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## Material Information- Title:
- Conduction mechanisms in low breakdown voltage silicon P-N junctions
- Creator:
- Lukaszek, Wieslaw Adam, 1946- (
*Dissertant*) van der Ziel, Aldert (*Thesis advisor*) Chenette, Eugene R. (*Reviewer*) Shaffer, Charles V. (*Reviewer*) Trickey, Samuel B. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1974
- Copyright Date:
- 1974
- Language:
- English
- Physical Description:
- v, 157 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Charge carriers ( jstor )
Diodes ( jstor ) Electric current ( jstor ) Electric potential ( jstor ) Electrons ( jstor ) Information retrieval noise ( jstor ) Ionization ( jstor ) Low noise ( jstor ) Noise measurement ( jstor ) White noise ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Electronic noise ( lcsh ) Junction transistors ( lcsh ) Silicon diodes ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- White noise spectra of diodes 1N746 to 1N752 have been used to investigate the reverse conduction mechanisms in silicon junctions breaking down by a combination of tunneling and impact ionization multiplication. The reverse voltages at the onset of multiplication noise indicate that, regardless of junction field strengths, carrier multiplication sets in only when the carrier energies exceed the threshold energies for ionization. Further examination of noise data yields their values, indicating that the multiplication process is dominated by the effects of the threshold energies for ionization. When they are taken into account, it becomes clear why Tager's and Mclntyre's theories of noise in avalanche diodes, and the conventional DC multiplication calculations, are not applicable to low breakdown voltage diodes. Consequently, a new method of performing DC multiplication calculations as well as a new theory of noise in low breakdown voltage diodes is developed.
- Thesis:
- Thesis--University of Florida, 1974.
- Bibliography:
- Includes bibliographic references (leaves 154-156).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- Wieslaw A. Lukaszek.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000869363 ( AlephBibNum )
AEG6388 ( NOTIS ) 014267458 ( OCLC )
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CONDUCTION MECHANISMS IN LOW BREAKDOWN VOLTAGE SILICON P-N JUNCTIONS By Wieslaw A. Lukaszek A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974 ACKNOWLEDGMENTS I am indebted to Dr. Aldert van der Ziel, my most inspiring teacher, for his generous assistance and encouragement as my advisor. I also thank Dr. Eugene R. Chenette for his encouragement and Dr. Karel M. van Vliet for his friendly advice and many helpful discussions during the final stages of this work. The friendly help of Dr. Charles V. Shaffer and Dr. Samuel B. Trickey is also most warmly appreciated. The aid of Mr. Don Estreich of Teledyne who supplied the diode samples and data, and of Mr. Sam Weaver of Texas Instruments who supplied the low noise JFET's employed in this study, is gratefully appreciated. Finally, my deepest thanks go to my parents, Antoni and Stefania, and my brother, John, whose encourage- ment and countless sacrifices I shall never be able to repay. TABLE OF CONTENTS ACKNOWLEDGMENTS. ABSTRACT . CHAPTER I INTRODUCTION ....... CHAPTER II EXPERIMENTAL METHODS .... CHAPTER III DISCUSSION OF EXPERIMENTAL RESULTS. . CHAPTER IV DETAILED ANALYSIS OF NOISE DATA . CHAPTER V CALCULATION OF DC MULTIPLICATION. .. CHAPTER VI CONCLUSIONS .. . . ....... CHAPTER VII RECOMMENDATIONS FOR FURTHER STUDY . . APPENDIX A MULTIPLICATION NOISE RESULTING FROM NO MORE THAN ONE IONIZATION PER CARRIER TRANSIT ACROSS DIODE SPACE CHARGE REGION .. . ..... APPENDIX B MULTIPLICATION NOISE RESULTING FROM NO MORE THAN TWO IONIZATIONS PER CARRIER TRANSIT ACROSS DIODE SPACE CHARGE REGION . . . . . APPENDIX C DC MULTIPLICATION SIMULATION PROGRAM. . BIBLIOGRAPHY . . . . . . .154 BIOGRAPHICAL SKETCH ........ . . . . . . . iv 73 87 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CONDUCTION MECHANISMS IN LOW BREAKDOWN VOLTAGE SILICON P-N JUNCTIONS By Wieslaw A. Lukaszek August, 1974 Chairman: Aldert van der Ziel Co-Chairman: Eugene R. Chenette Major Department: Electrical Engineering White noise spectra of diodes 1N746 to 1N752 have been used to investigate the reverse conduction mechanisms in silicon junctions breaking down by a combination of tunneling and impact ionization multiplication. The reverse voltages at the onset of multiplication noise indicate that, regardless of junction field strengths, carrier multiplication sets in only when the carrier energies exceed the threshold energies for ionization. Further examination of noise data yields their values, indicating that the multiplication process is dominated by the effects of the threshold energies for ionization. When they are taken into account, it becomes clear why Tager's and McIntyre's theories of noise in avalanche diodes, and the conventional DC multiplication calculations, are not applicable to low breakdown voltage diodes. Con- sequently, a new method of performing DC multiplication calculations as well as a new theory of noise in low break- down voltage diodes is developed. CHAPTER I INTRODUCTION As the reverse bias across a p-n junction is increased, the junction eventually begins to pass appreciable current. Typical voltage-current characteristics for commercially available diodes are shown in Figure 1. It is evident that the reverse current characteristics are considerably dif- ferent for p-n junctions which "break down" at different voltages. The type of breakdown typified by Figure la is characterized by (i) temperature sensitive pre-breakdown (saturation) current which approximately doubles for each 10C rise in temperature, (ii) a well-defined breakdown voltage, VB, and (iii) a unique, positive, breakdown voltage temperature coefficient, B, defined by VB = VBO[1+B(T-TO) where VBO is the breakdown voltage at room temperature, T . This type of breakdown, characteristic of relatively wide p-n junctions with VB>14 volts, is commonly known as "avalanche" breakdown since it can be understood in terms of an impact ionization mechanism resembling an avalanche (Tyagi, 1968). Reverse conduction of the type shown in Figure Ic is characterized by (i) temperature insensitive reverse current -- temperature change of 2000C changes the current by a factor of two or three, (ii) lack of well-defined breakdown voltage, and (iii) a negative temperature coefficient whose magnitude depends on temperature and reverse voltage. This type of breakdown, characteristic of narrow p-n junctions capable of passing considerable current densities at reverse voltages smaller than about 2.5 volts, can be understood in terms of internal field emission of valence band electrons, also known as Zener tunneling (Chynoweth et al., 1960; Tyagi, 1968). P-n junctions wniich break down between about 2.5 to 14 volts do so as a result of the combined effect of internal field emission and impact ionization multiplication (Tyagi, 1968). Since most of the measurements which formed the basis for the understanding of p-n junction breakdown were DC measurements, this range of breakdown voltages has never been carefully investigated. The reason for it was the severe difficulty encountered in distinguishing the contributions of each of the two mechanisms. The aim of this study was to investigate the reverse conduction mechanisms in diodes breaking down in this range of voltages. The difficulty which previous workers encountered in distinguishing the tunneling current from the impact ionization multiplied current was overcome by using noise as a diagnostic tool, the idea being that noise would more closely reflect the basic microscopic conduction mechanisms. This technique was first employed by Lauritzen (1966) in determining the onset of carrier multiplication in low breakdown voltage silicon diodes. Interest in avalanche diodes as photodetectors, micro- wave signal sources, and amplifiers has already led to extensive experimental (Haitz, 1965, 1966; Haitz and Voltmer, 1968; Baertsch, 1966, 1967; Kumo et al., 1968) and theoretical (Tager, 1965; McIntyre, 1966; Hines, 1966; Gummel and Blue, 1967) studies of noise in these diodes at low and high frequencies. Because noise measurements provide the experimental foundation of this study, Chapter II is primarily devoted to the experimental considerations and techniques involved in collecting the noise data. DC and capacitance measure- ments are discussed there also. A general discussion of the data is undertaken in Chapter III. It is found that Tager's (1965) and McIntyre's (1966) theories of noise in avalanche diodes are incapable of explaining the data. The reason for this becomes clear when the mechanism of the onset of carrier multiplication is inferred from the noise data. It is then also apparent that the conventional method of calculating DC multipli- cation is not applicable to low breakdown voltage diodes. When a new definition of the DC multiplication factor, and a new functional form of the expression for noise are introduced in Chapter IV, the noise data reveal in detail the influence of the threshold energies for ionization on the multiplication process. On basis of the inferences made in this and the preceding chapter, new theories of noise in low breakdown voltage diodes are developed and compared with experimental results. The details of the multiplication process inferred in Chapters III and IV are incorporated in Chapter V, which discusses a simulation program implementing a new method of performing DC multiplication calculations in low breakdown voltage junctions. The results of the con- ventional and the new calculations are then compared with data. Chapter VI summarizes the highlights. Chapter VII suggests topics for further study. CHAPTER II EXPERIMENTAL METHODS The p-n junctions investigated in this study are com- mercially available diodes, types 1N746 to 1N752, manufactured by alloying small aluminum buttons onto n-type silicon. The raised appearance of the aluminum buttons, shown in a vertical cross-section of the diode in Figure 2, is the result of acid etch used to delineate the junction geometry and eliminate surface effects around the button periphery. The samples chosen for study were those which exhibited very little flicker noise. In final samples, flicker noise was not noticeable at frequencies greater than 300 Hz. This selection criterion presumably eliminated from consideration those diodes in which surface or edge effects might have had significant influence on the reverse conduction mechanisms of the junctions. All final samples, whose DC V-I charac- teristics are shown in Figure 3, came from Continental Device Corporation, now Teledyne Semiconductor. In this study the reverse conduction mechanisms were inferred from two sets of diode terminal measurements: the DC V-I characteristics and noise. The DC characteristics provided information on the average behavior of carriers in the junctions, whereas the noise measurements provided statistical information on the transient, or microscopic, details of the conduction process. The noise measurements thus provided the groundwork details for the conduction model, which will be discussed in the following chapter. The noise measurements, which form the experimental foundation of this study, were performed with the aid of the noise measurement system shown in Figure 4. The GR 1381 Random Noise Generator coupled to the HP 350-D 600 Q attenuator supplied a 2 Hz to 50 KHz white noise calibration signal. The 600 0 resistor following the attenuator provided the proper impedance match for the attenuator, while the 1 pf capacitor in series with the 200 K1 resistor provided DC and impedance isolation from the rest of the circuit. The impedance isolation was necessary to maintain a constant impedance level at the attenuator terminals regardless of the impedance levels existing in the diode bias network, and to convert the noise calibration network into a high impedance, current-like source to prevent excessive loading of the diode. The diode bias network consisted of a large number of wire wound resistors, Rb. When used to adjust the diode reverse current, they were chosen so that Rb was more than ten times greater than the diode dynamic resistance, Rd. The low noise preamplifier used for its first stage a preselected sample of a Texas Instruments low noise JFET, the SFB 8558. The schematic diagram of the preamplifier is given in Figure 5 and the preamplifier's equivalent noise resistance referred to the input, Rn, as a function of frequency, is given in Figure 6. The preamplifier was followed by a selectable gain amplifier which amplified the noise signal to a level suitable for processing by the GR 1925-1926 Real Time Spectrum Analyzer. The oscilloscope was used for visual observation of the noise waveforms. The GR 1925-1926 Real Time Spectrum Analyzer contains a bank of 45 third-octave filters, ranging in center frequencies from 3.15 Hz to 80 KHz. The output of each filter is