.73E+09 2.47E-24 7.62E-24 5.15E-24 2800.9

By using identical bias magnetic field scan rate and magnetic field modulation amplitude as Chapter 4. Three peaks show clearly the power absorption first derivative at three magnetic fields that matched the theoretical predication from. The screen shot of the real time EPR measurement software is shown as Fig. 5-2.

80

Figure 5-2 Single Point Surface EPR Real Time Detection

Although the sample was outside of the EMM probe wire loop, the RF magnetic field is still strong enough to get the EPR signals. The screen shot shows three transitions occurred and the magnitude of reflected power is smaller than the one showed in Chapter

4, which indicates the number of spins on the surface is much less. One way to explain this phenomenon is that the number of electron spins centers decreases dramatically once the sample moved outside of the tip wire loop, i.e. the filling factor of the Microstrip resonator decreases dramatically.

81 5.2 EPR Surface Signal Analysis

The following plot shows the first EPR absorption occurs around 0.135 Tesla.

39

Crystal Detector Signal Gaussian Line Shape 29

19

9

-1

Voltage 1250 1270 1290 1310 1330 1350 1370 1390 1410 1430 1450

-11

-21

-31

-41 Magnetic Field H(Oe)

Figure 5-3 Comparison of the First Derivative of the EPR Absorption Line Shape and Gaussian Line Shape

By comparing the first derivative of the EPR absorption line shape, the Gaussian and Lorentzian line shape as shown in Fig. 5-3 and Fig. 5-4, the EPR absorption line shape may be reconstructed with the linear combination of both Gaussian and Lorentzian line shapes.

82

Figure 5-4 Comparison of the First Derivative of the EPR Absorption Line Shape and Lorentzian Line Shape

From the first derivative line shape, the line width ΔH pp is approximately 22.14

Gauss and the g factor is 1.9740.

At 3.73 GHz, the new gyromagnetic ratio is

gμ −24 ⋅×⋅ /102779.91.9740 WbJ γ B == = MHz /36.17 Gauss (5-2-1) ×1063.6 −34 h J − sec 2π

The estimated spin-spin relaxation time is,

2 −9 Trelaxation = ×= sec10004.3 (5-2-2) 3γ Δ⋅ H pp

83 5.2.1 The Total Number of Spins near the Surface

By integrating the first derivative of the absorption line, Fig. 5-5 shows three peaks with noise background. There exists a drifting due to instrument noises, which can be eliminated by subtracting a low order polynomial function as shown below.

Figure 5-5 The integrated EPR Absorption Line Shape

The first EPR power absorption line peak at 0.13 Tesla is shown as Fig. 5-6, along with two other experimental data sets.

84

Figure 5-6 The Integrated Line Shape of the Surface EPR Signal

Each EPR scan takes about 40 minutes and no overload on the preamplifier or the lock-in amplifier. Therefore the microwave power is sufficiently low and no saturation occurs. The area underneath the line shape can be approximated by

=ni =ni H 2 = )( ≈ i ()ii −1 Δ=− VHHHVdHHVA i (5-2-3) ∫H ∑ ∑ 1 i=1 i=1

85 5.2.2 The Minimum Number of Detectable Spins near the Surface

The peak of the EPR line shape is the maximum power absorption point, which is 231 above the noise floor. Since the measurement sensitivity range of the lock-in amplifier is

120 at 10 μV . Therefore the maximum power absorption is

V Measurement ⋅10μV 120 PAbsorption = = 0385.0 μW (5-2-4) SCrystal_Detector

Where the crystal detector sensitivity S Crystal_Detector = /5.0 μWmV . Since the preamplifier has the same 1000 X gain as in Chapter 4, the actual reflected power from the Microstrip resonator is ×1085.3 −5 μW . The equivalent EPR resistance is

ΔPEPR −9 REPR =Δ 2 Z 0 1044.5 Ω×=⋅ (5-2-5) P1

1 1 For a S −= to S = transition, the microwave energy absorption is 2 2 hν 1063.6 −34 sJ 9 Hz ×=××⋅×= 10473.21073.3 −24 J . The power required at resonance

−10 condition for one single electron is / thv Relaxation , which is ×10232.8 μW . Therefore, the maximum number of spin centers is

−5 PAbsorption ×1085.3 μW 4 N max = = −10 ×= 106768.4 (5-2-6) PElectron ×10232.8 μW

The signal to noise ratio is about 2.5:1, therefore, the minimum number of detectable spin

4 centers is N min ×= 10871.1 .

