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Mitchell, Terry Blake
MAGNETIC AND MAGNETO-OPTIC BEHAVIOR OF BISMUTH AND THULIUM SUBSTITUTED YTTRIUM IRON GARNETS
The Ohio State University Ph.D. 1986
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University Microfilms International
MAGNETIC AND MAGNETO-OPTIC BEHAVIOR OF
BISMUTH AND THULIUM SUBSTITUTED YTTRIUM IRON GARNETS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Terry Blake Mitchell, B.S., M.S.
The Ohio State University
1986
Approved by Reading Committee:
Dr. Philip E. Wigen
Dr. David G. Stroud Philipp. Wigen, Advrser Dr. James T. Tough Department of Physics To My Parents
ii ACKNOWLEDGEMENTS
I would like to thank my advisor, Professor Philip Wigen, for
the friendship and support he has given me during my work at Ohio
State. I am deeply grateful for his guidance and the encouragement
he has given me to fulfill my educational goals.
I would like to thank Dr. Ladislav Pust, Dr. Martha Pardhavi-
Horvath, and Dr. Sharat Batra for the stimulating discussions and help
in pursuit of solutions to problems that arose during the course of the
investigations. I would like to thank Dr. Andrea Lehmann-Szweykowska
for the discussion and help she gave during her visits at Ohio State.
I would like to thank the NSF grant //DMR-8304250 and the Department
of Physics of The Ohio State University for financial support of myself and the research projects.
I am also indebted to the technical assistance and professional quality of the staff I have worked with especially Tony Bernardo,
Vijay Sehgal, Roy Tucker, and Claire McDonald. I would especially like to thank Pat Kimball for the superb job of typing this manuscript.
There is no way of expressing the gratitude I feel for the friend ships I have made while at Ohio State. There is no way I can include all my friends that had a pronounced affect on this work. Don, Sharat,
Ramesh, Arvind, Gregg, Ewa, Martha, Ladia, and Andrea are especially acknowledged for the support only very close friends can give.
iii VITA
January 27, 1958 ...... Born, Ft. Worth, Texas
1981 ...... B.Sc., The Ohio State University Columbus, Ohio
1983 ...... M.Sc., The Ohio State University Columbus, Ohio
1981-1984 ...... Teaching Assistant, Department of Physics, The Ohio State University Columbus, Ohio
1984-1986 ...... Research Assistant, Department of Physics, The Ohio State University, Columbus, Ohio
PRESENTATIONS
"The Faraday Rotation In TmBl Substituted YIG Thin Films" T. B. Mitchell and P. E. Wigen. Presented at the International Colloquium on Thin Films and Surfaces in Asilomar, CA in September 1985.
"Anisotropic Magnetic and Magnetooptic Behavior of Thin Garnet Films with High Q-values" T. B. Mitchell, L. Pust, and P. E. Wigen. Presented at the March 1986 meeting of the American Physical Society at Las Vegas, Nevada and to be published in JAP.
iv TABLE OF CONTENTS
DEDICATION ...... ii
ACKNOWLEDGEMENTS ...... iii
VITA ...... iv
LIST OF T A B L E S ...... vii
LIST OF FIGURES ...... viiii
CHAPTER PAGE
1. INTRODUCTION ...... 1
2. T H E O R Y ...... 6
Crystal Structure ...... 6 Magnetization ...... 8 Ferromagnetic Resonance ...... 13 Effective g-value ...... 18 Twin Peak Effect ...... 23 Faraday Effect ...... 35
3. EXPERIMENTAL PROCEDURES ...... 43
Saturation Magnetization ...... 43 Ferromagnetic Resonance ...... 45 Sample Characterization ...... 50 Magneto-optic Measurements ...... 51
4. EXPERIMENTAL RESULTS AND CONCLUSIONS ...... 57
Sample Characterization ...... 57 Saturation Magnetization ...... 57 Ferromagnetic Resonance ...... 63 Twin Peak Results ...... 68 Faraday Rotation ...... 81
5. CONCLUSION...... 90
v APPENDICES
A. Computer Program to Calculate the Magnetization Using MET ...... 93
B. Computer Program to Calculate the Effective g-value, and Heff from F M R ...... 97
BIBLIOGRAPHY ...... 102
vi LIST OF TABLES
TABLE PAGE
I. Effects on B for Different Cases of a Assuming H a > 0 ...... 30
II. Sample Characteristics ...... 58
III. Faraday Rotation of Quartz Windows and Blank Substrate ...... 82
IV. Faraday Rotation Coefficients of the Sublattice Magnetizations ...... 88
vii LIST OF FIGURES
FIGURE PAGE
1. View of the domain structure using the magneto-optic Faraday Effect. From top left clockwise; ring pattern, stripe pattern, labrynth pattern, and bubble pattern. (Ref. 8) ...... A
2. The underlying symmetries of the cubic garnet system. An oxygen ion is located at the corners of each polyhedra. (Ref. 9) ...... 7
3. The temperature dependence of the total magnetization for pure YIG calculated by MFT using the molecular field coefficients of Equation 8 and the program in Appendix A ...... 11
A. The temperature dependence of the total magnetization for pure TmlG calculated by MFT using the molecular field coefficients of Equation 9 and the program In Appendix A ...... 12
5. The coordinate system used in the theoretical discussion...... 14
6. The temperature dependence of the effective g-value of pure YIG. The experimental data (H ) is from Ref. 18 and the solid line represents the spin wave theory predicted by Equation 27 ...... 20
7. The theoretical temperature dependence of the effective g-value of pure TmlG. Notice the g-value increases with decreasing temperature and the divergence at 70 K is an angular momentum compensation point...... 22
8. Thin film geometry for light propagation along the z direction. The sign of the Faraday rotation is determined by the direction of the magnetization in each domain...... 25
viii 9. Illustration of case i from Table I. The left vertical axis is the magnetization angle B, and the right axis is the light intensity. B diverges at Heff as predicted by Equation 42...... 31
10. Illustration of case ii from Table I. The light intensity, I, is proportional to B2 ...... 32
11. Illustration of case iii from Table I. This figure shows the origin of the twin peaks. The sign of B is initially positive and then changes sign to negative above Heff...... 33
12. Illustration of cases iv and v from Table 1 ...... 34
13. The state of polarization of initially linearly polarized light after passing through a garnet film...... 39
14. The Faraday rotation for all rare earth garnets except thulium with different Bismuth substitutions at room temperature and a wavelength of 633 nm. (Ref.2 6 ) ...... 42
15. Experimental apparatus for measuring the saturation magnetization. (Ref. 27) 44
16. Raw data for the saturation magnetization from Sample 1 at room temperature...... 46
17. Experimental apparatus for measuring the ferromagnetic resonance. (Ref.27) 48
18. Typical spectrum of ferromagnetic resonance at room temperature of Sample 3 at 9.2 GHz...... 49
19. Experimental setup to measure the light intensity and Faraday rotation...... 52
20. Typical data for the light intensity as the magnetic field is swept in the perpendicular direction...... 56
21. The temperature dependence of the saturation magnetization for Sample 1. The composition compensation temperature is 142 K ...... 59
22. The temperature dependence of the saturation magnetization for Sample 2. The composition compensation temperature is 52 K ...... 60
ix 23. The temperature dependence of the saturation magnetization for Sample 3. No composition compensation temperature since no gallium is substituted for the iron...... 61
24. The temperature dependence of the saturation magnetization for Sample 4 ...... 62
25. The temperature dependence of the difference between the "d" and "a" sublattice magnetization for pure TmlG (upper curve) and YIG (lower curve) using the result of the M FT...... 64
26. The effective internal field for Samples 3 (O) and 4 (H) as a function of temperature...... 65
27. The temperature dependence of the first order cubic anisotropy field of Sample 3 (O) and Sample 4 (H)...... 66
28. The temperature dependence of the effective g-value of Sample 3 (O) and Sample 4 (**). Notice the g-value decreases with decreasing temperature totally opposite to the behavior of the spinwave theory of Figure 7 ...... 67
29. Theoretical fit (□) of data from Sample 4 for the effective g-value using Equation 28 with A=-1400 and 5=3300. The theoretical fit comes from the experimental values of the total magnetization. The agreement is excellent...... 69
30. The light intensity plotted as the square of the applied perpendicular magnetic field...... 72
31. The light intensity as the magnetic field is swept slightly off the parallel orientation (a =3.0 deg) from ±6000 Gauss...... 74
32. Light intensity for various small values of a for small fields. Note the striking quadratic behavior...... 75
33. The light intensity plotted as a function of the square of the applied magnetic field at o=3.0 deg...... 76
x 34. The light intensity plotted as a function of the square of the angle from the parallel direction for magnetic fields between 200 and 1000 Gauss...... 77
35. The light intensity from ±6000 Gauss for • - 0< ar <0.7 degrees. Notice the formation of twin peaks in the light intensity on both sides of zero magnetic field as predicted in Figure 11...... 78
36. The range of twin peak formation as a function of
37. The temperature dependence of the specific Faraday rotation of Sample 1 for light wavelengths of 633 nm, 578 ran, and 546 nm. The solid curve represents the theoretical fit using Equation 74 and Table IV ...... 84
38. The temperature dependence of the specific Faraday rotation of Sample 2 for light wavelengths of 633 nm, 578 nm, and 546 nm. The solid curve represents the theoretical fit using Equation 74 and Table IV ...... 85
39. The temperature dependence of the specific Faraday rotation of Sample 3 for light wavelengths of 633 nm, 578 nm, and 546 nm. The solid curve represents the theoretical fit using Equation 74 and Table IV ...... 86
40. The temperature dependence of the absolute value of the specific. Faraday rotation. The dip occurs in Samples 1 (") and 2 (□) due to a compositional dependence through the film thickness...... 87
xi CHAPTER 1
INTRODUCTION
Magnetism has always been in the forefront of modern technology.
Centuries ago the merchant ships sailing the high seas relied on their
magnetic compasses to travel from one distant land to another. Today,
the high technology of magnetic recording is the single largest
industry in the world. The field of magneto-optics is still very young
and already has many applications in the offering. Computer storage 1 2 devices, computer display screens, and optical signal processing
are just the beginning of the list of what magneto-optics has to
offer the high technology community.
The origins of magneto-optics can be traced back to the discovery 3 of magnetic birefringence by Michael Faraday in 1845. He discovered
that the plane of polarization of polarized light is rotated when a magnetic field is applied to transparent paramagnetic materials and that the rotation was linearly proportional to the applied magnetic field, B, and the thickness of the material, 51:
0f = V(v)BS. , (1) where V is called the Verdet constant, a frequency, v, dependent constant of proportionality. The origin of this rotation is the paramagnetic response of the material to the applied magnetic field.
1 2
Ferromagnetic materials also rotate the plane of polarization of
light, but the origin is of a different nature. The ferromagnetic and
antiferromagnetic ordering inside the garnet structure gives rise to a
magnetic circular birefringence. While true Faraday rotation has its
origin in the paramagnetic ordering, the literature of today has
adopted the term Faraday rotation to include this magnetic circular
birefringence.
There are four transmission magneto-optic effects acting in
ordered magnetic materials; magnetic circular birefringence, magnetic
circular dichroism, magnetic linear birefringence, and magnetic linear
dichroism. Magnetic circular birefringence (MCB) and magnetic
circular dichroism (MCD) are transmission effects when the direction
of light propagation is parallel to the magnetization. Magnetic
linear birefringence (MLB) and magnetic linear dichroism (MLD) are
transmission magneto-optic effects when the direction of propagation
is perpendicular to the magnetization. These effects will be
explained in detail in the theoretical discussion. The reflection
magneto-optic effects, such as Kerr rotation, will not be considered
in this work.
