MUONIUM DEFECT CENTERS IN ALUMINUM
NITRIDE AND SILICON CARBIDE
by
HISHAM BANI-SALAMEH, M.S.
A DISSERTATION
IN
PHYSICS
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Approved
Roger L. Lichti Chairperson of the Committee
Charles Myles
Carl David Lamp
Stefan K. Estreicher
Accepted
John Borrelli Dean of the Graduate School
May, 2007
Copyright 2007, Hisham Bani-Salameh
Texas Tech University, Hisham Bani-Salameh, May 2007
ACKNOWLEDGMENTS
I would like to express my deep gratitude toward those who helped me throughout this experience. First and foremost, I would like to thank “Allah” for everything I managed to accomplish in my life so far, and for giving me the ability and the strength to get through this amazing learning experience, and giving me the chance to work with one of the best professors in the world, Dr. Roger L. Lichti. For you Dr. Lichti, I can’t find enough words to express my thanks to you for believing in me and for your support and patience; without you, none of this would have been accomplished. My father Nahar Bani-Salameh, my mother Rasmieh Bani-Salameh, my six brothers (Khalid, Jamal, Abdulsalam, Gazi, Mohamed and Ahmed) and my two sisters (Ahlam and Nesreen), I couldn’t have wished for better family, thank you very much for everything you’ve done for me and may “Allah” give me a chance to pay you back. My wife Kolthoom Alkofahi, I will never forget your unlimited support and patience; thank you for your sacrifice and I hope I’ll be able to help you fulfill your dreams and accomplish your goals in this life. My kids Layth and Layan, thank you for making me smile in the worst of days and keeping me fit by chasing you around. Please forgive me if one day I was too busy to give you the attention you need. I would like to thank the U.S. National Science Foundation and the Robert A. Welch Foundation for supporting this work. Last but not least, I would like to give a special thanks to Dr. Steve F.J. Cox for his significant contribution to the progress of the μSR research and for complimenting this work after reading through this thesis.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii TABLE OF CONTENTS iii ABSTRACT v LIST OF TABLES vi LIST OF FIGURES vii 1 INTRODUCTION AND MOTIVATIONS 1 2 FUNDAMENTALS OF μSR TECHNIQUES 7 2.1 Muons 7 2.2 μSR Techniques 11 2.3 Data Collection 15 3 THEORETICAL DESCRIPTION 18 3.1 Isotropic Muonium 23 3.2 Time Evolution of the Muoniun Spin Polarization 24 3.2.1 Longitudinal Field Case 29 3.2.2 Transverse Field Case 31 3.3 Anisotropic Muonium 32 3.4 Another Method to Calculate the Polarization 35 3.5 Muonium Dynamics 36 3.6 Charge exchange and Spin exchange processes 42 4 ALUMINUM NITRIDE EXPERIMENTAL DATA AND DISCUSSION 45 4.1 AlN Structure. 45 4.2 Hyperfine Decoupling Results. 47 4.3 Site Assignments for Mu0 in AlN. 50 4.4 Dynamics of Mu0 in AlN. 54 4.5 Discussion. 63 5 SILICON CARBIDE EXPERIMENTAL DATA AND DISCUSSION 71 5.1 Temperature Dependence of Hyperfine Constants in SiC 74
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5.2 Dynamics of Muonium States in 4H and 6H SiC 80 5.3 Discussions and conclusions 93 6 SUMMARY AND CONCLUSIONS 98 REFRENCES 102
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ABSTRACT
We report the results of μSR measurements on Aluminum Nitride (AlN) and Silicon Carbide (SiC). The importance of studying muonium states comes from its analogy to atomic hydrogen making it an excellent source of information on isolated hydrogen impurities in various materials. Neutral muonium exists in AlN to high temperatures, a large hyperfine constant of ~4450 MHz with a small temperature- dependent dipolar contribution indicating weak anisotropy is obtained from decoupling curves. Tentative site assignments and results on the diffusion of these Mu0 centers along with the associated conversion rates are presented. The low-energy location of neutral muonium in AlN lies off-axis in the unblocked c-axis channels at sites anti-bonding to Aluminum. Motion of Mu0 at low temperatures is due to tunneling and is dominated by thermally activated processes at high temperatures. Diffusion-limited conversion out of the mobile Mu0 state is observed in both low and high temperature regimes. All electrical types, high-resistivity, n-type and p-type, of the hexagonal 4H and 6H polytypes of SiC were studied. Two isotropic Mu0 states were found in 4H-SiC and a total of four Mu0 states were seen in the 6H-SiC samples. Temperature dependence of the
hyperfine constant (AHF) for each state is discussed. Data on the hyperfine interactions imply isotropic atomic-like states with no hint of any bond-centered Mu0 species in SiC. Temperature and field dependences of signal amplitudes and relaxation rates were studied. Tentative assignments for locations and some of the dynamical characteristics of the muonium centers have been reached; however, more work is needed to fully understand the nature of these centers in SiC.
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LIST OF TABLES
2.1 Muon and Proton Physical Properties. 2
4.1 Aiso and D values at different temperatures in AlN. 48 4.2 Fit results of the temperature-dependent hop rate and conversion rate in AlN. 61 5.1 Hyperfine Constant Parameters in SiC. 79 5.2 Activation energies from the diamagnetic amplitudes in 6H-SiC. 84
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LIST OF FIGURES
2.1 Decay of pions into a muon and a muon neutrino. 9 2.2 The angular distribution of emitted positrons from the muon decay. 11 2.3 Schematic of the position of the four positron counters. 13 2.4 Raw Time-dependent asymmetry data. 16
0 3.1 Breit-Rabi diagram for MuT in Si. 24
0 3.2 Field-dependent polarization of MuT in Si. 30 4.1 Part of the AlN wurtzite structure showing bond lengths and angles. 46 4.2 Experimental repolarization curves in AlN. 49 4.3 The Wurtzite structure of AlN showing site assignments for Mu0. 52 4.4 Simulated zero-time longitudinal asymmetry for possible Mu0 sites in AlN. 53 4.5 Muon spin depolarization raw data in AlN. 55 4.6 Longitudinal-field data indicating motion of Mu0 in AlN. 56 4.7 Temperature-dependent conversion rate and hop rate of Mu0 in AlN. 59 4.8 A segment of AlN wurtzite structure showing possible paths for motion. 64 4.9 Zero-field data on AlN. 65 4.10 LCR data on AlN. 68 5.1 The stacking sequence for 4H and 6H polytypes of SiC. 72 5.2 The muon spin precession spectra for SiC. 76
5.3 The temperature dependence of the hyperfine constant for the Mu2 in SiC. 78 5.4 A 3-D representation of the diamagnetic signal in n-type 4H-SiC sample. 81 5.5 The diamagnetic amplitude in n-type 6H-SiC sample. 82 5.6 Temperature-dependent amplitudes and relaxation rates in p-type 6H-SiC. 83 5.7 The full temperature-dependent diamagnetic signal amplitude in 6H-SiC. 85
5.8 Relaxation rates of Mu1 and Mu2 in p-type 6H-SiC. 86 5.9 TF data on the high-resistivity 6H-SiC sample taken at 15 gauss. 87 5.10 Muon magnetic resonance data of the three neutral signals in hr 6H sample. 89 5.11 The diamagnetic amplitude and phase in hr 4H-SiC sample at 80 Gauss. 91
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5.12 Amplitudes of the diamagnetic, Mu1 and Mu2 signals in p-type 4H-SiC sample. 92 5.13 A 3-dimentional representation of the 4H structure. 96
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CHAPTER 1 INTRODUCTION AND MOTIVATIONS
Because of their importance in modern electronics, semiconductors have been studied extensively. Impurities (whether intentional through doping or unintentional during growth or during processing) play a critical role in determining the properties of the semiconductor. The doping process usually results in an extra energy level in the bandgap that will modify the electronic properties of the material; therefore, as long as we have control over what kind and what amount of impurities are introduced, this process can be very useful. Unfortunately, some impurities (like Hydrogen) will enter the material unavoidably, most likely during the growth process or during various device processing steps. Since the early 1980’s, following the discovery that atomic hydrogen can passivate shallow donors and acceptors in Si, the structure and properties of hydrogen in semiconductors have received increasing attention. More intensive research in the following years revealed that hydrogen has the same important role as a passivating agent in a wide range of semiconductors [1-4]. Hydrogen will form complexes with dopants and other impurities in the host material forcing the corresponding energy level of that defect to move into or out of the bandgap modifying the host material’s electrical and optical properties as a result. With hydrogen being the most abundant element on earth, its presence in semiconductors is very difficult to avoid. Once inside, hydrogen can exist in three different charge states, H0, H+ and H- depending on the Fermi energy with respect to the hydrogen-related levels. Hydrogen is very reactive and very mobile and once inside the material, it will quickly tie up dangling bonds and form bound states with the existing defects. Because of its ability to change its charge state to act as a compensating defect in heavily doped materials, hydrogen can passivate both donor and acceptor dopants by tying up the extra charge carrier associated with the dopant atom. Experimental data on hydrogen-related complexes are abundant. Except for Si, Ge and GaAs, data on isolated atomic hydrogen on the other hand, are almost non-existent.
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Muonium (Mu = μ+e-), which is formed when positive muons are implanted into a material, provides a light ‘isotope’ of hydrogen that is far more accessible in its isolated form than hydrogen itself. Measurements of the time-evolution of the muon spin after it enters the sample will provide valuable information about the interaction with the host nuclei. Mu has a mass of ~1/9 that of H and a mean lifetime of 2.197 μs and is expected to form the same charge states and to occupy the same interstitial locations as hydrogen.
Since muonium and hydrogen have almost the same electron reduced mass (1/me + 1/mμ
= 0.995187 and 1/me + 1/mp = 0.999456), they have nearly identical electronic structures in vacuum. The hyperfine parameters are usually proportional to the magnetic moments, therefore; those parameters will be higher for muonium than hydrogen (the ratio of the magnetic moment of the muon to that of the proton is 3.183:1). A comparison between muons and protons is presented in table 1.1.
Table 1.1 Comparison in physical properties of muon and proton. Physical property Muon Proton 105.66 (MeV/c2) 938.28 (MeV/c2)
Mass = 0.1126 mp = 1 mp
= 206.7683 me = 1836.15 me Charge +e +e Spin ½ ½
Magnetic moment (μp) 3.183 1 Gyromagnetic ratio (MHz/ T) 135.54 42.58 Lifetime (μs) 2.19703 Stable
When trying to compare results on hydrogen and muonium, it is very important to keep in mind the similarities and differences between the two atoms. Let us first consider the fact that in muon spin research (μSR) experiments, only a few muons are implanted in the sample at any given time. Along with the short life time of the muons, this will allow us to neglect any muon-muon interactions and often the interactions with any other
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impurities in the sample. As a result, Mu almost always occurs as an isolated center making it easy to study Mu in its isolated state and thereby provides information about isolated hydrogen. Studies of hydrogen impurities and H-defect complexes are essentially always performed under thermodynamic equilibrium conditions. Because of the muon’s short lifetime on the other hand, the muonium states that can be studied are restricted to only those that are formed within few muon lifetimes (~10 μs). Those muonium centers may not have enough time to reach thermal equilibrium especially at low temperatures. As a result, multiple muonium states with the same charge can co-exist in the host material at the same time [2]. A second major difference comes from the fact that H and Mu have different masses; this will lead to different kinetics for physical processes such as diffusion and tunneling. With its lighter mass, the muonium energy levels in a given potential well are higher than those for hydrogen which will reduce the site-change barrier for muonium and may influence the stability of certain sites. In semiconductors with cubic tetrahedral structures, it is well established that
0 + 0 − there are four different types of muonium centers Mu BC , Mu BC , Mu T and Mu T [1,5]. This is the now commonly used notation to describe any muonium center with the superscript denoting the charge state and the subscript denoting the interstitial location. There are two types of interstitial sites, one is labeled bond-centered (BC) and the other is labeled tetrahedral (T). In the BC location, the muon resides at or near the center of the bond between two host atoms. In the T location, it resides within the largest open region of the diamond or zincblende lattice; the center of that cage has tetrahedral symmetry. In general, each one of these sites can support two separate charge states. In the early literature, they used Mu and Mu* to describe normal and anomalous paramagnetic centers that are now identified by their location as T and BC respectively. All of the materials we studied have hexagonal structures, this means more available sites for Mu and consequently, more complex systems.
0 The T-site neutral center Mu T is characterized by a large isotropic muon-electron
hyperfine interaction with HF parameter AHF roughly half that of the free muonium (4463.302 MHz). There is clear evidence that this center has high mobility at all
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temperatures, particularly in Si, Ge and GaAs and perhaps in other semiconductors [6-8].
0 The Mu BC hyperfine interaction is axially symmetric about a 〈111〉 bond direction and can be described by two parameters A|| and A⊥ which are roughly an order of magnitude
0 smaller than that of the Mu T center. From the available data on this center, it looks like it is stationary over the time scale of the experiment (~10 μs). I started my PhD research with the goal of studying the effects of hydrogen on Si- vacancy related defects in SiC. The techniques we were trying to use initially were electron spin resonance (ESR), electrically detected magnetic resonance (EDMR) and capacitavely detected magnetic resonance (CDMR). We never managed to get the signal to noise ratio to a level that allow us to separate resonance signals well enough to reliably identify the specific vacancy-related defect centers. Dr. Lichti was involved in muonium research, therefore we decided to use this technique to study the muonium equivalent of hydrogen in SiC and other wide-gap semiconductors. By the time I got involved in the mounium research in 2004, Dr. Lichti and co- workers had already collected some data on all three electrical types (except p-type 6H) of 4H- and 6H-SiC samples along with data for a single crystal AlN sample. My involvement started with analyzing the existing data and my goal became to characterize muonium centers in both SiC and AlN in terms of locations and motional dynamics to qualitatively model hydrogen in these materials. I was involved in all the measurements on the p-type 6H-SiC sample and follow up measurements on the other SiC samples to study several dynamical features identified for different spectroscopic signals. All results reported in this dissertation are from my own analyses of the data. Prior to the current project, there was a single report [9] of muon spin rotation (μSR) results for SiC published in 1986. In that report, the authors studied a 6H-SiC sample and confirmed the existence of three atomic-like Mu0 centers at low temperatures and only two around room temperature. There have been no previously published results using μSR on 4H-SiC or on AlN. Dr. Lichti and co-workers published their initial hyperfine spectroscopy results on SiC just before I got involved in this project [10]. There was also a conference report by Dr. Lichti on the results of a preliminary analysis of a
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subset of the AlN data [11]. In this report, the authors extracted hop rates and conversion rates for Mu0 in the temperature region above 300 K using slightly different fitting parameters than the ones used in this study. They also reported on the initial results from the level-crossing resonance study on this sample that provide supporting evidence for the final state of the Mu0 conversions resulting from the motion, as examined in this dissertation. In this dissertation we report the results of μSR measurements on Silicon Carbide (SiC) and Aluminum Nitride (AlN). The goal was to characterize the muonium centers (Mu = μ+e-) present in those two materials. All electrical types, including high resistivity as well as p-type and n-type samples, of the two most popular crystalline structures of SiC (4H-SiC and 6H-SiC) were studied. Two isotropic Mu0 states were found in 4H-SiC and a total of four states were seen in the 6H-SiC samples. Hyperfine constants for these centers were extracted from spin-precession measurements performed at a magnetic field
of 6 Tesla. The temperature dependences of the hyperfine constant (AHF) are well characterized by interaction with long-wavelength phonons having an effective Debye temperature of 800 K. All the data on the hyperfine interactions imply isotropic atomic- like states with no hint of any bond-centered Mu0 species in SiC. Temperature and field dependences of signal amplitudes and relaxation rates were studied. The results allow us reach tentative site assignments and extract dynamical characteristics for a few charge- state transitions of these muonium centers, although study of Mu dynamics in SiC remains far from complete. As for the AlN sample, the data were collected by Dr. R. Lichti and cooperators in 2003 before my involvement in this project. In this work we develop a consistent model for the data on single-crystal AlN to identify the locations of muonium centers and to assign the dynamical features to specific diffusion and conversion processes. An atomic– like neutral muonium center with a small anisotropy is inferred from hyperfine decoupling curves and longitudinal relaxation data. Comparison of simulated zero-time longitudinal asymmetry assuming a specific site geometry along with the experimentally observed curves lead us to conclude that the low-energy location of neutral muonium in
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AlN lies off-axis in the unblocked c-axis channels at sites close to Aluminum. As a result of fits of the experimental decoupling curves to an approximate repolarization function, we obtain hyperfine constants that where then averaged and then used in a simple model of the motional related relaxation rates to extract Mu0 hop rates. With the temperature dependence of the hop rate at hand, we then extracted several activation energies corresponding to different dynamical processes for Mu motion in AlN in different temperature regions.
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CHAPTER 2 FUNDAMENTALS OF μSR TECHNIQUES
This chapter serves as a brief introduction to the μSR techniques, detailed description can be found elsewhere [1,12,13]. μSR in its generic sense stands for muon spin research, the R in the acronym can refer to different things depending on the specific technique or application used. Some examples include the time-differential longitudinal field technique (LF-μSR), where in this case the R stands for relaxation, the time-integral radio (or microwave) frequency technique (RF-μSR), where the R stands for resonance and the time-differential transverse field technique (TF-μSR) in which the R stands for rotation. Longitudinal and transverse above refer to the direction of the applied field used with respect to the initial muon spin. First we concentrate on muons and their characteristics that make these kinds of experiments possible and then talk briefly about the various techniques.
2.1 Muons Muons were discovered in 1937 as a secondary radiation from the cosmic rays. Muons are naturally produced in the earth’s atmosphere by interaction of cosmic rays with gas molecules, but for research purposes they are produced using a particle accelerator. With a magnetic moment of 3.18 times larger than that of a proton, when implanted in any material, muons are a very sensitive magnetic probe. In μSR experiments, it is important to use muons with low energy that will stop in the sample under study, furthermore, the muon beam should be 100 % spin polarized so that all muons enter the sample with the same initial spin orientation. To produce such muons, pions are first produced from collisions of high-energy protons (> 500 MeV) with the nuclei of a light element such as carbon or brelium. The high-energy protons will react with neutrons and protons of the target producing pions as follows:
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π + ++→+ nppp
π + ++→+ nnnp (2.1)
π − ++→+ ppnp
All the work in this dissertation was accomplished using positive muons (μ+), therefore, we will concentrate on those alone. One product of the above reactions is the positive pion (π + ), this is an unstable spin zero particle that will survive for only 26 ns before it decays to a positive muon and a muon neutrino according to the following reaction:
+ + +→ νμπ μ (2.2)
Muons that are commonly used in μSR experiments are the so-called ‘surface muons’; these are produced from the decay of pions stopped near the surface of a primary production target. Surface muons have an energy of 4 MeV and when collected in a secondary beamline, are almost 100 % spin polarized. With the spin of the pions being
+ zero, the conservation of angular momentum requires that the μ and ν μ together are also in a state of zero angular momentum. The helicity of the neutrino produced by this parity-violating weak decay is negative (H = -1) meaning that the neutrino spin direction is antiparallel to its momentum; the emerging muons must then have negative helicity and have their spin direction antiparallel to their momentum, see figure 2.1. With the spin polarization maintained through the process of implanting the muons into the sample, the muonium state will be formed with a known direction for the muon’s spin and the detected signal will have maximum strength regardless of temperature or magnetic field values. This is a significant improvement over other techniques like nuclear magnetic resonance (NMR) and electron spin resonance (ESR) in which the spin polarization is small and depends strongly on temperature and field; therefore, achieving sufficient polarization will usually require operating at very low temperatures and strong magnetic fields.
