Double-Injection: High Frequency Noise and Temperature Dependence Citation Lee, Don Howard (1969) Double-Injection: High Frequency Noise and Temperature Dependence. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/964x-ab98. https://resolver.caltech.edu/CaltechTHESIS:10072014-144607202 Abstract Noise measurements from 140°K to 350°K ambient temperature and between 10kHz and 22MHz performed on a double injection silicon diode as a function of operating point indicate that the high frequency noise depends linearly on the ambient temperature T and on the differential conductance g measured at the same frequency. The noise is represented quantitatively by〈i^2〉 = α•4kTgΔf. A new interpretation demands Nyquist noise with α ≡ 1 in these devices at high frequencies. This is in accord with an equivalent circuit derived for the double injection process. The effects of diode geometry on the static I-V characteristic as well as on the ac properties are illustrated. Investigation of the temperature dependence of double injection yields measurements of the temperature variation of the common high-level lifetime τ(τ ∝ T^2), the hole conductivity mobility µ_p (µ_p ∝ T^(-2.18)) and the electron conductivity mobility µ_n(µ_n ∝ T^(-1.75)). Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Nicolet, Marc-Aurele Thesis Committee: Unknown, Unknown Defense Date: 8 May 1969 Record Number: CaltechTHESIS:10072014-144607202 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10072014-144607202 DOI: 10.7907/964x-ab98 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8676 Collection: CaltechTHESIS Deposited By: Bianca Rios Deposited On: 07 Oct 2014 22:01 Last Modified: 24 May 2024 18:28 Thesis Files Preview PDF - Final Version See Usage Policy. 21MB Repository Staff Only: item control page

Copyright ~ DON HCWARD LEE 1969 by DOUBlE INJECTION: HIGH FREQUENCY NOISE AND TEMPERATURE DEPENDENCE Thesis by Don Howard Lee In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1969 (Submitted May 8, 1969) ii TO MY PARENTS iii ACKNCWIEOOMEN'IB The author would like to express his thanks to the many people at Caltech who contributed to the work presented herein. In particular, the author is indebted to Dr. M-A. Nicolet whose direction and guidance underlie this work. Also, the help of Drs. H. Bilger, E. McCarter and Mr. N. Harlambis during the summer recesses is also acknowledged. A special note of thanks goes to Dr. Mr. L. c. A. Mead of Caltech and c. Hedrick of Tektronix Inc. for their constant encouragement and inspiration during the course of this work. The double-injection diodes were generously provided by Drs. J. W. Mayer, R. Baron and Mr . 0. J. Marsh, Hughes Research Laboratories, Malibu, California. The financial support of Tektronix Inc., Fairchild Camera and Instrument, t he Ford Foundation and the National Science Foundation through their Fellowship programs is gratefully acknowledged . This work was also supported in part by NASA Electronics Research Center, Cambridge, Mass. and the Jet Propulsion Laboratory by NASA Contract No. NAS 7-100. Mrs. Doris Schlicht, who typed the manuscript, deserves special thanks for her patience and assistance. I am also grateful to my wife for her constant support and self-denial. iv ABSTRACT Noise measurements from 140°K to 350°K ambient temperature and between 10kHz and 22MHz performed on a double injection silicon diode as a function of operating point indicate that the high frequency noise depends linearly on the ambient temperature differential conductance g T and on the measured at the same frequency. The noise is represented quantitatively by 2 (i ) = a•4kTg6f. pretation demands Nyquist noise with =1 in these devices at high frequencies. This is in accord with an equivalent circuit derived for the double injection process. static I-V illustrated. a A new inter- The effects of diode geometry on the characteristic as well as on the ac properties are Investigation of the temperature dependence of double injection yields measurements of the temperature variation of the 2 common high-level lifetime T(T ~ T ), the hole conductivity 2 18 mobility ~p (~p ~ T- • ) and the electron conductivity mobility ~nc~n ~ T-1.75). v TABLE OF CONTENTS CHAPTER I l.l. 1.2. HIGH FREQ,UENCY NOISE IN DOUBLE INJECTION Introduction DC Properties of the Semiconductor Regime 1. 2 .1. 1. 2 . 2 . 1. 2 . 3 . Current-Voltage Characteristics Influence of Geometry on Semiconductor Regime I-V Characteristics Experimental. Results on a Long Silicon p+rt · n+ Structure From 140°K to 350°K l l 3 3 7 16 1.3. Differential Step Response of Double Injection 20 1.4. AC Properties of the Semiconductor Regime 25 1.4.1. 1.4. 2 . 1.4. 3 . 1.5. Small Signal ac Equivalent Circuit Experimental Verification of Results Nonlinear Effects High Frequency Noise in Double Injection Silicon Diodes 1.5.1. 1.5. 2 . 1.5. 3 . Generalities Model of High Frequency Noise in Double Injection Measured Spectral Noise Density 25 30 36 37 37 41 45 1. 6 . Evaluation of Results and Conclusion 53 1.7. Discussion and Outlook 58 CHAPTER II 2.1. 2.2. 2.3 . 2 .4. 