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This is to certify that the thesis entitled RAMAN SCATTERING FROM TWO—PHONON EXCITATIONS AND FROM NITROGEN ELECTRONIC IMPURITY LEVELS IN CUBIC SILICON CARBIDE presented by Philip Allen Gaubis has been accepted towards fulfillment of the requirements for Ph.D. (19‘,er in Physics A U. fl Major pro ssor Date W7 0—7639 ‘\ RAMAN SCATTERING FROM TWO—PHONON EXCITATIONS AND FROM NITROGEN ELECTRONIC IMPURITY LEVELS IN CUBIC SILICON CARBIDE By Philip Allen Gaubis AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1977 A”) 6:2)? ABSTRACT RAMAN SCATTERING FROM TWO—PHONON EXCITATIONS AND FROM NITROGEN ELECTRONIC IMPURITY LEVELS IN CUBIC SILICON CARBIDE By Philip Allen Gaubis The two—phonon Raman spectra and nitrogen donor electronic Raman spectra of the 3C polytype (zinc—blende) of 810 are presented. 30 SiC is a IV—IV semiconductor having an indirect gap of 2.U eV with a conduction band minimum at X. The nitrogen impurity concentrations of both samples used in this study are found to be well below the Mott transition. The P component of the two—phonon spectra is the l dominant component. A number of spectral features are identified and assigned to critical point overtones and combinations, including overtones due to critical points 1 and TO(Z) at 737 cm_l, which had not TA(Z) at M76 cm- previously been measured. No difference combination scat— tering is observed, in accord with intensity estimates for difference combination bands in 3C 810. All feature assignments are consistent with theoretical selection rules and previously measured phonon energies. The close Philip Allen Guabis agreement of branch energies at X with previous luminescence measurements supports the assignment of the conduction band minimum to X. Sample nitrogen concentration, surface damage due to sample preparation, and exciting line wavelength are found to have negligible effects on the second-order spectra. The spectral feature assignments in conjunction with a lat- tice dynamical calculation are used to perform a critical point analysis, producing a set of critical point sector numbers which satisfy the two- and three-dimensional Morse relations. The low temperature spectra of both samples show the appearance of a line at 67.5 cm_1 of E symmetry. The thermal behavior, symmetry, concentration dependence, and theoretical intensity estimates strongly favor the identi- fication of this line as a lS(Al)+lS(E) transition in the valley-orbit split 18 nitrogen donor level. The thermal behavior of this line is investigated from 7°K to 180°K and indicates that the lS(Al) level is the true ground state. No other transitions are found, in accord with theoretical intensity estimates for the inter-manifold transition intensities and with the assignment of the conduction band minimum to X. ACKNOWLEDGMENTS The author would like to give special thanks to Professor P. J. Colwell whose patience and valuable direction made this project possible. Thanks are also due to Dr. w. J. Choyke of the Westinghouse Research and Development Center who kindly supplied the samples used in this study along with several useful preprints of his publications. The author gratefully acknowledges many instructive conversations with Dr. L. A. Rahn concerning silicon carbide and Raman scattering in general. ii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . iv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . V Chapter I. INTRODUCTION . . . . . . . . . . . . . . . . . 1 II. TWO—PHONON RAMAN SPECTRA OF 3C SiC . . . . . . 5 A. Background . . . . . . . . . . . . . . . . 5 B. Phonon Symmetries . . . 10 C. First— Order Polarization Selection Rules . 13 D. L0 and TO Phonon Selection Rules . . . . . 18 E. Raman Intensity Matrices . . . . . . 19 F. Critical Points . . . . . . . . . . . . . . 21 G. Two-Phonon Scattering . . . . . . . . . . . 29 H. Experimental . . . . . . . . . . . . . . . 3“ I. Results . . . . . . . . . . . . . . . . . . 39 J. Analysis . . . . . . . . . . . . . . . . . 51 K. Discussion . . . . . . . . . . . . . . . . 62 III. RAMAN SCATTERING FROM NITROGEN ELECTRONIC IMPURITY LEVELS IN 3C SiC . . . . . . . . . . . 66 A. Background . . . . . . . . . . . . . . . . 66 B. Donor Levels . . . . . . . . . . . . . . . 69 C. Raman Scattering . . . . . . . . . . 75 D. Experimental . . . . . . . . . . . . . 77 E. Results . . . . . . . . . . . . . . . . 78 F. Discussion . . . . . . . . . . . 83 IV. SUMMARY . . . . . . . . . . . . . . . . . . . . 88 A. Two-Phonon Spectra . . . . . . . . . . . . 88 B. Nitrogen Donor Spectra . . . . . . . . . . 9O Appendix Page A. EXPERIMENTAL GEOMETRY AND EQUIPMENT . . . . . . 91 B. ZINC—BLENDE INTENSITY MATRICES . . . . . . . . 95 C. THROUGHPUT AND WAVELENGTH CALIBRATION . . . . . 98 D. SECOND—NEIGHBOR IONIC (SNI) CALCULATION . . . . 106 REFERENCES......................109 iv Table 10. ll. l2. 13. 14. LIST OF TABLES Experimental lattice energies (cm—1) Decomposition of the polarizability tensor and Raman tensors for point group Td Zinc-blende intensity matrices Critical point topological index and weight Zone-center two-phonon representations for 3C 810 Thermal intensity ratios of two—phonon Raman spectra in 30 81C . . . . . Energies and feature assignments from the two-phonon Raman spectrum of 3C SiC . . . . Critical point analysis of 3C 810 Donor level symmetries . . . . . . . . . BC SiC nitrogen donor level spacing Zinc-blende backscatter intensity matrices Zinc—blende right-angle intensity matrices Neon calibration lines SNI parameters for 30 SiC Page 10 I6 20 28 33 ”9 56 61 71: 86 96 97 104 107 Figure 10. ll. 12. 13. LIST OF FIGURES Brillouin zone of 3C SiC and wavevector groups . . . . . . . . . . . . . . . . . . . Critical point density of states shapes . . X(ZZ)X spectrum of sample A, T— 77° K, .4579A laser line . . . . . . . . X(ZY)X spectrum of sample A, T: 77° K, A579A laser line . . . . . . . . . x(Y'§')x spectrum of sample A, T277° K, 4579A laser line . . . . . . . . . . . . . . X(Z'Z')X spectrum of sample A, T277o K, A579A laser line . . . . . . . . . . . . . X(Z'Z')X spectrum of sample B, T277° K A765A laser line . . . . . . . . . . . . . . X(Z'Z')X spectrum of sample A, T=300° K, 6471A laser line . . . . . . . . . . . . Growth face spectrum of sample A, T— 300° K, A579A laser line . . . . . . . . Comparison of second— order spectrum with calculated two- -phonon density of states of Reference 24 . . . . . . . . . . . . . X(ZZ)X spectra of sample A, A579A laser line . . . . . . . . . . . . . . Sample A spectra at T- 7° K, A765A laser line . . . . . . . . . . . . . Nitrogen donor line width and intensity temperature dependence . . . . . . . . . vi Page 12 26 NO 41 H2 43 “5 52 58 79 81 82 Figure I“. 15. l6. l7. l8. Raman backscattering geometry . . . System throughput measurement geometry System throughput, T(X) Wavelength calibration spectrum Second neighbor ionic (SNI) calculated dispersion curves for 3C SiC vii Page 92 100 102 103 108 CHAPTER I INTRODUCTION Silicon carbide is an indirect gap IV—IV semiconductor which can grow in a variety of different polytypes. The 30 or cubic polytype used in this inves— tigation has a zinc—blende structure with two atoms per primitive cell and space group T2 (FE3m). This structure d can be visualized as two interpenetrating face—centered cubic lattices, one of silicon and one of carbon, which are separated along one—quarter of a body diagonal. The zinc—blende structure is nothing more than the diamond structure with every other carbon replaced by a silicon. From this point of View, cubic SiC may be considered as a perturbed silicon lattice. $10 is of interest in that it may be considered to exist "between" the purely covalent IV-IV semiconductors (such as silicon and germanium) and the more ionic III—V zinc-blende semiconductors. This study is concerned with the first observation and analysis of two—phonon Raman host crystal spectra and Raman scattering from nitrogen donor electronic levels in 30 SiC. Since the topics of two-phonon spectra and donor spectra are not closely related, this thesis is organized into two nearly independent sections. The analysis of the two-phonon spectra presented in Chapter II yields detailed information concerning host crystal phonon dispersion curves, including a set of dispersion curve critical points. Nitrogen donor Raman spectra are reported in Chapter III and discussed in terms of effective mass theory (EMT) as it applies to weakly bound donor electrons. Relevant background is included in each chapter. Group theory is employed where possible, as it often gives clearer insight than brute force calculation. Many of the experimental and theoretical details of this study are deferred to the Appendices. The remainder of this section includes a very brief description of the Raman effect and its experimental utility. The Raman process may be more descriptively referred to as inelastic light scattering, wherein the incident light exchanges energy with a sample concurrent with a transition between quantum levels in the sample. The energy of the scattered light will be shifted by an amount equal to the difference in energy of the quantum levels involved in the transition. The scattered light will be shifted toward the violet if it gains energy during the interaction, and toward the red if it loses energy. The violet shifted light is referred to as anti—Stokes scattering while the red shifted light is called Stokes scattering. (All spectra throughout this study are Stokes spectra, with energy scales representing the energy shift between the incident and scattered light.) The scattered light intensity of a Raman process is quite weak (typically lO_8 to 10"12 of the incident intensity), and early experiments suffered from the lack of suitably intense, narrow band sources. Lasers are ideal sources for Raman spectroscopy, having highly monochromatic outputs of several watts, and are directly responsible for the widespread use of Raman techniques. A great variety of phenomena in semiconductors have been studied using Raman scattering, including one—phonon processes (called first—order scattering), two—phonon processes (called second—order scattering), donor and acceptor impurity levels, magnons, plasmons, defect modes, single-particle excitations, impurity—induced scattering, resonant processes, LO—plasmon coupling, and others. This list is by no means complete, and is intended only to show the broad applicability of Raman techniques to semiconductor studies. Infrared absorption spectra and Raman spectra can be complementary in the sense that selection rules are often different. For example, in materials with inversion sym- metry, where parity is a good quantum number, only odd- parity transitions are infrared—active. Conversely, in such systems only even—parity transitions are Raman—active, so Raman and absorption spectra are strictly complementary. In addition, Raman techniques can easily detect transitions of energy as low as ~l meV, which lie in an experimentally difficult region for absorption techniques. CHAPTER II TWO—PHONON RAMAN SPECTRA OF 3C 810 A. Background This chapter presents the two-phonon (second—order) Raman spectra of 3C SiC, an identification of spectral features, and a critical point analysis of lattice disper— sion curves. Several theoretical topics require development before the observed spectra can be interpreted. In this section, a preliminary overview of Raman scattering from lattice vibrations is presented, followed by a description of previous measurements of phonon energies in 30 SiC. In one—phonon or first—order Stokes Raman scat— tering, an incident photon of wavevector Ki creates a phonon of wavevector Rp and is scattered with wavevector RS. Wave— vector conservation requires that Ri—RS=E In a typical p' semiconductor, the wavevector of visible light is of order 5 10 cm“1 (depending on the dielectric constant of the cry— stal) while the width of the Brillouin zone is of order 108 cm—l. Thus the only phonons which can participate in first—order Raman scattering have a wavevector very near the center of the Brillouin zone and are therefore + referred to as "zone—center" or "k=O" phonons. This scattering process can be roughly visualized in a