Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1969 Optical properties of silver-palladium alloys Bruno Francis Schmidt Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Condensed Matter Physics Commons Recommended Citation Schmidt, Bruno Francis, "Optical properties of silver-palladium alloys " (1969). Retrospective Theses and Dissertations. 4149. https://lib.dr.iastate.edu/rtd/4149 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 70-13,629 SCHMIDT, Bruno Francis, 1942- OPTICAL PROPERTIES OF SILVER-PALLADIUM ALLOYS. Iowa State University, Ph.D., 1969 Physics, solid state University Microfilms, Inc., Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED OPTICAL PROPERTIES OF SILVSR-PALIADIUM ALLOYS by Bruno Francis Schmidt A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Solid State Physics Signature was redacted for privacy. In Charge of Ha^or Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University Ames, Iowa 1969 11 TABLE OF CONTENTS Page INTRODUCTION 1 , Preliminary Reinarks 1 Optical Constants 2 Free-Electron Gas 7 Electrons In a Real Solid 10 Band Structure of the Noble Metals 18 Band Structure of the Noble Metal Alloys 24 Band Structure of the Noble Metals Under Strain 29 EXPERIMENTAL PROCEDURE 38 Sample Preparation 38 Optical Measurements 43 Analysis of Data 4? RESULTS 53 Presentation of Results 53 Discussion of Results 55 SUMMARY 81 LITERATURE CITED 82 ACKNOWLEDGEMENTS 86 APPENDICES 87 Appendix A. R and T as Functions of n and k 88 Appendix B, Kramers-Kronig Analysis of T 93 Appendix C. Kramers-Kronlg Analysis of AR/R 102 1 INTRODUCTION Preliminary Remarks In the past decade there has been considerable interest in the electronic band structure of tne noble metals. Optical studies (along with other experimental work) have been useful in verifying theoretical band calculations as well as suggest­ ing improvements to be made in the calculations. In the present work the optical properties of silver and silver-palladium alloys were studied in the visible and ultra­ violet, where three basic types of electron excitations in silver occur; individual free-electron-like excitations, collective oscillations, and interband transitions, k fourth type of excitation of electrons in localized impurity levels of alloys is also possible. In silver the three types of excitations are reasonably well separated in energy. However, the interband transitions do affect the collective oscilla­ tions, In addition, there seems to be more than one interband transition occurring at about the same energy. The purpose of the present work is to learn more about these various over­ lapping excitations by altering slightly the energy bands in silver. Two techniques were used. First, the bands were altered by alloying palladium with silver (introducing the possibility of the fourth type of excitation). Information about the energy bands and the different excitations was gained by 2 measuring the reflection and transmission of thin films of silver and silver-palladium alloys. Second, the bands were changed by straining the films of silver and the alloys. The fractional change in reflectivity of the strained films was measured and this was related to the band structure. Optical Constants To gain knowledge of the electronic behavior of a materi­ al through its optical properties, it is necessary to use Maxwell*s equations to obtain certain optical constants from the measured optical behavior. These optical constants can then be related to the electronic behavior of the material. The photon energy range we consider is high enough so that only electrons contribute to the optical characteristics; lattice contributions and magnetic effects are insignificant. For a non-charged, non-magnetic, isotropic, homogeneous dielectric, the conduction current J le zero and Maxwell's equations can be written V • B = 0, (1-1) e ^ . E = 0, (1-2) (1-3) ^ X B = ~ , (1-4) C 3t where E and B are the electric and magnetic fields and e is 3 the dielectric constant which relates the displacement vector D to E according to the equation D = ef. (1-5) The standard solutions of these equations are transverse plane waves (except, perhaps, for a static, constant term that is not important) and are written as f = Eo e^'5 • r - lot)^ (i_6) B = S el(g - r - (1-7) q is in the direction of propagation and has the magni­ tude 2TTA, X "being the wavelength of the radiation in the dielectric, u) is