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Magnetic properties of sputtered CoCr films and magneto-optics of rare earth-transition metal multilayers Li, Zhanming 1988

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M A G N E T I C P R O P E R T I E S O F S P U T T E R E D CoCr FILMS A N D M A G N E T O - O P T I C S OF R A R E EARTH-TRANSITION M E T A L MULTILAYERS By Zhanming Li B.Sc, Zhongshan University, China, 1982 M . S c , University of British Columbia, 1984  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS  FOR T H E D E G R E E OF  D O C T O R OF PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A May 1988 ©Zhanming Li,  1988  In presenting this thesis degree  at the  in partial fulfilment  of the  requirements  University of British Columbia, I agree that the  for an advanced  Library shall make it  freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes department  or  by  his  or  her  may be granted  representatives.  It  is  by the  understood  that  head of my copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Abstract The goal of the thesis is to make contributions to the development of two new technologies for data storage: perpendicular recording and magneto-optic recording. CoCr and rare earth-transition metal multilayers are the most suitable media for perpendicular recording and magneto-optic recording technologies, respectively. In part A of the thesis, magnetic properties of CoCr thin films produced by dc magnetron sputtering are studied for various deposition conditions. Dielectric constants and extraordinary Hall effect are also studied to provide information complementary to magnetic properties. In part B, new methods are developed for theoretical analysis of the magneto-optics of rare earth-transition metal multilayers, which can be used to optimize the readout of the recording system. Part A For dc magnetron sputtered CoCr films the perpendicular and parallel magnetic coercivities are found to be mainly controlled by the substrate temperature during film growth. Substrate temperatures between 180 and 300 C are necessary to fabricate CoCr thin films for recording media. Films produced in this manner have magnetic anisotropy constants ranging from —1.0 to +0.5 10 erg/cc. The magnetic anisotropy has a complicated depen6  dence on a large number of deposition parameters and can be best controlled by the dc sputtering power and the target-to-substrate distance. Based on microstructural analysis film properties are interpreted in terms of the adatom diffusion during film growth. It is found that high adatom mobility and low deposition rate promote positive magnetic anisotropy. The dielectric constants measured by ellipsometry are found to depend on the film thickness because of the change in film morphology during film growth. The effects of asymmetric sputtering are analyzed, and the relationship between the extraordinary Hall effect and the magnetic properties is investigated. Part B The 4x4 matrix method proposed by Lin-Chung and Teitler[P. J. Lin-Chung and S. Teitler, J. Opt. Soc. A m . A 1 703(1984)] is applied to the magneto-optics of the rare earth-transition metal multilayer system. Based on a plane wave model, the above method enables one to calculate the sensitivity of the readout to the layer thicknesses as well as effects of oblique angle of incidence, anisotropy in the nonmagnetic part of the dielectric constants and misalignment of the magnetization. Finally, an improved model is presented to take into account the fact that the reading laser is a strongly focused beam instead of ii  a plane wave. This new model is used to optimize the magneto-optic multilayer system. When the focal spot size of the reading laser beam is less than about three wavelengths, significantly different results are obtained from the focused beam and the plane wave models.  iii  Table of Contents Abstract  ii  Table of Contents  iv  List of Figures  vi  Acknowledgements  viii Chapter 1 Introduction  1.1. Significance of Research on Magnetic Thin Films  1  1.2. Media for Perpendicular Recording and Magneto-Optic Recording  2  1.3. Organization of the Thesis  3 Part A  M a g n e t i c Properties of C o C r F i l m s Chapter 2 Sputtering of C o C r F i l m s 2.1. Experimantal Apparatus  6  2.2. Film Properties  8  2.3. Effects of Substrate Temperature  10  2.4. Demagnetization Field for CoCr Films  12  2.5. Substrate Effects  13 Chapter 3  T h e C o n t r o l of Magnetic A n i s o t r o p y in C o C r F i l m s 3.1. Effects of Sputtering Power and Target-Substrate Distance  38  3.2. The Roles of Adatom Diffusion and Covering Rate  40  3.3. Effects of Asymmetric Sputtering  44 Chapter 4  O t h e r P h y s i c a l Properties of C o C r F i l m s 4.1. Saturation Magnetization and Electronic States of CoCr  57  4.2. Dielectric Constants of CoCr Films  58  4.3. Anisotropy in Dielectric Constants  60  4.4. Extraordinary Hall Effect of CoCr Films  62  iv  Part B Optics of Magneto-Optic Recording Media Chapter 5 Use of the 4 x 4 Matrix Method for Magneto-Optic Multilayers 5.1. Plane Wave Model for Magneto-Optic Recording  85  5.2. Implementation of the 4x4 Matrix Method  87  5.3. Application to Quadrilayer Recording Medium  89  Chapter 6 Interaction of Focused Light Beams With Magneto-Optic Recording Media 6.1. Problems in Magneto-Optics Theories  98  6.2. Interaction of a Strongly Focused Beam With Magneto-Optic Medium  99  Chapter 7 Summary 7.1. Summary of Part A  104  7.2. Summary of Part B  105  Appendices A . A Computer Program for Reflection Matrix of Multilayer Systems  106  B. A n Article on Reflected Beam Fields of a Light Beam Focused on a Magneto-Optic Multilayer Structure  114  C. A n Article on Optimization of Readout From Magneto-Optic Recording Medium  135  References  144  V  List of Figures Fig. 1.1. Demagnetization fields in longitudinal recording  4  Fig. 1.2. Schematics of perpendicular recording  4  Fig. 1.3. Optics in the magneto-optic recording system  5  Fig. 2.1. Schematics of the sputtering system  15  Fig. 2.2. Details of the substrate holder assembly  16  Fig. 2.3. Torque magnetometer  17  Fig. 2.4. A typical T E M photo of a CoCr  film  18  Fig. 2.5. T E D photograph corresponding to Fig. 2.4  19  Fig. 2.6. Predicted hep T E D pattern  20  Fig. 2.7. A typical cross section S E M photograph of CoCr  21  Fig. 2.8. Another S E M photograph similar to Fig. 2.7  22  Fig. 2.9. Coercivity versus substrate temperature  23  Fig. 2.10. Remanence versus substrate temperature  24  Fig. 2.11. Anisotropy constants versus substrate temperature  25  Fig. 2.12. T E M of CoCr at 210 degree C  26  Fig. 2.13. T E M of CoCr at 300 degree C  27  Fig. 2.14. Grain size versus substrate temperature  28  Fig. 2.15. M  29  s  versus T  s  Fig. 2.16. Magnetization versus internal magnetic  field  30  Fig. 2.17. V S M curve with hard magnetization axis in plane  31  Fig. 2.18. Substrate effects on perpendicular coercivity  32  Fig. 2.19. T E D photograph of NiCr  33  Fig. 2.20. Trianglar lattices in fee  34  Fig. 2.21. Fit of T E D data of NiCr to fee  35  Fig. 2.22. Fit of T E D data of NiCr to bee  36  Fig. 2.23. Fit of T E D data of NiCr to hep  37  Fig. 3.1. Anisotropy constant versus power  46  Fig. 3.2. Perpendicular coercivity versus power  47  Fig. 3.3. Direction of magnetic axis versus sputtering power  48  Fig. 3.4. X R D intensity versus sputtering power  49  Fig. 3.5. T E M photograph for W=150 watts and d  ts  = 7.8 cm  50  Fig. 3.6. Same as Fig. 3.5 except W=750 watts  51  Fig. 3.7. T E D for the same sample as in Fig. 3.5  52  Fig. 3.8. T E M for W=150 watts and d =5.3  53  te  vi  cm  Fig. 3.9. Same as Fig. 3.8 except W=350 watts  54  Fig. 3.10. Same as Fig. 3.8 except W=750 watts  55  Fig. 3.11. M - H curve for a sample with 8 degrees of misalignment  56  Fig. 4.1. Electron density of states for two subbands  65  Fig. 4.2. Combined electron density of states  66  Fig. 4.3. ci for CoCr on glass  67  Fig. 4.4. 62 corresponding to Fig. 4.3  68  Fig. 4.5. Dielectric constants for CoCr sputtered on NiCr  69  Fig. 4.6. Same as Fig. 4.5 except T = 250 C  70  Fig. 4.7. Same as Fig. 4.5 except T = 250 C on T i  71  Fig. 4.8. Surface S E M of 230 nm CoCr on NiCr  72  Fig. 4.9. Same as Fig. 4.8 except the thickness is 400 nm  73  Fig. 4.10. Same as Fig. 4.8 except the thickness is 910 nm  74  Fig. 4.11. Surface S E M of 900 nm CoCr on heated NiCr  75  Fig. 4.12. Plots of Drude model  76  Fig. 4.13. Definition of Euler angles  77  Fig. 4.14. Complex reflection ratio versus <f>  78  Fig. 4.15. Sample and circuit diagram for Hall effect  79  Fig. 4.16. Schematics of the cryostat  80  s  8  Fig. 4.17(a). A typical Hall voltage output for T = 300 C  81  Fig. 4.17(b). Another output of Hall voltage for T = 210 C  82  Fig. 4.18. A n ideal Hall voltage output  83  Fig. 4.19. Measured Hall resistivity of CoCr  84  Fig. 5.1. Schematic for the iterative formalism  92  Fig. 5.2. Magneto-optic multilayers  93  Fig. 5.3. Contour plot of reflectance  94  Fig. 5.4(a). Same as Fig. 5.3 except n" reduced by 1/2  95  Fig. 5.4(b). Same as Fig. 5.3 except n " increased by 1/2  95  Fig. 5.5. Effects of anisotropy on reflectance  96  Fig. 5.6. Reflectance versus magnetic misalignment  97  s  e  Fig. 6.1(a). Contour plot of a Gaussian beam intensity  101  Fig. 6.1(b). Contour plot of a real beam  102  Fig. 6.2. Spectral range for a focused beam  103  vii  Acknowledgements The author thanks R.R. Parsons as a helpful supervisor for suggesting the thesis topic, providing the apparatus and for his valuable guidance. Thanks also go to J.F. Carolan, R . C . Thompson, E.P. Meagher for the use of apparatus and J.D. Affinito, M . J . Brett, N. Fortier, B . T . Sullivan, D. Burbidge, Z. Celler for discussions and assistance during the work. The financial assistance of the University of British Columbia as a graduate fellowship is also acknowledged.  viii  Chapter 1 Introduction  1.1. Significance of Research on Magnetic Thin Films With the popularity of computers it becomes more and more important to develop a means for fast storage and retrieval of large amount of data.  Market studies1 have  estimated that tens of billions of dollars are involved in the disk storage market. Kryder wrote 2  "The fact that the Santa Clara valley in California, regarded as the center of U . S. electronics industry, has been dubbed 'Silicon Valley' piques manufacturers of magnetic data-storage devices. After all, the companies based in the valley actually derive more of their revenue from magnetic devices than from semiconductor devices. A more appropriate sobriquet, the manufacturers suggest, would be 'Iron Oxide Valley,' after the commonest material from which magnetic-recording mediums are made."  The point is that magnetic data storage technology is as important and critical for today's computer as semiconductor technology. A powerful computer system is useful only if it can access data in large volume and at high speed. Research on the subject could be beneficial in terms of funding for universities. Good examples are the magnetics research centers at Carnegie Mellon University, the University of San Diego and Santa Clara University 3 . The subject of magnetic thin films is also an interesting subject for pure research. The question of how a small region of a magnetic medium reverses its magnetization direction may sound like a classical problem. However, even today no quantitave prediction exists for magnetization reversal of most thin magnetic films4.  1  1.2. Media for Perpendicular Recording and Magneto-Optic Recording In the most common recording media today, such as iron oxide, information is stored by impressing a bit pattern of magnetization on a thin film. The direction of magnetization is parallel to the magnetic film. As the storage density becomes higher, the magnetization is decreased due to the increasing demagnetization field to a point that the written pattern can no longer be distinguished (see Fig. 1.1). To decrease the demagnetization field one can reduce the film thickness; however the magnetic field on the surface becomes very weak and the readout of signal is difficult. One way out of this difficulty to achieve higher density is to write a bit pattern with magnetization perpendicular to the film so that the demagnetization field of one bit pattern is actually favorable for the magnetization in the next bit pattern (see Fig. 1.2). A sharpe region of magnetization transition is formed between adjacent bits and higher density can be achieved in this way. For the magnetization to have a stable direction normal to the film it must first overcome its own demagnetization field. In a simple model, a single, "perpendicular" magnetic domain in a homogeneous and perfectly smooth thin film has a demagnetization field of 4nM, where M is the magnetization intensity in emu units. The associated magnetostatic energy is 2irM . 2  To hold the perpendicular magnetization there  must be an internal force in the domain that rotates it towards the film normal. This force is described by the anisotropy field and the associated energy is the anisotropy energy. The usefulness of CoCr as a suitable perpendicular medium was discoverd by Iwasaki5 in 1977. Subsequently, CoCr films produced by rf diode sputtering were studied by many workers and found to give rise to a strong anisotropy field which favors perpendicular magnetization. CoCr was chosen as the thesis project because of its potential applications in perpendicular recording technology. Another reason was that, at the start of the project, relatively little work had been done on CoCr produced by dc magnetron sputtering, which is a suitable deposition method for massive production because of the process controllability and the readiness to scale up from the experimental size. The magneto-optic (MO) multilayer is another type of erasable recording medium which was proposed by Connel 6 . The rare earth-transition metal(RE-TM) (e.g. TbFe) materials are antiferrimagnetic with perpendicular magnetic anisotropy. Two dielectric layers sandwich the R E - T M layer, both to protect it and to optimize the optics of the multilayer system. Room temperature lies between the Curie temperature and the compensation tem2  ( perature of the R E - T M . As shown in Fig. 1.3, a focused laser heats up the bit and reduces coercivity of the region so that an external field can reverse the magnetization (writing). For reading, a laser with lower power is polarized in one direction and focused on the written bit. Depending on the direction of magnetization in the R E - T M layer, the reflected (transmitted) light will contain polarization in the other direction due to the polar Kerr effect (Faraday effect), which can be used to determine the recorded information. The second part of the thesis is concentrated on the optimization of the optics of the M O multilayer system, which is critical for achieving high readout signal. Previous works were limited to plane wave model calculation. The present work considers the real case of focused beams.  1.3. O r g a n i z a t i o n of the Thesis Part A consists of chapters 2, 3 and 4 and it describes the experiments on CoCr films. Chapters 2 and 3 detail the process used to deposit the films and the characterization of the magnetic and structural properties. Answers are given to the question of how one can control the magnetic properties in terms of deposition parameters. Chapter 4 discusses the electronic properties of CoCr and describes the measurement of the dielectric constants and the extraordinary Hall effect of CoCr films. Due to the lack of availability of necessary apparatus and/or time limitations, no attempt was made to carry out a comprehensive investigation for the work described in chapter 4. However, some results are interesting and lead to better understanding of the films. Part B includes chapters 5 and 6.  In chapter 5 , a new matrix method is used to  calculate the optics of the M O multilayers for a plane wave model. In chapter 6 a model for the optics is presented, which takes into account the focal nature of the beams, and is used to optimize the M O films. Comparison is also made between the plane wave and the focused beam models. Chapter 7 is an overall summary of the original contributions in this thesis. Figures are collected at the end of each chapter for convenience. Lengthy, but important, derivations and useful computer programs are presented as appendices.  3  D E M A G N E T I Z I N G FIELD  Figure 1.1.  Demagnetization fields in the conventional  ("longitudinal") recording medium.  The field depends on the thickness  and the recording density.  PERPENDICULAR-RECORDING  MEDIUM  turn i it [i[Tl  SUBSTRATE  Figure 1.2. Schematics of perpendicular recording.  4  (  5  Part A Magnetic Properites of CoCr Films Chapter 2 Sputtering of CoCr Films  2.1. Experimental Apparatus Sputtering is a thin film deposition process in which A r ions are accelerated towards a target made of material from which the films are to be made. The atoms of interest are ejected from the target by the energetic A r ions and land on a substrate. To keep the current of A r ions constant, the ions can be used to produce secondary electrons during target bombardment. The secondary electrons in turn ionize A r atoms.  In magnetron  sputtering, magnets are installed near the target so that the secondary electrons travel in a cycloidal path and, thereby, are localized near the target surface. In this way, high sputtering efficiencies and high deposition rates can be achieved. Since less secondary electrons are needed with the use of magnetic field confinement, one can work at a lower Ar pressure; e.g. less than 3 mTorr. Besides a higher sputtering rate, magnetron sputtering also has the advantage of easy scale up to manufacturing size, mainly because the plasma is well confined to the target region and is not affected by other parts of the sputtering chamber. Pure Co films cannot support perpendicular magnetization. As we mentioned in chapter 1, the demagnetization field of Co is much larger than its anisotropy field. There are two ways of getting around this problem. One is to reduce the magnetization M(hence the demagnetization field) by adding other elements. The other is to deposit polycrystalline films so that the shape of the grains reduces the effective demagnetization field. Adding 18 to 22 atomic % Cr to Co gives an CoCr alloy which, in thin film form, can support perpendicular magnetization. 7 - " 1 1 A planar dc magnetron sputtering system was used to deposit CoCr thin films. The main features of the sputtering coater are seen in Fig. 2.1. The target composition, 80 at.% Co and 20 at.% Cr, was selected on the basis of the best results previously r e p o r t e d . 7 - 1 1 The target diameter was 15 cm. Magnets behind the target confined the erosion track to a 7 cm diameter ring. The vacuum system was pumped by a helium cryopump to a base pressure about l x l O - 6 Torr. The substrate holder assembly was designed such that the  6  substrates were uniformly heated directly from behind by a quartz halogen lamp, as shown in Fig. 2.2. The substrate temperature, T , was monitored by a thermocouple attached e  directly to the substrate. In order to obtain accurate measurement of T , a hole was drilled 8  on one of the 6 glass substrates in Fig. 2.2, and the thermocouple was clamped between a glass washer and the glass substrate by a screw through the hole. About 20 minutes were required to reach a steady state value T before sputtering. By adjusting the heating a  power, T could be controlled to within 10 degrees C during sputtering. e  It is well known that at small target to substrate distance, d_ s , the position of substrate for optimal film uniformity is slightly off center. 