The Faraday rotation in garnet films was first observed by
Dillon in 1958. With the development of high quality single crystal magnetic garnet films the Faraday rotation has become a very important probe in the study of magnetic behavior of materials in general and garnet films in particular. The Faraday rotation allows for direct observation of the domain structure in the ferrimagnetic garnet films. In zero applied fields, the magnetic film will divide 3
into areas of oppositely magnetized regions termed domains. The
formation of domains lowers the total free energy by decreasing the
magnetostatic energy. Since the sense of the Faraday rotation depends
on the direction of the magnetization, a viewing of the sample between
two polarizers will show the domain pattern if the polarizers are
rotated such that the light intensity from one domain is a minimum.
This is shown in Figure 1.
The magnitude of the Faraday rotation in garnets can be enhanced 5 by as much as two orders of magnitude by the substitution of bismuth
at the dodecahedral site. This phenomenon is not well understood at
the present time and serves as the motivation for this work. The
present understanding of the theory will be discussed in some detail
in the theoretical section.
The rare earth element thulium has shown an anamolous behavior
in the effective g-value6 when substituted in the garnet structure.
The simple spin wave result for the effective g-value does not explain
the temperature dependence of the thulium iron garnet system. A 7 theory has been proposed to explain the anamolous behavior of the effective g-value which includes the effects of the single ion anisotropy energy and the spin wave damping. The results are in reasonable agreement with the experimental data.
A series of TmBi doped YIG samples were grown by Airtron, a
Division of Litton Corporation, containing various amounts of thulium and bismuth. These ions replace yttrium at the dodecahedral site in the garnet structure. Bismuth is nonmagnetic yet it has significant effects on the magneto-optical properties. Thulium is a heavy rare 4
m m m mmmmm
Plgure 1 View of the domain structure using the magneto-optic Faraday Effect. From top left clockwise; ring pattern, stripe pattern, labrynth pattern, and bubble pattern. (Ref. 8) 5 earth element that is magnetic and modifies the magnetic properties.
The magnetic properties investigated in this work include the effective g-value, the magnetization, the effective internal field, and the cubic anisotropy energy. The magnetooptic effects probed include the Faraday rotation (MCB) and Faraday ellipticity (MCD).
During the course of these investigations, a magneto-optic technique for measuring the cubic anisotropy energy was discovered.
A theory was developed to explain the experimental observations and the fit is extremely good. The theory predicted some effects that were not previously reported. The predictions of the theory were observed experimentally after some modifications were made in the equipment to aid in their detection. CHAPTER 2
THEORY
The materials investigated in this work are bismuth and thulium
substituted garnet films. This section details the origins of the
magnetic and magneto-optic behavior of these artificially grown
materials.
Crystal Structure
The garnet crystal structure is composed of three interpenetrating
sublattices that have an overall cubic symmetry. The three sublattices
have an underlying dodecahedral, octahedral, and tetrahedral site
symmetries. The garnet crystal structure is shown in Figure 2. The
unit cell consists of eight formula units of RgFe^O^,, where R is a
trivalent rare earth ion, Fe is trivalent iron ion, and 0 is divalent
oxygen.
The dodecahedral site, also known as the "c" site, has the largest
volume. The rare earth ions tend to be located in these positions as well as any ion with an ionic radius larger than the rare earth ions.
There are three dodecahedral sites per formula unit for a total of 24 sites per unit cell.
The octahedral site, also known as the "a" site, is next in decreasing size. This site is occupied by trivalent iron ions.
6 7
• Fes+ (a) • Fe3+ (d) o YJ+
Figure 2 The underlying symmetries of the cubic garnet system. An oxygen ion is located at the corners of each polyhedra. (Ref. 9) 8
There are two "a" sites per formula unit or 16 sites per unit cell
each surrounded by six oxygen ions.
The smallest symmetry site is the tetrahedral or "d" site. This
position is occupied by trivalent iron surrounded by 4 oxygen ions.
There are three "d" sites per formula unit or a total of 24 per unit
cell.
The model rare earth garnet film is when R=Y or YIG. The
materials investigated in this study are partial substitutions of
bismuth and thulium for yttrium, and gallium for some of the iron.
None of the films occur naturally, but are grown by liquid phase
epitaxy techniques. Films of 10 micron thicknesses are grown
artificially on host substrates. The standard substrate used is
Gd.Ga.O.„ or GGG. 3 5 12
Magnetization
The magnetic ordering of garnet films is ferrimagnetic. The
permanent magnetization is due to the magnetic moments of the inter penetrating sublattices. The type of exchange interaction responsible for the magnetic ordering is through the superexchange between the sublattices via the oxygen ions.
The microscopic origin of the magnetic properties lies in total angular momentum, J, of the electrons. The total angular momentum is the sum of the spin angular momentum, S, and the orbital angular momentum, L .
The trivalent iron ion has the electron configuration of 3d^ and is a transition metal ion. A half filled outer shell gives L=0 and S=5/2. The thulium ion has the electron configuration of 4f1 2 . This
gives a ground state of L=5 and S=1 or J=6. When a paramagnetic
material is placed in a magnetic field H, it has a macroscopic
magnetization10 given by:
M = | y J Bj (By JH) , (2)
where
Bj(X) = coth X) - coth (|j) , (3)
is the Brillouin function, y is the gyromagnetic ratio, N is the
number of ions, V is the volume and B is defined as the inverse of
Boltzmann's constant times temperature. This paramagnetic result is
the starting point for the mean field theory (MFT) for the magnetiza
tion in ordered magnetic materials.
The difference between the ferrimagnetic ordering of the
sublattices and the paramagnetic ordering described by Equation 3 lies
in the origin of the magnetic field H. In the paramagnetic case, the
magnetic field was an applied field from an external source. The
source of the applied magnetic field in the ferrimagnetic case is in
the exchange interaction. This interaction causes an effective
internal field at the ion site even in zero applied magnetic field.
This internal field, called the Weiss molecular field,11 is given by:
H = XM , (4) where X. is the nine component tensor: 10
where the components represent molecular field coefficients due to
the exchange interaction and M is the total magnetization whose compo
nents are the sublattice magnetizations:
- / MA Mt = Mc + Md + Ma; M = ( Md I . (6) \ M c /
The MFT result for the magnetization of these garnet films give
a series of equations that must be solved self-consistently. For the
ferrimagnetic garnet system Equation 2 becomes:
«i = J Yi Ji Bj (BYiJiHi) , (7)
where i=a,d,c. Equations 6 and 7 represent the equations that must
be solved self-consistently. A computer program was written to solve
these equations and numerically calculate the sublattice magnetiza
tions as a function of temperature. The program is listed in
Appendix 1 and the results for pure YIG and TmlG are plotted in 12 Figures 3 and 4. The molecular field coefficient used for YIG are:
M d = -30.4
(8) M d = 97.0
Mc = Ma = Md = Mc = Mc = 0 » and fo r TmlG13 are:
M d = -30.4 M e ~ —3.44
M a = —65.0 M d = 6.0 (9)
M d = 97.0 M e = 9 • 4irM (Gauss) 2500 2000 1500 1000 500 Figure 3 The temperature dependence of the total magnetization for magnetization total the of dependence temperature The 3 Figure 100 coefficients of Equation 8 and the program In Appendix A. Appendix In program the and 8 Equation of coefficients pure YIG calculated by MFT using the molecular field molecular the using MFT by calculated YIG pure eprtr (K) Temperature 200 300 400 500
600 i I
4rrM (Gauss) 1000 1500 500 Figure 4 The temperature dependence of the total magnetization for magnetization total the of dependence temperature The 4 Figure 100 coefficients of Equation 9 and the program in Appendix A. Appendix in program the and 9 Equation of coefficients pure TmlG calculated by MFT using the molecular field molecular the using MFT by calculated TmlG pure eprtr (K) Temperature 200 300 400 500
600 13
The "a" and "d" sites occupied by iron are solely responsible for
the magnetic behavior of YIG. Yttrium is a non-magnetic ion situated
on the "c" sublattice so Mc is zero. The magnetic interaction between
the "a" and "d" site is antiferromagnetic. This yields the three "d"
site moments opposing the two "a" site moments for a total of one iron
moment at zero temperature. One iron ion has the moment of 5pD which is O the moment of YIG at T=0.
Substitution of the rare earth ion thulium in place of yttrium
complicates the magnetization because thulium has a large orbital
angular momentum. This "c" sublattice magnetization makes a signifi
cant contribution to the magnetic properties of the material, such as
the magnetization, the effective g-value, and the first order cubic
anisotropy energy constant.
Ferromagnetic Resonance Theory
Ferromagnetic resonance is the experimental technique used to
measure fundamental magnetic parameters of the garnet films. It is
used to measure the effective g-values, uniaxial anisotropy fields,
and cubic anisotropy fields.
The coordinate system used for the analysis of the ferromagnetic
resonance theory is shown in Figure 5. The coordinates 0 and
the magnetization’s polar and azimuthal angles respectively. The coordinates 0 and
z [111]
■y [Ho]
X [115]
Figure 5 The coordinate system used In the theoretical discussion. (10) M2sin29,
where F is the total free energy of the magnetic system, w is the
resonance frequency and y Is the gyromagnetic ratio. The subscripts
indicate partial derivatives with respect to the coordinates i.e.
