PHYSICAL REVI E% B VOLUME 14, NUMBER 12 Thermoreflectance of white tin 15 DECEMBER 1976 C. Carotenuto, E. Colavita, S. Modesti, and M. Sommacal Istituto di Fisica dell'Universita di Roma, Rome, Italy (Received 5 January 1976) Thermoreflectivity measurements on thin films of white tin have been carried out in the 0.7-4.0-eV photon energy region at room temperature. The technique has allowed a good resolution of the optical structure arising from interband transitions involving the Fermi surface. Contributions to the modulated absorption were located at 0.83, 0.97âand 1.65 eV in good agreement with previous results of static reflectivity measurements. INTRODUCTION EXPERIMENTAL Modulation techniques, in which a parameter (temperature, field, etc. ) is varied periodicallyâ thus introducing a periodic component in the optical response of a material, have allowed quite good results in the understanding of the optical spectra of semiconductors. ' ' Applications of these meth- ods to metals have been extremely fruitful also4 ' even though comparatively much less common. The noble metals have been thoroughly investigated and, recently, thermomodulation results have also been obtained for transition metals. Very little effort has been put into using modulation techniques for studying the polyvalent simple metals, ' "and, despite their "simplicity, " the understanding of their electronic structure lags somewhat behind (especially so a few electron volts away from the Fermi surface). Several investigations of the static optical prop- erties of white tin have been carried out, " "some of them up to very high photon energies. Most of them, however, fail to reveal much structure in the low-energy region which is the most interesting for checking the existing band-structure calcula- tions. " ~0 The most complete results have been obtained by Schwarz" both on oriented single crys- tals and evaporated thin films at different tempera- tures. His spectra show several structures in the 0.5-5 eV photon energy region which were attrib- uted to features of Craven's'9 band-structure cal- culation. Several discrepancies were found how- ever. Thermomodulation experiments in metals can give valuable information especially on the loca- tion of optical transitions which either start from or end on the Fermi surface, "'"and on the ex- istence of critical points, "especially so when the deformation potential of the pair of bands in- volved is large. It was felt, therefore, that thermomodulation spectra could improve the understanding of the electronic structure of white tin and help resolve previous discrepancies. Opaque films of Sn were obtained by evaporating 99.999%%uo-pure tin in a 10 '-Torr vacuum onto a highly polished room-temperature quartz sub- strate. In order to obtain a well-crystallized sur- face, a very high evaporation rate had to be used" (-200A/sec). Before introduction in the vacuum system, the substrates were accurately cleaned using standard procedures and rinsed in deionized water and alcohol. A stainless-steel mask was used in front of the substrate so that the resulting sample was about 20X2 mm . The samples were roughly 2000 A thick and copper leads were indium soldered at each end to provide electrical con- tacts. The temperature modulation was obtained by a pulsating current of frequency 5 Hz dissipating about 0.8% inside the sample. The optical sys- tem was conventional. ' The monochromatic bght from a 0330 Hilger-%'atts monochromator im- pinged at near-normal incidence (e. , ~ 9 ) on the sample and was refocused on a 1P28 HCA photo- multiplier for the measurements between 2.0 and 4.0 eV; below about 2 eV, a cooled PbS detector was used. The output of the detector was fed to a lock-in amplifier and digital integration of the output was used. Integration times up to several hundred seconds were used in order to insure that the statistical uncertainty in aR/R would always be below about 31 over the whole energy range. To avoid excessive oxidation, the films were not annealed. RESULTS AND DISCUSSION The experimental spectrum of aR/R for an un- annealed sample is plotted in Fig. 1. The results have been Kramers-Kronig analyzed" in order to obtain the variation of the phase shift of the elec- tric field 4(9. The