semiclassical fashion. Given a crystal having polar- izability tensor P and incident field E, an oscillating dipole moment M = P-E is established in the crystal. This moment reradiates the incident beam, producing elastic Rayleigh scattering. However, as the nuclei move, the electrons follow adiabatically, and the polarizability changes. It is the change of electronic polarizability with nuclear motion that gives rise to Raman scattering. Thus, the moment M contains "sidebands" formed from the product of the incident field and the phonon—modulated polarizability tensor. This qualitative description can be put in more formal terms. Raman cross sections for first—order scat- tering can be expressed using straightforward perturbation theory. The process involves three steps: a photon— electron interaction, an electron—lattice interaction, and a scattered photon-electron interaction. The resulting expressions1 involve double sums over (usualy unknown) electronic states in the crystal. In order to obtain useful expressions for the cross section, one resorts to the "polarizability approximation"2 by making the following assumptions: a. Vibronic states are written as simple products of electron and phonon states, with the nuclear coordinates appearing as parameters in the electron states (adiabatic approximation). b. The incident and scattered light are of much higher energy than the phonon states. In other words, the crystal polarization is largely electronic in char— acter. For visible light, this is typically a quite good approximation. With these assumptions, the intensity of Stokes first—order scattering from a specific phonon becomes I . ~ + 2 IStokes “ lei -esl (nf+l) (1) where P is the polarizability tensor, |o> and Ivf> are the ground and excited vibrational states, respectively, and 61 and Es are the polarization vectors of the incident and scattered light. The term nf+l, where nf is the Bose popu- lation factor, results from the thermal average over initial states. A knowledge of the dependence of (l) on E1 and gs for a given Ivf>, known as "selection rules," may be obtained from group theoretical arguments. Selection rules will be discussed in depth in a following section. The traditional means of obtaining detailed infor— mation on lattice dynamics is through neutron scattering. This technique, while quite powerful, requires large single-crystal samples (on the order of several cm). To date, it has not been possible to grow single crystals of 3C SiC this large, thus precluding neutron studies. Raman scattering provides the most complete information available concerning the lattice dynamics of crystals that are inherently too small for neutron studies. Phonon energies of 3C 310 at several high—symmetry points in the Brillouin zone have been measured previously with a variety of techniques, including polytype analysis, luminescence measurements, and first—order Raman scattering. These techniques are now described. SiC can exist in a number of different polytypes which are all very similar in structure, differing mainly in the stacking sequence of hexagonal planes along the (111) or C axis (group label A. Subsequent group labels, to be defined later, follow Reference 7). As the number of planes in the stacking sequence varies, so also does the size of the unit cell along the C axis. As the unit cell size increases, the Brillouin zone size decreases. For example, if the unit cell is doubled in size, the Brillouin zone is folded in half. Modes that were at the zone edge before the folding now become zone-center modes and therefore Raman active. It is then possible to obtain (approximately) phonon energies along A in 3C SiC by examining the Raman spectra of other polytypes. It is not obvious a-priori that such a technique should work at all. Its success depends on the extent to which one may consider variations in stacking sequences as small perturbations on the structure of the 3C polytype. This technique has been successfully applied to 810,3 and provides not only a fairly complete determination of the 3C dispersion curves along A, but the phonon energies at the point L as well. Luminescence studies“ of 3C SiC have revealed an indirect gap of 2.A eV as measured from the conduction band minimum at X. The location of the conduction band minimum is fortunate in that the indirect phonon—assisted transi— tions observed in the luminescence spectra permit a determination of the phonon energies at X point. The zone-center optic phonons (at F) are strongly Raman active and have been determined from first—order Raman spectra.3 Two—phonon absorption bands, also observed in 3G SiC,5 were quite weak, allowing only very tentative identification of a few spectral features. The previous experimental measurements of phonon energies at P, X, and L are listed in Table I. These previous measurements at I, X, L, and along A have been the only input to lattice dynamical calculations of 3C SiC. A satisfactory assessment of model calculations cannot be made without more extensive experimental input. 10 Table 1. Experimental lattice energies (cm—1) F X L Branch (Ref. 3) (Ref. A) (Ref. 3) TA -- 373 266 LA —— 6A0 610 TO 796 761 766 L0 975 829 838 B. Phonon Symmetries The space group of 3C SiC is T§(Ffi3m). This group is symmorphic, having an associated point group Td' There are two atoms per primitive cell, giving rise to six normal modes, three optic and three acoustic. The wavevector of visible light is typically several orders of magnitude smaller than the size of the Brillouin zone. Due to wave- vector conservation, the phonons which can participate in first—order scattering are very nearly at the zone center. To the extent that these wavevectors can be considered as zero (typically a very good approximation), the zone—center modes must transform as representations of the full point group Td' For later purposes, the symmetry properties of modes at several high-symmetry points in the Brillouin zone will be needed. For any given k in the zone, there will be a set ll of symmetry operations from the space group T: which leave that k vector unaltered. This set of operations will form a subgroup of the original space group and is referred to as the group of the wavevector.6 Each such group will have associated representations and a character table which will characterize the symmetry of modes with that wavevector. Parmenter7 has conducted such an analysis for zinc—blende. Figure 1 contains a sketch of the Brillouin zone adopted by Parmenter, P character table, and wavevector groups of specific points. Subsequent notation in this chapter follows Figure I. If the atomic displacement representation of the zone—center modes are reduced into representation of F, the result is 2F That is, the zone-center acoustic 15' and optic modes are triply-degenerate, and belong to representation F15. In ionic crystals, however, there is a long—range electrostatic field associated with the longitudinal optic modes (see Reference 1). This field lifts the degeneracy of the optic branches, leaving a singly—degenerate longitudinal optic (LO) mode higher in energy than the doubly-degenerate transverse optic (TO) mode. It is for this reason that two lines appear in the first—order Raman spectrum of 3C 810 rather than one (as in silicon). T(Td) E 02 C3 qu JC2 r1 1 1 1 1 1 x2+y2+z2 r2 1 1 1 —1 —1 r12 2 2 —1 o 0 /§(x2—y2),322-r2 F15 3 -l O —l l xy,yz,zx;z,x,y r25 3 —1 0 1 —1 POINT WAVEVECTOR GROUP A 02V (2mm) A,L 03V (3m) 2 CS (m) K CS (m) X D2d (A2m) W SHUT) Figure l. Brillouin zone of 3C SiC and wavevector groups. Jaw-4w- 13 C. First—Order Polarization Selection Rules Experimentally, the apparatus used to perform Raman spectroscopy can easily be arranged to analyze the polarization selection rules of a spectral feature (see Appendix A). In this section, the origin of selection rules for first-order (zone-center) scattering are dis— cussed as well as the extra considerations required for zinc-blende crystals with LO—TO splitting. Within the context of the polarizability approximation, the Stokes intensity is given by (1). As with any tensor, the individual elements of P must transform as products of coordinates (XX, YY, etc.). This implies that one can find linear combinations of the elements of P which transform like representations of I. These combinations are obtained directly from an examination of the basis functions included with common character tables. The (zone—center) normal modes of the crystal will also belong to certain representations of the point group of the crystal. Using a general group theoret- ical result,6 the matrix element in (l), , will vanish unless P and |vf> have parts belonging to the same representation of the point group (the ground state is taken as totally symmetric). In other words, the only normal modes which can participate in first—order scattering are 1A ones belonging to a representation contained in the decomposition of the polarizability tensor. The polarizability tensor, which is modulated by the normal modes, may be expanded in terms of normal mode coordinates Q, P=P +2 3% Q(I‘ ——(——7 )+... (2) O FajaQ F3. 0 J where 3 denotes the partners of a given representation F. The first term gives rise to Rayleigh (elastic) scattering, the second term to first-order Raman scattering. In order that (2) be an equality, the second term must have the same transformation properties under all group operations as P. Therefore, the sum over f includes only those representa— tions which are contained in the reduction of the polar— izability tensor consistent with the general selection rules above. (The zero order polarizability tensor PO must be invariant under all point group operations of the crystal.) One demands that the interaction energy, Ei-P-ES (E1 and Es are incident and scattered electric fields), be invariant under all group operations of the crystal. Since normal coordinates belonging to different representations do not mix under group operations, the invariance of Ei-P-Es requires that for each F3, 8P E E. __.__._. 1 3Q(I‘J.)O s 15 transforms identically to Q(fj) (so that their product will contain the totally symmetric representation). The tensors 313 3Q(Tj) o are commonly referred to as Raman tensors, and may be determined by inspection of the character table basis functions. (A listing of Raman tensors for all crystal classes appears in Reference 1.) These concepts are now illustrated for zinc—blende. The decomposition of the polarizability tensor for zinc— blende and associated Raman tensors, obtained with the aid of the Td point group character table of Figure l, are shown in Table 2. The Raman tensors within a given representation are determined up to an overall multiplicative constant. Since the decomposition of P includes only representations Fl, T12, and F15, only modes belonging to these representa- tions can be Raman active. The phonon polarization direc- tion associated with each F15 Raman tensor is indicated in Table 2. This phonon polarization vector indicates the relative displacement of the two sub—lattices of the crystal at zero wavevector in zinc—blende. The utility of these Raman matrices is that they allow a determination of the symmetry of observed spectral features. Denoting the Raman tensors as R(Fj), the 16 Table 2. Decomposition of the polarizability tensor and Raman tensors for point group Td Polarizability Representation Basis function components Fl x2+y2+z2 Pxx+Pyy+Pzz P12 /§(X2'y2)s /§(Pxx‘Pyy)’ 2z2-x2—y2 2Pzz'Pxx—Pyy F15 xy,yz,zx ny’PyZ’PZX RAMAN TENSORS: J? 0 OF —1 o o _0 0 0 o 0 2 0 o 07 ’0 0 1 T ' c O O 1 ; c O O 0 ' 15 ’ o 1 o__ 1 o 0 Note: x, y, z denote phonon polarization directions. l7 intensity of a mode belonging to a particular representation F is given by 1(r) « 3'31 - fi(rj) - Es|2, (3) J where 31 and gs are unit polarization vectors of the incident and scattered light. A standard notation has been adopted which completely describes the experimental geometry of a given spectrum. This notation is Ei(éi,ES)ES, where R1 and Rs are wavevectors of incident and scattered light, respectively. For example, the notation X(YZ)X indicates incident light propagating in the direction -x with Silly, and scattered light propagating in the direction x with gsllz. (This geometry is called "back scattering" for obvious reasons.) This notation is employed throughout. A glance at Table 2 shows that only modes of F15 symmetry will be observable in this geometry. Thus, experimental geometries can be selected such that only modes having a desired symmetry will appear. 