32F „ 32F 32F f * 9 0 where 0O and determined by the energy minimum conditions: 0 . (12) The total free energy of the garnet film is the sum of four contributions: the Zeeman energy, the demagnetization energy, the uniaxial anisotropy energy, and the cubic anisotropy energy. The Zeeman energy is given by the expression: (13) This term represents the energy stored in the film due to the inter action of the permanent dipole moment and the applied magnetic field. Evaluated in the coordinate system shown in Figure 5, Equation 13 becomes: FZeeman = -MH[sin0 sin0 cos( The demagnetization energy is due to the dipole-dipole interactions in the garnet film. The basic equation for this energy term is given by: 16 FDemag (15) where HD = N»M , (16) where N is a geometric factor called the demagnetization tensor. Reference 15 shows that the demagnetization tensor for the thin film geometry is given as: (17) which gives a demagnetization energy of FDemag = 2irMz . (18) In terms of the coordinates of Figure 5, Equation 18 becomes: FDemag — 2itM*“Co s ^0 . (19) In thin garnet films, the uniaxial anisotropy energy is due to the deviation of the magnetization from the direction normal to the film plane which is the preferred axis or easy axis of magnetization. 2 The free energy can be expanded in terms of a sin 0 series and reduced in terms of the coordinates of Figure 5 as follows: Funiaxial = Kusin2® + higher order terms , (20) where Ku is a phenomenonological parameter called the uniaxial aniso tropy energy constant. The higher order uniaxial anisotropy energy contributions can be neglected in the garnet films of interest in this investigation. The uniaxial anisotropy energy comes from two sources; growth Induced anisotropy and stress induced anisotropy. The mismatch of the lattice constant of the GGG substrate and the garnet film is 17 the source of the stress induced anisotropy. The growth induced anisotropy comes from the layering of the atoms along the (111) direction during the growth process. The cubic anisotropy energy has its origin in the cubic crystal structure of the garnet system. Deviations from the cubic structure can be represented as a series expansion of the direction cosines from the cubic axes and causes an anisotropy energy of the form1^: Fcubic = I Fijk ala2a3 » <21) ijk where the oj, ag, and <*3 are the direction cosines from the (100), (010), and (001) cubic crystal axes respectively and i, j, and k can have any integer value. The cubic symmetry limits the allowed values of i, j, and k to even integers due to the invariance of rotations by 180 degrees. In the sum of Equation 21, it is necessary to include only the low power quadratic terms to give: Fcubic = F1 + a2a3 + a la 3> + F2 (af The garnet films used in this investigation are grown along the (111) crystalline direction. A coordinate transformation from the cubic axes, which Equation 22 represents, to the coordinate system of 17 Figure 5 yields a cubic anisotropy energy of the form : Fcubic = 12 £3-6cos20 + 7cos4e + 4vr?sin3 + r~r [✓?sin30 sin3 The total magnetic free energy of the garnet film can be written as: F = FZeeman + FDemag + Funiaxial + Fcubic • (2**) where the energy contributions are given by Equations 14, 19, 20, and 23 for the (111) crystal orientation. The resonance equation now becomes extremely complicated due to the anisotropy of the film. The parameters y, Ky, K^, and must be determined by this equation. The first and second partial derivatives must be calculated and substituted into Equation 10 and evaluated at the equilibrium angles of the magnetization. 18 A computer program was written to fit the angular dependence of the ferromagnetic resonance to Equation 10. It was found that the second order cubic anisotropy energy term, could be neglected arid the values of y, Ku , and determined by the fit of the theory to the experimental data. The Fortran program is listed in Appendix 2 and was run on an IBM 3081 mainframe high speed computer. Effective g-Value The effective g-value is related to the gyromagnetic ratio of the garnet film by the equation: Y* = EeffVB (25) where f> is Planck's constant divided by 2ir and geff is the effective g-value. 19 The Lande g-value for an ion with angular momentum J, L, and S is given by: „ _ 3J(J+1)-L(L+1)+S(S+1) B 2J(J+l) ’ ^ ’ 19 The g-value for a free electron, J=S=l/2 and L=0, is 2. The effective g-value relates the free ion Lande g-value with the garnet structure’s magnetic ordering. The garnet film has three sublattices and there fore the contribution of the magnetic sublattices must be taken into 20 account. The simplest theory, a spin wave theory, for the temper ature dependence of the effective g-value gives: Mda~Mc Seff = ; Mda = Md - Ma . (27) 6da Sc where M^, Mc , and Ma are the sublattice magnetizations of the "d", "c", and ”a" sites respectively, and g and g are the free ion Lande QSi C g-value given by Equation 26 of the ion situated on the different sites. The model rare earth garnet YIG has a non-magnetic "c” sublattice, i.e. Mc=0. Equation 27 then predicts that the effective g-value is temperature independent and equal to the free ion value of the iron ion of the "d" and "a" sites. The ground state of the trivalent iron of J=S=5/2 and L=0 gives an effective g-value of 2.0. The temperature 21 dependence of the effective g-value of pure YIG is shown in Figure 6 . The spin wave theory agrees very well with the experimental values. There exist two special conditions in Equation 27. The numerator and denominator can be independently zero. The numerator is zero at the magnetization compensation point. The magnetization compensation point occurs when the sum of the sublattice magnetizations are equal to zero. This temperature is not the same as the Neel point when each sublattice magnetization is equal to zero. The Neel point in ferrimagnets is the analogue of the Curie point in ferromagnets. Effective g-value 2.00 2.05 2.10 r 2.10 go o g i 1.95 10 0 30 400 300 200 100 0 Figure Figure 6 The temperature dependence of the effective g-value of pure of g-value effective the of dependence temperature The YIG. The experimental data (“ ) is from Ref. 18 and the and 18 Ref. from )is (“ data experimental The YIG. solid line represents the spin wave theory predicted by predicted 27. theory Equation wave spin the represents line solid eprtr (K) Temperature ______ 21 The denominator can be equal to zero at the angular momentum compensation temperature. This zero causes a singularity in the effective g-value. With a magnetic ion located on the "c" sublattice, the effective g-value can be calculated theoretically by knowing the temperature dependence of the three sublattices. The effective g-values of thulium iron garnets have shown anamolously low values.^ The ground state of the thulium ion is J=6 , L=5, and S=1 which produces a Lande g-value of gc=7/6. The sublattice magnetizations have been calculated numerically using Equation 7 and Figure 7 is the prediction of the spin wave theory for geff for TmlG. The actual temperature dependence from the FMR experiments performed showed the gef»f decreasing with decreasing temperature instead of increasing with decreasing temperature as predicted in Figure 7. The spin wave theory is clearly not the full picture in the case of the thulium substituted YIG. 7 A recent theory predicts that the effects of a single ion anisotropy field and spin wave damping will make significant contri butions to the g-value in the case of ions with a large spin-orbit coupling. This theory is applicable for the thulium substituted garnet films due to its large orbital contribution to the angular momentum J. The single ion anisotropy energy is a crystal field effect. The crystal field interaction is responsible for an enhance ment of the spin-orbit effects. This theory predicts the temperature dependence of the g e j.j. as: /M m ir!^a n » C i ( da-Mc)Cgda (1 * Md ) ~ gc3 geff = Mda A, _ Me 2 SHja 2 ' Sda «d gc gdaMd Effective G-value 10 5 5 0 Figure 7 The theoretical temperature dependence of the effective the of dependence temperature theoretical The 7 Figure 100 decreasing temperature and the divergence at 70 K is an is K 70 at divergence the and temperature decreasing nua oetmcmesto on. M point. compensation with momentum angular Increases g-value the Notice TmlG. pure of g-value eprtr (K) Temperature 200 300 400 500 600 N 5 23 where . _ 3gcBdMBHcf _ 3gdr & = T-T- ;-- and 5 - (29) 2 1 \ic I 2 IXdcI where Hcf is the single ion anisotropy field and r is the spin wave damping parameter. If the spin wave damping and single ion anisotropy field are neglected, Equation 28 reduces to the simple spin wave result of Equation 27. Equations 28 and 29 will be used later to fit the experimental results of the temperature dependence of the thulium substituted garnets. Twin Peak Effect Theory The twin peak effect theory was developed to explain the experi mental observation of a set of twin peaks in the light intensity when magnetic fields are swept near the parallel direction of the film plane. This effect can be used to determine the magnitude of the first order cubic anisotropy energy constant. This technique probes the magneto- static conditions of the equilibrium orientation of the magnetization in applied magnetic fields using the magneto-optic Faraday effect as a probe. The origin of the Faraday effect will be discussed in detail in the next section but its existence is assumed for now. A linearly polarized light wave will rotate the plane of polari zation when passed through a magnetized material. The magnitude and direction of rotation depend on the sign and component of the magneti zation along the propagation direction of the light wave. The sign dependence of the Faraday rotation allows for the probing of the 22 23 domain structure dynamically and statically. 24 The experimentally measured quantity is the light intensity which is given as: c c VS 2 1 = ^ (") |gtot| ( (30) where I is the light intensity, c is the speed of light, c is the dielectric constant, p is the relative permeability, and Etot is the combined electric field of both up and down domains. The up and down domains have a thickness SL and width and L2 respectively as shown in Figure 8 . Etot can be calculated by decomposing the electric field vectors of each region, and L2. Since the magnetizations are oppositely directed, the Faraday rotation will be oppositely rotated. The Faraday rotation is proportional to the component of the magnetization along the direction of propagation of light. The component of magnetization along this direction is determined by the magnetostatic equilibrium conditions given by Equation 12. The plane of the sample defines the x-y plane and the light is propagating along the z-direction (111) as shown in Figure 8 . The initial direction of polarization is the y-axis and after passing through the sample the y-axls polarization is extinguished by a polarizer. An expression for the polarized light intensity along the x-axis must be derived for the saturated and unsaturated cases. The z-component of the magnetization causes a Faraday rotation of Mz Qf = n 1 j^j- = 11 I cos0o , (31) where n is a constant of proportionality, Mz is the z-component of the saturation magnetization, and Qq is the equilibrium value of the Figure 8 Thin film geometry for light propagation along the z direction. The sign of the Faraday rotation is determined by the direction of the magnetization in each domain. N) Ln 26 magnetization's polar angle in the spherical coordinate system. The electric field in region i is given by *> rigX * n z x * Ej = E0sln(n9. x + E0cos(n8, — ) y , (32) where 1=1 ,2. The total electric field is obtained by weighting the regional electric field by the regional area. The total electric field with the y-component extinguished is given as n7i rj7p [LiSintnil - j p + L2sin(nfc - j p ] x . (33) Using Equation 30, the intensity of linearly polarized light after passing through a garnet film with the y-component extinguished is given as Equation 34 represents the basic theoretical result for the intensity of plane polarized light after passing through a thin magnetic film. This equation can be used to explain the light intensity obtained in saturated regions by taking the limit that L,,=0. The light intensity depends on the magnetostatic equilibrium direction of the magnetiza tion in applied magnetic fields. The equilibrium direction of the magnetization is determined by the energy minimum conditions given by Equation 12. The applied fields are close to parallel so the free energy is written in terms 27 of a and 8 defined in Figure 5 as follows: F = -MH[sina sinB + cos( -(Ku - 2irM2) sin2B K + — [3-6sin2a + 7sin^a + 4/2sin3 [/?cos3a sin3 + constant . (35) It is convenient to first solve for B without including the cubic anisotropy energy terms. This value of B is defined as Bq and is determined by the conditions: 3F = 0 . (36) 98 Bo ;Ki=K2=0 The solution to this equation is 2Hsin(80 - a) - Heff sin2B0 = 0 , (37) where Heff=Hu-4irM, and Hu=2 Ku/M. For the angles used in this experiment, the small angle approxi mation is valid for high applied magnetic fields and Equation 37 reduces to K z n ^ «H . . Note that at exactly parallel fields, a=0, this equation predicts a null in the light intensity given by Equation 34. The experimental light intensity was not a null at a=0 which indicates the cubic anisotropy energy must be included. 