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+ + νμ π μ
Pν Iν Iμ Pμ
Iπ = 0
Figure 2.1 Decay of pions into a muon and a muon neutrino. The symbols Iμ, Iπ and Iν represent the muon, pion and muon neutrino spins respectively and the symbols Pμ and Pν represent the muon and the muon neutrino momentum.
To ensure that a large number of the produced muons hit the sample, the highly polarized muons are guided down the beam-line by different kinds of magnets. Dipole magnets are used to bend the muon beam and reduce scatter, quadrupole magnets serve as a focusing tool for the beam so muons implantation into the sample is more effective. Contamination of the muon beam with other particles (like positrons) is hard to avoid. To eliminate or minimize contamination, the muon beam is forced through a set of crossed electric and magnetic fields that are both perpendicular to the beam. These fields serve as a velocity filter (therefore the name Wien filter) to eliminate contamination particles that have the same momentum but different mass or velocity than the muons such that, only particles with velocity v = E/B will pass through. Another use for the Wien filter is as a spin rotator: since the muon spin will precess in a magnetic field, it can be used to rotate the muon spin by up to 900 relative to the muon’s momentum. The last manipulation to the muon beam before it enters the sample is done through a set of collimators, which reduces the muon beam spot size and helps reduce any background signal. Upon entering the target sample, a positive muon has on average 2.197 μs before it decays into a positron, an electron neutrino and a muon anti-neutrino:
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++ e e ++→ ννμ μ (2.3)
Depending on the directions of travel of the two emitted neutrinos, the emitted positron’s energy will vary continuously from zero to a maximum of 53 MeV. If the two neutrinos are emitted in opposite directions, they will carry away all the kinetic energy available and the positron ends up with no energy. If on the other hand the neutrinos are emitted in the same direction opposite to the positron, the positron gets the maximum energy possible (53 MeV). Different aspects of the muon decay can be studied using the theory of weak interactions, one of which is the probability that a positron is emitted at an angle θ with respect to the muon spin direction and with an energy ε. This probability reflects the parity violation inherent in weak interactions and is given by [13]:
1 dW θε ),( a ⋅⋅⋅⋅+= ddn θεεθε )cos()()]cos()(1[ (2.4) τ μ
where ε is the positron energy scaled to the maximum possible energy the positron can
have from the muon decay (ε = E/Emax). The parameters a(ε) and n(ε) are defined as:
ε −⋅ 12 a ε )( = 23 ⋅− ε n 2 ⋅−= εεε )23()( (2.5) when all possible positron energies are sampled with equal probability (integrating equation 2.4 over the entire range of positron energies ε = 0 to ε = 1), the angular distribution of positrons from the muon decay will be given by:
1 θ )cos( W θ )( = 1[ + ] (2.6) 2 ⋅τ μ 3
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This angular distribution is plotted in figure 2.2. It is clear from the plot that the muon decay is not spatially isotropic and the positron is preferentially emitted along the direction of the muon spin. Keeping record of the direction of the positron emission will therefore help us determine the direction of the muon spin at the time of decay. It is this feature of positron emission along with the fact that the muon beam has a large natural spin polarization that make μSR experiments possible, and provides it with extreme sensitivity compared to other magnetic resonance techniques.
μ + I μ
Figure 2.2 Plot of the angular distribution of emitted positrons from the muon decay.
2.2 μSR Techniques There are many µSR techniques that are commonly used by researchers. General classification of these techniques is based on two main criteria: the time dependence of the data collected and the direction of the applied magnetic field with respect to the initial muon spin. There are two modes of detecting the muon decay after it enters the sample, one is called time differential (TD) where the time dependence of the muon polarization is recorded and the other is called time integral (TI) where the time evolution of the muon polarization is not observed and only the lifetime weighted change in polarization is monitored. These two different modes of detection, along with the ability to apply the magnetic field at different directions, give us a variety of techniques that can be used to extract all kinds of information about the muonium states inside the sample.
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To accommodate all the techniques, there are four major μSR facilities around the world that are categorized into two different categories according to their time structure: continuous wave (CW) facilities and pulsed beam facilities. In CW facilities (Paul Scherrer Institute (PSI) in Switzerland and TRIUMF in Canada), a nearly continuous source of protons is provided to produce muons that bombard the sample of interest at random times. Pulsed beam facilities on the other hand (The booster Muon (BOOM) facility at the high energy accelerator research organization (KEK) Meson Science Laboratory in Japan and ISIS at the Rutherford Appleton Laboratory (RAL) in the United Kingdom), provide intense pulses of muons such that a large number of muons (up to ~ 3000) enter the sample at approximately the same time. All the data used to produce this dissertation were collected at either TRIUMF or ISIS. The advantage of having a continuous beam of muons hitting the sample is the ability to achieve a very small time resolution (~100 ps), which allows for detection of rotation frequencies in larger magnetic fields and fast relaxing signals. If a TD experiment is performed in CW facility, only one muon can be allowed in the sample at a time to avoid any ambiguity in relating a decay positron to its parent muon. This requirement is very important since we have to record the time evolution of the muon polarization in such experiments, which is done through comparing the time of observation of the positron to the time of implantation of the parent muon into the sample. This puts a strict limit on the rate that muons can be implanted into the sample, which is reduced even farther by the electronics that work to reject events involving multiple muons or positrons. The other option we have is to use the TI mode to collect data. In this mode, the raw data are simply the positron rates along one direction normalized to the beam rate. We use this mode when it is not necessary to determine the time dependence of the muon polarization. The muon rate is not a restriction in this mode, therefore; we can use the maximum muon beam intensity available and with that comes the advantage of increased sensitivity in some types of experiments. Time-integral measurements are often used in
12 Texas Tech University, Hisham Bani-Salameh, May 2007 longitudinal field geometry if the goal is to get asymmetries or polarizations as a function of field or if one uses rf fields to derive the resonance, as in NMR. At a pulsed beam facility, the requirement for μSR experiments is that the time width of the muon pulse must be much shorter than the muon lifetime and any time dependence one wishes to measure while the time spacing between any two consecutive pulses must be much longer than the muon lifetime. This puts a limit on the time resolution of the experiment since it is governed by the beam pulse width (on the order of 10 ns). The advantage of these facilities is the ability to use all the muons in a given pulse and there is almost no background noise in the μSR signal that might arise from particles other than the muons in the muon beam.
L B F Iμ Pμ
sample μ+ TM R
Figure 2.3 Schematic of the position of the four positron counters (top view) used in different μSR setups to detect the emitted positrons from the decay of the muons.
To measure the muon spin polarization along one axis, we have to place two positron counters along two directions 180o from each other. Depending on the type of measurement, the number of positron counters used can be either two or four; the position of these counters is shown in figure 2.3. The labels on these counters are from the viewpoint of the muon beam just before it hits the sample, therefore; the labels are:
13 Texas Tech University, Hisham Bani-Salameh, May 2007 right (R), left (L), front (F) and back (B). In some occasions, two more counters (up and down) might also be used. Those counters are basically plastic scintillators, they will give off a flash of light when an ionizing particle passes through them. The photons are collected in a photomultiplier tube. After proper amplification, the signal is then carried by coaxial cables in a form of voltage pulses to electronics outside the experimental area for further processing. The thin muon detector (labeled TM in figure 2.3) has to be thin (~250 microns) in order to reduce scattering of the muon beam so it won’t miss the sample and the background signal will be minimum; also so it does not measure e+ from decay events. For the commonly used magnetic field geometries (longitudinal or transverse), a magnetic field can be applied parallel (longitudinal) or perpendicular (transverse) to the initial muon spin direction. In the case of TF measurement, four counters are used and they all contain precession information. Instead of changing the direction of the applied field, the muon spin can be rotated by 90o by mean of the Wein filter prior to implantation to meet the transverse field setup requirements; in this case, one uses the up and down plus the left and right counters while the front and back pair is not effective. When zero-field (ZF) experiments are needed, zero net magnetic field at the sample is ensured by mean of trim coils that are used to cancel out any stray fields. In both LF and ZF experiments, only those counters along the initial muon spin direction (B, F) are actually used. Longitudinal-field repolarization was the first technique used to detect the formation of muonium in solid materials. The idea is to apply a longitudinal magnetic field to decouple the muon from the associated electron and therefore quench the hyperfine interaction (hence the alternative names of “hyperfine decoupling curves” or “longitudinal field quenching”). Although this is a powerful technique and has an advantage over the spin precession technique in determining the hyperfine parameters for few special cases, time-differential precession techniques provide much more accurate hyperfine parameters. Measuring the muon spin rotation in a transverse field also allow the separation of different muonium states making it easier to determine their parameters.
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Accuracy in determining the hyperfine parameters gained with the transverse field technique comes with a price, if we are studying an anisotropic muonium state, it becomes important to use a large single-crystal sample as the precession frequencies in the transverse field will depend on the orientation of the sample with the applied field. If the sample at hand is only available in polycrystalline form (powder), or it is an amorphous sample, the precession signal will get too broad to be detected. In situations like these, one can clearly see the advantage of the LFQ technique, as it will still be possible to detect such signals.
2.3 Data Collection If data are collected in TD mode, time between implantation of the muon and its decay is used to generate a time histogram of decay events in each counter. Typically, histograms containing 5 to 20 million muon decay event are used to obtain a single data point. The number of events recorded in one particular positron counter (labeled as i in the following equation) has the form:
−t /τ μ i = i ii )](1[)0()( ++ btPAeNtN i (2.7)
where Ni(0) is a normalization factor, τμ is the muon lifetime, Ai is a constant that carries information about the muon decay and the solid angle of the counter, Pi(t) is the projection of the muon spin polarization on the symmetry axis of the positron counter
(the ith one in this case) and bi is a time-independent background noise. To isolate the signal of interest (the muon polarization), raw count rates from two opposing counters are combined to define the so-called “asymmetry” as follows:
−α tNtN )()( tA )( = i −i (2.8) i + α −i tNtN )()(
15 Texas Tech University, Hisham Bani-Salameh, May 2007
20
15
10
5
0
-5
-10
Asymmetry (a.u.) a) -15
-20 02468101214 4
3 b)
2
1
0
-1
-2 asymmetry (a.u.) -3
-4 012345678 Time (μs)
Figure 2.4 Time-dependent asymmetry for a) high-resistivity 4H-SiC sample with data taken at a temperature of 973 K and with a field of 80 gauss applied perpendicular to the initial muon spin direction (TF), b) AlN sample with data taken at 523 K and with a field of 350 gauss applied parallel to the initial muon spin direction (LF). The solid line represents the best fit to the data.
16 Texas Tech University, Hisham Bani-Salameh, May 2007
Ni(t) and N-i(t) are the count in the two counters labeled as i and –i respectively with any 0 background removed. The two counters have a 180 phase difference and therefore Pi(t) must equal – P-i(t). α is used to account for differences in the solid angle and efficiencies between the two counters and can be fit from the raw data for an oscillatory signal. Asymmetries of equation 2.8 have muon exponential decay evident only in the error bars of the counts, these are proportional to the square root of the number of counts and grow exponentially with time. This asymmetry is used to yield information about the muonium states and their behavior. Examples of this asymmetry from TF and LF experiments are shown in figure 2.4.
17 Texas Tech University, Hisham Bani-Salameh, May 2007
CHAPTER 3 THEORETICAL DESCRIPTION
The time evolution of the muon spin polarization is calculated from the relevant quantum Hamiltonian for the environment under study. All the systems considered are assumed to have only Zeeman and hyperfine interactions and any other type of interaction is neglected. In the muonium atom, the magnetic moments of the muon and the muonium electron interact with each other via the hyperfine interaction and they both interact with the applied magnetic field via the Zeeman interaction. Because these interactions are non-dissipative, the resulting time-dependent muon polarization will be a sum of un-damped oscillating components. The muonium atom can also experience some dissipative interactions arising from the motion of the muonium atom, in other words from fluctuations of the local fields seen by the muonium atom. This kind of interaction usually results in a loss of polarization, the effects of which is commonly referred to as muonium dynamics. Without any dissipative interactions, the time-dependent muon spin polarization is governed by the following Hamiltonian [1,12,13]:
μμμ −⋅−⋅⋅= μee ⋅ SBgIBgSAIH (3.1) where S and I are the electron and muon spins, A is the hyperfine tensor and B is the applied field. The term ⋅ ⋅ SAI deals with both isotropic and anisotropic hyperfine
0 interactions. One of the muonium states in cubic semiconductors is MuT where the muonium is sitting at a high symmetry interstitial site. The hyperfine interaction between the muon and the electron in this case is isotropic, described by a single hyperfine constant, and the matrix representing the hyperfine interaction has a single constant on
0 the diagonal and zeros as off-diagonal elements. If we are dealing with MuBC , then the interaction is anisotropic and we have to use more than one hyperfine constant to describe it. The goal here is to use the Hamiltonian to calculate the time evolution of the muon
18 Texas Tech University, Hisham Bani-Salameh, May 2007 spin polarization. The procedure we followed is to build a general Hamiltonian matrix that can apply for any special case whether it is isotropic or anisotropic and then solve it for eigenvalues and eigenvectors. Once the eigenvalues and eigenvectors are available, we can determine the evolution of the polarization. In building the Hamiltonian matrix [14-16], we deal with two different coordinate systems, one oriented along the applied magnetic field that defines the spin quantization, and one in which the hyperfine interaction tensor is diagonal given by:
⎡Ax 00 ⎤ A = ⎢ A 00 ⎥ (3.2) diagonal ⎢ y ⎥ ⎣⎢ 00 Az ⎦⎥
We use Euler angels to build a matrix to transform the hyperfine tensor to the coordinate system of the applied field. The transformation matrix we used is:
⎡ x y coscoscos ααα z ⎤ ⎢ ⎥ T γβα ),,( = ⎢ x y coscoscos βββ z ⎥ ⎢ ⎥ ⎣ x y coscoscos γγγ z ⎦
⎡ − + sincoscoscossinsinsincoscoscos − cossin γβγαγβαγαγβα ⎤ ⎢ ⎥ = ⎢− − − + sinsincoscossincossincossinsincoscos γβγαγβαγαγβα ⎥ (3.3) ⎣⎢ sincos βα sinsin βα cos β ⎦⎥ where α, β and γ are the Euler angles of the three rotations to transform into the applied magnetic field coordinate system. With this transformation defined, we can write the Hamiltonian of equation 3.1 in the applied field coordinate system. To do so we have to find the components of the hyperfine tensor in the field coordinate system starting from the system where the tensor is diagonal. The way to do this is:
19 Texas Tech University, Hisham Bani-Salameh, May 2007
T new = TA γβα ),,( diagonal ⋅⋅ TA γβα ),,( (3.4)
T where Anew is the hyperfine tensor in the applied field coordinate system and T γβα ),,( is the transpose of T α β γ ),,( . With the applied field assumed to be along the z direction, we found it easier to write the Hamiltonian in terms of the raising and lowering electron and muon spin operators ,,( ISS +−+ and I − ) . With B = Bzˆ , the Hamiltonian becomes:
−⋅⋅= μ μμ − μ BSgBIgSAIH zeez (3.5) and the term ⋅⋅ SAI is expanded and written in terms of the raising and lowering operators as follows. We define the hyperfine tensor terms in the transformed matrix by:
⎡ AAA xzxyxx ⎤ ⎢ ⎥ Anew = ⎢ AAA yzyyyx ⎥ (3.6) ⎢ ⎥ ⎣ AAA zzzyzx ⎦ the term ⋅⋅ SAI is then given by:
⎡ AAA xzxyxx ⎤ ⎡S x ⎤ ⎢ ⎥ =⋅⋅ IIISAI ⋅ AAA ⋅ ⎢S ⎥ (3.7) []zyx ⎢ yzyyyx ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ AAA zzzyzx ⎦ ⎣S z ⎦
++= + + + + + + SIASIASIASIASIASIASIASIASIA zxxzxzzxyzzyzyyzxyyxyxxyzzzzyyyyxxxx then we apply the relations
20 Texas Tech University, Hisham Bani-Salameh, May 2007
1 +⋅= SSS )( x 2 −+ 1 S −⋅= SS )( y 2i −+ 1 +⋅= III )( (3.8) x 2 −+ 1 I −⋅= II )( y 2i −+ to write ⋅⋅ SAI in terms of the raising and lowering operators. After tedious but straightforward calculations, we apply this to the basis set (|| i〉 = || μ mm e 〉 , i = 1,2,3 and 4) and finally obtain the Hamiltonian matrix:
⎡ ω− + zz zx − yz zx − yz yyxx −− 2iAAAiAAiAAA xy ⎤ ⎢ + iAA ω −− + AAA +− iAA ⎥ H = ⎢ zx yz + zz yyxx zx yz ⎥ (3.9) ⎢ zx + iAA yz + AA yyxx ω+ − Azz zx +− iAA yz ⎥ ⎢ ⎥ ⎣⎢ yyxx 2 xy zx yz zx −−−−+− iAAiAAiAAA yz ω− +− Azz ⎦⎥
where ω± is defined by ω± = ωe ± ωμ /)( 2 with ωe and ωμ being the Larmor precession angular frequencies of the electron and the muon respectively. The basis set used above consists of eigenstates for both the muon and the electron, denoted by the quantum numbers mμ for the muon and me for the electron. These can either be a spin-up eigenstate (+) or a spin-down eigenstate (–) giving a total of four different possible combinations as a basis set: 1|| =〉 || +〉+ , 2|| 〉 = || + −〉 , 3|| 〉 = || − +〉 and 4|| 〉 = || −〉− . To make the definition of the Hamiltonian matrix in equation 3.9 complete, we should list the values of the individual components in the hyperfine matrix in terms of the principal values and Euler angles:
21 Texas Tech University, Hisham Bani-Salameh, May 2007
1 ⋅= AA cos[ 2 α + A cos 2 β + A 2 γ ]cos xx 4 x yx zx x 1 ⋅= AA cos[ 2 α + A cos 2 β + A 2 γ ]cos yy 4 x yy zy y 1 ⋅= AA cos[ 2 α + A cos 2 β + A 2 γ ]cos (3.10) zz 4 x yz zz z 1 ⋅== AAA coscos[ αα + A coscos ββ + A γγ ]coscos xy yx 4 x x yy x yxzy 1 ⋅== AAA coscos[ αα + A coscos ββ + A γγ ]coscos zyyz 4 x y yz y zyzz 1 ⋅== AAA coscos[ αα + A coscos ββ + A γγ ]coscos zxxz 4 x yzx zxzzx
With the Hamiltonian matrix at hand, we get the eigenvalues by solving the secular equation | − hλIH | for λ , where I is a × 44 unit matrix. The eigenvectors can be expressed as a superposition of eigenstates of the basis set defined above, one example is:
Ui i1 ||| Ui2 || +−〉+++〉+=〉 Ui3 || − +〉 +Ui4 || − −〉 (3.11) where i = 1,2,3 and 4 corresponding to the four eigenvalues, and the U components represent normalization cofactors. With four of these equations, one can construct the
4 matrix: = ∑UU , ji where i and j represent specific eigenstates. Building this matrix is , ji one of the most important steps in determining the evolution of the spin polarization. Before we go any further, it should be pointed out that it is possible to get analytical expressions for the eigenvalues and eigenvectors using this method. But because of the complexities involved, this is done for only a few special cases. Some of these cases are (1) isotropic muonium, (2) anisotropic muonium with axial symmetry in zero field (3) fully anisotropic muonium in zero field (4) anisotropic muonium with one of its principal axes in the field direction (5) muonium in high field. In the following
22 Texas Tech University, Hisham Bani-Salameh, May 2007 section, as an example, we discuss the case of isotropic muonium in more details to get the analytical solution for the eigenvalues and eigenvectors.