18 D. L0 and TO Phonon Selection Rules As mentioned previously, the triply—degenerate F15 zone—center optic modes of the zinc—blende lattice are split by a long—range electrostatic field associated with the LO mode. The resulting singly—degenerate LO mode is higher in energy than the doubly—degenerate TO mode, yielding two spectral lines in the Raman spectrum. The selection rules of each individual line, first obtained by Poulet,8 require an extension of the group theoretical treatment given above. The assumption is made that the photon wavevectors R1 and is are small but finite, so that the phonon wavevector Ephonon equals ki—Rs is determined by the directions of the external fields. Consider first the case where the phonon wavevector is parallel to, say, x. Poulet asserts that the F15 Raman tensor associated with phonons polarized along x describes the LO phonon scattering, while those associated with y and z describe the TO phonon scattering. (Recall that the LO mode has phonon polarization parallel to its wavevector, while the TO mode has phonon polarization perpendicular to its wavevector.) For general phonon wavevectors, one rotates all three F15 Raman tensors such that the phonon polarization coordinate associated with any one of them (under the same rotation) is parallel to the phonon wave— vector. This "parallel" tensor then describes LO scattering, while the remaining two describe TO scattering. This LO—TO l9 scattering cannot be treated with conventional group theory since the phonon R vector direction, and hence the LO-TO selection rule, is determined by external fields and not by crystal symmetry. This procedure, put forth as a hypothesis by Poulet, accounts very well for observed LO-TO selection rules in zinc—blende crystals. E. Raman Intensity Matrices Relation (3) is valid for arbitrary orientations of 31 and gs‘ It is not convenient to work directly with Raman tensors since (3) must be evaluated for every exper- imental geometry employed to obtain the expected intensity. If one is willing to restrict the possible polarization vectors to directions along principal axes x, y, or z, only nine possible cases need be considered. These possibilities can be conveniently tabulated by defining intensity matrices (following Poulet) for each set of Raman tensors. The intensity matrix is defined by + ~ + 2 FaB(P) m glea ~R(fj) ~eBI ,d,B =x, y,z, (A) where F is P1, Fl2’ or F15 in zinc-blende. L0 and TO intensity matrices are similarly defined. However, from the discussion above it is clear that one must specify also the photon wavevectors R1 and is in order that Poulet's procedure may be carried out. 20 Selection rules to this point have been discussed under the assumption that the crystal axes and laboratory axes coincide (i.e., Raman tensors and polarization vectors all refer to a common set of axes). Experimentally, it is sometimes more convenient to employ other crystal orien— tations. For arbitrary crystal orientations, (3) and (A) still apply provided all Raman tensors involved are rotated to coincide with the laboratory system. All intensity matrices and corresponding coordinates employed in this study are listed in Table 3. The backscattering geometry is employed throughout. The unprimed and primed axes both represent laboratory coordinates, indicating different crystal orientations. Table 3. Zinc-blende intensity matrices X=(lOO)=X’ ’000‘ 0 1‘ I(LO)= 0 0 1 I(TO)= 1 0 0 Y = (010) Y’ = (011) _0 1 0_ 1 0_ z = (001) Z‘ = (Oil) _0 0 0' 0 1 1T I’(LO)= 0 1 0 I’(TO)= 1 0 0 p 0 1_ 1 0 0_ 1 0 0 ‘1 0 07 0 1 1‘ I(Fl)= 0 1 0 1(r12)= 0 1 0 1(rl5)= 1 0 1 _0 0 1 _0 0 1_ 1 1 0_ [‘1 0 o 17: 0 0‘ 0 1 1‘“ I’(Pl)= 0 1 0 I’(r12)=E 0 1 3 I (r15)= 1 1 0 0 0 1 _0 3 1_ 1 0 1_ 21 Appendix B contains a more extensive computer-generated list of zinc-blende intensity matrices for a variety of common experimental geometries, including right—angle scattering. The discussion of selection rules here is in the context of scattering from crystal phonons. Many of the concepts involved are actually more general, and may be applied to atomic, molecular, and crystal electronic scattering. Briefly, this generality has origin in the fact that Raman processes can often be described by second—order perturbation theory. The cross—section for Raman scattering involves matrix elements of the form <1|Ei 'rlm>’ where i and f denote initial and final quantum states, and m represents a complete set of quantum states.6 Because the operator r|m> « —eJ§i——-, <5) -é’|v6<fi>| where S is the surface in k-space such that w=v. There are certain points, known as "critical points," in the disper- sion curves where le(R)I will either vanish or change sign discontinuously. Van Hove9 first investigated "analytic" critical points and showed that they produce discontinuities in slope in the density of states, with accompanying charac— teristic shapes. Phillipslo later extended this work to include non-analytic or "fluted" critical points, and explicitly treated lattice dispersion curve critical points. Consider a general point wo(ko) on a non—degenerate branch of the dispersion curves. In the immediate vicinity of this point, the energy may be written to second order as + + —> w = w Vw 1 -g +-1 52:666 + . . . (6) °'+ k 2 E o . o -> ++ + where g = k-ko. A critical point is simply that k where the second term of (6) vanishes. The third term, representing a 23 second-degree polynomial of the components of R, determines the nature of the critical point. For critical points where degeneracy exists, an expansion such as (6) cannot be performed, since the energies are solutions of a secular equation. Thus a distinction is made between analytic critical points, associated with non-degenerate branches, and non—analytic or "fluted" critical points associated with degenerate branches. Certain points in the Brillouin zone are required by symmetry to be critical points. The set of solutions |j>, j=l, ..., n at an n—fold degenerate point R0 will transform according to some Pd, a representation in the I These groups for zinc— "group of the wavevector" at Ro.l blende are listed in Figure 1. The operator ka transforms like a vector, and will generate a representation TB of the group of R0 which, when reduced, will contain representa- tions labeled by components of vectors. Phillips has shown that K0 is a critical point if = 0, for 1, j=l, ..., n. Equivalently, R0 will be a critical point if the reduction of Ta x1"B xFa does not contain the totally symmetric representation. For general points in the zone, the group of the wavevector will consist of only the iden- tity element. The product fa xfB XFa must then contain the totally symmetric representation, so general points are not symmetry—required critical points. General critical points are called "accidental." 2A This procedure allows a systematic test of all high-symmetry points in the Brillouin zone to determine if they are required by symmetry to be critical points. Parmenter7 has conducted such an analysis for zinc-blende and has found points T, X, L, and W to be symmetry required critical points. There may be other accidental critical points not required by symmetry, as determined by the detailed lattice dynamics. The shapes generated in the density of states by various critical points depend on the behavior of the dispersion relations in the immediate neighborhood of the critical point. Phillips10 has presented a classification scheme which allows a check of the topological consistency of any supposed set of critical points. This scheme is based on "sector numbers," obtained by constructing a small sphere in k—space about the critical point Ro(wo) in ques— tion. The surface of the sphere is divided into sectors where w>w0 (positive sector) or w vanishes, (where vf is now a two-phonon state) then the transition is not Raman active. In other words, the matrix element vanishes unless P and |vf> have parts belonging to the same zone-center representation. In zinc—blende, the polarizability tensor has zone-center rep— resentation f P 1’ 12’ are of relevance in the reduction of zinc—blende two-phonon and T15' Only these representations states. The representations which occur (among these three) in the reduction of two-phonon states from several high— symmetry Brillouin zone points in zinc-blende have been determined by Nilseh16 and are listed in Table 5. Only actual dispersion curve representations are included. Since all two-phonon combination states from general points in the zone and all overtone states contain Pl, they will be symmetry-allowed in the Raman process. The two— phonon spectrum is therefore a continuous function of 33 Table 5. Zone—center two-phonon representations for 30 SiC Representation Reduction Representation Reduction [X5]2 rl+2rl2+r15 [L3]2 r1+r12 [X112 r1+r12 [L112 rl+r15 [X312 I‘14'1‘12 L3XL1 F12+P15 XSXXI P15 L3xL3 P1+P12+2P15 X5xX3 r15 leLl rl+r15 X5xx5 r1+2r12+r15 [Wl]2 rl+r12 XBXXI P15 Wixwi F1+P12 [r15]2 rl+r12+r15 wlij r12,1 # J frequency rather than a line spectrum. The two-phonon Raman cross section may contain contributions from all symmetry- allowed two-phonon pairs throughout the zone consistent with overall energy and wavevector conservation. The second— order spectrum can be roughly considered as being modulated by the two-phonon density of states. There will certainly be other energy dependence in the cross section. However, the important point is that the structure in the two-phonon density of states due to critical points will appear in the two—phonon Raman spectrum. This is the essential link between critical points and second—order Raman spectra. 3A The connection has been traced from dispersion curve critical points to their manifestations in second- order spectra. Experimentally, the inverse problem must be solved. That is, given measured spectra, one attempts to determine the positions and types of critical points in the Brillouin zone. This inverse problem is the more difficult and permits only a partial solution at best. Even with neutron scattering data (not available for SiC) a critical point assignment can never be made with absolute certainty. The inherent difficulties in applying the preceding theory to experimental analysis will be discussed in depth. H. Experimental It is quite difficult to grow large, single poly— type crystals of 3C SiC. Fortunately, Raman scattering requires samples only slightly larger than the focused incident beam diameter. Using samples of dimensions of ~1 mm or smaller presents problems in X—ray orientation, surface preparation, and general handling. The two samples used in this study have dimensions on the order of 5-10 mm. The crystal axes of both samples were determined using X—ray Laue back-reflection. Growth faces of 3C SiC tend to be [111] planes. In order to obtain a [100] face, a special goniometer was constructed which allows trans— ferring the sample from the X-ray apparatus to a diamond 35 saw while preserving the sample orientation. The larger of the two samples, sample A, was cut in this manner. The sample orientation obtained in this manner is accurate to within 1°, as judged from subsequent X—ray checks. The arrangement of optical components is shown in Figure 1“, Appendix A. The backscattered light is collected over a finite solid angle determined by the lens diameter and the distance between the lens and the sample. This arrangement has two important experimental consequences. First, the theoretical selection rules discussed previously are based on definite, well—defined wavevectors for both the incident and scattered light. The lens in effect integrates all final wavevectors in its solid angle. However, refrac— tion at the crystal surface tends to limit the effective size of this solid angle, so that experimental selection rules are better than might be supposed. Other limits on selection rule measurements include the polarization purity of the incident beam, the accuracy