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Effect of strains and electronic fields on the acceptor states in boron-doped silicon White, James Judson 1966

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The University of Brit i s h Columbia' FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of JAMES JUDSON WHITE B.Sc, Queen's University, Kingston, Ontario, 1960 M.Sc, Queen's University, Kingston, Ontario, 1962 THURSDAY, FEBRUARY 23rd, 1967, AT 2:30 P,M„ IN ROOM 301, PHYSICS (HENNINGS) BUILDING COMMITTEE IN CHARGE Chairman: W„ D. Finn R,Barrie K. ti. Harvey J, W. Bichard D.. L. Williams Ro E. Burgess L, Young External Examiner: S.. Zwerdling Lincoln Laboratory Massachusetts Institute of Technology Research Supervisor: R. Barrie EFFECT OF STRAINS AND ELECTRIC FIELDS ON THE ACCEPTOR STATES IN BORON-DOPED SILICON ABSTRACT In boron-doped silicon, optical excitation of bound holes from the ground state to the various excited states of the neutral acceptor impurity leads to an absorption line spectrum. By applying an external strain, the degeneracies of the acceptor ground state and the four lowest "observable" excited states were determined and were found to only partially agree with theory. By applying a uniform electric f i e l d to compensated samples, the "Stark effect" for the acceptor states was observed. The Stark shift of the excited states is second order in the f i e l d as was earlier predicted from symmetry consid-erations. The Stark broadening of the acceptor absorp-tion lines is attributed to an unresolved partial removal of degeneracy of the'excited states. The absorption line broadening mechanisms (phonon, dislocation, concentration, ionized Impurity) were deter-mined from new'half width measurements, which corrected an earlier study. The two theories of ionized impurity broadening, which is caused by the screened Coulomb "internal fields" of thermally ionized impurities in uncompensated samples, are compared. The effects of compensation on the boron absorption spectrum were measured and attributed to the unscreened Coulomb fields of compensationally ionized impurities. The properties of a weak new absorption line, which appeared in the compensated spectrum, are described. GRADUATE STUDIES G. M, Volkoff R. Barrie R.,E, Burgess M. Bloom J. B. Brown Field of Study: Physics Electromagnetic Theory Quantum Theory of Solids Stochastic Processes in Physics Advanced Magnetism Low Temperature Physics PUBLICATION "Degeneracy of Impurity States in Boron Doped Silicon", M, W, Skoczylas and J. White, Canadian Journal of Physics, 43, 1388, 1965. EFFECT OF STRAINS AND ELECTRIC FIELDS ON THE ACCEPTOR STATES IN BORON-DOPED SILICON - ' by JAMES JUDSON WHITE B.Sc., Queen's University, Kingston, Ont., I960 M.Sc., Queen's University, Kingston, Ont., 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1966 In presenting this thesis in pa r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that die Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission„ Department of T^ /y^ yo-Zc^ o The"University of B r i t i s h Columbia Vancouver 8, Canada ^ t e Ma^ 14, If67 i i ABSTRACT In boron doped silicon, optical excitation of bound holes from the ground state to the various excited states of the neutral acceptor impurity leads to an absorption line spectrum. By applying an external strain, the degeneracies of the acceptor ground state and the four lowest "observable" excited states were determined and were found to only partially agree with theory (Schechter 1962). By applying a uniform electric field to compensated samples, the "Stark effect" for the acceptor states was observed. The Stark shift of the excited states is second order in the field as predicted by Kohn (1957) from symmetry considerations. The Stark broadening of the acceptor absorption lines was attributed to an unresolved partial removal of degeneracy of the excited states. The absorption line broadening mechanisms (phonon, dislocation, concentra-tion, ionized impurity) were determined from new halfwidth measurements, which corrected an earlier study (Colbow 1963). The ionized impurity broadening is caused by the screened Coulomb "internal fields" of nearby ionized impurities which are present in uncompensated samples at temperatures greater than 50°K. A new theory of this broadening contribution (Cheng 1966) is in reasonable agreement with experiment% the earlier theory of the same effect (Colbow 1963) is shown to be inadequate. The effects of compensation on the boron absorption spectrum were measured and attributed to the unscreened Coulomb fields of ionized impurities present because of the compensation. The properties of a weak new absorption line which appeared in the compensated spectrum are described. i i i TABLE OF CONTENTS Abstract i i Table of Contents . . . i i i List of Tables . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . vi Acknowledgements v i i i Chapter 1 - INTRODUCTION 1 Chapter 2 - THEORY OF ACCEPTOR STATES 2.1 Hydrogenic Model 6 2.2 Effective Mass Model 7 Chapter 3 - GENERAL EXPERIMENTAL PROCEDURES 3.1 Apparatus 14 3.2 Transmission and Absorption . . . . . . . . . . 17 3.3 Sample Preparation and Mounting 18 3.4' Electrical Contacts 20 Chapter 4 - BROADENING OF ACCEPTOR ABSORPTION LINES 4.1 Introduction . . . . . . . . . . . . 23 4.2 True Halfwidths . 23 4.3 Broadening Mechanisms . 26 4.4 Ionized Impurity Broadening 31 4.5 Statistical Stark Broadening 38 4.6 Conclusions 39 Chapter 5 - APPLIED STRAINS 5.1 Introduction 41 5.2 Degeneracy of the Acceptor States 41 5.3 Abnormal Broadening in Strained Spectrum , 45 Chapter 6 - APPLIED ELECTRIC FIELD (STARK EFFECT) 6.1 Theory of the Stark Effect a) Effeotive Mass Approximation . . . . . . . . . . . 50 b) Breakdown of the Effeotive Mass Approximation . . . 53 i v Page Chapter 6 - (Cont'd) 6.2 E l e c t r i c a l Properties of Si(B,P) at 4 K a) Introduction . . . . . . . . . . . . . 54 b) Measurement of E l e c t r i c a l Properties . . . . . . . . 55 c; E l e c t r i c a l Characteristics 57 d) Nonuniform Fields - Contact Effects 61 6.3 Stark Effect on Si(B pP) - Boron Lines a) General Effects , 63 b) Peak Position . . . . . . . . . . . . . 66 c) Halfwidth . 69 d) Peak Height . . . . . . . . . . . . . 72 e) Area . . . . . . . o « » . . . . . . . . . • « « » 72 f ) Effect of the Internal Field on the Stark Parameters 76 6.4 Conclusions - Stark Effect . . . . . . . . . . . . . . . . 76 Chapter 7 - COMPENSATION EFFECTS 7.1 Compensated Si(B,P) Spectrum - General. . . . . . . . . . . . 79 7.2 Effect of Compensation on Boron Peak 3 . . 79 7.3 Internal E l e c t r i c Fields due to Compensation 83 7.4 Conclusions - Compensation Effects 86 Chapter 8 - NEW LINES 8.1 New Line due to Compensation . . . < . . . . . . . . . . . . 87 8.2 Effect of I n t r i n s i c Radiation on New Line a) Introduction . . . . . . . . . . . . . . . . . . . . 90 b) Present Experimental Arrangement 91 c) Reduced Intrinsic Radiation (Cold F i l t e r ) . . . . . 92 d) Increased Intrin s i c Radiation (Laser) . . . . . . . 93 8.3 Stark Effect on Si(B,P) - New Lines a) Effect of Applied Field on Peak Position . . . . . . 96 b) Effect of Applied Field on Peak Height . . . . . . . 96 e) Poor Contacts . . . . . . . . . . . . . . . . . . . 100 d) Good Contacts . . . . . . . . . . . . . . . . . . . 101 8.4 Conclusions - New Lines . . . . . . . . . . . . . . . . . . 103 Chapter 9 - GENERAL CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 106 Appendix A ABSORPTION RELATIONS . . . . . . . . . . . . . . . . . . . 109 Appendix B DONOR STATES . . . . . . . . . . . . . . . . . . . . . . . 112 113 V LIST OF TABLES Table Page 4.1 Halfwidth Samples . . . . . . . . . . . . . . . . . . . 24 4.2 Dislocation Broadening-Results 27 4.3. Ionized Impurity Perturbation Terms, 33 5.1 Degeneracy of the Acceptor States i n S i l i c o n . . . . 45 6.1 Stark Shift Coefficients for Si(B,P) 69 6.2 Stark Broadening Coefficients for Si(B,P) . . . . . . 71 6.3 Increase i n Neutral Boron with Field . . . . . . . . 75 7.1 Impurity Concentrations of Compensated Samples . . . 81 vi LIST OF FIGURES Figure Page 2.1 Valence Band Edge in Silicon 8 2.2 Acceptor States in Boron Doped Silicon 12 3.1 Experimental Layout (Schematic) . 16 3.2 Sample Mounting. . . . . . . . . . . . . . . 20 4.1 True Halfwidths of Si(B) Absorption Line 2 25 4.2 Broadening Contributions of Si(B) Line 2 25 4.3 Concentration Broadening Contribution ( Ah) of Line 2 . 29 4.4. Concentration Broadening of Si(B) Lines 29 4.5 Ionized Impurity Broadening Contribution of Line 2 . . . 30 5.1 Effect of Strain on the Absorption Spectrum of Si(B) . .' 43 5.2 Extra Broadening of Upper 2P-?. State by Lower State "X. . 49 6.1 Electrical Circuit 56 6.2 Electrical Characteristics and Resistivity . . . . . . . 58 6.3 Nonuniform Field Effects 62 6.4 Electric Field Distribution . . . . . . . . . . . . . . 62 6.5 Effect of an Electric Field on the Si(B,p) Absorption Spectrum a) Si(B) Peak 1 , 64 bj Si(BJ Peak 2 64 c) Si(B) Peaks 3 and 4 65 6.6 Shift of Peak Position with Applied Field . . . . . . . 6? 6.7 Stark Shifts of the Si(B,P) Absorption Lines 68 6^8 Stark Broadening of the Si(B,P) Absorption Lines . . . . 70 6.9 Peak Heights of the Si(B,P) Absorption Lines (Stark Effect) 73 6.10 Areas of the Si(B,P) Absorption Lines (Stark Effect) . , 74 7.1 Boron Absorption Peak 3 (Compensation Effects) 8D v i i Figure Page 7.2 Effect of Compensation of Boron Peak 3 . . . . . 82 8.1 Effect of Compensation on the New Peak 88 8.2 Electron and Hole Transfer Processes in Compensated Silicon 91 8.3 Effect of Intrinsic Radiation on the Si(B,p) Spectrum. . . 94 8.4 Stark Shift of P°(2P+) New Line . . . . . . . 97 8.5 Peak Heights of the New Lines (Stark Effect) . 98 v i i i ACKNOWIJEDGEWTS It is a pleasure to thank my supervisor, Dr. R. Barrie for his helpful advice and assistance both throughout this investigation and during the thesis preparation. I am indebted to Dr. J. ¥. Bichard for many valuable discussions concerning this research and to Mr. P. Y„ C. Cheng for providing his ionized impurity perturbation calculations prior to publication. The research for this thesis was supported by the Defence Research Board of Canada, Grant No, 9510-35. CHAPTER I - IUTRODUCTION A neutral acceptor consists of a positive hole (electron deficiency) loosely hound to a negative acceptor ion in a set of energy levels located near the valence band edge. Optical excitation of the bound hole from the ground state to the excited states of the neutral acceptor leads to an absorption line spectrum in the far infrared first observed by Burstein et al (1953). The original simple hydrogenic model of the impurity states was succeeded by the effective mass theory based on cyclotron resonance knowledge of the bands (Kohn 1957). With an accurate description of donor states and preliminary know-ledge of the acceptor states, Kohn predicted the effects of various perturbations (magnetic, electrical and strain) on both donor and acceptor energy levels. For donor impurities in both silicon and germanium, the observed unperturbed energy level scheme (see Kohn 1957) as well as the effects of applied magnetic field (Zwerdling et al. 1960a) and applied strains (Aggarwal and Ramdas 1965) agree with theory (Kohn 1957). Perturbation of the donor spectrum by an electric field (Stark effect) has not yet been reported. For the acceptor states, only approximate solutions of the wavefunctions and energy eigenvalues are possible in the effective mass theory because of the complex nature of the degenerate valence band. Variational solutions for the ground state and (what were thought to be) the four lowest excited states were obtained by Schechter (1962) for acceptors in both silicon and germanium. From symmetry considerations, a l l states have either 2-fold (including spin) or 4-fold degeneracy. In general, Schechter's energy level scheme agrees poorly with the observed spectrum (Hrostowski and Kaiser 1958). With a perturbing magnetic field, Zwerdling et al (l960a) attempted to observe the Zeeman splitting of the acceptor levels. Since the highest magnetic field available (39 Kilogauss) was insufficient to split the absorption lines into resolvable components, i t was 2 difficult to determine even the degeneracies, hut they appeared to agree with Schechter's ordering. For acceptors in germanium only, the variational calcula-tions were improved and extended by Mendelson and James (1964), with the major effect of introducing more low energy, 4-fold degenerate states into Schechter's energy level scheme. Mendelson's scheme is in fairly good agreement with the unperturbed experimental spectrum (Jones and Fisher 1965) but unfortunately applies only to acceptors in germanium. This thesis is a detailed experimental study of the acceptor states in boron doped silicon. The absorption line studies by Burstein et al (1953, 1956), Newman (l956), Hrostowski and Kaiser (1958), Colbow (1963) and others, has probably made boron the most extensively studied acceptor impurity. However, many of these studies of the positions and halfwidths of the absorption lines are remeasurements and reflect more the gradual refinement in measurement techniques, rather than the accumulation of new information. Other than these properties, experimental information is very meagre, especially of the acceptor excited states. The effects on the acceptor absorption spectrum of a perturbing electric field and a perturbing applied strain are the main topics in this thesis. The effects of compensation and the broadening mechanisms of the acceptor absorption lines are also reported. The application of a uniaxial compressive force (Skoczylas and White 1965) caused wide splittings of the acceptor states and the degeneracies of the ground state and the four lowest excited states were determined for the first time. Since these degeneracies did not agree with the ordering predicted by Schechter but resembled more the acceptor scheme by Mendelson for Ge, i t would be worthwhile to have Mendelson's treatment extended to acceptors in silicon. Similar independent experiments by Fisher and Hamdas (1965) came to the same conclusions. The absorption line intensities in the strained spectrum were observed to be strongly polarization dependent as expected (Kohn 1957). For large applied 3 strains, the observed abnormal broadening of several transitions was explained using the same phonon broadening mechanism (Nishlkawa and Barrie 1963> Barrie and Nishikawa 1963) responsible for much of the broadening of the unstrained absorption lines. By applying an external electric field to compensated samples, the acceptor states were observed to shift and broaden. The Stark shifts were quadratic in the field as predicted by Kohn (1957) and the corresponding Stark parameters were measured for the four lowest excited states. The Stark broadening of the absorption lines is consistent with an unresolved second order removal of degeneracy of the excited states (complementing the degeneracy information determined from the strain experiments). Significant Stark effects were observed only for large applied electric fields and the limitations of avalanche break-down due to the impact ionization of neutral acceptors by "hot" carriers (Bok et al. I960) was avoided by using compensated Stark samples. Compensation (with phosphorus) caused some of the boron absorption lines to shift and broaden measurably (at zero applied field). Calculation of these compensation effects, from the internal electric fields of the ionized impurities and from the Stark parameters measured earlier, were in reasonable agreement with experiment. A new weak line appeared in the compensated absorption spectrum (see also Pajot 1964) with no applied field and its energy corresponded to the strongest neutral phosphorus transition. The increased intensity of the new line with compensation (phosphorus), suggested that a small fraction of the phosphorus impurities were neutral. However, for equilibrium conditions at low temperatures, all compensating impurities should be ionized (Shock^y 1950). Neutral compensating impurities with long lifetimes are possible depending on the relative amounts of intrinsic radiation absorbed by the sample (Honig I960), A but present experiments show that this non-equilibrium mechanism is not res-4 ponsible for the observed new line. The above and various other properties of the new line are described. Although the new line appeared to be the strong transition of neutral phosphorus, the concentration of which depended on the intensity of the (internal and external) electric field, experimental difficulties prevented unambiguous confirmation of this result. The absorption line broadening mechanisms may be determined from halfwidth measurements at low temperatures (4 90°K). A recent such study by Colbow (1963) corrected earlier work which had neglected instrumental broadening (eg. Burstein et al. 1953)» and led both to an improved theory of phonon broadening (ETishikawa and Barrie 1963), and to a new (ionized impurity) broadening mechanism. Present experiments showed that many of Colbow's measured halfwidths were incorrect because his samples had been accidentally strained by their mounting. Conse-quently, the broadening mechanisms were redetermined from new (and more detailed) halfwidth measurements. Although the same four broadening mechanisms (phonon, dislocation, concentration, ionized impurity) were s t i l l present, the relative contributions were quite different. The Stark parameters (from the applied electric field experiments) were useful in comparing the two ionized impurity broadening theories (Colbow 1963? Cheng 1966) with experiment. The ionized impurity broadening effects are closely related to the compensation effects already discussed as both are caused by the perturbing of neutral impurities by the Coulomb fields of ionized impurities. The properties of the acceptor states described in this thesis f a l l into several separate (but easily inter—related) divisions, and these divisions are retained in the plan of the thesis. The general chapters on the introduction, theory, procedure and final conclusions (i.e. chapters 1, 2, 3» 9) apply to a l l .the results. The intermediate chapters are practically self-contained studies (with their own introduction, results and conclusions). Included are the effects of applied strains (chapter 5) and of applied electric fields (chapter 6 ) , 5 compensation effects (chapter 7 ) , the properties of the "new lines" (chapter 8 ) and the halfwidth study (chapter 4 ) . The original objective of the thesis was the observation of the Stark effect on the acceptor spectrum (chapter 6) but the use of compensated Stark samples and the observation of new lines necessitated the studies of chapters 7 and 8. These three chapters are thus closely related and constitute the central core of the thesis. The halfwidth and applied strain chapters both grew from side projects and are more independent. It is hoped that the division of the thesis into smaller, self-contained ohapters will make i t easier to read. 6 CHAPTER 2 - THEORY OF ACCEPTOR STATES This chapter discusses briefly the theory of the acceptor states and the conclusions drawn from a comparison of the theoretical and experimental energy level schemes. 2.1 HYDROSENIC MODEL A "neutral acceptor impurity" consists of a negatively charged "impurity ion" with a positive hole loosely bound to i t in a set of discrete states in the forbidden gap near the valence band edge. The impurity ion will polarize the semiconductor, and in the unit cell containing the impurity, the perturbation due to the ion is quite complex. However, at large distances, the presence of the ion is felt mainly by the perturbing electrostatic potential -e/Kr, where K is the static dielectric constant of the crystal. In this, hydrogenic model, the bound states for the hole are obtained from the same Schrodinger equation used for the hydrogen atom, except for the introduction of K and the effective mass m* of the hole. Lit. V 2 _ ef>(?) = (BVB-B) F (f) (2.l) V 2m* Kr ' The energy levels of the impurity center are given by E n - E ^ = Ej = l_m*_/ef.)2 n = 1, 2, 3, . . . (2.2) ~2 n 2 2n 2 * K ' n The effective Bohr radius of the IS ground state is a*="K2K/m*e2 (2.3) All the terminology associated with the hydrogen atom may be used to describe the impurity states; Ej is the ionization energy and n is the priaioalplo quantum number. By choosing m*/m = 0.46 to give the correct ionization energy (Ej = 44 meV ) for boron impurities in silicon (K = 12,0), the effective Bohr o radius of the ground state is 13.5 A. Thus even the ground state orbits are / o. large and extend over many orystal cells (unit cell edge for Si «= 5.4 A). 7 2 . 2 EFFECTIVE MASS MODEL The simple hydrogen model for the impurity centers can also be derived from quantum mechanical considerations. All states will be treated as electronic states. The Hamiltonian for a conduction electron in a periodic lattice with no impurities is ' „ H = - \r_ V + V (r) periodic. ( 2 . 4 ) 0 2m 0 The eigenfunctions of the Hamiltonian are the familiar Bloch functions of energy E(k). If a single charged negative acceptor ion is introduced substitu-tionally into the lattice, the Hamiltonian for an extra electron is taken to be H = H q + e2/Kr = E(k) + e2/Kr (2.5) Although the Schrodinger equation for this Hamiltonian may be solved using Wannier functions, most of the current solutions use Kohn-Luttinger functions which are some complete set of Bloch waves chosen near the band edges. These eigenfunctions have the form ^ ( ? ) = > ~ F . ( r )</).(?) ( 2 . 6 ) j=l 3 2 where the are the Bloch waves at the top of the valence bands in the un-perturbed crystal. The F.