28 The cubic anisotropy energy can be treated as a perturbation to the free energy. For external fields parallel or nearly parallel to the film plane, the equilibrium direction of the magnetization is changed due to the presence of cubic anisotropy energy to give B = B0 + 66 and where the perturbations 6B and 6(p are given17 as V7 sin3ip ... h 2. . . -sin6(p .. ___ 48 = " ~6 H-Heff (Hl + ^ ^ 5 Hi=2Ki/M and H2=2K2/M are the first and second order cubic anisotropy fields. The light intensity for parallel fields, Equation 34, depends on the value of Mz , therefore the change in the effect on M or on the light intensity, z The deviation of B due to cubic anisotropy energy can be written as 48 = H-Heff ’ <41) ■J7 Ho where Ha = — (H^ + — ) sin3 Equation 38, the additional term changes the equilibrium configuration of the magnetization's polar angle to aH-Ha s ■ * 3 5 ? • <,2) where Ha may be positive or negative depending on the sign of , H2 , or The light intensity in the unsaturated region of the applied magnetic field is due to the difference in domain widths L^-L^. The Zeeman energy will cause the width of the domain with magnetization 29 along the applied field to get larger at the expense of the magnetiza tion antiparallel to the applied magnetic field. This can be labeled an effective rotation, ai}d always follows the sign of a . There are five distinct cases of interest in applying Equation 42, depending on the value of a and the applied magnetic field. Table I summarizes the five cases assuming that H >0. The light intensity below the saturation magnetic field behaves like &e pp according to the value of the magnetic field and a. Figures 9 illustrates case i. The sign of the rotation in the saturated and unsaturated film is the same. Since the light intensity measures the square of the rotation, only one peak is theoretically predicted. Figure 10 illustrates case ii. The rotation is theoretically zero in the unsaturated case and negative in the saturated case. In case ii, the sign of the rotation still does not change and the theory predicts only one peak. Case iii gives quite a different behavior. Figure 11 shows case iii. In this case for positive a, there exists a range of a that yields a negative B in the high magnetic field region. The light intensity in the presence of domains follows Cases iv and v are shown in Figure 12. The sign of the rotation does not change in the transition from the saturated to unsaturated film and therefore predicts that only one peak should be formed. 30 TABLE I. Effects on B for Different Cases of a Assuming Ha > 0 CASE Beff B H conditions (i) a < 0 Beff < 0 B < 0 all H (ii) a = 0 Beff =0 B < 0 all H > Heff (iii) 0 < a < Beff > 0 6 < 0 HeffH < ? a (iv) a = rp3 - Beff > 0 B > 0 all H Heff4ef ^ Ha (v) a > rp3 - Beff > 0 B > 0 all H Heff j i -S p 0 I •s-1-3 a Peff H eff Magnetic Field Figure 9 Illustration of case 1 from Table I. The left vertical axis is the magnetization angle 8 , and the right axis Is the light intensity. 8 diverges at Heff as predicted by Equation 42. I H eff Magnetic Field Figure 10 Illustration of case 11 from Table I The light Intensity, 2 I, Is proportional to B . U> ro Peff CD c P 5 ■s a Magnetic Field Figure 11 Illustration of case iii from Table I. This figure shows the origin of the twin peaks. The sign of B is Initially LO positive and then changes sign to negative above Heff. U> 0 Peff Light Intensity Light Magnetic Field Figure 12 Illustration of cases lv and v from Table I. CO 35 The theory predicts that the two sets of peaks should be formed when 0 sin(3 follow the sin(3 of the range of twin peak formation, the cubic anisotropy energy constant can be determined magneto-optically. Faraday Effect The ferromagnetic Faraday effect can be traced to the inter action of the electric field of the polarized light with the orbital angular momentum contribution of the ions in the garnet film. The starting point for the theory is Maxwell's equations, and since garnet films are insulators, the equations become: B = 0 , (43) where B=jjH and D=£E. The magnetic permeability tensor, £, at the optical frequencies of interest can be set to the identity tensor because the spins cannot respond to such high frequencies. This means the light propagation behavior is completely dependent on the dielectric tensor £. The dielectric tensor for a ferromagnetic material saturated in the z-direction can be expressed as2**: (44) 36 where c^, and care frequency dependent constants. The form of this tensor can be derived from a simple microscopic theory and is shown in Reference 25. Using the relation -> *> *> *>•>*> V x V x E = V(V»E) - v 2e , (45) and Equation 43, the wave equation follows: -»o l 3 2 -> (46) E o = » • This equation can be solved by the ansatz Eox 2 . 2„ . # (47) Eoy which represents a wave propagating in the z-direction of angular frequency w and wavevector k. This ansatz is a transverse plane wave, no z-component of electric or magnetic fields, since k»D and k*B equal zero by Maxwell's equations. The wave equation gives the dispersion relation for k by substituting Equation 47 into Equation 46: o to2 ■* (k2 - — ) E0 = 0 . (48) This equation is a matrix equation and has a non-trivial solution if the determinant of the coefficients of E0 vanish: k2 - o — 0 k C1 c2 k2 - C1 — 0 (49) -i c 2 1 c2 0 * - This condition yields the solutions: 37 K2 = ^ [ci ± c 2 ] (50) These eigenvalues can be used to determine the eigenfunctions, i .e. normal modes, of the garnet film. Substituting the eigenvalues back into the matrix equation yields the normal modes of light wave propagation as 2 2 k+ = — [ c i + c 2 ] ; Eox = i E0y , (51) and 2 2 = — [ei - c 2 ] ; Eox = -i E0y . (52) C2 The normal modes represent right circular polarization (RCP) for K and left circular polarization (LCP) for k_. t E° i(k.z-wt) , 1. i(k_z-wt).. E = — [(i_) e + + (_i) e ' “ ] . (53) After passing through a garnet film of thickness SI, the electric field vector can be described by the equation: 3 Eo iwt _,1. ik.Sl ik_Sl . 1,, E = — e [(*) e + + e “ ( ) 3 (54) which can be recast into the form: ei(k+-k_)Sl/2 + e-i(k+-k_)!l/2 ■* Eo iwt i(k.+k_)St/2 E = — e e + (55) i(ei(k+-k_)!l/2 _ e-i(k+.-k_)SL/2 Equation 55 can be further simplified to give the result: cos[(k+-k_)Sl/2] (56) -sin[(k+-k_)Sl/2] If k+ and k_ are real numbers, then the state of polarization 38 predicted by Equation 56 is linear with the axis of polarization rotated from the initial direction by an angle, 0^., given by: ®f = 2 (k+-k-) % • (57) where 0f is defined as the Faraday rotation angle. The thickness of the film is usually removed from Equation 57 by defining the specific Faraday rotation and this convention is used from here on. Nature is never so kind, and in general c^, c2 , and c3 are complex which means that k+ and k_ are also complex. The complex behavior of k+ and k_ can be separated into real and imaginary parts by defining Of = 0f + i tyf , (58) where 0f = ~ Re (k+-k_) and qif = ^ Im (k+-k_) . (59) The complex behavior causes the initially linearly polarized light to become elliptically polarized as shown in Figure 13. The major axis of the ellipse is rotated from the initial state of polari zation by Oj. and the ratio of the minor axis to the major axis, the ellipticity, is given by tantyf. These effects are due to the birefringence and dichroism of the material and are not the same as the original Faraday rotation given by Equation 1. The convention remains to call this a Faraday rotation though the rotation is really a MCB and the ellipticity is a MCD. To summarize the Faraday rotation in garnet films briefly, a linearly polarized light wave is incident on a sample. Once the light enters the medium, the normal modes of wave propagation become RCP and 39 Figure 13 The state of polarization of initially linearly polarized light after passing through a garnet film. 40 LCP. The two normal modes propagate with different velocities and attenuations. Upon leaving the sample, the normal modes recombine out of phase to yield elliptically polarized light. Equations 51, 52, and 59 imply that the Faraday rotation is related to the dielectric response of the garnet film. The dielectric constant is related to the electric dipole transitions allowed in the garnet film. The dipole moment, p, of an ion is given as: p = -e where e is the electronic charge and the electron’s radius from the nucleus of the ion. The dielectric constant will depend on matrix elements involving the dipole moment of the ions. Since the dipole moment depends on the radius of the electron from the ion's nucleus, the important quantum numbers will be those of the orbital angular momentum. The orbital angular momentum of the electron depends on r as does the dipole moment. This means that the Faraday rotation has its origins in the orbital angular momentum states of the ions. The addition of bismuth in the "c" site of the garnet structure enhances the Faraday effect by two orders of magnitude. The reasons for this phenomena is still unknown. Since the Faraday rotation is a probe into the orbital states of the material and the magnetization is a probe of the spin and orbital states, information can be obtained by studying the spin-orbit coupling of the thulium and bismuth substituted garnet films. A simple connection can be made between the Faraday effect and the magnetization of the material. By combining the paramagnetic 41 Faraday rotation of Equation 1 and the mean field theory result for the internal molecular field from Equation 3, the following relation is obtained: Va^aa va^ad va^ac vdMa vdxdd vdxdc « {61) vc^ca vc^cd Vc\cc where 9^ is the Faraday rotation for the ith sublattice and is the Verdet constant for the ith sublattice. Temperature fits of the magnetization and Faraday rotation indicate that the "d" sublattice dominates the Faraday rotation in bismuth substituted garnet films and the rare earth ion contributes very little to the magneto-optic phenomena as seen in Figure 14. 42 05 10 15 — 1 " " i "I 20 * - 1 0- % E o» ■o® -20 in t O «fb u . 