of the sample orientation, and the quality of the analyser polaroid. The overall validity of experimental selection rules may be gauged by comparing the intensity of first-order spectral lines in allowed and forbidden geometries. This ratio is 50:1 or better for all spectra presented here. A second consequence of the experimental geometry is the need for a mirror-like sample surface. The 36 spectrometer is not an ideal instrument, in that intense light of any wavelength can appear as a continuous back- ground in a spectrum (due to imperfections and multiple reflections in the spectrometer). Intense monochromatic light entering the spectrometer can produce false structure in the spectrum, referred to as "grating ghosts." A rough sample surface can Rayleigh scatter a portion of the inci- dent laser beam into the spectrometer. A perfectly smooth sample, properly aligned, will reflect the incident beam directly back to the beam mirror, preventing its collection by the lens. This geometry at the same time guarantees the proper alignment of the sample. The [100] face of sample A was polished after sawing, using successively finer abrasive compounds, and finishing with quarter—micron diamond grit. Spectra from surfaces prepared in this manner show no grating ghosts and very low background levels. The question of surface damage is addressed later. After ultrasonic cleaning in ACS acetone, the samples were mounted in an optical "cold finger" liquid nitrogen dewar using thermally conductive grease. The cold finger is made of OHFC copper to enhance its thermal conductivity. The temperatures quoted in this study are approximate, since there is some unavoidable sample heating due to the incident laser beam. 37 Spectra are recorded using a spectrometer equipped with a stepper-motor wavelength drive, cooled phototube, and photon counting electronics. This equipment is detailed in Appendix A. The procedure is to count and record the photon flux for a set period of time, move the spectrometer a set wavelength increment, and begin another count. The spectral plots presented here represent a set of discrete points joined by straight line segments. Because the spectrometer is calibrated in wavelength, the wavenumber shift must be calculated for each point using _ 8 l l “"10 (if?) (9) where Xi and is are the incident and scattered wavelengths in Angstroms, and w is the spectral energy in cm-l. For true Raman processes, spectral features will have a constant w regardless of the wavelength of the incident light. Non- Raman processes, such as flourescence, emit constant wave- length light, and will appear at different wavenumber positions in the Raman spectrum as the laser wavelength is changed. This fact allows Raman and non-Raman spectral features to be identified. Several laser lines are employed in this study from both argon and krypton lasers for the above reason and other reasons to be described. The recorded spectra will have a wavelength dependence (in addition to that of the sample) arising 38 from the measurement system itself. The system response as a function of wavelength, or throughput, can be measured using a standard lamp with a known spectral output. The instrument throughput can then be removed from the spectra, as is done for all spectra presented here. Spectral wave- lengths are calibrated by superimposing discharge lines from a neon lamp on the two—phonon spectrum. Wavelength errors introduced by the spectrometer drive, which may be several cm-l, can be corrected. Feature energies quoted here are measured from the nearest identifiable neon line, and have an estimated uncertainty of i2 cm_l. Descriptions of the throughput and wavelength calibrations are included in Appendix C. Photon counting statistics have a Poisson distribution. That is, successive counts for fixed time intervals and constant input will have some average count N, and the distribution will have variance /fi. The quality of the signal may be roughly gauged as the ratio of the mean to the variance. In other words, the quality of the signal improves as /§. The time allowed for photon counting at each spectral point may be made as long as needed to obtain a suitable number of counts. The total recording time is then also made longer. The two-phonon spectra presented here require recording times of several hours each with a laser power of 150 mw. 39 The samples are both known to have nitrogen donor impurities in an unknown concentration. The relative nitro- gen concentration of the two samples is 2:1, as discussed in Chapter III. The larger, sample A, is the purer and is used to obtain most of the two-phonon spectra. Sample B is used to judge the effects of the nitrogen donors on the two- phonon spectra and as a consistency check. I. Results A brief study of the Poulet intensity matrices of Table 3 indicates the possibility of arranging the experimental geometry such that polarizability components P12 and I15 may be isolated. The main results, all of sample A, are shown in Figures 3 through 5. The geometry, allowed polarizability components, and allowed first-order scattering are indicated in each Spectra. Since the polarizability tensor in zinc-blende has only three parts, three spectra are sufficient to isolate the various compo- nents. The fourth geometry, in Figure 6, provides a con- sistency check. Experimentally, the f spectrum is found 1 to be much stronger than either the F12 or F15. For reasons of clarity, the relative scales of these four spectra have been multiplied by 1:5:5:l for Figures 3, A, 5, and 6, respectively. These four spectra were recorded using a 4579A (2.7 eV) argonion laser line and an instrument 140 .mcHH somma Momma .Mossue .< oHQEmm go ESLpOodm xANNVM .m ohsmflm 1.53 Elm 5630mm... oooN oom_ com. com . _ A... a a is 4 u d .1q.4 a... digimq 44.7.14 . ..n . islets _ l. _ d a 7.14 d I V vaOPN .l o m M M F I\ + «I W 1 a 1 1| 1 m. .1 L+L 1 XoNNnFr T XANNVm l O.._ < Mira—24m ”PIPLL. Lei-.p .L _ _ l. . b L _ b,u_.—.P . u LLL ...Plb _ L27..r.b .7; b AIISNBINI ’41 .ocHH Lemma Kmsm: “Momsue .< deEwm mo Sappoodm waNVM .: mpsmflm Ares Him 53:85 OOON oom_ CON. 00m d44...d44._444444442fi.a.AJaqmaadadJJad I scouixvoj 1 I “50...; l _ - . a, X 1 u... 1 m l NM 1 j J 03%: T m Vick...» 1 .. m oe xcflvm l bub-PP_L.EI—IFE__PL_LLF..P Ppbfibhmfibdwwlwflhzwm. AI8N31NI A2 .mcHH Lemma Mmsm: .mossue s< oHQEMm mo Esnpoon XA‘N‘wVM .m mhswfim Arse term 262.450me oooN com. com. com flld 4 a 41. d d n.7q..l_ d.d14.dxa.a a 474.44144 414.L_7.7114_d.474 a a 4.4 I m“. \1tfifl114 1fi)/\\1€\\J(lllu T TIV W 1 (mm H T + 01- ]. L .I ..quv .N m )H (m n... T n... H... n_. ( i m 0 .l NM NM 1 + w. I \1 4 I365 m N. v T I l L. m 1 v;okwkwflnPl T n:. xpwxam- 0:. < Uni—24m _._ h b b b u,u_._ p — _ b A . 1.1.7. b u h b u7__._ . 7; b _ 1..»7b bib b L AIISNBINI A3 .msfia somma Momma .xossue .< madamm mo Sappoodm xAtN‘NVM .w omswfim :sE ov.b&Im >uzm30mmu OOON OO®_ CON. 00m ddddiRARAd—dfidd«AdudfiJdddAAdAJ—ddddddd r l humans. u r 20:... A o.1 ..< flazfi »—Lth»ELF— bh--PFb-ble- bbh .th .. or xAZNvi... AIGNBINI AA resolution of 7 cm-l. A number of critical points are observed, many of which have been identified and noted in Figures 3, A, and 5. The first-order TO (797 cm-1) and L0 (975 cm-1) phonon peaks appear weakly in geometries where they are forbidden, such as Figure 3, due to the approximate nature of experimental selection rules. This is to be con— trasted with Figure A, where the LO phonon is allowed and dominates the two-phonon scattering. The addition of impurities to a crystal has the effect of disturbing the strict translational symmetry of the lattice. Wavevector conservation, which originates directly from crystal periodicity, may be somewhat relaxed by the presence of impurities. A consequence for light scattering is that previously forbidden lattice modes may become optically active.19 Impurity-induced scattering generates a continuous spectrum since modes from the entire zone may contribute to the scattering. Sample B, having roughly double the nitrogen concentration of sample A, is used to judge the effects of the nitrogen donors on the two—phonon spectra. The sample B spectrum appears in Figure 7. The geometry, chosen to include all polarizabil- ity components, is identical to the geometry of Figure 6. The scale of Figure 7 has been adjusted to facilitate a comparison of these two spectra, and they appear to be nearly identical. Although not shown, no features appear A5 .oQHH pommfi was: «monwue .m mHQEmm mo ESppoon xA\N‘NVM .s opswflm CON ple-bbbk 00m: ddeded4dd444¢J4Jfilfifiddd ufidJ.dR—qfidfid+4 2.53 Him 65:82“. oom_ oom bbPPpr—hpbb-b AIISNBINI A6 below A00 cm-1 and no features appear which can be interpreted as impurity-induced first-order scattering. The strength of the peak at 1630 cm_1 does appear to scale with nitrogen concentration. This feature cannot be a first—order impurity-induced feature, as its energy is well above any first-order modes. Electronic impurity level energies are also too low to account for this feature (see Chapter III). It may be an impurity-enhanced two- phonon feature, but remains unresolved. The remaining portions of these two spectra are identical (with the exception of a small feature immediately preceding the TO phonon, to be discussed later). For present purposes, sample A may be regarded as pure. When the energy of the incident light approaches the band-gap energy in a semiconductor, it is possible for some Raman processes to be resonance-enhanced (for example, see Reference 12). Both samples A and B exhibit broad luminescence in the energy range of the indirect gap (2.A eV) which obscures the two-phonon Raman spectrum precluding indirect gap resonance studies. The wavelength dependence of the two—phonon spectrum can be gauged using a 6A71A (1.9 eV) krypton-ion laser line, well below the indirect gap energy. This spectrum appears in Figure 8, and is to be compared with Figure 6, which has the same geometry. A7 .mcfia Lemma « (n(wl)+l)(n(w2)+l) for Stokes scattering, where n(wl) and n(w2) are Bose population factors for each phonon of the pair. For an overtone mode of energy 2w, the thermal factor is (n(w)+l).2 A9 The thermal factors for two temperatures and a given feature can be compared by forming the ratio I(Tl)/I(T2). Using Td_=300° K and T2==77° K, these ratios for several overtones in the 3C 810 spectrum are shown in Table 6. (Experimental ratios from Figures 6 and 8 have been normalized to the theory for 2TA(X).) Table 6. Thermal intensity ratios of two—phonon Raman spectra in 3C 810 1(300° K)/I(77° K) Overtone Theory Experiment 2TA(L) 1.8 2.0 2TA(X) l.A l.A 2LA(L) 1.1 1.0 These ratios cannot be taken too seriously, since matrix elements, background luminescence, crystal absorption, and other factors will have temperature dependence also. The important point is that the thermal intensity factors account for the major portion of the relative intensity differences between Figure 6 and Figure 8. The phonon energies in SiC are such that quite high temperatures would be required to significantly change the thermal factors for the structure beyond lAOO cm—l. Broadband 50 luminescence in these samples, which increases with temperature, precludes high temperature two-phonon studies. Difference processes, where one phonon is created and another destroyed, have thermal intensity factor I(T) « n(wl)(n(w2)+1) where ”l and w2 are the destroyed and created phonon energies, respectively. For Stokes processes, w2>wl. The difference in thermal intensities of difference processes versus overtone processes is used in some materials to determine the nature of a two-phonon spec- tral feature (for example, see Reference 16). Assuming room temperature, a feature energy of 600 cm-1, and dif- ference pair energies of 300 cm.1 and 900 cm-1, one obtains I(overtone==l.7 while I(difference) =.3. Using the same energies but T'=77K, one obtains I(overtone) = l and I(dif— ference)~l—-3. The example energies, typical for SiC, were actually chosen to favor I(difference). Higher feature energies generally give an even greater difference in the intensity factors. Assuming that matrix elements for dif- ference processes are roughly comparable to combination processes, the conclusion is that no difference processes will be observed in the spectra of Figures 3 through 6 (all at T==77K). 