(r) are slowly varying "envelope" functions which modulate the Bloch functions. In the simplest case of a single valence band edge at k = 0, and with energy given by, 0 0 E(k) = \ g - i n r/aa*, (2.7) the envelope functions satisfy the "effective mass" equations ( 2,l). Bands which have an anistropic effective mass m* can easily be treated, but the above approach s t i l l corresponds only to the approximation in whioh terms to order of k are retained in the expansion of the energy E(k). Kohn (1957) extended the effeotive mass approach to treat the more complex "many valley" conduction bands (for donors in Si and Oe) as well as the degenerate valence bands (for acceptors in Si and Ge), 8 Valence Band The valence band of Si is quite complex. In the absence of any spin orbit coupling and in the tight binding limit, the wavefunctions corresponding to the highest point go over into atomic 3P wavefunctions. The bands so formed from the 3P atomic levels of the isolated atoms should then have the threefold orbital degeneracy (neglecting spin) characteristic of P states. The spin orbit coupling splits the states into P^ and P^_ states and partially removes the degeneracy (figure 2.l). The top of the valence band remains at k = 0 and is fourfold degenerate (including spin), corresponding to atomic J =| states. These two bands, labelled the light and heavy hole bands, have anisotropic effective masses. The split-off band, corresponding to J = \t is twofold degenerate. For silicon, the spin orbit splitting \ is about 44 meV. Besides the acceptor states which are primarily associated with the Pi valence bands, there is a second set of "internal impurity levels" associated with the split-off P^  valence band (Zwerdling et al. 1960b). The acceptor and the internal impurity states are shown schematically in figure 2.1. Figure.2.1 Valence Band Edge in Silicon, 9 Acceptor States Since the spin orbit energy \ is comparable to the ionization energy of the neutral acceptor impurities in silicon, one might expect Bloch waves from all six valence bands to be involved in the wavefunctions for the acceptor states. Instead of a single partial differential equation for the envelope function, one has a set of six coupled partial differential equations of the form y = i where E is the acceptor state energy measured relative to the valence band maxi-mum ( .'. E > 0 for bound states). The parameters 3 1 1 4 6^  are related to the effective masses and are merely used to describe mathematically the energy shape of the valence bands. The specific form of these parameters as well as references to the derivation of equation (2,8) are given by Kohn (1957). Approximate solutions to equation (2,8), relying on the variational method and group theoretical considerations, were made by Schechter (1962) for acceptor states in both Si and Ge. In the presence of the spin orbit splitting, the degenerate wavefunctions corresponding to a particular energy level of the acceptor must form the basis for a or irreducible representation of the full double tetrahedral group. Consequently, the solutions have the degeneracies corresponding to representations fg (fourfold) and [~^ 9 (both twofold) of this group. Because the effective mass Hamiltonian in equation (2.8) is invariant to inversion in the origin, the eigenfunctions F.(r) have definite parity. Por this reason, Schechter expanded the envelope trial functions Fj(r) in spherical harmonics and kept only the terms with even or odd parity. The form of the spherical harmonics in the trial functions was determined by group theoretical arguments to be N j =11 F,(r)'(J>. (5) - £ V exp (- T\Z1 <>{M (e, 0) <k (r). j=l J  D 1 V 'P 1/ j,m x m ± m 3 (2.9) 10 By first doing a variational calculation for the acceptor energy levels in the extreme limits of zero spin orbit coupling ( A. = 0) and infinite spin orbit coupling ( X = co )t Schechter then made first order perturbation calculations in X and in "X"1 in order to estimate the energy levels corresponding to the actual, finite values of X for Si and Ge. The calculated ground state is labelled "IS" and is fourfold degenerate corresponding to the four linearly independent functions (J).. There are a total of twelve "2PW states grouped in four degenerate levels labelled 2P^ , 2P,,, 2Py 2p in order of decreasing electron energy; the subscript is merely to differentiate the levels. The symmetry and the degeneracy (including spin) of the states are IS ( fg, fourfold), ZP^ ( Q, fourfold), 2Pg (|~^» fourfold), 2P^ ( f~^, twofold), and 2P^  ( twofold). These states are labelled "IS" and »2p« because their envelope functions are predominantly S-like and P-like and, in the limit corresponding to spherical constant energy surfaces for each of the valence bands, these envelope functions become simple hydrogenic IS and 2P wavefunctions. Schechter did not calculate the energies of the other excited states which are 2S-like, 3P-like, etc. The intensities of radiative transitions between acceptor states is determined primarily by the form of the envelope functions. The matrix element of r associated with the transition between states a and b will be j t * .r.y b d5x = S Z J F J (r) . r . 7^ (?) d V (2.10) to the approximation in which the slow variation of the modulating functions over each orystal cell is neglected, and in which |(J)j (r)| 2 is replaced by its average value of 1 in that cell. Strong dipole transitions will only occur between states having envelope functions of opposite parity. If the modulating functions have the same parity, there may s t i l l be weak electric dipole transi-tions, since equation (2.10) is only an approximation, Schechter's odd parity 2P states were assumed to be the four lowest lying 11 excited states. The strong optical transitions to these levels from the IS ground state (even parity) should correspond to the four lowest energy acceptor absorption lines. Schechter's calculated energy level scheme is compared with the experimental (boron doped silicon) spectrum in figure 2,2. (jSince the energies of the acceptor states are calculated relative to the valence band edge, and measured relative to the ground state, a comparison of energy level schemes requires the ionization energy The value of Ej for boron is somewhat un-certain (see Staflin 1965) with experimental values usually in the range of 44 to 46 meV (see V.B, error bar). Therefore comparison of the energy levels is made relative to excited state 2, (E^ turns out to be 44.0 meV). The experimental energies are similar to those of Colbow (1963).] The calculated ground state energy is expected to be incorrect because of a breakdown of the effective mass approximation for S-like states. However, this approximation should be realized for the 2P-like excited states, and close agreement is expected for these states, especially since the present comparison emphasizes this agreement. Schechter's energy level scheme is, however, only in moderate agreement with experiment. While the 2?1 and 2Pg states agree reasonably with excited states 1 and 2, the 2P, state does not agree well with excited state 3. The 2P„ state could J 4 . correspond to either excited state 4 or 4A, The lack of agreement of some states was attributed to approximations in the calculations and/or to deviations from the effective mass approximation. Consequently, Schechter's wavefunctions are frequently replaced by simpler hydrogenic wavefunctions. The energy level scheme and the degeneracies are discussed further in section 5.2. An improved calculation for germanium only, using more general trial functions but similar methods was made by Mendelson and James (1964). The simplification of very strong spin-orbit coupling restricts the solutions to germanium. Since the trial functions allowed the excited states to have radial functions with nodes (3P-like states), the energy level scheme was more complete. 12 EXPERIMENTAL (White, 1966) THEORETICAL (Schechter, 1962) 4 K 4 - IT 1' absorption lines o + i t o tO • O 8 o + » o in • to OJ s O + 1 c~ to oo to to 8 o + l t o vo cn t o o o + 1 o » cn to IT O to CM LO AO o d + 1 OJ c-1 O J o + 1 IT co to cr (?) (6? ) (4) (2)-(2)_ (41—(4)-(4) O S CO I CD p Qi nj (4)_ •(4) 2P, 2P, 2P„ 2P„ CD •3 i-t is r8 r8 {-a? -5 -10 -15 Hole Energy (meV) -20 -25 -30 -35 -40 -45 Pigure 2.2 Acceptor States in Boron Doped Silicon. 13 The calculations concentrated on the excited states with odd parity envelope functions to which strong optical transitions might be observed. Mendelson*s states are labelled according to 1) the symmetry (ie. 8, 7, or 6 corresponding to the f~^ , or symmetry of the state, 2) the parity of the envelope function (ie. + or -), 3) the number of radial nodes in the envelops functions (ie. 0, 1 , 2, 3. . . )» 4) the root of the secular equation (ie. 1 , 2, 3 . . . . ) in order of increasing energy; this index is used only i f there is more than one root. Thus the state (8 - 13) would be the third lowest energy state in a series of states a l l of which had symmetry, odd parity, and one radial node in the envelope function. The energies of the nodeless states agree closely with those calculated by Schechter. The important new feature of Mendelson's spectrum was the increased number of low lying fourfold degenerate states (corresponding to states whose envelope functions contained nodes). Jones and Fisher (1965) have compared Mendelson's spectrum with experiment and obtained fairly close agreement. That Mendelson did not extend his calculation to silicon is unfortunate, because i t could possibly explain why a larger number of low lying fourfold degenerate states are observed (section 5.2) than are predicted by Schechter. 14 CHAPTER 3 - GENERAL EXPERIMENTAL PROCEDURES 3.1 APPARATUS Monochromatic infrared radiation was obtained from a grating spectrometer. Radiation from a Globar source, Operating at 250 watts, was chopped hear the entrance s l i t by a semicircular disc rotating at 13 cycles per second and was then dispersed i n a Model 83 Perkin-Elmer monochromator, modified to use a Bausch and Lomb grating with 30 grooves per millimeter blazed at 30 microns in the f i r s t order. The calculated resolving power was >/A>\ = 1920. By using NaP reststrahlen plates i n the entrance and exit optics and one sooted mirror, short wavelength scattered radiation, which was transmitted by a NaGl plate (i.e. )\ < 25 microns), was reduced to less than 5$ of the total beam intensity. Less than 3$ of the radiation transmitted by the silicon samples was short wave-length seattered radiation. After passing through the silicon sample, the chopped monochromatic radiation focussed on a thermocouple detector with a Csl window. The thermocouple ©utput was amplified with a phase-sensitive Perkin Elmer model 107 Amplifier and the rectified output signal, which is proportional to the transmitted infrared beam intensity, was displayed on a Brown chart recorder. Due to the short lifetime of holes in the higher excited states at low temperatures, the silicon sample w i l l return to equilibrium during the time that the infrared beam i s blocked. Hence no saturation effects are observed. The continuous background room temperature radiation falling on both sample and detector is not detected by the phase=sensitive amplifier. Some improvements to the detection system were made. The far infrared radiation, intensity .is quite weak because the Globar source, which acts like a black body, operates at temperatures below about 1400°K, Under intensity-limited conditions, large spectrometer s l i t widths are required with a corres-pondingly poor resolution. A new (Charles M. Reeder & Go*j) thermocouple detector, 15 which had a responaivity 4^ 5 times greater than the previously used Perkin Elmer detector, was installed. The new detector was found to be very vibration sensi-tive? the resultant increased noise was minimized by balancing the chopper system and by preventing vibrations of the pumping systems from reaching the spectro-meter. A substantial remaining source of vibration, and hence noise, i s due to the bubbling of the coolants in the dewar. The improved detectivity (Jones 1959) allowed the,spectrometer s l i t widths to be reduced, thus modestly improving the resolution (cf. Colbow 1963). The spectrometer was calibrated using atmospheric water vapour absorption lines (Blaine et a l . 1962)j Some interference fringe measurements served as a check of the calibration* During transmission measurements, the spectrometer was continuously flushed with nitrogen gas to reduce atmospheric water vapour absorption* The nitrogen gas was produced by putting about 300 watts of power into a heater immersed in liquid nitrogen. Since liquid oxygen is occasionally present i n the nitrogen, the heater system should contain as l i t t l e flammable material as possible and should not reach eloser than about 2 inches from the bottom of the nitrogen storage can. Also, since the liquid oxygen tends to accumulate, the storage can should be completely emptied regularly. By observing the above precautions, conflagrations can be avoided. A transmission polarizer (MLtsraisbi, et al.. I960), was constructed te provide linearly polarized light. The polarizer consisted of 10 sheets of polyethylene, each about 12 microns thick, set at the Brewster angle of 55.5 degrees for n = 1.46, The design principles of fanning the polarized sheets to reduce unwanted components and to prevent deflection of the beam (Bird and Sureliff 1959) were followed. If the intensity of radiation passing through two such polarizers 16 in their crossed and uncrossed positions are I min and I max, the percent polarization P for each of the two "imperfect" polarizers is measured to be microns. Since the "unpolarized" beam is partially polarized by the grating and by the reststrahlen plates, separate doped and intrinsic runs must be made for each polarizer position. The low temperature metal dewar was positioned in the spectrometer such that the cooled parallel-sided silicon samples were at a focus point of the infrared beam. Two Csl transmission windows allowed evacuation of the dewar to a pressure —6 of 10~ mm Hg. A doped and an intrinsic sample were mounted to a slotted copper block at the base of the inner coolant container, which could be rotated to alternately move either sample into the infrared beam. A nitrogen shield surrounding the inner coolant container blocked much of the incident room tem-perature radiation as shown schematically in figure 3.1. P = 21 max - I min x 100 = 94$ 21 max + I min The polarizer transmitted approximately 15% of a plane polarized beam at 30 outer dewar (300GK) nitrogen shield (77°K) inner coolant container (T0K) slotted copper block (T0K) silicon sample (T°K assumed) infrared beam mm • vacuum Csl window 300°K background radiation Pigure 3.1 Experimental Layout (Schematic) 17 Measurements were usually made at five temperatures corresponding to the boiling point (B.P.) of helium (4.2°K), solid nitrogen at 1mm Hg pressure (54°K)* triple point of nitrogen (63°K)» B.P. of nitrogen (77°K) and the B.P. of oxygen (90°K), The helium evaporation rate was reduced to 0.08 litres/hour by improve-ments in the dewar design and by reducing the size of the optical slots in the nitrogen shield. The remaining heat leaks are mostly heat conduction down the stainless steel supporting column of the inner container and room temperature radiation entering via the radiation ports. Since l i t t l e error is introduced, the sample and bath (i.e. the copper block) temperatures are assumed to be equal. [Using a gold (+ 0.03 at. fo iron) vs. normal silver (+ 0.37 at. $ gold) thermo-couple (Johnson Matthey & Co. Ltd. wires), with glued thermal contacts for both sample and 4.2°K reference junctions and with a calibrated sensitivity (Berman and Huntley 1963) of 14/*-V/°K in the 4-14°^ range, the measured temperature of a strain-free mounted silicon sample was 6 K. With a flat copper spacer between ' o o the sample and the 4 K copper block, the sample temperature was 11.5 K. Since both silicon and copper are good low temperature thermal conductors, the above temperature increases are attributed to the Kapitza thermal resistances of the ' silicon-copper and copper-copper interfaces. Sublimation of the soiid nitrogen refrigerant near its container walls may produce a poor coolant-coutainer thermal contact dominated by gaseous thermal conduction (at 1 mm pressure); the steady state temperature difference between the copper block and the 54°K refrigerant was assumed to be small J] 3.2 TRANSMISSION AMD ABSORPTION Radiation absorption in a solid is characterized by a monochromatic absor-ption coefficient Considering multiple reflections from both sample surfaces, the transmission (relative to vacuum) of normally incident, monochromatic radia-tion of frequency JJ through a parallel-sided specimen of thickness d is given by 18 T = ( l - R) 2 exp (-« d) (3.1) 1 - R 2 exp (- 2*d) where R i s the surface r e f l e c t i v i t y . I t i s assumed throughout, that T and =< are both functions of the frequency JJ. I f the material does not absorb (U =0), the transmission T q depends only on the surface r e f l e c t i v i t y , T Q = (1-R)7(1-R^) . (3.2) )2 / ( l - R 2 I f the temperature of an i n t r i n s i c semiconductor i s s u f f i c i e n t l y low that no free carriers are present, the transmission of extrinsic radiation i s T Q . [ i n the spectral region investigated, there are no l a t t i c e absorption bands a»id no nbisorp-•tion -bando and no absorption due to the presence of oxygen.] The surface r e f l e c t i v i t y R was measured to be 0.31 - 0.03 hy Bichard and Giles (1962) and Colbow (1963) using equation (3.2), Within experimental error, the r e f l e c t i v i t y for both i n t r i n s i c and boron-doped s i l i c o n remained constant over the frequency and temperature ranges of present interest. The transmission ratio of a doped relative to an i n t r i n s i c semiconductor i s (assuming no interference fringes are present), T _ ( l - R 2) exp ( -*d) (3.3) I 2 In the present experiments, «: i s calculated from measurements of T, T and d TQ 1 - T exp ( - 2* d) using equation (3.3)» the advantage of this expression being that T/T q and °C are not sensitive to R, The surfaces of thin samples were made s l i g h t l y l e n t i -cular to suppress interference.fringes 5 the variation i n a sample's thickness (a few percent) was minimized to prevent serious defoeussing and loss of the multiple-reflected rays. [Use of non-parallel sided samples, of average thickness d, could introduce a',maximum error i n T/T q of 9.6$ for zero absorption (i„e.°<d=0) and < ifo for moderate absorption ( ^ d ^ O . l to 2,0),] 3.3 SAMPLE PREPARATION AND MOUNTING A l l samples were cut from float-zoned single crystals obtained commercially and, except for the Stark samples, a l l samples were uncompensated. The sample thicknesses were chosen to give optimum measurements of the absorption c o e f f i -19 dent (T/T q must be > 10$). The sample surfaces were ground with #600 grit silicon carbide on glass, #800 alumina on glass, and finally polished with #600 silicon carbide on an Astromet cloth. Before mounting in the dewar, the samples were degreased in toluene and in alcohol using an ultrasonic cleaner. Impurity concentrations were determined from four-point probe resistivity measurements at room temperature, using the calibration by Irvin (1962). The intrinsic sample had a resistivity of 3500 .a cm, corresponding to an acceptor 12 3 concentration of about 5 x 10 atoms/cm . The very weak boron absorption lines, observable at 4.2°K in the intrinsic sample, were allowed for. The sample orientations were determined by light reflection (Schwuttke 1959) and/or X-ray methods. The infrared beam was always transmitted in the [ i l l ] orystallographic direction. Mounting: Since i t was found that a) external strains are easily intro-duced by the sample mounting and b) these strains strongly affect the observed absorption spectrum, alternative methods of mounting samples were tested. The "standard" mounting, used by Colbow (l963) and G-oruk (l964)» was to gently hold the samples against the copper block by means of a flat copper faceplate (figure 3.2a)$ to ensure good thermal contact, a grease containing silver powder was applied to the areas in which the sample and copper block were in contact (Burstein et al. 1956). Variations of this method Included greasing one side only, using an elastic band instead of a faceplate, etc. Even i f care was taken not to strain the sample when mounting at room temperature, strains would s t i l l be introduoed because the grease would freeze when the sample was cooled. An external strain was then caused by the difference in thermal contraction of the copper and silicon. For the applied strain experiments (chapter 5)> this difference in thermal contraction supplies most of the uniaxial strain to the samples in the strain mounting (figure 3.2b). 20 a) Standard Mounting b) Strain Mounting c) Strain-free Mounting Figure 3.2 Sample Mounting. For a strain-free mounting (figure 3.2c), the copper block was redesigned. Mounting consisted of gently clamping one end of the sample and viewing only the unconstrained portion of the sample. Several similar methods of mounting gave consistent results, and any remaining mounting strains were thus assumed to be small. The above method of strain-free mounting was used for all present results (except the uniaxial strain experiments). 