1 2 For this reason, a substrate of 5 cmx5 cm is located 2.4 cm from the axis of the target. Corning 7059 glass is used as substrate material for most of the films. To study the effects of the substrate material on the film properties, NiCr(50 at.% Cr) and T i films deposited on glass are also used as substrates for CoCr films. The choise of T i is based on a previous work 1 3 , which found T i to be a good substrate material for CoCr. As discussed below, NiCr is shown to be better than Ti. To optimize the the film properties the sputtering parameters are varied in the following range: argon pressure P A d  ts  T  from 1.5 to 10 mTorr; dc sputtering power from 150 to 900 watts;  from 5 to 10 cm; rf substrate bias from -200 to 0 volt; and T from 25 to 350 C . A s  deposition rate of 100 nm/min for CoCr is typical for the above experimental conditions. Most of the films are sputtered to a thickness of 500 nm. Larger thicknesses do not give rise to any significant change in film properties.  7  2.2. Film Properties The magnetization curves(M-H curve) of the deposited CoCr films are measured with a vibrating sample magnetometer(VSM). Depending on the deposition conditions, saturation magnetizations range from 350 to 450 emu/cc. For most films, both M-H curves with external field perpendicular and parallel to the film surface are measured, and the corresponding coercivities H ± and H \\ are determined. H ± and if c || are found to range c  c  c  from 100 to 1400 Oe. By an optimal choice of the deposition parameters, values as high as 1800 Oe are obtained for H ±. c  The magnetic anisotropy of CoCr is measured by the torque magnetometer as shown in Fig. 2.3. It consists of two dials connected by a torsion wire. The difference between the readings of the two dials gives the torque experienced by the magnetic film sample. Detailed instructions for building such a magnetometer are provided by Cullity 1 4 . The critical points include making the dials symmetric and choosing the proper diameter and length for the torsion wire. For CoCr films of thickness 500 nm and size 2.5 cmx2.5 cm, a tungsten wire of 38 cm in length and 0.013 cm in diameter is suitable. The constant of torsion is measured to be 7.15 dyne-cm/degree. Film orientation is referenced by the coordinate system in Fig. 2.2. The torque of a film is measured in a magnetic field of 13 kOe. It is found, in general, that the axis of the anisotropy is not normal to the film, which is not too suprising since the substrate is not located at a symmetric position with respect to the target. At large dt  s  (e.g. 9 cm) the  anisotropy axis is not significantly different from the film normal. The torque data fit very well to the torque derived from the anisotropy energy of the form: E  = K sin (a 2  a  u  - a ) + K sin (a 4  u  2  - a) u  (2.1)  where a defines the direction of the applied field with respect to the film normal in the torque measurement, and a  u  gives the direction of the magnetic easy(or hard) axis specified  with respect to the the target(see Fig. 2.2). The measured K 4.5 x 10 6 , and K  is typically -1 to 0.5 10 erg/cc, which is to be compared with K 6  u  u  =  = 1.5 x 10 6 erg/cc for a pure Co single crystal without contribution from  2  the shape anisotropy. For CoCr, K be larger than K . u  2  is typically 20% of K , u  but in some cases, K  2  can  It is so far unclear what controls the ratio of the two quantities and  how one can interpret it in terms of microstructural properties. However, a meaningful 8  (quantity is K  u  +  It is an indication of the total anisotropy and seems to be better  controlled than K  u  or K2 individually.  For example, we sputtered several films under  the same condition and found that all the films have about the same value of K  u  however, the ratio of K  u  + K2,  over K2 can be quite different for the films.  Microstructural analysis has been made with a scanning electron microscope(SEM), transmission electron microscope ( T E M ) with transmission electron diffraction(TED) capability, x-ray diffractometer(XRD), and texture goniometer(TG). T E M and T E D are found to be the most reliable and most reproducible of the above methods for the analysis of the sputtered CoCr films. About 100 nm of CoCr is deposited on a carbon coated copper grid for T E M and T E D experiments. The grid is mechanically clamped to the glass substrate and the temperature of the grid is assumed to be the same as that of the substrate. Typical T E M and T E D photograghs are shown in Figs. 2.4 and 2.5. The grain size is determined as follows: the sizes of the bright spots(single crystals that diffract the electron beam) are measured and averaged for each T E M photogragh. In the present study, the grain size is found to range from 20 nm to 70 nm. The crystal structure of CoCr determined by T E D is found to be hep. Since there has been reports of a transition from hep to fee phase for pure Co films15, great care has been taken to analyse the T E D results. So far, no phase transition has been found. A comparison of the predicted T E D pattern for the hep structure (see Fig. 2.6) generated by a computer program with the obverved T E D pattern in Fig. 2.5 is quite convincing. In the computer program the film is assumed to have c axes perpendicular to the film surface so that all the (hkO) diffraction peaks have strong intensities and the rest of the peaks have weak intensities. This preferred crystal orientation (or texture) is confirmed by X R D and T G methods as will be discussed later. S E M provides a good indication of surface roughness and a reliable measurement of the thin film thickness. But cross section S E M does not reveal the grain structure in a reproducible manner and very much depends on how the film is fractured. For example two films deposited in one run of sputtering (one run can sputter 5 films) under identical conditions appear very different in S E M photos as shown in Fig. 2.7 and 2.8. Because of this problem most of the discussion on grain structure in this thesis is based on T E M and T E D data. In X R D and T G measurements only the (002) peak is detected. This is mainly due to the texture of the film (c axis perpendicular to film surface) and also because of the fine grains. T G is useful only when the films are well crystallized and have a well defined peak  9  of (002). For those films a 650 value of 3.5 degrees is typical, where #50 is defined such that 8 ± 650 corresponds to half the peak intensity for (002), and 6 is the diffraction angle.  2 . 3 . Effects of Substrate T e m p e r a t u r e At the beginning of the experiments, no substrate heating system was installed. For rf diode sputtering, the substrate heating is not critical since the substrate is bombarded by electrons and ions, and, therefore, significant substrate heating occurs. Hoffmann et a l . 1 6 studied the case of dc magnetron sputtering of CoCr and found that the coercivity increases with the substrate temperature.  The present study provides more details on  the effects of substrate heating for dc magnetron sputtering. When no substrate heating was provided, no matter how hard the author tried for all combinations of sputtering parameters other than T s , almost all the CoCr films had the same magnetic properties. The films were nearly isotropic with a low coercivity of about 200 Oe, which is unsuitable for recording applications. When a substrate heating system was installed films with larger coercivity could be obtained 1 7 . As T increased to about 180 C , a large increase in both H ± and M x was s  c  r  observed as indicated in Fig. 2.9 and 2.10, where M denotes the magnetic remanence. The r  sputtering parameters other than T are as follows: P 8  Ar  = 3 mTorr, dc power W = 350  watts and df = 9.0 cm. The substrate was electrically floating. The film thickness was S  about 500 nm. This T dependence is quite general and observed for films deposited with s  other sputtering parameters. F i s h e r 1 5 , 1 8 has recently reported the use of sputtered CoCr films for the production of commercial, longitudinal-type hard disks. As shown in Fig. 2.11, negative anisotropy required for longitudinal recording is achieved in the present study. However, it is clear from the scatter of data points in Fig. 2.11 that T is not the best parameter to control s  the anisotropy constant. T E M photograghs of films deposited at du = 9.5 cm, T = 25, 210 and 300 C , respecs  tively, are given in Figs. 2.4, 2.12 and 2.13. The dependence of the grain diameter d as a function of the substrate temperature is summarized in Fig. 2.14. The grain diameter is determind from T E M photographs in the following way. A line is drawn across the photograph. When we count across the line, each bright spot is consider as a grain and each dark spot is also a grain. The dimensions of the grains measured across the line give 10  the average grain diameter and the standard deviation. As can be seen from Fig. 2.14, for unheated substrates the film consists of fine grains (20 nm). As T increases to 210 C , s  the grain size doubles. But as T increases further to 300 C , the grain size becomes finer e  again. The latter behaviour is probably related to the fact that if the heating is too high, the perpendicular coercivity starts to drop as shown in Fig. 2.9. There seems to be a strong correlation between the coercivity and the grain size: larger grain size corresponds to larger coercivity. Recently, Khan et a l . 1 9 also found a similar relationship between the coercivity and the grain size for CoCr produced by rf sputtering. The difference in grain size seen in Figs. 2.4(T S =25 C) and Fig. 2.12(T6=210 C) can be interpreted in terms of Thornton's 2 0 structural zone model as follows: At low T , because of low adatom mobila  ity and the shadowing effect fine grains with voids at the grain boundaries develop; As T  s  increases, the grain sizes become larger and the grain boundaries fuse together. The  explanation for the effect of heating at higher temperature (greater than 300 C) on the microstructure remains an unsolved problem.  How does one interpret the effect of substrate heating on the coercivities ? There are two possible mechanisms. One possible mechanism is based on the observation that a larger coercivity is associated with a larger grain size. In the extreme of very fine grains separated by voids, the exchange interaction between electrons in different grains are weekened and there is week coupling between the magnetic moments of the grains.  If the grain size  and the coupling between grains are so small that thermal flutuation can easily reverse the magnetic moment, the film will behave like a superparamagnet with low coercivity. 21 The other possible mechanism is based on the assumption of segregation which is still controversial. 1 9 It is well known to metallurgists that, upon heating, many solutes tend to segregate to the grain boundaries to reduce the free energy. Therefore, it is reasonable to think that the Cr atoms diffuse to the grain boundaries at high T and, thereby, increase 8  the compositional inhomogeneity. This inhomogeneity will in turn increase the coercivity since it is harder for the magnetic domain wall to move in the process of reversing its magnetization. Support of this interpretation is provided by Friedberg and P a u l 2 2 who have classified the coercivities of many alloys ranging from 0.002 Oe for supermalloy to 10 kOe for Co^Sm using a simple model in which the domain wall is blocked by planar defects such as grain boundaries, solute segregates, etc. The assumption of segregation is further supported by the obvervation that M  s  gets larger as  is increased (Fig. 2.15), a  point that will be discussed in more detail in chapter 4 in terms of the band structure of Co.  11  An attempt was made to obtain direct, experimental evidence for segregation of Cr at the grain boundaries with the use of an energy dispersive x-ray spectrometer. No variation in Cr content was observed by scanning the beam across the grains; however, because of the beam size(10 nm) was comparable to the grain size(about 40 nm), the E D X R S results were of little use. A composition analysis with higher resolution is required.  2.4. Demagnetization Field for CoCr Films This section will discuss some observations made during the V S M and T M measurements and their relation to microstructure. In general, for a thin film, K  u  can be written,  as: K where N± and  u  = K  x  - 27r(JV - JVy)M 2  (2.2)  ±  are the demagnetization factors in directions perpendicular and parallel  to the film, respectively, and Ky is the part of anisotropy constant internal to the grain (e.g. crystal anisotropy). The second term is the contribution from the microscopic shape anisotropy. For a single magnetic domain in a homogeneous and perfectly smooth thin film, the contribution from the shape anisotropy, or in other words, from the magnetostatic energy is —27rM 2 (i.e.  N± — 1 and iV|| = 0).  Therefore, the shape anisotropy of this  idealized model favours longitudinal magnetization. For a real film consisting of grains and voids, the contribution is smaller and one can obtain a film with positive macroscopic anisotropy. The above point is confirmed by plotting M perpendicular to the film plane versus the magnetic field within the film: H i app  — 4nN±M,  where H i app  is the applied field  and the second term is the demagnetization field. If one sets N± — 1 as in the idealized model, one gets an unphysical M-H curve with a negative slope as shown in Fig. 2.16; i.e. this would mean that the effective demagnetization factor is less than unity. Another interesting point is that, in the case of films deposited with T greater than s  200 C , a positive magnetic anisotropy necessarily means a large perpendicular coercivity, while the reverse does not hold. It is possible to support perpendicular magnetization even if the macroscopic magnetic anisotropy has a negative value. As seen in Fig. 2.17, the easy magnetization can be parallel to the substrate, but a significant coercivity in the perpendicular direction can be obtained.  12  2.5. Substrate Effects.  Three type of substrates, glass(Corning 7059), Ti-coated glass, and NiCr-coated glass, were used to investigate the effects of the substrate material on the magnetic properties of the CoCr films. Fig. 2.18 gives the coercivity, Hc±, for the perpendicular magnetization as a function of CoCr film thickness. In the case of glass substrate, the reduced value of  Hc±  for small film thickness has been previously associated 23 with the initial layer (about  100 nm) deposited, which has poor texture. As seen in Fig. 2.19, the initial layer effect is not obversed in the case of Ti/glass and NiCr/glass substrates. A possible explanation of this observation is that NiCr and T i films have textures and crystal structures which are similar to those for CoCr so that no initial layer can develop. Similar results have been recently observed by Tanaka and Masuya 1 3 for e-beam evaporated CoCr on T i substrates. T E D experiment has been carried out to analyze the structure and texture of NiCr and the data are shown in Fig. 2.19. The first structural model used to analyze the T E D data is fee of Ni. It has been well known that hep and fee are two different ways of close packing spheres. 24 Fig. 2.20 shows how to section the fee lattice to form two-dimensional trianglar lattices. If A , B and C represent different ways of stacking the two-dimensional trianglar lattice upon each other, the stacking sequence of hep is (...ABABAB...) and that of fee is ( . . . A B C A B C A B C . . . ) . 2 4 Therefore, fee has the same trianglar lattice structure in the ( i l l ) planes as hep in (002) planes. If NiCr has the the structure of fee, its (111) planes will match the (002) planes of hep of CoCr. Fig. 2.21 plots the theoretical reciprocal lattice constant Ktheo for fee versus the experimental T E D data  K . exp  The second structural model for NiCr is bee of Cr. The fit of bee to experimental data is presented in Fig. 2.22 in terms of the reciprocal lattice constants.  Because  of the similarity between fee and hep, the T E D data are also fitted to the structure of hep assuming that the c axis is normal to the film(i.e. only the strong peaks indicated in Fig. in Fig.  2.6 are seen in the T E D data of NiCr).  2.23 is much better than those for fcc(Fig.  The fit to hep as shown  2.21) and bcc(Fig.  conclude that the sputtered NiCr film has the structure of hep.  2.22). We  It is also interest-  ing to compare the nearest atomic distances for the following metals (taken from the periodic table) and alloyed films (determined from T E D data of the present study): Co(hcp):  a = 2.51 A 13  a' = 2.49 A Ni(fcc): a = 2.49 A Ti(hcp): a = 2.95 A  Cr(bcc):  CoCr(hcp):  a = 2.52 A  NiCr(hcp):  a = 2.35 A  where a is the lattice constant for the two-dimensional trianglar lattice in hep or fee, and a' is the smallest lattice constant for bee. One notices that the nearest atomic distance of NiCr(hcp) is smaller than those of Ni and Cr. One also finds that the match of NiCr lattice to CoCr lattice is better than that of T i to CoCr.  14  (  CRYOARRAYS FLEXIBLE HELIUM HOSE THROTTLE VALVE • •  GATE VALVE TO CAPACITANCE MANOMETER CHAMBER INSULATOR  SUBSTRATE HOLDER  F i g u r e 2.1.  A  Ar  HELIUM GAS COMPRESSOR  TARGET  SUBSTRATE^ SHUTTER^  f Jr^MAGNET feKcATHODE  Schematics of the dc magnetron sputtering system used  to deposit CoCr films.  15  CLAMP MAGNET TARGET CATHODE  INSULATOR  Z  N N S  PLASMA  *L  ts  SUBSTRATE  CHAMBER/ '  .SHUTTER  HOLDER HEAT LAMP  F i g u r e 2 . 2 . Details of the substrate holder assembly. T h e size of the substrate is 5 c m x 5 cm and the center of the substrate is located 2.4 c m from the axis of the target.  16  UPPER DIAL  F i g u r e 2.3.  Torque magnetometer. The tungsten torsion wire has a  length of 38 cm and a diameter of 0.013 cm.  17  Figure 2.4. A t y p i c a l T E M p h o t o g r a p h of a C o C r f i l m . T h e s p u t t e r i n g p a r a m e t e r s are as f o l l o w s : a n d dts — 9-5  c  m  T  S  = 25 C , P A = 3 m T o r r , W T  = 350 watts,  - T h e s a m p l e is d e p o s i t e d on a c a r b o n c o a t e d c o p p e r g r i d  d e s i g n e d for T E M s t u d i e s .  -  F i g u r e 2.5. T E D photograph for the same sample as in F i g . 2.4. T h e electron beam energy is 200 k V .  I) 1  F i g u r e 2.6. Predicted T E D pattern for the hep structure. The intensity of the rings are predicted assuming the film has an orientation with c axis normal to the film surface. The rings that have been seen in experiment and have been identified are labelled.  20  Figure 2.7. A typical cross section S E M photograph of CoCr sputtered on glass at T = 200 C , P s  Ar  = 3 mTorr, W = 350 watts and d  ts  = 8.0 cm.  The magnification is 40,000 and the dotted line at the lower right corner of the photograph represents 750 nm.  21  Figure 2.8.  Another S E M photograph of CoCr deposited on glass  under the same condition as in Fig.  2.7 in the same run of sputtering.  The magnification is 50,000 and the dotted line at the lower right corner represents 600 nm.  22  14  7?l  o : IN PLANE • : PERPENDICULAR  1.2-  1.0  0.6-  O  J T I IT  o  Hp  X 0.6 0.4-  ii  0.2  00 20  —  i  60  1  1  100  140  1  1  160  T  s  220  1  260  1  300  1  340  —  360  (DEG C)  F i g u r e 2 . 