51 Preceding discussions explain the choice of the A579A laser line for the main spectra of Figures 3 through 6. Because sample A is quite transparent to the 6A71A line, the experimental selection rules are obscured by multiple internal reflections from flaws in the crystal and from the crystal faces. Since the luminescence near the band—gap may be avoided with either line, and since identical spectra result, the natural choice is the AS79A line. To investigate the effects of surface damage which could result from surface polishing, a spectrum was recorded from a [111] growth face of sample A. This spectrum appears in Figure 9. While the geometry of this spectrum is dif— ferent than that of Figure 6, the allowed polarizability components are identical. There are no dramatic differences between the two spectra, indicating that surface preparation has not introduced new structure. J. Analysis The analysis of second—order spectra has several purposes. Among these are an accounting for the optical second—order spectra structure, extraction of phonon energies at several high-symmetry points in the Brillouin zone, and a determination of the general topology of dis— persion curves in the Brillouin zone. A complete analysis requires neutron scattering results, second—order infrared .mcfia momma gmsm: .mooomue n< oHdEmm mo Edhpomdm ooM0 mesonw .m opswflm 0 000m 000_ Arse Kim 5630mm... 00m. 000 lddSJquddd—udd-uJAJddddddddquddiqAaadJ 52 ‘04+oe 62.2.1. 8...: = N 55 __ > 2.: __ x XANNUIA No.4... Ik>>0m0 b-Pbb-b bbLbLbb—bL-p L AilSNEINI 53 and Raman spectra, and possibly lattice dynamical calculations. Neutron scattering and Raman scattering are complementary in several respects. Neutron scattering can give detailed energy and wavevector information from general points in the zone. Energies derived from Raman spectra are generally more accurate but are restricted to critical points. Raman results are also more difficult to interpret, since the spectra contain contributions from all parts of the zone at once. Optical spectra can give infor- mation, through selection rules and spectral shapes, on the symmetry properties of critical point phonons. Any analysis of second-order spectra without neutron data must neces— sarily be somewhat incomplete. Since there is no neutron data for 30 SiC, the approach here is to supplement the optical spectra with a reasonable lattice dynamical calculation. Direct application of critical point and selection rule theory, presented earlier, to the analysis of experi- mental spectra involves some practical limitations. The spectral shape of an overtone critical point from a non- degenerate branch is determined by the nature of the dispersion curve critical point, since adding a branch to itself is equivalent to a simple doubling and preserves the nature of the critical point. Consider next a combina- tion arising from two non—degenerate branches, which will 5A make a contribution to the two-phonon density of states given by D (u) a] dS » (10) 2 + S IV (wl+w2)l where “l and w2 are the branch energies and S is such that v=wl+w2. This relation, a natural extension of (5), shows that the shape appearing in the second-order spectrum will depend in detail on the relative curvatures of the two critical points. A lattice dynamical calculation can give the nature of critical points in the dispersion curves, where only the signs of curvature in the immediate neighbor— hood of the critical point are required. Determination of (10) places a much greater demand on lattice dynamical cal- culations, since the relative values of the curvature near the two points are required. Dispersion curve calculations are generally not reliable enough to permit an evaluation of (10). The situation is further complicated when one or both members of the two-phonon pair are from a degenerate criti- cal point, where non—analytic or "fluted" behavior is possible. A second complicating factor is that observed critical point shapes are often obscured by other scattering, sometimes to such an extent that even the discontinuity in slope does not appear distinctly. Determining the energy of such points from the measured spectra is of necessity somewhat subjective. 55 The limitations of experimental selection rules have been discussed previously. Selection rules indicate what components of the polarizability for a particular two-phonon critical point are symmetry-allowed, but not necessarily which of these components will be observed. Since the P spectrum is much stronger than the F12 or F15 1 spectrum, strong F features may appear weakly in geometries l where they are not allowed due to the approximate nature of experimental selection rules. As with critical points, selection rules of weak features are often obscured by other scattering. Since one does not strictly know what the intensity ratios should be between various polarization components of a given feature, experimental selection rules must be considered as approximate only. Their utility is mainly in approximately testing the consistency of supposed feature symmetries. A number of critical points appearing in the spectra of Figures 3 through 5 have been identified and are listed in Table 7. These assignments have also been indicated in Figures 3 through 5. The considerations leading to these assignments are detailed later. Features having an indis- tinct energy or selection rule, as discussed above, have been so noted. Energies of such features have uncertainties perhaps as high as :5 cm_l. The labels of Figures 3 through 5 have been placed in the figure where the associated 56 Table 7. Energies and feature assignments from the two- phonon Raman spectrum of 3C SiC —' Energy (cm‘l) Polarization Branch ObservedIPreviousa ObservedIAllowed 2TA(L3) 53A 532 r1 rl,r12 2TA(X5) 7A8 7A6 r1 rl,rl2,r15 LA(L1)+TA(L3) 883 876 -—b r12,rl5 2TA(2l or 22) 9A5 -- -—b rl,r12,rl5 TA(X5)+LA(X1 or X3) 1010 1013 -—b r15 TA(L3)+TO(L3) 1035 1032 -—b rl,rl2,r15 TO(X5)+TA(X5) 1133 113A r1 rl,rl2,r15 TA(X5)+LO(X1 or x3) 1201 1202 r15 r15 2LA(L1) 1212 1220 r1 r1,r15 2LA(xl or X3) 1275C 1280 r1 rl,r12 T0(x5)+LA(xl or x3) 1A00 lAOl r15 r15 L0(Xl)+LA(X3) or 1A60 1u69 r15 r15 LO(X3)+LA(X1) 2TO(Zl or £2) 1A73 -— Pl F1,Pl2,F15 2TO(X5) 1523 1522 r1 rl,rl2,r15 2TO(L3) 1530 1532 r1 rl,rl2 TO(X5)+LO(X1 or x3) 1588 1590 r15 r15 2TO(F15) 1593 1592 r1 rl,r12,r15 TO(L3)+L0(L1) 16100 160M r12 r12,r15 2LO(P15) 1952c 1950 r1 r1,r12,r15 a 0 0 These energies derived from prev1ous experimental measurements. bExperimental selection rules could not be obtained-—see text. CIndistinct feature--see text. 57 critical point energy appears most distinctly, and do not imply a particular selection rule. The energies of the second column are from these previous measurements. The symmetry species of Table 7 follow Reference 16, as determined from a reduction of the atomic displacement representation into species of the point in question. Some ambiguities in species assignment remain as indicated in the first column of Table 7. Lacking neutron data, the nature of dispersion curve critical points must be obtained by calculation. Several dispersion curve calculations have been performed for 30 Sic.21’22’23’2“ These models have variable param— eters which are adjusted to fit the available experimental energies at F, X, L, and along A. However, these calcula- tions are not reliable enough to extract energies for other points in the Brillouin zone. The calculation of Reference 2A includes a determination of the two-phonon density of states. The spectrum of Figure 6, which contains all com- ponents of the polarizability, is compared with this calcu— lation in Figure 10. The agreement below 1300 cm-1 is fairly good, while above 1300 cm—1, where the structure is due to the optic branches, the agreement is poor. Model calculations in general have the most difficulty accounting for the optic branches. The important point is that even extensive density of states calculations are not .zm monommmmm mo mmpmum mo mpfimcoo cocozdlozp oopmHSono spas sapwoodm Looho ocooom mo somfimeEoo .oa mpswflm 1753 Kim 5280mm... ooom oom. , cow. oom 58 qqddqdfii A u u a dddAAd uddddfil—d 144444 1 . A. ‘ _. _ u . L : 1 ._. L _ a . . I. l r T l ' I h l —_ h b -b — b —b p - pb — pL - — — - b _ -L b b b _ pb _ AIISNEINI 59 sufficiently detailed to permit a critical point analysis. A more complete analysis can be performed using the dispersion curves directly. A comparison of published dispersion curves for the calculations noted shows that they share certain qual- itatively similar aspects. Of concern here is the behavior of the dispersion curves in the immediate vicinity of crit— ical points. The topology near critical points in a few cases does appear to depend on which calculation is used. Some of the calculations show extra "kinks" which may or may not be realistic. In an attempt to determine a set of critical points and their sector numbers, the calculation of Reference 21 was duplicated. Details of this calculation, including dispersion curves, are presented in Appendix D. This model is referred to as a second-neighbor ionic or SNI model. In this study, sector numbers were obtained by inspection of a small grid of solutions surrounding each critical point. Sector numbers so obtained are peculiar to the SNI model and can serve as only a rough guide. The final set is determined by considering published dispersion curves from more sophisticated models and observed spectral shapes and energies. This process is not as arbitrary as it might at first appear, since the observed spectral shapes, observed energies, and sector numbers are all tightly inter- woven by the Morse relations, (7) and (8). For example, the 60 ordering of the energies of critical points in a given branch will have a profound effect on the topology of that branch, and hence also on the topology of the critical points. The results of this two- and three-dimensional critical point analysis are shown in Table 8. The planes [100] and [110] are used for the two—dimensional analysis. The number appearing next to each group label is the multi- plicity of that point in its associated manifold (plane or volume). The branches are labeled (in order of increasing energy) as TA, LA, etc., even though the phonons have these simple polarizations only in certain directions. As pointed out by Phillips, such a set does not necessarily represent the true set. It is always possible to add pairs of kinks (and their symmetry equivalences) to a branch without spoil- ing the topological consistency. However, the assignments here are consistent with the observed energies and spectral shapes, and satisfy the two- and three-dimensional Morse relations. As an example of the analysis process, including sector numbers, consider the shoulder appearing at 53A cm-1 (Figure 3). Table 1 shows this energy to be very close to twice the TA(L) energy, and no other combinations or over- tones can be formed from Table 1 that agree as well as 2TA(L). Most assignments are performed by considering the energy first. Figures 3 through 5 show this shoulder 61 Table 8. Critical point analysis of 3c Sica Branch r(1) X(2) R(l) W(A) 2(A) Two-Dimensional; (100) Plane: TA]. (1,0) (2,2) ([434) (031) "" TA2 (1,0) (2,2) (1,0) (0,1) (2,2) LA (1,0) (0,1 (0,1) (2,2) -— T01 (0,1) (1,0) (A,A> (2,2) (1,0) T02 (0,1) 1,0 (1,0) (2,2) -- LO (0,1) (2,2) (0,1) (1,0) (2,2) Branch r(l) X(l) L(2) 2(2) Two-Dimensional; (110) Plane: TAl (1,0) (0,1) (2,2) —- TA2 (1,0) (2,2) (2,2) (0,1) LA (1,0) 0,1) (2,2 —— T01 (0,1) (2.2) (2,2) (1,0) T02 (0,1) 1,0) (2,2 —— LO (0,1) 2,2) (2,2 (1,0) Branch r(l) X(3) L(A) W(6) 2(12) Three-Dimensional: TAl (1,0) (4,1) (1,2) (0,1) -- TA2 (1.0) (1,2) (1.2) (0,1) (2,1) LA (1.0) (0.1) (1,2) (2,1) -- T01 0,1) (l.A) (2,1) (1,2) (1,0) T02 0,1 (1,0) (2,1 (1,2) —— L0 , (2.1) (2, 1,0) (1,2) aNumbers in parentheses following critical point labels indicate point multiplicity. R is the point (110). Branches are listed in order of increasing energy. Point W has mixed phonon polarization. 