3.4 ELECTRICAL CONTACTS Electrical contacts were made to dumbell shaped Stark samples by plating gold to an etched silicon surface, alloying the gold-silicon bond, and bonding a doped gold wire to the alloy. The dumbell shaped samples were cut from polished silicon slices with a steel, ultrasonic cutting tool using #600 silicon carbide abrasive. The procedure to make gold contacts to the dumbell ends (see also Burgess 1964) was tos 1) Thoroughly clean samples in the ultrasonic cleaner. 2) Regrind the ends with #600 silicon carbide on glass, 3) Quickly wash off the abrasive with deionized water and place the freshly ground end in a hot concentrated (l0$) NaOH etching solution. 21 4) After vigorously etching for approximately 1 minute remove sample and without washing, quickly place i t in the gold cyanide plating solution and begin plating. 5) Plate gold at approximately 20 milliamps/cm for 2 minutes using a pure gold anode to replenish the solution. 6 ) Remove sample, wash with deionized water and place sample in the clamp of the alloying machine (in a certified-grade nitrogen atmosphere). 7) Heat sample (with radiation from the nichrome heater coil surrounding the sample) until the. gold-silicon eutectic is formed (sharp colour change from gold to a metallic grey), 8) Bond a 0.005" diameter, 0.7$ Ga-doped, gold wire to the eutectic when using p-type silicon, (for n-type silicon use a 0.5$ Sb-doped gold wire). 9) Allow to cool. Por the delicate side arms, which cannot be reground, step 2 is omitted. The cyanide plating solution is made by dissolving 2.1 gms. of 0.005" diameter, pure (99.999$) gold wire in a solution of 15.3 gms. KCN and 4.0 gms. of Wa_HP0_.12Ho0 in 1 litre of water. It takes over six months to dissolve the gold. The plating solution was cleaned by plating between two cleaned, pure gold wire electrodes.; the gold is continuously replenished from the anode, while the other positive ions in the solution are removed. Extreme cleanliness in handling samples, solutions, tweezers, etc. is essential. Also, the time from regrinding to the start of plating should be minimized (less than 2 minutes) in order to avoid oxidizing or contaminating the silicon surface. Often a thin white film is present on the sample surface after the contacts are completed. This film is from the NaOH etching solution and considerably cuts down the infrared transmission of the sample. Consequently, the samples were repolished after the electrical contacts were made. If gold was inadvertently plated between the side arm and the dumbell, i t was removed by careful sandblasting. 22 Although the electrical contacts were ohmic at room temperature, they were sometimes non-ohmic at low temperatures. Consequently, a l l Stark samples were made dumbell shaped in order to avoid contact effects (see section 6.2). Six 0.005" diameter Advance wires with teflon insulation, entering the dewar through kovars and terminating near the sample, facilitated the electrical connections to the gold leads of the sample. 23 CHAPTER 4 - BROADENING OF ACCEPTOR ABSORPTION LINES 4.1 INTRODUCTION Neutral impurities in silicon may be perturbed by lattice vibrations, crystal imperfections, and other impurities and these perturbations al l contribute to the half widths of the neutral impurity absorption lines. Earlier halfwidth measurements (Burstein et al. 1953? Newman, 1956) neglected the large instrumental broadening caused by the scanning of narrow absorption lines with infrared spectrometers of low resolution (Kane., I960), Recent halfwidth studies for boron acceptors (Colbow. 1963) and arsenic donors (Goruk 1964) in silicon were made with high resolution grating spectrometers and included instrumental broadening corrections (see also Pajot 1964). Present experiments showed that Colbow*s measured halfwidths were incorrect, because his samples had been accidentally strained by their "standard mounting". Consequently, with a "strain-free mounting" (figure 3.2) and slightly improved resolution, the halfwidths were remeasured. The absorption lines were symmetric, reproducible and sharp. New information regarding the broadening mechanisms is described. The ionized impurity broadening is treated in more detail because i t is related to the compensation and Stark effect studies of chapters 7 and 6. 4.2 TRUE HALFWIDTHS The impurity concentrations of the uncompensated, float-zoned samples are listed in table 4.1. Using standard experimental procedures (chapter 3), absorption spectra were obtained at various temperatures for each sample (eg. figure 5.l). Only transitions from the ground state to the five lowest excited states were viewed and the absorption lines are labelled 1, 2, 3» 4» and 4A in order of increasing energy (Colbow 1963; Fisher and Ramdas 1965). Most of the absorption between the peaks is attributed to free carrier absorption (Staflin 24 and Huldt 196l) and multiphonon absorption (Barrie and Nishikawa 1963, 1964). Having removed this "background" absorption, the "observed halfwidth" (full width at half maximum power) is measured. Table 4.1 Halfwidth Samples (boron doped silicon) 300°K boron resistivity concentration •'•.( JLcm)* ( x I O 1 4 cm-3)** 150 1.0 17 . 7.8 11 12 5.0 28 1.2 120 *four-point probe measurements at 300°K. **using the resistivity calibration by Irvin Since the general procedure for making the instrumental broadening correc-tions has already been described by Van de Hulst and Reesinck (1947) and Colbow et al (1962), only the differences in procedure are mentioned. The instrumental (or spectral) lineshape and halfwidth were determined from water vapour absorption lines in air at \ atmospheric pressure, to reduce the pressure broadening of the narrow gas lines (Burch et al, 1962). In the region of absorption line 2, the spectrometer s l i t width was 0.75 mm and the spectral halfwidth was 0.09^  meV (milli electron volts). The observed impurity halfwidths were then corrected for instrumental broadening, using the new Voigt function tabulation by Davies and Vaughen (1963). The resultant "true halfwidths" (h) of absorption line 2 are shown in figure 4.1. sample thickness (cm) 0.7181 0.1262 0.1054 0.0478 0.0201 (1962). 25 1.41-1.2 1.0 True Halfwidth 0.8 (meV) 40 60 Temperature (°K) ..2xl016cm~^ 15 L.2xl0 Jcm " Qxl014cm~3 0 cone. 100 Figure .4.1 True Halfwidths of Si(B) Absorption Line 2; the boron concentrations are indicated on the graph. i.4 h-1.2. 1.0 Halfwidth Contribu-tions o.8 (nieV) 0.6 0.4 N = 1.2xl016cm"*5 a total halfwidth .0 20 40 60 Temperature ( K) Figure 4.2 Broadening Contributions of Si(B) Line 2; the boron concentration is 1.2xl016cm"3. 26 4.5 BROADENING MECHANISMS Prom the true halfwidths (h) as a function of both temperature (T) and impurity concentration (N), the various broadening contributions ( A h ) , are separated according to their dependence on T and N to be, Broadening Contribution ( A h). Assumed to Limit of (Ah). 3 J Name Symbol Depend on as T, K->0 dislocation ( A h)^ neither T nor N £ 0 phonon ( A h) n T only jt 0 concentration ( A h ) N only. = 0 ionized impurity* ( A h).. both T and N =0 *called s t a t i s t i c a l Stark broadening by Colbow (1963). The procedure and assumptions involved i n this separation are discussed by Colbow (1965). The various broadening contributions to peak 2 (figure 4.2) w i l l now be discussed. Dislocation Broadening; Since the dislocation broadening and the 0°K phonon broadening contribution (h ) are both independent of temperature and concentra-o tio n , they cannot be separated. A l l halfwidth samples had a nominal dislocation 4- 2 density (n) of about 4 x 10 disloo/cm . In an attempt to determine experimentally the dislocation broadening contribution ( A h)^, the halfwidths of three low concentration samples of various dislocation density were measured (Table 4.2), Since the experimental error and the variation i n concentration broadening are comparable i n magnitude to the dislocation broadening only a very rough estimate of ( A h)^ i s possible, i.e. ( A h ) d ~ 0.02 meV (for peak 2; n * 4 x 104 disl/cm 2) This agrees reasonably with the theoretical estimates of about 0.04 meV (Kohn 1957; Bir et a l . 1963). Dislocations appeared to have a greater effect on donor 27 transitions (appendix B). Table 4.2 Dislocation Density * (#/ cm2) Resistivity at 300°K ( <J1 cm) Boron Cone. ** (xl014cm"3) Halfwidth of Peak 2 observed true+ normalized++ (meV) (meV) true (meV) low £ 2000 ~ 400 0.59 .17 .11 .12 4 avg. rv4 x 10 150 1.00 .21 .14 .14 high 10 5 - 10 6 ~ 200 0.59 .19 .12 c 5 .13 5 •manufacturer's nominal values, •determined optically (appendix A). +spectrometer halfwidth = 0,097 meV. .. _ ++the true halfwidths normalized to concentration of 1.0 x 10 cm" . The purpose of these dislocation measurements was not only to estimate ( A n )^» but also to show that some other larger broadening mechanism is opera-tive at these low temperatures and low concentrations. This contribution (hQ) is the temperature independent portion of phonon broadening (see equation 4.l). In the limits of zero concentration and 0°K, the true halfwidth of peak 2 becomes, lim ( hi = h + ( A h), = 0.08 meV. N, T-*0 1 > ° h -0,06 msV, o Phonon Broadening!, (see also section 5.3). The absorption lines are believed to be zero phonon transitions, with phonon broadening due to a " l i f e -time" effect (Kane 1960s Nishikawa and Barrie 1963? Barrie and Nishikawa 1963). For line 2, the good f i t of the experimental data to the theoretical phonon broadening contribution, ( A h) , = h \ l-exp(-T /T ) I - 1 (4.l) indicates that this (temperature dependent) broadening is primarily due to a state with, a (hole) energy at least 0,7 meV below the excited state of line 2. Independent evidence of such a lower energy state which strongly influences peak 2 28 is given in sections 5.3 and 6.3. Concentration Broadening; This broadening is caused by the overlap of wavefunctions of neighbouring impurities (Colbow 1963$ Baltensperger 1953) and thus should depend only on the impurity concentration. The concentration broadening contribution, ( A h ) , for peak 2 is shown in figure 4.3. The sharp increase in concentration broadening with impurity concentration, reported by Colbow (1963) and Goruk (1964), was attributed to the accidental straining of their samples by the mounting. In an earlier study of concentration broadening, Newman (1956) measured the observed halfwidths (h^) of peaks 2 and 7 at liquid hydrogen temperatures. Newman did not correct for the large instrumental broadening of his prism spec-trometer and consequently observed a large halfwidth (h^) even at the lowest concentration. £ Since only the ratio n 0 ^ / ^ c was given, a value of h ^ = 1.0 meV is assumed Newman's observed halfwidths (dashed curves of figure 4.4) there-fore represent correctly the concentration broadening contribution, (Ah) , only in the high concentration region where the effect of instrumental broadening and of other broadening mechanisms is small. Present experiments establish limits for ( A h) at low concentrations (bars in figure 4.4). For the entire concentration range, the best estimate of (A h) is made by interpolation (solid c curves of figure 4.4). The present results for peak 2 are included as a dotted line. The concentration broadening contributions of the boron lines, as summarized in figure 4.4,show that a strong broadening occurs at high concentrations. Recent photoconductivity experiments on Si(B) by Scott (1966) show that impurity-banding effects begin to occur at ~1 x lO^cnf 3 (peaks 5» 6, 2.* 8> 9), ~1 x 10 cm~^  (peaks 3» 4), ~5 x 10 cm"p (peak 2) and ^  5 z 10 cm"5 (peak l ) . These results suggest that, even at our low concentrations, the source of concen-tration broadening may be impurity-banding (cf. Newman 1956). (Ah) (meV) ^^14 Boron .^15 Concentration (cm-3) 1 ( J 6 _L 1.5 2.0 1 j - , . (Concentration)T (x 10 cm" ) Figure 4.3 Concentration Broadening Contribution ( A h ) of Line 2 2.5 Boron Concentration (cm ) 1 0 1 4 , 1 0 1 5 J.0 1 6 4 7 Halfwidth and ( A h)_ (meV) 2 4 i 6 (Concentration J* 18 (x lO^cm"1) 10 12 Figure 4.4 Concentration Broadening of Si(B) Lines 30 Ionized Impurity Broadening: The broadening which depends on both tempera-ture and concentration is attributed to ionized impurity broadening caused by the screened internal electric fields of ionized impurities. This broadening should be appreciable only in the temperature range (T > 50°K, for boron doped silicon) in which the concentration of ionized impurities is large (see Shockley 1950). This broadening contribution, shown in figure 4.5 for three different impurity concentrations, is discussed in the next section. 0 . 2 0 40 60 80 100 Temperature (°K) Figure 4.5 Ionized Impurity Broadening Contribution of Line 2. 31 4.4 IONIZED IMPURITY BROADENING (a) General Many concepts of atomic spectral line broadening have been included into the theory of impurity absorption line broadening in solids. The ionic broadening of atomic spectral lines by the electric fields of ions in a plasma was reviewed by Margenau and Lewis (1959), Baranger (1962) and Breene (l964). The ions were usually assumed to be fixed (quasistationary approximation), the spatially constant electric field of the ions was calculated and the broadening of the atomic transitions through both a first order "linear Stark effect" or a second order "quadratic Stark effect" was determined. A similar approach was used in the earlier theory (Colbow 1963) of ionized impurity broadening (called "statistical Stark broadening") of acceptor absorption lines in silicon. Muller (1965) extended the theory of ionic broadening of atomic lines by using better approximations to the Coulomb fields of the perturbing ions. The spatial inhomogeneity or gradient of the perturbing electric field was found to contribute significantly to the broadening, especially of atoms with zero dipole moment. A new theory (Cheng 1966) of ionized impurity broadening in silicon, includes treatment of the field inhomogeneity. Since Cheng's study is not yet complete, this section is restricted to a brief description of the change in transition energy due to the Coulomb field of a nearest neighbour ion. This result is sufficient for explaining the significance of the present results. (b) Ionized Impurity Perturbation Terms Consider a neutral impurity located at the origin (r = 0). An ionized impurity located at S will, through its screened Coulomb field, perturb the charge bound to the neutral impurity through the perturbing potential H' r-. - e 2 exp(- |?-S|/ A ) ^ -e 2 exp(-(r 2 + R2 - 2rRcos Q)^2/\ } (4.2) K|S-?| K(r 2 + R2 - 2rRcos Q)1^2 32 where K is the dielectric constant of silicon and X is the screening length. The implied assumption of nearest neighbour interactions only, should be reasonable for the low impurity concentrations of present interest. If the dimensions of the perturbed neutral impurity are small compared to the distance to the nearest perturbing ion (r < R), H' may be expanded in powers of r/R, i.l . H» = - e_ exp (- R/X) KR + r cos 0 (l + R_) R •> (4.3) R cosf© (3 + JR + R2. ) - 4-d + R) higher order terms The three terms in H' describe the perturbation due to the potential, due to the electric field and due to the gradient of the electric field - a l l evaluated at r = 0 (Huller 1966). If the perturbation H1 is rewritten as R ,k+l k=0 (4.4) the perturbed energy of a non degenerate level (in descending powers of R) is: 6 €° +H« + T Z H« / ( £ ° - £°) ^ vi -^ vi vi v* : mW — ^  ^ rn' nn £n mn f 0 +M0/rm - f ^ n n + <&2nn + >W Yt .1 i — m • • • 1 — 1 • I •• !• 1 I I . — n '"mi. * hv nn o 'm R R" R-' K* { n^ n 6 ° - £' The interpretation of the various terms is listed in table 4.3 nondegenerate acceptor states; the equivalent effects for degenerate states (not shown) are included. The constant potential shifts all states by the same amount. Hence, transitions from the ground state to the excited states of the same neutral impurity will not be effected. This shift |--(e2 /KR)exp(-R/>, )j\ is merely the potential at the neutral impurity (r = 0) due to the ion located at R. The perturbation due to the electric field has the form ( e F ^ r cos ©), where the "internal field" F int = - ef_ (1 + R) exp (- R/X), (4.5) KR represents the constant magnitude (at r = 0) of the Coulomb field. The linear Table 4.5 Ionized Impurity Perturbation Terms Interaction of neutral atom, located at r = 0 through its With the ionized atoms through their resulting Yields for a (non degenerate level) Por acceptor impurity states, the resultant broadening is from Non degenerate degenerate level (n) level (n, k) point charge dipole moment quadrupole moment first order polarization potential at r = 0 field at r = 0 gradient of field at r = 0 field at r = 0 . o/ R^nn 1'/ nn R ±M2' — n n R^  R4 5 .0 0 0 0 (linear Stark effect) -shift only -shift and splitting (field inhomogeneity) in 1st order -shift only -shift and splitting (quadratic Stark effect) in second order 34 Stark effeot (first order perturbation by the internal field) is zero for acceptor states in the effective mass approximation (see section 6.1). This is equivalent to describing the neutral acceptor as having a permanent electric dipole moment of zero. The quadratic Stark effect (second order perturbation) will shift non degenerate acceptor states and both shift and split degenerate states. The quadratic Stark effect may be attributed to an.induced electric-dipole moment (or a first order polarization) of the neutral acceptor. The inhomogeneity of  the field interacting with the quadrupole moment of the acceptor also shifts non degenerate states and both shifts and splits degenerate states (by removal of degeneracy in first order). In summary, to second order in r/R, only the field inhomogeneity and the quadratic Stark effect will contribute to the broadening of the impurity absorption lines. (c) Perturbed Acceptor Transition Energies Cheng (1966) made first order perturbation calculations using Schechter's detailed wave functions for the acceptor states and a screened Coulomb field of an ion located at R. The perturbation effects are illustrated schematically + V + A l i B l : + A i B 0 0 where E is the constant potential, i.e. E = -(e2/KR) exp(-R / A ) . A is small below} the degeneracies (including spin) are in brackets. ?:v-< <>\ • v N——<2) Vs, - V X ( 2 ) 2o 1 3 — — (*>\-. i E I S = V \ 35 compared to a l l other terms and may he neglected. A,, ^ have terms i n ( X ) ~ 2 and/or (R X B , B B 2 have terms i n ( R ) ~ 2 , (R X ) _ 1 and ( X )~2. Therefore, for no screening, A = ^  and A^ = A 2=A^ = A^ = o!j The changes i n the transition energies from the IS ground state therefore are, IS' - 2P • E = A. - B i B. 1 1 I 0 , 1 IS - 2P_ E„ = L i B i B, 2 2 i 2 - B p " ^ 2 \ (4.6) + 3 "3 ~ "3 ~ ~o IS - 2P„ E„ = A^ - B IS - 2P. E = A - B 4 4 4 o Details of these calculations and evaluation of the terms w i l l be published lat e r (Cheng 1966). Although the IS - 2P^ and IS - 2P^ transitions are associated with boron absorption lines 1 and 2, transitions IS - 2P^ and IS - 2P^ are not asso^-elated with boron lines 3 and 4 because the excited states of these absorption lines do not have the twofold degeneracy of Schechter's 2P^ and 2P^ states (see section 5.2), In discussing the effects of compensation on boron l i n e 3 (chapter 7), no detailed knowledge of the f i r s t order, f i e l d inhomogeneity terms i s required." Since the second order perturbation by the internal f i e l d (P^^) and by the external f i e l d (£) have the same formj the theory for the external Stark f i e l d (section 6,l) may be used. The perturbation to the ground state i s small com-pared to that of the excited states and may be neglected. The second order s h i f t (and s p l i t t i n g , i f the excited state i s fourfold degenerate) of the acceptor transitions i s expected (section 6,l) and observed (section 6.3) to be quadratic i n the f i e l d . The s h i f t i n transition energy to a non degenerate state w i l l be E = t(P. , ) 2 internal f i e l d , ") "I \ (4,7) and E = t( £) external f i e l d , J The parameter t i s the same for both f i e l d s , i„e„ for a non degenerate state, t = H , < i | - e r cos © |*n>|2/ (B - E.), (4.8.) i . The calculation of the second order perturbation by the internal f i e l d i s therefore simplified to a calculation of F^^, i f values of ,t are available from uniform f i e l d (Stark) experiments* 36 The changes in transition energy to a degenerate acceptor state due to the internal field, when similarly treated, are <JE = 4 i <T/2 (4,9) where, apparent shift = A = t F? . 1 I (4.10) apparent splitting = £ » \ ^ i n t J By taking experimental values of t and\t, from uniform external field results s ' n (tahles 6,1 and 6,2), the second order calculation again simplifies to a deter-mination of F^^.. The screened internal field (F^^) is known exactly for nearest neighbours and approximately for a l l ion neighbours (see Colbow 1963). £ The approximate sign in ( £ z ^^jxf} reflects the inaccurate nature of t^ due to neglect of both instrumental broadening corrections and lineshape effects,] That the theoretical calculation of t and t^ may be avoided, is fortunate because this calculation would be crude due to the approximate nature of Schechter's wave functions and due to the unknown energies of some (eg, 2S-like) states. The total pert^baticrt up to second order, of say the 1S-2P,, transition, therefore, is £®2~ (^ + t*^) - (\ + \ + i \ F!