sampled for up to 32 seconds and the dB of the RMS voltage of each filter is computed and displayed on the GR 1926, or is printed out by the MDS 800 tape printer. The measurements were performed as follows. First, the noise calibration signal, provided by the GR 1381 and the 600 Q attenuator, was removed by disconnecting the attenuator from the circuit and replacing it with a 600 0 resistor placed in parallel with the 600 2 attenuator termination resistor. The diode bias resistor, Rb, was then chosen to obtain the desired value of diode reverse current, and five, 32 second, diode noise data sets were recorded. Next, the 600 0 resistor in parallel with the attenuator termination resistor was removed, and the attenuator returned to the circuit. The attenuation level was then adjusted to obtain a calibration plus diode noise output about 20 dB higher than the diode noise output alone, and a second set of five, 32 second, readings was recorded. Given the system noise equivalent input circuit of Figure 7, it is a trivial matter to determine that the diode noise current spectral density, Sid, is then given by S. S Sig Sva Sid AdB=- Sir Sib (2.1) [10 -1] zt12 and -dBatt 1 2- ig 200KQ (1 Sv = (20- ) (10 )Svg (2.2) Sva = 4KTRn (2.3) 1 1 S = 4kT(-l )L (2.4) Sir 200K 200K) (2.4) 4kT Sib Rb (2.5) I tl2 = R2 1 ) (2.6) Sb 2+W 2 C2R2 1 Rt= 1 1 (2.7) 20Ki)KQ 200Kn R d R b where S is the spectral density of the noise generator out- vg put voltage, Rn is the equivalent noise resistance of the low noise preamplifier, Rd is the diode dynamic resistance, Rb is the bias resistance, Ct is the total input capacitance, dBatt is the attenuator setting, and AdB is the difference in dB between the calibration plus diode noise level, and the diode noise level alone. This calculation was carried out for each filter, and the final results were averaged over all filters processing a white noise spectrum. Typically, this resulted in averaging over the 25 filters of center frequencies greater than 250 Hz. The system accuracy was verified by measuring the thermal noise of resistors. When resistors of 200 : to 2 Mn were inserted in place of a diode, and the above measurement pro- cedure was repeated, the resistance values predicted from noise measurements agreed to better than 2% with values obtained from precision bridge measurements. Since the low noise preamplifier employed a JFET for the first stage, it sensed the voltage, rather than the current, at its input terminals. Consequently, the diode noise current was converted to a noise voltage by the diode dynamic impedance before the noise signal was amplified. The diode dynamic impedance, therefore, had to be determined, and was measured using the diode bias network, capacitively coupled to a Wayne Kerr B601 Radio Frequency Bridge utilizing a Wayne Kerr SR 268 Source & Detector, as shown in Figure 8. The 100 KHz signal applied by the bridge to the diode was adjusted until further signal level reduction produced no difference in the measured RC values. This typically occurred for applied signal levels of less than 40 mV RMS. The parallel equivalent RC values of the diode and its bias network were thus deter- mined to an accuracy of 1%. The capacitance values determined above were also employed in the C-V plots used in estimating the junction doping profiles and doping densities. The diode areas were obtained from junction photographs taken after the aluininum button was etched away in phosphoric acid. The error in determining the junction area in this manner was estimated to be about 15%. A typical junction surface is shown in Figure 9. Although the surface is full of ridges, it is assumed that the junc- tions are very nearly planar because they are extremely thin. On the scale of the several hundred angstrom thin space charge regions (SCR), the ridge contours should appear as gentle undulations in the junction topology. The DC V-I measurements were made using the test setup shown in Figure 10. It consisted of the diode bias network, a Keithley Model 615 Digital Electrometer operating as an ammeter, and a Fluke Model 801B Differential Voltmeter. The accuracy of the DC measurements was 0.5% for currents greater -7 -7 -9 than 107 amperes, and 2% for currents between 107 and 10 amperes. Voltage measurements were accurate to 10.05%. All measurements were taken at room temperature, 2220C. CHAPTER III DISCUSSION OF EXPERIMENTAL RESULTS In order to make the forthcoming discussions of noise and DC V-I measurements meaningful, it is necessary to know the junction properties and parameters. Some of them can be inferred from the ly vs. Vr plots for the four C most important diodes, the 1N749 to 1N752, shown in Figures 11 to 14. Since the 2 vs. Vr plots do not yield straight lines over the entire range of reverse voltages, it appears that 1 the junctions are not abrupt. Unfortunately, vs. V C r plots do not yield straight lines, either. Consequently, it seems that the junction profiles are composite of linearly graded and abrupt. If it is assumed, however, that to a first order approximation the diodes can be treated as step junctions (provided that the doping densities are evaluated from -- vs. Vr slopes obtained for large C r reverse biases, where the doping densities are most likely to be representative of the bulk doping densities) the results of Table I are obtained. The calculated doping densities on the n-side of the junctions are seen to lie within the range of doping densities predicted from the starting resistivities of Z 0 W 0 Hm El U E. 40'C OW 40 40- 3H oa 8:0 0M 0:0 P:0 H M aH 00M H / H- Z :0a 4-4: 400 0 ?-44 010. H< 40 ' 40-< c 1-4 a0 rn :0 PI 0 0 f~ 0 02 040>- Q 404 0 0 0 0 c0 0 CM CM (S 0 U 0 4 T-l 01 c 0 -0 -4) 0 -H 0 0 U- 4 ia I 40 4< n SX 0 m 0)> 0) 0) 0)i 40 0 4< 4l ~0 :01 :M -O 0) 0 04 1444 40 -o- u-t 04 enCl -4 m oo so ~o m u- c -a" m rLi> VI o0 a w 40 0 9z 0 0 I-I 1-4 1-1 0 U) +J > 0 ,- 0 enI 0 II 0 II II > 0 10 c 0 0 44 0 1 41 -i r~e >3 n3 > 4U1 w 0 c 0 0 *'- U -r-1 *-u 0 -0J 0 w cm 0 n33 0 fi 04 44 -a t 0 j0 40 0J 00 >1 0 C U~ 0 r1 S40 " -a U1 T (L-I 44r U oU rno o t- c (U > cu P0 C 414 S0 04 00 14 -i 4< the n-type wafers. This seems to indicate that for reverse voltages greater than about 1.5 to 2 volLs, the doping densities at the junction edges attain the bulk doping densities, and the junctions should begin to assume step junction characteristics. The changes in the 7 vs. Vr C slopes for smaller reverse biases are caused by the extreme thinness of the diode SCR's: as the reverse biases decrease, the SCR's narrow and enter the diode metallurgical transition regions where the doping densities are no longer constant. However, demonstrating that the doping densities at junction edges reach bulk doping densities does not con- clusively prove that for Vr greater than 1.5 to 2 volts the junctions may be approximated by step junctions. Another piece of evidence which strengthens this contention, though, will be obtained when the DC V-I characteristics are examined. This discussion will be most meaningful after the noise measurements are examined. The typical noise data obtained for diodes 1N746 to 1N752 are shown in Figures 15 to 21. The diode noise current spectral density, Sid, has been normalized with respect to the noise current spectral density of a saturated thermionic diode conducting a DC current equal to the p-n junction reverse current. The noise ratio, NR, is, therefore, given by NR = i- (3.1) r where Ir is the p-n junction reverse current. Since the noise current of a saturated thermionic diode arises from the reception at the anode of all individually, and randomly emitted, thermally excited cathode electrons, a noise ratio of unity indicates that the p-n diode reverse current is also due to collection of randomly and singly generated carriers, or carrier pairs. This is precisely what happens when the p-n junction reverse current arises from internal field emission in which thermally excited valence band electrons tunnel to the conduction band. The noise measurements thus furnish a foolproof method of determining the range of reverse voltages for which the diode reverse conduction arises from a single step tunneling process. (A few samples exhibiting multi-step tunneling were also observed. They are characterized by a noise ratio less than unity (van der Ziel, personal communication), as shown in Figures 21a and 21b. These samples, however, exhibited substantially 1 greater noise than the remaining samples and, therefore, were excluded from further study.) An independent method of verifying that over the range of reverse voltages for which the noise ratio is unity the reverse current arises from single step tunneling is obtained from the laws of internal field emission. Chynoweth et al. (1960) and Tyagi (1968) found that for a phonon-assisted tunneling process the diode reverse current is given by y 3/2 yE- (--) 4 2m I = AVrEe Y (3.2) r r 3qh 3ash where Ir is the diode reverse current, Vr is the diode reverse voltage, A is a constant for a given temperature, E is the electric field, E is the band gap energy, m is the effective mass of tunneling electrons, q is the electronic charge, T=-, where h is the Planck constant, 2-' yE 3/2 and n and n are constants. The factor e represents the phonon-assisted tunneling probability, En accounts for image fields or Coulombic forces corrections included in the theory, and V takes into account, in an empirical r way, the effect of electron tunneling from the conduction band back to the valence band so that for zero reverse bias the net current, Ir, is also zero. For abrupt junctions Tyagi claims n=l, n=1.5. YE 3/2 E no Compared to e AV En is a slow varying function of the reverse voltage so that the reverse current may be written as s Ir = Ie E (3.3) where Io and s are considered constant provided Vr 0, and E is an "effective" electric field such that _s w s e T= / e dx (3.4) o where the integration extends over the entire width of the diode space charge region, w. For an abrupt junction the main contribution to the integral comes from electric field values close to the maximum electric field value, E . m Thus, little error is made in assuming that the "effective" electric field is given by Em. Consequently, for an abrupt junction s Ir = I0e r (3.5) where s is a constant and 1 is the built-in potential of the junction. Thus, if a junction is indeed rtur-.L and the reverse current is due to tunneling, log i should give a straight line. Inspection of Figures 22 to 28, which show i:-i i 1 vs. for diodes 1N746 to 1N752, reveals trne .' -. Vr the range of reverse voltages for which the noi-:. rtci: is uir. log I vs. is, indeed, a straight li'-. 71- L.. vrr V-I data and the noise data are, therefore, cor.I r.r.i: and the contention that the Iuncr,i.: .: I a,. t ,: f r r. *.cird :Cc r.* i',:ti,:.., be c,:.i. .J-r.i rupr. uL'r.ber s t r.r r--r, r, . If the randomly emitted tunneling electrons undergo multiplication due to impact ionization, the randomness of the multiplication process generates additional noise, and the noise ratio becomes greater than unity. Comparison of Figures 15 to 21 with Figures 22 to 28 reveals that over the range of voltages for which the noise ratio increases from unity, log Ir increasingly departs from the straight line it followed at lower reverse voltages. This suggests that the failure of the reverse current to obey the tunneling relation is due to impact ionization multiplication of the tunneling current. If this contention is indeed correct, then it should be possible to verify it quantitatively with the help of noise theories of Tager (1965) or McIntyre (1966) relating the spectral density of noise current generated in impact ionization multiplication to the terminal DC current flowing through the diode. According to Tager, the low frequency spectral density of noise current produced in an impact ionization multiplication process in which the electrons and holes have the same ionization coefficients, is given by Si = 2qIt(M )3 (3.6) where It is the current initiating the multiplication process, which in this case is the tunneling current, and M is the DC current multiplication factor, defined by I M = (3.7) It where Ir is the terminal reverse current flowing through the diode. Thus, if it is assumed that the expression for tunneling current remains valid in the presence of impact ionization multiplication (which, in light of the current densities encountered, is a very reasonable assumption) then M is simply the ratio of the measured value of reverse current to the extrapolated value of the tunneling current. Thus, if Sid 2qltM (NR) log2- = log[ 2---t ] = log[M (NR)] (3.8) is plotted vs. log M as done in Figures 29 to 32 for diodes 1N749 to 1N752, a straight line with a slope of three should be obtained. It is clear that, with the exception of the 1N752 diode for which the agreement between experimental results and theory is reasonable, considerable discrepancies exist. Moreover, the discrep- Sid ancies are serious because the measured values of 2Si 2qIt are larger than the theoretical one, which represents the upper limit attainable in impact ionization multipli- cation for which electrons and holes have the same ioniza- tion coefficient. In addition, at low values of multipli- Si cation the rate of increase of 2 also exceeds the tqI theoretical, and, therefore,the upper limit value. At this point it might be argued that these discrep- ancies arise from the unequal ionization rates of electrons and holes in silicon. However, if the latest ionization rate data of van Overstraeten and de Man (1970), shown in Figure 33,are extrapolated to the field strengths encountered in diodes used in this study, the extrapolated electron ionization rates are not substantially greater than the extrapolated hole ionization rates. For the sake of completeness, however, the noise data was also analyzed in terms of McIntyre's (1966) theory of noise current spectral density for impact ionization multiplication in diodes for which the electron and hole ionization rates are not equal. According to McIntyre, if the carriers initiating the multiplication are electrons, the low frequency noise current spectral density is given by Si = 2qI (M )3[l-(l-k) (M-- )2] (3.9) M where It is the electron current initiating the multipli- cation, which in this case is the tunneling current, k= is the ratio of the hole ionization rate, 8, to the electron ionization rate, a, and M is the DC current multiplication I factor, M = Since 8 and a depend strongly on the It electric field, this equation is strictly cor-ect only if an appropriate "average" value of k is used (MlcIntyre, 1973). However, for the purposes of this discussion this is not an important consideration. S. According to McIntyre's (1966) theory, 1 is given by tqI S. S-= (M 3[1l- (-k) (M-1)2] (3.10) 2qt M which is plotted for several values of k in Figure 34. Comparison of Figure 34 with the experimental values of ..r tija'r.