62 appears only in the P1 geometry. Table 7 shows that 2TA(L) is allowed in F and P12 geometries, consistent with the 1 observed polarization. Table 8 shows both members of the degenerate TA branch at L have sector numbers (1,2). For an overtone, the two-phonon density of states will have the same shape as the one-phonon density of states. Figure 2 shows the shape of a (1,2) point to be identical to the shape in Figure 3. The energy, polarization, and critical point shape all agree with the assignment of this feature as 2TA(L). While this spectral feature is exemplary, the assignment procedure is well illustrated. K. Discussion Figures 3 through 5 show the F1 or totally symmetric component as the dominant component. This is generally the case for homopolar or slightly ionic materials. By compar— ison, second-order spectra of alkali halides may have other components comparable in strength to the symmetric component.25 The second neighbor ionic (SNI) calculation, used in part to determine critical point sector numbers, was chosen due to its computational simplicity and qualitatively typical results. This calculation consists of first and second neighbor forces and a long range Coulomb interaction. The ionic charge, which appears as a parameter, is adjusted A!" 53 to 1.05 to obtain the best fit with experimental phonon energies. It agrees well with the Szigeti effective charge of .9A. The suitability of a rigid-ion model for a IV-IV covalent compound may be debated. However, the SNI calcu- lation qualitatively explains a number of observed spectral features. None of the calculations can be assessed without extensive neutron data, so that calculated energies at points not eXperimentally measured cannot be taken as strictly reliable. A portion of the spectral structure is likely due to critical point W, but assignments are not possible without a more accurate knowledge of branch energies at W. The energies of Table 7 agree quite well with previous measurements. Phonon energies derived from luminescence spectra actually are energies of phonons having the same wavevector as the conduction band minimum. The conduction band minimum in 3C 810 is believed to be at X (see Reference A). Second-order Raman spectra measure phonon energies at X directly. Assuming an uncertainty in both sets of measurements of i 2 cm-l, and using the SNI branch curvatures near X, it can be inferred that the wavevector at the conduction band minimum is within 5% of X point. This experimental limit lends support to the assignment of the conduction band minimum to X point. 6A The structure at 952 cm"1 (Figure 3) has been assigned as 2TA(X) at E = .65 (1,1,0) (per SNI calculation). This assignment is based on the fair agreement in energy of the observed feature with calculated energies. The SNI calculation shows this point to have sector number (2,1), in agreement with the observed shape. The polarization properties of this feature could not be obtained, as they are obscured by the LO phonon in some geometries. Phonon density of states calculations for 3C SiC show a gap between the acoustic and optical branches. For overtone modes, this gap should be replicated in the two- phonon spectrum at double the energy. Some weak structure does appear in this region (~1280+1A70 cm'l) which may be due to combinations with critical points at W or three- phonon modes. This structure remains unidentified. 1 marks The strong scattering beginning at 1A7A cm- the onset of overtone scattering from the TO branch. This structure has an energy well below the energy of 2TO(X) (1522 cm-1) or 2TO(L)(1532 cm-l), indicating that the energy minimum of the TO branch is not located at X or L. The SNI calculation exhibits such a minimum along 2 at R 2 .7 (1,1,0). The feature has therefore been assigned to 2TA(X) with sector number (1,0), in agreement with the observed shape. Dispersion curves of other calculations do not show this minimum, to the credit of the SNI calculation. 65 The strong scattering beginning at ~1670 cm_1 is mainly from L0 overtones. Unfortunately critical points in the optic branches are closer together in energy than the acoustic branches, making assignments more difficult. Model calculations also have the most difficulty accounting for the optic branches. If the impurity-dependent feature at 1630 cm.1 (discussed previously) is disregarded, there appears to be a gap between the TO and LO branches. The SNI calculation shows only a very narrow gap between these branches, especially along 2 (see Appendix D). Assignment of structure from the optic branches must await more refined calculations or possibly neutron scattering data from only selected critical points. Knowledge of branch energies at W, for example, would be a great aid in assessing the validity of various model calculations and would allow for a more complete accounting of the observed spectra. CHAPTER III RAMAN SCATTERING FROM NITROGEN ELECTRONIC IMPURITY LEVELS IN 3C 810 A. Background An important aspect of semiconductor applications is the ability to modify host crystal electrical transport properties with the inclusion of various impurities. The thermal behavior of electrical transport properties in doped semiconductors can be accounted for by the presence of states existing just below the conduction band or just above the valence band, referred to as donor and acceptor states, respectively. This study and the following discussion are limited to donor states of substitutional impurities. If a group V impurity atom is substituted for an atom of a IV-IV semiconductor, the extra electron does not participate in the tetrahedral bonding. Instead, it is weakly bound to the impurity by the extra unit of nuclear charge. The ionization energy of the bound electron, as measured from the minimum of the conduction band, can vary from several meV to several hundred meV, depending on the impurity and the host semiconductor. The donor electron can be thermally liberated to the conduction band, where 66 67 it can participate in electrical conduction. Because semiconductor band gaps are usually considerably larger than the donor binding energies, the electrical properties of semiconductors are dominated by impurity electrons (except at very high temperatures). In this simple picture, the carrier concentration varies roughly as exp(-EO/2kT), enabling early workers to extract the ionization energy EO from transport measurements.26 Much of the early theoretical work on donor states and application to real materials was performed by Kohn and Luttinger,27 and the method is often called the Kohn- Luttinger effective—mass theory (EMT). Qualitatively, in the effective mass picture the extra unit nuclear charge polarizes the host lattice and creates an impurity poten- tial in the lattice. At large distances from the impurity cell, this potential is 2 -e 0(5) = E.— (11) where e is the static dielectric constant of the host crystal. For direct gap materials, effective mass theory assumes the impurity electron as weakly bound and has an orbit large enough that (11) applies. The resulting impurity states have a hydrogen-like spectrum (measured from the conduction band minimum) given by 68 l m* En —-;1_2IE2_Ry (12) where Ry is the hydrogenic Rydberg, e is the host crystal static dielectric constant, and m* is the isotropic effec— tive mass of the impurity electron. Experimentally, it is found that this simple picture can account fairly well for the excited state energy levels of weakly bound donors but is not satisfactory for ground state levels or for "deep" (tightly bound) impurities in general. This failing arises from the fact that IS ground states, or "deep" states in general, have wavefunctions with an appreciable amplitude in the impurity cell, where the potential cannot be described by (11). Excited states, on the other hand, have a much smaller amplitude at the impurity cell so that (11) and (12) become more realistic. These simple consid- erations provide the motivation for a more rigorous approach (described later) which treats indirect gap materials. Technological advances have made it possible to produce silicon and germanium with very closely controlled impurity concentrations, and most work concerning donor levels has focused on these two materials. The most direct method of studying impurity levels is with infrared absorp- tion or Raman scattering. The energies and symmetries of the donor levels can be obtained directly using optical methods in conjunction with uniaxial stress and Zeeman 69 effect studies.28 Raman scattering from donor levels was first observed in 81,29 and subsequently in GaP,3° Ge,31 some polytypes of SiC,32 and CdS.33 In this chapter, the observation of Raman scattering from nitrogen donors in 3C SiC is described. Much of the experimental and theoretical discussion of Chapter II is applicable. This study is limited to dilute nitrogen concentration, so that impurity wavefunction overlap is negligible. B. Donor Levels The original Kohn-Luttinger treatment of impurity levels in semiconductors has undergone considerable refine- ment since its introduction. The simple hydrogenic picture of donor levels predicts the same energy levels for all single donor impurities. For example, the donor level energies of a IV-IV semiconductor having a group V impurity are predicted by EMT to be independent of the actual impurity. Experimental results do not indicate identical energies for different group V donors, especially for ground state energies. However, EMT does account well for the excited states of "shallow" donors. The applicability of this theory to a particular impurity can only be established by comparison with subsequent experiment. In indirect-gap materials the conduction band minimum and its symmetry 7O equivalences give rise to n-fold degenerate levels in the EMT, where n is the number of minima or "valleys." This degeneracy is actually lifted by "intervalley mixing,"3“’3s which gives rise to the experimentally observed "valley orbit splitting."36 The EMT is obtained in the following manner. The host crystal wavefunctions satisfy the one-electron approximation 2 2 |(-h /2m)v +VO|T = E0? (13) where V0 is some effective potential having the symmetry properties of the host lattice. The solutions are Bloch functions, labeled with band index n and wavevector E, + Tn(k,;) and Eno(k). The impurity wavefunctions satisfy 2 2 l(-h /2m)v +VO+U|¢ = so (1A) where U is the perturbation potential due to the impurity and E is measured from the conduction band minimum. In the conventional EMT, the impurity wavefunction is expanded as N ¢(r) = 331 oij(r)W(kj,r) (15) where N is the number of equivalent minima or valleys in the conduction band at wavevector RJ, and the OJ are numerical coefficients. Fj(;) is called the envelope function, since 71 it modulates the Bloch function T(RJ,P) and produces the impurity electron binding. The Fj(¥) satisfy the effective mass equation37 |Boj(—iV)+U(F)|FJ(?) EFj(?) (16) where E0 is the energy of the host crystal at the jth J minimum with R replaced by -iV. Several key assumptions are made in obtaining (16). Note that in (15) there is no sum over bands. The EMT assumes that the perturbation potential 0(2) is too weak to couple to other bands. Typically, the energy between the valence and conduction band at the minimum, that is Ec(kj)-Ev(kj), is several eV, while the donor binding energy is of order .1 eV. Similarly, the potential U(;) is assumed to be slowly varying so that it has no strong high frequency Fourier components. Finally, (16) is basically a one- valley equation, neglecting the fact that the N valleys are coupled. The correct many-valley equation is given by38 oi exp|i(Ei-E )oFI[Boi(—iv)+U(?)-B|Fi(?) = 0. (17) 1 J IIMZ i The solutions of (16) are N-fold degenerate, while the solutions of (17) provide for the observed level splitting. The assumptions above were first investigated in a systematic, 72 quantitative way by Pantelides and Sah.39 Their principal result is that the assumptions concerning the impurity potential are realistic provided that the impurity atom has the same core structure as the replaced host atom. Such impurities are referred to as "isocoric," and their perturbation to the host crystal potential can be well represented by (11). Any agreement of EMT with experiment for non-isocoric impurities is strictly coincidental. The many-valley equation (17), while correct, does not yield much insight into the nature and symmetries of the impurity levels. Instead, the one—valley equation (16) and group theory can be used to obtain the level symmetries (but not their energies). In the vicinity of a conduction band minimum at, say, R1 = (1,0,0), and using (11), the one— valley equation (16) becomes 2 2 2 2 2 2 [_a_i._1_(.