Notes on early developments in ellipsometry - Leó Kristjánsson

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1 Notes on early developments in ellipsometry - Leó Kristjánsson Introduction This is a brief sketch of over a century of research in physical optics leading to a technique called (since 1945) ellipsometry. This technique has been important in a variety of fields including modern electronics and communication technology. Rhombs of Iceland spar were essential to the experimental part of its earliest development, starting with the discovery of polarization of light in The rhombs were from ~1840 replaced by Nicol prisms, supplemented with other optical elements made from Iceland spar, quartz, muscovite mica, or clear gypsum (selenite). The spar crystals mostly came from Iceland until ~1920. Polarization by reflection In 1808 E.L. Malus discovered the changes which occur in light on reflection from smooth surfaces of singly-refracting solids (like glass) and of liquids. He termed them "polarization". By passage through doubly-refracting crystals such as Iceland spar, a narrow light ray is completely split into two linearly polarized rays. This means that the oscillations in the light in each ray take place only in a single direction. The two directions are also at right angles to each other. Ordinary light is in general partially polarized after being reflected: the reflection coefficient for that component of its oscillations which is parallel to the plane of incidence, is always lower than the coefficient for the oscillationcomponent which is at right angles to that plane. A. Fresnel published in 1821 theoretical equations for both coefficients in the case of incidence from vacuum. See graphs in Fig. 1; on the x-axis is the angle of incidence (measured from the normal to the surface). The equations showed that the parallel component should reach 0 at an angle of incidence i B = tan -1 (n) where n is the material's refractive index. This special case had already been established experimentally by D. Brewster for many insulating materials several years earlier. The changes in the phase of the light oscillations on reflection from say glass or water are either 0 or 180 (Fig. 1). When the incident light is linearly polarized in a direction inclined at any angle from the plane of incidence, the reflected light is also linearly polarized in a different direction, varying with the angle of incidence. This and other evidence agreed well with predictions of Fresnel's equations. They were therefore generally accepted as being correct, although a direct quantitative measurement of the intensities in Fig. 1 had to wait until equipment of sufficient accuracy became available decades later. 1

2 Fig. 1. Left: Reflection coefficients for amplitudes of polarized light from glass at increasing angles of incidence. Negative values indicate a 180 phase change. a" and b" apply to light oscillating in the plane of incidence and perpendicular to it. Right: Intensities for the same reflected components. The curve for the intensity of ordinary light lies midway between these. For a linearly polarized ray impinging on the surface of a material from within, Snell's law of refraction tells us that the sine of the angle of refraction is >1 and its cosine is imaginary. It was known that total reflection takes place for both the components of polarization. In 1823 Fresnel brilliantly interpreted his equations in such a way that the components suffer unequal changes in phase (as also happens when a polarized ray passes through a thin transparent plate of doubly refracting crystal). This he went on to prove by an experiment with a specially cut glass rhomb. The totally-reflected beam is said to be elliptically polarized; this term means that if a light wave is travelling in the z-direction, the projection of the oscillation vector on the xy-plane describes an ellipse (Fig. 2). Fig. 2. Left: The diagram shows two sinusoidal light waves of the same frequency, travelling in the z-direction. As they are plane-polarized in the x- and y-directions and differ in phase, 2