^) U.ll) (d) Qualitative Results The relative magnitudes of the perturbation terms determine the dominant broadening mechanism. Since the field inhomogeneity and the quadratic Stark terms vary as l/R3 and l/R4 respectively (equation 4»3)» then for each absorption line, there is a csritical R above which (or a corresponding critical ionized impurity concentration below which) the field Inhomogeneity dominates. The characteristic temperature dependence of the ionized impurity broadening contribution (figure 4,5) may be explained qualitatively. Since the distance R is related to the ionized impurity concentration (which equals the free hole concentration p) by "*> -1 R"' «< H. , the field inhomogenei'ty and quadratic Stark perturbations will vary as 4/3 p and p respectively. Therefore, neglecting screening effects, the ionized impurity broadening contribution, ( A h)^, will also vary approximately as p 37 or p and w i l l thus have roughly the same temperature dependence as the number of free carriers. The onset of ionized impurity broadening should therefore occur at the temperature (T ~ 50-60°K) at which the number of free carriers i s becoming appreciable, as observed (figure 4.5). Por a given ionized impurity concentration, the relative sizes of the perturbations d i f f e r for each transition. The quadratic Stark contribution for peak 2 (with the smallest Stark coefficients t and t^) w i l l be much less than for peak 3 (with the largest coefficients). (e) The broadening of the.absorption lines may be calculated from the change i n transition energy, & (equation 4.1l)» of one neutral impurity due to the Coulomb f i e l d of a nearest neighbour ion. Since an absorption l i n e represents the average transition energy of many neutral impurities, each with a different nearest ion neighbour distance R, some suitable average of ^Eg over a l l R i s necessary i n obtaining the broadened absorption l i n e . In this average over R, both the s h i f t and s p l i t t i n g w i l l contribute to the broadening. The calculation of the ionized impurity broadening i s i n progress (Cheng 1966) but i s not yet complete. His preliminary results indicate that, for lin e 2s (1) The f i e l d inhomogeneity mechanism, rather than the quadratic Stark effect, dominates the broadening of boron l i n e 2. (2) The temperature and concentration dependence of this broadening contribution can be correctly predicted from theory. ( 3 ) The theoretical broadening contribution i s within a factor of two agreement with experiment! since the calculation involves no adjustable parameters, this agreement i s considered good. The absorption l i n e perturbed by the ionized .impurities might be expected to be not only broadened, but also to be shifted somewhat from i t s unperturbed 38 position. Experimentally, any such net shifting of the absorption lines, i f present at a l l , was swamped out by other broadening mechanisms operating at these temperatures and concentrations. No shifts of the acceptor absorption lines were observed. [^ The shift of the donor lines with temperature (see appendix B) is due to a different effect)^] (f) Unscreened Ionized Impurities (Compensation Effects) The positive and negative ionized impurities, present due to compensationj perturb the acceptor absorption lines through their unscreened Coulomb fields, In general, the absorption line should both broaden and shift as the compensation is increased,; Since these compensation effects can be measured at 4.2°K where phonon broadening is relatively small, the shift as well as the broadening is clearly observed for absorption line 3 (see chapter 7 ) . This shift of line 3 is explained solely in terms of the quadratic Stark effect because the quadratic Stark contribution is particularly large for peak 3 and because the shifts caused by the field inhomogeneity are expected to be zero i f the perturbing Coulomb field is unscreened and caused by equal concentrations of positive and negative ionized impurities (see chapter 7 ) . This result fortunately avoids associating Schechter's 2P^ state with the excited state of peak 3p an associa-tion which is criticized in chapter 5» 4.5 "STATISTICAL STARK BROADENING" The earlier "statistical Stark" theory (Colbow 1963) of ionized impurity broadening considered only the (linear and) quadratic Stark perturbation i.e. H» = e P. . r cos ©, The internal field (P. ,) is the screened electric field int • mt at the neutral impurity center (r =.0) due to the ion located at R. Por a more general theory, Colbow used for P. ,,, the total field at a neutral impurity 39 center due to al l ionized impurities, i;e. int I j int Approximate values of this sum had been determined by Ecker (1957). The perturba-tion was assumed to shift the acceptor transitions by &E = ^c^int' The broadening was then determined in a statistical manner, with the coefficient t^ . being retained as an adjustable parameter to be determined for each acceptor level from a f i t to the experimental ionized impurity broadening contribution. A uniform applied field ( £ ) is found experimentally (chapter 6) to shift 2 the center of gravity of each acceptor transition by an amount &E = t £ . s Since both t gand.:t^ represent the same theoretical quantity, a comparison of t g and t^ o then constitutes a test of Colbow's theory.: For line 2 in the 1,2 JTU cm Si(B) sample: \ C = 23 X 10"8 meV cm2/volt2 (Colbow's theory) |tg|= 0.5 x 10"8 meV cm2/volt2 (section 6.3) This large disagreement led to the re-examination of the theory of ionized impurity broadening just outlined. , The disagreement is attributed to the neglect, in Colbow's theory of, (a) the field inhomogeneity perturbation which,,for many acceptor states (and especially for state, 2), is believed to be substantially larger than the quadratic Stark perturbation, (b) the splitting of the acceptor states which should also contribute signi-ficantly to the; broadening. Since 'tj £ c is an adjustable parameter, the neglect of any broadening contributions will cause tv to become too large (i.e. t, > t:)j as observed. iCC ICC s 4.6 CONCLUSIONS Improved measurements of absorption line halfwidths were used to determine the various absorption line broadening mechanisms, with particular attention paid to ionized impurity broadening. The experimental Stark parameters of 40 chapter 6 were used to determine the quadratic Stark broadening and showed that the earlier ionized impurity broadening theory by Colbow (1963) was inadequate. A new theory by Cheng (l966) appears to explain adequately the ionized impurity broadening contribution; this theory indicates that the field inhomogeneity (Muller 1965) is more important than the earlier quadratic Stark effect in broadening many of the absorption lines. la. The discussion of the ionized impurity perturbation effects a#e also pertinent to the compensation effects of chapter 7. 4 1 CHAPTER 5 - APPLIED STRAINS 5.1 INTRODUCTION Theoretical calculations by Kohn (1957) and Schechter (1962) assign a degeneracy (including spin) of either two or four to a l l acceptor states as a consequence of the symmetry of the top of the valence band. The ground state is fourfold degenerate,"and the twelve lowest excited states are grouped (in order of increased separation from the ground state) into 4-fold, 4-fold, 2-fold and 2-fold excited states. In the observations of the Zeeman splittings in Si(Al) by Zwerdling et al (l960a), the absorption lines were not split into resolvable components even at the highest magnetic field available (38.9 kilogauss). Consequently, the degeneracies of the excited states were difficult to determine, but the results appeared to confirm Schechter's ordering of the states. The observation of a complex structure in the fourth excited state of a Si(B) sample which was accidentally strained by the mounting (see section 3.3) showed that Schechter's predicted 2-fold degeneracy of this state was incorrect and prompted the investigation outlined in this chapter. By applying a uniaxial stress to Si(B), the absorption lines were split sufficiently to determine the degeneracies of most of the acceptor states. The room temperature resistivity of a l l samples was 11 Si, cm, corresponding 15 i 3 to an impurity concentration of 1,2x10 boron atoms/cm . The uniaxial stress was applied by means of the differential thermal contraction between the sample and a metal holder (figure 3.2) similar to the techniques described by Rose-Innes (1958). 5.2 DEGENERACY OF THE ACCEPTOR STATES Infrared absorption spectra at 4°K were obtained for three different applied strains. Only the results for a "light" strain of unknown magnitude applied in the ( i l l ) plane and for a "heavy" strain, estimated to be about 10~ , applied in 42 the [lio] direction of the ( i l l ) plane are presented. Results for an even larger strain in the [lio] direction, complement the data presented and are not shown. Since the intensities of the acceptor transitions in a strained crystal are expected to depend strongly on the direction of polarization of the incident light (Kohn .1957)» absorption spectra were obtained for the infrared polarized both II and 1 to the applied strain, as.well as wunpolarizedM. In a l l cases, the infrared was propagated in the [ i l l ] crystal direction. The present "unpolarized" spectra were not corrected for the partial polarization caused by the grating (and to a lesser extent by the reststrahlen plates) because the corrections will merely change the relative heights of. the absorption lines. The zero, light, and heavy strain spectra for unpolarized light are shown in figure 5.1. The zero strain spectrum, using the strain-free mounting (figure 3.2), is shown for comparison. The observed transition energies, for a light strain applied parallel to the ( i l l ) plane (see figure 5.1b), are consistent with a ground state splitting, A Eg, of (0„62 - 0.03) meV. This splitting is of the order of 2kT, and there is an appreciable hole population in the upper ground state. The spestrum for a heavy strain applied parallel to the [lio] erystallographic axis is considerably simplified, because the ground state splitting ( A Eg ~ 1.8 meV) is sufficiently large that only the lower ground state is appreciably populated. The general shift of a l l absorption lines to a higher energy indicated that the splitting of the ground state is larger than that of the excited states. No ©enter of gravity shift of the strain-split acceptor states is expected to first order in strain (Kohn, 1957). Transitions which appear intense only when viewed with polarized light whose electric vector was parallel or perpendicular to the applied stress are marked || and 1 respectively. Transitions moderately dependent on the polarization have the dominant polarization marked in brackets and, i f independent of the polarization, the transitions are unmarked. Because publication of a more complete analysis of the polarization properties is 43 .1 •H •8 •8 o g •rl O n 22 20 18 16 14 12 10 8 6 4 2 0 8 7 h 6 5 4 3 2 I 0 5 4 3 2 I T (a) ZERO STRAIN (b) LIGHT STRAIN (x2) (c) HEAVY STRAIN Ka) Kb) J AJ L. , A . -5 6 7 4A 2(a) 2(b) 4(b) 4(c) 3fa) 3(b) 4(a) AA 5? 6? 7?. 30 32 34 36 38 40 42 44 Energy (meV) Figure 5..1 Effect of Strain on the Absorption Spectrum of Si(B). a) Zero .strain "b) ;Light strain :in the (ill.) plane c) .Heavy strain ;in^the '[IIP] -crystallpgraphic -direction. 44 expected, the polarization was used only to distinguish the various transitions. Since the excited states associated with lines 1, 2, 3 have each split into two components, the lowest three excited states are each 4-fold degenerate (including spin). With three heavy-strain component lines 4(a), 4(h) and 4(c), the unstrained peaks 4, 4A. are associated with six excited states. Based on analogy with lines 1, 2, 3 where, with heavy strain, one transition energy remained nearly unchanged and one broader line moved to higher energy, line 4(a) is identified with the unstrained line 4, and lines 4(b) and/or 4(c) are associated with the "other" broader line from peak 4. The excited state of peak 4 is then a minimum of 4-fold degenerate. The extra 2-fold excited state may be due either to peak 4A or to an additional twofold state accidentally degenerate with peak 4 (cf. Jones and Fisher 1964). In summary, the acceptor ground state and the lowest four excited states a l l appear 4-fold degenerate, with the possibility of an additional 2-fold accidental degeneracy for the fourth excited state. The maximum theoretical degeneracy of the acceptor states is four. Both the observed energy separation (figure 2.2) and the degeneracy of the two lowest excited states agree with Schechter's (1962) acceptor states in silicon. However, neither the third nor fourth lowest excited states agree in these respects with Schechter's 2-fold states, presumably because some 4-fold (3P-like) excited states, which were not calculated, are of lower energy than the 2-fold (2P-like) states. For acceptors in germanium, the improved calculations by Mendelson and James (1964) introduce more low energy, 4-fold degenerate acceptor states into the earlier theoretical spestra (Kohn 1957? Schechter 1962) and the subsequent agreement with experiment (Jones and Fisher 1965) i s quite good. The four lowest energy absorption lines in Ge are a l l associated with 4-fold degenerate excited states. That Mendelson*s treatment has not yet been applied to silicon 45 is unfortunate, because i t might correct the observed disagreements with Schechter1s theory. The above degeneracy results (Skoczylas and White 1965) are compared in table 5.1 with the independent results by Fisher and Ramdas (1965) in lightly strained Si(B) samples. Although the degeneracies for the most part agree, there is some uncertainty regarding the 3rd and 4th excited states. Since the present heavy strain spectrum has a simple interpretation and was verified at an even higher strain, the reported degeneracies should represent the minimum possible values, (it is possible that Fisher's samples were not strained sufficiently to remove completely, the degeneracy of these states). The conclusions of both papers agree in the important respect, that Schechter's labelling of the lowest four excited states is incorrect and that Mendelson's treatment should be extended to silicon. Table 5.1 State Degeneracy of the Acceptor States in Silicon  Degeneracy Theory Experiment Schechter Present Results + Other Results ground 1st excited 2nd 3rd 4th " 4 4 4 2 2 4 4 4 4 4 + 2? 4 4 4 2 * 4 *Since peak 3 was weak, this degeneracy is tentative. +Skoczylas and White (1965). ++Fisher and Ramdas (1965). 5.3 ABNORMAL BROADENING IN STRAINED SPECTRUM A large broadening of some transitions was observed in the heavy strain spectrum. Broadening by non-uniformity of the applied stress was assumed small because of the presence of several narrow lines (la, 3a, and 4a) with normal half-widths, and because of the absence of severe broadening of peak 4 which was the transition most broadened in an experiment in which a bending moment (non-uniform 46 stress) was applied to a Si(B) sample. Transitions to the higher level of each split excited state were broader than transitions to the lower level of the same excited state. This selective broadening occurred for a l l transitions, and is especially pronounced for peak 1. The observed broadening is attributed to the same phonon broadening mechanism (Kane I9605 Nishikawa and Barrie 1963? Barrie and Nishikawa 1963) that dominates the impurity line broadening in the unstrained spectrum (Colbow 1963; Goruk 1964). The zero phonon absorption lines are larger than the phonon-accompanied absorption which contributes to the continuous background. In the limit of zero electron-phonon interaction, the absorption lines are & -functions. Non-zero absorption line halfwidths arise from the finite lifetimes of the excited states due to the electron-phonon interaction. The contribution of the state A , of energy TX , to the halfwidth of the transition to the state (3 can be written as * K'-0 , f ( 7?x) ./ l + ^ ( y , J T P> T x (5.1a) I ^ (yf>) T p< T v (5.1b) where ^ = | T p - T j /C 2 (y p x) = ^exp (|T/- T j /kT) - l } " 1 Both the function f(y ), evaluated for the simple hydrogenic model, and values px of the constants. and Cg, are given by Barrie and Nishikawa (l963). Since the sample temperature is about 5°K, kT » 0.43 meV. The total halfwidth (h) of a phonon-broadened absorption line then consists of the sum of the contribu-tions (AW.) from a l l states j . If the applied strain splits some degenerate state into the states u and 1, where T^ > Tj , the "width" of the states u and 1 will be h = I Z A W ^ +AWi = C + A w f (5.2a) *i "1 1 ' T 1^ = Z Z AW. +AWU = h Q + AW^ (3.2b) 47 In the unstrained spectrum at 4 K, a l l the narrow absorption lines have approxi-mately the same halfwidths (i.e. h^&h^ £ 0.2meV); these halfwidths are primarily the result of phonon broadening of each excited state by al l other states (see chapter 4). Consequently, in the present strained spectrum, the phonon broadening by a l l other states j (j ^  u, l ) is assumed to be approximately the same for each of the split states u and l(i.e. h c U - h^ ^  0.2meV). This assumption will also be realized i f no abnormal increase in broadening by a l l other states (j / u, 1 ) occurs. Only the phonon broadening contribution between the states u and 1 will be considered. The difference in halfwidth of the states u and 1 , as determined from equations 3.1 and 3.2 xd.ll be,. A h s h u - 1^ AW* - Av/ « C 1 f ( 7 u l) (3.3) The width of the upper state u will always be larger than that of the lower state 1 since Ah^O (for a l l reasonable yu^ )» At low temperatures, the width of the lower state is not increased appreciably by contributions from the upper state, because either (yu^) o r ^(y^) will usually be small, [in the heavy strain case; <U » -j^ exp. (l.2/0.43)-l| _ 1 = O.O65]. However, the width of the upper state can be greatly increased by the lower state, the magnitude of the increase depending on the particular wave functions involved. The general features of the strained Si(B) spectrum can now be explained. Since the ground state and the four lowest excited states of the boron acceptors are all degenerate (section 5.2), the above discussion should apply to a l l these states. For the heavy strain case, only the lower ground state is populated, and this state has not been broadened much by the upper ground state (see also Feher et al, I960). The observed halfwidths should depend then, primarily on the broadening of the excited states. Absorption lines to the upper excited states (peaks lb, 2b, 3b, 4b and 4c) should a l l be broader than to the lower level (peaks la, 2a, 3a, 4a) of the same state. The narrow lines should have half-widths comparable to the unstrained halfwidths. (The influence of the "other" 48 states, which has been neglected even though the strain splittings of the excited states are large, could broaden a few of the narrow lines). Transitions to the upper excited state may be broadened appreciably. The above qualitative predictions are, in general, observed. The only exception is peak 2a, which is broad. (Part of the width of peak 2a may be apparent and caused by the nearby peak 2b). The broadening of this transition could be caused by the influence either of peaks lb or la, or by a nearby excited S-like state. Independent evidence of a lower energy state influencing peak 2 comes both from the halfwidth study of this line (chapter 4) and from the Stark experiments of chapter 6. (For a uniform applied electric field, peak 2 was the only peak to move to a higher energy; such a second order Stark shift indicates the strong influence of a nearby state of lower energy). The broadness of peak 2a was therefore not considered surprising. A quantitative estimate of the extra broadening of the upper state may he made i f the states u and 1 are both approximated by 2P hydrogenic wavefunctions. This should be a reasonable approximation at least for the excited states of peaks 1 and 2 which are 2P-like in Schechter's theory, and which should become more hydrogenic due to the strain perturbation (viz. Peher et al, I960), (in view of the degeneracy results in section 5.2, the excited states of peaks 3 and 4 might be 3P-like). The extra phonon broadening (Ah) of an upper 2P state by a lower state X is shown in figure 5.2 as a function of the energy separation (AE) of the two states. The result is determined from the tabulated data of Barrie and Nishikawa (1963). Por small excited state splittings (AE < 0.5 meV), the extra broadening of the upper state is small ( A h£ 0.03 meV). Therefore, li t t l e extra broadening should occur in the light strain spectrum, as observed (see also Fisher and Ramdas 1965; Aggarwal and Ramdas 1965). For splittings of (0,5 < AE < 2 meV), the extra broadening increases rapidly with AE up to a maximum value of Ah ~ 0,4 meV. For splittings AE>2meV, Ah slowly decreases. 49 0.5T ( A E ) Splitting (me?) Figure 5.2 Extra Broadening of Upper 2Pr State by Lower State Al, ; (Experimentally, fracture might occur before this splitting is reached). At the observed heavy strain splitting of A E « 1.2 meV, the extra broadening is Ah » 0.25 meV (theoretical; for X= 2P,. ) A h «= 0,8 meV (experimental; for peak l) With no adjustable parameters, and using only approximate wavefunctions, the order of magnitude agreement is satisfactory. This calculation shows that extra phonon broadening of the upper state is quantitatively significant. In summary, phonon broadening gives a reasonable explanation of the abnormal broadening of the acceptor transitions observed in the strained spectra. 50 CHAPTER 6 - APPLIED ELECTRIC FIELD (STARK EFFECT) 6.1 THEORY OF THE STARK EFFECT (a) Effective Mass Approximation The perturbation Hamiltonian H1 for a hole in a uniform external electric field ( £ ) applied in the z direction is, ' H' =+ H e z s+fer ~r±6. (6.l) In the effective mass theory of shallow impurity states, the wave functions of the bound carrier have the form ^ ( m ) (?) = C ^ m ) (?) 