§_2.+_§_)-s_-a] ale) = 0 (18) 8Z2 8r where EO(R) has been expanded to second order. The form of EO(-iV) in (18), appropriate for 3C 810, is determined by the symmetry of the zone and depends on which minimum is being considered. (The factors l/ml and l/mt are conven- tional effective masses along and transverse to the wave— vector of the minimum, respectively.) Equation (18) is very similar to the hydrogen equation, and the convention has 73 been adopted of labeling the first few levels with hydrogen—like labels. For m1 = mt (spherical valleys) the solutions of (18) are exactly hydrogen-like (18, 2S, 2P, etc.). For ml 2 mt, the hydrogenic P level splits into a singlet and a doublet, which are labeled PO and Pi. This labeling is somewhat misleading, since the solutions to (18) have prolate spheroidal symmetry rather than spherical sym- metry. Solutions of (18) have been obtained numerically for various ml/mt ratios.“° In the one-valley approximation, every term in the sum (15) has the same energy by virtue of the N—fold degen- eracy of (18). The degenerate set is referred to as the 18 manifold, 2PO manifold, etc. The wavefunctions ¢(;) are solutions of (1A) (where the Hamiltonian has the symmetry of the impurity site) and must belong to representations of the impurity site symmetry group. For zinc-blende this group is Td' The conduction band minima in 3C 810 are at X point,"2 (yielding three equivalent minima), and Bloch functions at X transform under the group of the wavevector at X, D Solutions of (18) must belong to representations 2d° of Doo Products of functions such as FJ(;)W(RJ,;) will h' belong to representations of the group formed from the operations common to both Dooh and D2d’ or D2d' The Frobenius reciprocity theorem"1 may then be used to determine the subduced representations in Td‘ This 7A analysis for 3C SiC is indicated in Table 9. Group representation labels follow Reference 8. The corre— spondence of the representation labels here with those of Chapter II for Td are A1(Fl),E(Fl2),Tl(T25),T2(P15). Table 9. Donor level symmetries Manifold I(Dmh) I(D2d) I(Td) S 2+ A A +E g l l 2+(P ) A A u o 1 1+13 P Hu(Pi) E T1+T2 The important result is that the threefold degenerate solu- tions of (18) have been split by taking proper account of the true symmetry of (1A). The states are commonly referred to as lS(Al), lS(E), 2PO(A1), etc. The same result can be obtained by realizing that the set of functions Fj(;)T(Rj,;) will transform into one another under the operations of Td’ generating a representation which can be reduced in Td' The coefficients 05 in (15) for each representation can be found using projection operators, giving a set ¢(;) having the proper symmetry (i.e., that of (1A)). The set ¢(P) so determined will have only approximate energies in that they arise from the one-valley approximation (16), which neglects 75 inter-valley mixing. These sets for the S manifold are given by (15) with . 1 a301,) /_3_ (1,1,1) o.(B) = 2L (1,-l,0),2L (l,0,—l) J /2 /2 where j =l,2,3 denotes the minimum along kx, k and k2, y’ respectively. Group theory does not give the ordering of the levels. The assumption is made that the totally symmetric lS(Al) level is the ground state since it has a non—vanishing amplitude at the impurity center and will have a greater overlap with the binding impurity potential. Higher mani- folds are not expected to exhibit significant splitting, since they have a smaller amplitude in the vicinity of the impurity cell. The ordering of the first few levels in 3C SiC, determined by luminescence spectra in conjunction with Zeeman studies,”2 is lS(Al), lS(E), 2P 2s, 2P 0’ i' The splitting of manifolds above the 18 manifold is experimentally found to be negligible, so that only the splitting of the 18 manifold need be considered. C. Raman Scattering Having obtained the symmetries of the donor levels, the Raman selection rules may be obtained by group theory. 76 The scattering intensity for given initial and final states li> and |f> is proportional to"3 + + + -+ + . —> -> + + . Z ( > ' (19) m Ei-Em-hw Ei-Em+hwi + where e ,w s s and gi’wi refer to the scattered and incident photons. The dyadic Operator FIm> is the totally symmetric Al ground state, (19) will vanish unless Pf=Fd. Exactly as in Chap- ter II, a set of Raman intensity matrices may be defined corresponding to the irreducible components of the dyadic Operator. Thus, the intensity matrices of Table 3 apply. By adjusting polarization vectors 61 and gs and using the zinc-blende intensity matrices, the symmetry of an excited level may be experimentally determined (assuming a transition from the ground state). By making assumptions about the energies of the intermediate states, (19) may be used to Obtain estimates 77 of the relative Raman scattering intensities of transitions between various impurity levels.““ Such calculations have been performed for donors in 6H 810.32 A general result Of Reference 32 is that transitions between manifolds (say lS+2S) are of order Eg/(Ec(kj)-EV(RJ))2 less intense than transitions within a split manifold (say lS(Al)+lS(E)), where EB is the ground state ionization energy and EC and EV denote conduction and valence band energies at k=kj, the conduction band minimum. This ratio for shallow donors (such as nitrogen in 3C SiC) is typically of order 10_u. Strong donor spectral lines are then expected to indicate transitions within a given split manifold. D. Experimental Samples A and B of Chapter 11, used in this study, have nitrogen donors in an unknown concentration. The color of pure 3C SiC is pale yellow due to weak intrinsic absorption in the blue. Nitrogen doping is known to shift the color to yellow-green, due to free carrier intraband absorption which absorbs preferentially in the red."5 Sample A visually appears pale yellow, with very little green observable. Sample B appears dark green-yellow, indicating a greater concentration of nitrogen donors than Sample A. Sample donor concentrations will be discussed further following the presentation of sample spectra. 78 The experimental geometry employed here, as in Chapter II, is back-scattering. The helium temperature spectra are recorded using a commercial optical dewar equipped with a temperature control system. Other exper- imental aspects, such as sample preparation, mounting, laser heating considerations, and experimental selection rules have been discussed in Chapter II. The spectra of this chapter are not corrected for throughput since they are over a limited spectral range. The experimental geometry and equipment are detailed in Appendix A. E. Results The room temperature first-order spectrum of sample A is shown in Figure 11a, recorded using a A579A argon-ion laser line and instrument resolution of 3.7 cm‘l. The indicated geometry allows components Al and E, but not the L0 or T0. The first-order phonons appear weakly due to experimental limitations. The two small lines near 50 cm—1 are laser flourescence lines, designated F. The background structure beyond A00 cm—1 is recognizable as two—phonon scattering. This spectrum is repeated in Figure 11b at 770 K, showing the appearance of a new line. This spectral line was investigated using a different laser line (A765A) in order both to avoid the small flourescence lines and establish the line as due to a Raman process. 79 VTTFTTTITrTrUUTUTFTI SAMPLE A 1 12300°K SRZZD< L0- IM+E (A) INTENSITY 0 Figure 11. SAMPLE A I=77°K X(ZZ)X A|+E (B) lllJLlLLli 500 IOOO FREQUENCY SHIFT (cm") X(ZZ)X spectra of sample A, A579A laser line. 80 These spectra, at 7° K, are shown in Figures 12a and 12b, with instrument resolution of 3.7 cm-l. The strengths of the line in these two geometries is in the ratio 3:1. The Poulet intensity matrices of Table 3 show that this line has E symmetry, meaning it appears in geometries apprOpriate to an A1+E transition. The symmetry of this line has been investigated in a number of geometries, and is found to be purely E within the limits of experimental selection rules as discussed previously. A high-resolution search for other structure was performed, especially in the region close to the laser shoulder, without success. (The laser shoulder limits this search to energies greater than ~5 cm—l.) Sample B, which is thought to be more heavily doped with nitrogen, exhibits the same line, but with roughly twice the absolute intensity of sample A. The energy of this line, measured from a nearby laser flourescence line, is 67.5 1.5 cm-l. The strength and deconvoluted width of this line in sample A as a function of temperature is presented in Figure 13. The linewidth in sample B is roughly 10% greater than sample A at 77° K. As mentioned previously, the sam- ples are unavoidably heated to some extent by the incident laser power. The temperatures of Figure 13 are measured by a thermocouple attached to the edge of the sample with thermal grease. The thermocouple does not measure the true 81 1. .I I SAMPLE A n ‘ r I27°K I I. X(Y Z )X 4 §E P 4 a I- ., >- J L t: I- —I 00 1 1 z r I Lu I— I- 4 .2. .. b SAMPLE A .. 1’:7°K . __ X(Z'Z')X .. I... A|+£E+Tz d F - 0 5O IOO FREQUENCY SHIFT (cm") 0 Figure 12. Sample A spectra at T27°K, A765A laser line. 82 .ooeeUCOQOU mLSmeOQEOp szWCOpCH one nape: mafia Locoo :omoppfiz On. A 3:; mmak<¢ma2u._. nXu. — on u G In. .mH osswea X v. .8 ms... mv MU. .OuA 13 H' )m m3 .N [Halo I; ..A 83 temperature of the sample at the point of the incident laser focus. To assess the effect of laser heating, the linewidth at 77° K was measured using incident laser powers of 50, 100, and 200 mw. The linewidths at 50 and 100 mw were identical to within experimental error, while the linewidth at 200 mw was roughly 5% greater. The linewidths of Figure 13 were therefore measured using 100 mw laser power to minimize sample heating errors. F. Discussion The observed line is attributed to a lS(A1)+lS(E) transition between nitrogen donor levels, referred to as a valley-orbit transition line. The symmetry and impurity concentration dependence of the line are the main reasons for this assignment. The intensity temperature dependence is also consistent with this assignment. Figure 13 shows the line intensity approaches a constant as the temperature is lowered, persisting to 7° K. If another impurity state existed which was significantly lower in energy than the two states involved in the Observed transition, the inten— sity would fall at low temperature as the donor electron returns to this hypothetical ground state. Thus to within energy kT(~5 cm-1 for T=7° K), the observed transition is from the true ground state. Preceding theoretical estimates of the inter- versus intra—manifold transition intensity 8A strongly favor the assignment lS(Al)+ls(E). With increasing temperature the ground state is depopulated, accounting for the intensity decrease with increasing temperature. The linewidth is thought to be due mainly to perturbations of the impurity site by acoustic phonons.1+6 The relative strength of the line in samples A and B indicates that sample B has approximately twice the nitrogen concentrations Of sample A. It is found in Si”7 and Ge"8 that the width Of valley-orbit lines is a sensitive function Of impurity concentration when the concentration is near the metal-nonmetal transition. Since the linewidths of samples A and B vary by only 10% while their concentrations differ by a factor of two, the concentrations must be far from the metal-nonmetal transition. If the conduction band minima in 30 SiC were not at X, but instead along A (as in Si), there would be six equivalent minima instead of three, so that the 18 manifold would contain six levels. Taking prOper account of the true impurity site symmetry, these six levels will split into lS(Al), lS(E), and lS(T2) levels. Assuming a specific two-band model, Colwell and Klein32 have calculated the Raman cross section for transitions between levels of the split 18 manifold. They find the cross section proportional to 85 (20) where N is the number of equivalent minima, 31 and gs are polarization vectors of the incident and scattered photons, w and v are representation labels, and M3 is defined by M a 3213(12) -> aB,j 8Ea§EB kj' ~ For the minima here, MJ is diagonal. If the transition lS(A1)+lS(T2) is observed, then the conduction band minimum cannot be at X. Using the a; appropriate to a sixfold min— imum along A,39 it can be shown that (20) vanishes for the lS(Al)+ls(T2) transition. Because Of the many EMT approx— imations invOlved in (20), its vanishing implies only that this transition is likely quite weak. This transition is allowed by group theory, since the dyadic Operator in (19) has parts belonging to T2. Thus failure to observe this transition does not preclude the existence of a lS(T2) level. Raman scattering in this case is not an effective method of establishing the position of the conduction band minima. However, the failure to observe the lS(A1)+lS(T2) transition here is consistent with the assignment of the conduction band minima to X. Recent luminescence spectra of 3C SiC1+9 have revealed several inter—manifold energies, and show no 86 observable valley-orbit splitting of manifolds above the IS manifold. These inter-manifold energies are listed in Table 10, with level energies measured negatively from the conduction band minima. Table 10. 