9 . Dependence of the coercivity of C o C r deposited on glass on the temperature of the substrate, T , during film growth. T h e sputtering 8  parameters other t h a n T  B  are as follows: PAT=3  m T o r r ; W = 3 5 0 watts,  d =9.0 c m . T h e C o C r film thickness is about 500 n m . t8  23  0 35  0.30  D : IN PLANE  0.25-  S  • : PERPENDICULAR  w 0.20-  "5  0 15  --  ()  1  0.10JL  0.05  0.00  20  i  60  l  l  100  140  F i g u r e 2.10. Ratio M /M r  8  i  180  T  s  1  220  (DEG C)  i  260  1  300  1  340  380  of CoCr deposited on glass as a function  of T , where M is the magnetic remanence and M the saturation magne8  r  s  tization. The films are sputtered under the same condition as for Fig. 2.9. The film thickness is about 500 nm.  24  0.5 0.4 H  • : K  u  03  c}5" a: Ed  :  *2  o.i-  I  o.o  CO  o co  I  T  260  300  rji t+j "d>—  ^  I  o.o-|  3 ^  T  -0.2-0.3-0.4-  -0.5 20  1  60  1  100  1  140  180  T  220  (DEG C)  s  F i g u r e 2.11. The anisotropy constants K  u  s  e  during film growth.  are given in Fig. 2.9.  25  380  and Ki of CoCr deposited  on glass as a function of the substrate temperature T Sputtering parameters other than T  340  F i g u r e 2.12. T E M photograph of C o C r deposited at T = 210 C . T h e s  other parameters are the same as for the T E M photograph in F i g . 2.4.  26  F i g u r e 2.13. The same as Fig. 2.12 except T = 300 C. s  27  • : d^=7.8 cm s  • : d^=9.5 cm s  i)  11  -j  50  100  150  200  T (DEG C)  250  300  350  s  Figure 2.14. The grain size of CoCr, d, as a function of the substrate temperature, T . e  The films are deposited at different target to substrate  distance du. Other sputtering conditions are the same as for the T E M photograph in Fig. 2.4. The grain size is determined from T E M photographs.  28  450 440430420|  410-  2 E CD,  400 H 390 380370360350340 50  100  200  150  250  300  T (DEG C) s  F i g u r e 2.15.  The saturation magnetization of CoCr deposited on  glass, M , as a function of the substrate temperature, 7^. The other sputB  tering parameters are given in Fig. 2.9. M is determined by V S M measures  ment and the error is mainly due to the measurement of the film thickness and the area of the sample. The thickness is measured with a profilometer and/or cross section S E M .  29  350  F i g u r e 2.16. The magnetization M versus H i app  - 4-nNM  for a film  of CoCr deposited on glass at T = 280 C . Other parameters are the same s  as for Fig. 2.9. N is the demagnetization factor. The curves correspond to N± = 1 and JVj| =0.  30  F i g u r e 2 . 1 7 . E x a m p l e of V S M result of C o C r on glass showing easy magnetization direction in the plane and a significant Hc±.  The sputtering  parameters are: T8 = 210 C , and other conditions are the same as for F i g . 2.9.  31  F i g u r e 2.18.  The substrate effects on perpendicular coercivity as a  function film thickness. The growth temperature is 300 C and the target to substrate distance is 8 cm. The other parameters are the same as for Fig. 2.9. The lines between data points are used only as a guide to the eyes.  32  F i g u r e 2.19. T E D photograph of NiCr. The beam energy is 200 kV.  33  F i g u r e 2.20.  How to section the fee lattice into the same two-  dimensional trianglar lattice as (002) planes of hep lattice.  34  9  0  1  2  3 K  4  5  6  7  8  9  theo (1/ANGSTR0M)  F i g u r e 2.21. Fit of the T E D data to the structure of fee.  Kh t  e o  reciprocal lattice constant calculated for fee. K  exp  is the  is determined from the  T E D experiment. A best fit straight line through the origin is also shown.  35  9  0  1  2  3 K  4  5  6  7  8  9  theo (1/ANGSTROM)  F i g u r e 2.22. Fit of the T E D data to the structure of bcc. Ktheo is the reciprocal lattice constant calculated for bcc. K  exp  is determined from the  T E D experiment. Also shown is a best fit straight line through the origin.  36  9  K  theo (1/ANGSTROM)  Figure 2.23. Fit of the T E D data to the structure of hep. the reciprocal lattice constant calculated for hep.  K  exp  Ktheo  is  is determined from  the T E D experiment. The film is assumed to have a texture with c axis normal to the film surface so that only the (hkO) peaks are seen.  37  Chapter 3 The Control of Magnetic Anisotropy in CoCr Films  3.1. Effects of Sputtering Power and Target-Substrate Distance The first stage of the study concentrated on the development of dc sputtering process for the fabrication of CoCr films, which would be useful for recording applications. As described in chapter 2 it was found that the perpendicular magnetic coercivity H ± is c  controlled mainly by the substrate temperature T during film growth. The situation is 8  more complicated for the magnetic anisotropy. The anisotropy field Hk is defined as Hk = 2K\jM  8  and, in practise, is estimated  from the M - H curve for a field direction parallel to film surface. have shown that Hk is dependent on T , P s  Ar  Other workers 2 5 ' 2 6 ' 2 7  and the base pressure  Usually low  Pb and P / i r , and high T promote positive anisotropy. One complication enters at this 8  point.  In general, when Hk is estimated from M - H curve, one has assumed that the  demagnetization factor is zero for magnetization parallel to the film. But this is not the case as was demonstrated in the last chapter. Therefore, JET* estimated in this way does not indicate whether the film favors perpendicular or longitudinal magnetization. A more direct approach is to characterize the anisotropy with a torque magnetometer, as will be described in this chapter. For the results to be reproducible, the Corning 7059 glass substrates are cleaned (or preconditioned, to be exact) with a method developed by Sullivan 2 8 . First, the glass substrate is cleaned by methanol and wiped by lens paper. Then it is cleaned with trichloroethylene followed by methanol again. Finally it is blown dry by a N2 gas flow. The experimental apparatus is shown in Figs. 2.1 and 2.2 and is described in section 2.1. To obtain the lowest base pressure possible during sputtering the pump is not throttled. T  s  is fixed at 210 C to optimize H ±, P c  gas flow. A lower P  Ar  Ar  is kept at 3 mTorr, by the control of input A r  might be better for the magnetic anisotropy; but, it would make  the plasma difficult to turn on. By varying the dc sputtering power, W , and the target-tosubstrate distance, dt , sputtering rates ranging from 0.3 to 3.5 nm/sec are obtained. The s  deposition time is controlled so that all the films have a thickness of 500 ± 50 nm. The shortest dt that can be used is about 5 cm. At this point, the substrate assembly starts 8  to interfere with the plasma as is evident from the change of the target voltage caused by 38  the movement of the shutter. After sputtering, the film orientation with respect to the target is marked. The reference coordinate system is shown in Fig. 2.2. The film is then measured with the torque magnetometer described in chapter 2. The data is fitted to the torque curve calculated from equation (2.1). M - H curves are measured by V S M for directions perpendicular and parallel to the film, and in some cases, other directions are also scanned. The film structure is characterized by X R D . Texture goniometer measurement is not always possible since some of the films are not well crystallized and do not show any peaks in X R D measurement. However, we introduce the ratio I(oo )/h, where /(002) 2  a  n  ^  h  a  r  e  the intensities of the  (002) reflection and the background intensity in the vicinity of (002) peak, respectively, which can be used as a measure of the degree of crystallization. T E M and T E D are also used to give additional information on the microstructure. The (100) peak and other low intensity peaks are not seen above the background, except in the case of films with large 60).  koQ2)lh  Three sets of samples with dt  s  characterized.  29  equal to 9.5, 7.8 and 5.3 cm were sputtered and  The anisotropy constant, K  u  + K, 2  discussed in the last chapter, as a  function of the sputtering power, W , is given in Fig. 3.1. For dt opposite dependence on W to that for df  a  s  = 7.8 cm, it has the  = 5.3 cm. Compared to the corresponding  .flex's in Fig. 3.2, which have large values for all three sets of data between W = 350 and W = 600 watts, the K  u  -f K  2  data is more complex.  The direction of the axis for the anisotropy, a u and /(oo2)/Jfc  a  r  e  presented in Figs. 3.3  and 3.4, respectively. One notices that, in general, | a u | increases with W . A l l the easy axes (K (K  u  + K  u  + K  2  2  > 0) point away from the target center ( a u < 0) and all the hard axes  < 0) towards it ( a u > 0).  One also finds that, at d  crystallized film (i.e. with maximum I(oo )/h) has large values of K 2  small j a u |.  39  = 7.8 cm, the best  ta  u  + K, 2  H ±, and a c  f  3.2 The Roles of Adatom Diffusion and Covering Rate In this section we attempt to interpret the complicated dependence of Ku + K2 and a u on the sputtering parameters. It is necessary to consider the effect of the stress anisotropy for the following discussion. It is well known 2 0 that films of high melting point material  (T/Tm  < 0.25, where Tm is the melting temperature) sputtered at low pressure(< 8 mTorr)  can have internal compresive stress as high as 109 Pa. This is a result of bambardment by energetic coating atoms and by A r ions neutralized and reflected at the cathode. The stress anisotropy constant 1 4 K^o)  is a function of the stress 0 and is the order of +10 6  erg/cc for Co under a compresive stress of 10 9 Pa, i.e. it is of the same order of magnetiude as the measured total anisotropy. However, when the stress is along the c axis, Ka = 0 due to the uniaxial symmetry, which is probably the reason why the effect of internal stress on anisotropy could not be measured before. 25 We assume that the internal stress in the film is due to the bombardment of energetic particles coming from an angle with respect to the film normal. The induced stress anisotropy has an easy axis along the direction of the compresive stress 1 4 for Co and we assume this is also the case for CoCr. For simplicity we further assume that the stress anisotropy only contribute to K\ in equation (2.2). Then, we have  Ku + K2 = K° + K2 0 + KC7where K° and K2  2ir{N - N^M  2  ±  (3.1)  arise from crystalline anisotropy.  The anisotropy constant can be related to the sputtering parameters in the following way: 1) K\ +  depends on the microstructure and texture of CoCr films; 2) the mi-  crostructure and texture are determined by the mechanism of film growth(e.g.  adatom  diffusion); 3) the film growth condition is controlled by the sputtering parameters of a specific sputtering system. Regarding point l) above we assume that K\ and K% depend on the degrees of crystallization and texture. This assumption is supported by the findings of Hwang et a l . 2 5 and Ouchi and Iwasaki 3 0 . They found that Hk = 2Kx/M  s  increases as #50 is reduced. There-  fore, it is reasonable to assume that a high degree of crystallization promotes positive magnetic anisotropy. We note from the last term in (3.1) that compositional inhomogeneity and the shape of the grains also affect Ku + K2, as discussed in section 2.4. To study the dependence of the microstructure and texture on the film growth condition (point 2) above), we consider what happens to a sputtered atom landing on the 40  growing film. Upon arrival on the growing film, the sputtered atom diffuses with a kinetic energy much larger than the thermal ernergy before it loses its kinetic energy and reach thermal equilibrium with the substrate. 31 We assume that the kinetic energy is equivalent to an effective temperature T e / / for the diffusion constant D: (3.2)  D ~ exp{-Q /k T ) d  b  eff  where Qi is the surface diffusion activation energy(typically 0.5 eV) and fcf, is the Boltzmann's constant. This adatom diffusion process affects very much the morphology of the growing film. In Thornton's structure zone model, 2 0 high adatom mobility results in well crystallized films with large grain size. Glocker et a l . 3 2 have modeled the growth of CoCr films and found that the preferential growth direction is along the c axis because of the geometry of the unit cell of CoCr(hcp). In this study, we assume that the film is grown layer by layer in the direction of the c axis. To take the deposition rate into account in the study of adatom diffusion, we assume that the adatom is considered to be immobilized if it is covered by a layer of sputtered material of thickness c, where c is the height of the hep unit cell. The above assumption is based on the fact that the surface diffusion coefficient is much larger than the bulk diffusion coefficient.31 The time, td, available for the adatom to diffuse before it is covered by the incoming sputtered atoms can be expressed as t  d  (3.3)  = c/R  Within tj, the adatom can diffuse over a distance, s: (3.4)  s = y/2Dc/R  In the following discussion we assume that, for the film to be well crystallized, the adatom must diffuse over a sufficiently long distance, s, before it is covered by the incoming atoms. We now consider point 3) above. parameters (dt  s  To study the relationship between the sputtering  and W) and the growth condition (i.e.  the distance s), we note that the  sputtering rate can be written as (3.5)  R = Wg(dt ) a  where the function, g , depends on the geometry of the sputtering system.  The kinetic  energy associated with each sputtered atom arriving on the substrate can be written as E  k  =  Eo{W)exp{-dt /\e) s  41  (3.6)  where Eo(W) is the energy of an sputtered atom ejected from the target. It is determined by the cathode voltage which is a weak function of W. For a sputtering voltage of 500 V , EQ is estimated 2 0 to be between 5 to 10 eV. Formulas for estimating the mean free path, A e , have been given by Westwood 3 3 . Including the effect of kinetic energy 3 3 , we find A e = 3.1 cm for the present study. During the film growth, the adatom diffusion is also promoted by the radiation from the target, plasma bombardment and neutral A r atom impingement. The energy flux of the radiation and the energetic particles can be approximated as  E  T  =A  W  (3.7)  where A depends on the specific geometry and must be determined empirically for each experimental set-up. Ek in (3.6) and ET in (3.7) cause the effective surface temperature T ff to increase, which in turn enhances the adatom diffusion coefficient in equation (3.2). e  Finally, s is related to W and d  ts  through (3.4) and (3.5) as follows: (3.8)  where D ( W , d t s ) is the given through (3.2), (3.6)and (3.7). Due to the large number of undetermined parameters in the above equations, it is impossible to make any quatitative predictions. However, the above formulas enable us to interpret the dependence of K + K2 u  as follows. At dts — 9.5 cm and low power range(~ 150 watts) the sputtering rate is the lowest and the adatom has the longest  to diffuse (tj. ~1 sec from (3.3)) before it is covered.  At this dts, it is assumed that Ek and ET are too small to cause significant diffusion of adatoms to the nucleation sites. A small increase in W causes D to increase through the relations in (3.2), (3.6) and (3.7), which then results in a large increase in K  u  shown in Fig. Z.l(dt  s  + K2 as  — 9.5 cm). When D is small, the films are not well crystallized,  resulting in small 1'(002) / ' h ratio and a negative anisotropy, in agreement with the data discussed above. If dts is decreased and W made larger, R plays a more important role (e.g. D=6xl0~  1 4  cm /sec, 2  if  as estimated by Hoffman et a l . , and R=30 A/sec, we obtain 16  s=9 A, i.e., the adatom hardly has a chance to move around before it is covered). To interpret the power dependence of K  u  + K2 at dt = 7.8 cm, we assume that the substrate s  is still "outside" the plasma, and the effect of ET in (3.7) to T jj e  42  is less than that of Ek in  (3.6) which has a weak dependence on W . The increase of D(W, d ) with W is slower than ta  that of R. As a consequence the diffusion distance, s, in (3.8) decreases with W. Therefore, we expect I(oo2)/h to decrease and K + K u  to drop with increasing W, in agreement with  2  the dts = 7.8 cm curves in Figs. 3.4 and 3.1. To confirm the above point of view, T E M and T E D measurements were made on films deposited at different sputtering powers at dt = 7.8 cm. T E M photographs for s  W = 150 and W = 750 watts are shown in Figs. 3.5 and 3.6, respectively. The data is in good agreement with the theory. At high W , the adatoms do not have sufficient time to diffuse and fine grains with relatively well defined grain boundaries are developed. As W is reduced, larger grains with grain boundaries fused together are formed which is a result of longer adatom diffusion time. The T E D data indicates that the c axes of the films are normal to the film surface except for the film deposited at W=150 watts, which is the best crystallized film obtained(Fig. 3.7). The c axis is found to be tilted away from the film normal in this case probably because it is much better crystallized and, therefore, more sensitive to texture measurement. When d  is  of  ET  to  T jf e  =5.3 cm, the substrate is almost in contact with the plasma, and the effect is larger than that of  Ek-  The increase of D(W, d ) ts  with W is faster than  that of R. Therefore, s in equation (3.8) increases with W . This explains the increase of K  u  + K  2  with power at the the above target-to-substrate distance.  T E M photographs corresponding to W=150, 350 and 750 watts, with du = 5.3 cm, are presented in Figs. 3.8, 3.9 and 3.10, respectively. For W=150 and 350 watts, T E M results agree with the theory: As W increase the adatoms are more activated by the plasma and larger grains with fused grain boundaries are seen at 350 watts.  But something  unexpected is seen for W=750 watts: The grains become finer and the grain boundaries are sharper again. A possible reason for this is that, at the small dt used(5.3 cm), the s  surface heating is high and causes the Cr atoms to segregate and to form finer grains. The relatively large magnetic anisotropy in this case is likely due for the most part to the smaller demagnetization factor arising from the fine grains. This effect of overheating has been discussed in section 2.3.  43  3.3 Effects of Asymmetric Sputtering The off-axis location of the substrate with respect to the target as shown in Fig. 2.2 enables one to study the effect of asymmetric deposition. Results in Fig. 3.3 indicates that the magnetic easy axis is tilted away from the center of the target and the hard axis towards it. This asymmetry in the magnetic anisotropy can not be accounted for with crystalline anisotropy since it was shown in section 3.2 that most of the films have c axes normal to the film surfaces. Possible causes of the asymmetry in the anisotropy are: l) stress anisotropy due to the bombardment of energetic coating atoms and Ar ions neutralized and reflected from the cathode; 2) shape anisotropy of grains grown towards the direction of the flux of the sputtered atoms. The following is a simple model to demonstrate how the compresive stress causes the axis of the total anisotropy to differ from the film normal and why a  u  changes sign when the total anisotropy K  u  + Ki does.  For simplicity we neglect the fourth order term, Kisin (a  — a u ) in (2.1) and assume  4  the shape anisotropy part has the hard axis normal to the film surface.  However, the  method of derivation below applies to the most general case. From the geometry of the sputtering system in Fig. 2.2 we conclude that the impinging particles from the direction of a < 0 are more energetic than those from other directions. If acr(< 0) is the direction of the compresive stress which is along the direction of the flux of the impinging particles from the direction of a < 0, then the total anisotropy energy can expressed as E  a  = [K? -  2TT(N  -  ±  7 V | | ) M ] 5 m ( a ) + K sin {a  - a)  2  2  2  a  (3.9)  a  where the first term is due to the crystalline anisotropy and the shape anisotropy, and the second term arises from stress anisotropy. In the small angle approximation (3.9) becomes E  Since K  = [if, - 2n{N  - N^M^a  0  a  ±  + K {a  2  a  - a )  (3.10)  2  a  > 0 (see Ref. 14), one can show that from (3.10) that the minimum(easy axis)  a  or maximum(hard axis) occurs at a  where K  u  = K° -  2TT(N±  -  N\\)M  2  = -K  u  + K. a  a  | CL„ | /K  (3.11)  u  (3.10) and (3.11) indicate that if K  a u < 0 is an easy axis (minimum in E ) and similarly for K a  u  u  > 0 then  < 0. The above simple model  for the origin and sign of a u agrees well with the experimental data. It is important to note that a u is sensitive to the orientation of the film with respect to the target.  