0 , ( r ) (6.2) The matrix element of H1 between two such states is H' =(t ( m ), H«vp(n))^IZ(FU), H'F W) (6.3) mn D 0 J Since the effective mass Hamiltonian of these states [equation (2.8)^ ] is invariant under inversion and there are no accidental degeneracies (such as the 2S, 2P degeneracy in hydrogen) the first order Stark effect vanishes (Kohn 1957). [ If the effective mass theory is seriously in error, the acceptor ground state may have an appreciable first order Stark effect (section 6.1b).] A non degenerate state (m) will, in the effective mass approximation (H'^= 0), experience only a second order Stark shift of 6E = ' IH* I 2 /(E - E ). (6.4) By applying the closure rule, Kohn (1957) obtained the simplified approximate result 6E =T l V ( n ) . H - y ( n ) l 2 m E - E m n * ( y ( m ) , (HO 2t ( m )) / AE * ( F ( m ) , (H-)2 F ( m )) / AE, (6.5) where AE is an average of the denominators (E m-E n). The shifts will be of a similar order of magnitude for degenerate states. Using simple hydrogenic wave functions for F v ;, the shifts oE are 0 m <$E = C (e£a*r / AE, m m ' 51 where the constants C equal 1 (lS state). 14 (2S state), 18 (2P state), m o 6 (2P+ state) and 69 (3S state). The Bohr radius (a*) is taken to he 13.5 A. For the ground state ( AE~30meV), the Stark shift is only about 0.001 meV for an electric field of 1000 v/cm. The smaller energy separations ( A E ~ 4 meV) and larger constants (C ~ 14) of the higher excited states will result in shifts of about 0.06.meV for 2P-like excited states experiencing the same electric field. A degenerate state will, in second order, be split and shifted by the uniform field perturbation. Most acceptor states were found (chapter 5) to be fourfold degenerate, including spin. In the effective mass approximation, a degenerate state (k, m) is neither split nor shifted, in first order, by the uniform field (i.e. = 0, = = o). However, in second order, the perturbation mixes in contributions from other states and the degenerate state (k, m) shifts and splits into two, by the amount < $ E = A i ( A 2 - p F A 2 > (3 (6.6) where A = 1 ^ 1 ^ " ' l * * ^ H'uc) . (6.7) °Vm\ E m - E l / C t 1 /H» . H« . H' . Hf „ -H1 H' ,-.H' .H« ., \ l r . \ ml lm ky] jk kl lnv mt1 ,ik (6.8) Z__i I (^  ~EL) (E -E ) / l^ m j A , l m l 3 ' The magnitude of the shift ( A ) and the splitting [2( A 2 - (3 )^] of the degenerate level will depend strongly on the positions (E^ - E^) of a l l other states (l) and on the perturbation matrix elements with these states. The direction of the shift (sign of A ) will depend strongly on whether the nearest state is located above or below the state being perturbed. [ If the major influence on a degenerate level (k, m) comes from a single nearby state, (3 is equal to zero and 6E = 0, 2 A , In this special case, the unresolved splitting (2 A ) is twice as large as the apparent shift ( A ) of the degenerate 52 state.J For a 'uniform applied field £ , both the shift and the splitting will be quadratic in the field. Because the acceptor state wavefunctions are only crudely known for silicon and because the energies of the "2S-like" excited states are unknown, detailed calculations of the above quadratic Stark effect were not considered worthwhile. In general, the nearest state will be a higher excited state (E^ < Effi), so that A is usually positive, and the perturbed state will shift toward higher energy (relative to the V.B.) and move toward the ground state. With no nearby states, the shift and splitting of the ground state will be small; Bir et al (1963) have estimated the ground state splitting in silicon to be about -3 / 1.5 x 10 meV for a field of 1000 v/cm. In view of the shallow acceptor level scheme for germanium (Mendelson and James 1964; Jones and Fisher 1965) no "2S-like" states should occur between the ground state and the "observed" state 1' (see figure 2.2), Therefore state 1 should split and shift only toward the ground state. Since state 2 is located midway between states 1 and 3, and may also be influenced by a "2S-like" state, the shift of the center of gravity of state 2 could be in either direction. The other (higher energy) excited states will, in general, have their center of gravity shifted toward the ground state because of the larger numbers of nearby states of even higher energy. While the energies of the excited states are calculated relative to the valence band edge, the absorption lines measure the (hole) energies of the same states relative to the ground state. Thus, the general shift of the excited states toward the ground state, corresponds to a shift of the absorption lines to lower energy (see figure 2.2). To avoid any confusion, only (hole) energies relative to the ground state will be discussed. In the present experiments, splittings were not resolved, and under such conditions, the applied electric field will appear merely to shift the center-line and broaden the observed absorption line. The apparent shift (with splitting) 53 of the degenerate state should vary quadratieally with the field; the apparent shift will be of the same order of magnitude as the real shift (with no splitting) of a non degenerate state already estimated. In summary, insofar as the effective mass approximation is correct, an external uniform electric field will cause fourfold degenerate acceptor levels to both shift and split into two and cause twofold degenerate levels to be shifted but not split. (b) Breakdown of the Effective Mass Approximation In the effective mass approximation, the potential of the impurity center 2 (e /Kr) has spherical symmetry. The symmetry of the impurity states is then determined by the Bloch functions, which have the "double tetrahedral"symmetry of the crystal, i f spirt-orbit effects are included. The major inadequacy of the effective mass approximation occurs in the neighbourhood of the impurity atom. The potential of the impurity ion, will be 2 2 e /r in the immediate vicinity of the ion, rather than e /Kr. States that have appreciable amplitude near the impurity are also expected to be greatly affected by the specific nature of the impurity, such as local deformations due to the size of the impurity. Both the fields associated with this deformation and the local fields of the surrounding silicon acorns have tetrahedral symmetry. Other effective mass limitations are described by Jones and Fisher (l965). Deviations from the effective mass approximation have present importance because they can introduce a first order Stark splitting. If the full Hamiltonian of the impurity problem has only tetrahedral ssinmetry and is then not invariant under inversion, states belonging to the Pg representation can have an appreciate first AStark effect (Kohn 1957). Since the ground state has a large amplitude at the impurity, i t will be greatly affected by the above breakdowns in the effective mass theory. For this reason, the observed ground state energies are different for various 54 acceptor impurities instead of being the same as predicted. Some first order Stark splitting of the acceptor ground state is therefore expected, but this splitting will be small unless the breakdowns in the theory are serious. The wavefunctions of the excited states to which optical transitions occur are predominantly ,,2P-like", and are zero at the impurity. This, along with the fact that their Bohr orbits are very large, should ensure that the perturbing potential in the vicinity of the hole is approximately e /Kr. In the effective mass theory, the energies of the excited states are independent of the type of impurity. Since this is approximately true for acceptors in silicon (Hrostowski and Kaiser 1958) and germanium (Jones and Fisher 1965), the effective mass approach seems largely justified for the excited states. In summary, for boron acceptors in silicon, breakdowns of the effective mass approximation might cause some first order Stark splitting of the ground state. Bir et al (l963) estimated the Stark splittings of the boron ground state For an applied field of 1000 v/cm, the second order splitting ( AEo=0.001_ meV) is larger than the first order splitting ( A E = 0.0001 meV). In infrared l absorption experiments, these small splittings cannot be resolved. 6.2 ELECTRICAL PROPERTIES OF Si (B. P) (a) Introduction In principle, the Stark experiment is quite simple. A uniform electric field is applied to a Si(B) sample, and the resulting shifts and/or splittings of the Si(B) spectrum are measured as the electric field is increased. As in many experiments however, complications quickly become evident. The magnitude of the electric field is limited by avalanche breakdown to approxi-mately 100 v/cm in uncompensated samples at low temperatures. Also, the shifts to be 55 involved at these low fields are quite small and are easily masked by extraneous effects. Measurements of small shifts were made possible by adapting a strain-free mount, and by improving the signal to noise ratio (allowing greater resolu-tion) as described in chapter 3. The position and lineshape were quite reproducible. However, only by using compensated samples, could large electric fields (up to 1500 v/cm) be applied, and large Stark shifts observed. Conse-quently, all Stark results were obtained from compensated samples. The electrical properties are described in detail because of the correlation between the electrical and optical properties. All Stark experiments were performed at liquid helium temperatures (4,2°K), with the electric field applied along the [lio] crystallographic direction. The nominal impurity concentrations in the compensated Si ( B , P ) samples were, N =-1 x 10^ boron atoms/cm3 = 5 x lO"*"4 phosphorus atoms/cm3 The nominal compensation ratio (^  /N^ ) is 2. The excess acceptor concentration was estimated,from the room temperature resistivity of 19 -n. cm and the calibration by Irvin (l962), to be (K - N.) = 7 x 10 1 4 cm"3. From the intensity a a of the infrared absorption lines, (N - N.) was estimated to be 8,2 x 10 cm , a d The dimensions of the central region of the dumbell-shaped samples (figure 6,l) were typically 1.5 cm long x 0.4 cm x 0.1 cm. The overall length, including dumbells, was approximately 2,3 cm. (h) Measurement of Electrical Properties The basic electrical circuit is shown in figure 6,1, A Fluke model 504B, regulated, high voltage (0-3K7) power supply with reversible polarity, fed power to a voltage divider made of high stability carbon resistors. The resistor R^  protected the sample from excessive heating due to avalanche breakdown and/or thermal runaway. A simple voltmeter-ammeter circuit measured the total voltage Y across the sample and the current I through the sample. Small currents were 56 0 -3KV R R. ® infrared beam in dewar 0 - 5 0 0 volts Fluke Figure 6 . 1 Electrical Circuit; sample drawn to scale. measured with a H.P. Microvolt-Ammeter (Model 4 2 5 A ) . Large currents, in the range of 2 to 50y*amps, and a l l voltages V were measured with recently recalibrated Avolmeters. Voltage drops of up to 500 volts across the side arms were measured with a Fluke Model 801 Precision Potentiometric DC Voltmeter. For higher voltage drops, 300 volt batteries were added at V^ to reduce the voltage to within the range of the potentiometer. At 1 $ off null, the input resistance of the potentiometer was 1000 to 10,000 M*n and did not load down the sample which had a resistance of about 50MJ1. . The infrared beam was transmitted only through the central region of the sample (figure 6 , l ) , and the electric field £ (in volts/cm) in this region, was simply given by the voltage V^  divided by the effective distance. between the sidearms. (L = 0 . 9 3 4 cm). To maintain the sample temperature as close to 4°K as possible, one end of the sample was put in direct contact with the 4°K copper block; a small amount of silver grease ensured good thermal contact in this strain-free mounting. Attempts to get both good thermal contact and good electrical insulation between the sample and the copper block, by using intrinsic semiconductor or sapphire spacers were not satisfactory. The decreased thermal conductivity was usually 57 sufficient to raise the sample temperature and thus lower the avalanche breakdown field . Hence one end of the sample had a low resistance to the grounded dewar at room temperature; at 4.2°K however, this resistance R was many megohms. To S avoid uncertainties, this end of the sample was permanently grounded and the ammeter was operated at high voltages. Power dissipation in the sample was less than 0.1 watts. All electrical contacts were ohmic at room temperature. Most metal contacts to silicon involve barriers at the metal-semiconductor interface, and although these barriers may be small at room temperature, they are often large compared to kT at 4°K. The similarity between the observed electrical properties and the bulk electrical characteristics already reported in the literature, indicated that contact effects were small. By using dumbell shaped samples, i t had been hoped that all contact effects would be eliminated since all the injected carrier recombination would occur in the dumbell regions. Larger area contacts should ensure a low contact resistance. Since the dumbell sample length is over 2 cms, i t was somewhat surprising to sometimes observe (optically) the effects of the contacts in the central region of the sample. (c) Electrical Characteristics The electrical characteristics (figure 6.2) show the resistivity ^ and the current density J as a function of the electric field £ in the central region of the sample. These are bulk electrical properties and were studied in compensated silicon by Kaiser and Wheatley (l959)» Bok et al (i960) and Sohm (l96l). To check for contact effects, the direotion of the applied field was reversed, with the two directions labelled (+) and (-); the effect of contacts on the electrical characteristics is small (i.e. 5^10$). Although at 4.2°K, the rate of thermal generation of carriers is very small, there will be a moderate density of free carriers (holes) in the valence band due to photoionization of neutral acceptor impurities. The extrinsic radiation 58 Electric Field (v/cm) Figure 6.2 Electrical Characteristics and Resistivity. producing this photocurrent is predominantly room temperature (300 K) radiation A which reaches the 4°K sample via the optical ports for the infrared beam. When the sample viewed only the 77°K radiation from the nitrogen shield, the conductivity was about 4 times smaller (see figure 6.2). Most conduction is thus attributed to the photocurrent of free holes in the valence band, and not to impurity conduction due to the "hopping" of carriers between acceptors (which is the dominant conduction mechanism in low concentration, compensated samples in the absence of free carriers). The electrical characteristics (J- £ curve) may be divided into four different field regions for the purpose of discussion (Kaiser and Wheatley 1959). i) At low fields ( £< 10 v/cm), the sample is ohmic. The hole mobility yu. and the free hole carrier concentration p are constant and the holes are in equilibrium with the lattice, Por y*5=10^cm2 (volt.sec)""'' estimated from Sohm's Q results, the low field resistivity of 1,6 x 10 Sh cm corresponds to a free hole 6 -3 concentration p ft 4 x 10 cm due to extrinsic radiation. i i ) Between about 10 and 100 v/cm, the superlinear region of the J- £ curve corresponds to an increase in both p and ^  with increasing electric field. The increase in p with field is attributed to the decreasing thermal recombina-tion rate between free holes and ionized acceptor states. The mobility increases because scattering by ionized impurities is less effective for hot carriers [see also Godik and Molchanov (l96o)], i i i ) Between about 200 and 800 v/cm, the sublinear region of the J- £. curve corresponds to a decrease in the mobility with field. This decrease is attributed to the interaction between the hot carriers and either aocoustical or optical phonons. The density of carriers is approximately constant in this region. iv) Near breakdown ( £. > 1500 v/cm for the present sample), JK decreases quickly and p increases very quickly with field. The hot holes acquire 60 sufficient energy to impact-ionize neutral impurities and a non-destructive electrical breakdown occurs. In compensated specimens, a negative resistance occurs at the beginning of the breakdown (McWhorter and Rediker I960). In the present samples, the breakdown field has not been reached, although breakdown was observed in lightly compensated samples. The observed electrical characteristics are thus qualitatively explained by bulk conduction effects. Since the electrical properties depend strongly on the impurity concentration, on the compensation and on the background radia-tion, a quantitative comparison is not practical (or necessary). That the resistivity at high fields is similar to, rather than smaller (ef. above references) than the low field resistivity, may be attributed to a relative decrease in either j\ or p due to the greater compensation of the present samples. Brown and Jordan (1966) observed a similar resistivity curve. From the rate equations for the creation of carriers, Bok et al (i960) obtained a breakdown field given by, = A(EIfyu.f T ) (N/l^ - l)""* (6.9) The function A is essentially a property of the material, where E^ is the impurity ionization energy,yu. is the carrier mobility, and T the temperature (the temperature dependence is small). From Bok's experimental plot of this relationship for silicon, the breakdown field (E^ > 1500 v/cm) for the present samples corresponds to a compensation ratio (u^/W^) less than 4. This compares well with the nominal value of 2. The fields in the present work were considered too small to cause significant-field emission from the valence band or from the acceptors. Also, the amount of intrinsic radiation was too small to cause significant electron-hole pair production (this point is discussed farther in section 8.2). The samples are sufficiently long and have enough free carriers and traps (ionized impurities) to ensure that the injected carrier current (Gregory and 61 Jordan 1964; Brown and Jordan 1966) is much smaller than the photocurrent. If injection effects are dominant, the current flow is usually space charge limited and varies as some power n of the voltage ( i V 1 ) . For both single and double injection, n is often equal to 2; for field dependent mobility, n may equal 3/2, 5/4, etc. depending on the scattering mechanism. The absence of this and other (see above references) observable injection effects in the electrical charac-teristics, means only the bulk characteristics predominate. Some injection may s t i l l be present and this injection is believed to be responsible for the inhomogeneous electric field distribution sometimes observed. (d) Nonuniform Fields - Contact Effects A test for contact effects is to measure the field distribution in the sample. A nonuniform field distribution is indicative of contact (injection) effects. The fraction of the total voltage dropped across the sidearms (V^/v) as a function of the electric field £ is shown in figure'6.3. If the resistivity was field independent, or at least was the same in the central region and in the dumbells, the fraction V /v can be calculated from the sample dimensions to be 1 56.5$. The observed dip in v ^/v is due to the field dependenoe of the resistivity in the intermediate field range. Since, at both low and high fields, the resistivity should have the same high value both in the dumbells and in the central region, the figure indicates that the field distribution is not uniform in the (-) direction. The field distribution was checked by measuring the voltage drops V , V^ , across the various regions of the sample (see figure 6.l). For the (+) field direction, the measured field distribution (figure 6.4) was symmetric for a l l fields tested; but for the (-) direction, the field distribution was asymmetric (V « V^) at high fields, indicating a space charge region at one of the contacts. For an asymmetric field distribution, the measured electric field (£= V,/L)'will be an average field. No large contact resistances were apparent 62 E l e c t r i c Field Z (v/cm) Figure 6.3 Nonuniform Field Effects; fractional voltage drop across central region. i.Qr Length Figure 6.4 El e c t r i c F i e l d Distribution. 63 from the field distribution measurements at low applied fields. The above electrical measurements illustrate how poor contacts were detected. If the electric field i s not uniform, the boron impurities viewed by the infrared w i l l be situated in regions of different field. Individual energy levels w i l l not be shifted by the same amount and the "averaged" absorption line observed will be broadened, and will perhaps appear asymmetric and shifted (from i t s uniform fi e l d value;. Sufficient data was obtained so that most results with nonuniform fields could be omitted. Some nonuniform field effects are discussed in section 8.3. 