-: r :' l ir, i : rrfar a *:I'-- r a ift- r-rc Lt ein chr r. .:L r.J L- F r. *:.l. tl c .r : Lr.r d -ri, i cL t.: r..: i.nr i z a c i :n r ate r r.,L r. an t i a l 1 I l arir 4 r.z.ar, tr-.e e ei c r.r :n :nir :r.i: rr a- In iiirr. .: L r e re -a :ured ar,.j c r~r ,p.:1- 'J 'luai .: ri.:,n.rz s :n r, r e .t f tilujre 3 ri r-. :. i :.: r. r r.: A r rr. i : .cr i L:ur.t-I d r :i i r r:, r r. .r- .r a ,ri rn c- .r ',- u-r: "3 ., ua:- *r, th-- n 'Cr I-*r,5 I ai.ula i. .:r,- ,f C.": irul r. ip l r. che riaua r. : r:ttrin,.J ir :ii. te h tr- rultr F i lica i :.i r r.i -u,. J : f.r r tr. ,,:,r,-z~r : alue :- t '.' r e : c Tr ,. i g l r. t l r ,n .: i ra .-: = :,.u ldJ .:I e a ., ich i. ". :C, r. _rr ,ad i tr. : I th ti r.. .p .:t r, n t l i r -._r F jr : I r-,- F t i .li, f f c u.: ir : r _-,e _, ri.-_ i r f t :.: t_ r :r ) r :r' i : ,i .:t r r :i. Sid magnitude and functional dependence of 2 on M or 2qIt require an unreasonable ratio of ionization rates to possibly achieve an acceptable agreement of theory and experiment; they fail to correctly predict the magnitude of the experimentally observed DC multiplication; and they completely fail to predict the existence of a reverse voltage threshold for the onset of impact ionization multiplication. Any theory of reverse breakdown transition from tunneling to impact ionization multiplication must be able to eliminate all of these discrepancies, while remaining consistent with all other observations of break- down phenomena in silicon p-n junctions. One of these observations, made by many workers investigating carrier multiplication in semiconductors, is the existence of an energy threshold for pair production (Shockley, 1961; Moll and van Overstraeten, 1963): the agent attempting to generate a carrier pair must be capable of supplying at least a certain minimum amount of energy if it is to succeed in producing a hole-electron pair. The latest values of threshold energies for impact ioniza- tion in silicon, as determined in ionization rate measure- ments and first reported by Moll and van Overstraeten (1963), are 1.80.1 eV for electrons and 2.40.1 eV for holes. Since the noise data of Figures 15 to 21 indicate that carrier multiplication, as judged from the departure of the noise ratio from unity, always sets in at a well- defined value of reverse voltage that is almost independent of the diode breakdown voltage (if the breakdown voltage is defined as that value of reverse bias which gives rise to some constant value of reverse current density), the author contends that the onset of multiplication is governed by the magnitude of the threshold energy for ionization. To substantiate this contention quantitatively, the following multiplication threshold model, described in terms of Figure 35, which shows the electric field and energy band diagrams for a reverse biased abrupt p-n junction, is proposed. Since the tunneling probability is strongly dependent on the electric field, most of the tunneling current originates at x=0. At x=x1 the tunneling electrons become free carriers, absorbing 1.12Er eV, where Er corresponds to the emission or absorption of one phonon whose energy may be 17.9 meV (TA), 43.7 meV (LA), 53.2 meV (LO), or 58.5 meV (TO), (Logan, 1969). Continuing towards x2 the electrons acquire energy from the electric field, and at x2 those which have made the fewest number of colli- sions with the lattice are sufficiently energetic to participate in ionizing collisions. The onset of impact ionization multiplication, therefore, occurs when x2=x . Denoting the reverse voltage at which this occurs by Vrth' the abrupt junction approximation yields xn (1.12Er)+(1.80.1) x +x V +- (3.11) n p rth where x NA-ND n -3 .A x +x N (3.12) n p A Consequently, NA Vrth = [(1.12Er)+(1.8+0.1)] (NA- )- (3.13) Table II compares the reverse threshold voltages, Vrth' computed using the doping densities of Table I and E =0.06 eV, with the reverse threshold voltages obtained from Figures 18 to 21. In light of the approximations made and the experimental uncertainties involved in determining the doping densities, they are in good agreement. The holes produced in ionizing collisions between x2 and xn travel in opposite direction, but they, too, ionize only after they have acquired at least the necessary threshold energy for ionization. The electrons produced in hole ionizing collisions then travel towards xn, and after acquiring at least 1.80.1 eV participate in more ionizing collisions. This process repeats itself, but since the number of ionizing collisions decreases with each successive set of ionizations, the process converges to a finite value of DC multiplication. 24 TABLE II REVERSE VOLTAGES AT THE ONSET OF IMPACT IONIZATION MULTIPLICATION: COMPARISON OF DATA WITH THEORY DIODE TYPE EXPERIMENTAL REVERSE COMPUTED REVERSE THRESHOLD VOLTAGE THRESHOLD VOLTAGE 1N749 2.650.1 V 2.99 3.45 V 1N750 2.70.15 V 2.39 2.78 V 1N751 2.70.15 V 2.31 2.71 V 1N752 2.70.15 V 2.27 2.65 V In light of this model it is not difficult to see why the existing theories could not explain the experi- mental data: none of the theories take the influence of the threshold energies for ionization explicitly into account. The DC multiplication, which will be discussed in more detail in Chapter V, has conventionally been calculated by integrating the ionization rates a(x) and B(x) over the junction SCR in accordance with the solution of the differential equation dJ (x) dJ (x) -- d-- a(x)Jn(x)+6(x)Jp(x) (3.14) modeling the process (Moll and van Overstraeten, 1963; Lee et al., 1964; van Overstraeten and de Man, 1970). J (x) and J (x) denote, respectively, the electron and hole current densities as a function of position in the junction SCR. This equation does not explicitly consider the effect of threshold energies on the multiplication process because it claims that the differential increase in current densities at point x is proportional to the current densities at point x, which does not take into account a "distance delay" over which the carriers are incapable of ionizing because they are acquiring the necessary threshold energy. For diodes which break down at several hundred volts of reverse bias this "distance delay" is very small in comparison to the width of the junction SCR, and the approximation involved in neglecting it is very good. For diodes breaking down at several volts of reverse bias, on the other hand, the "distance delay" could constitute the entire width of the diode SCR. The approximation of neglecting it in such cases and employing equation 3.14 is totally unacceptable it leads to calculation of finite DC multiplication when no multi- plication is experimentally observed. Application of conventional calculations of DC multi- plication to very low breakdown voltage diodes thus leads to the conclusion that multiplication must be taking place in these junctions because the electric fields in them are very strong. The author, on the other hand, contends that if the electric fields in these junctions are very strong, multiplication will occur, but only if the carrier energies exceed the threshold energies for ioniza- tion. This is precisely why no impact ionization multi- plication occurs in tunnel diodes, in spite of the extremely strong electric fields existing in these junctions. Because McIntyre's theory of noise in avalanche diodes employs equation 3.14 in its derivation, it is now clear why his theory is not capable of predicting the multiplication noise in diodes employed in this study. Moreover, Tager's theory is also unacceptable because it does not incorporate in its derivation an upper limit on the number of ionizing collisions a carrier can undergo LI in its transit across the diode SCR. Such a limit must be imposed if threshold energies for ionization are taken into account. Not explicitly incorporating the influence of the threshold energies for ionization in the existing theories of noise and DC multiplication, therefore, renders them inapplicable to the diodes employed in this study. The next two chapters will attempt to resolve these difficulties. CHAPTER IV DETAILED ANALYSIS OF NOISE DATA Figures 29 to 32 indicate that if the DC multiplication factor, M is defined conventionally by I M r (4.1) It it is impossible to obtain a unique slope which would define the exponent, x, in Tager's (1965) expression for the low frequency noise current spectral density Si = 2qIt(M ) (4.2) The author suspected that this difficulty arose because the DC multiplication factor was not appropriately defined. This can be seen most clearly when M = 1, which is per- fectly acceptable mathematically, but is physically meaning- less, for a multiplication factor of unity means that there is no carrier multiplication. Thus, it was the author's contention that the DC multiplication factor should be defined by I It+MIt or (4.3) I M = 1 (4.4) t This definition reduces to the conventional one for large values of M, but is more appealing physically for it allows M to go to zero when there is no carrier multiplication. However, if M is defined by M = (Ir/It)-1, the expression for the noise current spectral density has to be reformulated, for consistency, to Si = 2qIt[l+f(M)] (4.5) where f(M) goes to zero when M goes to zero. The results of plotting f(M) vs. M for diodes 1N749 to 1N752 are shown in Figures 36 to 39. These graphs contain several interesting features. First, even though all data points do not lie on one straight line, it is possible to fit most of them along two or three straight lines, the slopes of which increase monotonically with increasing reverse voltages. Second, the departures of data points from lines of lower slopes to lines of higher slopes occur for all diodes at roughly the same (although somewhat increasing) value of reverse voltage. The values of reverse voltages at which these transitions set in, as defined, whenever possible, by the intersection of the two slopes, are: Diode type Onset of Ist Onset of 2nd transition transition 1N749 3.95 V 4.35 V 1N750 4.1 V 4.65 V 1N751 4.4 V 4.95 V 1N752 4.7 V 5.1 V Finally, even though the slope of any line tangent to the data points does not exceed its maximum allowed value of three, as did the tangents to the data points of Figures 36 to 39, it is now possible to draw tangent lines whose slopes approach unity -- which appears to contradict the established results that, according to the old definitions, the values of the slopes should never be less than two. All of these observations can be understood in terms of the conduction model incorporating the carrier threshold energy for ionization, as proposed in the preceding chapter. Since the amount of noise produced in impact ionization multiplication increases as the maximum number of ionizing collisions per carrier transit across the SCR increases, and eventually approaches the upper limit value given by Tager's (1965) expression for the case of unlimited number of ionizing collisions on a carrier transit across the SCR, the breaks in the straight lines in Figures 36 to 39 and the monotonic increase in slopes are indicative of transi- tions from a lower to a higher maximum number of ionizing collisions that a carrier can undergo in a transit