If the substrate is not parallel to the target, a  u  44  changes significantly.  For example, one sample was intensionally misoriented from the parallel direction by +8 degrees, and with W=350 and dt = 5.3 cm; as a consequence a s  degrees while K  u  and K  2  u  changed from -12 to -40  remained unchanged. The effect of this large | a  u  | to the M-H  curve is shown in Fig. 3.11. One can see a significant difference when the applied field direction a changes from 45 to -45 degrees. We conclude that the anisotropy constants can be controlled by the sputtering power and the target to substrate distance. It is important to orient the substrate during sputtering in order to control the easy (hard) axis orientation of the magnetic anisotropy. Once the films have been made, we should carry out some more experimental and theoretical studies on their physical properties such as the electronic states, dielectric constants and the Hall effect. Such a study is the task for the next chapter.  45  x  200  0  5.3 cm  x  400  600  800  1000  W (watt) F i g u r e 3.1.  A n i s o t r o p y constant K  u  + K  of C o C r on glass as a  2  function of the sputtering power W at various dt . 8  P  Ar  = 3 mTorr, T  8  Other parameters are:  = 210 C . T h e film thickness is 500 ± 50 n m . T h e lines  are used only to identify the d a t a sets.  46  1.5 CD O  —I o  9.5 cm 0.5  A  K  0 0  200  400  7.8 cm  5.3 cm 600  800  W (watt)  F i g u r e 3.2. Perpendicular coercivity H ± versus the sputtering power c  W at various target-to-substrate distance dt . e  Other parameters are the  same as for Fig. 3.1. The lines are used only to identify the data sets.  47  1000  0  200  400  600  800  W (watt)  Figure 3.3. The orientation of the easy (hard) axis of the ansisotropy a defined in equation (2.1) and Fig. 2.2 versus the sputtering power for the same sets of samples as in Fig.3.1. The straight lines are used only to identify the data sets. u  48  1000  t  70  60 h  50  cm * 7.8 c m * 5.3 c m 9.5  -Q  40  o  &  *  i i  30  • • •  *  20  5.  10  »«  0  =63-  0  200  400  800  600  W (watt)  Figure  3.4.  Ratio of (002) peak intensity /(002) t  o  t n  e background  intensity h from X R D versus the sputtering power W , for the same sets of samples as in Fig. 3.1. The lines are used only to identify the data sets.  49  1000  F i g u r e 3.5. T E M photograph for W=150 watts and d  u  Other parameters are the same as for Fig. 3.1.  50  = 7.8 cm.  Figure  3.6. S a m e as F i g . 3.5 e x c e p t W = 7 5 0 w a t t s .  51  • -  n a  n  n  o -i j  F i g u r e 3.7. T E D photograph for the same sample as in F i g . 3.5.  52  F i g u r e 3.9. S a m e as F i g . 3.8 e x c e p t W = 3 5 0 w a t t s .  54  55  - 2 0 0  1  0  1  1  •  1  '  1  6  4  2  •  1  8  H(k 0e)  F i g u r e 3.11.  M - H curve for a sample oriented +8 degrees from the  usual direction parallel to the target surface during deposition (see also F i g . 2.2).  a is the direction of the applied field defined i n F i g . 2.2 in torque  magnetometer measurement. Sputtering parameters are d  te  W = 3 5 0 watts.  56  = 5.3 c m and  1  1 1 0  Chapter 4 Other Physical Properties of CoCr Films  4.1 Saturation Magnetization and Electronic States of CoCr The saturation magnetization M  8  temperature T : In Fig. 2.15 M 8  is observed to increase with increasing substrate  increases from 360 emu/cc at T = 30 C to 420 emu/cc  8  8  at T = 250 C . s  Haines 3 4 has developed a model for the saturation magnetization and derived the following semiempirical formula: M  = M {l  8  e0  - C )(l Cr  -  (4.1)  4C ) Cr  where M o is the saturation magnetization for pure Co crystal and Ccr is the concentras  tion(i.e. atomic percentage) of Cr. The model assumes: l)Only Co atoms contribute to M s ; 2) Cr atoms are uniformly distributed and interact only with Co atoms; and 3) By transferring the outer electrons to the Co atoms, one Cr atom destroys the magnetic moment of 4 neighbouring Co atoms as estimated from the fact that bulk CoCr alloy becomes nonmagnetic at about C o = 25 at. %. He further argued that the observed M  s  should  be larger in thin films than in homogeneous bulk samples since the Cr atoms may not be uniformly distributed in the films, in which case the (1 — 4Cc>) factor should be replaced by a larger value. The above model explains his data, 3 4 and fits to the values of M  s  in Fig.  2.15 for Ccr = 20 at. %. A detailed study of the saturation magnetization as a function of Ccr, film thickness and sputtering parameters other than T was recognized as being 8  worthwhile; however, such a study could not be fit into the timeframe allowed for the main scope of this thesis. We notice that there is no report of a detailed study of the electronic states to justify the semiempirical formula (4.1). In the following discussion we attempt to interpret (4.1) in terms of a previous band structure calculation 3 5 for pure Co crystals. Singal and D a s 3 5 calculated the band structure for pure Co crystals using a hybridizedtight-binding-plane-wave method and were able to predict the saturation magnetization. For future reference in this thesis, the electron density of states for majority- and minorityband states are reproduced in Figs. 4.1 and 4.2. Singal and D a s 3 5 also derived the electron distribution in various bands as follows: Nl  d  = 5.033,  NL 57  = 3.490,  '  Nj  = 0.248,  ree  JVJ r e e = 0.230  (4.2)  where '3d' and 'free' denote for 3d bands and free electron-like 4s bands, respectively, and the arrows refer to majority- and minority-bands. We conclude from (4.2) that, approximately, 1.5 electrons are needed to fill up the magnetic, 3d bands and, consequently, to destroy the magnetic moments. If a Cr atom transfers all of its 6 (3d 5 4s 1 ) outer cell electrons to the 3d bands of Co, it can just make 4 Co atoms nonmagnetic. This explains the (l-4Ccr) factor in Haines' model.  4.2 Dielectric Constants of CoCr Films A n automatic spectroscopic ellipsometer described in detail by Sullivan 2 8 has been used to measure the dielectric constants of the CoCr films. It has been well known within the framework of the effective medium t h e o r y 3 6 - 4 0 that the measured dielectric constants are very sensitive to the surface morphology as well as to the dielectric constants of the material itself. Qualitatively, for a rough surface(described by a terminology "microroughness") or a material containing voids full of air or dielectric material (described by a terminology "porosity") the depolarization effects due to the microroughness or porosity will reduce the absolute value of the measured e\ and 6 2 , the real and imaginary part of the dielectric constants, respectively. In the present case air and chromium oxides and/or cobalt oxides fill the voids between the grains and reduce the absolute values of the measured dielectric functions. Dielectric constants of CoCr sputtered on an unheated glass substrate, with PAT — 3 mTorr, W=350 watts and dt  s  = 8 cm were measured and presented in Figs. 4.3 and 4.4.  Previous measurements on single crystal 4 1 and thin  film42'43  Co are also shown in the  figures for comparison. | t\ | and | e | of CoCr films decrease as the films become thinner, 2  indicating that larger microroughness or porosity are developed for smaller t h i c k n e s s . 3 6 - 4 0 Similar increase in microroughness or porosity with reduced thickness is observed for CoCr sputtered on unheated and heated (T — 250 C) NiCr coated glass as indicated in s  Figs. 4.5 and 4.6. To study the effect of substrates on the film morphology, the dielectric constants for CoCr deposited on heated (at T = 250 C) T i coated glass are also presented s  in Fig. 4.7. To clarify the relation between dielectric constants and the surface morphology, S E M surface photographs are shown in Figs. 4.8, 4.9 and 4.10, respectively, for CoCr films on unheated NiCr coated glass with thicknesses of 230 , 400 and 910 nm. A S E M 58  photograph for CoCr deposited on NiCr coated substrate at Tt = 250 C is also presented in Fig. 4.11. Common to Figs. 4.3 to 4.7 is that the substrate does not affect the dielectric function (or the surface morphology) significantly if the films are thicker than about 500 nm. The S E M photographs in Figs. 4.8, 4.9 and 4.10 indicate that a rough surface develops into a smoother surface as the film thickness is increased. Further inspection reveals that high  Ts increases the surface microroughness as confirmed by S E M photographs in Fig. 4.11. We interpret the increase in surface microroughness or porosity with reduced thickness as attributed to the growth of an initial layer with poor texture, as discussed in section 2.5. Since the preferential growth direction is parallel to the c axis, 3 2 grains with randomly distributed c axes will give rise to a rough surface. As the c axes become better aligned with larger thickness, a smoother surface results. As for the substrate effect, Inspection of data in Figs. 4.3 to 4.7 suggests that the NiCr substrate yields the smoothest CoCr film surface, or the weakest thickness dependence compared to other substrates under the same deposition conditions. We have found in section 2.5 that the similarities between the structure and texture of CoCr and NiCr weakens the thickness effect on the magnetic properties of CoCr films. The substrate effects on the dielectric function discussed above indirectly support the point of view regarding the effect of substrate on film growth as discussed in chapter 2.5. Sullivan 2 8 investigated the effects of substrate temperature on the surface morphology of Pd films, using ellipsometry techniques. He interpreted the increase in microroughness in terms of Thornton's 2 0 structure zone model. According to the model, as the substrate temperature is increased, small grains combine and grow into larger ones at the T-zone as a consequence of increased adatom mobility. Larger grains will, of course, make the surface rougher as confirmed by S E M photograph in Fig. 4.11. Let us consider the spectral dependence of the dielectric constants. From the above discussion, we conclude that the best samples for the study of the spectrum of the material are the thickest, unheated CoCr films because they yield the smoothest surfaces and the measured dielectric constants for those samples best represent the dielectric constants of the material. The first model that comes to mind is the Drude model of free electrons, which works well for noble metals 4 4 up to 2 eV. If the Drude model is valid, plots of —1\ and €2/2A versus A 2 , respectively, should yield straight lines as the model suggests42. 59  However, such plots as shown in Fig. 4.12 for a 870 nm thick CoCr film sputtered on unheated glass substrate do not give any hints of straight lines. One might think the long wavelength end could still be represented by Drude model, but the slope of the line at the long wavelegth end for the plot of —t\ versus A 2 yields an optical mass 360 times larger than the rest mass of the electron. We conclude that the Drude model is inadequate in the case of the above data and, therefore, the dielectric constants have a significant contribution from interband transitions.  In the case of pure Co, it was found that the  interband transition is the dominant mechanism 4 2 . Indeed the electron density of states in Fig. 4.2 indicates high density of states in the range of interest for this study. We also conclude from Figs. 4.3 and 4.4 that great care should be taken in the interpretation of measured dielectric constantes of Co and CoCr because they are very sensitive to the condition of the surface.  4.3 Anisotropy in Dielectric Constants. As discussed in chapter 3, CoCr films sputtered at small dt and located in an asym3  metric position have a magnetic easy (hard) axis tilted away from the film normal. What is the direction of the optical axes? In general the dielectric tensor for a material with magnetization M = 0 can be written as45  /<? e = A{4>,0,ib) 0  Vo  where  a n c  *  E  3a  r e  ° e° 0  0\ 0  A"1^,*,^)  (4.3)  4 )  the three principal values of the dielectric tensor and A is the  rotation matrix given by Goldstein 4 6 in terms of the three Euler angles: <f>, 0 and rp. The rotations defining the Euler angles 4 6 are shown in Fig. 4.13. CoCr samples 500 nm thick were deposited on glass substrates for measurement in the spectroscopic ellipsometer described in section 4.2. The deposition conditions were: PAT = 3 mTorr, dts — 8 cm and T = 210 C and the substrate is glass. A n angle of 20 8  degrees is found between the magnetic easy axis and the film normal as shown in Fig. 3.3. Accurate measurement of the Kerr effect requires the installation of a magnet near the sample holder in the ellipsometer since the error due to the misalignment of the film sample with respect to the incident light is comparable to the Kerr effect. A detailed study of the Kerr effect in CoCr is out of the scope of the thesis. To avoid the complication due 60  to the Kerr effect(which is negligible in the study of spectral dependence), the sample is "demagnetized" with an A C magnetic field that decreases slowly in amplitude, i.e. the magnetic remanence is brought down to zero. A special sample holder was made such that we can rotate the sample around the film normal without taking the sample off the holder. The sample is then rotated through the first Euler angle ^>(see Fig. 4.13) about the film normal while the complex reflection ratio  p is measured at 2 eV. $ = 0 correspond to such an orientation that the X-axis in Fig. 2.2 is in the plane of incidence. The complex reflection ratio, p , is defined as rp/ra  P = r  p  =  p/ t>  E  E  rs = KIK  (4-4)  where Ea and Ep are components of the electric field of the light perpendicular and parallel to the plane of incidence, respectively, and superscripts T and V denote for incident and reflected light, respectively. The result of p as a function of (j> is given in Fig. 4.14. Using a new matrix techique that will be described in detail in the next chapter, we are able to calculate p as a function of <j) and fit the experimental data to the nine parameters of the dielectric tensor (<f>, 0 and xb; real and imaginary part of e ° ,  and £ 3 ) . In practice,  only three or four parameters are needed to obtain a good theoretical fit to the data. When the nature of the optical anisotropy is unkown a priori,  the goal of the fit is to  obtain a combination of the smallest number of uncorrelated unknown-parameters and a reasonable fit to the data. The first model is the isotropy model. The dielectric constant (fo) obtained this way can be regarded as the average of the three principal values of the dielectric tensor. In the example of Fig. 4.14, over a dozen models with different number of parameters (usually 2 to 4) and different combinations have been tried. The best fit is as follow: e? = £ 0 ( l + « i - t " f c )  £° = £ 0 ( 1 - h + ih) 4 = <o  0 = $ = 0, <j> = 3 5 . 3 ° where € 0 = -14.17 + :'20.21 61  (4.5)  8.16 x 10- 3 3.69 x 10- 3 Where £  l5  (4.6)  6 and <f> are the fitting parameters. Therefore, such a sample can best be 2  described as being slightly biaxial with one optical principal axis normal to the film. It is so far unclear how this biaxial model is related to the uniaxial magnetic anisotropy model described in the last chapter.  4 . 4 Extraordinary Hall Effect of CoCr Films Extraordinary Hall effect is characterized by a Hall voltage in ferromagnetic materials about two orders of magnitude larger than that in nonmagnetic materials. This voltage is proportional to the magnetization of the ferromagnets and is a consequence of the spinorbit coupling of the conduction electrons. We neglect the contribution from the ordinary Hall effect and express the Hall resistivity pjj as p  H  -  47  (4.7)  AirR M s  where R is the spontaneous Hall coefficient and M is the magnetization of the sample. s  It was well known that the experimental data of R can be related to that of the a  electrical resistivity, p , by r  48  (4.8) where (l < n < 2). The theories of extraordinary Hall effect represent some of the most complicated and sophisticated theories in solid state physics. Early works include those of Karplus and Luttinger , Smit and Luttinger . The basic model is that the magnetically 49  50  51  polarized electrons move under the influence of an external electric field and scattered by impurities and phonons. The spin-orbit coupling gives rise to a transverse current which is the origin of the extraordinary Hall effect. Luttinger  51  found that the Hall coefficient R  s  can be divided into two parts. The  first part is the higher order corrections to the Born approximation for the scattering of polarized electrons by impurities. This part is the skew scattering part and was found to be proportional to the density of the scattering centers and, therefore, to the electrical resistivity. '  51 52  The second part arises from the off diagonal elements of the density matrix  describing the electronic states and does not have a classical analogue. The corresponding 62  Hall current was found to be independent of the density of the scattering centers. Therefore, this part of the Hall coefficient is proportional to the square of the resitivity 5 1 ' 5 2 and is often refered to as the side jump part. Irkhin and Shavrov 5 3 used the same theoretical point of view as Luttinger and showed that, as the temperature is raised, phonon scattering becomes important and enhances the side jump part significantly. The exponent n in (4.8) can be interpreted in terms of the above theories as follows: If n = 1 the mechanism of the extraordinary Hall effect is the skew scattering; if n = 2 the mechanism is the side jump. In the present study ac Hall measurement was chosen since it was found that the dc measurement had a large noise level due to thermoelectric and thermomagnetic effects. The dc method was used to determine the sign of the Hall voltage. The ac Hall voltage was found to be independent of the ac current frequency in the range from 10 Hz to 1000 Hz. During the measurement the frequency was kept at 83 Hz. A photolithographic technique was used to make the CoCr film into the four point probe pattern and a transformer was used to match the impedance of the sample to that of the lock-in amplifier (Fig. 4.15). The applied magnetic field, H , was scanned from -10 to 10 kGauss. A cryostat shown in Fig. 4.16 was used to provide ± 0 . 5 K temperature ranging from 4 K to 300 K . Three 500 nm thick CoCr films deposited on glass substrate at T  = 100, 210 and  B  300 C were studied. The other sputtering parameters are: P and W=350 watt. fiClcm.  