3.1 Isotropic Muonium In the case of isotropic muonium, we need only one hyperfine constant to describe the hyperfine interaction between the muon and the electron, therefore; Ax ≡ A y ≡ Az ≡
AHF. As a result, the Hamiltonian will simplify to the following:
⎡ω− + AHF 04/ 0 0 ⎤ ⎢ 0 ω −− A A 2/4/ 0 ⎥ H = ⎢ + HF HF ⎥ (3.12) ⎢ 0 AHF 2/ ω+ − AHF 04/ ⎥ ⎢ ⎥ ⎣ 0 0 0 ω− +− AHF 4/ ⎦ upon solving this Hamiltonian, we get the following eigenvalues:
ω0 ω0 ω+ E1 / += ω− E2 / h +−= h 4 2cos4 ξ ω ω ω E / 0 −= ω E / 0 −−= + (3.13) 3 h 4 − 4 h 2cos4 ξ and the following eigenvectors:
1| =〉 || +〉+ 2| 〉 = + −〉 + cs |||| − +〉 3| =〉 || −〉− 4| 〉 = + −〉 − sc |||| − +〉 (3.14) where c and s are defined as:
x x c 2 1( += 2/) s 2 1( −= 2/) (3.15) x 2 +1 x 2 +1
23 Texas Tech University, Hisham Bani-Salameh, May 2007
with x being defined as = Bx γ μ + γ e /)( ω0 where ω0 is defined in terms of the muon hyperfine constant as ω0 = 2πAHF . And finally the angle ξ in the eigenvalues is defined as cot 2ξ = x .
4000 0 α = β = γ = 0 3000 A = 2006.3 MHz HF E 2000 1
1000 E 2 0
E 3 -1000
(MHz) Energy E 4 -2000
-3000
-4000 0 500 1000 1500 2000 Magnetic Field (Gauss)
0 Figure 3.1: Breit-Rabi diagram showing the isotropic MuT muonium state energy levels in Si. These are labeled in descending order in energy at each field.
With these field-dependent eigenvalues, we can construct the so-called ‘Breit- Rabi diagram’ showing the muonium state energy levels. An example of this diagram is
0 shown in figure 3.1 for the neutral muonium center at a tetrahedral site ( MuT ) in Si with an isotropic hyperfine interaction constant of 2006.3 MHz.
3.2 Time Evolution of the Muoniun Spin Polarization To get the time evolution of the muonium spin polarization, we first have to consider the initial state for the particles involved, the muon and the electron. The initial direction of the muon spin can be precisely specified in 3-dimentions by the use of a unit
24 Texas Tech University, Hisham Bani-Salameh, May 2007
vector uˆ whose components are the direction cosines α0 β0 cos,cos,(cos γ 0 ) . We shouldn’t confuse the angles α0, β0 and γ0 with Euler angles defined above which specify the orientation of the muonium hyperfine tensor. For an arbitrary muon spin direction specified by uˆ , the muon spin state can be written as:
γ γ φ cos 0 | ++〉=⋅= em iδ0 sin 0 | m −〉=⋅ (3.16) μ 2 μ 2 μ
where δ 0 = α0 sin/coscos γ 0 and δ 0 = β0 sin/cossin γ 0 . The same thing can be done to describe the electron spin state. When muonium is formed, we know that we have 100% spin polarized muons, but the electrons involved are not spin polarized. This leads to two possible Mu spin states, one has both the muon spin and the electron spin parallel (the unit vectors describing their directions are the same Mup) and the other has them anti- parallel; Mua. Attempting to get the time evolution of the muon polarization, we assume unpolarized electrons leading to equal probabilities of forming either of the two states mentioned above. For a muonium center that has both the muon and the electron pointing in the same direction, we use equation 3.16 to write its initial state as:
γ γ γ γ φ [cos)0( 0 | ++〉=⋅= em iδ0 sin 0 | m ][cos 0 | ++〉=⋅−〉=⋅ em iδ0 sin 0 m −〉=⋅ ]| p 2 μ 2 μ 2 e 2 e (3.17) Equation 3.11 can be written in a more compact form as:
4 | ∑ ij || jUi 〉=〉 (3.18) j=1 and the reverse relation as:
4 * || ∑ ij | iUj 〉=〉 (3.19) j=1
25 Texas Tech University, Hisham Bani-Salameh, May 2007
where the star denotes the complex conjugate. We then reach the following simplified expression:
φp = C1 1|)0( 2 3 〉+〉+〉 + CCC 4 4|3|2| 〉 (3.20)
where the coefficients Ck are given by:
γ 1 δ = UC cos2* 0 ++ ** eUU iδ0 sin)( γ + * eU 2iδ0 sin2 0 (3.21) kk 1 22 kk 32 k 40 2
Recall that the matrix U is readily available upon diagonalizing the Hamiltonian matrix. As an example, for the isotropic muonium case, the U matrix becomes:
⎡ 0001 ⎤ ⎢0 cs 0⎥ U = ⎢ ⎥ (3.22) ⎢ 1000 ⎥ ⎢ ⎥ ⎣0 − sc 0⎦ which defines the initial muonium state. Since the state | i〉 as defined above is an eigenstate of the Hamiltonian with eigenvalue hωi , the time-dependent Schrodinger equation gives the spin state at a later time t to be:
− ω1ti − ω2ti − ω3ti − ω4ti φp 1 1|)( +〉= 2 2| 3 3| +〉+〉 eCeCeCet C4 4| 〉 (3.23)
Since we are after the spin polarization, we need to calculate the expectation value of the
μ muon spin operator σ . If the muonium atom is in the state φp (t) , then the expectation value of σ μ is:
26 Texas Tech University, Hisham Bani-Salameh, May 2007
μ p 〈= p ttP φσφ p (t)||)()( 〉
ω jk ti * μ = ∑ C kj σ || kiCe 〉〈 (3.24) jk
where ω ω −= ωkjjk is the energy difference between the states | j〉 and | k〉 . The
μ μ μ μ operator σ can be σ z , σ + or σ − , in each case it is defined in terms of the cofactor matrix U as [17]:
μ * * * * 〈 σ z || =〉 1 U jkj 21 jk 32 U −−+ 43 UUUUUUkj kjk 4
μ * * σ + =〉〈 (2|| Ukj j1 + 23 UUU kjk 4 ) (3.25)
μ * * 〈 σ − =〉 (2|| 3 + 41 UUUUkj kjkj 2 )
We can follow the same procedure to get the polarization from the other possible muonium state φa (t) , the polarization in that case will be:
ω jk ti * μ a )( = ∑ kj σ || kiDDetP 〉〈 (3.26) jk
where Dk is:
γγ γ γ D ( * −= * eUU 2iδ0 cos) 0 sin 0 −⋅ U cos( 2* 0 −U sin 2* 0 )eiδ0 (3.27) 1 kkk 4 2 2 k 2 2 k3 2
At this stage, we can make the assumption that both muonium states are produced with same probability, therefore; the average polarization will be:
27 Texas Tech University, Hisham Bani-Salameh, May 2007
1 μ )( += tPtPtP )]()([ 2 p a
ω jkti 1 * * μ ∑ [ kjkj σ ||] kiDDCCe 〉〈⋅+⋅= (3.28) jk 2 the middle term represents the initial state given by:
1 1 γ 1 γ [ * * ] =+ ( * + UUUUDDCC 2* (cos) 0 ) + ( * + UUUU 2* (sin) 0 ) 2 kjkj 2 jkjk 2211 2 2 33 jkjk 44 2 1 1 + ( * + UUUU * ) (cos + i βα )cos 2 31 jkjk 42 2 0 0 1 1 + ( * + UUUU * ) (cos − i βα )cos 2 13 jkjk 24 2 0 0
〈= k φ α β γ 000 |),,(| j〉 (3.29) therefore; our final time-dependent muon polarization is:
ω jkti μ )( ∑ 〈⋅= ketP γβαφ 000 σ |||),,(| kjj 〉〈⋅〉 (3.30) ,kj
This polarization function consists of two parts, the amplitude, which is completely specified by the matrix U, and the time dependent term, which is in the exponential. With all of the above components included in our program, we use it to simulate repolarization curves for all kind of muonium centers without any approximations. With knowledge of the hyperfine constant and the direction of the applied field, one can simulate the zero- time amplitudes as a function of field for any isotropic center. If the center has axial anisotropy, knowledge of the dipolar contribution is also needed. The program we developed is in the most general form and applies for any muonium center whether it is isotropic or anisotropic, and regardless of the initial orientation of the muon spin or the magnetic field with respect to the hyperfine tensor
28 Texas Tech University, Hisham Bani-Salameh, May 2007 axes. Two special cases usually get the most attention, transverse and longitudinal field named this way to indicate the magnetic field direction relative to the initial muon spin direction. With the field along zˆ , for transverse field we define the xˆ along the initial o o o spin direction, thus we have α ,β00 and γ 0 equal to 0 , 90 and 90 respectively, and in the longitudinal field case these angles will be 90o,90o and 0o. This will simplify the calculations significantly and ma ke the analytical solutions for some special cases less complicated. In the following, we go into more details to find the polarization function for the two common cases of transverse and longitudinal field using this method.
3.2.1 Longitudinal Field Case In this section, we’ll use the method explained above to calculate the polarization function for the case of isotropic muonium in longitudinal field experiments. For
0 0 longitudinal field experiments we have βα 00 == 90 and γ 0 = 0 , therefore:
1 〈k γβαφ |),,(| =〉 [ * + UUUUj * ] (3.31) 000 2 11 jkj 2 k 2
μ with σ z || kj 〉〈 given by equation 3.25, the polarization function is:
ω jkti 1 * * * * * * )( ∑ [ kjkj [] 3322112211 −−+⋅+⋅= jkjkjkj 4UUUUUUUUUUUUetP k 4 ] (3.32) ,kj 2 each piece contains only elements from the U matrix given in equation 3.22 for the isotropic case. For our isotropic muonium center, only six terms survive; four have zero frequency and two have non-zero frequency.
29 Texas Tech University, Hisham Bani-Salameh, May 2007
1.0 0 α = β = γ = 0 A = 2006.3 MHz HF 0.9
0.8
0.7
0.6 Polarization
0.5
0.4
0.3 100 1000 10000 Magnetic Field (Gauss)
0 Figure 3.2: Field-dependent polarization of the isotropic MuT muonium center in Si showing the recovery of the muon polarization from 50% at low fields to a full 100%
centered around the field B0B .
The zero-frequency terms represent the static (non-oscillatory) component of the polarization that is actually observed in a longitudinal field experiment. The non-zero frequency terms on the other hand are the oscillatory components that are usually unresolved unless the hyperfine constant is very small. With the use of equation 3.22 and with straightforward manipulations, equation 3.32 becomes:
1 1 1 )( 222 )( +−+−+= 22222 ()( ω24 + eecsscccsstP − ω24titi ) (3.33) 2 2 2
30 Texas Tech University, Hisham Bani-Salameh, May 2007
x returning to the definitions of c and s in equation 3.15, we find sc 22 =− , this x 2 + 1 leads us to the following final simple polarization function:
+ 21 x 2 cosω t tP )( = + 24 (3.34) x 2 + )1(2 x 2 + )1(2
where x is a normalized field defined earlier as x = B/B0 where 0 AB γ eHF += γ μ )/( with
AHF being the muonium hyperfine constant in MHz; γ e = π × .280242 6 MHz/T and
γ e π ×= 54.1352 MHz/T are the gyromagnetic ratios of the electron and the muon respectively. The muon polarization goes from 50 % at low fields to a full 100 %
centered around the field B0B as shown in figure 3.2 above.
3.2.2 Transverse Field Case The main difference between longitudinal and transverse field experiments is that
0 in the latter case, all terms are oscillatory. In transverse field, we have α0 = 0 and
0 γβ 00 == 90 , therefore:
1 〈k γβαφ |),,(| [( +=〉 * * +++ )(())( * + UUUUUUUUj * )] (3.35) 000 4 131 kkjj 3 242 kkjj 4 and the polarization is then given by:
jktiw 1 * * * * )( ∑ [( +⋅= 131 kkjj 3 +++ )(())( 242 + UUUUUUUUetP kkjj 4 )] ,kj 4
* * * * [ 1 21 32 −−+⋅ 43 UUUUUUUU kjkjkjkj 4 ] (3.36)
31 Texas Tech University, Hisham Bani-Salameh, May 2007 in this case, there are more surviving terms and the calculation is more tedious but still straightforward. Using eqs. 3.15 and 3.22, we reach the following:
1 )( = cos[ ω + cosω + cosω + ω tbtatbtatP ]cos (3.37) 2 12 23 34 14
with ++= xxa 2 )12/2/1( and +−= xxb 2 )12/2/1( . As clearly seen from this polarization function, it contains no static terms, the polarization will have only oscillating components representing the precession of the muon spin polarization about the applied field which occurs at certain frequencies. The TF- μ SR technique is one of the earliest and most commonly used techniques in this field, it serves as a quick check of existing muonium centers and their characteristics. With these precession patterns being dependent on the hyperfine interactions of the muon, they provide a signature of the muonium center present under the specific experimental conditions.
3.3 Anisotropic Muonium If we now consider the case where there is some anisotropy in the hyperfine interaction, the Hamiltonian of equation 3.1 still applies and can be used to determine the polarization function. The difference is that the hyperfine coupling tensor is no longer isotropic and at least two values (like A|| and A⊥ ) are needed to describe it. The origin of axial anisotropy is assumed to be dipolar coupling between the muon spin and the electron spin. In the following example, we consider the situation where transferred nuclear hyperfine interactions can be neglected because either neighboring nuclear spins are non existent in the muon environment; or if they do exist, their direct and indirect couplings to the muon are much smaller than the anisotropic part of the electron-muon hyperfine coupling. As mentioned earlier, studying the evolution of the polarization is very complicated in such systems and an analytical solution is almost impossible. The problem can be significantly simplified when the applied magnetic field is along a principal axis of the hyperfine tensor.
32 Texas Tech University, Hisham Bani-Salameh, May 2007
If the field is applied along the dipolar axis (parallel to Azz, the largest component of the hyperfine tensor A), Euler’s angles of this center for this particular case are all zeros. This will simplify the calculations to a point where it is possible to get an analytical expression for the spin polarization. The Hamiltonian matrix of equation 3.9 becomes [14]:
⎡ω− + Azz 0 0 − AA yyxx ⎤ ⎢ 0 ω −− + AAA 0 ⎥ H = ⎢ + yyxxzz ⎥ (3.38) ⎢ 0 + AA yyxx ω+ − Azz 0 ⎥ ⎢ ⎥ ⎣⎢ − AA yyxx 0 0 ω− +− Azz ⎦⎥
Assuming the magnetic filed is applied along the initial muon polarization (longitudinal field) so that both are along the z-axis, one can diagonalize equation 3.38 for eigenstates and eigenvalues to get the following polarization function:
x 2 y 2 cosω t cosω t tP )( = + − 23 − 14 (3.39) x 2 + y 2 + )1(2)1(2 x 2 + )1(2 y 2 + )1(2 with x and y being normalized fields defined as:
B B γ + γ )( x == e μ B0 + AA yyxx 2/)(
B B γ − γ )( y == e μ (3.40) B1 − AA yyxx 2/)(
Axx and Ayy are the two perpendicular components of the hyperfine tensor to the applied field with Azz being parallel to it. The obvious difference in this polarization function from that for the isotropic muonium center is the two-step repolarization, the first step
occurs at the field value B1B that depends on the anisotropy of the hyperfine interaction
33 Texas Tech University, Hisham Bani-Salameh, May 2007
perpendicular to the field and the second one occurs at B0B , which is proportional to the average hyperfine coupling perpendicular to the applied field. With the source of the anisotropy being the dipolar coupling, we now turn to the special case where the dipolar contribution is axially symmetric and can be described by a single parameter D. The hyperfine tensor is expressed as the sum of an isotropic part Ai and a traceless dipolar part
D. By further assuming that the principal axis of A along the field direction is Azz and is actually a symmetry axis (along the “bond” direction), then the hyperfine tensor becomes:
⎡Axx 00 ⎤ ⎡ i − DA 002/ ⎤ A = ⎢ 0 A 0 ⎥ = ⎢ 0 − DA 02/ ⎥ (3.41) ⎢ yy ⎥ ⎢ i ⎥ ⎣⎢ 00 Azz ⎦⎥ ⎣⎢ 0 0 i + DA ⎦⎥
where the relationships || = i + DAA and ⊥ = i − DAA / 2 were used in equation 3.41. in this notation, the subscripts on A (|| and ⊥ ) are with respect to the “bond” direction not to the applied field. In this special case, the polarization function of equation 3.39 becomes:
+ 21 x 2 cosω t tP )( = − 23 (3.42) x 2 + )1(2 x 2 + )1(2 with the normalized field x given by:
B B γ + γ )( x == e μ (3.43) B0 i − DA 2/)2( this polarization function shows only one step (much like the isotropic case (equation
3.34)) at a field that depends on both Ai and D.
34 Texas Tech University, Hisham Bani-Salameh, May 2007
If D is oriented along the x-axis perpendicular to the applied field (z-direction), the polarization function will still be given by the general form of equation 3.39 but with x and y defined as: B B γ + γ )( x == e μ B0 i + DA 2/)2/2(
B B γ − γ )( y == e μ (3.44) B1 D 2/)2/3( again showing two steps of repolarization at two different characteristic fields needed to recover the lost polarization.