3 they combine to produce elliptically polarized light. Right: J. Jamin's 1850 instrument. Light coming from the right is polarized by a Nicol prism before it is reflected from a surface. An observer analysed the polarization state of the light by means of another Nicol prism and a newly invented pair of movable quartz wedges to compensate for the phase difference in the left diagram. Such wedges, known as a Babinet compensator, were used extensively for many decades. Modified versions of this "grand circle" equipment which could be adapted for studying reflection from liquids, were manufactured commercially until 1910 at least. Puzzling properties of reflected light Researchers attacked successfully the problem of reflection of light from anisotropic crystals, which will not be considered here. Two aspects however turned out to be puzzling for the early researchers. One had to do with surfaces of metals and of semiconductors (like metal sulfides). As might be expected, the reflection coefficients for both the parallel and the perpendicular component of light were high for all angles of incidence on these surfaces. Less expected was the fact that plane-polarized light became elliptically polarized upon reflection from these opaque materials (except when the plane of polarization was also the plane of incidence or at right angles to it). The other puzzling aspect was the fact that Fresnel's laws did not quite fit for insulating materials, even when their surfaces were carefully polished flat. The complete polarization of incident unpolarized light that was anticipated at the Brewster angle i B of Fig. 1, sometimes occurred at a slightly different angle or was not quite achieved. In case of an incoming plane-polarized ray, the reflected ray was also elliptically polarized to a minor extent, an effect which was most marked near the Brewster angle. This circumstance, possible reasons for which will be discussed below, was for ever a drawback in the use of glass plates as practical means of polarizing light. J. Jamin confirmed in 1850 that reflections from liquids also showed similar anomalies. The puzzles in metallic reflection The oblique reflection of polarized light from metals was studied experimentally to some extent by pioneers like Malus around 1810, F. Arago in 1811 (published in 1817) and J.B. Biot in vol. 4 of his Traité de Physique Extensive measurements were reported by D. Brewster in 1830, and in 1840 by H. de Senarmont who developed mathematical formulas for the use of a quarter-wave plate in the analysis of elliptically polarized light. Various types of such plates (often made from mica or gypsum) are still valuable for this purpose. In 1847 de Senarmont also employed a double quartz plate in his study of reflections from semiconducting crystals (e.g. antimony sulfide). 3

4 The matter of reflection from metals was treated theoretically by F.E. Neumann in 1832, by J. MacCullagh in 1837, by the famous mathematician A. Cauchy in papers from 1839 to 1848, and by B. Powell around Both MacCullagh and Cauchy were inspired by Fresnel's interpretation of imaginary numbers in terms of phase changes, so they empirically accounted for the opacity of the metals by adding an imaginary term to the refractive index. This method was placed on a more solid theoretical foundation by J.W. Strutt in Other contributors to the theory of metallic reflection in the mid-19th century included F. Eisenlohr in 1858 (and 1877), and S. Haughton in Fig. 3. The mathematics of elliptical polarization is not simple. These equations describe certain parameters of a light-ellipse obtained by reflection from a metal. Some papers on the subject contain many pages of formulas like these which are taken from a 1928 textbook. Measurements of the polarization state of light reflected from surfaces of transparent solid substances as well as metals were carried out in by J. Jamin. He designed a polarimetric instrument (Fig. 2) for this purpose and he also used a Nicol prism to compare intensities of the light reflected from glass and a metal at the same angle of incidence. He could make his observations agree with empirical formulas based on Cauchy's theory. Similar studies were undertaken by A. Kurz on flint glass in G. Quincke in investigated reflections from metals as well as properties of light transmitted through very thin sheets of gold or other metals. Jamin and Quincke also attempted to find explanations for the colors which are characteristic of some metals. This was pursued further in 1874 by E. Wiedemann who in the process improved de Senarmont's methods of analysing elliptically polarized light. The incomplete agreement with Fresnel's reflection equations Gradually, suggestions emerged to explain why experimental results tended to disagree slightly with Fresnel's equations for the reflection of light from flat insulating isotropic surfaces. Below is a simple list of the present writer's impression of these suggestions: he has not investigated who originated or first tested any of them, or when. - A longitudinal standing wave being excited in the material (in addition to the transmitted transverse wave), taking up a small part of the incoming energy 4