A r  — 3 mTorr, dt = 9 cm s  The resistivities of the above three films were found to be about 80  The measured Hall voltage as a function of the applied field appears very different  for different samples. For example, Vu versus H plot for the film with T = 300 C in Fig. s  4.17(a) appears very different from that with T = 210 C in Fig. 4.17(b). However, careful s  inspection shows that they both can be explained by the same equation, given below: V„ = (R + 4nR M/t)I 0  (4.9)  s  where Ro is the resistance due to the misalignment of the leads where the Hall voltage is picked up. Ideally, if Ro = 0, the output signal would be like Fig. 4.18 and the highest sensitivity for the equipment would be achieved. When 4irR M/t s  be like that in Fig. 4.17(a), and 4nR M/t s  Ro »  4nR M/t, s  > Ro, the output will  < Ro yields an output like Fig. 4.17(b). If  a large off-set VJJ will decrease the sesitivity of the measurement of  pH- The above reasoning justifies the use of photolithographic technique to fabricate well aligned samples in order to optimize the sensitivity of the measurement. The temperature dependence of the Hall effect in the above CoCr films was measured 63  /  'using the cryostat in Fig.  4.16 from 4 K to 300 K. Fig. 4.19 gives />#, at the state  of saturation magnetization, as a function of the temperature for the three CoCr film samples,  pn seems to increase with T  the dependence of H on T c  pH to H  c  is as follows: H  c  s  as seen from Fig. 4.19, which is a reminder of  8  in Fig. 2.9. One possible explanation for the correlation of depends on Cr segregation as discussed in section 2.3. The  Cr segregates act as scattering centers for the magnetically polarized electrons and the spin-orbit coupling of the scattered electrons gives rise to the extraordinary Hall effect. McAlister and H u r d 5 2 ' 5 4 developed methods to determine the signs of the skew and the side jump part of the spontaneous Hall coefficient for various metals.  They found  that the sign for the side jump part varies from metal to metal. No explanations had been given for the signs of the side jump part.  According to McAlister and H u r d 5 4 , a  pure Co single crystal has a negative Hall voltage. At temperatures greater than 78 K, pn increases with temperature, indicating a positive side jump contribution to pn- The reason is that phonon scattering is more important at higher temperatures.  From Fig.  4.19, one sees that the sign of Hall voltage is positive, which is different from pure Co. Since p {T) r  increases with temperature due to scattering, Fig. 4.19 also implies that pn  of CoCr has a positive side jump part, just like a pure Co single crystal. The explanation of the difference between the signs of pn for CoCr and Co remains a theoretical problem. The electrical resistivity of the CoCr film with T = 200 C was also measured in the a  temperature range of 4 to 300 K and the following relation was deduced for the data: oc  R (T) S  \p (T)]  n  r  n = 1.5 ± 0.1  (4.10)  This result indicates that both types of mechanism, skew scattering and side jump, contribute to the extraordinary Hall effect of CoCr.  The relation (4.10) for CoCr is to be  compared to that 5 2 for Ni and Fe: Ni:  R (T)cx [priT)} -  Fe:  R (T)  1  5  s  S  oc [p (T)} -° 2  r  (4.11)  Further studies of CoCr films, such as a theoretical treatment of the extraordinary Hall effect would be very useful; however, as outlined in the introduction, we have decided to devote the remaining part of the thesis to another, equally important, subject related to emerging recording technologies: magneto-optic media.  64  F i g u r e 4.1. Electron density of states for majority- and minority-band electron states, taken from Singal and D a s . The Fermi energy and the 3 5  occupied states(shaded area) are shown.  65  F i g u r e 4.2. Combined eiectron density of states for both the majorityand minority band state, obtained from F,g. 4.1.  66  E (eV) F i g u r e 4.3. Real part of the dielectric constants t i for CoCr sputtered on glass. Sputtering parameters for the samples are: T = 30 C , PAT = 3 S  mTorr, W=350 watts and du — 8 cm. Data for single crystal Co are from Ref. 41. |i and _L correspond to electric field parallel and perpendicular to the crystal c axis of Co, respectively. Data for film 1 and film 2 are from Ref. 42 and 43, respectively.  67  40  35  30  25-  N  20H  Co SINGLE CRYSTAL (||) SINGLE CRYSTAL (1) FILM 1 FILM 2 CoCr THICKNESS 870 nm 560 nm ._ 250 nm  15-  10-  5-  E (eV) F i g u r e 4.4.  Imaginary part of the dielectric constants £2 for the same  set of sample as in F i g . 4.3. D a t a for pure C o are also from the same sources given in the previous figure caption.  68  E (eV)  F i g u r e 4.5. Dielectric constants for CoCr sputtered on unheated NiCr coated glass substrate. Deposition parameters are the same as those for Fig. 4.3.  69  THICKNESS:  i  1  F i g u r e 4.6.  ,  2  1  1  3  4  E(eV)  Same as Fig. 4.5 except T = 250 C . e  70  — i  5  1  6  F i g u r e 4.7. Same as Fig. 4.5 except Ts = 250 C and the substrate is T i coated glass substrate.  71  F i g u r e 4.8. S E M p h o t o g r a p h o f surface o f 230 n m t h i c k C o C r  sput-  t e r e d o n u n h e a t e d , N i C r c o a t e d g l a s s s u b s t r a t e . T h e m a g n i f i c a t i o n is 5 0 , 0 0 0 a n d t h e d o t t e d line at t h e l o w e r r i g h t c o r n e r o f t h e p h o t o g r a p h 600 n m . S p u t t e r i n g c o n d i t i o n s a r e t h e s a m e as for F i g . 4 . 3 .  72  represent  F i g u r e 4 . 9 . Same as Fig. 4.8 except the thickness is 400 nm.  73  £0KV  F i g u r e 4.10.  X5@!0K"'600nm  Same as F i g . 4.8 except the thickness is 910 nm.  7-1  20KV  X50.8K  6@0nm  F i g u r e 4.11. S E M photograph of surface of 900 n m thick CoCr sputtered on NiCr coated glass substrate at T  s  parameters are the same as for Fig. 4.3.  75  = 250 C. Other sputtering  22  oH 0.0  1  1  1  1  1  1  0.1  0.2  0.3  0.4  0.5  0.6  X (yixm ) 2  2  F i g u r e 4.12. Plots of - c i and £ 2 / 2 A versus A 2 , respectively, for the 870 nm thick sample in Figs. 4.3 and 4.4.  76  1 0.7  F i g u r e 4.13.  The rotations defining the Euler angles, taken from  Goldstein 4 6 .  77  -0.492  -0.506 j 0  i SO  I 100  I 150  I 200  I 250  I 300  I 350  <(> (DEGREE)  F i g u r e 4.14. Complex reflection ratio p versus <f> for a CoCr film, where <f> is the first Euler angle defined in Fig. 4.13. The deposition conditions for the film are specified in text. The solid line and the dashed line are theortical fits for the real and imaginary part of the complex reflection ratio, respectively.  78  I 400  145 kfl REFERENCE  cH 0.20 m m I  1  100 8.0 m m  N3-t  I  •  LOCK-IN AMPLIFIER TO RECORDER  0.52 m m  F i g u r e 4.15. measurement.  P a t t e r n of sample and circuit diagram for H a l l effect  O n l y two leads of the four point probe type of sample are  used in the measurement.  T h e applied magnetic field is along the  normal.  79  film  LIQUID N  DIFFUSION  g  PUMP  LIQUID He  F i g u r e 4.16. Schematics of the cryostat system for measurement of Hall voltage as a function of temperature ranging from 4 K to 300 K. He gas provides a thermal link between the sample and the liquid He reservoir. An electrical heater and a Si diode thermometer on the sample holder are used to control the sample temperature to ± 0 . 5 K.  80  0.8 " . 0.7-  H (kG) F i g u r e 4.17(a). Hall voltage, V#, as a function of the applied field, H , for CoCr deposited on glass substrate with T  s  sputtering parameters are: P  Ar  — 3 mTorr, d  ts  = 300 C. The other  = 9 cm and W=350 watts.  The applied field scans from -10 to 10 kGauss, but only the magnitude of the field is recorded in this experimental setup.  81  2.2-i  "1  1  0  2  i 4  H (kG)  1  1  1  6  8  10  F i g u r e 4.17(b). Same as Fig. 4.17(b) except T = 210 C . This figure e  appears very different from Fig. 4.17(a) because of different values of Ro, where Ro is the resitance due to the misalignment of the leads where the Hall voltage is picked up.  82  F i g u r e 4.18. Predicted Hall voltage, V „ , as a function of the applied field, H , for an ideal sample with Ro = 0.  83  1  0.9-  0.3 H 0  1  ;  50  |  1  100  1  150  200  1  250  T (DEG K)  F i g u r e 4.19.  Hall resistivity of CoCr as a function of temperature  for films deposited at different growth temperature T . 8  Other sputtering  parameters are given in Fig. 4.17(a). The lines are intended to help the reader separate the data sets.  84  300  Part B Optics of Magneto-Optic Recording Media Chapter 5 Use of the 4 x 4 Matrix Method for Magneto-Optic Multilayers 5.1. Plane Wave Model for Magneto-Optic Recording Media Erasable multilayer magneto-optical recording media were proposed by Connell 6 . The working principle of the system is that the optical interference of the multilayer medium enhances the reflectivity component induced by the polar Kerr effect in the magnetic layer 6 and, therefore, increases the signal-to-noise ratio (SNR). In practice the recording medium consists of four layers on a substrate; the overcoat layer, which acts both as an anti-reflectance layer and a protection layer for the subsequent layers, the magnetic layer, which causes the polar Kerr effect, an intermediate phase matching layer, and finally a reflectance layer. Nakamura et al.  studied experimentally the relation between the Kerr  55  rotation angle and the thickness of the intermediate layer and found that by choosing the proper material and thickness for the phase matching layer, the Kerr rotation angle could be significantly enhanced. Mansuripur et al  derived the expression for the SNR and used  56  an iterative formula to optimize the Kerr component of the reflectivity. If ret is the complex refractive index for the ith layer, then the iterative formula can be written as follows 5 7 :  l + r . ^ r ^ e  1  2  ^  *  >  where rj is the contribution from the reflectivity of all the layers below the tth layer as shown in Fig. 5.1 and /?, = 27r£,-nt-/A is the film phase thickness for the tth layer assuming an e~ ri-i i t  lult  time dependence in the electric fields. The complex Fresnel reflectance coefficient,  , for the interface between the (t — l)th and the tth layer is given by: i-i,i  r  ni-i - rti = ~—~zr  (-) 5  2  for normal incidence. The magneto-optical effect can be calculated by considering different indices of refraction for left-handed and right-handed circularly polarized light in the magnetic layer. Gamble et al.  58  have simplified the iterative formula by considering the  Kerr effect only to first order and showed that the gain of an optimized mutilayer system depends on the ratio n"/n',  where n' and n" are the real and imaginary parts of the in-  dex of refraction of the magnetic layer, respectively. This index of refraction is associated 85  with the nonmagnetic part of the matrix elements of the dielectric tensor (or the diagonal matrix elements in the case of an isotropic medium). The iterative formula, however, is valid only for isotropic media, perpendicular magnetization and for normal incidence. Although it is usually sufficient to consider the above restricted situations, a more general solution may be required in order to characterize the recording medium with oblique incident light in cases such as ellipsometry studies 5 9 ' 6 0 . As well, in practice the magnetization may not be perfectly normal to the film and may affect the SNR and, similarly, some of the thin film layers may be anisotropic. However, an analytical solution for a more general case is very complicated algebraically even for the case of a bare substrate 6 1 . Therefore, it is desirable to have a computational method to treat any arbitrary multilayer structure. Various methods have been described by Berreman 6 2 , Y e h 4 5 , and Lin-Chung and Teitler 6 3 and the last method has been found to be the most convenient.  86  '5.2. I m p l e m e n t a t i o n o f t h e 4 x 4 M a t r i x M e t h o d In general, light propagating through a medium has two eigenmodes (or two degrees of freedom) so that the electromagnetic field can be specified by two eigenpolarizations of the electric or magnetic fields, with a different index of refraction associated with each eigenmode 6 4 .  For a multilayer system the electromagnetic field for each layer has two  additional degrees of freedom due to the reflected light and must therefore be described by four independent variables. The boundary conditions on the interface connect the four variables of adjacent layers and leads to the 4 x 4 matrix method. Y e h 4 5 developed the 4 x 4 matrix algebra in a very compact form; however, his method requires the evaluation of the polarization vectors for the four eigenmodes to construct the matrix. This leads to computational difficulties in cases such as an isotropic medium where the two transmitting (or reflecting) modes are degenerate and the two associated polarization vectors are arbitrary. The expression for the polarization vectors of the eigenmodes given by Y e h 4 5 can  then not be normalized due to this degeneracy. While the above special case can be handled separately, we find it more practical  to implement the method of Lin-Chung and Teitler 6 3 . In this method they chose the four components of the electromagnetic field to be the four independent variables and constructed a column matrix ^(not to be confused with the third Eulerian angle): / E \ Hy Ey \-H J x  (5.3)  x  for  a right-handed Cartesian coordinate system where the z-axis is along the normal to  the multilayer structure.  This column matrix remains always well defined and can be  determined from the the matrix equation 6 3 , n tb = Atb  (5.4)  z  where n is the eigenvalue of the refractive index associated with the z component of the z  wavevector and the 4 x 4 matrix A is a function of the dielectric tensor of the medium. For a magnetic medium the complex dielectric tensor can be separated into magnetic and  nonmagnetic components: e = e° + e  (5.5)  m  where e° is the nonmagnetic component of the dielectric tensor and was expressed by Y e h 4 5 in terms of the three principal axis values c 0 , e and e% (see also equation (4.3)). 2  87  The magnetic part of the dielectric tensor can be written as ,  / e  m  =  66z i z i  0 0 -—Sgi 6zi Syi  -f>yi\ 6 6 x ix i  0  -Sxi  (5.6)  0  where S x , 6 y and 6Z are complex off-diagonal matrix elements responsible for the magnetooptical effects. Furthermore, 6 oc M , the magnetization vector. 6 4 After solving (5.4), the total field can be written as a linear combination of the four eigenvectors,63 or xb = ^<f>  (5.7)  where ^ is a 4 x 4 matrix consisting of the eigenvectors from (5.4) as columns and <j> is a weight vector chosen to be the electric field components. 6 3 The matrix # is calculated in a computer program as follows. After (5.4) is solved, the transmitting modes are separated from the reflecting modes by the signs of the real part of the eigenvalue n . Then the ratio, z  I C  p  = \E  X  E  I  2  \>+\E,  P  ( 5  -  8 )  is evaluated for the four modes. The eigenvectors of the transmitting modes with the large and small C occupy the first and third columns, respectively. Similarly, the eigenvectors p  of the reflecting modes with the large and small C will occupy the second and fourth p  columns, respectively. Then all the columns are normalized such that * n = 1, ^33  = ^34  = —1,  = 1- The above arrangements and normalizations, although not explicitly  stated by Lin-Chung and Teitler 6 3 , are very important to ensure a smooth transition from anisotropic to isotropic layers. The final result can be expressed in terms of the reflection matrix:  relating the incident and the reflected electric fields. The program was first tested on systems with known results: 1) substrate of isotropic nonmagnetic medium with oblique angle of incidence; 2) substrate of anisotropic nonmagnetic medium with oblique angle of incidence, for which Lettington 6 5 has given the analytical expression; 3) substrate of anisotropic and magnetic medium with oblique angle of incidence 6 1 ; 4) multilayers of isotropic and nonmagnetic medium as described by equation (5.1); 5) isotropic medium with magnetization direction normal to the film as 88  calculated by the iterative formula in equation (5.1) for left-handed and right-handed circularly polarized light; and 6) The gain function calculated by Gamble et al. .  In all cases  58  the agreement was good to four significant figures.  5.3. Application to Quadrilayer Recording Medium In order to optimize the Kerr component of the reflectivity with respect to the thickness of the dielectric layers of the recording medium, we have applied the above 4x4 matrix method to the magneto-optic multilayer structure 6 6 . The multilayer structure is shown in Fig. 5.2, with an overcoat and phase matching layer consisting of SiC*2 , and with M n B i and A l as the magnetic and the reflector layers, respectively. been studied by Mansuripur et al.  This system has  using the iterative formalism and their values for the  56  complex refractive indices for these layers were used in our calculations, corresponding to a wavelength of 840 nm. In addition, we further investigate the effect of anisotropy, an oblique angle of incidence, and misalignment of the magnetization on the SNR of the optimized quadrilayer system. We fix the magnetic layer thickness at 90 A , which is typical for mutilayer magnetooptical recording media. Mansuripur et a/. 5 6 and Gamble et al. 58 have shown previously that the readout of the recording system is insensitive to the magnetic film thickness in this thickness region. To optimize the SNR one calculates | r thickness of the overcoat layer t  c  | 2 as a function of the  sp  and that of the intermediate layer ti ( Fig. 5.2), and  presents the results in a contour plot shown in Fig. 5.3. The maximum of | r  sp  at ti ~ 0.225A, and t  c  ~ 0.238AC, where A t = A / n , and A c = A / n c are the wavelengths  of light in the intermediate and the overcoat layer, respectively. of 2000 | r  | 2 ~ 0.9 and the values of t  sp  | 2 occurs  c  The maximum value  and U corresponding to the maximum are in  reasonable agreement with those obtained Mansuripur et al.  56  From Fig. 5.3 , we find  that the maximum occurs at ti ~ A,-/4 as predicted by Mansuripur et al. 56 . To study the sensitivity of | r  8p  | 2 to the nonmagnetic component of the dielectric  tensor of the magnetic layer, we vary the imaginary part of the refractive index of the magnetic layer while leaving the magnetic part (i.e. the 6 vector) unchanged. The results are shown in the contour plots in Figs. 4 (a) and (b). We find that if n" is reduced by 1/2, | r  sp  | 2 is increased by about a factor of 3 and the peak of | r  | 2 moves to lower U  sp  and higher t as shown in Fig. 5.4(a). If n" is increased by 1/2 , | r c  sp  | 2 decreases by 2/3  and the peak moves to higher U and lower t , as seen in Fig. 5.4(b). The above trends c  89  are consistent with the findings of Gamble et al.  that the key factor to the gain of the  recording system is the ratio of the imaginary part of the refractive index of the magnetic layer to the real part. These contour plots can also be useful in recording disk structural design and can be fitted to experimental data when studying the effect of the thickness of the dielectric layers on the S N R . 5 5 ' 6 0 It can be seen from Fig. 5.4(b) that | r  8p  | 2 is very sensitive to t,- in the  range from |A t - to A,/2 , which is undesirable in system design as it is sometimes difficult to control this thickness. 