3.4 Another Method to Calculate the Polarization Another method to calculate the polarization function is discussed in references [4, 9, 10], this method uses the density matrix approach where:
σ μ =〉〈= tTrttP σρ μ ])([)()( (3.45) in this expression, ρ is the density matrix and σ μ is the muon spin operator. Initially, the muons are almost 100 % polarized and the electrons involved in forming the muonium atoms are unpolarized, therefore:
1 ρ )0( += σ μ ]1[ (3.46) k where k represents the dimensionality of the Hamiltonian or the number of the energy levels available for the muonium atom. Now two representations can be used to get the time evolution of the muonium polarization, the Heisenberg and the Schrodinger representations. In the Heisemberg picture, this is done through the calculation of the time evolution of the spin operator where:
35 Texas Tech University, Hisham Bani-Salameh, May 2007
μ = TrtP σρ μ t)]()0([)( (3.47) in the Schrodinger picture on the other hand, one usually uses the time evolution of the density matrix equation 3.45. Following Schrodinger representation, one finds the time evolution as follows:
ρ = iHt / h ρ )0()( eet −iHt / h (3.48) upon the assumption of 100 % polarized muons and considering only the component of the polarization that is along the initial muon spin direction, the final polarization along the rth direction can be calculated from the following:
2 μ 2 r tP 1)( ∑ σ r ji −⋅〉〈−= ωij t)]cos(1[|||| (3.49) K , >iji using equation 3.49 to calculate the time evolution of the polarization for different cases will yield exactly the same results as the method we discussed earlier.
3.5 Muonium Dynamics As mentioned earlier, paramagnetic muonium states have been detected in semiconductors at low temperatures. As the temperature goes up, dynamical changes take place and cause the precession signal from such centers to weaken and sometimes disappear. The environment that hosts the muonium state causes most of these changes by some kind of perturbation that affects the evolution of the spin polarization. Fluctuation of the local magnetic field and modulation of the hyperfine interaction of the muonium are two forms of this perturbation. Fluctuations in the local magnetic field sensed by the muons, can arise from motion of the muonium atom. Diffusion of muonium among inequivalent sites, trapping of diffusing muonium at a crystal defect as well as
36 Texas Tech University, Hisham Bani-Salameh, May 2007 repeated change of the charge state of the muonium will modulate the hyperfine interaction and modify the spin polarization. One of the dynamical changes that is well documented and well studied is muonium ionization, in this process, muonium losses its electron and therefore changes its charge state from paramagnetic to diamagnetic. The quantity of interest in μSR experiments is the time evolution of the muon spin polarization; this is greatly influenced by random perturbations of the surrounding environment [4]. Such perturbations are present in the case of a mobile isotropic Mu0 center in the form of fluctuations in the transferred hyperfine interaction. Fluctuations in the local magnetic fields originating from the randomly oriented nuclear spins of the host atoms are the source of such perturbations. With its magnetic moment being much larger, the muonium electron is much more sensitive to those fields than the muon. The electron spin will relax (or flip) due to this random variation in nuclear hyperfine fields causing the muon spin to relax via the hyperfine interaction. Such effects of Mu0 motion are most reliably observed in the relaxation rates of the muon spin polarization under longitudinal field conditions where the magnetic field is applied parallel to the initial muon spin direction. If an isotropic muonium center is stationary under longitudinal field conditions, the time-evolution of the muon spin polarization is given by equation 3.34. It consists of two components; one oscillating and unobservable in standard μSR experiments and averages to zero because its frequency is above instrument resolution, and the other is the static component that displays the hyperfine decoupling features and any relaxation related to dynamics. In equation 3.34, the oscillating component represents transitions
2 between the net M = 0 states with a frequency ω24 defined as 24 = πω HF 12 + xA . By fitting the observed polarization to the first term of equation 3.34, one can extract the hyperfine constant AHF. In the case of weakly anisotropic muonium center, the decoupling curve shows a second step at low fields, this step gives information on the dipolar contribution to the hyperfine constant, thus giving both Ai and D (see section 3.4). When the muon-electron hyperfine interaction is isotropic, a low-field decoupling step may still occur for a static muonium center due to transferred nuclear contributions
37 Texas Tech University, Hisham Bani-Salameh, May 2007 from overlap of the muonium electronic wavefunction with neiboring host atoms. Any motion of the Mu0 will then cause its unpaired electron to experience a fluctuating nuclear hyperfine interaction originating from the randomly oriented nuclear spins. This is due to the fact that the electron has a 207 times larger magnetic moment than that of the muon which makes it much more sensitive to fluctuations in the magnetic fields and allow us to neglect the muon interactions. This will cause fluctuations in the spin Hamiltonian that governs the muonium and induces transitions between the spin states. As a result; the electron spin relaxes (or flips) and the effect of that is communicated to the muon via the primary hyperfine interaction. Muonium is a four-level system consisting of one electron and one spin-1/2 nucleus (the muon). The problem of spin-lattice relaxation is complicated by the fact that the electron and muon spin states are mixed and coupled by the hyperfine coupling. Different perturbations caused by the environment that host the muonium can induce transitions between these mixed spin-states, the transitions will have different magnetic field dependencies and different selection rules depending on the mechanisms that cause the perturbation. The total polarization of the system will return to its thermal equilibrium value after it experiences any kind of perturbations with a characteristic rate constant or
−1 inverse relaxation time, T1 , that is equal to the sum of transition probabilities among the levels involved. Therefore; the problem of spin-lattice relaxation can be formulated in terms of simultaneous rate equations for the level populations [18]:
⎡ . ⎤ n1 ⎥ ⎢ ⎡ ( ++− WWW 141312 ) W21 W31 W41 ⎤⎡n1 ⎤ ⎢ . ⎥ ⎢ ⎥⎢ ⎥ n2 W ( ++− WWW ) W W n ⎢ ⎥ = ⎢ 12 242321 32 42 ⎥⎢ 2 ⎥ (3.50) ⎢ . ⎥ ⎢ W13 W23 ( ++− WWW 343231 ) W413 ⎥⎢n3 ⎥ ⎢n3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ . ⎥ ⎣ W14 W24 W34 ( ++− WWW 434241 )⎦⎣n4 ⎦ ⎣⎢n4 ⎦⎥
where ni being the time dependent occupancies of each of the four levels. The individual transition probabilities are defined by [18,19]:
38 Texas Tech University, Hisham Bani-Salameh, May 2007
2 2 τ c ijij ωij )( MJMW ij ⋅=⋅= 22 (3.51) + τω cij )1(
In equation 3.51, M ij represents the perturbation strength responsible for the transition and J (ωij ) is the spectral density function corresponding to that particular transition. The most important quantity for us is τc, it represents the motional correlation time to be extracted from the data and translates into the hop rate. Equation 3.51 is valid only under two conditions: first the transition probabilities are smaller than the hop rates (relaxation times must be longer than the correlation times) and second the transition probabilities are considered small (in frequency units) compared with the frequency (ωij ) of that transition.
4 In muonium, there are four states involved with the constraint ∑ ni = 1, i=1 therefore, the 4× 4 matrix of equation 3.50 has only 3 non-vanishing eigenvalues, and this means that the time evolution of the population of each level is basically a sum of three exponential decays and a constant. The general solution of equation 3.50 [18] is:
⎡ 1 tn )( ⎤ ⎡a1 ⎤ ⎡b1 ⎤ ⎡c1 ⎤ ⎡d1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 tn )( a2 b2 c2 d 2 ⎢ ⎥ = A⎢ ⎥ λ1t + Be ⎢ ⎥ λ2t + Ce ⎢ ⎥ λ3t + De ⎢ ⎥ (3.52) ⎢ 3 tn )( ⎥ ⎢a3 ⎥ ⎢b3 ⎥ ⎢c3 ⎥ ⎢d 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 4 tn )( ⎦ ⎣a4 ⎦ ⎣b4 ⎦ ⎣c4 ⎦ ⎣d 4 ⎦ each of the column vectors is an eigenvector of the × 44 matrix of equation 3.50, the prefactors are determined by initial and thermal equilibrium conditions. Only rarely, in the limits of high temperatures and fast hop rates, acquiring an analytical solution is possible, more generally, numerical solutions are required. In μSR experiments, the observable quantity is not the population difference of different levels, it is rather the overall muon polarization monitored via the asymmetry in
39 Texas Tech University, Hisham Bani-Salameh, May 2007 the muon decay. The time evolution of the muon polarization is defined in terms of the level population as:
4 z = ∑ i z ||)(2)( iIitntP (3.53) i=1 this expression for the polarization along with equation 3.52 shows that the polarization is in general a superposition of three exponential terms. The three exponents are the same for each ni(t) , equation 3.52, and therefore the general expression for the relaxation function will have the form:
−λ1t −λ3t −λ4t z )( +++= dcebeaetP (3.54)
therefore, in order to define a relaxation time T1 = /1 λ , one of these terms has to dominate or all of them must be very similar, in other words, the muon relaxation function is in practice indistinguishable from a single exponential.
To develop a general expression for the T1 relaxation, we assume only isotropic hyperfine interactions are present. We also use the high temperature limit so that the muonium electron is equally likely to have its spin parallel or antiparallel to that of the muon. Along with equation 3.14, this lead to initial occupancies of the four levels as 2 2 follows: n1(0) = ½, n2(0) = s /2, n3(0) = 0 and n4(0) = c /2 where c and s are defined earlier in equation 3.15. There are several mechanisms that can drive the relaxation, a commonly encountered one, most relevant to us, is relaxation by motional perturbation of the magnetic environment hosting the muonium atom; this mechanism is commonly expressed as fluctuation of effective local field seen by the muonium electron [18]. This is the mechanism responsible for the relaxation experienced by a muonium diffusing through a solid lattice. The local field in this case originates in the dipolar or superhyperfine interactions with host spins and is fluctuating due to the change in
40 Texas Tech University, Hisham Bani-Salameh, May 2007 magnitude and/or direction with each hop. Whether it is dipolar or superhyperfine interactions, the results are mathematically equivalent and except for some differences in
2 the magnetic field dependence of M ij , the same expression of relaxation can be used for both. If muonium is diffusing through a lattice, it experiences fluctuation of the effective local field. Five transitions ( → 21 , → 41 , → 32 , → 42 and
→ 43 ) are induced with comparable strength at low fields. At high fields, hyperfine interactions become decoupled and some energy levels are farther separated resulting in very different transition strengths. In longitudinal field, the transition that dominates the spin-lattice relaxation at high fields is between energy levels 1 and 2. The energy difference between level 1 and 2 is small compared with differences between other levels.
Therefore, the relaxation rate will be dominated by a single frequency ω12 because of the spectral density function in equation 3.51. This leads to a polarization function that approximates to exponential decay of the non-oscillatory signal given by:
+ 21 x 2 tP )( = − Tt )/exp( (3.55) z + x 2 )1(2 1
the spin-lattice relaxation (λ = 1/T1) is defined in equation 3.51 with a perturbation strength (M 2) being proportional to the transition probability. The transition probability x in this case ( → 21 ) is proportional to s 2 1( −≈ ) , (see equation 3.15) and 1+ x 2 therefore:
1 x τ x 2τδ λ 1( −∝≈ ) c 1( −= ) c (3.56) 2 22 2 22 T1 1+ x 1+ 12τω c 1+ x 1+ 12τω c
41 Texas Tech University, Hisham Bani-Salameh, May 2007 with δ 2 being the constant of proportionality which represent the perturbation strength. This is the form that is usually cited in literature [6,19]. Celio (1987) and Yen (1988) [20, 21] were able to calculate the theoretical polarization z tP )( by making the assumption that the nuclear terms in the Hamiltonian can be replaced by ( n = δ ⋅ ⋅ (tTSH )) where δ is the effective nuclear hyperfine interaction strength and T(t) is a unit vector randomly fluctuating with correlation time
τ c . They treated the nuclear hyperfine interaction as an effective magnetic field acting on the Mu electron, and were able to calculate the longitudinal spin relaxation using
Redfield’s theory. With this definition, the fluctuating rate (1/τ c ) can be interpreted as the rate with which the Mu is changing site (the hop rate). According to Celio and Yen,
z tP )( is a sum of four exponentials that can usually be replaced by an average exponential with decay rate 1/T1, leading to the polarization function of equation 3.55, with the T1 relaxation described by:
1 x 2τδ 1( −≈ ) c (3.57) 2 22 T1 1+ x 1+ 12τω c this basically the same as equation 3.56 with nuclear hyperfine interaction term,
2 12 AHF += ωωπω +− +− xx )1)/(1( and ω± = γ e ± γ μ )( B . These relationships serve as a good approximation from which one can obtain information about hop rates of the mobile muonium center.
3.6 Charge Exchange and Spin Exchange Processes If the material under consideration is n-type, there will be a significant number of free electrons in the environment of the muonium. Under such conditions, the ionized muonium will have a chance to recapture another electron from its surroundings, which it may loose again and capture another one and so on, this process is called cyclic charge exchange “CE”. Another process called spin exchange scattering “SE” might occur under
42 Texas Tech University, Hisham Bani-Salameh, May 2007 similar circumstances, in this process, the muonium can’t ionize for some reason, such as low temperature, but it can still interact with those free carriers through scattering off of them causing both the muonium electron and the conducton electron to flip their spin direction. Therefore, due to the SE process, the muonium is rapidly changing its electron spin state between the two available states, spin up and spin down. Both of these exchange processes lead to relaxation of the spin polarization with characteristic field and temperature dependencies. An isotropic neutral muonium center in longitudinal field will have the time evolution of its spin polarization given by equation 3.34, if the field is applied along the z-axis, then this represents the polarization along the z-axis. If we compare this with the spin polarization of a diamagnetic center (Mu+ for example), we’ll find that it is along the applied field direction and it is not evolving with time (Pz(t) = 1, Px(t) = 0 and Py(t) = 0). If the isotropic muonium undergoes a charge exchange process, one can qualitatively predict the behavior of the muon polarization. First at t = 0, an isotropic neutral muonium + is formed and then at t1 it ionizes and become Mu . The lost electron will carry away some of the polarization of the muonium depending on the ionization rate compared to the frequency of the oscillatory part of the polarization. If the ionization rate is much less than the hyperfine frequency then the muonium will undergo a large number of oscillations before it finally ionizes, in this case the polarization loss is field dependent and is proportional to the amplitude of the oscillating component of the polarization and we are in the so-called slow CE region. In contrast, if the ionization rate is much greater than the frequency, the muonium will ionize before it finishes one oscillation and in this case the loss of polarization will be much less than the previous case and will depend on the lifetime of the neutral muoium. This region is known as the fast CE region. After the first ionization, the muonium will be in a diamagnetic state and its 0 polarization stays constant till it captures another electron at t2 and becomes Mu again with less polarization than at t = 0. This process of capture and release will continue for the lifetime of the muonium until it finally decays where the observed polarization is an average value of the remaining polarization after each cycle. In general, if the process of
43 Texas Tech University, Hisham Bani-Salameh, May 2007
CE takes place, the spin polarization will decrease with time visible as a relaxation of P(t) in longitudinal field, in nuclear magnetic resonance (NMR) language; the muon is experiencing a 1/T1 relaxation. As the temperature increases, both rates (ionization and recapture) will increase, if the magnetic field stays the same, this means that we are going to the fast CE regime. The
1/T1 relaxation will peak with temperature when the relaxation rate is approximately equal to the frequency (ω24 ) where the process crosses over to the fast CE regime, after that point there will be a decrease in the relaxation rate. A cross over in the opposite direction (from fast to slow) can be accomplished by increasing the field, which will 2 result in increasing the frequency of oscillation (ω24 is proportional to B ). The nature of the hyperfine interactions involving the muonium will determine the field-dependence of the 1/T1 rates. For isotropic muonium, if the charge exchange process involves Mu0 and a diamagnetic center MuD, a very good approximation [12] for the decay rate is:
2 11 λλ DD 00 ω0 ≈ ( )( 2 2 ) (3.58) T1 2 0 + λλ DD 0 0D + ωλ 24
where λ0D and λ0D represent transition rates from paramagnetic to diamagnetic and from diamagnetic to paramagnetic states respectively. At low field where we will be in the fast CE regime, the relaxation rate is independent of the field and will stay constant (but depends on the temperature) till the cross over to the slow regime at higher field. In the slow CE regime, the relaxation rate decreases as the field is raised where 1/T1 relaxation is proportional to B-2. If the process is a SE rather than a CE, there will be a repeated cycling between two states (as discussed above), one with the muon and electron spin parallel, and the other with them antiparallel. The behavior of the spin polarization under this process is similar to its behavior under the CE process, therefore; we can expect the same kind of dependece on field and temperature for the 1/T1 relaxation.
44 Texas Tech University, Hisham Bani-Salameh, May 2007
CHAPTER 4 ALUMINUM NITRIDE EXPERIMENTAL DATA AND DISCUSSION
The group III-nitride binaries such as indium nitride (InN), gallium nitride (GaN), and aluminum nitride (AlN) possess direct band gaps of 0.7, 3.4, and 6.2 eV respectively [22]. This makes these alloys excellent materials for electro-optical devices with optical absorption or emission in a wide range of optical spectrum. The strong chemical bond makes these materials very stable and tolerant to harsh environments such as high power and high temperature. AlN has the largest bandgap and is the best material to extend the wavelength range well into the ultra-violet region. Hydrogen has an important role in these materials, in particular by helping to incorporate p-type dopants into GaN which has been the primary alloy constituent for most applications. Hydrogen functions as a shallow donor dopant in InN [23, 24] which has the narrowest gap of the nitrides, but shows the more standard deep compensating electrical properties in GaN and AlN.
4.1 AlN Structure The group III atoms form compounds with N that have four covalent bonds for each atom. AlN can crystallize in two crystal structures [25]; Wurtzite and Zinc blende with the stable one being the wurtzite structure. The wurtzite structure has a hexagonal unit cell with two lattice parameters a and c with an ideal ratio of ac == 633.13/8/ . Four nitrogen atoms surround every group III atom, and four group III atoms, which are arranged at the edges of a tetrahedron, surround every nitrogen atom. For an actual nitride, the structure is distorted from the ideal hexagonal packing of the basic molecule oriented along the stacking direction (c-axis). Because of this small distortion, there are two slightly different bond lengths in the wurtzite structure given by:
|| ⋅= cuB
a 2 1 B ( ) ⋅−+= cu 22 (4.1) ⊥ 3 2
45 Texas Tech University, Hisham Bani-Salameh, May 2007
Where u represents the cell-internal structural parameter (ideal value of 3/8), a and c denote the length of the lattice vectors of the structure. B|| represents the length of bonds parallel to the c-axis and B⊥ represents the length of bonds pointing in any other direction. The wurtzite structure with greater c/a value than the ideal value of 1.633 is unstable, while less than 1.633 is stable.
N
B ||
0 108.1 c B ⊥
0 110.8 Al
a
Figure 4.1 Part of the AlN wurtzite structure showing the two different bond lengths and the angles between bond directions.
46 Texas Tech University, Hisham Bani-Salameh, May 2007
In a perfect hexagonal wurtzite structure, the bond lengths between atoms in any tetrahedral cell are all the same and the angle between any two bond directions is also the same (the standard tetrahedral angle of 109.6o). In AlN, this structure is slightly distorted [26-28] with two different bond lengths and two different angles. X-ray experiments performed on AlN [29, 30] to measure lattice parameters a, c and u gave 3.1115 Ao , 4.9798 Ao and 0.3821 respectively. Upon using these values in equations 4.1, the two different bond lengths in AlN were found to be 1.9028 Ao (parallel to the c-axis) and 1.8899 Ao (otherwise). This leads to two different angles between bond directions that can be calculated using simple geometry rules (see figure 4.1 above). We got 108.10o for the angle between the two bonds when one of them is along the symmetry axis and 110.81o otherwise. With two different bonds between Al and N, there are two different bond center (BC) sites and two different anti-bonding (AB) sites associated with each atom. In one of those AB sites ( AB|| ), the direction of the bond Mu-Al or Mu-N is parallel to the c-axis and Mu resides in the blocked cage region of the structure. In the other three equivalent o sites ( AB⊥ ), these directions are at an angle of ~ 108.1 with the c-axis providing that the basic structure is not further distorted by the impurity and Mu resides within the wurtzite channels.