3C 810 nitrogen donor level spacing Levels Energy difference (cm-l) 2Pi_2po -38 2P,—2s ~25 2Pi—3PO +25 2Pi-3Pi +A5 Several important 3C SiC parameters are Obtained in Refer- ence A9. The effective mass mt is extracted from the Zeeman splitting of the 2P,_r level giving mt==.2A. Since the EMT is expected to apply quite well for excited levels, the inter- manifold energies above in conjunction with the EMT calcu— lations of Reference A0 give 6 =10, nu_=.67, and an EMT lS manifold energy of A7.1 meV (380 cm-l). Energy difference between donor-acceptor pair and free electron to bound acceptor luminescence components gives an experimental estimate of the lS(Al) binding energy of 5A meV (A36 cm-l). Most importantly, observation of two-electron satellites of 87 the 2Pi state directly yields the lS(Al) ground state energy of 53.6 meV (A32 cm‘l) (see Reference A9 for details). The transition energy observed in this study for the lS(Al)+lS(E) transition is 67.5 cm-l. A rough estimate of the lS(Al) binding energy can be Obtained by assuming the IS manifold valley-orbit splitting approximately preserves the "center of mass" (as in degenerate perturbation theory). Making this assumption, and using the EMT lS manifold energy 1 of 380 cm— , the ground state lS(Al) energy becomes 1 380 cm"1 +(2/3) x67.5 cm’ = A25 cm‘l. This value is consistent with the lS(Al) energy of A36 cm-1 from luminescence measurements. CHAPTER IV SUMMARY A. Two—Phonon Spectra The intensity of the second-order spectra of 3C 810 is dominated by the P1 or totally symmetric component of the polarizability, in agreement with spectra of other IV-IV semiconductors. Feature energies and assignments are con- sistent with previously measured phonon energies at F, X, and L. Analysis of the spectra indicates that the minimum of the TO branch is not at X but along 2, at an energy of 737 cm—1. Critical point TA(Z) (which appears in the dis- persion curves Of all published calculations for 30 SiC to date) is found to have an energy of A76 cm'l. The Observed spectra in conjunction with a model calculation are used to arrive at a set of critical point sector numbers. This set is consistent with the observed spectral shapes, energies, and selection rules, and satisfies the two- and three- dimensional Morse relations for all phonon branches. The close agreement between Raman and luminescence measurements of X point phonons supports the assignment of the conduction band minimum to X point. Results from the two—phonon spec- tra agree quite well with previous experimental results and 88 89 may serve as additional input for more refined lattice dynamical calculations. At present, there is insufficient information to make feature assignments to critical point W. These assignments require either neutron scattering data or reliable lattice calculations. It may be practical to perform neutron scattering on the small 3C 810 samples available at selected points in the Brillouin zone. Knowledge of phonon energies at W, for example, would greatly aid the interpretation of the second-order spectra. The effect Of impurities on second-order spectra could be better assessed if high-purity samples were avail- able. Neutron—activation analysis in 810 is not practical because of the long decay times of activated host atoms. Transport measurements require samples large enough to be cut into well-defined shapes, which has not been possible with 30 SiC. Impurity concentration at present can only be gauged in a very qualitative fashion. The variation Of 2:1 in nitrogen concentration in the samples here is found to have a small effect on the second—order spectra, and a negligible effect on the identified critical points. Purer samples have less luminescence, which might allow higher temperature second-order spectral recordings. Surface damage due to sample preparation has no effect on the second—order spectra. 90 B. Nitrogen Donor Spectra The low temperature spectra of nitrogen doped 3C 810 show an additional Raman line at 67.5 1.5 chl. This line is identified as a transition between donor impurity levels. The thermal behavior, symmetry, theoretical estimates, and concentration dependence of this line strongly favor its assignment to the lS(A1)+lS(E) valley-orbit transition. No other structure is Observed, consistent with the assignment of the conduction band minimum to X point and with theoret- ical estimates of cross sections of other transitions. The thermal behavior indicates that the lS(Al) level is the true ground state. The Observed valleyAOrbit lS manifold splitting agrees with luminescence measurements and EMT. The measured 18 manifold splitting may stimulate a cal— culation of the isocoric donor levels in SiC:N along the lines of Reference 39. If more heavily doped samples of known concentra- tions become available, the behavior of this line can be studied as a function of donor concentration, up to and through the Mott transition.“7"+8 Transitions to higher manifolds are expected to be very weak, but might be observable using modulation spectroscopy. The 2P: levels will be split by a magnetic field. A modulated magnetic field in conjunction with a synchronous detector might produce detectable signals. APPENDICES APPENDIX A EXPERIMENTAL GEOMETRY AND EQUIPMENT APPENDIX A EXPERIMENTAL GEOMETRY AND EQUIPMENT The geometry employed throughout is direct backscattering. The arrangement of optical components is shown in Figure 1A. The incident beam from the laser is passed through a prism and iris to remove laser plasma flourescence lines from the main beam. The laser beam is highly linearly polarized in the plane of the drawing as it emerges from the laser. The polarization rotator permits the polarization of the incident beam to be adjusted per- pendicular to the plane of the paper. The beam is focused to a small spot on the sample by the small lens. A small dielectric coated mirror diverts the incident beam onto the sample surface. Experimentally it is found that this mirror will produce a reflection of mixed polarization unless the incident polarization is in or perpendicular to the plane Of incidence. Only these configurations were employed in order to maintain the polarization purity of the incident beam. The sample is adjusted to reflect the main beam back to the mirror so it is not collected by the lens. Although not shown, the dewar windows are adjusted so that window reflections are also blocked by the small mirror from entering the collection lens. The remaining scattered 91 .mhpoEoow wQAAOppwomxomn cmemm .za opswfim E¢mm BZMQHOZH EmHma Salli T.. moeaeom oneaNHmaqoa ...JL 6, mzmq 22mm HHUv x qum . Ifiu mmemzomeomam \ N mqazam / mmqmzamom . mommHz 2222m 0H0m uaoco LMCDO I two (pcnc> UJOCh OLUU) OP4£= (Dunn erwA ouooa 112 110 111 HR): [UH-I: OWL/I) C\UUO WO‘IO kuvro C)! 0+2»: aware IIO 111 112 NPJFJ +403N cser FJOFO nicwa Fauna NHU‘I U'II\)I\.) Note: "6" e 5.8; "—" = .2. APPENDIX C THROUGHPUT AND WAVELENGTH CALIBRATION APPENDIX C THROUGHPUT AND WAVELENGTH CALIBRATION The spectral response of the apparatus is wavelength dependent. Over a limited spectral range, the instrument may be considered to have an approximately constant response. The second-order spectra of Chapter III extend over a broad spectral range, necessitating a correction for system response variations. Given a true spectrum M(A), the measured spectrum M'(A) is given by M'(1) = T(A)M(1) where T(A) characteristizes the instrument throughput with wavelength. T(A) can be determined by measuring the spec- trum of a source with a known spectral intensity. Referred to as standard lamps, these sources have a true intensity 8(1). From the measured intensity, S'(A), the system throughput can be determined as T(1) = S'(A)/S(l). Having T(1), the true spectrum M(l) is given by M(1) = M'(l)/T(1). 98 99 The standard lamp used in this study, a General Electric 6.6A/TAQ/lCL-200W quartz-iodine lamp, has an output which has been characterized by the National Bureau of Standardss° for use as a spectral standard. In recording the standard lamp spectrum, it is essential to duplicate as nearly as possible the conditions under which the correctable spectrum M'(A) was recorded. The technique used in this study to simulate the actual experimental conditions is shown in Figure 15. The iris simulates the laser spot of an actual experiment and is placed in the same position as an actual sample. The MgO slide, used to provide a diffuse reflecting surface, is placed as closely as possible to the iris. The slide is prepared by coating a glass slide with smoke from a burning magnesium ribbon. The reflectance of slides prepared in this manner has a variation with wavelength of less than 1% over the entire optical spectrum.51 A light shield prevents the lamp output from entering the spectrometer directly. The effective intensity may be adjusted by varying the lamp to slide distance. There are several important reasons for attempting to simulate a point source (as is the case in an actual experiment). First, the lens, polaroid, and scrambler must be filled, meaning as much of their area as possible should participate in the light transmission. In particular, the 100 .hameoow psoEoLSmmoE useswsopnp Eopmhm .mH ohzwflm mzm3 .NH Ohsmflm Arse Kim Szmaommu 000m 000_ 00m_ 000 a. I. RIqlql—dAA-dMSJJAJJdd—qudq4_14—dJ4deJ XA.N.NVM % 45.12% OLHE..PL._. ...bb.._._bf ...L+_p+ht mmz: zomz + l x622"... 1 AIISNEINI 10A Table 13. Neon calibration lines Measured wavelength (A) Mit tables (A) A61A.82 A61A.39 A656.82 A656.39 A70A.8A A70A.A0 A750.06 A7u9.58 A810.5A A810.06 A863.7A A863.08 A892.72 A892.10 A957.68 A957.03 5005.70 5005.16 105 to be ~i.3A. A systematic mean offset of ~.5A is easily corrected for. Random errors can only be minimized by measuring spectral features from the nearest calibration line. APPENDIX D SECOND—NEIGHBOR IONIC (SNI) CALCULATION APPENDIX D SECOND—NEIGHBOR IONIC (SNI) CALCULATION The SNI calculation21 employs a simple rigid-ion model with first and second neighbor force matrices and a long-range Coulomb part. An advantage of the SNI cal- culation is that force constants may be simply related to experimental lattice energies at F, X, and L. For x, y, and z directed along the C2 axes in zinc-blende, the most general forms of the force matrices allowed by the symmetry of the lattice are (first neighbor), Fi== *112 III (second neighbor) i QUDUO V0707 '(DQ'O UOUDQ 0961: Ov'CC where i.=l or 2 for Si or C second neighbor interactions. The SNI calculation assumes 6 =0, and u =0. The last parameter of the calculation is an effective ionic charge Z. The parameters which result in the best fit for 30 SiC are given in Table 1A. The calculation is straightforward, and further details may be found in Reference 21. 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