5 5 ' 6 0 We should point out that in some theories 5 8 ' 6 0 the intermediate layer is taken to be Aj/2 thick, while Figs. 3 and 4 indicate that this value does not optimize | r notices that the gain function defined by Gamble et al.  58  sp  | 2 . One  has such a special form that it is  unchanged whether one uses A,/2 or A t /4 for the intermediate layer, provided the magnetic layer is thin enough. However, the thickness will affect | r  sp  | 2 . The gain function G is  defined a s 5 8  G= l Ir where R =| rj  l^ ~ sp  |  m s  (1 -  /j. JQX  R)  | 2 and the subscript 'ms' denotes the corresponding quantity in the case  of a substrate of the same magnetic material. One finds that in the case of £, = A,/2, the system is not well matched, i.e. it does not have a minimum, R, so that the increase in R cancels the decrease in | r  sp  | in (5.10). Therefore one must be very careful in using the  gain function as the criterion for optimization, since it is meaningful only when the system is well matched. To optimize the readout of the system, t{ should be approximately A^/4 as predicted by Mansuripur et al.  56  , with the exact value depending upon the dielectric  constants and the thickness of the magnetic layer. Based on the optimized quadrilayer system corresponding to Fig. 5.3, we vary the refractive index of the intermediate layer to study the effect of anisotropy and oblique angle of incidence. In Fig. 5.5 | r  8p  | 2 is plotted as a function of the incident angle 6Q for  different degrees of anisotropy as described by K{ , where e3  Kie°  x  (5.11)  for the intermediate layer. Here we have chosen the principal axis of € 3 to be parallel to the z axis. We find that the effect of the oblique angle of incidence is small up to about 45 degrees. This is expected since | r  sp  \ is relatively insensitive to the phase mismatch in the 90  vicinity of the maximum as can be seen in Fig. 5.3. The anisotropy of the intermediate layer is quite significant as shown in Fig. 5.5. Finally, we study the effect of misalignment of the magnetization vector M. Fig. 5.6 shows | r  8p  | 2 as a function of 0 , the angle between the direction of magnetization and the m  normal to the film in the case of normal incidence. The Kerr component of the reflectivity | r  s p  | gradually decreases to zero at 90 degrees as expected. In practical situations, one  treats the magnetization as having a Gaussian distribution with a distribution function f(9m)  Fig. 5.6 shows (| r  sp  (5.12)  =  | 2 ) averaged by the Gaussian distribution while varying the standard  deviation, 0 , which is a measure of the misalignment. From these results, we find that a  even in the case of a very serious misalignment, the decrease in SNR is not significant.  91  Fig.  5 . 1 . Schematic for the iterative formalism. r,-_i - is the Fresnel )t  coefficient between i — 1th and tth layer and rj  is the contribution from  the reflectivity of all the layers below the tth layer.  92  (  OVERCOAT: SiO. n=1.50  90 A n =3.77+3.921 n =3.56+3.791 MAGNETIC FILM: MnBi INTERMEDIATE  n=1.50  LAYER: SiO, >500 A  n=2.00+7.10i  REFLECTOR: Al  SUBSTRATE  Fig.  5 . 2 . T h e structure of a typical, quadrilayer magneto-optical  disk. It consists of S i 0 2 as the overcoat layer and the intermediate phase m a t c h i n g layer, M n B i , as the magnetic layer, and A l as the reflector. T h e refractive indexes are taken from Ref. 56, at a wavelength of 840 n m . T h e superscript  and '—' denote the refractive indices for left-handed and  right-handed circularly polarized light, respectively.  93  F i g u r e 5.3. Contour plot of 2000 | r  8p  | as a function of the thickness 2  of the intermediate layer £, and that of the overcoat layer t . c  94  Fig.  5.4(a). Same as Fig. 5.3 except that the imaginary part of the  reflective index, n", is reduced by 1/2.  t,<\> Fig.  5.4(b). Same as Fig. 5.3 except that the imaginary part of the  reflective index, n " , is increased by 1/2.  95  F i g u r e 5.5. Plot of 2000 | r  ep  | 2 as a function of the incident angle  6 , showing the effect of anisotropy. Based on the quadrilayer in Fig. 5.2, 0  with optimized thicknesses for the layers, the principal axis value of the dielectric tensor, e0,, is varied for the intermediate layer. K{ = cpj/e? and e  i  =  £  2» where e°, e% a n d c°j are the principal values of the dielectric tensor.  We have chosen the axis of C3 to be the film normal.  96  0 (deg) a  0  10  10  0  20  20  30  30  40  50  60  70  80  90  0 (deg)  F i g u r e 5.6.  Dependence of 2000 | r  sp  | 2 on 0 , the angle between m  the direction of magnetization and the normal to the film. Also shown is 2000{|  l  ) averaged over a Gaussian distribution as a function of the  standard deviation of the distribution 0 . a  97  Chapter 6 Interaction of Focused Light Beams With Magneto-Optic Recording Media  6.1. Problems in Magneto-Optics Theories For a long time it has been realized that the reading laser in a magneto-optic recording system is a strongly focused light beam instead of a plane wave (see Fig. therefore, a Gaussian beam model,  56  1.3), and,  which has a Gaussian amplitude distribution, is  expected. However, no one seems to have carried out a detailed, theoretical study for a Gaussian beam interacting with the magneto-optic multilayers, probably due to the lack of a practical approach for this problem. Reviewing the literature on Gaussian beams, we found the following. In the 60's and 70's, most of the theoretical work were concerned with how a Gaussian beam was generated and how it propagates in space. 6 7 The medium that interacts with the Gaussian beam was approximated as a lens.  This approximation is good as long as the beam focus is not  too small or the beam rays are nearly parallel to each other. 6 8 Recently , researchers have started to look at the interaction of a Gaussian beam with a multilayer medium 6 9 and found that there was a shift in both the position of the focal spot and the angle of reflection for the reflected beam. But, the treatment is based on paraxial approximation and the interest is in nonspecular phenomena. For the ease of computation the Gaussian beam is often assumed to be a scalar field. This does not cause any difficulty in the paraxial approximation since light rays propagating parallel have only one field component. Simon et a l . 7 0 were the first to notice that, even in the paraxial approximation, the longitudinal component of the field in the direction of the beam axis, though small, is essential for a consistent description of the Gaussian beam. To find out what happens beyond the paraxial approximation, one has to solve the wave equation under a realistic boundary condition. Such a task was accomplished by Takenaka et a l . 7 1 based on some previous works of others. This is the starting point of the present work. In order to make use of the beam field solution derived by Takenaka et a l . 7 1 , we consider the following: First, it must be decomposed into plane waves so that the technique developed in the last chapter can be used. This turns out to be nontrivial task and requires a considerable amount of mathematical skill.  Second, for a strongly  focused light beam there is significantly large component with propagation vector in an 98  oblique angle of incidence, which implies that the field component parallel to the beam axis (longitudinal) is important. Therefore, one must include the longitudinal field in the theory, while Takenaka et a l . 7 1 only treated a scalar field. Third, the final result must be easy to apply for numerical studies so that optimization of a thin film system is possible.  6.2. Interaction of a Strongly Focused B e a m W i t h M a g n e t o - O p t i c M e d i u m The goal is to derive an expression for the reflected or transmitted beam given the incident beam field. For a focused beam propagating in the +z direction, the wave equation and a realistic boundary condition for the incident beam are as follows: (V  2  +  E {r,0)  2  0  =  x  (6.1)  k )E (r,<f>)=0 x  (6.2)  exp(-r /w ) 2  2  (6.2) is regarded as realistic since it emphasizes the important fact that the energy of the focused beam is concentrated on a finite area with symmetric intensity distribution in the focal plane. The Fourier transform of the solution to (6.1) and (6.2) as the incident beam, and the expression for the beam field reflected from a magneto-optic multilayer recording medium, have been described in an article 7 2 by the author for publication purposes. Since the article is up to date and quite self-contained, the author would rather include it as Appendix B than go through the nightmare of re-typing the mathematic equations. This section discusses some points that are not included in Appendix B, but quite important to understanding the physics behind the equations therein. To demonstrate the importance of going beyond the paraxial approximation when the focal spot size is small, we plot the intensity of a Gaussian beam (corresponding to s=0 in equation (3) of Appendix B) in Fig. 6.1(a), which is to be compared to the solution to (6.1) and (6.2) as shown in Fig. 6.1(b). In both Figs. 6.1(a) and 6.1(b), a spot size wo = A/2 is used. Another interesting point is that one can never produce a beam that satisfies (6.1) and (6.2) exactly. This point is demonstrated as follows. In equation (11) of Appendix B, the spectrum of the beam field contains 6(k + k /2ko 2  z  — ko) which relates the z component of  the propagation vector k to the transverse components k = y k\ + z  r  as indicated by the  solid and dashed line in Fig. 6.2. A n exact solution to (6.1) and (6.2) requires a spectrum that extends to the whole range (both solid and dashed line in Fig. 6.2). However, a real light beam only propagates in one direction and can never have k 99  z  < 0(i.e. only the solid  line in the figure is realistic). This is not very discouraging since the k  z  > 0 part of the  spectrum represents a very good approximation to the solution to (6.1) and (6.2) which well justifies the effort. To optimize the readout of the multilayer magneto-optic recording medium, one has to have an expression for the reflectance and transmittance for the focused beam. This is also included in another paper 7 3 written for publication purposes and is presented in Appendix C . The numerical integrations associated with equation (21) of Appendix B and (11) and (12) of Appendix C are quite time consuming. To minimize the cost of computation, the integrands in the above-mentioned integrals are approximated by polynomials of order 12. This concludes the major part of the thesis and an overall summary is presented next as the concluding chapter.  100  0  F i g u r e 6.1(a).  0.2  0.4  0.6  0.8  C o n t o u r plot of a Gaussian beam intensity near the  focal spot. T h e beam propagates in the +z direction and r = \ / x + y . z 2  and r are expressed in units of wavelength A .  101  2  F i g u r e 6 . 1 ( b ) . Contour plot of the beam intensity near the focal spot, given by the solution to equation (6.1) and (6.2) in the text, z and r are in units of wavelength A.  102  F i g u r e 6.2.  Spectral range for a focused beam as indicated by the  solid line and the dashed line. k  and k are the longitudinal and trasverse  z  r  components of the wavevector, respectively, and ko = 27r/A, where A is the wavelength. 6 in the figure is the angle of incident for a plane wave component with wavevector (k ,k ). r  z  k  z  film normal.  103  axis is assumed to be along the  Chapter 7 Summary  7.1. S u m m a r y of P a r t A  1. CoCr(20 at.  % Cr) films with potential for perpendicular or longitudinal recording  application have been fabricated using a dc magnetron sputtering technique.  The  magnetic coercivity of the films is found to increase with the substrate temperature, T, s  during film growth. Temperature T in the range from 180 to 300 C is essential a  for the fabrication of films with high coercivities. Through microstructural analysis, an account has been given regarding the effects of substrate temperature on the microstructure. The basic theoretical model is that higher adatom mobility upon heating results in larger grain size and, possibly, Cr segregation to grain boundaries. 2. Sputtered NiCr films can be used as a substrate material to eliminate the effect of initial layer growth, which occurs with glass substrate. A NiCr underlayer was shown to give a higher perpendicular coercivity than T i , which were previously reported. The improved magnetic property with NiCr (or Ti) substrate is attributed to the similarity of the structure and texture of NiCr(or Ti) to those of CoCr. 3. A broad range of magnetic anisotropy (-1.0 to 0.5 106erg/cc) has been obtained by varying the target-to-substrate distance and the sputtering power. The dependence of the anisotropy on the above two parameters is interpreted in terms of the adatom diffusion constant and the deposition rate. A balance between the two effects determines the anisotropy. 4. When the target-to-substrate distance is small or the deposition rate high, the effect of asymmetric sputtering becomes important. Asymmetry in both magnetic and optical axes of the CoCr films due to this effect has been analyzed. 5. Ellipsometry and S E M experiments indicate that CoCr film surface has larger microroughness as the film thickness is reduced. This microroughness is associated with the poor texture of the initial layer. 6. Models for the analysis of optical anisotropy measured by ellipsometry are described and demonstrated for CoCr produced by asymmetric sputtering. It is found that the 104  optical axes are not necessarily coincident with the axes of magnetic anisotropy. 7. The extraordinary Hall voltage for CoCr has been measured as a function of the applied magnetic field at temperatures ranging from 4 K to 300 K . The Hall voltage is found to be positive which is opposite to the sign for pure Co. The Hall resistivity increases with the growth temperature T . This increase is attributed to the scattering of conduction s  electrons by Cr segregation. The Hall resistiviy is found to be related to the electrical resistivity by pjj <x p l '  5 ± 0 A  , where both pu and p are varied by the temperature. r  7.2. S u m m a r y o f P a r t B 8. A 4 x 4 matrix technique has been implemented which makes the analysis of optical anisotropy and calculation of optics for magneto-optic multilayers routine activities. The technique has been applied to ellipsometry data of CoCr with optical anisotropy and the calculation of readout from magneto-optic recording medium in a plane wave model. It also provides input data for a more realistic model calculation as described below. 9. Using a Fourier transform technique, the reflected beam field of a strongly focused light beam from magneto-optic multilayers is calculated and found to have an asymmetric distribution in intensity. 10. The reflectance and transmittance of a focused beam interacting with a multilayer system have been derived and used to optimize the readout of the magneto-optic recording medium. It is shown that when the spot size is less than about three wavelengths, significant difference arises between the plane wave model and the focused beam model.  105  Appendix A A Computer Program for Reflection Matrix of Multilayer Systems  A . l . Purpose The purpose is to calculate the reflection matrix for a plane wave of light reflected from a multilayer system. The structure and the dielectric tensor for each layer is totally arbitrary. The reflection matrix is defined as: (A.1) The convention for definition of symbols is the same as those used through out the previous chapters. Note that the axis for reflected p-component of the electric field is defined such that it has the same direction as the axis for the incident p-component at normal incidence. This definition is different from that of some other authors. 7 4 To change to the convention of others, one just has to change the signs of r  ps  and r . pp  The dielectric tensor for each layer can be written for a time dependent factor exp(—iut) as follows:  6 is proportional to the magnetization in the magneto-optic medium, and the reader is refered to section 4.3 for the definitions of the above symbols. Comments are written in the computer program for those who want to understand it or improve it. The comments refer to equations appearing in three references as follows: ' L ' for Lin-Chung(Ref. 63); ' B ' for Berreman(Ref. 62); ' Y ' for Yeh(Ref. 45). The subroutine is named ' B E R R ' in honour of D . W . Berreman (Ref. 62) who introduced the four components for the electromagnetic fields of the eigenmodes in a manner most convenient for application.  106  A . 2 . H o w to U s e IMPLICIT  R E A L * 8 ( A - H , 0 - Z)  C O M P L E X * 1 6 RSS,RSP,RPS,RPP C A L L B E R R ( N L A Y E R , THO, RSS, RSP, RPS, RPP) In the calling program, a common area must be created to provide information about the structure and delectric tensor for each layer before calling the subroutine 'BERR', as follows: COMMON  / C R Y S T / ISO, E V , H , A N , PHI, T H E T A , PSI, E D , D E L T A  INTEGER*4  ISO(6)  C O M P L E X * 1 6 ED(3,6), DELTA(3,6),AN(6) REAL*8  E V , H(6), PHI(6), T H E T A ( 6 ) , PSI(6)  - N L A Y E R : On entry it must be set to the number of layers above the substrate. - T H O : On entry it must be set to the angle of incidence. - R S S : Complex*16 variable. On entry it need not be specified. On exit it gives the first element of reflection matrix defined in (A.l) above. Similarly for RSP, RPS, RPP. - I S O : If ISO(I)=0, the Ith layer(layers are numbered so that the 1st layer is the top layer and the substrate is ( N L A Y E R + l ) t h layer) is isotropic and one only needs to specify H , A N for the layer. If ISO(I)=l, the Ith layer is optically anisotropic or magnetic, and one must specify H , PHI, T H E T A , PSI, E D , D E L T A for the layer. - E V : On entery it must be set to the photon energy of the incident light in unit of eV. - H : O n entry H(I) must be set to the thickness of the Ith layer in unit of A . - A N : On entry, if ISO(I)=0, A N (I) must be set to the complex refractive index for the Ith layer. It need not be specified if ISO(I)=l. - P H I : On entry, if ISO(I)=l, PHI(I) must be set to the first Eulerian angle of the rotation matrix in (A.2) for th Ith layer. If ISO(I)=0, it need not be specified. Similarly for T H E T A , PSI, the second and third Eulerian angles, respectively. - E D : On entry , if ISO(I)=l, ED(J,I) must be set to the Jth principle value of dielectric constants (see (A.2) above) for the Ith layer. If ISO(I)=0,it need not be specified. - D E L T A : On entry , if ISO(I)=l, ED(J,I) must be set to the Jth component (J=l,2,3 correspond to x,y,z component) of the vector 6 in (A.2) for the Ith layer. If ISO(I)=0, it need not be specified.  107  C  C C  SUBROUTINE BERR(N, THO. RSS. RSP. RPS. RPP) IMPLICIT REAL*8(A - H.O - Z) C0MPLEX*16 RSS,RSP.RPS.RPP COMMON INTEGER*4 C0MPLEXM6 REAL'S  /CRYST/ ISO. EV. H, AN. PHI. THETA. PSI, ED, DELTA IS0(6) ED(3.G). DELTA ( 3 .6 ) . AN( 6 ) EV. H(6), PHI(G). THETA(G). PSI(6)  COMMON /PSI/BPSI, BPSIN , EIG COMPLEX*16 BPSI(4,4),BPSIN(4.4),EIG(4) C0MPLEX*16 CTHO.CITHO .PHCON ,AI.FT, 1 BK(4).F(4,4),YM(4,4). YMM(4,4). YMDEL PARAMETER (AI=(0.0D0.1.ODO),PI=3.141592653589793DO)  C C C C  PHCON=DCMPLX(O.DO. -2.DO*PI*EV/12400.DO) INVERSE PSIO AND INITIALIZE YMM FOR MATRIX MULTIPLICATION*, L14) DO 5 11=1.2 DO 5 Jd=1,2 YMM(JJ,11 + 2) (0.DO,0.DO) YMM(Ud+2,II)=(0.D0,0.D0) CONTINUE CTHO=DCMPLX(DCOS(THO)*0.5DO,O.DO) CITHO=DCMPLX(0.5DO/DCOS(THO).0.D0) YMM(1,1)=(O.5D0.0.DO) YMM(1,2)=CTH0 YMM(2.1)=(-0.5D0,0.D0) YMM(2,2)=CTH0 YMM(3,3)=(0.5D0.0.D0) YMM(3,4)=CITH0 YMM(4.3)=(0.5D0,0.D0) YMM(4.4)=-CITH0 B  5  C C  CALCULATE PSI MATRIX(L2) DO 200 LAYER = 1, N + 1 IF(ISO(LAYER).E0.0) THEN CALL ISOPSI(LAYER.THO) ELSE CALL ANIPSI(LAYER.THO) END IF  C C  IF (LAYER .LE. N) THEN CALCULATE THE PHASE FACTOR DO 160 1 = 1 . 4 BK(I)= CDEXP( PHCON*EIG(I)*DCMPLX(H(LAYER),0.DO) 160 CONTINUE C C CALCULATE THE F MATRIX ( L 6 ) DO 190 K " 1 , 4 DO 190 1 = 1 . 4 FT = (0.ODO.0.ODO) DO 180 J • 1. 4 FT = FT + BPSI(I.d) * BK(J) * BPS1N(J,K) 180 CONTINUE  C  108  )  F(I,K) FT CONTINUE 190 C C MATRIX MULTIPLICATION (L11.14) CALL CDMULT(YMM, F. YM, 4, 4, 4, 4. 4, 4) CALL CDCOPY(YM. YMM, 4, 4, 4. 4) ELSE CALL CDMULT(YMM, BPSI, YM. 4. 4, 4, 4. 4. 