variation of the imaginary part of the dielectric constant ~&, has been evaluated taking &, from Schwarz's" data on white-tin film 14 5328 CAROTENUTO, COLAVITA, MODESTI, AND SOMMACAL 14 10's JR 20â 10â -10â -20â 0 4.5 ~~(eV) FIG. 1. Thermoreflec- tance spectrum of white tin. Dashed lines repre- sent the extrapolations used in the Kramers- Kronig analysis. -40â -50â 1,0-(e.u.j (I I 0.5- I J I t ( I I ( I I I GO \ \ 1.0 I 30 %u(eV) FIG. 2. Variation of the imaginary part of the dielec- tric constant of white tin. The dashed lines show a tentative separation of the 6e& spectrum into three sep- arate contributions. at 295'K; &, has been obtained combining the data of Golovashkin et al." and MacRae et al." Figure 2 reports Ac, and a tentative decomposi- tion of the leading structures between about 0.6 and 2.5 eV into three separate contributions. Each of these shows the well-known and characteristic line shape of Fermi-surface-involving optical transitions'" thus confirming their important role in thermomodulation spectra of metals. A modulated free-carrier contribution may be present in our data, but we cannot evaluate its amount at present. Further data below the edge of interband transition would be needed to clarify this point. The absorption edge falls approximately between the first peak and the zero crossing of each struc- ture" and would occur according to our decomposi- tion at 0.83, 0.97, and 1.65 eV. Further structures in Ae, not involving Fermi-surface transitions are present at about 3.7 eV. The band structure of Sn as calculated by Craven" is shown in Fig. 3. Our first two struc- tures are certainly to be related to the very strong absorption peaks which Schwarz" finds in his 6, spectra at about 1.15 and 1.30 eV for both light polarizations (parallel and perpendicular to the tetragonal axis). He interprets the first peak as arising from parallel band absorption between the third a,nd fourth bands near & for light polarized parallel to the tetragonal axis (E ~~ Cj. According to the selection rules this transition is instead forbidden for E&C, so that the peak occurring with this light polarization was interpreted as due to transitions from the fourth band to the fifth and sixth bands near W. The second peak in Schwarz's data was interpreted again as parallel band absorp- tion occurring between the fifth and sixth bands along PV. These transitions a.re allowed for both light polarizations so that the origin of the peak should be the same in both spectra. . Recently Ament and de Vroomen" have performed 14 THERMOREFLECTANCE OF WHITE TIN 5329 1.2 1.0 0.8 = 0.6 0.4 P~+ P5) P7 0.2 0 r W LX FIG. 3. Band structure of white tin as calculated by Craven (Ref. 19). a relativistic APW band-structure calculation and using a linear extrapolation method have also cal- culated the joint density of states. According to their calculations, only the partial joint density of states arising from the fourth and fifth bands has a pronounced structure near 1 eV. They conclude accordingly that this pair of band (in the region near 1'W} is responsible for the strong low-energy absorption for both light polarizations. Since our sample is a polycrystalline film with the crystallites randomly oriented, we ascribe all the possible transitions at each photon energy, independently from the selection rules associated with different light polarization states. Our L&, thermomodulation spectrum confirms that the ab- sorption near 1 eV is indeed a twin structure as already found by Schwarz, and both the absorptions originate from Fermi-surface-involving interband transitions. The origin in K space of these absorp- tions however remains somewhat puzzling: The Ament and de Vroomen calculation accounts for only one absorption peak in this energy region and the fifth and sixth bands along PP (to which Schwarz attributed the origin of the second ab- sorption peak) are separated by more than 2 eV in the band- structure calculations. Allowing for nonperfect parallelism of the third and fourth bands along the I'X direction, the trans- itions marked by arrows in Fig. 3 occur at slightly different energies. It may be surmised, therefore, that the two main contributions to our thermomod- ulation spectrum may both arise from the same pair of bands. It