4.2 Hyperfine Decoupling Results The single crystal AlN sample studied in this work was on loan from Crystal IS. Data were collected at several different temperatures with the crystallographic b-axis of the wurtzite structure oriented parallel to the applied magnetic field and initial muon spin direction (longitudinal field setup). Even though the spin-precession signals have not been observed in AlN, an atomic-like neutral muonium center Mu0 with a small anisotropy is inferred from hyperfine decoupling curves and longitudinal relaxation data. The raw decay asymmetry data were fitted to two signals, the results indicate one is rapidly relaxing and the other is slowly relaxing with dependence of the amplitudes of both signals on the applied magnetic field and temperature.
47 Texas Tech University, Hisham Bani-Salameh, May 2007
Low-field step in the hyperfine decoupling curves (because of the weak anisotropy) allowed us to get information about the dipolar part of the hyperfine interaction. This was done by fitting decoupling curves to an approximate function provided by Pratt [31]. If the Mu0 were stationary, this observed second step in the decoupling curves might also be due to transferred (super) hyperfine contributions from the nuclear spins of neighboring host atoms. The fact that we see evidence of fast motion of this center makes us believe that this step is due to the small anisotropy rather than a transferred hyperfine contribution. Hyperfine constants from all repolarization curves were extracted by fitting to the approximate function [31]:
1 cos 2 θ +〉〈 y 2 ( + x 2 2 1+ y 2 P = (4.2) 1+ x 2
With x γ e += γ μ /)( AB i , and y = γ e + γ μ /)( DB , where Ai and D are the isotropic and dipolar hyperfine constants, γ e and γ μ are the electron and muon gyromagnetic ratios, and θ is the angle between the applied magnetic field and the axis of D. An average isotropic hyperfine constant Ai and an average dipolar contribution D were then calculated and used to simulate the hyperfine decoupling curves for the expected Mu0 locations in AlN for purposes of comparison to the experimental data.
Table 4.1 Aiso and D values at different temperatures acquired by fitting decoupling curves to the approximate function of equation 4.2.
Temperature Ai Uncertainty Average D/ Ai D Uncertainty Average (K) (MHz) (MHz) (MHz) (%) (MHz) (MHz) (MHz) 5 4816.5 330.0 2.4 116.3 15.3 4449.9 166.9 30 4044.6 500.0 2.58 104.3 21.4 ± ± 40 5065.6 436.9 4.25 215.3 38.0 435.5 28.6 225 3873.9 456.9 5.98 231.7 33.7
48 Texas Tech University, Hisham Bani-Salameh, May 2007
The fitted zero-time amplitudes as a function of field at temperatures of 5, 30, 40 and 225 K are shown in figure 4.2. The hyperfine interactions are close to the vacuum isotropic value but include a small dipolar contribution indicating a weakly anisotropic hyperfine interaction. The average value for the isotropic part of the hyperfine constant is
Aiso = 4449.9 ± 436 MHz compared to 4463.2 MHz for a free Mu atom, and the dipolar contributions are 116.3 at 5 K, 104.3 at 30 K, 215.3 at 40 K and 231.7 at 225 K with an average value of 166.9 ± 28.6 MHz (see table 4.1).
1.1 5K 1.0 30K 40K 0.9 225K
0.8
0.7
0.6 polarization 0.5
0.4
0.3
1 10 100 1000 field (Gauss)
Figure 4.2 Experimental repolarization curves at 5 K (squares), 30 K (circles), 40 K (triangles) and at 225 K (stars), along with fitting curves to the approximate function.
49 Texas Tech University, Hisham Bani-Salameh, May 2007
The low field decrease in visible asymmetry could equally well be interpreted as a nuclear hyperfine contribution if the Mu0 center were stationary. However, the relaxation data imply rapid motion which we argue should average out contributions due to overlap with neighboring nuclei, with those rapidly fluctuating interactions providing a motional depolarizing mechanism. Motion among a set of equivalent sites for which a dipolar hyperfine contribution repeatedly takes only a small number of orientations may not be completely averaged away. The fluctuating parts of the dipolar hyperfine interaction will also then contribute to depolarization due to Mu0 motion. Therefore, any transferred nuclear contributions to A should average to zero for long range diffusive motion and at least some fraction of a dipolar contribution to the direct muon-electron interaction will survive rapid switching among a very small number of orientations for D. If the motion were local motion or tunneling among a very small number of sites, both components might survive. Our conclusion is that the low-field feature in repolarization curves is due to a small anisotropy rather than a transferred nuclear hyperfine interaction.
4.3 Site Assignments for Muo in AlN Based on theoretical models, H or Mu is a negative-U defect with deep level states and compensating electrical properties in AlN. This implies that a neutral charge state will never dominate under equilibrium conditions. Theoretical predictions for hydrogen in GaN place H0 at the center of the open channels of the wurtzite structure parallel to the c-axis [32]. Earlier predictions suggest an off-center AB⊥ (N) location with strong N-H bonds expected [33]. Neutral hydrogen is predicted to diffuse along the c-axis within the channels. The repolarization curves for AlN imply the existence of an atomic-like neutral state with nearly the full vacuum hyperfine constant as shown in the previous section. In making a tentative Mu0 site assignment in this sample, we considered a number of theoretical and experimental results. Any anisotropy in AHF, other than that imposed by the dielectric properties of the hexagonal crystal structure, excludes the preferred theoretical site at the center of the unblocked c-axis channels as a possible location for
50 Texas Tech University, Hisham Bani-Salameh, May 2007 neutral muonium in favor of an off-axis site. The fact that we have a large hyperfine constant (nearly the full vacuum value of 4463 MHz) suggests that Mu0 has not reacted with the lattice. These two arguments imply an off-center location if Mu0 resides in the channels, suggesting one of the anti-bonding sites but with no covalent bonding to a host atom. In compound semiconductors, the location and stability of the hydrogen (muonium) atom will be affected by the fact that the host atoms have different electron affinities. Different factors such as the ionic characters of the host atoms, relative electronegativity of each atom and the relative bond strength will play crucial role in determining the stable site for H (Mu). According to Estreicher and coworkers [34-36], the more stable location for an unreacted H0 is near the least electronegative host. If H (Mu) is to reside in a location that is anti-bonding to the least electronegative atom (the Al atom in the case of AlN), it will be surrounded by four positively charged NN’s. This will cause an increase in the electronic wavefunction overlap between H (Mu) and its host
[34] and result in farther decrease in AHF constant.
If a Mu-N bond were to be formed, ⊥ NAB )( site would be the more likely location for Mu0. Without a covalent bond formation, the above arguments imply that the preferred site should be close to the Al host atom, thus the better assignment is ⊥ AlAB )( for the simple trapped Mu0 atom location. The small observed anisotropy is then due to the distortion of the Mu first atomic wavefunction due to the fractional positive charge of the nearby Al, thereby modifying the electronic wavefunction and providing the weak dipolar contribution to the hyperfine interaction. Figure (4.3) shows a section of the wurtzite structure of AlN with the four anti- bonding sites surrounding a single aluminum atom depicted. The site with Al-Mu oriented along the c-axis should be a higher energy location which is most likely metastable with a fairly large barrier for Mu0 to escape the blocked “cage” region of the lattice. Therefore, 0 || AlAB )( should be static and metastable site for Mu if it would be occupied at all. The three equivalent sites ⊥ AlAB )( represent the low-energy mobile state.
51 Texas Tech University, Hisham Bani-Salameh, May 2007
Al
AB N ||
Mu AB ⊥
Figure 4.3 The wurtzite structure of AlN showing our site assignments for neutral muoium indicated by the smallest sphere (the c-axis is vertical).
With the magnetic field applied along the crystallographic b-axis, the angle θ to the field direction for Al-Mu at each ABAl site is easily determined. This angle represents the orientation of the dipolar hyperfine contribution D for that site. The site that is parallel to the c-axis has a 90o angle with the field, one of the three AB sites lies in the bc plane at an angle of 161.9o to the b-axis and the other two are in mirror positions giving identical decoupling curves with θ = 61.624o, these were calculated assuming the c-axis is along the z-direction and the applied field is along the negative x-direction. With these angles, we simulated repolarization curves (see also ref. 37) for each site using the approximate function used to fit the experimental data.
52 Texas Tech University, Hisham Bani-Salameh, May 2007
1.0
0.8
0.6 Average polarization
from four sites 0.4 Polarization
0.2
0.0 1.0
0.8 161.9o 0.6
0.4 61.624o X 2 Polarization
0.2 90o 0.0 1 10 100 1000 10000 Magnetic Field (Gauss)
Figure 4.4 Simulated zero-time longitudinal asymmetry for all four possible sites of neutral muonium in AlN (bottom) and their average (top).
53 Texas Tech University, Hisham Bani-Salameh, May 2007
The hyperfine constants used in the simulation were average values from the fits discussed above, Aiso = 4450 MHz and D = 167 MHz, results of these simulations for the four ABAl sites are shown in figure 4.4. The average of these four curves produces a field dependence that agrees well with the data. This procedure assumes equal hyperfine interactions and equal populations for all four sites. This is reasonable for AB⊥ sites, but probably should not include AB|| . The data require some AB|| occupancy to match the two decoupling step amplitudes. To justify the use of the approximate function (equation 4.2), we simulated repolarization curves for all four sites using our simulation program mentioned above. Without any approximations made in the program, the simulations should reflect the actual repolarization curves expected from the theory. The hyperfine constants used in this simulation were the same average values from the fits discussed above, A = 4450 MHz and D = 167 MHz. The agreement between our program’s results and the results from using the approximate function was excellent and proves the validity of this function in our case. This gave us more confidence in the results from using Pratt’s function to extract A and D.
4.4 Dynamics of Mu0 in AlN In figure 4.5, we present a series of muon spin depolarization curves for different values of applied magnetic field at the same temperature (40 K). Both the rapidly relaxing (fast rate constants) and slowly relaxing amplitudes show dependence on the applied field, this by itself imply the existence of a paramagnetic free-atom like Mu0 that is experiencing some form of fluctuation dynamics that involve only that Mu0 state along with relatively slow conversion to a diamagnetic muonium final state [38]. Evidence for the kind of dynamics involved comes from the details of the magnetic field and temperature dependence of the fast rate constants. Mobility of Mu0 in AlN is evident [6] from the data as we see field-dependent longitudinal relaxation rates at all the temperatures tested [39], two examples are shown in figure 4.6. Temperature- dependent relaxation rates at 100, 350 and 1000 Gauss are also shown in figure 4.6 with
54 Texas Tech University, Hisham Bani-Salameh, May 2007 data collected to higher statistics to reduce errors for the very fast relaxation. The shift in peak relaxation rate with field implies that hop rates increase as the temperature increases above 300 K and also as it decreases below 300 K. If the basic hyperfine constant and fluctuating component are independent of temperature, those preliminary results require a
25
40 K 150 G 1000 G 20 2500 G 4000 G
15
Asymmetry
10
5 02468 time (μs)
Figure 4.5 Muon spin depolarization raw data in AlN at 40 K along with fits to extract various parameters including the rate constants that are used to extract hop rates and conversion rates.
55 Texas Tech University, Hisham Bani-Salameh, May 2007
15 AlN Crystal 12 B || b-axis LF 9
6 112 K
3 225 K Relaxation Rate (MHz)Relaxation 0 0.1 1 10 100 Magnetic Field (mT) 20 10 mT 35 mT 15 100 mT
10
5 Relaxation RateRelaxation (MHz) 0 0 200 400 600 800 Temperature (K)
Figure 4.6 Longitudinal-field data indicating motion of Mu0 in AlN: field-dependent relaxation rates (top) at 112 and 225 K. Temperature-dependent relaxation rates (Bottom) at 1000 G (squares), 350 G (triangles) and 100 G (circles).
56 Texas Tech University, Hisham Bani-Salameh, May 2007 transition out of the neutral state above roughly 400 K that appears to be limited by thermal diffusion to a defect related trap site. The field dependence for the rapidly and slowly relaxing fractions below about 200 K implies a relatively slow conversion from the neutral state to a diamagnetic one in this regime as well, with the maximum effect on repolarization data occurring near 100 K. The full data set collected on this sample includes curves at various other fields spanning the low temperature relaxation peak. These data imply rapid motion of Mu0 at all temperatures with minimum hop rates around room temperature. The onset of fast longitudinal relaxation with field seems to very closely follow the decoupling step associated with the dipolar component of the hyperfine interaction [37] seen in the decoupling curves. This supports our supposition that reorientation of D with Mu0 motion is the primary source of the observed relaxation rather than wavefunction overlap with neighboring atoms and transferred nuclear contributions. With the clear evidence of the existence of a mobile Mu0 in AlN at all temperatures, our initial goal was to extract hop rates from the depolarization data. To do so, we used a simple model for the longitudinal relaxation rate including a fluctuating contribution to the muonium hyperfine interaction and a correlation time for those fluctuations that translates into a hop rate in this model (equation 3.57). An average isotropic hyperfine constant Ai and average dipolar contribution D were obtained from fitting the hyperfine decoupling curves at different temperatures to the approximate repolarization function from Pratt as discussed above. In the attempt to extract hop rates; the weak dipolar part of the hyperfine interaction is included in a single random fluctuating component, along with the transferred contributions from neighboring nuclei. The resultant temperature-dependent hop rates show clear evidence of different diffusion processes in the various temperature regions. We analyzed the complete set of motion-related longitudinal relaxation data with the possibility of conversion included for all temperatures. This was accomplished by
−1 fitting the fast longitudinal relaxation rates T1 , obtained from the raw asymmetry data at specific temperatures (see figure 4.5 for an example), to equation 3.57 plus a conversion
57 Texas Tech University, Hisham Bani-Salameh, May 2007
−1 rate ( 1 λ )()( += CBBT ), this provides the needed information about the hop rate
1/τ c and the conversion rate C at that specific temperature with the field dependence entering via both x and ω12 in equation 3.57 for λ. Fits of equation 3.57 to the fast relaxation rates in the temperature region below 100 K display strong bimodal character. Hop rates extracted from the fits were either considerably larger or much smaller than the frequency of the dominant transition ω12. In this analysis, when including all the depolarization data available, only the hop rate at 40 K were actually in the high branch, this is the main reason why we chose the lower branch. If the Mu0 spin precession signals are observed, there would be a way to determine which fit branch is correct; in the high hop rate limit, the relaxation rates for these signals (under transverse field conditions) should be equal to the rates obtained
1 −− 1 under longitudinal field conditions, that is to say 1 = TT 2 in the standard magnetic resonance terms. This argument is not valid in the slow hop rate regime. In magnetic resonance studies, one usually observes longitudinal relaxation times (T1) that are much longer than transverse relaxation times (T2) in the slow fluctuation regime and become equal to it as the fluctuations reach the fast limit [40]. This kind of test would allow us to say for sure that we are in the right branch of the hop rates only if, in the high hop rate regime, the two rates are equal. It won’t, on the other hand, confirm that it is the wrong branch if the two rates are different since there might be other reasons for this difference. We are unable to do such a test for the present case because, unfortunately, the spin precession signals have never been observed in AlN. This is to be expected since the
−1 dephasing rates T2 for the precession signal are expected to be very large for the intermediate hop rates; therefore, the precession is rapidly damped and not visible. Equivalently, because the hyperfine interactions are anisotropic, with each hop, there is a change in the precession frequency which makes the precession harder to detect unless the hop rate is either very fast or very slow compared to the precession frequency.
We used the average value for the hyperfine constant AHF of 4450 MHz. The random fluctuating component, which may include both the weak dipolar part and the transferred superhyperfine contribution from neighboring nuclei, was set to an average
58 Texas Tech University, Hisham Bani-Salameh, May 2007
8
4
Conversion Rate (MHz) Rate Conversion 0
10000
1000
100 Hop Rate (MHz)
10 10 100 1000 Temperature (K)
Figure 4.7 Temperature-dependent conversion rate (top) and hop rate (bottom) of neutral muonium centers in AlN. Solid lines represent best fits to extract activation energies.
59 Texas Tech University, Hisham Bani-Salameh, May 2007 value of δ = 204.5 MHz. This was obtained from fits with that parameter free in regions where the hop rate is close to δ and thus the relaxation is most sensitive to the strength of the fluctuating interaction. With these numbers, one can get a rough estimate on the superhyperfine contribution of about 118 MHz reached with the assumption that the fluctuating component is equal to the square root of the fluctuating part of the dipolar contribution squared plus the fluctuating part of the superhyperfine contribution squared.
Keeping the AHF and δ fixed throughout the fits and letting the hop rate and the conversion rate free, we were able to extract the full temperature dependence of the hop rates and conversion rates shown in figure 4.7. The hop rate is minimum (30 MHz) around room temperature, this marks the crossover between quantum tunneling behavior which dominates at lower temperatures and thermally activated processes which take over at higher temperatures. Below 300 K, our analysis shows an increase in the hop rate with decreasing temperature down to roughly 100 K where it reaches a maximum tunneling rate of about 4000 MHz. Below 0 100 K the rate at which Mu tunnels among equivalent sites decreases again and levels off at about 250 MHz below 25 K. Above room temperature, there are two activated regions as evidenced by the kink in slope seen in figure 4.7 at roughly 550 K; rather clearly implying two different thermal processes or perhaps two separate diffusion paths. In fitting the temperature dependence 0 of the Mu hop rates, we have chosen to split the data into two regions. From 100 K upward, we fit to two thermally-activated functions plus an inverse power law. For temperatures below 100 K, our fit function consists of a constant plus one activated function. The conversion rates follow similar trends as the hop rates in general, except for the 100 to 300 K region where the hop rates are decreasing with temperature. We thus chose to fit the conversion rates to two activated functions above 100 K and to one activated contribution below 100 K, plus a constant in each case. In each region, there 0 appears to be a direct correlation between the motion of the Mu center and the conversion rate out of that state. More precisely, when left free, the activation rates
60 Texas Tech University, Hisham Bani-Salameh, May 2007 obtained for the conversion results in each region were very close to those obtained for the motion. We can therefore conclude that all three of these activated regions represent long range diffusive motion and that in these regions the conversion is diffusion limited, suggesting either a reaction with some other defect leading to a Mu-Defect bound complex, or a charge transfer interaction with another impurity or defect resulting in + - either a Mu or Mu ionic charge state. These comments represent our general initial conclusions; more details are given in the following discussion of specific processes in separate temperature regions.