4) END IF 200 CONTINUE C YMDEL • (1.D0, 0.D0)/( YM(1,1) * YM(3,3) - YM(1,3) * YM(3,1) ) RSS « (YM(4,3)*YM(1,1) - YM(4.1)*YM(1.3)) • YMDEL RPP = (YM(2,3)*YM(3,1) - YM(3,3)*YM(2.1)) * YMDEL RSP = -(YM(3,3)*YM(4,1) - YM(4.3)*YM(3,1)) * YMDEL * DCOS(THO) RPS = (YM(2.3)*YM(1,1) - YM(2.1)*YM(1.3)) * YMDEL / DCOS(THO) RETURN END SUBROUTINE ANIPSI(LAYER,THO) IMPLICIT REAL*8(A - H.O - Z) COMMON INTEGER»4 C0MPLEX*16 REAL*8  /CRYST/ ISO, EV. H. AN. PHI. THETA. PSI. ED. DELTA ISO(G) E D O . 6 ) , DELTA ( 3 ,6 ) . AN(G ) EV, H O ) , PHI(G). THETA(G), PSI(6)  COMMON /PSI/BPSI, BPSIN. EIG COMPLEX*16 BPSI(4,4),BPSIN(4,4),EIG(4) C0MPLEX*1G 1 2 REAL*8 1  C C C C  PARAMETER DATA ANX  EE(3,3), AI. BM(G,6). BD. BA3(6), BA6(G), DD(4.6). BDEL(4,4),DET,COND,DDET,DCOND,BPNOM1,BPN0M2, BPN0M3,BPN0M4 A O . 3 ) . S0R(2.4), C0MEX(4), V I ( 4 . 4 ) , AR(4,4), A I I ( 4 , 4 ) , ER(4), E I ( 4 ) . VR(4,4) (AI=(0.0D0,1.ODO).PI=3.141592653589793D0) BA3. BA6 /12*(0.0D0,0.0D0)/. BM /36*(O.ODO.O.ODO)/  = DSIN(THO)  CALCULATE ROTATION MATRIX A (Y2) SPHI = DSIN(PHKLAYER)) CPHI = DCOS(PHI(LAYER)) SPSI - DSIN(PSKLAYER)) CPSI ' DCOS(PSI(LAYER)) STHETA = DSIN(THETA(LAYER)) CTHETA = DCOS(THETA(LAYER)) A(1,1) • CPSI * CPHI - CTHETA • SPHI * SPSI A(1.2) » -SPSI * CPHI - CTHETA * SPHI * CPSI A(1.3) • STHETA * SPHI A(2.1) = CPSI * SPHI + CTHETA * CPHI * SPSI A(2,2) = -SPSI * SPHI + CTHETA * CPHI * CPSI A(2.3) » -STHETA * CPHI A(3.1) » STHETA * SPSI A(3.2) « STHETA * CPSI A(3.3) « CTHETA  109  C  TRANSFER BY A MATRIX TO FORM DIELECTRIC TENSOR EE (Y1) DO 30 11 = 1. 3 DO 20 J J « 1 . 3 EE(II.iJJ)  DO  10 20 30  = (O.ODO.O.ODO)  10 K 1 , 3 E E ( I I . J J ) * E E ( I I . J J ) + A(II.K) * A(dJ.K) * ED(K,LAYER) CONTINUE CONTINUE CONTINUE £E(1.2) = EE(1.2) + A l * DELTA(3,LAYER) EE(2.1) = EE(2,1) - A l * DELTA(3,LAYER) EE(1.3) «= EE(1.3) - A l * DELTA(2. LAYER) EE(3.1) = EE(3,1) + A l * DELTA(2,LAYER) EE(2.3) = EE(2,3) + A l * DELTA(1.LAYER) EE(3.2) = EE(3.2) - A l * DELTA(1,LAYER) c  C C  INPUT OPT CONST INTO BM MATRIX (=M MATRIX IN B3) DO 40 J = 1 . 3 DO 40 I = 1 . 3 BM(I,d) = E E ( I . d ) 40 CONTINUE DO 50 I = 4 . 6 BM(I.I) » (1.ODO.O.ODO) 50 CONTINUE C C BA3 AND BAG ARE USED TO EXPRESS THE EZ.HZ C COMPONENTS OF THE TOTAL FIELD (B20.21) BD = (1.D0.0.D0)/ ( BM(3.3) * BM(6.6) - BM(3,G) * BM(6,3) ) BA3(1) = (BM(6.1)*BM(3,6) - BM(3,1) *BM(6, G ) ) * BD BA3(2) = ((BM(6.2) - ANX)»BM(3,G) - BM(3,2)*BM(6,6)) * BD BA3(4) = (BM(G,4)*BM(3.6) - BM(3,4)*BM(6,G)) » BD BA3(5) = (BM(G,5)*BM(3.G) - (BM(3.5) + ANX)*BM(6,6)) * BD BA6(1) = (BM(6,3)*BM(3.1) - BM(3,3)*BM(6.1)) * BD BAG(2) = (BM(6,3)*BM(3.2) - BM(3,3)*(BM(G,2) - ANX)) * BD BA6(4) = (BM(6.3)*BM(3.4) - BM(3,3)*BM(6,4)) * BD BA6(5) = (BM(6.3)*(BM(3.5) + ANX) - BM(3,3) *BM(6.5)) * BD C C DD MATRIX IS USED TO ARRANGE (B15-18) INTO (B23) DO 60 J = 1. 6 DD(1.J) «= BM(5.d) + (BM(5,3) + ANX) * BA3(U) + BM(5,G) * BA6( 1 J) DD(2,J) « BM(1,J) + BM(1,3) * BA3(J) + BM(1,G) * BA6(J) DD(3.d) = -(BM(4,J) + BM(4,3)*BA3(J) + BM(4;6)*BA6(J)) DD(4.vJ) = BM(2.0) + BM(2.3) * BA3(J) + (BM(2,G) - ANX) * BA6( 1 J) GO CONTINUE C C THE BDEL MATRIX FORMS EIGN EQUATION FOR (EX,HY,EY.-HX) C OR (GAMMA 1,GAMMA5,GAMMA2,-GAMMA4) (B23) DO 70 I = 1. 4 BDEL(I,1) = DD(I.1) B D E L U . 2 ) ' DDU.5) BDEL(I,3) = DD(I,2) BDEL(I,4) * -DD(I.4) 70 CONTINUE C C SOLVE EIGENVALUE PROBLEM (L1) DO 80 I = 1, 4  110  80  DO 80 J • 1. 4 AR(d.I) = DREAL(BDEL(d,I)) A I I ( J . I ) « DIMAG(BDEL(J,I)) CONTINUE CALL OCEIGN(AR. A l l . 4. 4. ER. E I . VR. V I . 1ERR0R. 1. 1) IF (I ERROR .NE. 0) THEN WRITE (6.*) I ERROR, 'TH EIGN VALUE NOT FOUND' END IF  C C C C C  SORT EIGEN VALUE AND EIGEN VECTOR ACCORDING TO LINCHUNG'S CONVENTION: SORT MODES INTO I N C I D E N T ( 1 1 , 1 2 ) AND REFLECTED(d1,J2) MODES ACCORDING TO THE SIGNS OF THE POSITIVE PART OF THE EIGNVALUES. DO 90 1=1,4 SOR(I.I) = ER(I) S0R(2.I) = DFLOAT(I) 90 CONTINUE CALL SSORT(SOR, 2, -4, 4) 1 1 » S0R(2. 1 ) 1 2 = S0R(2,2) J1 = S0R(2.3) 02 = S0R(2,4) C C CALCULATE EX**2/(EX**2+EY**2 ) FOR EACH MODE TO DETERMINE C WHETHER EX IS ITS DOMINATE POLARIZATION DO 100 1=1,4 COMEX(I) = (VR(1,I)**2 + V I ( 1 . I ) * * 2 ) / (VR(1.I)**2 + VI(1.I)*» 1 2 + VR(3,I)**2 + V I ( 3 . I ) * * 2 ) 100 CONTINUE C C ARRANGE ORDER OF MODES ACCORDING TO THE C SCHEME OF LINCHUNG (L12) BASED ON COMEX OF EACH MODE IF (C0MEX(I1) .GE. C0MEX(I2)) THEN EIG(1) = DCMPLX(ER(I1),EI(11)) EIG(3) = DCMPLX(ER(I2).EI(I2)) DO 110 1 = 1 , 4 B P S K I . 1 ) = DCMPLX(VR(I.I1).VI(I.I1)) B P S K I . 3 ) = DCMPLX(VR(I,I2),VI(I.I2)) 110 CONTINUE ELSE EIG(3) = D C M P L X ( E R ( I 1 ) . E I ( I D ) EIG(1) = DCMPLX(ER(I2).EI(I2)) DO 120 I = 1. 4 B P S K I . 3 ) = DCMPLX(VR(I,I1).VI(I.I1)) B P S K I . 1 ) = DCMPLX(VR(I.I2).VI(I.I2)) 120 CONTINUE END IF IF (C0MEX(J1) .GE. C0MEX(J2)) THEN DO 130 I = 1. 4 B P S K I . 2 ) = DCMPLX(VR(I.J1).VI(I.d1)) B P S K I . 4 ) = DCMPLX(VR(I.J2).VKI.J2)) 130 CONTINUE EIG(2) = DCMPLX(ER(J1 ) . E I ( J 1 ) ) EIG(4) = DCMPLX(ER(J2).EI(J2)) ELSE DO 140 I • 1. 4 B P S K I . 4 ) • DCMPLX(VR(I,d1).VI(I.cM)) B P S K I . 2 ) = DCMPLX(VR(I .J2) ,VI(I ,J2) )  Ul  140  C C  CONTINUE EIG(4) • DCMPLX(ER(J1),EI(J1)) EIG(2) - DCMPLX(ER(J2).EI(J2)) END IF  NORMALIZATION ACCORDING TO (L12) BPN0M1=BPSI(1,1) BPN0M2=-BPSI(1,2) BPN0M3=BPSI(3,3) BPN0M4=BPSI(3.4) DO 150 1 = 1 . 4 BPSI(1,1) = B P S I ( I . I ) / BPN0M1 BPSI(1.2) = BPSI(1,2) / BPN0M2 BPSI (1,3) = BPSI(1.3) / BPN0M3 BPSI(1,4) = BPSI(I,4) / BPN0M4 150 CONTINUE  C C  CALCULATE THE INVERSION OF BPSI 00 170 I • 1, 4 DO 170 U = 1. 4 BPSIN(J.I) = B P S I ( J . I ) 170 CONTINUE CALL CDINVT(BPSIN. 4. 4. DET, COND) RETURN END  C  C C C C C  SUBROUTINE ISDPSI(LAYER.THO) IMPLICIT REAL*8 (A-H.O-Z) COMMON INTEGER*4 C0MPLEX*1S REAL'S  /CRYST/ ISO, EV, H. AN. PHI. THETA. PSI. ED. DELTA IS0(6) ED(3.6). DELTA(3.6).AN(6) EV. H ( 6 ) . PHI(6), THETA(6), PSI(6)  COMMON /PSI/BPSI. BPSIN . EIG COMPLEX*16 BPSI(4,4),BPSIN(4.4),EIG(4) COMPLEX*16 CTH.ANL ANL=AN(LAYER) CTH=CDSORT((1.DO.O.DO)- (DSIN(TH0)/ANL)**2 ) EIG(1)=CTH*ANL EIG(3)=EIG(1) EIG(2)=-EIG(1) EIG(4)=EIG(2)  C  5  DO 5 11=1.2 DO 5 JJ=1.2 BPSIN(JJ.I1+2)=(0.DO.O.DO) BPSIN(JJ+2.II)=(0.D0.0.D0) CONTINUE BPSIN(1,1)=(0.5D0.0.D0) BPSIN(1,2)=(0.5D0.0.D0)*CTH/ANL BPSIN(2.1)=(-0.5D0.0.D0) BPSIN(2.2)=BPSIN(1.2) BPSIN(3.3)=(0.5D0.0.D0)  112  BPSIN(3,4)=(0.5D0.0.D0)/(ANL*CTH) BPSIN(4,3)=(0.5D0,0.D0) BPSIN(4.4)=-BPSIN(3,4) C  65  DO 65 11=1.2 DO 65 JJ=1.2 B P S K d J . 11 + 2) = (0.DO.O.DO) BPSKJd+2.II)=(O.D0.O.D0) CONTINUE BPSI(1.1)=(1.DO.O.DO) B P S K 1,2) = (-1 .DO.O.DO) B P S K 2 . 1 )=ANL/CTH BPSI(2.2)=BPSI(2.1) BPSI(3.3)=(1.DO.O.DO) BPSI(3.4)=(1.DO.O.DO) BPSI(4,3)=ANL*CTH BPSI(4.4)=-BPSI(4.3) RETURN END  113  Appendix B An Article on Reflected Beam Fields of a Light Beam Focused on a Magneto-Optic Multilayer Structure  114  B-2  ABSTRACT  G e n e r a l f o r m a l i s m f o r r e f l e c t i o n o f a s t r o n g l y focused beam from m a g n e t o - o p t i c m u l t i l a y e r t h i n f i l m s i s e s t a b l i s h e d u s i n g the F o u r i e r technique.  transform  I t i s most u s e f u l when the focused spot s i z e i s o f the o r d e r o f  the w a v e l e n g t h , i n w h i c h case the p a r a x i a l a p p r o x i m a t i o n i s not v a l i d . s i g n i f i c a n t asymmetry i n the f i e l d d i s t r i b u t i o n i s found f o r the beam a t normal i n c i d e n t a n g l e .  A  reflected  T h i s asymmetry can be e x p l a i n e d i n terms o f  c o n t r i b u t i o n s from p l a n e wave components w i t h d i f f e r e n t  115  incident  angles.  B-3  1.  Introduction  I t I s o f t e n u s e f u l t o focus a c o l l i m a t e d beam t o v e r y s m a l l spot o f s i z e of a wavelength.  One such example i s the new e r a s a b l e  recording technology * . 1  2  magneto-optic  P r e v i o u s o p t i c s t h e o r i e s t r e a t the i n c i d e n t  beam i n the r e c o r d i n g system as p l a n e waves » »**. 1  3  Such a t r e a t m e n t  c o n v e n i e n t i n o p t i m i z a t i o n o f the r e c o r d i n g m e d i a .  the  laser  is  However, a p r o p e r  theory  s h o u l d take I n t o account the f a c t t h a t f o r s t r o n g l y focused beams, even the G a u s s i a n beam d e s c r i p t i o n i s i n s u f f i c i e n t and one has t o go beyond the paraxial approximation . 5  U s i n g a scheme proposed by Lax e t a l . , Takenaka e t a l . were a b l e 6  to  f i n d the beam f i e l d d i s t r i b u t i o n beyond the p a r a x i a l a p p r o x i m a t i o n . 5  C o r r e c t i o n s t o the fundamental G a u s s i a n beam were e x p r e s s e d i n terms o f complex-argument L a g u e r r e - G a u s s i a n beams.  Zauderer  7  the  l a t e r o b t a i n e d the same  c o r r e c t i o n terms u s i n g a p e r t u r b a t i o n e x p a n s i o n t e c h n i q u e .  It  is  interesting  t o s t u d y how a system o f m u l t i l a y e r f i l m s responds t o the beam f i e l d s mentioned a b o v e . I n t h i s w o r k , the i n c i d e n t beam i s d e s c r i b e d by the beam f i e l d s d e r i v e d by Takenaka e t a l . . 5  F o u r i e r t r a n s f o r m ( F . T . ) t e c h n i q u e i s used t o decompose  the beam f i e l d i n t o p l a n e waves so t h a t the respond o f the m u l t i l a y e r t h i n f i l m s t o p l a n e waves can be a p p l i e d .  The r e f l e c t e d beam o b t a i n e d I n t h i s way,  i n g e n e r a l , has an asymmetric d i s t r i b u t i o n over the beam c r o s s s e c t i o n .  It  w i l l be shown t h a t t h i s asymmetry a r i s e s from d i f f e r e n t responds o f v a r i o u s p l a n e wave components i n the i n c i d e n t beam.  116  B-4  2.  (  F o u r i e r Transform o f Beam F i e l d s Beyond P a r a x i a l A p p r o x i m a t i o n  A model f o r a s t r o n g l y focused beam can be d e s c r i b e d as f o l l o w s .  A  c o l l i m a t e d G a u s s i a n beam i s p o l a r i z e d i n the x d i r e c t i o n and then focused by a p e r f e c t l e n s down t o a spot s i z e W on the f o c a l p l a n e z » 0 as shown i n q  F i g . 1.  The e q u a t i o n and boundary c o n d i t i o n f o r the x component of the  e l e c t r i c beam f i e l d i s e x p r e s s e d i n c y l i n d r i c a l system as  (V  2  + k )E (r,z) o x  «= 0  2  (1)  -r /w2 E (r,0) = e ° 2  (2)  x  where k  Q  =  w/jIeT and  a time dependent f a c t o r e  is  depressed.  The s o l u t i o n f o r (1) and ( 2 ) was g i v e n by Takenaka  E (r,z) = X  Z  I n  I  n  ( - 1 ) * C» n  c< > f 2s  P  2 s  w"  2 m  v-  °  P  m  5  as  *<°>  e  1  V  (3)  OS+p  8=0 p=0 m=0  where  C = E}_ p (p-m)! m!  (4)  m  c  „( p c  2 b  ) =  ( o )  - 1  (-1) (2s)I s(p-l)!(s-p)!(s+p)! S + P  117  »  B-5  s -  1,  2, . . . , p • 0,  s.  1,  (5)  where one has used t h e c o n v e n t i o n t h a t a term c o n t a i n i n g a f a c t o r i a l of a n e g a t i v e number s h o u l d be s e t t o z e r o . And  f  1  k w o o  w (l + iz/z ) o o 2  z  - k w /2 o o  (6)  2  o  4 / ° ^ i s t h e complex-argument L a g u e r r e - G a u s s i a n beam d e f i n e d  J-0 (n-j)l(j!)  5  as  °  2  F o u r i e r t r a n s f o r m i s d e f i n e d by  ,# U ( t ) = 'In)'  where x  z  + y  = r  2  9  JJJ E(x,y,z)e  -ik x-ik y-ik z dxdydz X  y  Z  (8)  has been used and i n t e g r a t i o n over a l l p o s s i b l e v a l u e s o f  the v a r i a b l e i s i m p l i e d throughout  t h i s work.  From Appendix A , one f i n d s  2j - y r - i k x - i k y l i r e dxdy 2  x  118  7  B-6  -kr /4y  |  2  JL  -(j+JH-l),2A  (9)  Jl»0  Coefficient F.T.  Q. i s g i v e n i n Appendix A and k i s d e f i n e d j " r r e q u i r e s the f o l l o w i n g r e l a t i o n :  ;  (-if* ^) . 1  q=A  * „  2  P  (n-q)!(q)  2  as k  2  2  QH  6  (  1  0  )  ^  q  Using (9) and ( 1 0 ) ,  which i s a l s o derived i n Appendix A.  - k + k . r x y  2  the d e s i r e d F . T . can  be w r i t t e n as:  U (k , k ) = X  x  r  k  Z  „ . . 2(8+p)  E I I s=0 p=0 m=0  ( 2 n )  m m (2s) 2s 2(s+p-m4-l) 8+p p p o 8+p  -3/2 _ (  c  c  f  Q  w  - k / 4 v - i ( k - k )z e ° dz 2  J  y"  111  r  Z  " r s p ... , -k w /4 k.z Z Z Z (2n)G(8,p,m)k^ P) e ° s=0 p=0 m=0 2  00  2  N  1 / 2  8 +  r  o  .m 6(k. )  b  e  dk  (11)  m  where  k k  = k  b  z  +  2  2k  - k o  o  r s _ m ( 2 s ) 2(s+p+l) -m s+p G(s,p,m) = f C c 'w ' z Q .5 * ' p p o o s+p 2  x  v  v  v  r  r t  n  x  119  ,  1  v (12) 0  B-7  and 6 f u n c t i o n i s used such t h a t :  J  F(x)  4— dx  3.  6(x-x )dx - ( - l ) °  m  m  m  (13)  ^ — F(x) dx m  R e f l e c t i o n from Magneto-Optic M u l t i l a y e r F i l m s  U s i n g the 4 x 4 m a t r i x t e c h n i q u e * , 3  8  one i s a b l e t o f i n d the  r e f l e c t i v i t y from a m u l t i l a y e r t h i n f i l m system w i t h a r b i t r a r y tensors.  dielectric  I f t h e f i l m system i s p l a c e d a d i s t a n c e d from the f o c a l p l a n e , the  spectrum f o r the r e f l e c t e d beam can be expressed  i n terms o f r e f l e c t i o n  matrix  f o r plane w a v e s : 3  i2dk  r z r  ss ps  (£) '  r (£) sp '  (£)  r (£) pp  v  v  v  v  D ({) B  (14)  o (S> _ p  where the s u p e r s c r i p t ' r ' denotes the r e f l e c t e d beam and the d i r e c t i o n of s and  p components a r e d e f i n e d i n F i g . 2. Since k  = - k , one needs t o r e p l a c e z i n the i n v e r s e F.T. by z z r  z  (see F i g . 1 ) . From Maxwell's e q u a t i o n : V»E • 0, one can r e l a t e U  k U + k U - 0 XX z z  Using  to U  = -z by:  (15)  ( 1 4 ) , (15) and the c o o r d i n a t e system i n F i g . 2, the spectrum o f  the r e f l e c t e d beam f o r the model o f F i g . 1 can be expressed as  120  B-8  i2dk U  v  "  e  2  U  x  l  ( R  c o s 2  *k  +  B±n2  2  R  \  +  R  3  B  i  n  \  c  o  s  \>»  v « x, y  i2dk D  z  =  6  Z  x  D  l  ( R  C O S < | )  k  +  R  2  8  i  n  V  (  1  6  )  where  R  x  1  - r  , R • r , R - -(r /cos9 + r pp' 2 ss' 3 sp ps x  v  'X • r /cos8, 1 sp  R  l  =  r  p p  U s u a l l y , the r e f l e c t i o n R^(9,k). j  cos9) '  x  t  8  9  *  R  2  R^ • - r cos9, 2 ps '  =  "'PS  8  l  n  3  = r  9  pp  -  r  ss  (  matrix i s only a function  k , k can be r e l a t e d t o 8 by r z  1  7  )  v of 9 and k : R . ( k , k ) » j r z  tan 8 = k / k r z  k  The i n v e r s e F . T . r e q u i r e s an  2  - k  integral  121  2  r  + k  2  z  (18)  B-9  k. z b O  (  „V  r  .m 0  1  \  r / i  i ( k -k ) z Z  o'  ,.  i ( k -k /2k ) z 2  (-1)  e  (19)  » . (k ) jm r  where  .m (k j  m  ik,(z-iz  )  ) v  e  2  v  ftk b  r  (20)  R ( k , k.+k - k / 2 k ) ] j r ' t> o r o'  b m  J  V  0 K  o  From (16) and ( 1 9 ) , one f i n a l l y o b t a i n s the r e f l e c t e d beam f i e l d s as f o l l o w s :  s z E s=0 p=0  E;(r,<t>,z') -  x /- 2(--h»). r o k  " tJ  < 1. " 2 m W  W  p E m=0  r  1 2  { 1  ) c O S  2  *  f  1N  ik z * G(s,p,m)e  2  v im  ( v  7  +  m  W  +  w  1 „v 3m s  i  v 2m' o  n  2  v  r'  * ]J  ( r 2  V>W  v = x,y  °> s p . .m S E E s=0 p=0 m=0 N  E'(r,<|>,z ) = r  f 2(s+p) -k /4y * •'or e  i k z' G(s,p,m)e °  2  k  (W^sin*  122  - W cos<|))iJ (rk )k dk Z  m  1  r  r  r  (21)  B-10  where 4> i s d e f i n e d  through  x • r cos <J> y - r s i n 4>  (22)  and  z«  = z  r  + 2d  y' -  (23) w [l 2  In  4.  (21) J , J , and J„ a r e B e s s e l o 1 2  + i(z +2d)/z ] r  functions,  R e f l e c t e d Beams a t F a r F i e l d s  From (3) one can show that  z^  at z »  t  o n l y the f i r s t  G a u s s i a n beam) i s s i g n i f i c a n t and behaves l i k e l / ( z - i z ) . o  much f a s t e r . in  The q u e s t i o n a r i s e s  ( 3 ) when c a l c u l a t i n g  naturally:  term i n (21) ( t h e A l l the r e s t  decays  Can one n e g l e c t a l l other terms  the r e f l e c t e d beams a t f a r f i e l d ?  Or does the o t h e r  beam f i e l d s mix w i t h the Gaussian beam a f t e r i n t e r a c t i o n w i t h the m u l t i l a y e r system? At  farfield,  Wj  the l e a d i n g term i s  ffl  (k ) = [i(z+2d - i z ) ] r  o  123  m  R*(k., k - k 2 / 2 k ) o  o  (24)  B-ll  S i n c e R j and J ^ ( r k ) have upper bounds, the i n t e g r a l s i n (21) can be w r i t t e n r  as  -k /4Y' 2  g( + ) j  (z+2d - i z )  f f l  - i<-2ik )  8 + P + 1  o  o  a o  ^  (  8  +  P  )  e  1  k dk r  g(4>) ( s + p ) ! / ( z + 2 d - i z )  S + p  o  where g i s a bounded f u n c t i o n .  -  r  (25)  m + 1  One c o n c l u d e s i m m e d i a t e l y from (25) and (21)  t h a t o n l y s = 0 ( G a u s s i a n beam) i s i m p o r t a n t a t f a r f i e l d .  T h e r e f o r e , i n such  a model the s t r o n g l y focused beam can be approximated by a G a u s s i a n beam a t far  fields. I t i s i n t e r e s t i n g t o f i n d from (21) t h a t , i n g e n e r a l , the r e f l e c t e d beam  has a s i g n i f i c a n t dependence on  T h i s dependence w i l l be l e s s s i g n i f i c a n t  as the spot s i z e g e t s l a r g e r , as e x p e c t e d .  (21) has been a p p l i e d t o  erasable  m a g n e t o - o p t i c m u l t i l a y e r r e c o r d i n g media f o r system o p t i m i z a t i o n and the d e t a i l s w i l l be p u b l i s h e d e l s e w h e r e .  One example i s g i v e n below t o  demonstrate the asymmetry mentioned above. A t y p i c a l n u m e r i c a l a p e r t u r e f o r the l e n s o f the r e c o r d i n g s y s t e m N.A. = 0.5.  I f one t a k e s the d i a m e t e r o f the l e n s t o be i r w ( z ) (99%  criterion ),  where  9  w(z)  the e s t i m a t e d spot s i z e i s  9  w  - W / 1 + (z/z )' q  2  is  (26)  o  • ( / 3 / 2 ) \ , where \ i s the w a v e l e n g t h .  For  o convenience d i s set to zero s i n c e i t only Introduces a s h i f t i n z  124  T  I n the  B-12  present model. The parameters  a r e c a l c u l a t e d from the same m u l t i l a y e r s t r u c t u r e  i n R e f . 3 w i t h the t h i c k n e s s and r e f r a c t i v e i n d e x e s l i s t e d i n T a b l e 1.  as  The  w a v e l e n g t h i s t h a t o f a l a s e r d i o d e ( 0 . 8 3 0 um). W i t h the system parameters i n (21)) f i e l d  s p e c i f i e d above, the G a u s s i a n beam (s=0 term  i n t e n s i t y 2000* |E*(r,<|>,z ) | / | E ( o , z ) | r  2  r  2  x  I n F i g . 3 as a c o n t o u r p l o t .  for z  = 5 Z  q  i s shown  The n o r m a l i z a t i o n f a c t o r i s chosen so t h a t  the  r e s u l t can be compared t o the r e s u l t s i n R e f . 3 . v  The asymmetry a r i s e s from a n o n - v a n i s h i n g w ^  y o  y  and ( w ^ Q  One can see from ( 1 7 ) , t h a t t h i s i s due t o a nonzero a n g l e 6. the c o n t r i b u t i o n o f E t o E I n c i d e n t a t x y of (9,-<|)) w i t h a n o n - v a n i s h i n g r - r . PP 88 r  2o^  i  ^  2 1  ^*  F o r example,  fundamental  on the r e f l e c t e d beam, w h i l e I n a r e c o r d i n g system d and the  spot s h o u l d be p o s i t i o n e d on the b i t s .  n  (6,<1>) w i l l have o p p o s i t e s i g n t o t h a t  The weakness o f the model i s t h a t d does not have any effects  w  T h e r e f o r e , a model i n c l u d i n g  focused the  v a r i a t i o n o f the d i e l e c t r i c c o n s t a n t o f the magnetic l a y e r i n the x - y plane n e c e s s a r y f o r a complete d e s c r i p t i o n o f the r e c o r d i n g s y s t e m . One o f the a u t h o r s  ( Z . L . ) thanks D r . T . T i e d j e f o r  discussions.  125  interesting  is  B-13  Appendix A  The purpose I s t o e v a l u a t e  the i n t e g r a l i n ( 9 ) .  u v  o o  Define:  - I k /2y x - ik  y  /2y  u • x + u v = y +  V  o (Al)  q  The d e s i r e d i n t e g r a l can be w r i t t e n  2p - y r - i k x - i k y I = / / r e dxdy P 2  x  -k /4y  7  2  2  i  r  //  (u +v -hi 4v -2u _ o o o 2  2  2  2  u  ) e" o P  2 v  v  Y ( u  2\ >dudv  (A2)  I n t r o d u c i n g a n o t h e r set o f v a r i a b l e s :  u ° V cos $ v = V s i n <J> u v  o  = V cos 4 o o  o  » V sin $ o o  (A2) becomes  126  (A3)  B-14  P  j  (p-j)!  m  x cos ((|)-(t» ) e ~ m  ( j - m ) ! m!  2  J  o  J  do) VdV  YV  o  (A4)  The summation i s over a l l p o s s i b l e n o n - n e g a t i v e i n t e g e r s as bounded by t h e factorial.  