86 5.3 Comparison between the Surface EPR and Enclosed EPR

By plotting the first peaks of the surface and enclosed EPR together, Fig. 5-7 shows clearly resembles of the two line shapes.

Figure 5-7 The Comparison of the First Derivative Line Shapes between Enclosed and Surface Measurement

The two first derivative line shapes have different zero crossing fields or different g-factor, because of the different Electron Resonance peak fields.

87

Figure 5-8 The Comparison of the Integrated Line Shapes between Enclosed and Surface Measurement

The comparison of the line shapes shows the total number of spin centers dropped when the sample is moved outside of the Microstrip Line Resonator tip wire. However, because the magnetic field is the strongest near the probe tip either inside or outside the tip wire, the surface detection has smaller minimum detectable spin centers.

5.4 Magnetic Field Distribution

At off EPR resonance condition, the load matches the transmission line perfectly, which results half of the source microwave power delivered to the load, assuming the

88 transmission line is lossless. Therefore, the power delivered to the load is half of the 3dB, which is approximately 0.706 mW. The resistance of the load is approximately 50 Ω and the current through the tip can be approximated by,

P I L ≈= 76.3 mA (5-4-1) R

By using magnetic field analysis software, the field distribution for 3.76 mA at

3.7 GHz is shown in Fig. 5-9. The magnetic field strength is at the maximum at the surface of the copper wire and it decreases exponentially. The EPR average power is proportional to the square of the RF magnetic field strength. Therefore, the power absorption decreases even faster with the distance.

Figure 5-9 Magnetic Field Distribution near the EMM EPR Probe tip

89 References

[1] M. Tabib-Azar, X. Li, J. Adin Mann Jr., “Local Molecular Spectroscopy Using the Evanescent Microwave Probe Technique: Electron Spin Resonance”, IEEE sensor conf. 2005

[2] V. Zevin, J. T. Suss, “ESR in Layer-substrate Structures: the Line Shape and Nondestructive Contactless Measurements of the layer Conductivity”, Phys. Rev. B, Vol. 34, No. 10, November 1986

[3] T. Wei and X. D. Xiang, “Scanning Tipe Microwave Near-field Microscope,” Appl. Phys. Lett. 68 (24), June 1996

[4] V. V. Talanov, L. V. Mercaldo, “Measurement of the Absolute Penetration Depth and Surface Resistance of Superconductors and normal Metals with the Variable Spacing Parallel Plate Resonator,” Rev. of Sci. Instruments, Vol. 71, No. 5, May 2000

[5] J. Adin Mann, X. Li, M. Tabib-Azar, "A Near-Field Microwave Microscope and Electron Spin Resonance Detection", Optical Society of America: 2004 Photon Correlation and Scattering Conference, Van der Waals Institute Amsterdam, Aug. 16-18, 2004.

[6] F. Sakran, A. Copty, M. Golosovsky, N. Bontempsb and D. Davidov. “Electron spin resonance microscopic surface imaging using a microwave scanning probe”, Appl. Phys. Lett., Vol. 82, No. 9, March 2003.

90 Chapter 6 Conclusions and Future applications of EPR

6.1 Summary of the EPR measurement on the Ruby Surface and future works

A unique technique for few electron paramagnetic resonance detections on the Ruby surface has been developed to spatially detect the electron spin centers along with typical

EPR information, such as EPR line shapes, g-factor, number of spins and spin-spin relaxation time, etc. The following are the summary of the experiments,

• Developed Real time EPR software to automatically control the magnetic field

scanning rate and scanning magnetic field step, collect and display the EPR signal

in real time.

• Constructed a Magnetic Dipole Microstrip Line Resonator with 3.77 GHz

resonant frequency and overall resonant system Q of 5000.