across the diode SCR. In particular, if the built-in voltage of the junctions is taken into account, and an electron threshold energy of 1.80.1 eV is assumed, as given by Moll and van Overstraeten (1963), then it becomes clear that the onset of the first break in the slopes of Figures 36 to 39 is due to a transition from a maximum of two to a maximum of three ionizing collisions per electron transit across the diode SCR. This is easily verified by dividing 1.80.1 eV into the total diode potential, the sum of the built-in potential listed in Table 1, and the potential at the onset of the first slope transition, listed above. The results are: Diode type Total potential divided by 1.80.1 eV 1N749 2.92 2.62 1N750 3.01 2.69 1N751 3.19 2.85 1N752 3.36 3.01 When the procedure is reversed and the total potential at the onset of the first slope transition is divided by three, the following electron threshold energies for ionization are obtained: Diode type Electron threshold energy for ionization 1N749 1.66 eV ln750 1.7 eV 1N751 1.8 eV 1N752 1.9 eV This is in excellent agreement with Moll and van Over- straeten's (1963) value of 1.80.1 eV. The apparent increase in the threshold energy for ionization with increasing diode breakdown voltage is also reasonable. Both Moll and van Overstraeten's (1963) and these values include the "intrinsic" threshold energy as well as energy lost in phonon collisions. Since the rate of energy acquisition with distance is smaller in higher voltage breakdown diodes (the electric fields are weaker), but the rate of energy losses is the same (because it depends on mean free path between phonon collisions, which, presumably, is not strongly dependent on field strength), it can be expected that an increasingly larger number of phonon collisions would occur between successive ionizations, resulting in increasingly higher observed threshold energies in higher breakdown voltage diodes. An analogous analysis can be made for the second slope transition in Figures 36 to 39. Since the threshold energy for ionization for holes is, according to Moll and van Overstraeten (1963), greater than the threshold energy for ionization for electrons, the second slope transition may be suspected to arise from a transition from a maximum of two to a maximum of three ionizing collisions per hole transit across the diode SCR. When the total potential at the onset of the second slope transition is divided by three, the following values of threshold energy for ionization for holes are obtained: Diode type Hole threshold energy for ionization 1N749 1.79 eV 1N750 1.89 eV 1N751 1.99 eV 1N752 2.04 eV These values are considerably smaller then the 2.40.1 eV obtained by Moll and van Overstraeten (1963). Thus it may be suspected that the second slope transition is due to a transition from a maximum of one to a maximum of two ionizing collisions per hole transit across the SCR. This, however, results in hole threshold energies that are substantially higher than those given by Moll and van Overstraeten (1963): Diode type Hole threshold energy for ionization (assuming transition from a maximum of one to a maximum of two ionizations per transit) 1N749 2.69 eV 1N750 2.84 eV 1N751 2.99 eV 1N752 3.06 eV Moreover, if the hole threshold energies obtained under this assumption are used in the simulation of DC multiplication, a serious difficulty arises which does not occur when the hole threshold energy of 1.90.15 eV is used. As a con- sequence, it is suggested that the value of 2.40.1 eV is probably too high. An immediate application of this interpretation of noise data can be made in conjunction with the conduction 1 model proposed in the preceding chapter and the -- vs. V C data to compute the doping densities on the n and p sides of the junctions. With the help of Figure 35, the analysis proceeds as follows. At the onset of impact ionization multi- plication V +V NA-ND 9 A D (4.6) rth A where V = E /q, Ve E /q and E is the bandgap energy and E is the electron threshold energy for ionization. 1 Moreover, from -- vs. V data C (NA-ND)ND AV A D D 2 -()p (4.7) NA Eq A- C2 Therefore, PV N = (4.8) D Vg+V g e Moreover, N 2 NA NDP or (4.9) PV2 NA = h (4.10) S V (V +V )-(V +V ) rth g e g e AV Since r and, therefore, P, can be determined from the 1 vs V A- C C data, and Vrth and V can be determined from the noise data, rth e ND and NA can be calculated. A comparison of ND and NA obtained in this manner with the values of ND and NA given in Table I is shown in Table III. The agreement is quite good, which lends further credence to the proposed conduction model and the subsequent interpretation of noise data. Z u M 1 0 I a o4 Co m if (N ( tn C co r 0 co (N (N <( s 0 X rE El S II HZ O 0 E 0 F:4 u 0 M HH Z H H H H o OH OZH H * en n in H --H -- H r~ -^ r~~ r- 2 2 S Z r-1 r-1 -1 i- S'an mn 0 e H E- 0 w H r0 HElo Z HO N N rH H- I H l00 O H rH O 0 C c rCl r N 0 0 40 0 o 0 E C U > (N in H co en 3> en N rH ' H I U Co H I I I Co SH oo (N r en o Co (N (N Hl Hl ' *H 41 .t 0 0 0 41 c, 0 0 0 q *' 43 0 -H O e- -Sb Moreover, if the proposed conduction model is truly valid, then a suitable derivation incorporating the essential features of the model should yield expressions for the mag- nitude of the noise current spectral density which should be in good agreement with experimentally observed results. From the point of view of noise as a statistical fluctuation phenomenon, the most important feature of the proposed con- duction model is the limit it imposes on the maximum number of ionizing collisions a carrier can experience in one transit across the diode SCR. Consequently, any derivation of the noise current spectral density must incorporate this constraint. This complicates the problem considerably for it requires that the multiplication process be examined on basis of individual, successive transits of carriers across the diode SCR, rather than on monitoring the end product of the multiplication process at the device terminals, as was done by Tager (1965) and McIntyre (1966). To obtain some insight into how the analysis of multiple ionizing collisions might be carried out, the author began by considering the limiting case of a maximum of one ionizing collision per carrier transit across the diode SCR. This is more than just an exercise. According to the author's interpretation of Figures 36 to 38, about half of the data points correspond to multiplication resulting from no more than two ionizing collisions per carrier transit across the SCR. Moreover, even when up to two ionizing collisions are energetically possible, the probability of the second collision will be considerably smaller then the probability of the first collision because the space charge regions of the diodes examined in this study are very narrow, whereas the mean free path between ionizing collisions, as quoted by previous workers (Moll and van Overstraeten, 1963; Lee et al., 1964) is relatively large. Thus the approximation of no more than one ionizing collision per carrier transit across the diode SCR should yield reasonable results, especially for the case of the diode with the narrowest SCR, the 1N749. The analysis of the noise current spectral density is best carried out by referring to Figure 35. The tunneling electrons, originating at x=0 at the rate of n0 per second, are accelerated from x=x1 to x=x2, and undergo ionizing collisions from x=x2 to x=xn, producing n, electron-hole pairs. The n tunneling electrons and the nl ionization produced electrons are swept to the right and out of the SCR, The nl holes travel to the left and, after acquiring the necessary threshold energy, undergo ionizing collisions, producing n2 electron-hole pairs. The (nl+n2) holes are then swept to the left and out of the SCR, while the n2 electrons travel to the right and, after acquiring the necessary threshold energy, ionize, producing n3 electron- hole pairs. This process continues, so that the total number of carriers, N, collected at the device terminals becomes N = n. (4.11) i=0 Letting nl=aln0, n2=b2nl=b2aln0, n3=a3n2=a 3b2an0, etc., where the denotes averages and ai and bi represent the probability of ionization for electrons and holes, respectively, on the i-th transit across the diode SCR, i N = n0 E H aj oddbj even ;a01, b01 (4.12) 0i=o j=0 This converges to a finite number because a. since no more than one ionizing collision per carrier transit across the SCR is allowed. A reasonable simplification which can be made at this point is that a.=a and b.=b. This means that the probability of ionization differs for holes and electrons, but remains the same on all transits. In this case, N = n0(l+a+ab+a25+ab+ ...) = n ( +a ) (4.13) 1-ab and the DC multiplication factor, M = 1, becomes n0 a(l+b) M = a (4.14) The low frequency spectrum of the current fluctuations resulting from the multiplication process can be obtained from SN = 2var N (4.15) where SN denotes the magnitude of the spectrum of N, the number of events occurring during a unit time interval (van der Ziel, 1970). Since the diode current, I is given by Ir=qN, SI = 2q2var N (4.16) r where var N = N2 -2 2 = (n )2 ( n) i=0 i=0 = i Z (nnj-n.nj ) (4.17) i=0 j=0 1 ] 1 3 In order to evaluate Sr it, therefore, remains to evaluate nin This is accomplished by representing ni by 0 b. ; i even n = a b a b r (4.18) 1 r=ir 2r3r4r. air; i odd where ajr and bmr are either 1 or 0 (depending on whether or not an electron or hole ionizes on the j-th or m-th transit across the SCR) and ensemble averaging over the products n.n.. This analysis is discussed in detail in Appendix A. The resulting expression for var N is given by -- 2- var N = n l+3a+ab+a b} (4.19) S(1-ab)2 I-I and, therefore, 2- 1+3a+3ab+a b2 Sr = 2q n0o (1-a,)2 *r (-ab) = 2qIt l+3a+3a+2 (4.20) (1-ab) If, for the moment, it is assumed that the probability of ionization is the same for holes as it is for electrons, a=b, then M (4.21) 1-a and (l+a) Sr = 2qIt (1l) (4.22) r (1-a) Eliminating a, S = 2qIt(1+3M+2M2) (4.23) r which is, indeed, in the previously postulated form S = 2qIt[l+f(M)] .(4.24) r This result provides a more formal justification of the previously proposed redefinitions of the DC multiplication factor and the functional form of Si, and lends greater credence to the interpretation of noise data based on the new definitions. When equation 4.20 is rewritten in the form Sr = 2qIt[l+f(a,b)] (4.25) r and f(a,b) is plotted as a function of M for various values of k=-, Figure 40 is obtained, which contains the somewhat a surprising result that f(a,b) is rather weakly dependent on k. This result is exploited in the derivation of the magnitude of the noise current spectral density for the case of up to two ionizations per carrier transit across the diode SCR. Moreover, since SI = 2qlt[l+f(a,b)] = 2qlt(1+3M+2M2) (4.26) r over a fairly wide range of k= for low values of M, it a demonstrates that the slope of f(M) vs. M can, indeed, approach unity, as it does for the data of Figure 36. To compare the results of this analysis with experiment, Figure 41 shows, along with f(M)=3M+2M2 plotted versus M, the experimental data for the 1N749 diode. Although the theoretical curve shows the same functional dependence on M as do the experimental data, it is about a factor of three too low. However, in view of the fact that the derivation of the theory is strictly statistical, taking no account of any junction parameters, this result was most encouraging. At this point the author suspected that the discrepancy between theory and data resulted from dismissing the possibility of the second and third ionizations, and proceeded to extend the theory to more than one ionization per carrier transit across the SCR. Although this is easily done in principle, it becomes extremely tedious algebraically because the analysis must take into account the various branching possibilities, and the correlation between the branches. In fact, the extension of the theory to the case of up to two ionizations per carrier transit across the SCR is a composite of an infinite number of the up-to- one-ionization analyses. In order to restrict the algebraic details to manageable size, the author chose to analyze the case of up to two ionizations per carrier transit under the additional restriction that the probability of the second ionization is much smaller than the probability of the first ionization, and, therefore, that terms containing -2 a2, where a2 is the probability of the second ionization, could be neglected. It was felt that this would be a reasonable approximation for the noise data of Figures 36 to 39 below the first slope transition. The analysis of this case under the additional approxi- mation that the electron and hole ionization probabilities are the same (as justified by DC calculations of Chapter V and by the relative insensitivity of the results of Figure 40 to k=b)is given in Appendix B. The algebraically tedious a 2 results, evaluated for 2 = 0.3, which is the highest ratio al of the probability of the second ionization to the probability of the first ionization that the author felt could be assumed under