0» -50 Figure 14 The Faraday rotation for all rare earth garnets except thulium with different Bismuth substitutions at room temperature and a wavelength of 633 nm. (Ref.26) CHAPTER III EXPERIMENTAL APPARATUS This investigation required the measurement of the films* magnetic and magneto-optic parameters. The magnetic properties include the saturation magnetization, the uniaxial anisotropy field, the cubic anisotropy field, the effective internal magnetic field, and the effec tive g-value. The magneto-optic parameter measurements include the Faraday rotation constant and the Faraday ellipticity. The experi mental techniques used to measure the magnetic parameters are a vibrating sample magnetometer, ferromagnetic resonance and a magneto- optic set-up designed and constructed to measure the magneto-optic effects. Saturation Magnetization The saturation magnetization was measured by a PAR model 155 Vibrating Sample Magnetometer. A schematic drawing is shown in Figure 15. The principle behind the magnetometer is Faraday's law of induction. A quartz rod with the sample mounted on the tip is vibrated perpendicularly to the applied magnetic field at 80 Hertz. Two pairs of search coils are mounted around the sample. The oscillation of the sample's magnetization induces an emf in the search A3 VIBRATING VIBRATION OSCILLATOR HEAD AMPLIFIER VIBRATING PLATES ■vr FIXED PLATES ~ \ . LOCK-IN X-Y SAMPLE ROD- AMPLIFIER RECORDER DIFFERENTIAL AMPLIFIER SAMPLE :k u p c o i l s MAGNET MAGNET POWER SWEEP SUPPLY CONTROL MAGNET Plgure 15 Experimental apparatus for measuring the saturation magnetization. (Ref. 27} 45 coils. The induced emf is directly proportional to the magnetization of the sample. The magnetometer is calibrated by using a nickel standard sample which has a saturation magnetization of 55.01 emu/g. Raw data from the magnetometer is given in Figure 16. The data is the magnetic moment from the magnetometer plotted simultaneously with the applied magnetic field and is called the hysteresis loop. The magnetometer measures the film's magnetization and the paramag netic response of the GGG substrate. The data in Figure 16 represents the superposition of the two magnetic moments. The effect of the substrate must be taken into account. Since the paramagnetic moment is linearly proportional to the applied magnetic field, extrapolating the saturated film to zero applied magnetic field will cancel the paramagnetic signal leaving only the ferromagnetic saturation magnetization. The sample is placed in a variable temperature cryostat which enables the saturation magnetization to be measured as a function of the temperature. The temperature is controlled by a Lake Shore Cryogenic Temperature Controller Model 520. This controller can stabilize the temperature to ±.1 K between 6 K and 300 K using GaAs temperature sensitive diodes as the sensor units. Ferromagnetic Resonance Ferromagnetic resonance is a standard technique employed in the investigation of magnetic properties. The g-value, the effective internal uniaxial field, and the cubic anisotropy field can all be measured by the angular dependence of the magnetic resonance of the films. Magnetic Moment (memu) -25 -5Q r 0 5 25 20 10 -100 -150 -200 i Figure 16 Raw data for the saturation magnetization from Sample 1 Sample from magnetization saturation the for data Raw 16 Figure at room temperature. room at antc il (G) Field Magnetic 5 0 0 100 50 0 -50 L. 150 200 47 Figure 17 shows the experimental arrangement for the ferromagnetic resonance data obtained during this investigation. Figure 18 shows a typical spectrum from an FMR experiment. The data shows the derivative of the absorbed power as a function of the applied magnetic field. The electon's spin will precess about an applied magnetic field with an angular frequency given by the Larmor precession frequency. The Larmor precession frequency for magnetic fields of a few kiloGauss lies in the microwave region for free electons. The principle behind FMR is supplying a microwave b-field power to the electrons to induce the spin precession. When the spins of the system go through resonance, the power absorbed by the garnet film is a maximum. The applied magnetic field is modulated by a set of coils mounted such that the magnetic field generated by the coils is parallel to the dc magnetic field. A PAR Model 124 Lock-in Amplifier is used to drive the modulation coils at 400 Hertz. A crystal detector measures the amount of microwave power absorbed in the sample and is the voltage measured by the lock-in amplifier. The modulation of the magnetic field causes the first derivative of the absorbed microwave power to be measured. At the maximum power absorption, ferromagnetic resonance, the derivative is zero. Experimentally, resonance occurs when the output of the lock-in amplifier passes through zero. The frequency used in this investigation was X-band or 9.2 GHz. This frequency places the applied magnetic field in the range of 2000 Gauss to 6000 Gauss. These fields are attainable by electromagnets in the lab. The magnet is a 15 inch Varian electromagnet. The range of 48 circulator nasnet eweap control Figure 17 Experimental apparatus for measuring the ferromagnetic resonance. (Ref.27) Absorb. Power Deriv. (V) -.6 -.2 Figure 18 Typical spectrum of ferromagnetic resonance at room room at resonance ferromagnetic of spectrum Typical 18 Figure 4000 temperature of Sample 3 at 9.2 GHz. 9.2 at 3 Sample of temperature antc il (G) Field Magnetic 50 5000 4500 5500 6000 -P* viD 50 magnetic fields varies from 0 to 18,000 Gauss. The entire electro magnet pivots 360 degrees. This allows the angular dependence of the ferromagnetic resonance to be measured. The microwave cavity in which the sample is placed during the experiment is located inside a variable temperature cryostat. This allows the measurement of the temperature dependence of the ferro magnetic resonance. The same temperature controlling unit is employed in the resonance experiments as in the saturation magnetization experiments. The temperature range of interest was 6 K to 300 K. Sample Characterization The samples in this investigation were grown at Airtron, a division of Litton Corporation. The films were prepared by liquid phase epitaxial growth techniques and are single crystals. The thicknesses of the films were measured by Airtron by optical inter ference patterns. The composition of the films were measured by electron microprobe analysis. This technique uses x-rays from electrons scattered from the ions in the garnet film. The electron energies used are in the 25 kev range. The JEOL/35 and EDAX machines at the Electron Optics Facility at the Metallurgical Engineering Department of The Ohio State University was the electron microscope used for the analysis. This machine can determine the presence of any material in the garnet film with an atomic weight above nitrogen. The raw data from EDAX gives the composition in atomic weight percent and this data can be converted into the stoichemetric quantities. 51 Magneto-optics An apparatus was constructed to measure the Faraday rotation constant and the Faraday ellipticity. The set-up in Figure 19 shows the necessary equipment to make the magneto-optic measurements. The basic need is a method for measuring the light intensity and the necessary optics to analyze the polarization state of light. The light intensity measurements were made by using a RCA model IP21 photomultiplier tube. A PAR model 124 lock-in amplifier is used to measure the voltage signal from the photomultiplier tube. A light chopper is used as the modulator of the light source at 110 Hertz. The frequency dependence of the magneto-optical effects were made by using two different light sources. The first source is a Melles Griot Helium Neon laser. The laser provides a monochromatic light beam at 633 nm and a power of 5 mW. The second light source is a Bausch and Lomb Monochromator. The monochromator is a device with a diffraction grating that can be rotated. Since the wavelength of the diffracted light depends on the angle of the diffraction grating, any wavelength from a multiline source may be selected by rotating the grating. The light source for this investigation was a high pressure mercury lamp. The high pressure in the lamp Doppler broadens the spectral lines giving more wavelengths for the monochromator to select. The mercury spectrum has strong lines at 436 nm, 546 nm, and 578 nm. The line at 436 nm is below the transmission limit for the samples in this work there fore unsuitable. The other two lines, 546 nm and 578 nm, are trans- mittable and have bright intensities and are reported in the analysis. Optic Dewar Attenuator Chopper Comp Analyzer Quartz ^Window, Aperture Laser Lens PMT / Sample Polarizer ref 110 Hz Magnet Lock in Amp Hall Probe Magnet Controller y-axis tem p Controller Gaussmeter x-y recorder x-axis Pigure,19 Experimental setup to measure the light Intensity and Faraday rotation. Ln ro 53 The remaining components in the magneto-optic set-up are for measuring the polarization state of the light. The polarization of any light beam is completely described by specifying the angle of the major axis of the elliptically polarized beam and the ellipticity of the light. The components needed to specify these properties are polarizers and a Babinet-Soleil compensator. The polarizers are Melles Griot dichroic sheet polarizers. The Faraday rotation is a measure of the rotation of linearly polarized light as it passes through the sample. A polarizer is used to polarize the light source. The laser is initially polarized but the light from the mercury source is unpolarized. The first polarizer ensures the light beam is polarized. A Babinet-Soleil compensator is an optical device that can introduce any phase difference between two orthogonal directions of polarization. The compensator is a Karl Lambrecht model BSA-13-6 mounted on a rotation stage that allows for a rotation of 360 degrees. The compensator uses the birefringent property of quartz to introduce the phase difference. The compensator’s micrometer screw shifts the quartz wedges so any phase difference may be selected by changing the path length each orthogonal direction travels. The phase difference also depends on the frequency of the light. The compensator is calibrated for wavelengths between 250 nm and 1200 nm. The calibration curve yields the distance the micrometer screw must be translated in millimeters to retard one wavelength between the orthogonal directions. 