5 - Slight roughness of the surface, of amplitude perhaps reaching a similar order of magnitude as the wavelength of the light - The reflected wave emanating not only from particles at the very surface of the material but from a layer of finite thickness (much less than a wavelength), with a delay in the contribution from deeper parts of that layer - The optical properties of a thin top layer in the material having been altered, by sawing it and polishing the surface with wet abrasives - Permanent strains of mechanical (cf. e.g. the previous suggestion) or thermal origin in the material, making it doubly refracting to some extent - Adsorption of water vapor or other gases from the surroundings L. Lorenz published in 1860 a paper where he applied a general theory to the specific case of Jamin's experiments by assuming the existence of thin (< 1/10th of a wavelength) surface layer on his samples of transparent materials. The index of refraction was supposed to increase gradually through this layer, and Lorenz found that its presence could explain some of Jamin's anomalous results. Obviously, some of the above suggestions do not apply to liquids. However, it was known that small quantities of for instance oily, fatty or soapy substances from the environment around any liquid sample might have got into it before reflection measurements were performed, to form a very thin surface film. The first suggestion in the list had some theoretical justification while the aether was thought to possess elastic properties, but its basis vanished with the advent around 1865 of J.C. Maxwell's electromagnetic theory of light. Many scientists however ignored that theory for the next 30 years or more. H.A. Lorentz was in 1877 the first to derive Fresnel's light-reflection equations for insulators from Maxwell's theory. In the following year he extended his derivations to metals, where the agreement with measurements was only qualitative because the relevant properties of the metals at optical frequencies were not known. Some further studies on reflected light in the late 19th century Scientists continued investigating light reflection in the late 19th century, using polarimetry and also to some extent photometry or interference techniques. Among these scientists were J. Conroy who experimented on metals in and on glass (including effects of polishing) in W. Wernicke published papers of both experimental and theoretical content about insulators (such as glass), metals and semiconductors in , among other things paying attention to the possible presence of surface films caused by polishing or by hygroscopic adsorption. An electromagnetism-based theoretical treatment of reflection from an insulator covered by such a film was carried out in 1883 by 5

6 A.C. v. Rijn v. Alkemade. Their contributions did to some extent explain the observed deviations from Fresnel's reflection formulas. Fig. 4. A. Cornu's equipment for studying reflections from various materials. This redrawn diagram is simplified from his 1889 paper. Visible and near-ultraviolet radiation is produced at E. Light in selected spectral intervals is polarized by the Iceland spar rhomb P. After reflection from the surface M it is analysed by the phase-compensator B and the spar rhomb A. It is then recorded photographically at G. Further work in the 1880s included a theoretical discussion of the optical constants of metals by W. Voigt in Experiments were also carried out on reflected light as well as light transmitted through gratings, thin metal sheets and wedges. Those reported by O. Wiener in 1887, A. Kundt in 1888, G. Meslin in 1888, H. du Bois and H. Rubens in 1890 made use of various methods which mostly did not involve polarization and sometimes produced unsatisfactory results. A. Cornu in 1889 employed a setup (Fig. 4) resembling those of Jamin and of de Senarmont, but incorporating an arc light source and a spectrograph. His measurements on oblique reflections from four insulators and from silver deposited on glass, showed that phase differences between the components parallel and perpendicular to the plane of incidence increased in all cases with decreasing wavelength, for visible light and into the ultraviolet range. This result agreed with theoretical considerations presented by A. Potier in 1872, rather than with Cauchy's idea of a constant phase difference for each material. Drude enters the field P. Drude (Fig. 5) was a student of W. Voigt, who in turn had been a student of F.E. Neumann. Drude was a very gifted and productive physicist, whose work in his tragically short career mostly concerned theoretical and experimental optics. Several papers by him in deal with the polarization of light reflected or refracted by insulators (including Iceland spar), metals, and thin films. In "Ueber Oberflächenschichten" (Fig. 6) in he presents in a general form what is often called the fundamental equation of ellipsometry. It is tan e i where 6