• Natural Ruby sample was placed inside the Microstrip Line Resonator tip and the

Maximum Number of spin centers detected is about 20,100.

• Natural Ruby sample was placed outside the Microstrip Line Resonator tip, where

the probe tip is touching the ruby surface. EPR real time measurement was

performed and the Maximum Number of spin centers detected is 50,000, the

minim number of detectable spin centers is about 20,000. Meanwhile the probe

tip location on the Ruby surface can be accurately located.

With the current EMM EPR probe, both study objectives in the chapter 1 are met.

91

Electron Spin P.C. overlaping Topography Reference Imaging force + PID Laser - Control Displacement Activ Q detector Control Signal Generator Energy absorption Detecting Unit Position (1GHz-10GHz) Control signal

Probe

Sample

Enhanced EMM EPR B= 50 - 600 mT Station

Figure 6-1 Schematic of Future Enhanced EMM EPR Setup

For future works, the EMM EPR can be combined with the Atomic Force microscopy (AFM) to detect even fewer electron spin centers as shown in Fig. 6-1.

Unlike the current EMM EPR tip, The AFM tip size is in micro-meter level, therefore, the space is more confined and the minimum number of detectable spin centers can reduce to one signal electron spin center. Other advanced control such as Active Q control, fuzzy logic control can be utilized for the precise control of the AFM tip locations.

92 The proposed micro-fabricated AFM tip combined with the Carbon Nano Tube is shown in Fig. 6-2,

Coaxial Wave guide

Carbon Nano Tube

Micro- Fabricated AFM Tip

Figure 6-2 Future Micro –Fabricated AFM Tip with Carbon Nano Tube

with more spatially confined magnetic field and larger filling factor, the Micro-Fabricated

EMM AFM should increase the sensitivity dramatically.

93 6.2 Applications of surface EMM EPR

Material Defect Detections Quantum Computation Material Identification Aging ESR (Surface or Embedded) (Single Electron) (Chemical Constitution) (Free Radical)

Enhanced EMM AFM

Active Q Control Optical Frequency Pulsed EPR Carbon Nano Tube

Microwave Evanescent Atomic Force Microscopy Probes

Figure 6-3 Future Research Road Map

As shown in Fig. 6-3, the future research will be based the combination of EMM ERP probe and the AFM techniques along with other advanced techniques [1-2]. In addition to the continuous wave EPR, pulsed microwave power can be used for pulsed EPR detection. Also the EMM ERP tip short wire can be replaced with the Carbon Nano Tube, which reduces the size of the wires tremendously. The following sections describe some of the promising future applications.

94 6.2.1 Quantum Computing

Electron spin is the smallest switch nature offers and with one single electron spin represents a quantum bit which is the fundamental building block of a quantum computer.

By flipping a single electron spin upside down in an ordinary commercial transistor chip, and detected that the power absorption changes when the electron flips.

Quantum computing [3-4], which holds the promise of nearly unlimited processing power, secure network communications, also the power to decode encrypted data. With high density memory devices such as Spin Net, the storage of entire libraries in the world could be in the size of a small coin. The flipping of a single electron, the tiny magnetic moment, may seems to be straight forward, it is hard to realize in the real world because of the overwhelming noise background. Also the spin to spin relaxation time is extremely short for most unpaired electrons, which makes the bandwidth of the electron very narrow. Different research groups have been conducted different approaches to increase the signal to noise ratio and the spin to spin relaxation times. New material may also emerge to conquer all obstacles to make the single electron spin as a quantum bit in the future.

6.2.2 Defects in Material

In material research, defects may also be detected using this technique because they trap unpaired electrons. The most popular local point defect is the F-center which causes

95 color effects caused by an electron in an anion (a negatively charged ion) defect. The localized EMM EPR probe will also be very effective in detecting material defects [5-8] with high resolution, such as stress cracks, premature oxidation and defects under a surface coating.