the approximation that a2< with the results of the up to one ionization per transit theory, evaluated for k--=l. Although the two curves inter- a sect giving rise to a slope transition, in accordance with the previous interpretation of the slope transitions in the experimental data, the results are somewhat disappointing in that f(al,a2) is not sufficiently larger to give sub- stantially better agreement between theory and the noise data of Figures 36 to 39. A careful examination of the various assumptions and approximations made in the theoretical analysis revealed that the source of the final discrepancy between the theoret- ical results and the experimental data lay in the simplifying approximation that the carrier ionization probability is the same on all transits. A typical plot of the number of carriers generated on a given transit per one tunneling electron, obtained from DC calculations for the 1N752 diode operating at reverse bias of 5.2 volts, and shown in Figure 43, demonstrates that the ionization probability for electrons on the first transit across the diode SCR is considerably smaller than the ionization probability for carriers traversing the SCR on succeeding transits. This may be deduced by comparing the ratio of consecutive terms for transit numbers greater than about five which, in the case of Figure 42 is 0.88, with the value of the first term,0.516. If the ionization probability were identical for all transits, the value of the first term would have been 0.88. In retrospect, it is easy to see why the ionization probability on the first transit is less than on succeeding transits. The electrons under- going ionizations on the first transit come from tunneling, which, in terms of Figure 35, means that they do not become free carriers until x=xl. Consequently, they ionize over a much smaller portion of the SCR than do the succeeding holes or electrons originating as free carriers closer to the edges of the SCR. To determine what effect the smaller ionization probability on the first transit would have on the noise theory, the derivation of the up-to-one-ionization-per: SCR-transit theory was repeated for the case of a=b, -=r where al is the ionization probability on the first transit. The result, derived in detail in Appendix A, is given by Si = 2qlt(1+3M+2M2/r) (4.27) clearly indicating that theoretically predicted noise output will increase for r This analysis was also repeated for the up to two ionizations per SCR transit theory. Indeed, this is the derivation that appears in Appendix B. Because the results are algebraically tedious, they are evaluated numerically once the values of all, al and a2 have been determined. Here, all denotes the probability of the first ionization on the first transit, and al and a2 are the first and second ionization probabilities on succeeding transits. For the range of voltages where the up to two ionizations theory is valid, the probability of second ionization on the first transit is zero. The details of evaluating all' a, and a2 from the DC multiplication program are discussed in Appendix C. The final results of the up to two ionizations per SCR transit theory for the case of all with the experimental data for the 1N749 to 1N751 diodes in Figures 44 to 46. In view of the many approximations made in the derivation of the theory, the agreement is quite good, even for those values of M where the probability of a third ionization is no longer zero. In the case of the 1N751 diode, where at low values of M the theory predicts a larger noise output than appears to have been observed, the discrepancy is probably due to incorrect values of M, which at low values, can be subject to substantial error. The analysis of the noise data was terminated at this point for it was felt that even though an up-to-three- ionizations-per-SCR-transit theory would undoubtedly yield better agreement between theory and experiment, it would probably offer no new insights into the details of the multiplication mechanism, and, therefore, in view of the algebraic tedium required for its derivation, would not be worthwhile. CHAPTER V CALCULATION OF DC MULTIPLICATION As has already been mentioned in Chapter III, the calculation of DC multiplication in avalanche diodes has conventionally been done by integrating the electron and hole ionization rates, a and B, over the junction SCR, in accordance with the solution of the differential equation dJ (x) dJ (x) ---d (x)Jn(x)+3(x)J (x) (5.1) modeling the process. Since this equation incorporates the effect of the threshold energies for ionization implicitly in the ionization rates and, therefore, is not capable of modeling those segments of the carrier trajectories over which multiplication does not occur because the carriers do not have sufficient energy to ionize, it is not applicable to calculation of DC multiplication in low breakdown voltage diodes in which these "inactive" segments of carrier trajectories con- stitute a very sizeable portion of the diode SCR. These "inactive" segments could be taken into account in the above equation by incorporating "distance delays," L1(x) and L2(x), so that (5.2) dJ (x) dJ (x) ~- x x = a(x)Jn[x-L (x)]+S(x)J[x+L2(x)] . dx dx 1 pt2 However, if Ll(x) and L (x) are taken so that x V = -qf E(x) dx (5.3) e x-L (x) and x+L2(x) Vh = -qf E(x) dx (5.4) x where Ve and Vh represent, respectively, the threshold potentials for ionization of electrons and holes, then the magnitudes of J and J must be increased from their n p actual values at xL(x) since, in general, there will be some carriers with non-zero energies which will not need to travel the full distance L(x) to acquire the necessary threshold energy for ionization. On the other hand, if J n and J are left unaltered, then L1(x) and L2(x) need to be evaluated as suitable averages over all carrier energies. Since neither correction is easy to make at the outset, and since the modified differential equation would still need to be solved numerically, the author chose to abandon this approach in favor of a numerical simulation of the mul- tiplication process. The simulation, whose program listing is given in Appendix C is accomplished as follows. An abrupt junction doping profile is assumed, and the electric field triangle is cut into a large number of variable width segments such that the integral of the electric field is the same over each segment. (In the results shown in Figures 47 to 50, x+Ax dv = -qf E(x) dx = 0.01V, so that the electric field x triangle was cut into several hundred segments.) The integral of the ionization rates, which represents the probability of an ionization, and the fraction of the tunneling current originating in a given interval are then evaluated approximately for each interval by x k+Ax dPn(k) = x a dx .[a(xk+tAxk)+a(xk)]Axk (5.5) xk+Axk dPh(k) = I B(x) dx = [B(xk+AXk)+6(xk)]Axk (5.) Xk and xk+Axk s x e dx dIt (k) = k w s I e-t dx 0 s s [e E(xk+Xk) + e Exk)Axk s' s (5.7) ,1 L-FTx Ax) r ETxk [e k+ + e- ]Axk 2k where dPn(k) and dPh(k) represent, respectively, the probability of ionization for electrons and holes in the k-th interval, and dIt(k) represents the normalized tunneling current originating in the k-th interval. The multiplication'process is then simulated by following the motion of electrons and holes back and forth across the diode SCR, and allowing them to participate in ionizations after they acquire their respective threshold energies, which can be kept track of very easily by counting off the appropriate number of intervals. Thus, the tunneling current originating in the k-th interval becomes a free V electron current in the (k+m)-th interval, where m= , which may then ionize starting in the (k+m+nl)-th interval, V where n1= and where V and Ve denote, respectively, the bandgap potential and the electron threshold potential for ionization. That fraction of the tunneling current which ionizes in a particular interval, and the ionization produced electrons originating in that interval, are then not allowed to ionize again until they have traveled at least another n1 intervals. The tunneling electrons are traced in this manner across the entire SCR, while the holes produced in ionizations occurring in a given interval are traced in the opposite direction, not being allowed to ionize until they V have traveled at least n2= intervals, where Vh is the hole threshold potential for ionization. The fraction of the hole current ionizing in a particular interval, along with the ionization produced holes originating in that interval, are then not allowed to ionize again until they have traversed at least another n2 intervals. In this manner the hole current is traced across the entire SCR. This procedure is alternately repeated for electrons and holes, and the contribution to the terminal current due to ionizations on each carrier transit across the SCR is recorded. When the contribution on the p-th transit is less than 0.005 of the contribution on the first transit, the simulation is stopped and the contributions to the terminal current from all transits are added. The same procedure is followed for the multiplication process initiated by the holes left in the valence band when the electrons tunnel to the conduction band. The total multiplication is then the sum of the contributions from the tunneling electron initiated multiplication and the typically much smaller hole initiated multiplication. The simulation thus circumvents both of the previously discussed difficulties which would be encountered in the differential equation approach to the computation of DC multiplication. It incorporates unambiguously the threshold energies for ionization and, by keeping track of the past history of the carriers, automatically computes the fraction of the total current crossing a given interval which is sufficiently energetic to ionize in that interval. This is equivalent to automatically adjusting the magnitudes of Jn [-L (x)] and J [x+L2(x)] to account for the fact that not all carriers need to travel the full distance L(x) in order to acquire the necessary threshold energy for ionization. Unfortunately, the simulation creates a different problem which, too, is difficult to resolve at the outset. Namely, it assumes that when an ionizing collision occurs the primary carrier and the secondary carriers are scattered in a way which, on the average, does not impart any excess energy to either the primary or the secondary carriers, so that all carriers will be accelerated again essentially from rest. Since the scattering produced by an ionizing collision is probably not isotropic, this assumption may not seem very reasonable. However, in the actual calcula- tions the non-isotropic effects should, at least partially, cancel, for if the scattering favors the direction of the ionizing carrier, then the carriers of the same polarity as the ionizing carriers will not need to accelerate the full distance L(x) to acquire the threshold energy, whereas the carriers of opposite polarity will need to accelerate farther than the distance L(x) to acquire their threshold energy. Since the simulation is performed by alternately tracing the motion of electrons and holes across the SCR, these effects should alternately cancel to some extent. However, without knowing the exact details of the ionization and scattering mechanism, it is difficult to estimate how much cancelling actually takes place. Although the simulation program was written especially for computation of DC multiplication occurring in low break- down voltage diodes, where it is necessary to take the threshold energies for ionization explicitly into account, it may also be used to perform DC multiplication calculations in the conventional manner simply by letting the threshold energies for ionization, and the bandgap energy, approach zero. In actual calculations the bandgap and threshold potentials were set at 0.01 V. The ionization rates employed were those of van Overstraeten and de Man (1970). From the results obtained for diodes 1N750 to 1N752, shown in Figures 47 to 49 under the label of "conventional calculations," it is apparent that the conventional method of performing DC multiplication calculations is inadequate for it predicts much greater multiplication than is experi- mentally observed, even to the point of predicting finite multiplication when no multiplication actually occurs, as evidenced by the noise data of Figures 15 to 21. Although, as shown in Figures 47 to 49 under the label of "conventional calculations, including tunneling," the agreement between the experimental and the calculated results improves somewhat when, on the first transit, the bandgap energy for tunneling is taken into account, the calculated values of DC multiplication are still considerably larger than what is experimentally observed, and finite values of multiplication are still calculated when no multiplication actually occurs. When the threshold energies for ionization are finally taken into account, the calculated values of DC multiplication do go to zero when no multiplication is observed in the