Table 4.2 Fit results of the temperature-dependent hop rate and conversion rate to extract activation energies. Hop Rate Conversion Rate Temperature Activation Uncertainty Activation Uncertainty range (K) Energy (meV) (meV) Energy (meV (meV) < 100 14.8 1.1 22.2 7.7 300-550 106.1 11.4 150.6 6.3 > 550 818.1 120.0 815.9 100.8
Kinetic processes, in general, deal with mass and charge transport through or over some potential barrier. At high temperatures, one of the possible ways for a particle to overcome a barrier is to gain the needed energy through its interaction with lattice excitations (phonons for example). At low temperatures, on the other hand, the environmental excitations are no longer an option, being frozen out, and the only way to overcome a potential barrier is to tunnel through it. This phenomenon is known as quantum diffusion QD and was originally introduced to deal with diffusing particles that are heavier than the electron. Muonium has a mass of about 200 times more than the electron and about 1/9 that of hydrogen, this enhances its tunneling probability over that of H and makes it an excellent candidate for QD studies. Because of this unique mass, quantum diffusion of Mu has been observed in a wide temperature range, this is
61 Texas Tech University, Hisham Bani-Salameh, May 2007 considered rare since atoms are usually too heavy to tunnel and the electrons are too light such that they are usually delocalized in the lattice [41,42]. There are two kinds of tunneling, coherent and incoherent, depending on whether the interaction with the environment leads to spatial localization of the wave function or to band-like motion. At low temperatures, the density of lattice excitations is very low and their effects are very small such that diffusing particles will propagate freely in a band-like motion without any loss of phase coherence of the wavefunction. Coherent tunneling of the particle typically occurs when energies of the levels are the same for adjacent sites and there is significant wavefunction overlap between the two sites. Such tunneling motion will not be effective however if these sites are well separated. As the temperature increases, the density of lattice excitations increases. This will increase the possibility of scattering of a diffusing particle and therefore affecting its motion. In the temperature region where scattering occurs, there will generally be a decrease in the diffusion rate with increasing temperature as the wavefunction of this particle becomes spatially localized at one site. Quantum tunneling of the particle through the barrier can still occur in this region but scattering will result in loss of phase coherence for the wavefunction. Therefore, this kind of tunneling is referred to as incoherent quantum tunneling. Another important effect when talking about quantum diffusion processes is barrier fluctuations. Potential barriers between any two interstitial sites are defined by positions of host atoms in the immediate environment. Thermal motion of these atoms will greatly influence potential barriers by increasing or decreasing their heights and widths. This effect is more pronounced at high temperatures where optical phonons become active. Long-wavelength acoustic phonons shift whole regions of the lattice without changing the relative positions of atoms, thus do not modify the barriers. There have been numerous explanations offered for tunneling related features in μSR on various materials at low temperatures. Commonly, there is an expected change from incoherent to coherent tunneling characterized by a region of constant motion. Various other features have been attributed to scattering of the coherent state by
62 Texas Tech University, Hisham Bani-Salameh, May 2007 impurities, a delocalization of the tunneling muon's wavefunction, and creation of a band- like tunneling state. In most cases, there has not been complete agreement as to the precise nature of many of the observed low-temperature quantum properties of muon or muonium motion in various solids. As a general rule, an increase in motion as the temperature decreases is a clear indication of quantum tunneling that is interrupted by phonon scattering. This regime is
−n typically characterized by an inverse power-law temperature dependence ν h = AT , with experimental results for muonium in a number of materials [42] giving typical exponents of roughly n = 3 to a high of about 7. A large shift in the energy of a diffusing particle due to absorption of single phonons will prevent tunneling of that particle, quasi- elastic phonon scattering on the other hand will leave the energy almost unchanged. This kind of absorption-emission disruption mechanism to the Mu diffusion is expected to be the dominant mechanism. Recent experimental results on Mu quantum diffusion [43-46] offered the direct evidence that the quasi-elastic phonon scattering is the dominant mechanism that breaks up the tunneling coherence but still allows tunneling motion. As the temperature is increased beyond the point where we have the minimum hop rate, thermal activation processes dominate. Different signatures like activation energies and prefactors help in sorting out the kind of process dominating the particle’s motion in that region. Relatively low activation energy coupled with a low prefactor is characteristic of phonon-assisted tunneling. The primary process changes to ordinary thermally activated diffusion if both the prefactor and the barrier are relatively large.
4.5 Discussion 0 In the present case of Mu in AlN, we offer a fairly simple explanation of the features observed below 100 K. If one examines the wurtzite structure, there is a series of coplanar AB⊥ sites that provide a straightforward picture of what could occur to give a constant tunneling rate at low temperatures followed by an activated region as the temperature increases, prior to the onset of phonon disruption. There are three AB⊥ sites for each Al atom, with Al-Mu oriented into three separate channel regions, and within
63 Texas Tech University, Hisham Bani-Salameh, May 2007
Al
Mu N
Figure 4.8: A segment of the AlN wurtzite structure showing the two possible paths for the localized tunneling motion for Mu0 in AlN at low temperatures. each channel, there are three closely spaced sites associated with different Al atoms (see figure 4.8 above). A coherent localized tunneling state is possible among each of these sets of three sites (in the channel among the three sites from three different Al atoms, or among the three sites surrounding one Al atom) and both localized tunneling paths appear likely with better chances for the in-channel path because of the lack of Mu-Al bond. We therefore propose that one (or perhaps both) of these local tunneling states is active below 25 K and that a small barrier between the two gives rise to the activated behavior below 0 100 K. With both of these local tunneling states active and transitions between them, Mu will be confined to a single ab plane but diffusively mobile in two dimensions. In this picture, we interpret the 14.8 ± 1.1 meV activation energy obtained in these fits as the energy required to hop from one such region of local motion to a neighboring one via the
64 Texas Tech University, Hisham Bani-Salameh, May 2007 other localized tunneling state. Alternatively, one can think of this as thermally assisted tunneling between two adjacent localized tunneling regions. Either picture implies that two-dimensional long-range motion will occur.
0.4
AlN Crystal
0.3
+ 0.2 Mobile Mu Trapped Mu+
0.1 Fractional Amplitude
0.0 0 200 400 600 800 Temperature (K)
Figure 4.9 Depolarization measurements on the AlN crystal in zero field. Two diamagnetic states are present and assigned to Mu+, one is weakly relaxing and mobile at all temperatures and the other one is stationary to roughly 750 K and correlates well with LCR spectra assigned to Mu-defect complex.
The fact that we observe a conversion rate in this low temperature region which shows a very similar small activation energy gives some additional justification for this
65 Texas Tech University, Hisham Bani-Salameh, May 2007
0 interpretation. Once Mu is diffusively mobile, even in a two-dimensional sense, it can move through the lattice to encounter another impurity or defect and interact with it in some manner. When left free, we obtain an energy of 22.2 ± 7.7 meV as an activation 0 energy for the conversion from Mu to a diamagnetic state below 100 K. There are several data sets using different μSR techniques which show an increase in diamagnetic fraction near 100 K or slightly higher, including the weakly-relaxing amplitudes in the present longitudinal data. In zero-field measurements on this sample (figure 4.9) which separate various diamagnetic contributions, this small increase appears to be primarily in a signal that we have assigned to a mobile (small relaxation rate in zero + + field) Mu state. This is the lowest energy state for a Mu , which in this temperature range is interpreted as tunneling locally among the three ⊥ NAB )( sites for a single nitrogen atom to which it is strongly bound [47]. + + Assuming that Mu in its N-Mu ground state is the correct final-state assignment for the low-temperature transition out of Mu0, we suggest that a charge-transfer scattering with another impurity is the most likely conversion mechanism. The ratio of conversion 0 rate to weakly activated Mu hop rate is roughly 1300, giving an estimate of the number of 2-D hops required on average before a conversion occurs. This also yields some estimate of a minimum density for the responsible charge-transfer scattering center. It is not immediately clear why this conversion process should turn off just as the tunneling motion enters the regime where it is limited by interactions with lattice phonons; however, that is what these data imply. Perhaps the other impurity changes its charge- state at roughly the same temperature, thereby turning off that transition route. For the temperature region between 100 and 300 K, where we see a decreasing hop rate with increasing temperature, various fits to the results in figure 4.6 yield n = 5.5 0 to 5.9 (0.1), depending on how the data points are weighted. We interpret the Mu motion in this region as incoherent tunneling transitions from one site to the next with significant residence time at each individual site. We now turn to the two high-temperature thermally activated regions. In the temperature range of 300-550 K, we have concluded that thermally-assisted tunneling is
66 Texas Tech University, Hisham Bani-Salameh, May 2007 the dominant process. The relatively low activation energy coupled with a low prefactor is characteristic of phonon-assisted tunneling. Upon fitting the hop rates and the conversion rates to extract activation energies in this region with the energies free, we find the barrier for hopping motion to be 106 ± 11 meV and the barrier for conversion of muonium to a new state is 115 to 150 meV depending on weighting choices. In the higher temperature range above roughly 550 K, the primary process changes to ordinary thermally activated diffusion as evidenced by a much larger prefactor and larger barrier. 0 Mu is diffusing rapidly in this region with a barrier of 818 ± 120 meV and is converting to a new state with a barrier of 816 ± 100 meV in fits with data weighted using a constant fractional uncertainty. We note that at the highest temperatures, the extracted conversion rates do not follow the hop rates; however, as discussed below, this apparent discrepancy is to be expected. The prefactors in this analysis indicate that there are only about 40 to 50 steps in the diffusion prior to trapping for both of the high-temperature activated processes. It should be pointed out that if one uses higher δ in the fits, the hop rates will be higher and will result in higher number of steps prior to trapping [48]. The direct correlation between the conversion and hop rates in this analysis and additional information on the nature of the final state, lead to the conclusion that the conversion of neutral muonium to another state above room temperature is due to 0 diffusion-limited trapping of the mobile Mu at defect sites giving rise to a bound Mu- Defect complex [23, 47]. The specific evidence for a diamagnetic bound complex comes from muon level-crossing resonance (LCR) data [11]. Basically, we observe an increasing trapped (immobile) diamagnetic Mu fraction, as shown in figure 4.9, and a LCR spectrum (figure 4.10) which both grow in a manner that is correlated with the high-temperature 0 Mu conversion rates. In both cases, we see the peak with temperature is at roughly 800 K where we see the peak in conversion rate. The LCR spectrum is identical with the one seen already at much lower temperatures in highly stressed and defective thick AlN films grown by a variety of techniques. That LCR spectrum was assigned [23] to a bound Mu-Defect complex originating from a high-density defect in the AlN films. The equivalence of the resonance
67 Texas Tech University, Hisham Bani-Salameh, May 2007
0
-1 (a) -2 1033 K
556 K -3 935 K 694 K -4 837 K -5 051015 QLCR Amplitude (% dP) Magnetic Field (mT)
7 AlN Crystal (b) 6 Main LCR Resonance R = 4.52 ± .06 mT 5 0 4
3 2 1 4.5 mT LCR Intensity 0 400 600 800 1000 Temperature (K)
Figure 4.10 (a) Level-crossing resonance spectra for trapped Mu centers in AlN at several temperatures, (b) the intensity of the main LCR resonance seen at 4.5 mT as a function of temperature.
68 Texas Tech University, Hisham Bani-Salameh, May 2007 spectra in these films and the single crystal, along with essentially identical dissociation dynamics, means that we observe the same complex in both cases. The different formation dynamics suggests that the implanted muon finds the other defect during thermalization in the films. A good candidate for this defect is dislocation. Since the responsible defect is present at much lower concentrations in single crystal AlN, muonium must be diffusively mobile to reach the same defect. While it was concluded 0 + [23] that both Mu and Mu undergo long range diffusion above 350 K in AlN, the overall 0 results indicate that most of the formation of this complex originates from the mobile Mu state [11]. The resulting complex dissociates above 750-800 K with an energy of almost exactly 1.0 eV based on thick film results [49]. Data for the crystal are consistent with that, but a large variation between two temperature sensors makes it difficult to obtain an accurate barrier due to the uncertainty in temperature. The reduction in net conversion rate seen at the highest temperatures in figure 4.7 is a consequence of the fact that the complex formed in the conversion process does not remain bound at those temperatures. Due to the large uncertainties in the rapid depolarization rates in some important temperature regions, one might ask how robust the reported results on Mu0 diffusion are? The general reported features of a minimum hop rate near 300 K, thermally activated diffusion at high temperatures and tunneling behavior below 300 K are extremely robust. These general features persist in several different analyses we did with slightly different fitting parameters and different number of points in the relaxation curves used in the fit. There is a strong correlation between relaxing amplitude and very fast rate constant; accuracy in any of the extracted parameters depends on how this is handled in fits to the raw data. This should affect the non-relaxing part of the asymmetry but not the fast relaxing one therefore won’t contribute to uncertainties in hop rates or conversion rates. The temperature dependence in relaxing amplitude was smoothed to reduce the scatter in rate constants in the reported analysis, thus quoted statistical uncertainties in barrier energies, for example, do not include any systematic uncertainties introduced by this
69 Texas Tech University, Hisham Bani-Salameh, May 2007 smoothing process. We estimate that overall, the various barriers probably should be assigned about 20 % uncertainties. Our results on neutral Mu in AlN represent, at least qualitatively, an experimental model for the analogous neutral hydrogen defect center. Because of the lighter mass of muonium compared to hydrogen or deuterium, the quantum tunneling features in Mu motion will be much less prominent for hydrogen, although local tunneling motion and phonon-assisted tunneling are both known to occur for hydrogen in common semiconductor materials. As a consequence of zero-point energy differences, the barriers for thermally activated diffusion are expected to be larger for H by a few tenths of an eV (based on theoretical calculations) compared to the values obtained for Mu. With this in mind, reasonable estimate of the barrier for H0 diffusion in AlN can be 1.1 or 1.2 eV based on the ~ 0.8 eV for Mu0. Detailed modeling of the potential energy surface within which both Mu and H would reside and move is required in order to make a more accurate adjustment in translating the present results to model the diffusion of isolated neutral hydrogen impurities [50]. One more thing to keep in mind is the fact that both H and Mu are negative-U defects, thus the neutral charge state of either one will never be the dominant state in thermodynamic equilibrium. However, there are typically rather rapid charge-state transitions present, especially in doped materials, so that diffusion can still be dominated by motion of a very mobile neutral center particularly in the absence of an electric field to derive motion of the ionic charge states. The μSR data on AlN films imply that the Mu+ center is also quite mobile, but the results show large variations from one sample to another, so we are not yet able to properly compare diffusion parameters for Mu0 and Mu+.
70 Texas Tech University, Hisham Bani-Salameh, May 2007
CHAPTER 5 SILICON CARBIDE EXPERIMENTAL DATA AND DISCUSSION
Silicon carbide has received increasing attention in recent years because of its ability to tolerate high temperatures and high powers without major changes in performance. Material properties of silicon are no longer sufficient to meet the rapidly increasing demands for high-power, high-temperature and high-frequency electronics. Silicon carbide is now replacing silicon in such applications because of its larger band gap, larger thermal conductivity and larger break-down fields. These properties make it an excellent candidate for use in electronics applications under extreme conditions [51- 53]. Silicon carbide is known as a wide bandgap semiconductor existing in many different polytypes. In all polytypes, the carbon atom is located above the center of a triangle of Si atoms from one layer and underneath a Si atom from the next layer. The lattice constant a (the distance between any two neighboring silicon or carbon atoms) is approximately 3.073 Ao for all polytypes. The distance between the C atom to each of the four neighboring Si atoms (C-Si bond length) is the same where the carbon atom is positioned at the center of mass of the tetragonal structure outlined by the four neighboring Si atoms. Simple geometrical calculations reveal this distance (C-Si bond length) to be a(3/8)1/2 which is approximately 1.89 Ao. The height of a unit cell, c, is different for different polytypes and therefore the ratio c/a is also different [54]. Silicon carbide has over 200 identified crystal structures [55] resulting from different stacking sequences of the fundamental Si-C unit. The difference between different polytypes is in the stacking order of succeeding layers of carbon and silicon atoms. Among all the known crystal structures of SiC, the two structures 4H and 6H have received the most attention and the growth of which has become most advanced. These two structures are both hexagonal with a repeated stacking order of either 4 or 6 layers thus labeled as 4H and 6H respectively.
71 Texas Tech University, Hisham Bani-Salameh, May 2007
The stacking sequence is shown (figure 5.1 below) for the two most common polytypes, 4H and 6H. If the first Si-C double layer (where each one of these layers is basically a planar sheet of Si atoms coupled vertically to a planar sheet of C atoms) is called the A position, the next Si-C double layer can be placed at the B or the C position according to a closed packed structure. The different polytypes can be constructed by different stacking orders at these three positions. For instance, 6H-SiC polytype has a stacking sequence of ABCACB. In the given name for any polytype, the number denotes the periodicity and the letter represents the symmetry of the resulting structure.
B
C
A C
C A
B B
A A
6H 4H
Figure 5.1 The stacking sequence for the two most popular polytypes of SiC. Positions shown are for one atom type only as the base of a vertical molecule.
72 Texas Tech University, Hisham Bani-Salameh, May 2007
The presence of hydrogen in most epitaxially grown semiconductors is difficult to avoid, and SiC is no exception. Unfortunately, it is almost impossible to get experimental information on atomic hydrogen as an isolated impurity because of its high reactivity. Studying muonium in SiC is our only hope to get experimental insights on the behavior of H in this material. For all the measurements reported in this chapter, ~100 % spin polarized muons were implanted into commercially available 4H- and 6H–SiC wafers of each electrical type; high-resistivity, n-type, and p-type. The majority of spin precession measurements to extract hyperfine constants were performed in a nominal field of 6.0 T using the HiTime spectrometer in the M15 beamline at TRIUMF. This particular spectrometer is designed to provide the very high timing resolution required for high-frequency μSR spin precession measurements, and magnetic fields up to 7 T. Low-field diamagnetic precession measurements were done at ISIS, and longitudinal depolarization studies related to dynamical features are ongoing at both facilities. In the next few sections, we’ll present results on the temperature dependence of the hyperfine constants first published in 2004 [10], signal amplitudes and relaxation rates seen in all the samples we studied. We also briefly discuss some of our own preliminary conclusions regarding the relatively complicated muonium dynamics in these materials study of which is ongoing. The initial report [10] was on the hyperfine spectroscopy of muonium in 4H and 6H samples where all electrical types of the two samples (except the p-type 6H) were studied. I was not involved in that initial work; my involvement started by finishing the initial survey of precession measurements by doing the experiments for the p-type 6H sample. All the results reported in this chapter are from my own re-analysis of the old data plus analysis of new data. My results are in complete agreement with the initial report [10] regarding the hyperfine constants and their dependence on temperature as reported in the following section.
73 Texas Tech University, Hisham Bani-Salameh, May 2007
5.1 Temperature Dependence of Hyperfine Constants in SiC In the spin precession measurements performed to extract hyperfine constants, spin polarized muons were implanted into the samples with the initial spin direction normal to the magnetic field (transverse field technique). Precession spectra were recorded as time-dependent positron emission rates collected from four small counters placed symmetrically around the sample to detect emissions normal to the applied field. The positrons are emitted preferentially in the same direction as the spin (as discussed in chapter 2) and therefore the counters will each record a peak in the emission rate as the spin precesses, seeing an oscillatory signal with relative phases that indicate the precession direction. As the applied magnetic field is increased, the electron Zeeman interaction increases and dominates the hyperfine interactions at high fields. In this regime, two spin-precession signals will be observed from a paramagnetic Mu0 center with an isotropic hyperfine constant AHF, one below and one above the diamagnetic frequency. The main reason for the two frequencies at high fields is the selection rules on the muon and electron spins ( Δmμ = 1 and Δme = 0 ), this requires that the muonium precession in a transverse field occurs at only four different frequencies (ω12 , ω23 , ω34 and ω14 ) with values given by differences in the hyperfine energy levels (equation 3.13).