One of electrically active defects in semiconductor material is the damaged bond with free unpaired electrons. Also it is more likely when the unpaired electron of the defect center interacts with nearby nuclei that have a magnetic moment giving rise to hyperfine structure in the EPR spectrum. At the important Si/SiO2 interface, for example,

Silicon dangling bond defects are readily formed and must be passivated with hydrogen in order to achieve the electrical characteristics required for metal-oxide-semiconductor

(MOS) transistors.

6.3.3 Aging Process Detection

When an atom loses an electron, it becomes a free radical with unpaired electrons. A small membrane-enclosed region of a cell, which is called mitochondria, produces all the necessary chemical energy the cell needs. The electron is the main transport, which passes through different molecules to produce energy. In abnormal situation, the electron interacts with oxygen and produces a free radical. According to the free radical theory [9-

10] of aging, the cell in most organisms produces free radicals which damage and eventually kill the cell that is how the organism ages.

High spatial resolution detection of electron spin transitions can also be used to map free radicals in biological tissues that cause aging and cancer. Additionally, ESR

96 signal may be used in detection explosives and food freshness. Even with its currently low resolution, the probe is unique and can be used in micro-fluidic channels and biological tissues to detect free radicals. We are pursuing many applications of the probe in Alzheimer and other tissue studies where mapping the spatial distribution of free radicals may shed some light on the cause and diagnosis of disease.

References

[1] K. L. Ekinci, Y. T. Yang, X. M. H. Huang and M. L. Roukes, “Balanced Electronic Detection of Displacement in Nanoelectromechanical Systems,” Applied Physics Let., Vol 81, No. 12, Sep., 2002

[2] S. Sahoo, T. Kontos, and C. Schonenberger, “Electrical Spin Injection in Multiwall Carbon Nanotubes with Transparent Ferromagnetic Contacts,” Applied Physics Let., Vol 86, 2005

[3] M. Tabib-Azar, S. R. LeClair, J. F. Maguire and W. C. Fitzgerald, “Microwave Imaging with Atomic Force Microscope and its Applications in Molecular and Nano Electronics,” Microwave Nondestructive Evaluation and Imaging, 2002, pp. 25-46 ISBN: 81-7736-081-7

[4] P. Petit and E. Jouguelet, J. E. Fischer, A. G. Rinzler and R. E. Smalley, “Electron Spin Resonance and Microwave Resistivity of Single-Wall Carbon Nanotubes,” Phys. Rev. B., Vol. 56, No. 15, October, 1997

[5] R. M. Willett, G. A. Ybarra, W. T. Joines, J. A. Bryan, “Microwave imaging of Mammary Tumors: Phantom Development, System Design, and Image Reconstruction,” Science News, Vol. 155, No. 9, February 27, 1999, p. 140.

[6] W. T. Joines, R. L. Jirtle, M. D. Rafal, and D. J. Schaefer, “Microwave Power Absorption differences in Mornal and Malignant Tissue.,” Int. J. Rad. Oncl., Bio., and Phys., Vol. 6, no. 6, 1980, pp. 681-687

[7] K. R. Foster and H. P. Schwan, “Dielectric Properties of Tissues and Biological Materials: A Critical Review,” Critical Reviews in Biomedical Engineering, Vol. 17, No. 1, 1989, pp. 25-104

97

[8] M. Tabib-Azar, J. L. Katz, and S. R. LeClair, “Evanescent Microwaves: A Novel Super-Resolution Non-contact Nondestructive Imaging Technique for Biological Applications,” IEEE Transactions on Instrumentation and Measurement, Vol. 48, No. 6, December 1999

[9] P. S. Timiras, “Physiological Basis of Aging and Geriatrics”, CRC Press LLC, 3rd Ed, 2002, ISBN-0-8493-0948-4

[10] E. Cadenas, L. Packer, “Understanding the Process of Aging: the Roles of Mitochondria, Free Radicals, and Antioxidants (Antioxidants in Health and Disease, 8)”, Marcel Dekker Inc., New York, 1999, ISBN 0824717236

98 Appendix A 3-D Schrödinger equation

The Schrödinger equation is a 3-D description of vibrational states of electrons around the nucleus and it is the fundamental equation of physics for describing quantum mechanical behavior, which describes how the wave function of a physical system evolves over time. In modern physics, electrons are not orbiting particles and an electron can be describes as a wave and its spatial distribution can be determined by solving the

Schrödinger equations, which also gives the size and shapes of the electron orbital.