noise data, but the predicted magnitudes of the DC multiplication are smaller than those experimentally observed. This, however, is understandable for the integral of the ionization rates over a given distance represents, by definition, the probability of ionization over that distance. Therefore, eliminating large portions of the diode SCR from integration by explicitly taking the threshold energies for ionization into account, but employing conven- tional ionization rates derived from integration over the entire SCR, reduces the calculated ionization probabilities and, therefore, the calculated DC multiplication factor. It, therefore, seems appropriate to increase the ionization rates to compensate for this shortening of the ionization rates' integration region. Unfortunately, this is not a trivial matter since the functional dependence of the ionization rates on the electric field implicitly reflects, along with other parameters, the influence of the threshold energies for ionization on the ionization rates, and, therefore, should probably be modified when the threshold energies for ionization are explicitly taken into account. Since there exists no theoretical groundwork to indicate how this should be accomplished, the author chose not to modify the functional dependence of the ionization rates, but merely introduce scaling constants cl and c2 so that the adjusted ionization rates a and B are given by a =cle and B =c2 where a and B are the "reference", conventional ionization rates given by van Overstraeten and de Man (1970). The constants cl and c2 are adjusted to give the best possible agreement between the calculated and the experimentally obtained values of DC multiplication. Since the adjusted ionization coefficients would be useless if cl and c2 were determined individually for each diode, the author chose to adjust cl and c2 on diode 1N752, whose DC calculations are most sensitive to the values of cl and c2 because they need to attain the largest values of DC multiplication without diverging. Once this was done, calculationswere performed on the remaining diodes, using their respective values of the threshold energies for ionization, as determined in Chapter IV. The results of these calculations are shown in Figures 47 to 49 under the label of "calculated". The discrepancy between the calculated and the experimental values of DC multiplication occurring for low values of M is attributed to the tri- angle approximation for the junction electric field, which gets progressively worse with decreasing reverse voltage since the diode SCR narrows and gets progressively closer to the changing doping densities of the junction metallurgical transition region. In that region the triangle approximation predicts stronger electric fields than are actually present. This, in turn, leads to larger calculated values of ionization probabilities and, hence, larger than observed values of DC multiplication. In spite of this discrepancy, the DC calculations performed by explicitly taking the threshold energies for ioniza- tion into account and employing the adjusted ionization rates are in considerably better agreement with the experimental data than are the results obtained by con- ventional calculations. The values obtained for cl and c2 are quite reasonable also: c1=3, c2=5. In fact, it was on the basis of the magnitude of c2 that the author dismissed as unlikely the 2.850.2 eV value for the hole threshold energy for ioniza- tion, for when this value was employed in the DC calcula- tions it was not possible to obtain reasonable agreement between the calculated and the experimental values of DC multiplication for values of c2 as large as twenty. It was also observed that when the best agreement between the theoretical and experimental results was obtained, the electron and hole ionization contributions to the DC multiplication decreased monotonically on successive transits, as shown in Figure 43. Since the ratio of successive terms remained essentially constant regardless of which terms were picked (with the exception of the beginning few) this implies that the ionization probabilities for electrons and holes were essentially the same. This information is used in Appendix B to simplify the derivation of the expressions for the magnitude of the low frequency spectral density of noise current produced in a multiplication process resulting from no more than two ionizations per carrier transit across the diode SCR. CHAPTER VI CONCLUSIONS The principal result of this study is a detailed under- standing of the mechanism of reverse conduction transition from tunneling to impact ionization multiplication in low breakdown voltage silicon p-n junctions. It is concluded that the transition occurs when the energy of the tunneling electrons exceeds the electron threshold energy for ionization. If the DC multiplication factor is redefined by M = (Ir/It)-1, and the functional form of the expression for the magnitude of the noise current spectral density is changed to Si = 2qIt[l+f(M)], then the electron and hole threshold energies may be obtained from slope transitions in the plots of log f(M) vs. log M. The electron threshold energy for ionization determined in this manner is found to be in excel- lent agreement with the value of 1.80.1 eV found by Moll and van Overstraeten (1963). However, the hole threshold energy of 1.90.15 eV disagrees with their value of 2.40.1 eV. Moreover, it is found that the threshold energy for ionization is not constant, but increases slightly with increasing junc- tion width. The noise data may also be used to estimate the average minimum distance between ionizing collisions. This is accomplished by obtaining the junction width at the reverse voltage corresponding to the onset of the transition from no more than two to no more than three ionizing collisions per electron or hole transit across the diode SCR, and dividing that distance by three. The results are: Diode type Average min. distance Average min. distance for electron ionizing for hole ionizing collisions collisions 1N749 192 A 199 A 1N750 213 A 224 A IN751 223 A 234 A 1N752 242 A 250 A When the threshold energies for ionization, the total potential at the onset of carrier multiplication, and the value of (NA-ND)ND/NA are interpreted in terms of the transi- tion model (Figure 35), the values of NA and ND can be determined. The doping densities on both sides of a step junction may thus be evaluated from diode noise and C-V measure- ments. This technique could be used in some cases for the evaluation of impurity solubilities in silicon. Conversely, when the junction doping profile and the threshold energies for ionization are known, the transition model may be used to predict the reverse potential for the onset of impact ionization multiplication. This furnishes a method for distinguishing the tunneling section from the multiplication section of the diode reverse V-I characteristics. Besides determining the onset of carrier multiplication, the threshold energies for ionization dominate the mechanism of the multiplication process. Because they allow no more than two or three ionizations per carrier transit across the diode SCR, theyrender Tager's (1965) and McIntyre's (1966) theories of noise in avalanche diodes inapplicable to the diodes of this study. This is amply demonstrated by the severe discrepancies between their theoretical results and the experimental data, and by the considerable improvement in agreement between the data and the theoretical results obtained when an upper limit of two ionizations per carrier transit across the SCR is explicitly imposed. Indeed, for low multiplication values, a noise theory based on the explicit constraint of no more than one ionization per carrier transit across the diode SCR gives considerably better agreement with the experimental data than do the theories of Tager (1965) or McIntyre (1966). Moreover, unless the effect of threshold energies for ionization is explicitly taken into account, a finite value of DC multiplication is calculated when no multiplication is experimentally observed. This discrepancy does not arise when a simulation program incorporating the threshold energies for ionization is used for calculating the DC multiplication. The many details of the transition mechanism thus lead to the conclusion that the conventional treatment of carrier multiplication phenomena, based on the equation 61 dJ (x) dJ (x) d- d= = a(x)Jn(x)+B(x)J (x) (6.1) is not applicable to carrier multiplication in low break- down voltage diodes. CHAPTER VII RECOMMENDATIONS FOR FURTHER STUDY The first undertaking in further work on this topic should be the removal of the triangular electric field approximation from the DC multiplication simulation program. The electric field should be obtained from the Poisson equation and the impurity doping profile. If this correction does not remove the discrepancy between the experimental and the calculated values of DC multiplication at low reverse voltages, the cause of the discrepancy should be investigated. When good agreement between the experimental and the calculated values of DC multiplication is obtained, the cal- culations should be extended to a case of greater practical importance: the reverse breakdown characteristics of base- emitter junctions of bipolar transistors. Because of the very heavy doping densities encountered in base-emitter junctions, it will probably be necessary to take bandgap narrowing into account. Moreover, if the reverse DC V-I characteristics of the base-emitter junctions or of the low breakdown voltage diodes are to be calculated directly from impurity doping profiles, it will also be necessary to refine the theory of tunneling breakdown (Chynoweth et al., 1960; Tyagi, 1968). Furthermore, the program should be modified to calculate carrier multiplication resulting from a single pass across the space charge region. This would permit calculation of carrier multiplication in short channel FET's and other small devices in which one carrier type is somehow removed so that it does not participate in the multiplication process. Since the threshold energies for ionization play an important role in the simulation program, accurate calcula- tions of DC multiplication in structures of arbitrary lengths and arbitrary impurity doping profiles will require accurate knowledge of the threshold energies under these conditions. Consequently, it will be necessary to carry out additional DC and noise studies to establish means of predicting the threshold energies for ionization in such structures. The effect of different junction widths and different electric field profiles could be investigated further with the aid of the recently developed replacement series for the 1N746 to 1N752 diodes employed in this study. The new series of diodes, manufactured by alloying aluminum onto the lightly doped p layer of a p-n diode, break down more abruptly than their older counterparts. The more abrupt breakdown is obtained from the resulting p -p-n doping profile (author's conjecture!), which, for any given reverse voltage, widens the space charge region and lowers the peak electric field. The lower peak electric field decreases the tunneling current and the wider junction sets up a more favorable environment for impact ionization. Thus, when impact ionization multi- plication finally sets in, it occurs more abruptly than in the conventional p -n alloy diodes employed in this study. Since the new diodes should, for a given value of DC multi- plication, exhibit more noise than their older counterparts, this conjecture should not be difficult to verify. Moreover, it would be possible to fabricate narrower junctions by employing stronger p-type material, such as boron doped aluminum (Chynoweth et al., 1960). The electric field profiles in such p -n diodes would be more asymmetrical, which should permit impact ionization multiplication at lower reverse voltages. This should reduce the likelihood of multiple ionizations per carrier transit across the space charge region, and make the noise data more amenable to interpretation in terms of the already developed noise theory. Due to phonon involvement in the carrier transport and energy balance mechanisms, the threshold energies for ioniza- tion should also be determined as function of temperature. Low temperature measurements might also shed some light on the mechanism of phonon collision involvement in the ioniza- tion and multiplication processes. Tunnel diode DC V-I characteristics measured at 4.20K revealed structure which reflected the cooperation of different energy phonons in the tunneling process (Chynoweth et al., 1962; Logan and Chynoweth, 1963; Logan et al., 1964). Such structure might also be observed in the DC or noise data of these diodes. In conjunction with the experimental work, the noise theory should be extended to the case of a multiple, but finite number of ionizing collisions per carrier transit across the diode space charge region. This might be possible by combining the author's method with the approach