6.3 STARK EFFECT ON Si (B.P) - BORON LIMES (a) General Effects The effect of a uniform electric field on the Si(B,P) absorption spectrum is shown in figures 6.5a, b, c. Each of the three spectral regions was always scanned with the same s l i t width and under the same experimental conditions. Only transitions to the four lowest excited states (transitions 1 to 4) were studied; peak 4A was too weak for a quantitative 3 t u d y . The shifts of the transitions due to the electric field can easily be seen. The centerlines of the absorption peaks are shown both for zero applied field (solid curve) and for a high applied field (dashed curve). The positions of two water vapour lines which were not completely removed in the analysis are indicated. A l l Stark samples were compensated and the internal fields due to compensa-tion both perturbed the boron spectrum slightly (see chapter 7) and caused the appearance of a new line (chapter 8). That the internal Coulomb fields due to compensation have l i t t l e net effect on the Stark parameter measurements is discussed in section 6.3f. The new line (dotted curve), located under the low energy shoulder of boron peak 4 at zero fiel d , i s due to compensation. When a strong electric f i e l d i s applied, two new lines are observable. The strongest line (at 39.1 meV), Figure 6,5a, b. Effect of an Electric Field on the Si(B,P) Absorption Spectrum (T 4.2°K). 66 labelled ?°(2P±), is associated with the zero field new line which has grown and shifted slightly with applied field. The weaker new line (at 34.1 meV), labelled P0(2Po), also grows and shifts slightly with applied field. Since the new lines are not associated with the main boron impurity spectrum, their properties are described separately in chapter 8. Stark effects are reported for applied fields up to 1300 v/cm. At higher fields, a field dependent noise (isolated spikes recorded on the infrared detection system) was one factor which prevented meaningful observations being extended up to the breakdown field. (b) Peak Position The uniform electric field ( £ ) affected the peak position of the absorp-tion lines (figure 6.6). Although the absolute value of the energy is known only to within i 0.01 meV, the relative shifts of each line should be accurate to within i 0.003 meV. The points represent experimental values obtained from the center lines of the absorption peaks. The solid curves are the least squares quadratic regression lines, with the standard errors of estimate represented as error bars on each curve. For transitions 3 and 4» with the largest shifts, the correlation coefficient for a quadratic f i t was substantially larger than for a linear f i t . Since transitions 1 and 2 shifted by smaller amounts, the scatter in the data points was sufficiently large to give similar correlation coefficients for linear and quadratic fits. Peak 2 was the only line to shift to a higher energy. The Stark shifts for a l l lines are summarized in figure 6.7. The relative shift of the strong new line, to be discussed later, is represented by the dashed curve. Only the quadratic regression curves and the standard errors of estimate from figure 6.6 are shown. Since the ground state should not be perturbed appreciably by the field (section 6.l), the peak shifts of figures 6.6 and 6.7 should represent only the shifts of the excited states. Since a broadening of 39.64 39.62 38.34 38.32 38.30 h Peak 38.28 Position ,39.64 (meV) 38.26 \-38.24 38.22 38.20 34.52 34.50' 30.36 -30.34 h 30.32 -439 .60 -^ 39.58 Peak 2 Peak 1 J I L _54.52 34.50 Jo. 36 -50.34 J30.32 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Electric Field (KV/om) Pigure 6.6 Shift of Peak Position with Applied Field. 68 +0.04 +0.02 L_ 0.00 -0.02 -p -0.04 I -0.06 £ -0.08 -0.10 U -0.12 h" -0.14 0,2 0.4 0.6 0.8 Electric Field (KV/cm) Figure 6.7 Stark Shifts of the 8i(B,P) Absorption Lines. 69 the absorption lines was attributed to an unresolved splitting of the energy levels by the electric field ( £. )» all peak shifts discussed are "apparent" Stark shifts. For a quadratic Stark effect, the peak shift is given by The experimental values of the Stark coefficient (t ) and the correlation s coefficients for the quadratic regression fits are listed in table 6.1. Table 6.1 Stark Shift Coefficients for Si(B.P) Transition (Peak #) Stark coefficient (t ) (x 10°"^  meV cm^/volt^) Correlation Coefficient 1 2 3 4 P°(2P±) -1.6 +0.5 -14.0 -3.2 -4.4 -0.96 +0.78 -1.00 -0.99 -0.97 (c) Halfwidth (hi) T The observed halfwidths of the four boron lines depend on the electric field (figure 6.8). All measurements on any one peak were made at the same spectrometer slit width and under the same experimental conditions. The observed halfwidths (hi) have not been corrected for instrumental broadening; both the "2 instrument and spectral slit widths are listed in table 6,2. The best regression line fits (solid lines) of the increase in observed halfwidth (or broadening) were quadratic in the field ( £ ), i.e., h ± ( £ ) - h i ( £=0) = t £ 2 . 2 2" n The Stark broadening coefficients (t^) and the correlation coefficients of the line fits are listed in table 6.2. In general, the quadratic f i t is not as good as for the peak position results (for the standard error of estimate, see error 70 Electric Field (KV/cm) Figure 6.8 Stark Broadening of the Si(B,P) Absorption Lines. 71 bars in figure). Since peak 3 is very weak at high fields, its measured half-width is inaccurate} this inaccuracy results in a large scatter in the halfwidth data (figure 6.8) and a small correlation coefficient for this peak. [For clarity, only a single, zero field, halfwidth data point is shown. It represents an average of the (4 to 6) observed halfwidths used in the regression f i t , A similar data presentation is used in figures 6.6, 6.9 and 6.10.] Table 6.2 Stark Broadening Coefficients for Si(B.P) Transition Stark coefficient (t, ) Correlation Slit width Ratio (Peak #) (x 10""8 meV cm2/volt2) coefficient instrumental (mm) spectral (meV) 1 2.0 +0.89 1.20 0.11 1.25 2 1.2 +0.69 1.00 0.12 2.40 3 30. +0.95 0.80 0.13 2.14 4 10.2 +1.00 0.80 0.14 3.19 Since the halfwidths of all the peaks increase, the possibility of some splitting of the ground state cannot be excluded. An upper limit on the ground state splitting, estimated from the small increase in halfwidth of peak 2 at 1000 v/cm is 0.012 meV. The portion of this increased halfwidth attributable to ground state splitting is not known. The broadening is different for the various peaks, indicating that most, i f not a l l , of the excited states are being split by the electric field. This unresolved splitting corresponds to a second order partial removal of degeneracy (section 6.l). Since transitions 3 and 4 broaden the most, their excited states must be at least fourfold degenerate (including spin)} this confirms the degeneracy labelling of chapter 5 (cf. Fisher and Ramdas 1965). The splitting of the excited states should be the same order of magnitude as the increase in observed halfwidth (figure 6.8). (To accurately determine the splitting from the observed broadening of the absorption lines requires first 72 that the observed halfwidths be corrected for instrumental broadening. The increase in true halfwidth may then be related to the splitting; this relation will depend on the lineshape.) In section 6.1, i t was shown that a degenerate acceptor level, i f perturbed through only one nearby state in the quadratic Stark effect, will split by 6E = 0, 2 A . For this simple case, the apparent 2 Stark shift of the absorption line is A = t £ and the splitting is s 2A« :h 1( £ ) - h i ( £ = 0 ) = t, £ . Consequently the ratio (t,/t ) of the Stark 2 2 h n s parameters should be approximately equal to 2. Experimental values of t./t II s (table 6.2) range from 1.25 to 3.19. (d) Peak Height (°<m) The field dependence of the observed peak heights of the boron lines are shown in figure 6.9 by the solid curves. The heights have not been corrected for instrumental broadening. From the field dependence of the integrated absorption (o< h i ) in the next section, the neutral boron concentration was m 2 observed to increase slightly with field. Consequently, the peak heights norma-lized to the zero-field neutral boron concentration (dotted curves) are more meaningful. The point by point normalization used here, does not involve any particular form for the field dependence of the area. The best regression line fits to the normalized data were quadratic in the electric field. All the normalized peak heights decrease with increasing field. This is to be expected, since a perturbation which removes degeneracy should only decrease the peak height of the absorption line (see appendix A), and not increase i t , as might have been inferred from the original peak height curve for peak 2. (e) Area (<* . h i ) m Y The last two graphs showed that both o< and hi are field dependent. m 2 Figure 6.10 shows the field dependence of the product (c* hj_), which is directly m 2 proportional to the integrated absorption or area under the absorption line. The 73 Electric Field (KV/cm) Figure 6.9 Peak Heights of the Si(B,P) Absorption Lines (stark Effect). 74 Figure 6.10 Areas of the Sl(B,P) Absorption Lines (Stark Effect). 75 least squares linear regression f i t was chosen for simplicity. (Due to the large scatter, the standard error was approximately the same for both the linear and quadratic fits). For a l l four peaks, the observed area increased. Such an increase (with field) can be attributed to an increased absorption cross-section and/or to an increased number of neutral boron impurities. Assuming the cross sections for each transition to be independent of the perturbing electric field, the increase in "integrated absorption" (c^h^) with field was attributed to an increased neutral boron concentration. From the slope and intercept of each regression line, the quantity R, which is the percentage increase in boron concentration per unit electric field was determined (table 6.3). Table 6.5 Increase in Neutral Boron with Field Transition R . (Peak #) (x 10""4 cm/volt) 1 0.7 2 1.1 3 (2.8??) 4 1.5 The values of R for the three strongest lines are considered to be the same (i.e. within the estimated errors). Since the field dependence of a l l three peak ( -height curves is explained by the same neutral boron concentration increase, the initial assumptions appear satisfied (see also section 8.3D). The product (c< hi) has the same proportionality to the integrated area at a l l fields* only m ~2 i f the lineshape does not change appreciably. This should be so for peaks 1 and 2 which split only slightly; consequently, their values of R should be accurate. The correlation of (°< hi) with field was sufficiently poor for peak 3 (due to m g • scatter in both °( and hi), that its value of R is unreliable. The assumption m "g-of constant cross section, the methods of determining the impurity concentrations 76 from the absorption data, and related matters are discussed in appendix A. The average value of R is 1.1 x lO""4 cm/volt. For a 1000 v/cm increase in field, the percent increase in neutral boron concentration is 11$ (from 8.0 x 1 0 ^ cm""^  at zero field to 8.9 x 10^ cm~^  at 1000 v/cm). This small increase in concentration does not affect the peak position or halfwidth, but does affect the peak height, and necessitated the normalization performed in the last section. (f) Effect of the Internal Field on the Stark Parameters Since compensated samples were used for the Stark experiments, the acceptor states will also be perturbed by the internal fields due to ionized impurities. The magnitude of the internal fields (section 7.3) is fairly large (~ 500 v/cm), but is essentially independent of the applied field. Even including the observed change in neutral (and hence also in ionized) boron concentration with field, the internal field decreases sufficiently slowly (14?°) over the range of external fields studied, that i t is s t i l l approximately constant. Since a l l Stark effect measurements are made relative to their zero field values, the approximately constant internal field should have li t t l e net effect on the measured Stark parameters. The effects of the internal field perturbation (chapter 7) are determined by comparing the spectra of samples with different compensations. That the internal field had a measurable effect only on peak 3, while the external field significantly affected a l l peaks, is indicative of the relative sizes of the perturbations (and of the quadratic nature of the field dependence). 6.4 CONCLUSIONS - STARK EFFECT The results of this chapter represent the first observation of the Stark effect on either donor or acceptor levels. Although the boron acceptor states were studied with the electric field applied only in the [lid] direction, measurements may be extended to other crystal directions, to other acceptors, 77 and to the donors using the same techniques. Now that a l l three perturbations discussed by Kohn (l957) have been applied to the acceptor states, an applied strain appears to be the most sensitive perturbation on the acceptor excited states, followed by the Zeeman and Stark effects. The Stark effect results for the boron acceptor states agree with the predictions of the effective mass theory (section 6.l). The effective shifts of the absorption lines appear quadratic in the electric field indicating a second order Stark effect. The magnitude of the peak shift of 2P states, calculated to be about 0.06 meV at 1000 v/cm, agrees reasonably with experiment. In general, the lines shift to a lower energy as expected; the slight shift of peak 2 to a higher energy means only that the excited state of peak 2 is being influenced primarily by a nearby state of lower (hole) energy. j^This nearby state could be either the excited state of peak 1 or a "2S-like" state (see Jones and Pisher 1965). A similar conclusion was drawn from the halfwidth study (chapter 4) and the abnormal broadening study (section 5.3).] The higher excited states, with greater Bohr orbits, undergo the largest shifts as expected. The broadening of the absorption lines also appeared quadratic in the electric field, indicating a second order partial removal of degeneracy. The upper limit of the ground state Stark splitting was 0,012 meV (at 1000 v/cm); the calculated splitting by Bir et al (1963) of 0,0015 meV is well within this upper limit. This limit indicates that the first order Stark effect, resulting from a breakdown of the effective mass approximation, is small even for the ground state. A detailed comparison of the Stark results with theory is difficult because of uncertainties in Schechter's wavefunctions for the acceptor states (chapter 5). The measured Stark parameters should be useful for testing any future acceptor state wavefunction calculations and were already used in 78 studying the ionized impurity broadening mechanism (section 4.4). Although most Stark results were not surprising, the observation of the new lines and the apparent increase in boron concentration with field were unexpected. These features are believed to be related and are discussed again in chapter 8. 79 CHAPTER 7 - COMPENSATION EFFECTS 7.1 COMPENSATED Si(B*P) SPECTRUM - GENERAL For equilibrium conditions at 0°K, Si(B,P) compensated samples with total concentrations of N& boron (B) and N^  phosphorus (P) impurities should contain (Shockley 1950): (HN-Nj) neutral boron (B°), N^  ionized boron (B"), Nd ionized phosphorus (P+) and zero neutral phosphorus (P°) impurities. Since optical transitions are possible only for neutral impurities, only the absorption spectrum from the (N -N,) boron impurities is expected. Since no systematic study of the effects of compensation on the impurity absorption spectrum has been published, the present experiments xirere made before measuring the Stark effect (chapter 6) using compensated samples. The observed infrared spectrum of compensated Si(B,P) is very similar to the uncompensated Si(B) spectrum as expected. However, some of the compensated boron lines (especially peak 3) have been shifted slightly and a shoulder on the low energy side of boron line 4 has appeared (figure 6„5e). This shoulder is attri-buted to a new line (at 39.11 meV) whose properties are discussed in chapter 8. The present chapter treats only the 4»2°K compensation effects on the main boron absorption spectrum. For al l boron lines except peak 3» the shift in transition energy ( < 0„01,_ meV), the increase in halfwidth ( < 0,01 meY) and the decrease in peak height ( < 2fo) were small even for the most heavily compensated (2/l) sample„ 7.2 EFFECT OF COMPENSATION ON BORON PEAK 3 Significant compensation effects were clearly observed for boron peak 3, whose compensated and uncompensated absorption spectra are compared in figure 7.1. The nominal compensation ratios label the various samples (table 7.l). The vertical lines are the center lines of peak 3 with the dotted lines refering to other compensated spectra which are not shown. Since both the neutral boron Absorption Coefficient Energy (me?) Pigure 7.1 Boron Absorption Peak 3 (Compensation Effects). 81 Table 7.1 Impurity Concentrations of Compensated Samples Resistivity (w -N) Concentrations ** Nominal at 300°K * (cnr3) Compensation (jLcm) (xlO om ) Boron Phosphorus Ratio=N/N Comments 17 8.2 + 17 9.3 + 19 8.6 + 11.5 12 * 17 7.8 * 150 1.0 * *from room temperature resistivity using the calibration by Irvin (l962). **nominal impurity concentrations from manufacturer's specifications. +from infrared absorption lines (see appendix A), 1. , ++this phosphorus concentration is greater than 1.4x10 cm (see sections 8.2d, 8.3d). ixio15 5xl0 1 4 IxlO 1 5 IxlO 1 4 IxlO 1 5 5xl0 1 3 12 10 io 1 2 io 1 2 2/1 10/1 20/l 1000/1 compensated Si(B,P) uncompensated Si(B) concentration (N -N^ ) and the s l i t width are approximately the same for the two spectra, figure 7*1 is illustrative of compensation effects. The individual compensation effects for peak 3 are summarized in figure 7.2. With similar experimental conditions and the same sli t width (0.65 mm), the results may be directly compared even though the lineshapes were not corrected for instrumental broadening effects. An uncertainty by a factor of two in the nominal phosphorus concentration was arbitrarily chosen for an error bar. Since the nominal boron concentration was well, within this factor, the bars should represent the maximum errors. The straight lines indicate the trend of the data points. The shift to lower energy for the transition was accurately measured. In the three uncompensated Si(B) samples of Table 7.1, the transition energy was 38.374 - 0.003 meV. In the Si(B,P) sample, compensated 2/l, the transition energy was 38,326 i 0.003 meV for six measurements. The shift of the transition energy therefore is 0.048 i 0,006 meV, For the other compensations, only single runs xrere made. Only the measurement errors for the relative shift are shown; •f the absolute energy errors are limited by the calibration to - 0.01 meV. That Phosphorus Concentration l.xl.0 cm ; 0 2 4 6 8 10 0 2 4 6 8 10 Phosphorus Concentration (x 10 em"') Pigure 7.2 Effect of Compensation on Boron Peak 3 (T = 4.2°K* s l i t s = .0.65 am for a l l runs). 83 the shift as well as the broadening is clearly observed, is partly because the compensation effects are measured at 4.2°K where other broadening mechanisms are relatively small. The halfwidth increases and the normalized peak height decreases with increasing compensation (figure 7.2). The uncompensated sample used for these two curves was the 17 cm Si(B) sample of Table 7.1. The neutral boron concen-trations (N -N,), determined optically from the peak heights of the other boron 14 -3 lines, varied only from 7.8 to 9.3x10 cm ' for the.four samples of figure 7.2 and the observed peak heights were therefore normalized to a constant concentra-tion of (Na-Nj) = 8.6 x lO^cm™^ to show the compensation effects. Since the concentration broadening contributions are similar in a l l four samplesj the observed increase in halfwidth with compensation is attributed only to "compensation broadening". The above compensation effects are attributed to the perturbing potential H' due to the unscreened Coulomb fields of the compensationally ionized imparities, i.e. K ( &=1 j=l 3 m J th — where the Coulomb fields from the i ionized donor located at r. and from the I "bh "fch j ionized acceptor located at r. perturb the m neutral acceptor hole located at r . The summation over a l l Kf^  ionized donors and al l ionized acceptors is included. The effect on the observed spectrum will involve some average of E' over all (N -H,) neutral acceptors. 