54 The light beam is elliptically polarized when it leaves the sample as shown in Figure 13. The polarization of the light is converted to linear polarization by introducing a phase difference of ir/2 between the major and minor axis of the ellipse. The direction of linear polarization is rotated by an amount ^ from the major axis. The function of the Babinet-Soleil compensator is to convert the ellipti cally polarized light into linearly polarized light. The compensator is used essentially as a variable wavelength quarter wave plate. Once the elliptically polarized light is converted into linearly polarized light, it can be extinguished by using another polarizer (analyzer). A minimum in the light intensity corresponds to the condition that the axis of the compensator is colinear with the major axis of the ellipse and the analyzer is along the direction of the resultant linearly polarized light. The function of the magnet is to ensure that the film is saturated in the direction of the light propagation. The sign of the Faraday rotation depends on the direction that the magnetization points. The direction of the Faraday rotation changes sign if the magnetization is reversed. The Faraday rotation can be measured simultaneously with the Faraday ellipticity. The film is saturated in one direction. A minimum in the light intensity is achieved by simultaneously rotating the compensator and analyzer to their respective positions. The angles are recorded and are listed as the initial conditions. The applied magnetic field is then reversed and the Faraday rotation also changes direction. The compensator and analyzer are again rotated to 55 give a minimum in the light intensity. The new angles are recorded and along with the initial conditions yield the Faraday rotation and ellipticity. The Faraday rotation is given as: Qf = \ ABS (62) where flgg is the difference between the initial and final rotation angle of the Babinet-Soleil compensator. The Faraday ellipticity is given as: where is the difference between the initial and final analyzer rotation position. The recorder is used to monitor the light intensity as the magnetic field is swept. The Faraday rotation must be measured when the sample is completely saturated. The magnet can rotate 180 degrees which means the sample can be saturated parallel or perpendicular to the film plane. Typical data is shown in Figure 20 for the light intensity as a function of perpendicularly applied magnetic field. When the light intensity becomes flat, the film is saturated. A special sample holder was designed for the twin peak experiment that would allow for a rotation of the sample into the plane of the applied magnetic field. The sample could be rotated 70 degrees in the azimuthal direction to probe the CHAPTER IV EXPERIMENTAL RESULTS AND DISCUSSION This chapter discusses the results of the magnetic and magneto-optic experiments described in the experimental and theory chapters. Sample Characteristics The results of the electron microprobe analysis is given in Table II. The bismuth content is the same in all films investigated within experimental error of the measurements. The microprobe analysis is able to determine the stoichiometry to within 15%. Within this accuracy the bismuth concentration is 0.6 in the films with a thulium concentration ranging from 1.2 to 2.A per formula unit. The thicknesses are on the order of 10 microns and are also given in Table II. Magnetization The temperature dependence of the saturation magnetization is given in Figures 21-24 for samples 1-4. In the samples containing gallium, the gallium ion replaces the iron ion of the "d" site preferentially over the "a" site, with the result that the net moment of the "d" and "a" sublattice will be reduced making the moment of the "c” site dominant. 57 58 TABLE II. Samples' Characteristics Sample No. Composition Thickness (pm) 1 (187-2-13) Tm2.4Bi0 .6Fe3 .7Gal .3°12 12.29 2 (1464-1-16) Y0 .5Tml .9b 10 .6Fe3 .7Gal .3°12 12.10 3 (1522-1-29) Y1 .0Tml .45Bi0 ■55Fe5°12 13.51 4 (1522-1-28) Y1 .2Tml .2b 10 .6Fe5°12 11.49 5 (M533-4-9) (YGdBi)3Fe5012 12.24 4trM (Gauss) 0 • 300 SOOr 200 400 QQl Q iQ * * * * x ' ■.w . ■-..w '1 — ■ ■ ■ - x Figure 21 The temperature dependence or the saturation magnetization magnetization saturation the or dependence temperature The 21 Figure for Sample 1. The composition compensation temperature is temperature compensation K.142 composition The 1. Sample for 0 20 300 200 100 x * x # * #* X * X #* * eprtr (K) Temperature *x ------s ------i ____ vO U1 4trM (Gauss) 250 0 - * X * * - 200 5 h * h150 100 50 * r i Figure Figure x X 22 ------X The temperature dependence of the saturation magnetization magnetization saturation the of dependence temperature The X for Sample Sample for x * 52 K.52 0 20 300 200 100 ... l ------eprtr (K) Temperature 2. The composition compensation temperature is temperature compensation composition The x —. :— ------_ i ------x * - o ' O O' Temperature (K) 100 100 200 300 ***** * * * * * * * ***** x * since no gallium is substituted for the iron. for Sample 3. No composition compensation temperature * Pigure 23 The temperature dependence of the saturationmagnetization * 500 1000 1500 2000 (9) N»fr ro (T* a Temperature (K) 100 100 200 300 ******** for Sample 4. Figure 24 The temperature dependence of the saturation magnetization ***** ***** * * 500 1500 1000 2000 r (9) 63 The substitution of thulium in the place of yttrium ion greatly changes the total magnetization, but has an insignificant effect on the "d" and "a" sublattice magnetization. This can be seen in Figure 25 which shows the theoretical calculation of the difference between the "d" and "a" site sublattice magnetizations for pure YIG and TmlG. The calculations are made using the program of Appendix I. This result is very important since now the "c" sublattice magnetization can be calculated in Tm:YIG by knowing the total magnetization and the difference between the "d" and "a" sublattices. The "c" sublattice is given by: Mc = MT - Hda . (64) The "c” sublattice magnetization must be known in order to fit the theory of the effective g-value with the experimental results. Ferromagnetic Resonance The angular and temperature dependence of the ferromagnetic resonance was measured to determine the effective g-value, the effective internal field and the cubic anisotropy field. The results shown in Figures 26-28 are from samples 3 and 4. Figure 26 shows that the effective internal field is negative in these samples which means that the magnetization lies in the film plane in zero applied magnetic field. Figure 27 shows that the first order cubic anisotropy field increases with decreasing temperature. This is to be expected since the cubic anisotropy field is related to crystal field effects. 2500 2000 co 1 5 0 0 ZD ru CD ~ 1000 (U T 3 1 500 100 200 300 400 500 600 Temperature (K) Figure 25 The temperature dependence of the difference between the "d" and Ha" sublattice magnetization for pure TmlG (upper curve) and YIG (lower curve) using the result of the HFT. O' -P* 3 -2000 CD X -1500 * * # # * o » i H □ 0 0 a 0) » D • r i □ U- -1000 4-> C HH *500 03 > • r t 4 -> U 03 0 100 200 300 ID Temperature (K) Figure 26 The effective internal field for Samples 3 (□) and 4 {*) as a function of temperature. ON Ui Temperature (K) Figure 27 The temperature dependence of the first order cubic anisotropy field of Sample 3 (O) and Sample 4 {"). Temperature (K) Figure 28 The temperature dependence of the effective g-value of Sample 3 (□) and Sample 4 (”). Notice the g-value decreases with decreasing temperature totally opposite to the behavior of the splnwave theory of Figure 7. 68 The primary results of the FMR experiments is shown in Figure 28. The effective g-value decreases as the temperature decreases totally opposite to the spinwave theory result plotted in Figure 7. The theoretical fit to the effective g-value is obtained from Equation 28. The "c" sublattice magnetization is determined by Equation 64 and the "d" and "a” sublattice magnetizations are determined by the MFT results plotted in Figure 25. The "c" sublattice magnetization can only be determined at temperatures where the total magnetization was measured. The value of 5 used to fit the theory with experiment in Figure 29 yields a T from Equation 29 of 4.76xl07 Hz and the FMR 7 linewidth data gives a r of 3.64x10 . The agreement between the theory and experiment is reasonable for the effective g-value. Twin Peak Results The theory of light propagation in thin garnet films was outlined in the theory chapter. First, the basic equation for plane polarized light intensity, Equation 34, is verified for various applied magnetic fields. There are three cases of interest in applying Equation 34; perpendicular, parallel, and slightly off parallel magnetic fields. Second, the twin peak in the light intensity is shown and the agree ment between the theory and experiment discussed. The films of interest have Q>1 where Q is defined as follows: Q = Hu/4irM . (65) Having Q>1 is equivalent to Hu-4irM>0. This means the film's magnetization will point along the direction perpendicular to the film's plane. Sample 1 has a Q=23 which is very high for garnet films. Effective g-value .0 r 2.00 0.00 1.50 1.00 5 [• .50 * * * 10 0 300 200 100 0 i Figure 29 Theoretical fit (□) of data from Sample 4 for the for 4 Sample from data of (□) fit Theoretical 29 Figure xeln. ^ excellent. auso h oa antzto. h gemn is agreement The magnetization. total the of values and A=-1400 with 28 Equation using g-value effective 5=3300. The theoretical fit comes from the experimental the from comes fit theoretical The 5=3300. eprtr (K) Temperature 1 □ 1 & 70 The uniaxial anisotropy field is very large due to the large amount of bismuth causing a large lattice mismatch and the gallium substitution which causes a smaller magnetization. The fact that Q is large implies that large fields are required to pull the magnetization from the film normal direction. This requirement is essential for the viewing of the twin peaks. The first case analyzed is when the applied magnetic field is directed along the film normal. In zero applied field, the sample minimizes the total energy by forming regions of oppositely magnetized areas called domains. The magnetization in each domain is equal in magnitude but opposite in direction: «zl = -«Z2 = M . (66) This condition simplifies Equation 34 to: ) sin2(nS-) (67) As the perpendicular magnetic field is increased, the domains with the magnetization directed parallel with the applied magnetic field have their widths increase while the other domain have their widths decreased. This is due to the lowering of the Zeeman contribution to the free energy at the expense of slightly increasing the demagnetiza tion contribution. Equation 67 shows that the light intensity of the unsaturated magnetic film depends on the difference L^-L,,. Once the magnetic film becomes saturated, h^=0, the light intensity is 27 constant. The condition that for small perpendicular fields : L t -Lp (68) can be substituted into equation 67 to yield: I ~ H* . (69) The light intensity measured as the perpendicular field was swept from -100 to 100 Gauss is shown in Figure 20. The perpendicular saturation field can be measured by this method and is 24 Gauss in this sample. The saturation point is easily seen as the place where the light intensity becomes constant. To check that the light intensity varies quadratically in the applied magnetic field, the light intensity is digitized and plotted as a function of the squared applied field. This result is shown in Figure 30. The light intensity for small applied magnetic fields is clearly a quadratic dependence. The low field off parallel case is very similar to the perpen dicular case. As stated previously, the uniaxial anisotropy field is much larger than the magnetization so the magnetization is not signi ficantly pulled from the film normal direction for a wide field range in such high Q films. This implies that M^s-M g=M and Equation 24 may be used again, now to characterize the light intensity for small H||. If the magnetic field is deviated from the plane by a small angle a, then there is a small component of magnetic field in the perpendicular direction given by: H± = H||Sina « H||a . (70) Since the normal component of the magnetization is not significantly affected by the parallel component, the film behaves as if a perpendi cular component is the only applied field. By substituting Equation 70 into 69, the light intensity for off parallel low fields becomes: Light Intensity (arb. units) 15.0 12.0 9.0 6.0 3.0 0.0 Figure 30 The light intensity plotted a s .the square of the applied the of square .the s a plotted intensity light The 30 Figure 0 80 20 1600 1200 800 400 epniua antc field. magnetic perpendicular sur Gauss) (square 73 I ~ H\cl2 . (71) This equation indicates the low field behavior should follow a quadratic field dependence as did the perpendicular case, but also a quadratic dependence on a. Figure 31 shows the intensity of light for the case when the applied magnetic field is 1 degree off parallel. The field range is from 0 to 6000 Gauss. Figure 32 shows the experi mental low field dependence for various small angles off parallel. The low field dependence is quadratic. Figure 33 shows the linear behavior by plotting the light intensity versus the magnetic field squared for a=l degree. The intensity is clearly linear from 0 to 2 about 0.80 (kG) . This behavior is in agreement with Equation 71. The data obtained at 0, 0.2, 0.4, 0.6, 0.8, 1.3, 1.8, 2.8 degrees off parallel is plotted as a function of the deviation angle in Figure 34 