One n o t e s :  2%  r  2n J (  o  2k cos 4>d<t> - 2 • * r(k + l / 2 ) / k ! Z K  2k+l . . . cos q>d<p « 0  , , (A5)  n  A  N  U s i n g ( A 5 ) , (A4) becomes  ' r k  S  "  /  4  J  j £  6  x  (-l) "  Y  k  r^T 2 " p!(p-j+k)!r(k + I) 4k  2j  (p-j)!(j-2k)!(2k)!k!  -(p+j-k+1) 2 ( J - k )  (A6)  k  I f one d e f i n e s H = j-k and exchanges  I  the o r d e r o f summation i n ( A 6 ) , one has  - e " ^Z V A=0  with  127  Q  P  ( P +  *  + 1 )  k  2 A  (A7)  B-15  n*  « v (-D t  w h i c h i s the d e s i r e d (A7)  A  / * 2 " p!(p-A)!T;(k+l/2) (p-A-k)!(A-k)!(2k)!k! 2 ( k  J l )  (A8)  coefficient*  and (A8) have been t e s t e d n u m e r i c a l l y f o r a l a r g e range of p and  Y« G i v e n ( A 8 ) , one can prove r e l a t i o n  r  (10) as f o l l o w s :  <-l)*V)» * Q  p (n-p)!  (p!)  2  P  j - C4?, !L ;' ;! ( 1  One n o t i c e s  2  fc  )  r<tfl/2>  i c>c- >%- p-*- >...cp-*- > 1  W  1  k+1  <A> 9  that  x" .(l-x) A  = £ p  n  dx  cP(-l) (p-A)(p-Jl-l)...(p-Jl-k+l)x " " P  P  J L  k  (A10)  and i t can be r e a d i l y shown t h a t  I C (-l) (p-A)(p-A-l)...(p-A-k+l) P P  P  h  d~k X  X  (  1  _  X  0 , H < n and k < n  )  x-1  From ( A 8 - 1 1 ) , one o b t a i n s  128  (All)  B-16  " i^snl J . q  6  o»  (A12)  where  Q« = ( - ! ) «  129  2  "  2  N  *  (A13)  B-17  References  1.  M . M a n s u r i p u r , G . A . N . C o n n e l l and W . J . Goodman, " S i g n a l and n o i s e i n m a g n e t o - o p t i c a l r e a d o u t " , J . A p p l . P h y s . , 5 3 , 4485-4494  2.  Y . A o k i , T . I h a s h i , N . Sato and S. M i y a o k a , " A m a g n e t o - o p t i c system u s i n g TbFeCo", IEEE T r a n s . , MAG-21, 1624-1628  3.  (1982). recording  (1985).  Z . - M . L i , B . T . S u l l i v a n and R . R . P a r s o n s , " A p p l i c a t i o n o f the  4 x 4 -  m a t r i x method t o the o p t i c s o f m u l t i l a y e r m a g n e t o - o p t i c a l r e c o r d i n g m e d i a " , t o appear i n A p p l i e d O p t i c s . 4.  G . J . Sprokel, "Reflectivity, film structures",  5.  r o t a t i o n , and e l l i p t i c i t y o f  A p p l i e d O p t i c s , 23,  3883-3989  magnetooptic  (1984).  T . Takenaka, M . Yokota and 0 . F u k u m i t s u , " P r o p a g a t i o n o f l i g h t beams beyond the p a r a x i a l a p p r o x i m a t i o n " , J . O p t . S o c . A m . , A2, 826-829 (1985).  6.  M . L a x , W . H . L o u i s e l l and W . B . M c K n i g h t , "From M a x w e l l t o p a r a x i a l wave o p t i c s " , P h y s . R e v . A l l , 1365-1370 ( 1 9 7 5 ) .  7.  E . Z a u d e r e r , "Complex argument H e r m i t e - G a u s s i a n and L a g u e r r e - G a u s s i a n beams", J . O p t . S o c . A m . , A3, 465-469  8.  (1986).  P . J . L i n - C h u n g and S. T e i t l e r , " 4 x 4 m a t r i x f o r m a l i s m s f o r o p t i c s i n s t r a t i f i e d a n i s o t r o p i c m e d i a " , J . O p t . S o c . A m . , A l , 703-705  9.  (1984).  A . E . Siegman, i n " L a s e r s : p h y s i c s , systems and t e c h n i q u e " , W . J . F i r t h and R . G . H a r r i s o n e d . , P 8 2 - 8 5 , The S c o t t i s h U n i v e r s i t y Summer S c h o o l i n Physics  (1983).  130  B-18  Table 1 Parameters o f t h i c k n e s s and r e f r a c t i v e Indexes o f the m a g n e t o - o p t i c m u l t i l a y e r t h i n f i l m s f o r the i n t e n s i t y c a l c u l a t i o n as shown i n F i g . 3 . S u p e r s c r i p t s ' + ' and ' - ' denote f o r l e f t - h a n d e d and r i g h t - h a n d e d c i r c u l a r l y p o l a r i z e d l i g h t , r e s p e c t i v e l y . The w a v e l e n g t h X. • 0.830 um.  Layer  refractive index  1  n  2  n+ - 3.77 + 3.92 i n~ = 3.56 + 3.79 i  110 A  3  n  -= 1.50  0.225  4  n  r  - 2.00 + 7.00 i  >500 A  n  g  = 1.50  substrate  c  ±  = 1.50  thickness  0.2375  131  (X/n ) c  (X/n ) 1  ^  POLARIZER  FOCUSING  LENS  LAYER 1  t  z  LAYER  2  LAYER  N  SUBSTRATE  F i g u r e 1.  Schematics o f the model system and c o o r d i n a t e system f o r the beam fields.  The i n c i d e n t beam propagates a l o n g +z d i r e c t i o n .  132  (  Figure 2.  C o o r d i n a t e system o f p and s l i g h t f o r I n c i d e n t and p l a n e wave components, r e s p e c t i v e l y . r e f l e c t e d waves.  133  Superscript  reflected  » r ' denotes f o r  1.0  -1.0  -0.8  -0.6  -0.4  -0.2  0.0  0.2  0.4  0.6  0.8  1.0  X ( w(z) )  Figure 3.  Contour p l o t o f 2000 |E (r,<t>,z ) | 1 |E ( 0 , z ) | y x r  r  2  r  Parameters f o r the i n c i d e n t beam I s d e s c r i b e d  2  at  z  r  = 5 z . o  i n t e x t and t h o s e  f o r the m u l t i l a y e r t h i n f i l m s a r e s p e c i f i e d i n T a b l e 1. spot s i z e d e f i n e d  in  (26).  134  w(z)  is  f  Appendix C An Article on Optimization of Readout From Magneto-Optic Recording Medium  135  Abstract Reflectance and transmittance of a light beam strongly focused on multilayer thin films are derived at far field. Using the Fourier transfer technique, the reflectance (transmittance) can be expressed in terms of the reflection (transmission) matrix for plane waves. The readout from magneto-optic multilayer films is optimized for a strongly focused beam. It is found that when the focal spot size two is less than three wavelengths, the readout is significantly different from that calculated for plane waves.  136  Erasable magneto-optic recording technologies have received increasing amount of attention because of the high storage density.1 The magneto-optic recording medium consists of a thin layer of rare earth-transition metal alloy sandwiched between two dielectric layers. Mansuripur et al.  have optimized the readout from the multilayer film system  2  against the thicknesses of the cover and intermediate layers.2 As for the maximum readout, Mansuripur 3 has derived a figure of merit as the upper bound. Using the 4x4 matrix technique, L i et al.  were able to study the sensitivity of the readout with respect to the  4  thicknesses of the dielectric layers. In all the above theories the incident wave was assumed to be a plane wave for convenience while in real recording systems, the reading laser beam is focused down to a spot size of the order of a wavelength, in which case even the conventional paraxial Gaussian beam theory failed to describe the beam fields properly. 5 ' 6 We are going to derive the reflectance and transmittance of a beam focused on the magneto-optic recording medium.  Since the formalism is very general, it will also be useful for other  thin-film designers dealing with strongly focused laser beams. The model is as follows. A collimated Gaussian beam propagating in the +z direction is polarized in the x direction and then focused by a perfect lens on the focal plane 2 = 0. Assuming that the beam intensity on the focal plane has a Gaussian distribution with spot size wo, Takenaka et al.  5  expressed the beam fields in terms of a Gaussian beam  and correction terms of increasing order in l/kotvo, where ko — u^/fle. Recently L i and Parsons 6 were able to derive the k-space spectrum for the beam fields and expressed the reflected fields in terms of the reflection matrix for plane waves. In the following reasoning, only reflectance is considered since the derivation of transmittance is completely analogous. The reflectance is given by: _Ref[iE r*x(VxE )} dxdy r  R  z  Re J[iE* x (V x E)\ dxdy z  '  where only E rx and E rz are needed for v = x and similarly for u = y. The surface integral is over a plane at far field, and the superscript r denotes for the reflected light. Using a spherical coordinate system, (k, 0, <f>), the k-space spectrum for the reflected beam fields can be written as 6 :  Ul = U r , x  r  u = x, y, z,  (2)  = RiCos <f> + R^sin ^ + R^sin<f>cos<f>, 2  v  v  2  v = x, y,  r = R\cos<f> + R^sirKJ),  (3)  z  where Ux is the spectrum for the incident field and R% is a function of the elements of the reflection matrix: I * \ rps r  i  R  8  r  * r  p  pp  = PP> r  j and is given by 6 : J &2 ~ ss, r  RI = -{r /cos0 sp  137  +  r cos0), ps  Rf = r /cos6,  R% = r  Rl = -r cosO,  sp  ps  R* = r tanO,  -  pp  r, ss  (4)  R\ = —r sinO.  pp  ps  Taking advantage of the 6 function 6 in U , eq. (l) can be transferred into integration of x  the corresponding spectral functions with respect to dk dk . x  e  2^T j  Rv=  0  ~ " K ° - r/ o) fc>  /2  fc  k  I"  2k  r  The result is  y  I -k Re{rlr )]dk dk , 2  v  z  x  (5)  y  with v = x, y, and /„ = /  - * >/2*„ r  +  T^gj^Kik,,  (6)  where A;2 = k + fc2 and integration over all possible value is implied. In r , k is no longer x  u  z  independent and must be expressed as k = k — k*/2k . 6  z  0  0  If the reflection matrix is independent of <j> (e.g. isotropic thin films), eq. (5) can be further written as: R  u  =  i h j  ~  e  k  W  o  '  2  ^ -  fc2  /2*o) <l r " !  2 >  -krG ]k dk , u  T  r  (7)  where u = x, y, and  <| r | 2 >= 3 | R\ | 2 +3 | R | 2 + | iE£ | 2 +2i2e(i?ri^), u  v  2  G  x  — Re(3R}* R^ + R *R\ 2  G y = RefiR^R*  +  + JEg*^^), +  flfJEf).  (8)  Since a powerful technique has been developed for calculating the reflection and transmission matrices, 4 ' 7 (5) or (7) enable one to calculate the reflectance and transmittance of a light beam strongly focused on a thin film system with arbitrary structure and dielectric tensors. For the readout (a R )  of a magneto-optic medium with a magnetic moment M,  y  | [ r y ( M z ) — r (—M )\ y  z  and ^\r (M ) z  z  — r {—M )\ are used to eliminate a background arising z  z  from the diagonal matrix elements of the reflection matrix.  The integrands in (7) are  evaluated by the 4 x 4 matrix technique 4 ' 7 for incident angle 0 ranging from 0 ° to 8 7 ° . For comparison purposes, the same film material studied by Mansuripur 3 are considered (i.e., the intermediate and cover layers have refractive indices n, = n c = 2.0; TbFe has dielectric constants e = —1.4 + t'28.3 and e' = 0.12 + i0.57 and substrate is A l ; the wavelength A = 8200 A). We found that for plane waves, the optimal thickness for TbFe layer is 175 A and the readout reaches a maximum 95% of the figure of merit 3 at f, = 0.175 A, 138  and t = 0.213 A c , where c  and t are the thicknesses of the intermediate and cover layers, c  respectively, and A,- = A/n^, A c = A / n c . The sensitivity of the readout power (scaled to 10 4 x R ) y  to ti and t  e  is shown as a contour plot in Fig. 1 for 175 A  thickness of TbFe  layer. To study the effect of the spot size on the readout, 10 R 4  y  is shown as a function of  wo in Fig. 2 for a film structure optimal for plane waves. When wo < 3A, R significantly. The dependence of 10 R 4  y  on  y  decreases  and t is also pressented in Fig. 3 as a contour c  plot for a typical wo = A/2 and the same TbFe layer thickness as for Fig. 1. In this case, not only is the maximum readout power reduced to 92% of the figure of merit, but the optimal ti and t also shift 18% and 13%, respectively, to larger values than those of the c  plane wave model. Another interesting difference between Fig. 1 and Fig. 3 is that the readout is less sensitive to <t anf t for focused beams than for plane waves. The reason is c  that the distribution of plane wave components in the beam smoothes out the response of the film system to a single component. In conclusion, the reflectance (transmittance) of a focused light beam has been calculated in terms of the reflection (transmission) matrix. A plane wave model calculation is insufficient if good accuracy is required. The small focal spot size makes it harder to reach the figure of merit derived by Mansuripur 3 .  139  References 1. M . H . Kryder, Scientific American O c t . , 117(1987). 2. M . Mansuripur, G . A , N . Connell and W . J . Goodman, J. Appl. Phys. 53, 4485(1982). 3. M . Mansuripur, Appl. Phys. Lett. 49, 19(1986). 4. Z.-M. L i , B . T . Sullivan and R.R. Parsons, Appl. Opt. 2 7 , 1334(1988). 5. T . Takenaka, M . Yokota and O . Fukumitsu, J. Opt. Soc. A m . A 2, 826(1985). 6. Z . - M . L i and R.R. Parsons, to be published in J. Opt. Soc. A m . A , sept(1988). 7. P.J. Lin-Chung and S. Teitler, J. Opt. Soc. A m . A 1, 703(1984).  140  0.0  0.1  0.2  0.3  0.5  0.4  FIG. 1. Contour plot of the readout (scaled to 10 J2 ) for a plane wave (A = 8200 A) as a function of the intermediate and cover layer thicknesses t{ and t in units of A,- = A/n,and A = A/n , respectively, where n,- and n are the corresponding refractive indices. TbFe layer thickness is 175 A. 4  y  e  c  c  c  141  F I G . 2. Reflectance 10 R 4  y  as a function of the beam spot size wo. The thicknesses for  the layers are: t, = 0.175 A and t = 0.213 A and TbFe layer thickness is 175 A. t  c  c  142  F I G . 3. Same plot as Fig. 3 except the incident wave is a focused beam with spot size to = A/2. 0  143  References 1 J. McLeod, Market study report of Electronic Trend Publications(1985).  Also,  R.R. Parsons, private communication 2 M . H . Kryder, Scientific American, Oct.,117(1987) 3 R . M . White, Physics Today, Nov.,89(1987) 4 R . H . Victora, Phys. Rev. Lett., 58, 1789(1987) 5 S. Iwasaki and Y . Nakamura, I E E E Trans. Magn., M A G - 1 3 , 1272(1977) 6 G . A . N . Connell, Appl. Phys. Lett., 40, 213(1982) 7 H.S. Gill and T . Yamashita, I E E E Trans. Magn., M A G - 2 0 , 776(1984) 8 J . A . Thompson and P.B. Mee, I E E E Trans. Magn., M A G - 2 0 , 785(1984) 9 R . D . Fisher, V.S. Au-Yeung and B.B. Sabo, I E E E Trans. Magn., M A G - 2 0 , 806(1984) 10 S. Kadokura and M . Naoe, I E E E Trans. Magn., M A G - 1 8 , 1113(1982) 11 P.J. Grundy and M . A l i , J. Magn. Magn. Mater., 40, 154(1983) 12 R . K . Waits, in Thin Film Processes, J.L. Vossen and W . Kern ed., Academic Press(1978) 13 T . Tanaka and H . Masuya, Jap. J. Appl. Phys., 26, 897(1987) 14 B . D . Cullity, Introduction to Magnetic Materials, Addison-Wesley, 1st. edn.(1972) 15 R . D . Fisher, private communication 16 H . Hoffmann, L. Kochanowsky, H . Mandl, K. Kastner, M . Mayer, W . D . Miinz and K. Roll, I E E E Trans. Magn., M A G - 2 1 , 1432(1985) 17 Z . - M . L i , J.F. Carolan, R.C. Thompson and R.R. Parsons, Thin Solid Films,  154,  431(1987) 18 R . D . Fisher, J.C. Allen and J.L. Pressesky, I E E E Trans. Magn., M A G - 2 2 , 352(1986) 19 M . R . Khan and J.L Lee, J. Appl. Phys., 63, 833(1988) 20 J.A. Thornton, in Deposition Technologies for Films and Coatings, R . F . Bunshah ed., Noyes Publications (1982) 21 M . R . Khan and J.L Lee, J. Appl. Phys., 57, 4028(1985) 22 R. Friedberg and D.I. Paul, Phys. Rev. Lett., 34, 1234(1975) 23 E.R. Wuori and J.H. Judy, I E E E Trans. Magn., M A G - 2 0 , 774(1984) 24 N . W . Ashcroft and N.D. Mermin, Solid State Physics, chapter 4, Saunder College(1976) 25 U . Hwang, Y . Uchiyama, K. Ishibashi and T . Suzuki, Thin Solid Films, 147, 231(1987) 26 R. Ludwig, K. Kastner, R. Kukla and M . Mayr, I E E E Trans.  Magn.  MAG-23,  94(1987) 27 M . Mayr, K. Kastner, R. Ludwig, R. Kukla and R. Ludwig, I E E E Trans. M A G - 2 3 , 131(1987) 144  Magn.  28 B . T . Sullivan, Ph. D. Thesis, The University of British Columbia(1987) 29 Z . - M . L i and R.R. Parsons, submitted to J. Vac. Sci. Technol. A 30 K. Ouchi and S. Iwasaki, J. Appl. Phys., 57, 4013(1985) 31 K . L . Chopra, Thin Film Phenomena, McGraw-Hill (1969) 32 D . A . Glocker, W . E . Yetter and J.-S. Gau, I E E E Trans. Magn., M A G - 2 2 , 331(1986) 33 W . D . Westwood, J. Vac. Sci. Technol., 15, 1(1978) 34 W . G . Haines, J. Appl. Phys., 55, 2263(1984) 35 C M . Singal and T . P . Das, Phys. Rev., B16, 5068(1977) 36 D . E . Aspnes, SPIE, 276, 188(1981) 37 D . E . Aspnes, J.B. Theeten and F. Hottier, Phys. Rev., B 2 0 , 3292(1979) 38 J.R. Blanco, P.J. McMarr and K. Vedam, Appl. Opt., 24 , 3773(1985) 39 R . W . Collins and C.J. Tuckerman, J. Vac. Sci. Technol., A 4 , 2343(1986) 40 B . T . Sullivan and R.R. Parsons, J. Vac. Sci. Technol., A 5 , 3399(1987) 41 J . H . Weaver, E . Colavita, D . W . Lynch and R. Rosei, Phys. Rev., B19, 3850(1979) 42 P.B. Johnson and R . W . Christy, Phys. Rev., 39, 5056(1974) 43 A . Y - C . Yu, T . M . Donovan and W . E . Spicer, Phys. Rev., 167, 670(1967) 44 P.B. Johnson and R . W . Christy, Phys. Rev., B6, 4370(1972) 45 P. Yeh, Surface Science, 96, 41(1980) 46 H . Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA(1965) 47 C M . Hurd, The Hall Effect in Metals and Alloys, Plenum Press(1972) 48 See, for example, C. Kooi, Phys. Rev., 95, 843(1954) 49 R. Karplus and J . M . Luttinger, Phys. Rev., 95, 1154(1954) 50 J. Smit, Physica, 21, 877(1955); Physica, 24, 39(1958) 51 J . M . Luttinger, Phys. Rev., 112, 739(1958) 52 C M . Hurd, in The Hall effect and its applications, C L . Chien and C R . Westgate E d . , Plenum Press, New York and London(1979). See also, L. Berger and G . Bergmann, in the same book. 53 Yu. P. Irkhin and V . G . Shavrov, Soviet Phys.-JETP(Engl, trans.), 15, 854(1962) 54 S.P. McAlister and C M . Hurd, J. Appl. Phys., 50,7526(1979) 55 K. Nakamura, T . Asaka, S. Asari, Y . Ota and A . Itoh, I E E E Trans.  Mag-21,  1654(1985) 56 M . Mansuripur, G . A . N . Connell and J.W. Goodman, J. Appl. Phys. 53, 4485(1982) 57 O.S. Heavens, Optical Properties of Thin Solid Films, Dover, New York(1965) 58 R. Gamble, P.H. Lissberger and M.R. Parker, I E E E Trans. Mag-21, 1651(1985) 59 D.J. Smet, Surface Sci. 56, 293(1976) 60 J. Kranz and Ch. Schrodter, Appl. Phys. B 34, 139(1984) 145  61 S. ViSfiovsky, Czech. J. Phys. B 34, 155(1984) 62 D . W . Berreman, J. Opt. Soc. A m . 62, 502(1972) 63 P.J. Lin-Chung and S. Teitler, J. Opt. Soc. A m . A l , 703(1984) 64 L . D . Landau and E . M . Lifshitz, Electrodynamics of continuous media., Oxford, New York, Pergamon Press(1960). 65 A . H . Lettington, Optical Properties and Fermi Surfaces of Zinc,in Optical Properties and Electronic Structure of Metals and Alloys, F. Abele ed.(North Holland, Amsterdam 1966) 66 Z.-M. L i , B . T . Sullivan and R.R. Parsons, Appl. Opt.(in press), April(1988) 67 See for example, W . J . Firth and R . G . Harrison ed., Lasers: physics, systems and techniques, The Scottish University Summer School in Physics(1983) 68 See for example, H . Kogelnik, Appl. Opt., 12, 1562(1965) 69 See for example, T . Tamir, J. Opt. Soc. A m . A3, 558(1986) 70 R. Simon, E . C . G . Sudarshan and N . Mukunda, J. Opt. Soc. A m . , A3, 536(1986) 71 T . Takenaka, M . Yokota, and O. Fukumitsu, J. Opt. Soc. A m . , A2, 826(1985) 72 Z.-M. L i and R.R. Parsons, accepted by J. Opt. Soc. A m . A 73 Z.-M. L i and R.R. Parsons, submitted to Appl. Phys. Lett. 74 See, for example, E . Hecht and A . Zajac, Optics, page 74, Addison-Wesley(l974)  146  Publications 1. Z . - M . L i and R.R. Parsons, 'Interaction of focused light beams with multilayer films: optimization of magneto-optic recording media', to be published in Appl. Phys. Lett. 2. Z . - M . L i and R.R. Parsons, 'Reflection of strongly focused light beams from magnetooptic multilayer films', to be published in J. Opt. Soc. A m . A , Sept(l988) 3. Z . - M . L i and R.R. Parsons/Controlling the magnetic anisotropy of D C magnetron sputtered CoCr films', submitted to J . Vac. Sci. Technol. A 4. Z . - M . L i , B . Bergersen, P. Palffy-Muhoray and D. Beigie, 'Optical properties of monolayer NbS  2  suspensions', to be published in Can. J . Phys.  5. Z . - M . L i , B . T . Sullivan and R . R . Parsons, 'Application of the 4x4-matrix method to the optics of multilayer magneto-optical recording media', Appl. Opt. 27,1334(1988) 6. Z . - M . L i , J . F . Carolan, R. C . Thompson, and R. R. Parsons, 'Magnetic properties of D C magnetron sputtered CoCr thin films', Thin Solid Film 154,431(1987) 7. Z . - M . L i and B. Bergersen, 'Evaluation of two-centre dipole matrix elements of Slater s, p, and d orbitals; application to the optical absorption spectrum of NbS J . Phys. C 19,7281(1986)  2  monolayers',  

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