9. CL The screened Coulomb field of a nearest neighbour ion perturbs the neutral acceptor transition energy to fourfold degenerate (including spin) states through both the quadratic Stark and field inhomogeneity mechanisms (see section 4.4), i.e. "-U+.VLjiCB + B ^ . + y ^ ( W ) 84 Since the excited state of boron peak 3 was found experimentally to be fourfold degenerate (section 5.2), i t cannot be associated with Schechter's twofold 2P^  state and the field inhomogeneity contributions (A and B) for peak 3 cannot be calculated. (The contribution B q arises from the first order ground state splitting.) For unscreened Coulomb fields and nearest neighbour interactions only, Gheng (1966) first order field inhomogeneity shift (i.e. A^=^^=A^=0) of the transition energies among Schechter's states, but only a splitting (section 4.4c), For boron peak 59 the excited state wavefunctions are unknown and the field inhomogeneity may, even for unscreened fields, shift the transition energy (i.e. A=0). However, for nearest neighbour interactions only, the direction of the first order transition energy shift of any particular neutral impurity depends on whether its perturbing ionized impurity is positively or negatively charged. Compensation introduces equal concentrations of ionized donors and acceptors which are assumed to be randomly distributed throughout the crystal. Since the absorption line averages the transition energies of the (wa=°w^) neutral boron impurities, i t will not be shifted by the field inhomogeneity because the average first order transition energy shift i s zero. Thus, the unscreened Coulomb fields of compensationally ionized impurities should shift the absorption lines only through the second order quadratic Stark effect which i s independent of the sign of ionized impurity charge. The quadratic Stark effect ( 6E = t i ^ ) is particularly large for peak 3 since tg(=14.0x10°°^ meV 2 2 cm /volt ) is large (section 6,4). This result fortunately avoids associating the excited state of peak 3 with Schechter's 2P^  state (see Skoczylas and White 1965) and requires only the calculation of the internal field. [The more complicated "compensation broadening" of the absorption lines presumably includes both field inhomogeneity and quadratic Stark contributions and will not be discussed.] Although the positions of the ionized donor and acceptor impurities are 85 correlated (Price 1958), the array of ionized impurities is usually assumed for simplicity to be random. For a random array of similarly ionized impurities of concentration EL, the probability distribution of the internal electric field |F| is given by the Holtsmark distribution, ¥ ( If I ) = H(p)/Po where P = 1.002 (eA^ £ )r~ 2 o o and H(p) = (2Tr/J9 ) j x sin x exp [- (x/f ) 5/ 2] dx. o The function H( p), which is tabulated by Chandrasekhar (l943)» peaks at the most probable \f\ equal to approximately 1.6 F Q. The "normal field" F Q is nearly equal to the field caused by a single ionized impurity located at the distance r where o> 4 TT r^ N /3 H 1. For silicon (K =€/ £ =12): c PQ = 3.13 x IO**"8 N ^ 5 volts/cm. For the low impurity concentrations of the present samples, the ionized impurities are sufficiently widely spaced that nearest neighbour interactions predominate. Hence, the internal fields, produced by equal concentrations of positively and negatively ionized impurities, may also be approximated by the Holtsmark distri-bution i f for TS t the total ionized impurity concentration is used (i.e. ^ =21^). In the present compensated samples, F equals 313 v/em (2/l), 100 v/em (lO/l) and 70 v/cm (20/l sample). For the (2/1) sample, the most probable value of the internal field is 1.6 F o (^500 v/cm). The shift of peak 3 in the (2/1) sample then is <£E = t (1.6 F ) 2 = 0.035 MeV (calculated shift) S O ' ' <fE - 0.048 meV (experimental shift). The calculated shift involves no adjustable parameters and agrees well with the experimentally observed shift. Both the internal field (by compensation) and external field (by the Stark effect? see section 6.3') affect the neutral boron 86 spectrum similarly, in that peak 3 shifts, broadens and decreases in height substantially more than the other lines. This similarity occurs because the two electric field perturbations, although different, act primarily through the same quadratic Stark mechanism, as illustrated by the calculated shift of peak 3 due to compensation. 7.4 CONCLUSIONS - COMPENSATION EFFECTS Most of the measurable compensation effects on the boron spectrum were restricted to one absorption line (peak 3 ) which shifted, broadened and decreased in height. These effects were attributed to the unscreened Coulomb fields of compensationally ionized impurities. This perturbation is expected to shift the acceptor absorption lines only through a quadratic Stark effect and the' shift, so calculated for peak 3 » agrees well with experiment. 87 CHAPTER 8 - HEW LINES The observation of a new line in the compensated Si(B,P) spectrum, and of two new lines in the Stark experiments, was reported in chapters 6 and 7. Since these new lines are not part of the main boron spectrum, a description of their properties was postponed to the present chapter. It was noted earlier that the energy of the new line coincided with the strongest neutral phosphorus transition. However, for equilibrium conditions at 0°K, Si(B,P) compensated samples with total concentrations of N boron (B) and N, phosphorus (P) impurities should contain £L CL (Shockley 1950): N^  ionized phosphorus (P+) 0 neutral phosphorus (P°) N, ionized boron (B~) d N - N, neutral boron (B ) a d Consequently, part of this chapter is an attempt to determine whether the new lines are neutral phosphorus transitions and, i f so, why such transitions are observed. 8.1 NEW LINE DUE TO COMPENSATION In the compensated Si(B,P) absorption spectrum (figures 6.5c and 8.l), the shoulder observed on the low energy side of boron line 4 was attributed to a new line due to compensation. Since the new line is weak and is located so near the strong boron peak, its position and height are somewhat uncertain. However, within the experimental error, the averaged peak energy of the new line (39.11 -0,05 meV) coincided with the strongest neutral phosphorus (P°) transition (39.15 -0.01 meV). When the amount of compensating phosphorus is increased, the intensity of the new line also increased (figure 8.l). Since its position did not change appreciably with compensation, the new line does not originate from a state splitting away from the boron peak 4. (The internal field Stark shift is only about 0.01^  meV for peak 4 in the 2/l sample). That the compensating donor is 88 38.2 38.4 38.6 38.8 39.0 39.2 39.4 Energy (meV) Figure 8.1 Effect of Compensation on the New Peak. 89 phosphorus is known both from the manufacturer's specifications and from a laser experiment (section 8.2d). The above properties indicate that the new line is the strongest P° transi-tion. An alternate explanation is that the new line is a B° transition which is usually forbidden (eg. to a "2S-like,, state, etc.), but which becomes allowed in the presence of the internal electric field. It might be a coincidence that the only such forbidden transition in this region occurs at the energy of the strongest P° transition. For the time being, the new line is labelled the P°(2P+) transition (after the lS^)—>• 2P+ P° transition), but this should be regarded merely as a label. [ The presence of neutral phosphorus transitions would be confirmed i f the next strongest P° line (the lS^)—>2P q transition) could also be observed. This second P° line, i f present, should be located at 34.09 meV which is in the wing of the strong boron peak 2, Since this second P° line is weaker by a factor of about 3 (compared to strongest P° line), i t would be on the limit of detectability even at the highest compensation (2/l)„ Consequently, the inability to observe this second P° line was not considered conclusive. In the Stark experiments (section 8.3)» a transition at this energy-was observed The new T°(2P±) line was also observed by Pajot (1964) in Si(B,P) specimens under similar conditions. His observed line was much stronger (l,95 and 1.85 cm""1) t 14 ~3\ for two samples with a constant phosphorus \ 1.1x10 cm ) and different boron / 14 15 -3\ / \ concentrations (.1.8x10 and 1.3x10 cm j. The fraction of (.assumed; phosphorus which is neutral, is only 3 or 4$ in the present 2/l sample, compared to the 100$ reported by Pajot. ["Due to an uncertainty of i 0.1 cm""1 in peak height, the fractions for the other samples (ll$ - 10/lj YT?o - 20/l) are less reliable,] Pajot attributed the new line to a P° transition, and proposed that phosphorus impurities were neutral because of a non-equilibrium electron distribution caused by intrinsic background radiation. There was no mention of searching for other P° lines or of experimental verification of the background radiation mechanism. 90 8.2 EFFECT OF INTRINSIC RADIATION ON NEW LINE (a) Introduction Absorption of intrinsic radiation can cause a steady state non-equilibrium situation in which some of the minority phosphorus impurities in the Si(B,P) sample are neutral. In this section, present experiments indicate that the new line is not due to non-equilibrium caused by intrinsic radiation. Many of the important aspects of the intrinsic radiation mechanism were illustrated by Honig (i960) in electron spin resonance (esr) experiments on n-type compensated Si(P,B). Initially both intrinsic and extrinsic radiation were prevented from reaching his Si(P,B) sample at 4.2°K, and an esr signal proportional to the P° concentration (= N.-N ) was observed. By illuminating the sample with intrinsic radiation (i.e. X< 1.03 microns), large numbers of electron-hole pairs were produced. The electrons were predominately captured by the positive phosphorus ions P+, and the holes by the negative boron ions B~, hence neutraliz-ing the impurities (see figure 8.2). The esr signal indicated that the P° concentration had increased very nearly to N^ . The intrinsic radiation was then shut off and, in the absence of extrinsic radiation, P° remained constant (» N^ ) for up to several hours, indicating a very small rate of electron transfer from neutral donors to neutral acceptors. [This radiative electron transfer has been seen by Honig and Enck (1964) in the recombination radiation spectrum of more highly doped, compensated, n-type silicon.] By then illuminating the neutralized Si(P,B) sample with extrinsic radia-tion, Honig delocalized the impurity electrons by exciting them into the conduction band (figure 8.2). The ratio of trapping cross sections for conduction electrons by P+ traps to that by B° traps was found to be 0.032. The small fraction of conduction electrons that do f a l l across the gap, gradually decreased the P° concentration to its equilibrium value of (N,-N ). 91 1 1 ex 1 C f T hJJ Q © @ pO p* \ h i ) . • \ in \ B° B~ \ © © © 0 © t — T — * ex—• • . 1 C 9- 1" Extrinsic Radiation (hjj ) Intrinsic Radiation ( h ^ . ) Electron transfer ionizes minority impurities neutralizes impurities donor -*• acceptor Figure 8.2 Electron and Hole Transfer Processes in Compensated Silicon Honig's results show that the concentration of non-equilibrium neutral minority impurities depends on the relative magnitudes of intrinsic and extrinsic radiation. Since Honig's impurity concentrations (N^ = 6.0 x IO 1 5 P/crn^, •i r 7 N = 2,0 x 10 B/cm ) are similar to those used in the present experiments, one would expect similar behaviour, except that the roles of the phosphorus and boron would be interchanged, [other neutralization processes associated with compen-sated semiconductors are described by Ryvkin (1964).] (b) Present Experimental Arrangement The two major sources of intrinsic photons that can reach the Si(B,p) sample under the present experimental conditions are from room temperature background radiation and from scattered radiation. The scattered radiation originates from the globar source, is scattered through the monochromator and past the reststrahlen f i l t e r s , and reaches the sample via the infrared beam. This chopped radiation i s measured experimentally with the phase sensitive detector. Most of the 4$ of the total detected signal, which was scattered radiation and transmitted by a NaCI f i l t e r ( }\<20 microns), was also intrinsic radiation ( X < 1 micron) 92 because i t was absorbed by an intrinsic silicon transmission filter. Knowing the detector sensitivity, the maximum rate of absorption of intrinsic scattered 9 / radiation by the sample was measured to be 10 photons/sec. The calculated component of room temperature background radiation which is intrinsic ( X< 1 micron) and which could reach the sample under the present experimental geometry was found to be negligible in comparison (10^ photons/sec). Prom the maximum intensity of the new line (assumed to be the strong phosphorus transition), the steady state concentration of neutral phosphorus o 13 ""3 12 (P ) was estimated to be 2,2 x 10 cm (i.e. 1.5 x 10 neutral impurities). Under these conditions, the average lifetime 1^ of P° would have to be 25 minutes to account for the observed non-equilibrium P° concentration. Lifetimes of this magnitude are possible in view of Honig's results, but are not too probable because of the large amount of extrinsic radiation also incident on the sample. The effect of intrinsic radiation was checked experimentally both by decreasing and by increasing the amount of intrinsic radiation absorbed by the sample while approximately retaining a l l other experimental conditions. (c) Reduced Intrinsic Radiation (Cold Filter) Most of the intrinsic radiation was absorbed by a thin (0.010") germanium filter at 77°K mounted over the light slot on the source side of the nitrogen shield (see figure 3.l). The filter was sufficiently thin to prevent serious loss of light intensity due to the germanium lattice absorption bands. Since the filter will strongly absorb radiation of energy greater than the germanium band gap (0.7 ev.), nearly a l l the intrinsic scattered radiation will be absorbed before reaching the silicon sample. The extrinsic radiation will not be reduced by more than 10% since i t is nearly entirely background radiation, and reaches the sample primarily through the opposite port on the detector side. The absorption spectrum was found to be unchanged when the intrinsic radiation was so reduced. (Blocking, in the same manner, the intrinsic radiation 93 reaching the sample from the detector side, also left the spectrum unchanged. The radiation loss due to Fresnel reflection was too great to use two germanium filters simultaneously). It was concluded that the large number of incident extrinsic photons (about 10^ photons/sec) was sufficient to reduce the average lifetime of any neutral minority Impurities. The new line is thus not due to non-equilibrium minority impurities caused by absorbed intrinsic radiation as proposed by Pajot (1964). Although the new line is apparently due to some equilibrium process, this experiment does not indicate that the line is a neutral phosphorus transition. (d) Increased, Intrinsic BMiation JLaser) When the sample was illuminated with intrinsic (l.960 ev) radiation from a He-He gas laser, photons were absorbed sufficiently fast to neutralize many of the minority impurities. With the laser on, two phosphorus transitions (identified by their energy positions and relative peak heights) were clearly observed as shown in figure 8,3. That the new line (dotted curve) corresponds in energy to the strong phosphorus transition is also apparent. Because the ex-perimental geometry prevented a uniform laser illumination of the sample and because of the short P° lifetime, saturation was not reached and the P° concen-14 -3 i o \ tration of 1.1 x 10 cm (determined from the maximum P peak height; must be regarded only as a lower limit on the total phosphorus concentration. \_ The present technique of monitoring the infrared impurity absorption lines while neutralizing all impurities with intrinsic radiation, allows the compensation ratio K in both n and p type silicon to be easily, determined. This "1R method" is similar to the "ESR method" (igo 1965) which accurately determines K . J With the laser on, small vibrations of either the sample or laser will modulate the reflected portion (~ 55$) of the. laser beam and any 13 cycle components Energy (meV) Energy (meV) Pigure 8,3 Effect of Intrinsic Radiation on the Si(B,P) Spectrum: (T=4.2 K). 95 in the modulated reflected beam which reach the detector will then appear as a noise signal. This noise made the peak heights of the strong boron lines 2 and 4 somewhat uncertain. An upper limit on the average lifetime of the neutral phosphorus caused by the non-equilibrium absorption of intrinsic radiation was also measured. The infrared beam was used to monitor the height of the strong phosphorus transition. After, switching the laser on or off, the height of this transition reached its new steady state value within the time constant of the recording system, indi-cating that the average lifetime of neutral phosphorus under the present operating conditions was less than 2 seconds. Using the optical constants of Moss (1959) and Phillip and Taft (i960) for silicon in the laser range, a l l of the estimated 10 laser photons/sec which enter the silicon will be absorbed; the calculated P° lifetime of 5 x 10"°^  seems reasonable. The laser experimental results complement the previously described cold filter results and the same conclusions were made, namely that the new line is not caused by intrinsic radiation. 8.3 STARK EFFECT OH Si(B.P) - MEtf LIMES In the presence of an external electric.field, two new lines are observed (see figure 6.5). The stronger new line is associated with the new peak due to compensation which appears to shift and increase in intensity as the external field increases. Since this new line occurs, at the position of the strongest P° transition, i t is labelled the P°(2Pj;) transition. A second weaker new line, that is observed at the energy of the second strongest P° transition, is labelled the P° (2PQ) transition. The intensity of the second new line also depends directly on the external field and, in the limit of zero applied field, is too weak to observe. 96 (a) Effect of Applied Field on the Peak Position The observed shift of the new P°(2P±) line with increasing electric field is shown in figure 8.4. The experimental points, quadratic regression line and the error bar have the same meaning as for the equivalent boron data (figure 6.6). The zero field data have a large scatter due to uncertainties in peeling, and for clarity, these data were omitted. The observed position (39.16 meV) of the equivalent transition from the uncompensated Si(P) spectrum is indicated by the arrow. The P°(2P±) peak shifts to lower energy, as did most of the boron lines (figure 6.7). [Due to the general theoretical similarity, between the shallow donor arid acceptor states, the effects of an electric field perturbation should be similar. The nearby higher energy 3PQ state could be responsible for the shift of a 2P+ phosphorus transition to lower energy Uncertainties in peeling introduced a larger scatter in the measured peak position of the weaker P°(2PQ) new line, and this scatter masked the electric field dependence. The low field energy of the weaker new line was 34.09 - 0.03 meV; the energy of the equivalent line in an uncompensated Si(p) sample is also 34.09 meV.' Since the energies of both new transitions agree closely with uncompensated Si(p) transitions, the presence of neutral phosphorus impurities (while the external field is applied) is confirmed. Unfortunately though, part of neutral phosphorus appears to be due to contact effects. (b) Effect of Applied Field on Peak Height Since the heights of the new lines were sensitive to contact effects, data from both good and poor contacts are included. The peak heights (°< ) of the two new lines as a function of applied electric field are shown in figures 8.5a and 8,5b for samples with poor contacts (dashed curves a, b, c) and with good contacts (solid curve). The uniformity of the field distribution (section 6.2d) was used as the criterion of good and poor. The fact that contact effects 97 39.18T Electric Field (KV/cm) Figure 8.4 Stark Shift of P°(2P+) New Line. 98 weak new line P (2F ) o / (cm""1) fl -©-© 0 0.8 1.0 (EV/cm) (cm"1) 3 0.4 0,6 Electric Field 0.8 1.0 (XV/cm) Figure 8.5 Peak Heights of the New Lines (Stark Effect); the peak heights of both new lines are shown for poor contacts (dashed curves a, b, c) and for good contacts (solid curves). 99 are large for the new lines and small for boron lines, in itself indicates that the new peaks are not boron transitions. That the new lines are P° transitions is further shown by comparing the ratio of peak heights of the new lines to the ratio for the equivalent two P° transitions in uncompensated Si(p); (1.50/0.75) = 2.0 (new lines - average high field values) (4.7 /1.87) = 2.5 ( new lines - near maximum U ) (13.5/5.2) = 2 . 2 (uncompensated Si(P) lines) This agreement is considered to be within the experimental errors. The peak height of an absorption line should vary linearly with impurity concentration, as long as the halfwidth and lineshape remain constant (see appendix A), Consequently, the concentration scale of figure 8.5b will accurately convert peak height to P° concentration only at low fields, where li t t l e Stark broadening of the P° line has occurred. . Most observations in this section (including the data for figure 8.5) were obtained during the main Stark effect study of chapter 6. Since the Stark samples were cut from adjacent slices of the same uniform Si(B,P) crystal, they should have the same bulk electrical properties. The field was applied in the [lio] orystallographic direction in a l l cases. When the field on the first sample was reversed (i.e. s t i l l a [lio] direction), the intensity of the new lines had changed, even though the magnitude of the (apparent) field was the same. The homogeneity of the field was checked for each direction of field. For an inhomogeneous field, both new lines were quite intense (curve a - of both figures); when the field was reversed, the field distribution was much improved and both new lines were weak (curve c - figure 8.5b only). On a second sample, the field distribution was poor for the field applied in the (+) direction (see section 6.2d) and the new lines were moderately intense (curves b); for the field reversed and in the (-) direction, the field distribution was uniform (i.e. within 5$) and the intensity of the new lines were given by the "good contact" curve. [All the boron Stark data presented in chapter 6 were obtained on the second sample under 100 the "good contact" conditions. Since the boron lines were insensitive to contact effects, inclusion of a l l the boron data including that obtained with "poor contacts" does not change the reported boron results by more than 10$. The at the same time from the same sample and may be directly compared. Since, under "good contact" conditions, the field distribution was uniform and the electrical characteristics showed l i t t l e apparent evidence of injection, i t was assumed that any remaining contact effects were small. No further improvements of contacts were attempted, since perfect low temperature contacts have a reputation of being elusive. In the present analysis, the good contact results are treated as a separate bulk effect (section 8.3d) and only the differences between the various "poor contaot" curves and the "good oontact" ourve are attributed to contact effects (section 8.3o). The conclusions are appraised in section 8.4. (o) Poor OontaotB The oontaot effects are attributed to electron injeotion from the cathode end on the sample. If the Injected electrons neutralize phosphorus impurities, P° transitions may occur until the P° is again Ionized (by extrinsio radiation, hot oarriers, direot recombination, etc,), Initially, i t had been thought that all injected electron (minority carrier) recombination would take place in the large dumbells, but the observation of P° transitions in the center of the sample (due to contact effects) means that the injection is very deep (l or 2. cms). At low fields (say < 10 v/cm), no appreciable injection into the central region of the sample should occur. Por electric fields in the range 10-100 v/cm, a decreased recombination rate between carriers and ionized impurities is believed (Kaiser and Wheatley 1959) responsible for the increase in conductivity with field (section 6.2c). The same mechanism would cause reduced recombination of minority carriers in the dumbells and permit deeper injection into the crystal good contact P° data (figure 8.5) and the boron results (chapter 6) were obtained 101 in the present work; evidence of this is the similarity in the field dependences of resistivity (figure 6.3) and P° concentration (or peak height - figure 8.5). Since the field in the dumbells is lower (by about x3.4) than the apparent field o £ in the central region of the sample, the value of c at which maximum P due to injection is observed ( ~ 250 v/em) should be higher than the field £. for maximum conductivity ( *~ 125 v/cm). The maximum value of ot.