for magnetic fields between 200 and 1000 Gauss. The plot shows that the light intensity is linearly related to the angle squared, i.e. quadratically related to the off parallel angle which is also in agree ment with Equation 71. This phenomenom is seen because the Q value is so large and should not be seen in materials with a small Q value. Figure 35 shows the region where the twin peaks in the light intensity is formed. The small range of twin peak formation, designated as ar is shown for the case when ip=30 degrees. The theory of the twin peak formation given earlier predicted that the range of twin peak formation is given by: ✓? - H1 “r = e sin3* W ' (72) where the second order cubic anisotropy field is neglected. This iue 1 h ih Itniya temgei fedI swept Is field magnetic the as Intensity light The 31 Figure Light Intensity in arb. units o.o o ±6000 Gauss. ±6000 lgtyofteprle retto ( 30dg from deg) =3.0 (a orientation parallel the off slightly antc il i KGauss in Field Magnetic (V n m 6.0 74 Figure 32 Light Intensity for various small values of a for small for a of values small various for Intensity Light 32 Figure Light Intensity In arb. units o.o ils Nt tesrkn udai behavior. quadratic striking the Note fields. antc il InGauss Field Magnetic 75 Light Intensity (arb. units) 10.0 r 10.0 6.0 . ■ 4.0 8.0 2.0 Figure 33 The light Intensity plotted as a function of the square of square the of function a as plotted Intensity light The 33 Figure the applied magnetic field at a=3.0 deg. a=3.0 at field magnetic applied the qae Mgei Fed (kG Field Magnetic Squared 2 ) O' Light Intensity (arb. units) 12.00 r 12.00 .0 ■ B.00 00 .0 4 Figure 34 The light intensity plotted as a function of the square of square the of function a as plotted intensity light The 34 Figure the angle from the parallel direction for magnetic fields Gauss. 1000 magnetic for and 200 between direction parallel the from angle the BOO Gauss BOO 600 Gauss 600 00 Gauss 1000 200 Gauss 200 Gauss 400 78 5 mV a • -,3‘ J Q *N 5 mV a -.2* I— p L cn 2 mV a ■ .4* a *.5' a ■ .6' & 5 4 3 2 I 0 123456 MAGNETIC FIELD (KG) Figure 35 The light Intensity from ±6000 Gauss for 0< ar <0.7 degrees. Notice the formation of twin peaks in the light intensity on both sides of zero magnetic field as predicted in Figure 11. 79 equation predicted that the twin peak formation follows a sin(3 behavior which was not experimentally observed before. A new sample holder was constructed to probe this behavior and data was taken over a range of -10 to 60 degrees. The results of the experiment are shown in Figure 36. The solid line represents a least square fit to the data. The curve follows sin(3(j>) exactly as predicted by the theory and the amplitude is 0.67 degrees. The amplitude of the sin(3 Hi/Heff . (73) Knowing the amplitude and the effective internal field of 2270 Gauss, the first order cubic anisotropy field can be measured magneto- optically and is observed to be 110 Gauss in sample 1. This yields a 3 first order cubic anisotropy energy constant of 5500 ergs/cm . The final verification of the theory lies in checking the magnitude of Hj magneto-optically with the value obtained from FMR computer fits. The problem with sample 1 is that the large amount of thulium produces a FMR linewidth of 800 G. The computer program cannot be trusted to produce accurate results for such wide linewidths. Another sample 5 was obtained from Airtron that has a very high Q value, and the identical experiments performed. The result from the magnetooptic experiments is H^=144±10 Gauss and the FMR result is ^=150130 Gauss. The value of He^ magneto-optically is 1.87 Guass and the FMR value is 1.92 k Gauss. The agreement is very good. The error in the twin peak experiment of the value of the first order cubic anisotropy field is smaller than the error in the FMR computer fitting experiment. 0 20 6040 cp (degrees) Figure 36 The range of twin peak formation as a function of 9 . The solid curve is a least square fit to sin (3 Faraday Rotation The temperature dependence of the Faraday rotation and Faraday ellipticity was measured using the technique outlined in the experi mental section. This section reports the results of the experiments on Samples 1, 2, and 3 at the light wavelengths of 633 nm, 578 nm, and 546 nm. These samples have the same bismuth concentration, but widely separated in the thulium concentration. The light intensity when the applied magnetic field is in the perpendicular direction should be a flat line when the film is satura ted. The light intensity shows a slight slope in the saturated region which indicates that the rotation angle of the plane of polarization is not constant in magnetic field. This phenomenom is due to the para magnetic response of the GGG substrate and the quartz windows. The solution to this problem was to place a blank substrate in the dewar and measure the Faraday rotation of the quartz and substrate as a function of temperature, wavelength, and magnetic field. The results are shown in Table III. This paramagnetic Faraday rotation was observed to be temperature independent, as well as having a strong dependence on frequency and magnetic field. These rotations must be subtracted from the total Faraday rotation measured in the garnet film sample to obtain the Faraday rotation of the garnet film itself. There is an arbitrary direction to be chosen when dealing with a relative rotation. The direction chosen in this work, which is 28 standard for Bi:YIG, is that the paramagnetic rotation of the substrate is positive. Using this convention, all of the TmBi:YIG films had negative Faraday rotations at room temperature. 82 TABLE III. Faraday Rotation of Quartz Windows and Blank Substrate Wavelength (nm) 20f/H (deg/kG) 633 .310 578 .410 546 .470 436 .860 83 The errors involved in the measurement of the Faraday rotation is 0.2 degrees which is on the order of 1%. The largest error source in the specific Faraday rotation comes from the accuracy of the measure ment of the thickness of the thin film which is 556. This uncertainty is larger than the error in the Faraday rotation results. The errors involved in the measurement of the Faraday ellipticity is even larger. The Faraday ellipticity is taken as the difference between two experi mentally determined values which doubles the absolute error to 0.4 degrees which corresponds to a relative error of 50% . This means that the Faradayi^ellipticity is so small in these garnet films that it can be neglected for these analyses. The temperature dependence of the Faraday rotation is shown in Figures 37-39 for light wavelengths of 633 nm, 578 nm, and 546 nm. The Faraday rotation changes sign at the magnetization compensation temperature in Figures 37 and 38. This supports the conclusion that the moment at one of the sublattices dominates the Faraday rotation since at the magnetization compensation temperature the sublattice magnetizations changes directions therefore changing the sign of the Faraday rotation. The change in sign is catastrophic but the amplitude of the Faraday rotation remains a smooth function of temperature. This is observed by plotting the absolute value of the Faraday rotation as a function of temperature which is shown in Figure 40 for a wavelength of 633 nm. Samples 1 and 2 show dips in the amplitude of the Faraday rotation near the compensation point. This result is due to an inhomogeneity in the film composition through the film thickness. j c C- o Faraday Rotation (deg/mi c - - -3.0 2.0 3.0 0.0 2.0 1.0 1.0 Plgure 37 The temperature dependence of the specific Faraday rotation Faraday specific the of dependence temperature The 37 Plgure -» a o a a --- 0 0 0 D 0 0 0 D 50 k -- sn qain7 n al IV. Table and 74 Equation using fSml o ih aeegh f63n, 7 m and nm, 578 nm, 633 of wavelengths light for 1 Sample of 546 no. The solid curve represents the theoretical fit theoretical the represents curve solid The no. 546 * --- 100 a a x -- eprtr (K) Temperature —je-£— a 150 o a B a 200 a 546 nm *633 no a 578 no a □ 250' a a o D a a 300 CO Faraday Rotation (deg/micron) -3.0 - -2.01- 3.0 r 3.0 0.0 2.0 2.0 f -x fc 0 - i 1.0 k BHHQQaaaaD Figure 38 The temperature dependence of the specific Faraday rotation rotation Faraday specific the of dependence temperature The 38 Figure B D0 a 546nm 00 D 0 B OB *633 nm *633 OB B 0 0 10 0 20 300 250 200 150 100 50 using Equation 74 and Table IV. Table and 74 Equation using and no, 578 no, 633 of wavelengths light for 2 Sample of 546 no. The solid curve represents the theoretical fit fit theoretical the represents curve solid The no. 546 dddodooddd eprtr (K) Temperature □578 nm 00 U! 00 c c o c_ o Faraday Rotation (deg/mi 3 o -3 -2.5 - i.5 - - 2.0 -.5 1.0 Figure 39 The temperature dependence of the specific Faraday rotation rotation Faraday specific the of dependence temperature The 39 Figure fSml o ih aeegh f63 m 68n, and nm, 678 nm, 633 of wavelengths light for 3 Sample of 546 nm. The solid curve represents the theoretical fit IV. theoretical Table and the 74 Equation represents using curve solid The nm. 546 100 eprtr (K) Temperature 150 200 56 nm 546 Q d # 633 633 # 578 578 nm nm nm 300250 Faraday Rotation (deg/micron) 1.25 1.00 0.00 .50 25 Figure 40 The temperature dependence of the absolute value of the of value absolute the of dependence temperature The 40 Figure 50 film thickness. film and pcfcFrdyrtto. h i cus nSmls1 “) (“ 1 Samples in occurs dip The rotation. Faraday specific 2 (□) due to a compositional dependence through the through dependence compositional a to (□) due 100 eprtr (K) Temperature 150 200 5 300 250 350 00 88 Various parts of the film are passing through compensation at different temperatures causing the rotation of the Faraday effect to change discontinuously throughout the bulk of the film. The total Faraday rotation described by Equation 61 can be summed from the separate contributions of the different sublattices to yield the result: Of = CMc + DMd + AMa , (74) where A, C, and D are frequency dependent but temperature independent quantities. The sublattice magnetizations are calculated by the MFT and the constants determined by the best fit to the Faraday rotation data at 633 nm. This wavelength was chosen because the light source is a helium-neon laser, the only true monochromatic light source used in this investigation. The results of the fit are given in Table IV. TABLE IV Sample A (deg/G*vjm) C 1 0 -2.2110-4 0 2 0 -1.8910-4 0 3 0 -1.1110-4 0 The fit to the theory shows that the "a" and "c” sublattices have minimal effects on the Faraday rotation and that the "d" sublattice dominates the Faraday rotation in TmBi:YIG. Similar results have been 28 reported in other rare earth and bismuth substituted YIG. The large spin-orbit coupling on the "c” sublattice appears to have no effect on the Faraday rotation. In Samples 1 and 2 the gallium effects only the sublattice magnetizations. However the magnitude of the Faraday rotation with and without gallium has the same value within 89 experimental results suggesting that the constant D must change as shown in Table IV. The slight spread in the magnitude of the observed Faraday rotation can be attributed to the error in the sample thickness and the measured bismuth concentrations. The agreement between the theory and experiment is good. The experimental observation that the Faraday rotation depends only on the Bi concentrations but does not depend on the gallium concentration is surprising. This has several consequences. First, the bismuth ion may effect only the nearest neighbor ions, i.e., it is a local effect and not a long range interaction. This supports the evidence of the temperature dependence of the tetrahedral magnetiza tion of the Faraday rotation since the tetrahedral site is closer to the bismuth site than the octahedral site. Second, this also implies that the iron ion preferential site near bismuth ion rather than the gallium ion. The origin of the