is, however, recognized that at least a imple model calculation would be needed to corroborate our suggestion. It should be re- marked also that our proposed explanation does not account for the origin of the second peak in the Schwarz E & t." spectrum. Further efforts, both theoretical and experimental, seem to be needed to clarify this point. Our third resolved structure indicates the ex- istence of a third Fermi-surface-involving inter- band transition with the absorption edge located at about 1.65 eV. This is certainly to be related to the broad absorption band that Schwarz finds in his parallel polarization absorption spectrum and attributed to transitions between the fourth and fifth bands along I'X. According to Ament and de Vroomen, the most likely candidates are transi- tions between the third and fifth bands whose par- tial joint density of states shows a jump at the cor- rect energy. Our data cannot discriminate the two proposed explanations since both are interband transitions involving the Fermi surface. Our last structure is somewhat washed out (espe- cially in the ae, spectrum), but its line shape ex- cludes transitions involving EF and seems instead to indicate the presence of a critical point located at about 3.75 eV. The transition between the fourth and fifth bands at I' is a very likely candidate to explain this absorption and similar conclusions are also drawn by Schwarz to explain the structure observed at this photon energy. It should be noted, however, that this conclusion is at variance with 5330 CAROTENUTO, COLA VITA, MODESTI, AND SOMMACAL 14 Ament and de Vroomen" who assign the structure to transitions between the fifth and seventh bands near W. CONC LU SION S We have shown that thermomodulation spectra, even when carried out at room temperature, of polycrystalline films of anisotropic materials like tin, can be very helpful in elucidating aspects of their electronic structure. ACKNOWLEDGMENTS We are greatly indebted to Dr. R. Rosei for help- ful discussions and encouragement during the whole course of this work. The technical help of M. Pacifici is also gratefully acknowledged. 'M. Cardona, in Solid State Physics, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1969), Suppl. 11. Semiconductors and Semimetals, edited by R. K. Wil- lardson and A. Beer (Academic, New York, 1972), Vol. 9. 3Proceedings of the First International Conference on Modulation Spectroscopy, Surface Science Vol. 37. U. Gerhardt, Phys. Bev. 172, 651 (1968). W. J. Scouler, Phys. Rev. Lett. 18, 455 (1967). 6C. G. Olson, M. Piacentini, and D. W. Lynch, Phys. Rev. Lett. 33, 644 (1974). R. Rosei, C. H. Culp, and J. H. Weaver, Phys. Bev. B 10, 484 (1974). R. Rosei and D. W. Lynch, Phys. Rev. B 5, 3883 (1972). ~A. I. Golovashkin, K. V. Mitsen, and G. P. Motulevich, Fiz. Tverd. Tela 14, 1704 (1972) [Sov. Phys. -Solid State 14, 1467 (1972)]~ ' J. H. Weaver, D. W. Lynch, and R. Rosei, Phys. Rev. B 5, 2829 (1972). "A. I. Golovashkin and G. M. Motulevich, Zh. Eksp. Teor. Fiz. 47, 64 (1964) [Sov. Phys. -JETP 20, 44 (1965)]. '~H. R. Hunter, in OPtical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles (North-Holland, Amsterdam, 1966), p. 136. '3B. A. MacRae, E. T. Arakawa, and M. W. Williams, Phys. Rev. 162, 616 (1967). B. Haensel, C. Kunz, T. Sasaki, and B. Sonntag, Appl. Opt. 7, 301 (1968). 5R. H. W. Graves and A. P. Lenham, J. Opt. Soc. Am. 58, 884 (1968). ' H. Schwarz, Phys. Status Solidi B 44, 603 (1971). 'YG. Weisz, Phys. Rev. 149, 504 (1966). ' M. D. Stafleu and A. B. De Vroomen, Phys. Status Solidi 23, 683 (1967). S. E. Craven, Phys. Bev. 182, 693 (1969). R. Rosei, Phys. Rev. B 10, 474 (1974). 'J. C. Lemonnier and S. Robin, C. R. Acad. Sci. (France) B 265, 661 (1967). The calculation was performed with the same computer program used in a recent paper by A. Balzarotti et al . [Appl. Opt. 14, 2412 (1975)], dealing with errors in- troduced by Kramers- Kronig analysis. A careful analysis of the origin of the various structures in 6&2 (and in particular of the twin structure at about 0.85 eV) makes us confident that they are real and not an artifact of the calculation, even though their amplitude is somewhat dependent on the extrapolations used. M. A. F. . A. Ament and A. B. De Vroomen, J. Phys. F 4, 1359 (1974).