Z

Electron

θ

Nucleus Y

φ

X

Figure 6-4 Coordinates of an Electron

2 ∫ ψ ( ) dt 3rr = 1, (A-1) timeSpace, D-3 timeSpace,

By solving the Schrödinger equation, a three-dimensional representation of the probability of finding an electron at certain distance from the nucleus can be easily visualized. The time independent Schrödinger equation is:

99 ⎡ ∂ 22 ∂ 12 ⎛ ∂ 2 ∂ 1 ∂ 2 ⎞⎤ − h + + ⎜ + cosθ + ⎟ ψ rv + )( = ErV ψψ rr vv (A-2) ⎢ 2 2 ⎜ 2 2 2 ⎟⎥ () () () 2μ ⎣∂r ∂rr r ⎝ ∂θ ∂θ sin ∂φθ ⎠⎦

Assume the solution of the wave function is ψ rv = rR Υ θ φ),()()( , substitute the solution

in equation (A-2) and separate the

2 1 ⎛ ∂ 2 2 ∂ ⎞ 2 11 ⎛ ∂ 2 ∂ 1 ∂ 2 ⎞ − h ⎜ + ⎟ rR )( − h ⎜ + cosθ + ⎟ φθ )(),( =+ ErVY ⎜ 2 ⎟ 2 ⎜ 2 2 2 ⎟ 2μ rR )( ⎝ ∂r ∂rr ⎠ 2 r Y φθμ ),( ⎝ ∂θ ∂θ sin ∂φθ ⎠

⎛ ∂ 2 ∂ 1 ∂ 2 ⎞ ⎜ + cosθ + ⎟Y ⋅= Y φθλφθ ),(),( ; (A-3) ⎜ 2 2 2 ⎟ ⎝ ∂θ ∂θ sin ∂φθ ⎠

⎛ ∂ 22 2 ∂ ⎞ − h ⎜ + ⎟ + = rERrRrVrR )()()()( (A-4) ⎜ 2 ⎟ 2μ ⎝ ∂r ∂rr ⎠

2 Υ sin),( dd φθθφθ = 1, (A-5) ∫∫all angles where θ φ F θ Φ=Υ φ)()(),(

100

Figure 6-5 Spherical Electron Harmonics for s and p Orbital

By using Matlab, plot the solution of the e-D Schrödinger equation, which is shown in

Fig. 6-4, the spherical electron harmonics are symmetric plots for one s orbital and three p orbital.

101

Figure 6-6 Spherical Electron Harmonics for d Orbital

Fig. 6-6 shows the electron harmonics for the d orbital, five orbital divides the energy level into two different energy states, with three lower energy levels and two higher energy levels. In this study, the Cr3+ ion in the single Ruby crystal has three unpaired 3d electrons, which occupy the lower three energy states.

102 Appendix B Magnetic Dipole moment

An electron possesses both an electrical charge and its spin, which produce a magnetic moment which has both magnitude and direction and behaves analogously to angular momentum. The electron is the lightest elementary particle that has magnetic moment and the electron also be treated as waves in Quantum Mechanics. When an unpaired electron is placed within in a magnetic field, it attempts to align its direction with the magnetic field.

The product of the current and the area is the magnetic moment of a loop. A current loop can be viewed as bunch of electrons traveling in a circle; therefore, the electron spin can be defined in the similar manner. = −1 pTB

B B1

I(in) F I1(in) F θ θ θ θ Ft Ft

Ft F1t θ θ F I(out) F1 I1(out)

Current Loop (Macroscopic Magnetic Moment) Electron Spin (Microscopic Magnetic Moment)

Figure 6-7 Macroscopic and Microscopic Magnetic Moments

The macroscopic magnetic moment is = ⋅ AIm , the torque exerted on the current loop is m ⋅= BT sinθ . While for he microscopic magnetic moment is given by