preference is not known. CHAPTER V CONCLUSION A series of magnetic and magneto-optic experiments have been performed on TmBi:YIG. The films analyzed have various thulium concentrations with a constant bismuth substitution for yttrium in the iron garnet system. A new magneto-optic technique was discovered to measure the first order cubic anisotropy energy constant. A theory was developed to explain the experimental observation of a pair of twin peaks in the light intensity as the magnetic field is swept parallel and slightly off parallel the film plane with the film fixed between crossed polarizers. The twin peaks are observed in the two films investigated and the agreement between the magneto-optical cubic anisotropy field and the FMR result is excellent. Once the theory was developed, it predicted that the range for the twin peak formation should follow a sin(3q>) dependence. After modifying the experimental apparatus in order to probe the dependence, the sin(3 90 91 larger than 1, i.e. an easy axis film; and the Faraday rotation constant needs to be large enough to produce a significant change in the light intensity. The larger the Faraday rotation, the larger the amplitude of the twin peaks. The Bismuth films in this investigation were ideal due to the Bismuth enhancement of the Faraday rotation. The thulium in these films cause a large cubic anisotropy field which yielded a large range of twin peak formation. The thulium iron garnets have shown anomalous low effective g-values at room temperatures. The temperature dependence of the TmBi:YIG g-value shows that the g-value decreases with decreasing temperature. This behavior is contrary to the simple spinwave theory for the effective g-value. A theory has been proposed to account for this anomalous behavior and presented is the first experimental data to verify this theory. The fit of the experimental g-value with the theory is accomplished by calculating the sublattice magnetizations by a MFT. The agreement between the experiment and theory is good. The Faraday rotation has been measured as a function of temperature and wavelength of light. A simple sublattice theory of the Faraday rotation shows that the "d" sublattice is the dominant sublattice. The "c" sublattice has no effect on the Faraday rotation even though the concentration of the thulium in the "c" sublattice varied from 1.2 to 2.A. The gallium had no effect on the magnitude of the Faraday rotation but produced a magnetic compensation temperature at which point the Faraday rotation changed sign. The enhancement of the Faraday rotation in the BiTm:YIG can be traced solely to the effect of the Bi ion. This is consistent with the results in other 92 rare earth and bismuth substituted garnet films. The absence of any gallium effect on the Faraday rotation indicates that the bismuth ion affects only the nearest neighbor tetrahedral iron ions and that the iron ions are preferentially near bismuth ion over that of the gallium ions. The Faraday ellipticity was measured and shown to be very small and negligible. This project still has some questions to answer. The theory for the effective g-value predicts that below the magnetic compensation temperature, the effective g-value should change sign and curvature. This presents experimental difficulties because the samples with a compensation point have so much thulium which causes the FMR linewidth to be broad. The temperature dependence of these spectra are very difficult to follow as the temperature decreases. The Faraday rotation can be probed by obtaining films with constant Tm concentration and various Bi concentrations. The twin peak experiment can be investi gated as a function of temperature and is the logical extension of that effect. The thulium iron garnet system has not been previously studied in detail because the large spin-orbit coupling is difficult to handle theoretically. The thulium and bismuth system deserves further investigation in ar. effort to uncover the origin of the bismuth enhancement of the Faraday rotation. APPENDIX A COMPUTER PROGRAM TO CALCULATE MAGNETIZATION 93 10 CLEAR 20 REAL m s (4,250) 30 REAL ma(100),md(100),mc(100),mt(100) 40 b=9.2741E-21 \ Bohr magneton ! 50 ga=2 ! Lande g—value of sublattice "a" ! 55 gc=2 ! Lande g-value of sublattice "c" ! 60 gd=2 ! Lande g—value of sublattice "d" ! 70 N=6.02217E23 ! Avogadro's number ! 75 lc=.00000012378 ! lattice parameter of YIG ! 76 sc=32*PI /N/lc/lc/lc ! converts emu into Gauss ! 80 x=0 ! No gallium on "a" sublattice ! 90 y=0 ! No gallium on "d" sublattice ! 100 DISP "Do you want to store the data(y/n)"; 110 INPUT a$ 120 IF a $ = " y " THEN flagl=l ELSE flagl=0 130 ka=x/2 135 kc=0 140 kd=y/3 150 k=1.38062E-16 ! Boltzmann constant ! 160 naa=-(65*(1—1.26*kd)) ! Exchange between "a" and " ■ 170 ndd=-(30.4*(1—.87*ka)) ! Exchange between "d" and " ! 175 ncc=0 ! Exchange between "c" and "c" ! 180 nad=97*(l-.25*ka-.38*kd) 183 nac=-3.44 187 ncd=6 , 190 sa=5/2 ! Spin on "a" site ! 200 sd=5/2 ! Spin on "d" site ! 202 jc=7/2 ! Spin on "c" site ! 210 Zd=(2*sd+1)/2/sd 220 zd=l/2/sd 230 Za=(2*sa+l)/2/sa 240 za=l/2/sa 243 Zc=<2*jc+l)/2/jc 247 zc=l/2/jc 250 ms(4,0)=0 ! ms(4,i) is the temperature ! 260 ms(l,0)=3*gd*sd«b*N*(l-kd)*(l-.l*ka) 270 md(0)=ms(1,0) ! ms(l,i) is the 'd' magnetization ! 280 ms(2,0)=2*ga*sa*b*N*(1-ka)*(l-kd^S.4) 285 m s (3,0)=3*gc*jc*b*N*(1-kc) 290 ma(0)=ms(2,0) ! ms(2,i) is the 'a' magnetization ! 295 m e (0)= m s (3,0) ! ms(3,i) is the 'c' magnetization ! 300 ms(0,0)=ms(1,0)-ms(2,0)-ms(3,0) 310 m t (0)= m s (0,0) ! ms(0,i> is the total magnetization 320 FOR t=l TO 2 ! Loop for the first two increments ! 95 330 ms(4,t)=2.3*t 340 T=2.3*t 350 FOR j=l TO 100 360 ma(j)=ms(2,0)*(FNBra(Za)-FNBra(za)) 370 md (j)=ms(1,0)*(FNBrd 720 m d (0)= m d (j) 730 RETURN 740 -found2: 750 ms(O,t)=mt(j) 760 mt(j)=md(j>—ma(j)—me COMPUTER PROGRAM TO CALCULATE Heff, geff, and H FROM FMR 97 //* // HSGCLASS=A. REGI0N=512K, TIME=1, //PROCLIB DD DSN=HEP.PROCLIB,DISP=SHR /*ROUTE PRINT ROBLAB //SI EXEC FTVCG,PARM.FORT= ’IIOSOURCE, NOMAP’ //FORT.SYSIN DD * C *** PROGRAM FIT5.FORT *** (..VCG,PARM.FORT=‘IIOSOURCE,NOMAP ’) COMMON /ABC/ HR,TR,FRQ,T,MIMS DOUBLE PRECISION X(5) ,FVEC(50) ,V/A(500) ,T0L,HR(5O) ,TR(50) , . FRQ,T,MIMS,HTHEO,HI,HU,H2,MIMEF,PI,SUM.ERR INTEGER IWA(5) EXTERNAL FCN 2 FORMAT (’ PROGRAM FIT5.F0RT *** FIT OF FMR DATA USING’, . ’ EXACT FMR SOLUTION. *** (C) L.PUST ***’/’ VERSION’, . ’ OF ** THURS-DAY ** NOVEMBER 21, 1985, 4:05P’, . ’ ** USING MINPACK PROGRAM LMDIF1’/’ CALCULATING OF K1’, . ’ KU. G.DELFI AND DELTT. EVERYTHING ELSE IS FIXED.’/) M=5 SUM=0.DO PI=3.141S926535898DO READ(5,*) M,FRQ,MIMS,T DO 1 JJ=1,M READ (5,*) TR(JJ),HR(JJ) 1 CONTINUE C ***** X(l) = Kl.CDD (ERG/CM3), X(2) = KU (ERG/CM3), X(3) = G, C X(4) = DELFI (DEC), X(5) = DELTT (DEC) X(l)=~16000.DO X(2)=-5580.DO X(3)=l.97D0 X(4)=l.DO X(5)=1.DO T0L=1.D-6 LWA=500 CALL LMDIF1 (FCN, M, N ,X, FVEC, TOL, INFO, IV/A, V/A, LV/A) C *** Hl=8.D0*PI*X(1)/MIMS HU=8.D0*PI*X(2)/MIMS MIMEF= MIMS-HU V/RITE (6,2) WRITE (6,3) T.FRQ,MIMS,TOL,INFO,X(l),H1,X(2),HU,MIMEF,X(3) . ,X(4),X(5) 3 FORMAT (/’ SAMPLE ** MS33-4-9 ** T =’,F5.0,’ K, ’, . ’ FREQUE’!CY= ’ ,F7 .1, ’ MHZ, 4PIMS = ’ ,F5.0, . ’ G, TOL=’,E8.1, ’ INFO = ’,13///’ BEST FIT IS *** ’, . ’ K1(ERG/CM3) HI(C) KU(ERG/CM3) HU(G)’, . ’ MIMEFF(G) G DEL-FI(DEC)DEL-TT’/20X,5F9.0,F 9 .3,2F9.2 . ///’ N THETA H EXP’, H RES HE-HR’) DO 8 JJ = 1,M HTHEO=HR(JJ)+FVEC(JJ) SUM=SUM+FVEC(JJ)**2 V/RITE (6,66) JJ.TR(JJ) ,HR(JJ) , HTHEO, FVEC (JJ) 66 FORMAT(16,F7.0,3F9.1) 8 CONTINUE ERR=DSQRT(SUM/M) V/RITE (6,4) ERR 4 FORMAT (/’ ******* LEAST SqUARE ERROR IS \F8.2,' G . ’) STOP END C ======&======SUBROUTINE FCN(M.N,X,FVEC,IFLAG) COMMON /ABC/ HR,TR,FRQ,T ,MIMS DOUBLE PRECISION B2.B1,ME.OMG,FHCD,THCD,MIMS,H.PI,K1,K2,FRQ, . KU,HR(50),TR(50),X(5),FVEC(50).MIMEF,ETA,T PI=3.1415926535898D0 K2=0.DO FHCD=X(4) MIMEF=MIMS-8.DO*PI*X(2)/MIMS ME=MIMEF/MIMS OMG=FRq/(l.3996108D0*X(3)*MIMS) B1=PI*X(1)*4.DO/MIMS/MIMS B2=4.D0*PI*K2/MIMS/MIMS C ...... DO S JJ=1,M THCD=TR(JJ)+X(5) CALL REZN(B1,B2,ETA,ME,OMG,THCD,FHCD,MIMS,H) FVEC(JJ)=H-HR(JJ) 5 CONTINUE RETURN END C c ======SUBROUTINE REZN(B1,B2,ETA,ME,OMG,THCD.FHCD,MIMS,H) C **** ALL ONLY DIMENSIONLESS CALCULATION **** DOUBLE PRECISION B2,ETA,ME,OMG,TH,FH,FI,TT,FF,FT,FFF,FTT,FFT, . D4,D5,FHCD,THCD,B1,DLTT,DLFI,FHD,THD,EE,CT2,DS2,DS3,OMGNC,ALF, . FHC,THC,AXH,AYH,AZH,RO,MIMS,H,EPSL,PI . ,CD,DETA,CH,CP,ETCOM,D2,D3,ETAD,AT2,ETAH,EP EE=1.D-6 EPSL=1.D-6 PI=3.1415926535898D0 IERET=0 IERFT=0 IERAL=0 IERV/=0 C *** FROM DEG TO RAD ** THC=THCD*PI/180.DO FHC=FHCD*PI/180.DO C **** DIRECTION COS OF H IN CRYST. AXES DS2=DSqRT(2.DO) DS3=DSqRT(3.DO) AXH=(-DCOS(FHC)/DS2-DSIN(FHC)/DS2/DS3)*DSIN(THC)+DC0S(THC) ./DS3 AYH=(DCOS(FHC)/DS2-DSIN(FHC)/DS2/DS3)+DSIN(THC)+DC0S(THC) . /DS3 AZH=2.DO*DSIN(FHC)*DSIN(THC)/DS2/DS3+DC0S(THC)/DS3 100 RO=DSQRT(AXH*AXH+AYH*AYH) C C *** TRANSFORMATION FROM KART COORD. TO SPHERICAL COORD(FH.TH) C ** CALC OF TH (AT TH=0 END), TH BETWEEN (0 AND PI) IF (DABS(RO).GT.EE) GO TO 110 WRITE(6,3) 3 FORMAT(’ *** TH=0 *** IMPOSSIBLE TO CALCULATE’) GO TO 69 110 IF (DABS(AZH).GT.EE) GO TO 120 TH=PI/2.DO GO TO 140 120 TH=DATAN(RO/AZH) IF (AZH.LT.O.DO) TH=TH+PI C **** CALC OF FH, FH BETWEEN -PI/2 AND +3/2*PI INCL 140 IF (DABS(AXH).LT.EE) GO TO 152 FH=DATAN(AYH/AXH) IF(AXH.LT .0.DO) FH=FH+PI GO TO 170 152 FH=PI/2.DO IF (AYH.LT.O.DO) FH=1.5D0*PI 170 FHD=FH*180.DO/PI THD=TH*180.DO/PI CT2=DC0S(THC)**2 AT2=(21.D0*CT2-1.DO)*(CT2-1.DO) OMGNC=DSQRT(0MG*0MG+AT2*ME*ME/4.DO) ALF=ME/0MGNC/2.DO ETA=ME*(3.D0*CT2-1.DO)/2.DO+OMGNC C ***** BLOCK = CALCULATION OF RESONANCE FIELD C *** ZERO APPROXIMATION FI=FH TT=TH IJ=0 ETCOM=ETA DETA=.4D0 C **** SOLUTION OF THE RESONANCE EQUATION *** FINDS ETA 149 1=0 IJ=IJ+1 IF (IJ.LT.40) GO TO 150 IERAL = 1 GO TO 500 150 ETAD=ETA-DETA ETAH=ETA+DETA 155 ETA=ETAD CALL DER(2,OMGPETA,ME,B1,B2,FI,TT,FH.TH,D4,D5,FT,FF,FFF,FTT,FFT) CD=D5 ETA=ETAH CALL DER(2,OMG,ETA,ME,B1,B2,FI,TT,FH.TH,D4,D5,FT.FF .FFF,FTT,FFT) CH=D5 ETA=ETAH-CH*(ETAH-ETAD)/(CH-CD) IF (ETA.LT.-.1D0) GO TO 69 1=1+1 IF (I.LT.30) GO TO 205 IERET=IERET+1 GO TO 200 205 IF ((CD*CH).GT.O.DO) GO TO 150 CALL DER(2,OMG,ETA,ME,B1 ,B2,FI,TT,FH,TH,D4 ,D5,FT ,FF,FFF,FTT,FFT) CP=D5 IF(DABS(D5).LE.(0MG*0MG*2.DO*EPSL)) GO TO 200 IF ((CD*CP).LE.0.DO) GO TO 160 ETAD=ETA GO TO 155 160 ETAH =ETA GO TO 155 200 IF (DABS(ETCOH-ETA).LT.(ETA*1.5D0*EPSL)) GO TO 500 ETCOM=ETA C C ***** BLOCK - CALC OF EQUILIBRIUM ANGLES FI AMD TT ** J=0 CALL DER(1, OMG.ETA,ME.Bl,B2,FI, TT,FH,TH,D4,D5,FT,FF,FFF,FTT,FFT) 310 J=J+1 IF (J.LT.30) GO TO 350 IERFT=IERFT+1 GO TO 149 350 D2=FT*FFT-FF*FTT D3=FF*FFT-FT*FFF FI=FI+D2/D4 TT=TT+D3/D4 CALL DER(1,OMG,ETA,ME,B1,B2,FI,TT,FH ,TH,D4 ,D5,FT ,FF,FFF,FTT,FFT) DLTT=TT-TH DLFI=FI-FH EP=OMG*EPSL IF ((DABS(FF).GT.EP).OR.(DABS(FT).GT.EP)) GO TO 310 GO TO 149 C C **** COilCLUS101i OF SUBROUTINE ***** 500 CALL DER(1.OMG,ETA,ME,B1.B2,FI,TT,FH ,TH,D4 ,D5,FT ,FF,FFF,FTT,FFT) H=ETA*MIMS 69 RETURN EtID C C ======SUBROUTINE DER(IPAR,OMG,ETA,ME,B1,B2,FI,TT.FH,TH,D4,D5, . FT,FF,FFF,FTT,FFT) C ** IF IPAR=2, IT CALCULATES ONLY THE SECOND DERIVATIVES DOUBLE PRECISION B2,ETA,ME,OMG,TH,FH,FI,TT,FF,FT,FFF,FTT,FFT, . D4,D5,B1,CFI,SFI,CTT,STT,CTH,STH,SFH,CFH,SQ3,Z2,Z3,Z4,Z6 CFI=DCOS(FI) SFI=DSIN(FI) CTT=DCOS(TT) STT=DSIN(TT) CTH=DCOS(TH) STH=DSIN(TH) SQ3=DSQRT(3.DO) CFH=DCOS(FH) SFH=DSIN(FH) Z2=2.DO Z3=3.DO Z4=4.DO Z6=6.DO IF (IPAR.EQ.2) GO TO 90 C C ***** FIRST DERIVATIVES C ** FT *** FT=(Z3*ETA*CTH*STT-Z3*ETA*CTT*SFH*SFI*STH-Z3*ETA*CTT*CFH*CFI .*STH+ME*CTT**2*SFI+ME*CTT**2*CFI+Z2*ME*CTT*SFI*CFI*STT-ME*SFI .*STT**2-ME*CFI*STT**2+Z6*B1*CTT**3*STT+Z2*Z6*B1*CTT*SFI**2*CFI**2 ,*STT**3-Z6*B1*CTT*STT**3+Z2*Z6*B2*CTT**3*SFI***2*CFI**2*STT**5)/Z3 C ** FF *** FF=(STT*(-Z3*ETA*SFH*CFI*STH+Z3*ETA*SFI*CFH*STH-ME*CTT*SFI+ME .*CTT*CFI-ME*SFI**2*STT+ME*CFI**2*STT-Z6*B1*SFI**3*CFI*STT .**3+Z6*Bl*SFI*CFI**3*STTFI**3*STT**3+12,D0*B2*CTT*SFI**3*CFI*STT**5-12.DO .*B2*CTT*SFI*CFI**3*STT**5)/Z3 C C *** JACOBIAIJ D4 *** DEVIATION FROM RES. COND. 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