^ for the strong line (figure 8.5h) corresponds to o 14-3 a P concentration of about 1.4 x 10 cm . This new lower limit on the total 14-3 phosphorus concentration is similar to the value of 1,1 x 10 cm determined by the laser experiments (section 8,2d). (d) Good Contacts For good contacts, the P° peak height increases with field, then flattens off; the increased peak height indicates a higher P° concentration, [in analysing the Stark effect on the boron lines (section 6.2), the neutral boron concentration was found to increase (assumed linearly) with field. After normalizing to a constant concentration, the peak heights of a l l neutral boron lines decreased with field (due to Stark splitting), as expected.] The field dependence of the peak height for the P° transitions will be explained in the same manner. As long as the free carrier concentration is sufficiently small, as i t should be in the present work, the concentrations of ionized boron and phosphorus impurities will be the same. The increases in concentration of neutral phosphorus and neutral boron impurities should then be equal i n order to maintain charge neutrality. This equality will serve as a test of the method of analysis. The increase in P° concentration cannot be determined from the area («* h<) m 2" as in the case of the boron peaks because the P° halfwidths are too uncertain (due to peeling). The P° halfwidths should increase roughly quadratieally with field due to splitting of the degenerate 2P and 2P+ states. However, at low 102 fields where the halfwidth is s t i l l approximately constant, the peak height ) is directly proportional to the concentration. The P° concentration was thus assumed to be linear in the field and given by the straight (dotted) line in figure 8.5b. The zero field P° concentration was calculated from the observed 1*3 3 o peak height and the equivalent Si(p) data to be 2.0 x 10 cm . The total P / 13-3 o concentration at 1000 v/cm is then about 9 x 10 cm . The increase in P of 7 x 10 cm agrees closely (within the experimental error) with the equivalent i o \ 13—3 increases in neutral boron B^ ; of 9 x 10 cm calculated from the increasing area («* hi) of the three strongest boron absorption lines. The agreement between these two independent measurements of concentration increases (of P° and B°), satisfies the initial assumptions. The same concentration increase explains the dissimilar field dependences of peak heights; the large relative increase in peak height for the P° transitions is due to the small initial (zero field) phosphorus concentration. The assumption that the P° concentration Increases linearly with the field is only approximate. Due to the small percentage increase in neutral boron with field, the form of the field dependence was uncertain. Both a linear and a quadratic field dependence gave equally good fits to the data, and were con-sistent in that the. same concentration increase must explain the data from a l l three boron lines. Prom the phosphorus data, a linear dependence was preferred. Since the external applied f i e l d caused the P° concentration to increase from its zero field to high f i e l d value, perhaps the internal field (due to com-pensation) caused the initial increase up to the zero field value. The PQ concentration extrapolates to zero (figure 8.5b) at an electric f i e l d intercept of about 280 v/cm. This field might then represent the homogeneous electric field that has an effect (on the P° concentration) equivalent to the inhomogeneous internal field. The most probable value of the internal f i e l d for the present 2/l sample (section 7.3) was 500 v/cm and this internal f i e l d roughly corresponds with the 280 v/cm field intercept. 103 The normalized P° peak height (i.e. normalized to the constant zero field concentration) will, with the present analysis, decrease monotonically with the field as expected. Assuming the absorption cross section for this P° transition to be constant, the calculated normalized halfwidth, i.e. hi( t )/hi (£=0), . 2 2 increased smoothly with the field in a manner similar to the equivalent boron results. These corrected peak height and halfwidth results (curves not shown) were obtained only for the P°(2P+) transition for which adequate data were available. 8.4 CONCLUSIONS - HEW LINES In compensated Si(B,P) samples with no applied field, a new line (see also. Pajot 1964) was observed in addition to the normal boron transitions. The new line was not due to a non-equilibrium situation caused by intrinsic radiation or contact effects. Reasons for attributing the new line to a neutral phosphorus (P°) transition are: (a) The energy of the new line corresponded, to the strongest P° transition, (b) Phosphorus is the compensating Impurity. (c) The intensity, of the new transition increased as the concentration of compensating phosphorus increased. In the presence of an applied electric field, the above new line became more intense and a second weaker new line appeared at the energy of the second strongest P° transition. With the correct ratio of peak heights, there was no doubt that both lines were P° transitions. The results were complicated because the new lines, unlike the boron transitions,, were sensitive to contact effects. Because of the association of neutral minority impurity transitions with a nonuniform field distribution i n the sample, i t was assumed that contact effects were causing minority carrier injection. Since the P° transitions were observed in the central region of the dumbell shaped samples, the injection must be very / 104 deep (l or 2 cms). This monitoring of neutral donor and acceptor impurities in the presence of an applied field, as in the present experiments, is a sensitive method of studying directly the elementary conduction and contact processes at low temperatures. The "good contact" properties of the new lines were attributed to bulk  effects, rather than to residual injecting contact effects, because? (i) The high field peak heights (figure 8.5) appear to approach a constant value, independent Of contact effects. (ii) The field distribution was uniform and the electrical characteristics showed li t t l e apparent evidence of contact effects. ( i i i ) Even with no applied field, P° appears to be present due to a bulk effect in compensated samples. If this zero field P° concentration is associated with the internal Coulomb fields of ionized impurities due to compensation, and increases with the compensation, then the P° concentration might also be expected to depend on the external field in a similar (bulk effect) manner, (iv) A simple analysis adequately explained most "good contact" results. The similar increases in neutral phosphorus (P°) and in neutral boron" (B°) with field explained the dissimilar field dependences of peak height for the B° and P° lines. After normalizing to a constant concentration, the P° lines depend on the applied field in a manner similar to the equivalent B° Stark results, as expected. In summary, most properties of the new lines could be simply explained by intro-ducing a mechanism which neutralizes a small fraction of the phosphorus ions as the electric field (both external and internal) is increased. Since the observations were complicated by the low intensity of the single new line at zero field and by the contact effects, alternative interpretations are possible. Consequently, the present description of the properties of the 105 new lines is intended only as the basis for further experimental work on high quality samples with higher compensation ratios and with improved contacts. Other types of compensation should also be checked. The presence of P° in the present Si(B,P) samples is currently being checked with ESR (X band) experiments. 106 CHAPTER 9 - GENERAL CONCLUSIONS This thesis i s a study of the effects of perturbations by applied strains, by applied ele c t r i c f i e l d s and by internal Coulomb f i e l d s on the acceptor states of boron-doped s i l i c o n . Many of the present results should be characteristic of other acceptor states i n both s i l i c o n and germanium. An applied uniaxial strain s p l i t s the acceptor states s u f f i c i e n t l y far to determine the degeneracies of the ground state and the four lowest "observable" excited states (labelled 1, 2, 3, 4 i n order of increasing hole energy). A l l states were fourfold degeneratej with the p o s s i b i l i t y of an additional 2-fold accidental degeneracy for excited state 4. The degeneracy of the third and fourth excited states.disagree with the theoretical (twofold) predictions of Schechter (1962). The calculations for germanium by Mendelson and James (1964) have the correct effect of introducing more low energy fourfold degenerate states but unfortunately these calculations have not yet been extended to the acceptor states i n s i l i c o n . An electric f i e l d was applied i n the [lio] crystallographic direction of compensated Si(B,P) s i l i c o n and the Stark effect on the boron acceptor states was observed. The s h i f t of the absorption l i n e s , i n general to a lower energy, appeared quadratic i n the ele c t r i c f i e l d (second order Stark effect). The Stark broadening of the l i n e s , also quadratic i n the field<, was attributed to a second order unresolved p a r t i a l removal of degeneracy of the excited states. The Stark parameters describing the sh i f t i n g and broadening of the four lowest excited states are given. These second order Stark effects cannot be accurately calculated using the approximate and incomplete wavefunctions of Schechter (1962), however, the observations are consistent with the general predictions of the effective mass theory (Kohn 1957). The internal f i e l d s are due to the Coulomb field s of ionized impurities. A recent theoretical study (Cheng 1966) indicates that the internal f i e l d s may 107 s h i f t or s p l i t the acceptor states through the gradient of the internal f i e l d (i.e. the f i e l d inhomogeneity at the neutral impurity) as well as through the quadratic Stark effect (due to the average internal f i e l d at the neutral impurity s i t e ) . The relative magnitude of each contribution depends on the particular (excited) state involved and on the ionized imparity concentration. The quadratic Stark contribution, which can be calculated using the Stark parameters measured ea r l i e r , w i l l be smallest for excited state 2 and largest for excited state 3. Compensation affects the boron states i n Si(B,P) samples, through the unscreened Coulomb f i e l d s of the ionized boron and phosphorus impurities. The boron l i n e s , especially peak 3, shifted and broadened (i.e. unresolved s p l i t t i n g ( of the excited states) as the compensation increased. The s h i f t of peak 3 due to compensation was explained i n terms of the quadratic Stark effect. Ionized impurity broadening of the absorption lines i n uncompensated Si(B) samples, i s caused by the screened internal f i e l d s of ionized boron impurities. Recent measurements of the absorption l i n e broadening' mechanisms i n Si(B) by Colbow (l963) were found to be derived from incorrect experimental halfwidths. (Colbow's samples were being accidentally strained by the mounting). The broadening mechanisms were redetermined from new absorption l i n e halfwidth measurements. The same broadening mechanisms (phonon, dislocation, concentra-tion, ionized impurity) were s t i l l operative, although the relative magnitudes were different. Por boron l i n e 2 , the ionized broadening contribution due to the f i e l d inhomogeneity (Cheng 1956) dominated that due to the quadratic Stark effect. The earlier " s t a t i s t i c a l . Stark" theory (Colbow 1963) of ionized impurity broadening was shown to be inadequate. Various results could be related. The abnormal broadening of selected absorption lines i n the heavy strain'spectrum was explained by the same phonon broadening mechanism (Nishikawa and Barrie 1963? Barrie and Nishikawa 1963) that broadens the unstrained absorption lines. Three independent pieces of 108 evidence indicated that the boron excited state 2 was being strongly influenced by an excited state of lower (hole) energy; this lower energy state might be a "2S-like" excited state. The broadening of the absorption lines by the applied electric field corresponded to a second order removal of degeneracy; this degeneracy information supplements the degeneracy labelling from the present strain experiments (cf. Fisher and Ramdas 1965). In compensated Si(B,P) at 4°K, a weak new line was observed at the energy of the strongest neutral phosphorus transition. However, a l l compensating impurities (i.e. phosphorus) should be ionized (Shockley 1950). With an applied electric field, two neutral phosphorus transitions were observed; both transi-M auc-tions were fairly sensitive to contact effects. 2fee observations could be explained by introducing a mechanism which neutralizes some of the phosphorus ions as the internal (due to compensation) and external electric fields are increased. The neutral phosphorus results were somewhat obscured by the weakness of the zero field new line and by uncertainty as to whether the injection effects had been eliminated. The present results should, however, provide a base for further study of the origin of the new lines. 109 APPENDIX A - ABSORPTION RELATIONS (a) Definitions (Lax 1954)and Units of the expressions related to absorption are. absorption coefficient = ( ^  ) cm-1 integrated absorption coefficient = cm meV 2 absorption cross section = a " ( ^ ) = »<-(^ )/N cm integrated absorption cross section = X = ^ )d(h ^  )/N cm^ meV In the present experiments, the absorption coefficient °^ ( >^ ) is measured as a function of the energy (h >W ) of the incident monochromatic radiation. [For convenience, ^ ( ) was written merely as ^ j • The integrated absorption (coefficient) is obtained by integrating the area under an absorption line. For ease of evaluation, this integral is often approximated by the product (^ hi_) m 2 of maximum peak' height (°< ) and full width at half power, or halfwidth (hi_). m 2 r C( TT/ln2)* (o< hi)/2 = 1.07(^ hi) for a Gaussian lineshape "TT (o< hi)/2 = 1.570* hi) for a Lorentzian lineshape L m Y m Y The absorption cross section Q~( <W ) and integrated absorption cross section (x) are defined so as to be independent of the concentration N of absorbing centers. In the present work, X did not change appreciably in the boron concen-tration range of 10 to 10 cm . (b) Effect of Instrumental Resolution on the Integrated Absorption . The finite band pass of a spectrometer causes distortion of spectral absorption lines. The assumption in this thesis that instrumental broadening will not, however, affect the integrated absorption coefficient, is not strictly true and so will now be examined. Dennison (1928) showed that the total area under the fractional absorption curve is independent of the resolving power CO . oQ (l - T ^ d U . = J (1 - T^ ) dJJ , (A.1) o o where T ^ is the percent transmission recorded by the instrument at the 110 frequency and T^ is the percent transmission for a spectrometer of infinitely high resolving power, i.e. O  T*: = J?AG») exp ( - o ^ d U = exp (- <*.d) (A.2) fov(*0 di3 T w = exp (- o^d). (A.3) where '-<<v and °< are the observed and true absorption coefficients respectively. Dennispn's result (equation A.l) is quite general (see also De Prima and Penner 1955) and assumes only that the sli t function (^-u) is symmetric about ^ . In the limit as c*^—»-0 , equation (A.l) becomes S .e< M . ~ J o c ^ di) : as - o ^ . — • 0 o o Therefore, only in the limit of infinitely weak absorption peaks, are the observed and true integrated absorption areas exactly equal. The experiments of KostkowsfcL and Bass (1956) show that, even for the large sl i t widths and strong absorption lines of the present experiments, the assumption that the integrated absorption coefficient is independent of the instrumental resolution will introduce an error of only a few percent. (c) Assumption of Constant Integrated Absorption Cross Section The integrated cross section (x) was assumed to be independent of the per= vt/^bing.uniform-electric field (chapters 6 and 8). If a l l transitions are allowed between two levels, each of which is twofold degenerate, the observed absorption line i s the sum of four superimposed component lines. In the presence of a small perturbation which slightly shifts and splits both levels, the observed absorption line becomes the unresolved sum of four slightly displaced component lines. It is then assumed that as long as the concentration (H) of absorbing centers has not changed, the observed absorption line should have the same integrated absorption, even though the peak height ( is decreased and the halfwidth (hj_) increased. If the concentration is changed by the perturba-I l l tion, the integrated cross section X should s t i l l remain constant. The above assumption should not be true in general. However, the uniform f i e l d causes only a small second order perturbation of the acceptor states, and i f the perturbation i s sufficiently small that the acceptor wavefunctions are not changed appreciably, the resultant change in the integrated absorption cross section should be small, (d) Estimation of Concentration The concentration of absorbing centers may be estimated from the observed absorption line i f the integrated cross section (x) is constant, X = V <<(.*) )a(h AJ )/N = constant (<< , h i ) / l = constant N = constant ( °< ,hi) (A,4) m "2~ If the peak height ( °^ 'm)> the halfwidth ( h p and the concentration (N) of a reference sample are known, the unknown concentration of the same absorbing centers in a second sample can then be estimated from i t s observed area (°< „hi) m 2 using equation A,4. (This assumes the lineshapes are similar. If the lineshapes are different, the integrated area must be used in place of o < m . h p . . • If the observed halfwidths of the reference and 'onknown sample are approxi-mately the same, the concentration can be estimated from the peak height using equation A. 5. This is often the case for shallow impurity absorption lines at 4°K, which have similar lineshapes and halfwidths, 'usually due to instrumental broadening. N = constant ©<. (A.5) 112 APPENDIX B - DONOR STATES The donor absorption lines i n Si(As) were reported to s h i f t with temperature (Goruk, 1964). This behaviour was confirmed i n Si(P) i n which the 2P and the 0 2p£ transitions moved 0.09 * 0.01 meV to higher energy when the temperature was raised from 4°K to 54°K. These shifts are attributed to the electron-phonon interaction (Cheung 1966), The smallest absorption line halfwidth yet observed, was measured i n a low dislocation density ( 4 2000 disloc/cm 2), low concentration ( < l O 1 ^ atoms/cm^), Si(p) sample at 4°K. Most of the observed halfwidth ( ~ 0.11 meV) was due to instrumental broadening; the true halfwidths of both the 2P Q and 2P± transitions were £ 0,04 meV, Since the instrumental broadening correction i s so large, the true lineshape i s uncertain. The deorease In halfwidth (cf. Bichard et a l , 1962; Goruk. 1964? Pajot, 1964) i s attributed to decreased dislocation broadening and improved mounting. An order of magnitude Improvement i n resolution i s required before accurate halfwidth and lineshape measurements of these narrow lines are possible. 113 BIBLIOGRAPHY Aggarwal, R.L. and Ramdas, A.K. (1965). Phys. Rev., 137., A602. Baltensberger, W. (1953). Phil. Mag., 4±, 1355. Baranger, M. (l962). 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