the angular frequency, and and BQ deter­ mine the amplitudes of the waves, E, B, and "q are mutually perpendicular with B = g q X E. (1-8) Now by substituting these solutions into Maxwell's equa­ tions, one can obtain the relation , =^ . (1-9) Then it is clear that = n , (1-10) where n is the standard index of refraction, the ratio of the radiation wavelength in vacuum to that in the dielectric. Now for metals where 3^ is no longer zero, the standard procedure is to introduce the complex dielectric constant c. 4 defined so that Maxwell's fourth equation becomes ^ X B = i ^ . (1-11) C Bt Solutions for metals are thus identical to those for dielectrics, with s replaced by e. Since e is clearly com­ plex, Equation (1-9) must be replaced by VT = R = n + ik, (1-12) where n is called the complex index of refraction and n and k are real, n tJtill is the ratio between the vacuum and metal wavelength and k gives the damping of the wave as it passes through the metal, €, being complex, will be written as e = + iGg. (1-13) From Equation (1-12) then, = n^ - k^, and (1-14) Gg = 2nk. (1-15) A more detailed development of the preceding relations can be found in many discussions of optical properties (1,2). Most measured optical characteristics of a material are expressed as functions of n and k. Equations (l-l4) and (1- 15) show how then to obtain values of which can more easily be used to discuss electronic behavior than can n. The physical significance of ê will now be investigated. First, one of Maxwell's equations is rewritten. Since B = H, Equation (1-11) can be written 5 ÎT = B - 1 SB = 15:11 sus c at c 3t By taking the dot product of both sides of Equation (1-3) with C/^TT B and the dot product of both sides of Equation (l-l6) with -C/kn E, adding, and making simple vector manipu­ lations, one obtains ov'(ÊxB) + a E2+B2_ (6-1) a , ag ,, — * St W ® Now the first term is the rate of flow of the field energy density of an electromagnetic wave; the second term is the rate of increase of the field energy density. By energy conservation» the right hand term must be the rate of transfer of energy density from the electromagnetic field to the medium. To find the average rate of energy transfer, one wants to find the real part of the time average of the right hand term of Equation (1-17), Using the standard relation that Real <I ' B> = i Real (k • %, (1-18) where < > denotes time average, one obtains for the average rate of energy density absorption, 3W/Bt, the expression II = I Real ( « ^ E . If ). (1-19) Since S has an e"^^^ time dependence, # = . (1-20) Thus («@2 is a measure of the rate of energy absorption. Equation (1-20) is a valid expression for the energy loss 6 of a transverse electromagnetic field, with E perpendicular to the propagation vector q. However, for a longitudinal field, such as that associated with a moving charged particle, it can be shown (3) that the rate of energy transfer in an infinite medium is proportional to -Im(l/ë), which is commonly called the loss function, A simple method of deriving the loss function is to re­ write Equation (1-19) (4). E is the fundamental quantity for transverse waves. However, for a charged particle passing through a medium, D is the field associated with that particle from the equation V . D = 4np , (1-21) where p is the charge density of the particle. Thus, when discussing longitudinal fields, such as those due to moving charges. Equation (1-19) should be written |ï = è HeaK- f • fi li'- (1-22) or M ^ I D„|2 . (1.23) Although it will be assumed that the transverse waves used in this experiment will not be absorbed by mechanisms that absorb only longitudinal perturbations, it is still pos­ sible to jLearn something about such mechanisms by calculating the loss function from optical data. Doing this requires that ^ be the same for both transverse and longitudinal fields. 7 However, for long wavelength fields (such as visible light) the response is essentially the same, and one K is adequate to describe the system (5)- This can be made plausible by the following argument; for long wavelengths, q % 0, and it be­ comes impossible to distinguish between vectors perpendicular . and parallel to q, Free-Electron Gas It is instructive to consider first the electrons in a metal in terms of a free-electron gas, which is a collection of point electrons imbedded in a uniform sea of positive charge so that the net charge is zero.