employed by van Vliet in the derivation of avalanche photodetector noise (van Vliet, 1967). Incorporating the quantum efficiency and the Boson factor in the primary current would also make the theory useful for evaluating avalanche photodetector per- formance. Finally, the improved multiple ionization noise theory could be used to extract from the noise data information on the ionization probabilities. This would be of considerable help in undertaking a theoretical study of the adjusted ionization coefficients. Such a study might also lead to new approaches to the treatment of other aspects of carrier transport in small devices for which the conventional, differential equations approach, with its implicit assumption of unlimited scattering, is not valid. APPENDIX A MULTIPLICATION NOISE RESULTING FROM NO MORE THAN ONE IONIZATION PER CARRIER TRANSIT ACROSS DIODE SCR The derivation of expressions for the magnitude of the low frequency spectral density of noise current produced in impact ionization multiplication resulting from no more than one ionization per carrier transit across the diode space charge region proceeds as follows. As indicated in Chapter IV, the total number of carriers, N, collected at the diode ter- minals in one second is given by N= n. i=0 ' where n0 is the number of electrons tunneling in one second, and ni, i=l, 2, 3, ... is the number of hole-electron pairs produced in one second on the i-th carrier transit across the diode SCR. If it is assumed that nl=an0, n2= bn=aEn0, n3=an2=a2 n0, etc., where the denotes averages and a and b represent the ionization probabilities of electrons and holes, respectively, then ----------- -2------2-2 N = n0(l+a+ab+a b+a b +...) and the DC multiplication factor, M = N/no-1 becomes M = a(l+b) 1-ab The low frequency spectrum of the current fluctuations can be obtained from SN = 2var N where SN denotes the magnitude of the spectrum of N, the number of events occurring per unit time interval (van der Ziel, 1970). Since Ir = qN, SI = 2q2var N r = 2q2 (N N2) S 2 = 2q2{(( ni)2-( ni) } i=0 i=0 = 2q2{ Z Z (n.n.-n.n.)} i=0j=0 3 To evaluate ninj, it is necessary to have explicit expressions for n. and n.. These may be obtained as follows. If the emission of tunneling electrons is pictured as a sequence of l's, then the terms n., j=0, 1, 2, ... can be pictured as n0: 1 1 1 1 1 1 1 1 ... 1 ... 1 nl: 0 1 1 0 1 0 0 1 ... a ... 1 n2: 0 1 1 0 1 0 0 0 ... alrb2r ... 0 n3: 0 1 0 0 1 0 0 0 ... alrb2ra3r ". 0 n4: 0 1 0 0 0 0 0 0 ... a lrb2ra3rbr 0.. etc. where 1jk j=l, 2, 3, ..., k=l, 2, 3, ... no, denotes the occurrence of an ionizing collision on the j-th transit, initiated by the k-th tunneling electron, and where the O's denote the absence of ionizing collisions. The transitions from 1 to 0 occur independently and at random. From the above tabulation, it is apparent that n. is given by n0 ajr; j odd n alrb2ra3r ... or r=lr2r3r4r bjr; j even where ajk and bjk, j=l, 2, 3, ... k=l, 2, 3, ... n0 are 1 or 0, for eventhough ajk and bjk are allowed to fluctuate independently and at random, H akrbk retains the two essential k=l r r features of the above table, namely, that (i) if all preceding entries in the k-th column are 1, then the next entry can be 1 or 0, and (ii) that if any of the preceding entries is 0, then the next succeeding entry must be 0. The average nO air no a. nin- = ( alrb2r .. r )( p alpb2p r=l ir p= bjp may now be evaluated by ensemble averaging. Averaging over the mN subensembles containing n0 = N elements mN N air N ajp n.n. = { ( E alr b ... or )( a b ... or )} 3 N m=l r=l b.i p=l p b. 2 ajr 1 N N 2 ir (i+l)r ...or mN Z I a b ...or m=l r=l 2 ir (i+l)r ...ajr or bjr 1 N N N air ap + { Z(a br.. or )(ab2...or) mN =l r=l p=l bir bjp r/p Since all a's and b's are equal, then if both i and j are even -- N 2 (7-) (1) 2) -2'2 n.n. = N(a2) (b2) (a) ) j>i +N(N-1)(a) 2 (b) 2-2 Moreover, since a and b are 1 or 0, a =a and b2=b. Therefore, -N =N() ()(i)( ) n N .= Na b +N(N-l)a b j>i Finally, averaging over all subensembles ( 4) b(-) i (1+') i+J, -2 2 + 2 2 n"in = na b + (n0-nO)a b j>i Thus, iFF -ii. ii. 2 b +) -2 nin-nin = n0a + (ng-nO)a b j>i (i+_ ) ("+) -2- 2 -n0a b ) () 2 - = n0ga + [(n0-n0)-n0]a b However, since the tunneling electrons have a Poisson distri- bution, for which var no = no (n.n.-n.nj) = na ; i,j even j>i The same expression is obtained if j is even and i is odd. If j is odd and i is odd or even, _ (3+1) (JJ.- (n n -n n) = n0 If i j, the above expressions hold provided j is replaced by i. Consequently, var N = E Z (ninj-ninj) i=0 j=0 -5 * __ _ S (n. -ni ) + 2 Z Z (n.n.-n.n.) i=0 i=0 j=0 3 j>i -2- -2-2 = n0 (1+a+a+a b+a b +... 2- -2-2 +2n 0(a+ab+a b+a b + --2- -2-2 +a2 +a22+ .. S2)2 1+3a+3ab+a2i = no0 (abl- } Therefore, SI = 2q2var N r 2- 1l+3a+3ab+a b = 2q no{ L---- = q1{l+3a+3ab+a b j (1-ab)2 If a=b S = (l+a) r (l-a) and M a (1-a) Therefore, Sr = 2qIt(l+3M+2M2) If, however, the probability of ionizing collision on the first transit, al, is not equal to a, the probability of ionizing collision on succeeding transits, then (n n -n n )= --al-(j-1) (ninj-nin) = n0ala1 ; j
and 2 -2 (n0-n0)= Consequently, var N = z i (n.n.-n.n.) i=0 j=0 (l+a)2+a (l-a)+2a = n{ (1-a) 2 and M a (1-a) al Therefore, if = r a Sr = 2qt (l+3M+2-) APPENDIX B MULTIPLICATION NOISE RESULTING FROM NO MORE THAN TWO IONIZATIONS PER CARRIER TRANSIT ACROSS DIODE SCR The derivation of expressions for the magnitude of the low frequency spectral density of noise current pro- duced in impact ionization multiplication resulting from no more than two ionizations per carrier transit across the diode space charge region proceeds as follows. It is assumed that carrier multiplication taking place during one transit across the diode SCR is, in general, of the form al ala2 al 1 \ a \ ala2 Sx x 1 2 -xx /I a / 1 where "l" denotes the "primary" carrier initiating the multiplication process, "x" denotes an ionizing collision, al and a2 denote the probabilities of the first and second ionizing collision, and the dashed lines indicate carriers of opposite charge to the "primary" carriers. If the above diagram is extended to include several transits across the diode SCR, it takes on the appearance of a "tree" from which it becomes apparent that the multi- plication process can be decomposed into an infinite number of "branches" of the form S-2- --2-2 -3-2 n nal na nala nala2 nala2 --P x-- 1x -L x -x x... etc. which is precisely the up to one ionization per carrier transit across the SCR process for the case of unequal probabilities for holes and electrons. (It should be observed, however, that in this analysis, the ionization probabilities for electrons and holes are assumed to be the same. The evidence supporting this assumption is dis- cussed in Chapter V. Even if this assumption were not strictly valid, however, Figure 40 indicates that the results obtained for unequal ionization probabilities do not differ substantially from the result obtained for equal ionization probabilities in the one ionization per carrier transit case. And since the up to two ionizations per carrier transit across the SCR case is a composition of the one ionization per carrier transit cases, the final result should not be very sensitive to the difference in ionization probabilities for holes and electrons.) To reduce the number of "branches" that need to be considered, it will also be assumed that a2< branches beginning with allala2, k>2, will be neglected. This assumption is justified in Appendix C Moreover, since the probability of ionizing collision is smaller on the first transit than it is on succeeding transits, the probability of the first ionizing collision on the first transit across the diode SCR will be denoted by all. According to the conduction model of Figure 35, the prob- ability of the second ionizing collision on the first transit, a21, for the range of voltages where the up to two ionizations per transit across the SCR theory may be considered valid, is zero. This follows from equation 3.13, with 1.80.1 replaced by 2V The results of this calculation are Diode type Reverse voltage for which a21>0 1N749 4.89-5.06 1N750 4.85-5.00 1N751 4.90-5.06 1N752 4.95-5.10 The reverse voltages given above are greater than the voltages corresponding to the first slope transition in Figures 36 to 39, thus confirming the contention that for the range of voltages where the up to two ionizing collisions per carrier transit theory is valid, a21=0. Under the condition stipulated, the "tree" for this multiplication process is given in Figure 50. The heavy arrows denote electron ionizations, the light arrows denote hole ionizations, and the ]ij denote "branch" numbers. To avoid excessive clutter, only the beginnings of each "branch" are indicated. Thus, "branch" Il is the branch given by -3-2 allala2 x --4-2 allala2 etc. All branches, therefore, continue vertically downward, and their continuation is indicated by dashed arrows. The beginning of each new branch is indicated by a horizontal translation to the right, followed by the downward arrows of that branch. The analysis of this multiplication process proceeds as follows. Let N be the total number of carriers collected at the diode terminals as a result of multiplication initiated by n0 tunneling electrons. Then, N = Zm. where mi denotes the number of carriers produced by the i-th branch. Hence, var N = E var m.+2ZE (mimj-mi mj) i 1ij j>i mi = E nik k=0O where nik is the number of carriers produced in the i-th branch, on the k-th transit across the SCR. Following the procedure of Appendix A, nik is, in turn, represented by n0 nik = i (product of all, all's and a2's)r r=l where all, all's and a2's are 1 or 0, and the product function of all, al's,and a2's reflects the past history of the r-th primary carrier, in the i-th branch, on the k-th transit across the diode SCR. Thus the problem of up to two ionizing collisions per carrier transit across the SCR is reduced to a sum of an infinite number of up to one ionization per carrier transit analyses. To obtain the expression for M, the branch contributions are summed as follows: N = m = 10 + mlj + Z m0j i j=l j=o + E Z mOij + i: m. + i: m.. i=2 j=l j=2 i=2 j=l resulting in 1+al N = n0[l + all - 1-ala2 a lal a2 (l+aI) -2 2 (1-ala2) (1-a )2 a llal (l+a2) (1-ala 2) (1-a ) + 11 1 S-3- (+2) allaa2(l+a2) -- 2 (l-ala2) (1-a) 2 -4- allala2 (1+a2) (1-ala2) (1-a 1 a a^ ( 1-a 1) Hence, M = 1 becomes n0 ala2 1+a1 M = al [ (1+- ) (--- ) 1-al 1-ala2 -2 a(1-a+2ala2) (1+a2) + (1-l) 2(--a2 (1-a1) (1-aYa2) To calculate the results for var N, the procedure of Appendix A is duplicated to obtain - -2- a11(3+5al+al 2-ala2) var mi = 1 + - (1-ala2) and the variances of the branches b, and b2 whose averages are given by b +a +a 2- 2 2+a 22+... bl: 1+al+ala2+al a2+a a + and b- a+- -a +2- 1 2-2 .. b2: al+ala2+al a2+a a2 + They are, respectively, 2- l+3al+3ala2+a 2a2 var b_ 2 (1-ala2) and al (l+3a2+3ala2+ala2) var b2 = a 2)2 (1-ala2 2 Observing that the variances of branches whose averages are represented by alala 2 (1+al+ala2+a2 ) or k- --- -2- -2-2 alla1 a2a 1 +ala2+al a2 +a a2+ ...) are given, respectively, by j- k allal a2 var b1 or -j- k allal a2 var b2 and repeating the summation performed to obtain N = Em. i I 2- all(3+5al+ala2-ala2) Z var mi = 1 + 2 i 1 (1-ala2) -2- all1l12 1+3a1+3ala2+ala2 + ---- (1-a1) (1-ala2) -2 ------ 2 a la2(1-a+2aj 2) 1+3a2+3ala2+ala2 (1-a l) (1-ala2)2 It, therefore, remains to compute 2 IE (mm.-mm.). ij j>i Fortunately, it is necessary to compute only seven (mm-mm.)'s, since the remaining can be obtained by inspection. The pro- cedure employed in calculating these is illustrated for the case of (m0mlmll-m0 mll) (m ml lmlm1) i = E 1(nloinllj-nl0inllj) i=l j=l 1 where "n0 nl0i = 1 (1 all a a2 a a2 a )r r=l i terms and nn0. "llj = E [1 all al a2 al (a1 a2 a .. .)]s s=1 j terms 81 Here, denotes identically the same factor in nl0i and nllj, and the factors bracketed by are to be treated as one, since they correspond to i=l or j=l. Since this approach is identical to that employed in Appendix A, it may be observed there that (nl0inllj-nl0inllj) =(nlOinllj r=s 2 -2- since n -n0 =no 2a -- 2- 2- Moreover, when r=s, a 1=a a =a and =a2. Consequently, (ml01ll-mlmll) = j I(n"l0ilj11 r=s i=1 j=1 --3- -(+24i2 Sallaa2 (l+a2) (5+a2-4aa2) (= n0 l-al a2)2 The other six (mm.