7 is the ratio between reflection coefficients for light with parallel and perpendicular polarization. and are parameters (Fig. 3) describing the shape of the polarization ellipse (Fig. 2). They depend on properties of the materials involved, the wavelength of the light, and the angle of incidence. Voigt had already applied this equation to a specific theoretical situation in Rather than being a gradual transition from one material to the other as assumed by earlier investigators, the layers of Voigt and Drude could have optical properties quite different from both. The thicknesses of the films were supposed to be a small fraction of the wavelength of the light. Drude's papers on reflection in the above interval were based on the elastic-aether theory, but in his 1900 textbook of optics he treats this and related subjects in terms of Maxwell's electromagnetism. Fig. 5. Left: Paul Drude ( ). Right: Drude's instrument for measuring the changes taking place in polarized light upon reflection from S (from his Lehrbuch der Optik, 1900). 7

8 Fig. 6. Beginning of Drude's 1888 paper on surface layers in the Göttinger Nachrichten. It is not intended to describe here the theory of Drude, but it prompted research by others into the optical properties of metals and thin films. Thus, Rayleigh showed in 1892 that Jamin's results on reflection from liquids had been severely affected by surface layers. R.S. Minor studied metallic reflection of ultraviolet light in 1903, and G. Betz, H. Fritze and B. Pogany carried out measurements on thin metallic layers in 1905, 1915 and 1916 respectively. R. Kynast observed phase changes in light reflected from coated insulators in 1907, H.v. Wartenberg measured the optical constants of several metals at high temperatures in 1910, and H. Hauschild investigated the effects of thin insulating films on metal reflections in (published in 1920). All these scientists employed polarimetry. The instrumentation for such measurements kept improving, for instance with the introduction of new spectral lamps and new accessories for analysing elliptically polarized light. A book by R.C. MacLaurin in 1908 presents a general theory of some aspects of ellipsometry, and L.B. Tuckerman published in 1909 a thorough mathematical treatment of the available phase-compensators and half-shade devices, Progress was also made in different experimental approaches to the field of reflection: as an example, H. Rubens and his collaborators carried out valuable work on the reflection of infrared radiation from metals and insulating crystals, using thermo-electrical sensors. Research on thin films from around 1930 The above research efforts seem to have been mostly undertaken as typically pure science, without specific practical aims. Not much seems to have happened in the field of thin films between from WW I to around Occasional papers appeared on polarimetric measurements of reflections from various metals, e.g. by R.F. Miller in 1925, G. Pfestorf in 1926 and J. Ellerbroek in The latter two mention the effects of surface contamination on results. L. Tronstad and his collaborators used polarimetry in in studies of the so-called passive oxide films that are deposited on metals in contact with air or electrolytes. They later investigated for instance fatty films on mercury, finding good agreement with Drude's theory. A.B. Winterbottom of this group presented in 1937 a new instrument where the light incident on a metal surface could be made elliptically polarized in such a way that the reflected light would be planepolarized. Readings were taken at several wavelengths and several angles of incidence, as single values of these variables will not provide sufficient information about the optical characteristics of a surface film and its substrate. 8

9 Fig. 7. Left: A simple diagram to explain ellipsometry, showing polarization states. Right: A modern desk-top spectroscopic ellipsometer. Both are taken from Google Images. This field of research soon expanded rapidly, because polarimetric analysis enabled people to measure the properties of very thin films rapidly and more accurately than any other available method. This technique has been of great importance in solid state physics, electronics and photonics, which depend much on surface phenomena and finely layered structures; an article in the Physics Today magazine (May 2009) claims that ellipsometry speeded up the development of modern computer technology by 15 years. Manufacturers offer nowadays many models of automated ellipsometers (Fig. 7), which can measure non-absorbing surface films of thickness from less than 1 nm to over 10 µm. Dichroic sheets are probably used nowadays in the production and analysis of polarized light. However, review papers on the subject of ellipsometry always acknowledge the fundamental contributions of Paul Drude and other pioneers in their research where Iceland spar was indispensable Completed in Dec. 2015, partly from sect of the report "Iceland spar and its influence in the development of science and technology in the period ", 4th ed Many references can be found there and in "Selected Papers on Ellipsometry" ed. R.M.A. Azzam, published by SPIE in A paper by A.V. Hall in Surface Science vol was particularly useful. Rósa Ólafsdóttir re-drafted Fig. 4. 9