μ S g μ BS ⋅⋅= S and the magnetic moment associated with orbital momentum is

103 μL g μ LL ⋅⋅= L , where S is the spin angular momentum operator, L is the orbital

momentum operator and μ B is the Bohr magneton:

×1063.6 −34 10602.1 −19 C ××− J − sec e μ h == 2π −24 J ⋅×−= 2 /102779.9 Wbm . (B-1) B 2m ×× 1011.92 −31 kg

The magnetic moment of the electron is much stronger than the one of the nuclear spin, whose nuclear magneton is

eh 2 μ K −×== /271005.5 WbJm (B-2) 2m p

In this study, only unpaired electrons are considered, therefore, the orbital moment of the electron is ignored. The spin magnetic moment of an electron is independent of the spin angle or magnetic field, which is shown as,

μ S = gμ BS (B-3)

I = current μ=IA A=area

Figure 6-8 Magnetic Moment of One Single Electron

For one single electron with ½ spin, the magnetic moment is about one Bohr magneton. For an effective radius of 1 nm, the relative current is about 3 μA, which

−12 generates a magnetic field of μ0 IB ×≈= 108.3 Tesla .

The paired electrons inside an atom have exactly opposite spins, therefore, they tend to cancel each other and no net spin angular momentum or magnetic moment exists.

For the unpaired electron, the electron spins along its own axis and it resists forces that

104 tend to change its state of motion. For an analogues, a bar magnet that is placed in a magnetic field, its north pole seeks the south pole of the external magnetic field and it comes to rest if it aligned with the external magnetic field. Works have been done to change the orientation of the bar magnet.

105 Appendix C the Atomic Force Microscopy (AFM)

In the fall of 1985 Gerd Binnig and Christoph Gerber [1-2] used the cantilever to examine insulating surfaces and discovered the force between tip and sample could be measured by tracking the deflection of the cantilever. After the measurement of atomic structure of boron nitride by using a silicon micro-cantilever, the AFM became the most important tools for the world of surface science.

The most common modes for the AFM are the non-contact, tapping mode [3] and the contact mode where the tip scans the sample in close contact with the surface. The force on the tip is kept as a constant during the scanning of the surface. Therefore, the deflection on the cantilever can be detected and recorded for the surface topography [5-6].

When using contact mode, Fig. 6-9 is obtained by scanning a 10 mm x 10 mm sample using the Thermomicroscope Explorer TM AFM system.

Figure 6-9 AFM Imaging of silicon (Si3N4)

106 References

[1] G. Binnig, C. F. Quate, and Ch. Gerber, “Atomic force microscope,” Physics Rev. Letter. 56, 930, 1986 .

[2] G. Meyer and N. M. Amer, “Novel optical approach to atomic force microscopy,” Appl. Physics Rev. Letter. 53 1045, 1988

[3] T. Sulchek, R. Hsieh, J. D. Adams, G. G. Yaralioglu, S. C. Minne, and C. F. Quate, J. P. Cleveland, A. Atalar, D. M. Adderton, “High-speed Tapping Mode Imaging with Active Q Control for Atomic Force Microscopy”

[4] T. Tran, J. N. Nxumalo, Y. Li, D. J. Thomson, G. E. Bridges and D. R. Oliver, “Quantitative Two-Dimensional Carrier Profiling of a 400 nm Complementary Metal- Oxide_Semiconductor Device by Schottky Scanning Capacitance Microscopy,” J. of Appl. Phys., Vol. 88, No. 11, December 2000

[5] M. Guthold, M. Falvo, W. G. Matthews, S. Paulson, J. Mullin, S. Lord, D. Erie, S. Washburn, R. Superfine, F. P. Brooks Jr. and R. M. Taylor II, “Investigation and Modification of Molecular Structures with the NanoManipulator,” J. of Molecular Graphics and Modelling 17, 1999, pp. 187-197

[6] L. M. Fok, C. K. M. Fung, Y. H. Liu, and W. J. Li, “Nano-scale Mechanical Test of MEMS Structures by Atomic Force Microscope,” Proceedings of the 5th World Congress on Intelligent Control and Automation, June 2004

107