At high fields, only two of these (ω12 and ω34 ) will survive with sufficient strength and the other two won’t show up. The diamagnetic frequency ωd is the Larmor precession frequency (ωd = γ μ B ) due only to the moun’s magnetic moment. These hyperfine frequencies are:
1 ωω 2π ⋅⋅±Δ+= A (5.1) d 2 HF
where Δ is the shift of the center of the hyperfine splitting away from ωd and is given by
22 =Δ AHF μ + γγπ e )/( B .
74 Texas Tech University, Hisham Bani-Salameh, May 2007
When an atomic-like Mu0 resides in any material, there will be an increase in the electronic wavefunction overlap between Mu and the host atom resulting in a decrease in the electron occupation of the muonium 1s wavefunction, thus decreasing the hyperfine interaction sensed by the muon and therefore reducing the hyperfine constant from the value for the free mounium atom (A0 = 4463.3 MHz). Due to the scarcity of Si and C isotopes with non-zero nuclear moments, nuclear hyperfine contributions due to overlap onto neighboring atoms -usually seen as satellite lines near each main Mu0 hyperfine frequency- are too small to detect and therefore can be neglected, especially at the statistics collected in the current measurements. At a magnetic field of 6 Tesla, where most of this data was collected, the diamagnetic signal is at approximately ωd = .813 2 MHz. Two or more paramagnetic frequencies were observed, in addition to the diamagnetic line, for each of the hexagonal SiC samples. Figure 5.2 shows the Fourier transform spectrum obtained for all the samples of SiC we studied. The diamagnetic frequency, produced by Mu+, Mu− or any bound state with no unpaired electrons, can in principle yield a precise measurement of the magnetic field; we used the location of the diamagnetic line to calibrate the magnetic field accurately. With an applied field of 6 T and hyperfine constant AHF > 2000 MHz, the high-frequency ω34 hyperfine line is unobservable because of the spectrometer resolution limit of about 1500 MHz. The observed lines are in fact negative frequencies that fall below ωd and represent the ω12 hyperfine lines. As the applied magnetic field was increased, there was a shift of the frequencies of these lines toward lower values, this implies that all hyperfine lines seen above 4 T correspond to the negative rotation direction of the ω12 muonium transition for a large hyperfine constant. This negative rotation direction (opposite precession direction to that of the diamagnetic signal) was also confirmed by analysis of the phases for signals from each positron counter. Equation 5.1 provides a good estimate for the hyperfine constant of the frequency lines observed at high fields (π ⋅ AHF = ω12 |(| +ωd + ∇) ), but instead of using this approximate expression in the actual analysis, we did the following: we used a well-known exact
75 Texas Tech University, Hisham Bani-Salameh, May 2007
0.06 0.16 (a) (d) Mu 2 6H-SiC 4H-SiC Dia High-Resistivity High-Resistivity 9.6 K 6.0 T Dia 0.12 10 K 6.0 T 0.04
0.08
Mu Mu 1 Mu 1 Mu 2 1b 0.02
Fourier Amplitude Fourier 0.04 Mu 3
0.00 0.00
0.012 0.048 (b) (e) 6H-SiC Mu 0.010 0.040 4H-SiC 2 n-type n-type 10 K 6.0 T 10 K 6.0 T 0.008 0.032
Mu Mu 1 0.006 0.024 1
Dia Dia 0.016 0.004 Mu Fourier Amplitude Fourier Mu 3 2
0.008 0.002
0.000 0.000
(c) (f) 0.10 Mu 0.032 2 4H-SiC 6H-SiC p-type p-type 0.08 11.5 K 6.0 T 7 K 6.0 T Mu 0.024 1
0.06 Mu 1 0.016
0.04 Mu 2
Fourier Amplitude Fourier Dia Dia 0.008 0.02 Mu 3
0.00 0.000
500 550 600 650 700 750 800 850 500 550 600 650 700 750 800 850 Frequency (MHz) Frequency (MHz)
Figure 5.2 The muon spin precession spectra obtained at 6 T for (a) high-resistivity 4H- SiC, (b) p-type 4H-SiC, (c) n-type 4H-SiC, (d) high-resistivity 6H-SiC, (e) p-type 6H-SiC and (f) n-type 6H-SiC.
76 Texas Tech University, Hisham Bani-Salameh, May 2007 results [1, 13, see also discussion in section 3.1 on the diagonalization of Hamiltonian matrix for isotropic muonium] to calculate ω12 for an isotropic muonium center and then matched this value to the experimentally observed value by varying the hyperfine constant AHF. This was done at each temperature point to insure a more accurate hyperfine constant determination corresponding to the measured ω12 at that temperature. TF-μSR signals are usually analyzed online with fast Fourier transforms, some examples are shown in figure 5.2. This serves as a quick and easy way to see the number of signals at that particular temperature and field and obtain an initial estimate of their precession frequencies. To get more accurate estimates of the experimental parameters (precession frequencies, asymmetries, phases and relaxation rates), we performed time- domain fits (see section 2.3). Since the muon precession frequencies are very high at 6 T, we used a reference frame set to rotate at a frequency of 0.5–1.0 MHz below the signal frequency. The data were binned so that frequencies far from the reference frame frequency were not observable. No indication of an anisotropic hyperfine interaction was detected since no frequency variations outside of statistical uncertainties were found upon changing the c-axis orientation with respect to applied field. These results give isotropic hyperfine constants that fall roughly midway between those assigned to a ‘normal’ Mu atom located at T-sites in Si and diamond. For the temperature dependence of the hyperfine constants of all the signals, we investigated the temperature region from 2.5 K to about 300 K. The hyperfine constant is nearly constant at low temperatures (up to 70 K) and falls gradually as temperature increases (see figure 5.3). Each of the observed Mu0 centers in SiC is atomic-like and is expected to reside in a relatively open interstitial region of the hexagonal structure with reduced hyperfine constant from its free atom value because of overlap of the Mu 1s wavefunction with neighboring host atoms. Long-wavelength acoustic phonons modulate the effective size of the region occupied by the Mu0 atom and therefore modulate this overlap. As the lattice vibrations increase, the overlap increases causing the hyperfine constant associated with the central muon to decrease. Another factor that might affect the hyperfine constant is the lattice expansion, this effect would reduce the overlap with
77 Texas Tech University, Hisham Bani-Salameh, May 2007
neighboring host atoms and therefore increase AHF. The interaction with long-wavelength phonons is stronger than the effects of lattice expansion and therefore the net effect is reduction in AHF.
3030 4H-SiC Mu 1 3025
High-Resistivity 3020 p-type
A = 3029.958 MHz 3015 0 θ = 824.250 D C = 0.136
Hyperfine Constant (MHZ) Constant Hyperfine 3010
3005 0 50 100 150 200 250 300 350 Temperature (K)
Figure 5.3 The temperature dependence of the hyperfine constant for the Mu1 signal seen in all three samples of 4H-SiC as marked in the figure. The curve is a fit of the data points to equation 5.2 leading to the listed parameters.
By treating the phonons within the Debye model [10], one obtains a temperature- dependent hyperfine interaction that can be described by:
θ T xT 3 −= CATA )(1[)( 4 dx] (5.2) 0 ∫ x θ 0 e −1
78 Texas Tech University, Hisham Bani-Salameh, May 2007
This basic model was used very successfully to treat the temperature-dependent hyperfine constants for the neutral muonium centers in several other common semiconductors [1]. When comparing the results for the effective Debye temperatures θ from this model, it is usually lower, approximately 2/3 of θD obtained from specific heat measurements, suggesting that only transverse phonon modes are involved in modulating the hyperfine interaction.
Table 5.1: Parameters acquired by separately fitting the hyperfine constants to equation 5.2 for each of the Mu0 signals found in the 4H and 6H SiC samples.
Sample Signal AHF (T = 0) MHz θD (K) C
Mu1 3029.930 ± 0.012 815.805 ± 1.5 0.133 ± 0.0009 4H (HR) Mu2 2824.742 ± 0.165 777.337 ± 8.9 0.136 ± 0.0057
4H (p- Mu1 3029.930 ± 0.165 839.894 ± 1.5 0.143 ± 0.0011 type) Mu2 2825.278 ± 0.116 744.070 ± 5.7 0.133 ± 0.0024 4H (n- Mu (3029.8) 1 type) Mu2 (2824.6) Mu1 3011.152 ± 0.018 807.823 ± 1.8 0.134 ± 0.0012
6H (HR) Mu1b 3026.158 ± 0.093 762.680 ± 2.7 0.113 ± 0.0011
Mu2 2768.249 ± 0.033 781.805 ± 1.3 0.157 ± 0.0007 6H (p- Mu1 3007.589 ± 0.018 739.842 ± 2.7 0.104 ± 0.0012 Mu 2767.907 ± 0.033 833.508 ± 1.8 0.190 ± 0.0012 type) 2 Mu3 2800.372 ± 0.033 910.801 ± 3.0 0.229 ± 0.0012 6H (n- Mu1 (3008.1) Mu (2768.2) type) 2 Mu3 (2801.1)
Low-temperature hyperfine constants along with parameters for the temperature dependence of AHF for each of the samples studied are shown in table 5.1. These results are from our own recent analysis and they agree well with the results from the earlier report [10]. The parameters shown in the table are results of separate fits of equation 5.2
79 Texas Tech University, Hisham Bani-Salameh, May 2007 to each Mu0 signal seen in the samples. The Mu0 signals in n-type samples of both 4H and 6H significantly broaden beyond detection above 100 K as the nitrogen donors ionize. Because of this, there weren’t enough data to yield reliable parameters for the temperature dependence. However, the low temperature results for AHF in n-type samples are consistent with data from the other two electrical types. To get more accurate hyperfine constants for the various signals, we put data for the same signal seen in different types of the same material together and fit to equation 5.2. We didn’t use the data from the n-type samples for the reason discussed earlier. Figure 5.3 shows the parameters from fitting the data from the high-resistivity and the p- type 4H-SiC samples together to equation 5.2. The same thing was done for the second signal seen in the 4H samples (Mu2) and the results were: A0 = 2824.938 MHz, θD = 756.207 and C = 0.132. Despite the increasing interest in SiC in the recent years, there has been very little experimental work aimed at studying Mu in any of the common structural phases of this material. We were able to locate only one report on Mu centers in SiC published in 1986 [9]. This report shows three atomic-like Mu0 centers at 20 K in 6H-SiC and only two at 310 K. The low temperature results were collected with the magnetic field applied perpendicular to the hexagonal c-axis and gave A⊥ hyperfine constants of 2767.85, 2797.32 and 3005.73 MHz for the three Mu0 states, which agree well with the current results on the n-type and p-type 6H samples except for a small shift. Due to difficulty in keeping N out during the growth process of these older samples, they most likely were n- type; the electrical type of our samples is much better controlled. There had also been a very brief attempt (unpublished, 1990 [56]) to look at Mu0 in 4H-SiC that shows two hyperfine signals at low temperatures but only one near room temperature.
5.2 Dynamics of Muonium States in 4H and 6H SiC In this section, we report on some of our experimental results on the different dynamical features observed in these samples. Final conclusive picture of muonium
80 Texas Tech University, Hisham Bani-Salameh, May 2007 behavior in SiC is still not clear and might need a significant amount of additional work. Our goal here is to report what we’ve seen and to discuss some possible explanations.
13
12
e
d u
t 11
i
l p
m 10
A
c
i t
e 9
n g
a 8
m
a i
D 7
6 400 0 300 20 40 200 (K) Mag 60 100 ture netic 80 era Field 100 0 mp (Ga Te uss)
Figure 5.4 A 3-D representation of the diamagnetic signal amplitude dependencies on temperature and magnetic field for the n-type 4H-SiC sample.
We begin with figure 5.4 that shows a 3-D representation of the magnetic field and temperature dependencies of the low-field diamagnetic precession signal in n-type 4H–SiC. At low temperatures (< 50 K), the amplitude of this signal is decreasing with the applied field strength; this is characteristic of a slow-forming state. The increasing amplitude above 50 K implies a second slow forming state which is supported by a dip in the phase with temperature at any given field; together these signatures are a clear indication of a transition into that state. The evidence for two slowly forming states is even clearer in the n-type 6H sample. In figure 5.5 below, the diamagnetic amplitude at several fields is displayed as a function of temperature. The decrease in amplitude with
81 Texas Tech University, Hisham Bani-Salameh, May 2007 increasing field is evident at all temperatures. At low temperatures, the amplitude decreases before it starts increasing, implying two separate diamagnetic states. The results displayed in figure 5.4 suggest the presence of two diamagnetic states, assumed initially to be Mu+ and Mu − , and imply transitions into and out of each state. Because this is an n-type sample, if these are in fact the isolated ionic states, the low-T state is likely Mu+ which converts to Mu0 near 50 K closely following donor ionization, and the increase above 70 K should be electron capture by a neutral state to form Mu − . The temperature offset would be consistent with the different capture cross sections expected for these processes.
9
n-type 8 6H-SiC
7
6 100G 70G
5 50G 40G 4 2G 10G Diamagnetic Amplitude 5G 3 2G
2 0 40 80 120 160 200 240 280 320 Temperature (K)
Figure 5.5 The diamagnetic amplitude at several magnetic field as a function of temperature for the n-type 6H-SiC sample.
82 Texas Tech University, Hisham Bani-Salameh, May 2007
0.024
0.018
0.012 Asymmetry
0.006
0.000
100
p-type 6H-SiC
10
1 Relaxation Rate Relaxation
0.1
0 50 100 150 200 250 300 Temperature (K)
Figure 5.6 Temperature-dependent amplitudes (top) and relaxation rates (bottom) in p- type 6H-SiC acquired at 6 T: diamagnetic (filled squares), Mu1 (open squares), Mu2
(circles) and Mu3 (triangles).
83 Texas Tech University, Hisham Bani-Salameh, May 2007
The diamagnetic signal at 6 T in both 4H- and 6H-SiC shows a hint of two unresolved frequencies, most obvious as a small upward shift in frequency below 80–100 K for a single signal fit. An undergraduate student working with the group managed to separate two components in one 4H sample. Based on tentative assignments of Mu+ and Mu − in n-type material as discussed above, we suggest that Mu+ might have a higher frequency than Mu − by about 85 ± 15 kHz at 6 Tesla. This is an order of magnitude larger shift than should be expected [57], thus might imply that one of these signals has a different origin, probably a Mu-defect complex. All high frequency spin precession signals (6 T) detected in high-resistivity and p- type 6H samples show similar temperature dependence (see figure 5.6 for the p-type signals). The diamagnetic state shows a weak amplitude peak near 150 K, the same temperature region where all of the Mu0 signals have a strong amplitude dip. The relaxation rate of all four signals has a peak at around 120 K. There was no obvious connection in the p-type 6H data between the diamagnetic signal and any of the neutrals in terms of the activation energy. The increase in relaxation rates for the neutral signals above 200 K probably indicates the onset of ionization, but could be related to Mu0 site changes as well, this alternatively might signal the onset of diffusion.
Table 5.2 Activation energies from the diamagnetic amplitudes seen at low applied field in all three electrical types of 6H. Temperature region Activation Energy
(K) (meV) 50-150 36.7 ± 4 n-type 550-1000 545.9 ± 34 200-350 232.8 ± 19 p-type 600-1000 886.2 ± 63 300-500 275.6 ± 36 High-resistivity 700-1000 862.4 ± 168
84 Texas Tech University, Hisham Bani-Salameh, May 2007
1.0 p-type n-type 886 meV 0.8 , hr
546 meV 0.6 862 meV 276 meV 0.4 36 meV Fractional Amplitude 0.2 233 meV
0.0 0 200 400 600 800 1000 1200 Temperature (K)
Figure 5.7: The full temperature-dependent diamagnetic signal amplitudes in p-type (squares), n-type (circles) and hr 6H-SiC (triangles and stars). The curve for the hr sample contains data acquired at two different applied fields; 15 G (triangles) and 100 G (stars).
Figure 5.7 displays the full temperature dependence of the low-field diamagnetic precession signal for p-type, n-type and high-resistivity 6H samples. We extracted activation energies for various features related to the amplitudes of the diamagnetic signals in different temperature regions (see table 5.2). The first step increase for the n- type sample in the temperature region 50-150 K yields an energy of 36.7 ± 4 meV consistent with donor ionization and electron capture by a Mu0 acceptor state. The two
85 Texas Tech University, Hisham Bani-Salameh, May 2007
100 P-6H-SiC TF=6T
10
Mu 1
Relaxation Rate Relaxation 1
Mu 2 0.1 0 50 100 150 200 250 300 Temperature (K)
Figure 5.8 Relaxation rates of Mu1 and Mu2 in p-type 6H-SiC along with fits to extract activation energies. steps for the p-type sample give 233 ± 19 meV and 886 ± 63 meV as activation energies. We were able to correlate the 233 meV charge-state transition with at least one of the two neutral muonium signals Mu1 and Mu2 detected in this sample at an applied field of 6 T
(figures 5.6 and 5.8). The activation energy from the increase in relaxation rate for Mu1 at
6 T (figure 5.8) is 228 ± 13 meV and that for Mu2 is 189.8 ± 5 meV. To obtain these numbers, we added a Lorentzian function to account for the peak in relaxation around 150 K in the fits shown in figure 5.8. These energies are consistent with hole capture in p-type 6H-SiC, where Ea for the Al acceptor is between 220 and 260 meV [58]. The similarities in activation energies for the two steps in the p-type and the hr samples suggest that the transition processes may be the same for these two samples.
86 Texas Tech University, Hisham Bani-Salameh, May 2007
0.08
0.07 a) 0 Mu Dia 0.06
0.05
0.04
Asymmetry 0.03
0.02
0.01 40 b) hr-6H-SiC 30 TF=15G
20
Mu0 10 Relaxation Rate
100 200 300 400 Temperature (K)
Figure 5.9 Results from TF measurement on the high-resistivity 6H-SiC sample taken at 15 gauss: a) The amplitudes for the diamagnetic (squares) and triplet Mu0 signals (circles) along with fitting line to extract energy for the diamagnetic amplitude. b) Relaxation rate for Mu0 along with fitting line to extract energy.