-im.im.)'s of importance are i 3 iJ _____ l+a -4- l+a2 2 (mllm12-mllml2) = a 2 -2 1-ala2 _1 a a2 (l+a.) a2(l+al) '^IT0o-"o-i = -[3+]-- ] S-(~ +a2) (l-ala2) 13- 2 l2+ a 2) (ml0m021-mllm021) = a a22( _ 11l-a-a (1-ala2) (mllm020-mllm020) = (m020m021-m020m"21) - -3-2 - alla a2(l+al) (1+a2 = 2 l+a2 +a Sall ala ( ) (+ l ) l-ala2 1-ala2 4- l+a2 l+a2 (m221-m20m21) = all a2 ( ---) (2+ ) 1-ala2 1-ala2 All other (mimj-mimj)'s may be obtained from these by observing that, for example, (mimlj-mlimlj) = al (mliml (j-1) -mlml(j1) i>l, j>3 Generating additional (mim -mi )'s in this manner and adding the results - -3- n0allala2 (l+a2) (5+a2-4ala2) (1-al) (1-aa)2 iZ (mlimlj,-m imlj ) i=0 j=1 1 j>i -- -4- nOallala2 1 + 0 1( )2 (1-al) 1-ala2 3-22 2) n,,llala (l+al)(3+a-2aaa) B: (m10m02j-m10m02j) j=0) 1-al)(1-al 2) n0allala2 (l+al) (3-2ala2+a2) (1-ala2 2 i1 j (m lim02j-lim02j - -3-2 ) (- n0allala2(1+a (1I+a2 (-al)(-a 22 -5-2 - 0allala2 1+a2 2 + 2 l- 2 Snalal(l+a2) (3+a2-2ala2) i= 10 2 1- -a1) la)2 -2- (l-al+ala2) (1-al) (1-ala2)2 - -4- -2- E 0 --i an0allaa2-a +aa2a ) 1+a 2 E: Z (m m2j -mlim2 ) 2 -2 i=1 j=0 (-al) l1-a a2 - -3- - --- _Pn a aa (1+a ) (2+a2-1a2) (m02im02j-m02im02j) n0allala2+a2) i 02 02 0 (1-al) (1-ala2 j>i 2 - -4- nallala2 1+a2 2 + (1-al) 2 1 2 (1-a1) 2lala2 i Z (m r n m Sj=l 21 2j 2im2j) j>i - -4- - allala2( 1+a2 (2 +a2 (l-ai -a- )a(2+ -aa2 (1-al) l-a a2 1-ala2 - -5- - + nallala2 1+a2 2 (1-al) 1-ala2 -2- H: Cm n0allala2(l+al) (l+a2) j=0 (m2002j -2002j) (-ala2i2 -4- n0allala2 +a 2 (1-al) 1-ala2 I:il 2im2j-m2imo2j) i=1 j=0 - 4-2 (1-al) (l-ala2 - -6-2 + n0allala2 1+a ) (1-al) 2 l-ala2 85 _______ ______ _2_2 1+a - -2-2 1+al J: (m020m03j-m020m03j) = nallala2( - j=0 l-ala2 -4-2 Snllala2 (1+al) (1+a2) (1-al) (1-ala2)2 S_ 4-2 - n 0n aalla a2 (1+al) (1+a2 iK: =l j (m02im03j-m02im03j (1-a) (l-aa1 )2 - -6-2 n0allala2 1+a2 )2 + ( 2 ia (1-al) 1-ala2 L (m -a2 (1+al)(+a2)(1-a 1+a 2) j=0 (1-a1) (1-ala2) - -5- -2- : (m2nallala2 (1-aala2 iZ=l j=O(mo2im~ j -m oij (l3alm2 1+a 2 (la_2 )2 l-ala2 -2- n0allala2 (1+ 1 (l+a 2 N: (m20m03j-m20m03j) (1 ala 2 - -4- n+ allala2 l+a2 2 (1-al) l-ala2 - -2- n0allala2(1+a1) (l+a2) 0: E E (m2imj-03j m2im03j ()i 2 i=l1 j=0 l-a -4- n"011a1a2 1-a n0allala2( a2 )2 n_ alal(la-a a ) l+a P: Z (m20m3i-m20m3j) - j=o (1-a0) 1-ala na n0a 11la a2 (l-a1+aa 2) Q: (m2im3j -m2im3j)2 i=1 j=0 (1-al 1+a2 2 l-ala2 Finally, 2Z E (m-mm.m) = 2[A+ (B+C+D+E+F+G+H+I) i j 1-a j>i + ---=--- (J+K+L+M+N+O+P+Q)] (1-a1) After all, al, and a2 are calculated using the DC multiplica- tion program of Appendix C, the above results are evaluated, and (var N) 1 is plotted vs. M in Figures 44 to 46. .0 APPENDIX C DC MULTIPLICATION SIMULATION PROGRAM The DC multiplication simulation program performs calculations of carrier multiplication resulting from impact ionization in p-n step junctions. Its input variables are: NSETS : number of diodes whose DC V-I characteristics will be calculated NVREV : number of reverse voltages for which the DC multiplication will be calculated XNA : NA-ND = net doping density on p-side (cm-3 XND : ND = net doping density on n-side (cm-3) ALPHAl electron ionization rate constants: adjusted I B1_ B1 ionization rate is given by a =ALPHAl e E ALPHA2 hole ionization rate constants: adjusted ioni- B2 zation rate for E < 4 x 105 V/cm is given by B2 = ALPHA2 er ALPHA3 hole ionization rate constants: adjusted ioni- B3 zation rate for E > 4 x 105 V/cm is given by B3 2, = ALPHA3 e E PHI : 4, junction built-in potential Xk+Axk DV : dV = -q fk E(x) dx, where Axk is the length S k of the k-th interval in the E(x) vs. x diagram TV : tunneling potential EV : electron threshold potential for ionization HV : hole threshold potential for ionization SLOPE : the tunneling current, It, is given by SLOPE It= Ioe Fr VREV : reverse voltage for which DC multiplication will be calculated The different portions for the program perform the following computations. Statements 29 to 66 divide the p and n sides of the electric field triangle into equipotential intervals, DXN(K), DXP(K), and compute the electric field values, EN(K), EP(K), at the end points of the intervals. Statements 67 to 95 number these quantities consecutively, starting on the p side with the first interval and the first electric field value. Statements 96 to 164 compute the approximate values of the ionization probabilities for electrons and holes, dP (k) = XINTN(K) and dPn(k) = XINTP(K), in each interval, DX(K). The spatial distribution of the tunneling current, dIt(k) = TI(K), is computed in statements 173 to 219. The electron initiated multiplication is calculated in statements 220 to 241, 251 to 266, and 276 to 287. TV Statements 220 to 225 shift the tunneling current by DV intervals to obtain the free electron current. (In "con- ventional calculations" TV is set to one DV.) The book- keeping algorithms which trace the electrons and holes back and forth across the SCR and compute the DC multi- plication on each transit are contained in statements 237 to 241 and 262 to 266. (If the available storage is sufficiently large, the encremental increase in current, DIE(N) and DIH(N), resulting from carrier multiplication in each interval can be replaced by DI(K,N), where K is the transit number and N is the interval number. Combining I DI(K,N) with the interval number as function of distance k then yields the distribution of carrier multiplication as function of distance across the SCR.) Statements 242 to 250 and 267 to 275 retrieve and save the electron multiplication in each interval on the 10-th and 12-th transits, and the hole multiplication on the Il-th and 13-th transits. These quantities are used later in the program to compute the values of ala2, and a3' the first, second, and third ionization probabilities. (There is nothing special about these transits. Any other corresponding set of transits for which the multiplication has achieved a steady state condition would do as well.) The multiplication initiated by holes left in the valence band when electrons tunnel to the conduction band is calculated in statements 293 to 338. They are essentially mirror images of the statements handling the electron initiated multiplication. The total DC multiplication is then the sum of the electron initiated and the hole initiated multi- plication. It is denoted by SUM and computed in statement 342. This ends the DC multiplication simulation. The remainder of the program is devoted to the calcula- tion of the ionization probabilities, al, a2 a3, and to the evaluation of the final results of the up to two ionization per SCR transit noise theory. The electron ionization probabilities are computed in statements 345 to 421. The probability of the first ionization, al, is computed by taking the multiplication due to holes on the 11-th and 13-th transits and calculating the electron ionizations on the 12-th and 14-th transits, allowing no more than one ionization. This is performed in statements 345 to 356 and 379 to 390. The square root of the ratio of the first electron ionizations on the 14-th transit, El, to the first electron ionizations on the 12-th transit, Dl, is al. If this calculation is repeated starting with the first electron ionizations on the 12-th and 14-th transits (the results of the computation just completed), as done in statements 358 to 367 and 391 to 401, a2 is obtained. Repeating the calculations once more in state- ments 368 to 378 and 402 to 412 leads to a3. The corresponding hole ionization probabilities are computed in the same manner, but starting with the electron multiplication on 10-th and 12-th transits. These calcula- tions are carried out in statements 452 to 519. Because the two ionizations noise theory does not distinguish between electron and hole ionization proba- bilities, it is now necessary to evaluate the "average" ionization probabilities to be used in the noise theory. They are simply the square root of the product of the electron and hole ionization probabilities, as given in statements 558 to 560. According to Appendix B, the probability of the second ionization on the first transit is zero for all reverse voltages for which the probability of the third ionization is zero. Hence, all = S(2), the probability of ionization on the first transit. Now, that the ionization probabilities all, al and a2 are known, the theoretical expressions for noise resulting from up to two ionizations per carrier transit across the SCR may be evaluated. This is performed in statements 561 to 577. This completes the program. The calculations described above yield the following ionization probabilities for the 1N751 diode: Diode reverse voltage all al a2 5.05 0.485 0.909 0.114 5.0 0.474 0.897 0.107 4.95 0.462 0.886 0.0994 4.9 0.451 0.874 0.0926 4.8 0.428 0.851 0.0796 4.7 0.403 0.829 0.0676 Diode reverse voltage all al 2 4.6 0.378 0.807 0.0567 4.5 0.353 0.786 0.0469 4.4 0.327 0.766 0.0381 4.3 0.299 0.746 0.0304 4.2 0.273 0.728 0.0237 4.1 0.245 0.710 0.0180 4.0 0.218 0.693 0.0133 3.9 0.190 0.677 0.0094 3.8 0.163 0.661 0.0064 The range and functional dependence of the above values are typical of the other diodes employed in this study. Consequently, the approximation that a2 << al, invoked in the derivation of the up to two ionizations per SCR transit noise theory, is justified. 0001 DIMErNlIN XN(600),VN(600),EMAX(C600),DXN(0OO)EN(600,DrP(600), SP(60)rVP(6OO)EP0(600),t( 900iINITN ( o00), tNTP( 9q0).0X( 900), VBEPv30)I 910. 3(301),3 IN(0900),A80I900),SSDI0900), lDIE(1200)s6TH(?,no1, 0002 REAO 13 SFTS,NVREV 003 13 FnRMAT( ) 00004 on o00o Nq=SNSE 0005 RF A i, N, l 0 IPMA1Rl,ALPFA2,R2,ALPHIA3,93 000 810 t FRiT(2E .9 i E7').. 0007 REOA 11 PHI,0V,TOV,EV.3VSLOPE 0008 11 FRB"tTF5.FI"' 0,7l., 0000 REAO 1, 1(EV 2I=l,NVREV) 0010 12 FnP"AT(l6F5 3) 00 1 POINT 1,C(VEAy(1),11.NVPFV 00 1 FORMAT(l1VOEV /,1C(IX,FS.3.2X)) 0011 ES'P :.=1.17 0014 X SrP W -Z, 001. l=tbE-19 001* *Il=P TC(2.*EPS3EL*EPSOI/)e((SXNtXNO)/(XNA*WND))) 001 n 000 NV.lVREV t% .;5ltf*SPT(PI*+VREV(NV)) 0019 G2.SLPE/r 0026 EAX=(2.*S99T(PM,*VR9VfV)))/I 0001 IF(F/0y.rE. 1.2Ei) Gn TO 98 0 PIT 97 00 24 STOP 0027 98 XENZ(TXA/(Xm0A+X0 D)*y 0024 VXPp=W-YNN C CnMPUTE OX AND E FOR 510E 0029 XN(l)sXNN 0030 G( IY V'9 S033 KaVNk/t)V 3 IIFK- 101,99,99 0036 JJ=+1 003 DOXN! 1)=P(DCII)*(1,-9OT(1,-DV/V1(II)) 00 7EW(T I)FMlx(II)*(l.-DYN lI)/XH(lI)) o00 V"(JJ)=VN(IT)-DV 0039 XO JJ)NC(IIT)-0rN(II) 000 EM X(JJ)=rNCIII 00 4 100 CONTINUE 00 0N(K+1t )=2.*(VNM-W*DV)/EN(K) 003 GO TO 102 009 11ot CONTINUE 00 5 31.1 0006 DXNJJ3)=22*V0N/FMAXY C CnMPUTE OX AND F OR P SlOE 00 102 XP()=Xpp 004 VP(1)=VPP 00 9 EMAX(I=)EEAXX 0050 M VPP/OV 0051 IF M;-l) 201,199.199 005? 199 On 00 a =1,I 0o5 DXPcI)=Wpci)*(I.- 7(i,-y-D/VPt(I))) EP(I)=EMAYi '(I.-o.p(i)IxP(l)) 0050 ED"APi(J)1=EF 9I) 0059 200 CONTINUE 0000 Gn TO 202 0061 201 Jl1 0062 OYP(J)=2 *VPP/EMAXX 0i 0 202 CNT0NiiE 0069 DoP(*l )=?.([VPP-M*OV)/EPCMl 0066 203 CONTINUE C RENIMliEP OX 0067 3- 0069 3 = 0060 I(N D IPJI 8898YO 8?M 1 0071D NJM JAJJJ 007T DMcXNl-nDYN~i) 00310 COiNTIU C RFgOiPE0 E 00 31 JIs 00832-a 08 o 320 NE IAJO 00P8 ENEr)EPJSN 008o E P! :?o " 1011, 1: 3 k: JO 00 32 +I)EMAXX 00 IF (K C 330,330,322 000 322 JF 3 2 0096 JEErM+K+I 00o0 9 2=3l NE:JE,JEF 0091 E( NE)cEN(JP 0094 JP1J p 009 330 C~:NTNUF COM 3PUTE A NT XTNTP T ZEGRILO FOR ELE T ALUS 0098 331 On30 l 1, 1 344 KE S TV,, B!E=(N E)E!(J P) 0100 EsTaEtI+ 010 I 8atTEI 3 -K40 E: ESTTS.I Eo TO T41 010 331 0n IA? 0:o1b11 0 00 M RTOStI 010 342 CNTILNUF C SFT3 INTNIS AN XSNTP'S TO ZERO FOR FIELD VALUES WHICH ARE TOO SMALL 0 13 4 XINT (II)AR 30118 35 I IIKETOP* F .J-LF. ,T O ) GO TO 345 1 NT Ia It, 23 XINTTPII)O 011 6, 733-1 S.LT.03 GO TO 347 8012 346 TCn1TI UE C TPIANGLF OPPPnxTMATTON FOR FIRST 0ND LAST NON ZERO XINT. 817 347 SINTNCMIKE1TAB)I(nXES7ARI):ALPHl.tXP C( 4/EI(EftARII)/7 C TBAPECnTO MFTOVn [s3FD Fnh BFPia'ING XI4T07S 0129 7 IFEJT.PJK5T4 .LT. 1 0) GTn 401 1on u00 IoKESTAr.II XINTNC(J)IDXIJ]*LPMA1I*(EXP.hBIEII)]IEOPC-81/ECJI /2. 01003 K j SPU 0 0 C UTF F ST 1n 24O XXTNTP 2r F( F .T .0) L .E Ol GU T O 3 02 o311 7 INTP(KFSTIA (xKESTA* ALPHA 3*EXP(-B3/, CKETAR)))/2. 01302 XNTP(KEST0P)=(oX(KFSTA)*ALPHA2*EXPC(-. /E(KESTAR) i/2. C COMPUTE LIST NON 7ERO XINTP 0139 03 IF(F(FESTnp+lI .LF. o ELP GO TO a0 OI XITPCESTOPl) X(l STp P ) ALPHA3EXP(-B3/tEKESTnp)))/2. 0G TO P( 014A 400 X KNTP(K3TOP+I)=OX KRFSTUP+I1*ALPHA2*EXP(-B2/E(KESTOP)))/2, S USE TRPAP ZOIO APPROSIATION FOR REMAINING XINTPIS W4ILE CONSIDERING FIELD 0113 012 IFFMt TnP KESTAR .LT. 1) GO Tn 501 61isESTOP *I 01450n 50T D=KE TAR,II 0146 JtI-i 7IF(FI) .LE. .E5) CGO TO 451B3 A 450 XINTP(J)D J),'ALH 43(C P(B3 ))EX(3/IFJ))]/2. S50 451 XTNTP(J)=OX(J)ALPHA2*(EXP(-B2/E(I))+EXP(.62/E(J)))/2. p19 F T sIy ,a v = ',F5,3) 0150 to FORHT07,'1',7EUrF a lF5.3)I OIf'T 65i XP^DALP"*l..BltLPHA2.B2,ALPMt3tB3 015 PN1T 5i. PH,6Tv V l'.F, SSX3IHV = I.F.3) PPIlnf 18f 0161 P1rT 7. FAX 016 17T FRTT1/,ItX,'PEIK FIELD IN JUNCTION ',EIO.,3X, IVOLTS/CmI) BRA 32 FmbIT?: IX.'JUIWCTION WIDTH x 'EIO ,l ,' CM ') C PRINT STATEMENT FnR DYIINTNXINTP,IND E GCES HERE 0165 NTO=TV/DV+I. 0166 NTT.= TO.1 0167 NFO=EV/OV+1. 0168 NETONEO*A 01691 NHOeHV/DVI. 0170 JWOnJ.,IJr*n 0170 Jm0HJJJ-3NH 0172 jMH5Cj-1 C CnMPUTE OISTRTIBITON nF TUNNELING PROBARILITY C COMPUTE STAT AND STOP POINTS FOR TPAPEZOID METHOD nF APPrOX SUBINTEGOALS 0175 On 502 T=1.II 0176 KESTA =I 0177 BnE=G/E(I) 7AF (RnE LE.50.) Gn TO 503 79 502 CJNTINUF 0180 503 on 50a T=,.IIJ 010 SrE I/E(KFSTOP DI OIF (IPE LE.O0U. Gn TO 505 0180 50 CoNTIUE C8S T TUNNELING PPORARILITY TO ZERO FOR SMALL FIELD VALUES 015 505 ITESTAR- 016 I(IIILE.0) GO TO 507 01 7 KKxs+t 01 R Dn 506 T=1,KK 1T1 1$ t-l 5 IF0 F .LE.) GO TO 507 0506 LTINUE 01 3 507 CnNTINUE 0 I9 IIsF5TnP+2 01 9 IF (JO-KESTOP.LE.0 GO TO 509 0196 KK =K+ 0107 DO SD K IsIRs 019H TI9II)=0. 020 IF (JJJ-II.LT.0) GO Tn 509 0201 508 NTINUF 0202 509 CnN,TINUE C TRndldLF APORnXIMATIONM FUR FIRST AND LAST TUNNELING SUBINTEGRAL 8?80 TltK[AA';(0fIT)*F P-/(ETA lP)/)2, 838r1 V STnp*l)-(D( E-T p.1)Ep(-G/KEET) |