87 Texas Tech University, Hisham Bani-Salameh, May 2007
The processes are clearly different in the n-type sample. We are still not so sure about the final diamagnetic charge-state for either step in figure 5.7. We nevertheless are tempted to assign these charge-state transitions in the p-type sample to electron and hole ionizations from the Mu donor and acceptor sites in 6H–SiC. More work is required to verify this and to make specific site assignments. Another correlation we managed to establish is in the high-resistivity 6H-SiC sample. The high frequency spin precession data show that all signals have similar temperature dependence as shown above for the p-type sample. Again, there was no obvious connection between the diamagnetic signal and any of the neutrals in terms of activation energy. The low-field data we took later to explore the composite Mu0 dynamics above 300 K reveal the kind of correlation we were looking for in the high frequency data. The diamagnetic signal is almost flat till about room temperature and then starts increasing (see figure 5.9), the energy associated with this increase is 268.8 ± 31 meV. At the same temperature, there is a big drop in the amplitude of the neutral signal associated with an increase in its relaxation rate from which we obtained an energy of 196.3 ± 35 meV. This suggests that at least partial ionization of Mu0 states is taking place above room temperature, but unfortunately, the triplet transitions probed by this low-field technique do not distinguish which Mu0 state is involved. Another technique, called muon magnetic resonance, might identify which Mu0 state is involved in the transition to the diamagnetic state above room temperature. In this technique, one applies a strong static magnetic field parallel to the initial muon spin and then applies a weak oscillatory magnetic field (radio frequency or microwave frequency) perpendicular to the static field to induce transition among energy levels. The resonance is detected by monitoring the emitted positrons from the muon decay. Recent data using 0 this technique show that all three Mu signals (Mu1, Mu1b and Mu2) are actually disappearing in the temperature range 280-380 K, this is evident from the rapidly decreasing amplitudes (see figure 5.10 below). The relaxation rate for Mu2 show a decrease with temperature but for the other two signals, it was almost flat. This tells us that only Mu2 is involved in the transition to the diamagnetic state and the decrease in the
88 Texas Tech University, Hisham Bani-Salameh, May 2007 other two signals amplitude is simply due to the disappearance of the source that produces them. The correlation seen in fig. 5.10 suggests this source is probably Mu2.
120
100
80 hr 6H-SiC
60
40
20 Relaxation Rate Relaxation
0
40 ) -4 30 Mu 2
Mu 20 1 Mu 1b 10 Amplitude (x10 Amplitude
0 280 300 320 340 360 380 Temperature (K)
Figure 5.10 Relaxation rates (top) and amplitudes (bottom) of the three neutral signals detected in the hr 6H sample via muon magnetic resonance.
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From experimental data on n-type 6H–SiC sample at 6 T, the Mu0 signals disappear by 100 K and the amplitude of the diamagnetic signal is constant up to 70 K then rises with temperature consistent with a conversion from Mu0 to diamagnetic states; however the onset for the rise is rather abrupt at 70 K. This rise in the diamagnetic signal amplitude gave an energy of 28 ± 14 meV which was not correlated with anything from the neutral signals due to scatter in the data. This energy however, might be consistent with donor ionization in this low-temperature region and an electron capture that feeds into the existing diamagnetic state, supporting the low-field results. The same two paramagnetic states are present in all 4H samples regardless of electrical type. Hyperfine constants are essentially identical for all three samples as we reported above, 3029.958 ± 0.01 MHz for Mu1 and 2824.938 ± 0.17 MHz for Mu2 from combined data for 4H–SiC samples. From the 6 tesla TF data on n-type 4H–SiC, both of the Mu0 signals broaden above roughly 50 K and are nearly gone by 100 K. The diamagnetic amplitude increases almost linearly up to 70 K and then is roughly flat to 300 K. The high-resistivity 4H data suggest a relatively slow transition from paramagnetic to diamagnetic states above roughly 150 K. This is related to the detected increase in relaxation rates of the two neutrals above 150 K and the continuous decrease in their amplitudes. There is also an evidence of new slowly-forming state around 650 K from the large dip in the phase of the diamagnetic signal seen at that temperature in this sample at 80 gauss transverse field (see figure 5.11). This dip in the phase is occurring at the middle of increase of the amplitude for that signal suggesting that the new forming state has a different charge on it or is located at a different site.
/−+ In the p-type 4H sample, the results imply a possible 1 → MuMu → Mu2 conversion sequence between 150 and 220 K. In this region we see a decrease in the amplitudes of the diamagnetic and Mu1 signals associated with what looks like a slight increase in the Mu2 amplitude (figure 5.12). The diamagnetic signal also show a narrow peak in its relaxation rate at roughly 200 K which indicates that part of this signal is disappearing and converting to something else.
90 Texas Tech University, Hisham Bani-Salameh, May 2007
8.0 hr-4H-SiC TF = 80G
7.5
Amplitude 7.0
6.5
-2 -4 -6 -8 -10
Phase -12 -14 -16 -18 200 300 400 500 600 700 800 900 1000 Temperature (K)
Figure 5.11 Amplitude (top) and phase (bottom) of the diamagnetic signal seen in hr 4H- SiC sample at a transverse applied field of 80 Gauss.
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8 7 p-4H-SiC 6 TF = 6T 5 4
0.6
0.4
Relaxation Rate 0.2
0.0 0.040
0.035 Mu 0.030 1
0.025 Mu
0.020 2
0.015
Asymmetry Dia. 0.010
0.005
0 50 100 150 200 250 300 Temperature (K)
Figure 5.12 The relaxation rates (top) and amplitudes (bottom) for the diamagnetic
(squares), Mu1 (circles) and Mu2 signals (triangles) from a TF- μ SR data on the p-type 4H-SiC sample taken at 6 tesla.
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We have not found any indication of rapid cyclic charge-state conversions in SiC, as is commonly found for semiconductors with gaps below about 1.5 eV. However, there are several features (especially in the p-type 4H sample) which appear to indicate one- way charge-state transition sequences, either from one neutral to another via an intermediate diamagnetic state, or from one ionic charge-state to the other via an intermediate Mu0 state. This kind of transition must be accompanied by a site change in some part of the sequence in each case. The details are not yet completely clear for any of these one-way multi-step conversion sequences.
5.3 Discussions and Conclusions In the early 1990’s, Estreicher and coworkers [35,59] worked on hydrogen defects in SiC. They modeled hydrogen in the 2H and 3C structures and gave what appeared to be a convincing set of site assignments for muonium centers in 4H- and 6H-SiC. Available sites for H or Mu in 4H and 6H structures are a mix of those found in the 2H and 3C structures in terms arrangement of nearest neighbors. Their calculations indicate that there are two nearly identical 3C-like metastable sites in 6H-SiC (only one in the 4H structure) and a lowest-energy location centralized in the 2H-like unblocked channel region of the 6H structure labeled R. The two 3C-like sites both have tetrahedral local
\ symmetry and Si nearest neighbors thus were labeled TSi and TSi . Accordingly, they assigned the two lower AHF signals seen in the older data for 6H-SiC [35,59] to the two
TSi sites, and the higher AHF state to the R location. The R site was found to be the lowest 0 energy location for Mu in 4H or 6H-SiC with the TSi sites metastable at relatively low energies. More recently, in 2001, Aradi [60] investigated a number of hydrogen-related defect centers in 3C-SiC; he found the carbon antibonding (ABC) site to be the most stable H0 location. The localized electronic energy level for this site lies within the conduction band, therefore; losing the e- to the bottom of the band stabilizes that site and leads to shallow-donor behavior for H at this site. Although this group didn’t pursue detailed modeling of H in the hexagonal structures of SiC, they suggested that the TSi location,
93 Texas Tech University, Hisham Bani-Salameh, May 2007 which they found to be next lowest in energy for H0 in 3C, would probably be more stable in 4H or 6H due to the larger bandgap. This group’s assignment for the hydrogen + – ionic states placed H at ABC as the stable location and TSi as the metastable one. For H ,
TSi is the more preferred location over TC. Two years after Aradi [60], new theoretical results by Kaukonen and coworkers 0 + – [61] confirmed these site assignments placing H and H at ABC and H at TSi in 3C-SiC. This group also did some calculations for the hexagonal structures and found both H– and 0 H to be located at TSi with various inequivalent bond-centered (BC) sites at relatively 0 low energy for H and ABC at much higher energy. With these predicted theoretical locations in mind, we now try to make a comparison with our observed signals. We start by immediately eliminating any of the BC sites from consideration as a possible location for any of the neutrals. All of the observed hyperfine signals imply nearly isotropic atomic-like centers, if these were to reside at a bond-centered location, the Mu0 would have a much smaller and highly anisotropic hyperfine interaction oriented along bond directions as the well established cases of MuBC in Si or diamond. This leaves the TSi sites and the centralized channel site R as well as carbon related locations as possible locations for Mu0. 0 - From the theoretical results by Kaukonen, both H and H are stable at TSi in 4H and 6H polytypes which means that H has (deep) acceptor character at this location. Since both H and Mu typically have compensating electrical properties, it is possible to 0 - assume that under equilibrium conditions, the TSi site should switch from Mu to Mu as the dominant charge state when going from p-type to n-type material. This trend in the Mu signal amplitudes is in fact observed in most semiconductors even though site and charge-state populations for Mu are not expected to reach equilibrium within the μSR timeframe of a few microseconds. For the 4H samples, the Mu1 signal amplitude appears to be transferred to the diamagnetic amplitude in going from p- to n-type, while this trend is not observed in the Mu2 signal. Therefore, the Mu1 signal in 4H-SiC was re- assigned at least tentatively to TSi [10]. Because of the similarity in frequencies and local
94 Texas Tech University, Hisham Bani-Salameh, May 2007
structure, one should also assign Mu1 and Mu1b in the high-resistivity 6H sample to the two slightly different TSi locations.
There are three different types of ABC sites in hexagonal structures. One of these is located within the blocked 2H-like ‘cage’ and represents a high-energy site, therefore is not important as we discuss possible locations for Mu states. One of the other two sites lies in the cubic-like regions with the C-Mu bond pointing toward a TC location. The last site is in 2H-like regions with C-Mu bond oriented into an open channel roughly toward the location labeled as R [10], it is this site that is most likely referred to as ABC in the theoretical discussions. In the 2H-like region in 4H and 6H structures, there is a point where three ABC orientations intersect and a similar point for three ABSi orientations, the R site is located midway between these two points (see figure 5.13). It’s been suggested that the sites giving both lower AHF signals may lie in the 2H-like channel region [10]. Figure 5.13 shows the 4H-SiC structure with the anti-bonding sites, tetrahedral sites and the R site indicated. This representation of the 4H structure was constructed based on the lattice parameters a = 3.073 Ao and c = 10.053 Ao [62], assuming that the C-Si bond length is the same in all four directions.
Muonium has a large zero-point motion, therefore, part or all of the ABC/R/ABSi region is likely to merge into a single site. Because of charge separation in the Si-C bond, Mu+ is more strongly attracted to slightly negative C atoms and Mu0 prefers sites that are closer to the slightly positive Si atoms [35]. From this argument the conclusion is that if one of the Mu0 centers resides in the 2H-like channel, the most likely location is + somewhat displaced from the R site toward the Si-plane. If a Mu located at ABC (or on the channel axis offset from R toward the C-plane) captures an electron after it stops, there maybe a relatively small barrier for movement to the preferred nearby Si-related location, thus leading to two separate Mu0 signals from this region at very low temperatures. Depending on the exact Mu0 formation processes, delayed e- capture is more likely for n-type materials, suggesting that the weaker Mu3 signal seen in the n-type 6H sample might arise from the C-plane side of R in the channel region.
95 Texas Tech University, Hisham Bani-Salameh, May 2007
Si
C
T Si
c
AB C AB R Si
T C
Figure 5.13 A 3-dimentional representation of the 4H structure showing the positions of the anti-bonding sites, tetrahedral sites and the centralized channel site, R.
Proper understanding of the different dynamical features observed in these samples may contribute to a more accurate site assignments of Mu signals. Our investigation of Mu dynamics in SiC is at an early stage, with a number of features identified for further study. These dynamics appear to be far more complicated for 6H- SiC compared to those seen for 4H samples, as might have been anticipated due to additional sites. The fact that we observe the same set of Mu0 signals up to 300 K in the high-resistivity samples implies that these atomic-like centers are not mobile in SiC. This observation put a question mark on the suggestion of motional averaging of two Mu0 sites as a possible explanation for the disappearance of one Mu0 signal in the earlier data [9] and dynamics seen near 120 K in our data on 6H-SiC.
96 Texas Tech University, Hisham Bani-Salameh, May 2007
Based on results acquired from spin precession measurements, we confirm at least four neutral muonium states in 6H–SiC and two in 4H–SiC. There is evidence for two diamagnetic states, at least one slowly formed, in both 4H and 6H samples. All hyperfine interactions for Mu0 in SiC imply isotropic atomic-like states. Unlike the data for Si or diamond, there is absolutely no hint of any bond-centered Mu0 species in SiC. Based on this simple and evident result, the reasonable conclusion is that the BC site becomes unstable for Mu0 due to charge transfer in the Si-C bond. The theory results cited above suggest BC sites to be at higher energy but still meta-stable. If that is true, then why are they not seen at low temperatures? Relying on recent theoretical predictions of the sites for all three charge states of hydrogen in hexagonal SiC [61], and observed acceptor-like amplitude variations for the
Mu1 signal among the three 4H samples, we propose [10] that Mu1 is located at the TSi sites at one end of short c-axis channels in the 4H and 6H structures. There are two such sites in 6H and a single one in 4H, thus the double signal in that region for the high- resistivity 6H sample supports that assignment. However, the single Mu1 signal in both n- and p-type 6H–SiC, calls that assignment into question. We also suggest that Mu2 is likely to be at the Si anti-bonding (ABSi) sites near the center of these channels, and that
Mu3 may be from the ABC site in the same region, a donor location in hexagonal SiC. With the p-type 6H results, we are now somewhat less certain of these assignments. In the end, site assignments must be consistent with transition processes and the states involved. However, there is a considerable amount of work remaining in order to identify the precise nature of the complicated muonium dynamics in SiC.
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CHAPTER 6 SUMMARY AND CONCLUDING REMARK
In this dissertation, results of μSR measurements on Silicon Carbide (SiC) and Aluminum Nitride (AlN) have been reported. The goal was to characterize the muonium centers (Mu = μ+e-) present in those two materials in terms of locations and dynamical features. A variety of μSR experimental techniques were utilized for this purpose including LF-µSR and TF-µSR. Despite some serious efforts, the spin-precession signals have not been observed in AlN, thus the majority of our results on AlN reported in this dissertation were extracted from hyperfine decoupling curves and longitudinal relaxation data. From this data, we confirm the existence of an atomic-like neutral muonium center Mu0 with a small anisotropy. This conclusion was reached as a result of fits of the experimental decoupling curves to an approximate repolarization function by Pratt [31]. Hyperfine constants acquired from the fits were close to the free neutral muonium value with weak dipolar contribution. Based on the anisotropy and lack of covalent bond formation with host atoms, we propose that the most likely location for Mu0 in AlN is off-axis (toward the Al atom) in the unblocked c-axis channels in the so-called antibonding location (ABAl). Mobility of Mu0 in AlN is evident from the data as we see field-dependent longitudinal relaxation rates at all the temperatures tested. To extract Mu0 hop rates and conversion rates, we used a simplified treatment developed as an approximation for isotropic Mu0 centers by Celio [20] and Yen [21] to fit the field-dependent rate constants at different temperatures. Our final results on the temperature-dependent hop rates indicate that Mu0 motion in AlN is dominated by five different processes in different temperature regions. Minimum hop rates were seen around room temperature which marks the crossover between quantum tunneling behavior, dominating at lower temperatures, and thermally activated processes which take over at higher temperatures. In the temperature region below 100 K, we suggest that there is a switch from local tunneling motion below 25 K to two-dimensional tunneling-based diffusion with a barrier
98 Texas Tech University, Hisham Bani-Salameh, May 2007 of about 15 meV. The decrease in hop rates as the temperature increases above 100 K fits well to an inverse power law relationship of about T–6, this is a clear indication of quantum tunneling that is interrupted by phonon scattering. Above 300 K, phonon- assisted tunneling followed by (at around 550 K) standard three-dimensional thermal diffusion successively dominate the Mu0 motion. Energy barriers obtained for diffusion of Mu0 and its conversion to another state are very similar suggesting a diffusion-limited transition process at both high and low temperatures. Based on supporting evidence from LCR measurements, we conclude that there are two different mechanisms for conversion at low and high temperatures. While charge transfer interactions between Mu0 and another impurity is responsible for the conversion at low temperatures, it is trapping of the thermally diffusing Mu0 and formation of a Mu-Defect complex that is the main mechanism of conversion at high temperatures. We performed a variety of μSR measurements on all types of the 4H-SiC and 6H- SiC samples. We found two isotropic Mu0 states in 4H-SiC and a total of four states in 6H-SiC. We used well-known exact results [1, 13, see also section 3.1] to calculate the hyperfine constant AHF of individual signals at all temperatures tested. Temperature dependences of the hyperfine constant are well characterized by interaction with long- wavelength phonons having an effective Debye temperature of 800 K. All the data on the hyperfine interactions imply isotropic atomic-like states with no hint of any bond- centered Mu0 species in SiC with hyperfine constants that fall roughly midway between those assigned to a Mu atom located at T-sites in Si and diamond. The BC location presumably becomes unstable due to charge transfer in the Si–C bond; this leaves the TSi sites and the centralized channel site R as well as carbon related sites as possible locations for Mu0. Temperature and field dependences of signal amplitudes and relaxation rates were studied. At low temperatures, the amplitude of the diamagnetic signals is decreasing with increasing the applied field which is characteristic of a slow-forming state. These results, along with a small upward shift in the amplitudes below 80–100 K, suggest the presence
99 Texas Tech University, Hisham Bani-Salameh, May 2007 of two diamagnetic states, assumed initially to be Mu+ and Mu-, and imply transitions into and out of each state.
One of the ABC sites is located in 2H-like regions with C-Mu bond oriented into an open channel roughly toward the location labeled as R. We suggest that the sites giving both lower AHF signals may lie in the 2H-like channel region. Because of charge separation in the Si-C bond, we can farther conclude that one likely location for a Mu0 center is in the 2H-like channel somewhat displaced from the R site toward the Si-plane. Based on recent theoretical predictions [61] and observed acceptor-like amplitude variations for the Mu1 signal among the three 4H samples, we propose [10] that Mu1 is located at the TSi site in the 4H and 6H structures. In the 6H structure, there are two slightly different TSi sites, therefore, one expects to see two signals from this site. This assignment was supported by the two signals seen in the high-resistivity 6H sample, but the single Mu1 signal seen in both n- and p-type 6H–SiC, calls that assignment into question. We also suggest that Mu2 is likely to be at the Si anti-bonding (ABSi) sites near the center of the c-axis channels, and that Mu3 may be from the ABC site in the same region. In order to be certain of any site assignment, one should establish a consistent correlation with transition processes and the states involved. At this stage, there is a considerable amount of work needed to identify the precise nature of the complicated muonium dynamics in SiC and therefore, we can’t be certain of our site assignments. Our results on Mu centers in AlN and SiC represent, at least qualitatively, an experimental model for the analogous hydrogen defect center in these materials. The most important contribution of μSR to the understanding of the nature of the Hydrogen defect in various semiconductor materials is the information it provides on the electronic structure. Dynamical features in Mu motion, such as quantum tunneling, will be much less prominent for hydrogen because of the mass difference. As a consequence of zero- point energy differences, the barriers for thermally activated diffusion are expected to be larger for H by a few tenths of an eV compared to the values obtained for Mu. With all this in mind, the results from μSR experiments can still provide researchers working on
100 Texas Tech University, Hisham Bani-Salameh, May 2007 hydrogen with a very good starting point in terms of possible site assignments and transitions involved.
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