2 .5. TEMPERATURE DEPENDENCE OF THE CCMMON HIGH-IEVEL LIFETIME AND CONDUCTIVITY MOBILITY OF CARRIERS IN SILICON FROM DOUBLE INJECTION 64 Introduction Temperature Dependence of the Current-Voltage Characteristic for Double Injection 65 Common Hi gh-Level Lifetime Versus Temperature in Si 66 Majority and Minority Carrier Mobility Versus Temperature in High Resistivity Silicon 68 Conclusion 76 64 APPENDIX A SOLUTION FOR GEOMETRICAL FACTOR 79 APPENDIX B APPENDIX C SOLUTION OF AC SMALL SIGNAL IMPEDANCE 81 THEORY OF NOISE MEASUREMENT AND EXFERIMENTAL PROCEDURE C.l . Noise Spectral Density C.2 . Measurement System C. 3. Small Signal AC Equivalent Circuit and Notation 88 88 89 89 vi C. 4. Measurement Procedure C.5. Analysis APPENDIX D 93 93 REALIZATION OF NOISE MEASUREMENT APPARATUS 96 D.l. Description of Amplifier and Input Circuitry D.2. Amplifier and Wave Analyzer Characteristics D.3. High Frequency Effects 96 102 104 D.4. Low Frequency Effects 106 D.5. Measurement of Gain Ratio D.6. Comments on Construction 108 D.7. Active Integrator 108 APPENDIX E ~ CALIBRATION MEASUREMENTS 10( ll2 E.l. Wave Analyzer Detector 112 E.2. Verification of Nyquist's Theorem E.3. Verification of Schottky's Theorem ll3 ll4 E.4. Resistor Calibration ll5 F.l. General Considerations F.2. Description of Thermostat ll9 ll9 ll9 F .3. Operation and Procedure 121 F.4. Calibration 122 APPENDIX F APPENDIX G TEMPERATURE ENVIRONMENT AND THERMOOTAT CURVE FITTING AND ERROR ANALYSIS 125 G.l. Introduction 125 G.2. 125 G.3. Least-Squares Curve Fitting of a Function I eq = y(f, Cv ) Determination of the Weights wi G.4. Computer Program 130 REFERENCES 127 132 1 CHAPTER I HIGH FREQUENCY NOISE IN DOUBLE INJECTION 1.1. Introduction. Within the last few years, several experimental and theoretical studies directly related to the present subject of high frequency noise in double injection diodes have appeared in the literature. Two of the experimental investigators report that the mean square current fluctuations can be represented by (1.1.1) with a ~ 1, even though the two devices and their operating condi- · . . ·1ar. ( l.l, 1. 2 ) are d lSSliDl t lons I n one case, the d evlce . . a 1 ong lS germanium diode operated in a square law range, whereas in the other case the device is a thin commercial germanium photocell biased in the v3 range. Another study presents noise measurements on a long double injection silicon diode operated in the square law range.(l.3) The value of the limiting white noise level is stated there in terms of the low frequency conductance oi/oV (1.1.2) * Here k is Boltzman's constant; T the temperature in °K; 6f the frequency interval and g the conductance of the device. 2 where ~ = 0.52 ± 0.1 is not given. but the value of g at the measuring frequenc,y These results obtained on dissimilar devices operating under different conditions seem to indicate a general fact, namely, that double injection exhibits thermal noise at high frequencies. The present investigation has therefore been undertaken to clarify the matter. In general, the I-V characteristic is an unreliable means with which to confirm the presence of double injection. This is aptly demonstrated by Rose in his "comparative anatomy of models for double injection" in which two carrier injection encompasses a host of char1 4 acteristics, including that of single injection.< • ) Fortunately, by determining such physical parameters as geometry, doping level, mobilities and lifetimes a specific mode of double injection can be established. The investigation presented here is primarily concerned with a long silicon (semiconductor) structure. Section 1.2 thus deals with the pertinent analysis and resulting features of this special case. de The effects of geometry on the current voltage charac- teristic are also examined. Experimental results on a planar silicon diode at room temperature confirm the analysis. The + p 1C n + I-V characteristics are determined as a function of temperature, and a detailed description and analysis of the temperature behavior is presented in Chapter II. transient and small signal In Sections ac 1.3 and 1.4, which deal with behavior, an equivalent circuit is established for a long semiconductor double injection diode. The experimental verification of the results provide further confirmation of double injection. considered. Nonlinear and temperature effects are also Using the results from the transient and ac properties, 3 a model of the high frequency noise in double injection is proposed in Section 1.5. This model is verified by noise spectral density measure- Section 1.6 contains an estimate of the magnitude of the low ments. frequency noise spectral density which originates from generationrecombination effects. 1.2. DC Properties of the Semiconductor Regime. 1.2.1. Current-Voltage Characteristics. The current-voltage charac- teristics resulting from the simultaneous injection of electrons and holes (double injection) into insulator and high resistivi ty material have been dealt with in detail primarily by M. Lampert and A. Rose.(l.5, 1 • 6,l.7) In fact, the latter has compiled and discussed some fourteen different models for two carrier injection.(l. 4 ) Recently, J. Mayer, R. Baron and 0. Marsh mve experimentally verified three distinct double injection regimes predicted by Lampert and Rose.(l.B,l.9,l.lO) These 2 consist of the "insulator" (J o: v3 /L5 ), the "semiconductor" (J o: v /L3) and the negative resistance regimes. In conjunction with these experi- mental studies, the effects of diffusion and thermal generation on the I-V characteristics are investigated for both the insulator and semi- cond uc t or cases. (1.11,1.12) The basic time-independent equations which describe double injection current consists of (i) the current equations for the electrons and holes, Jn = <W n [nE + ~(\7n)] (1.2.1) (1..2.2) ..... ..... J = J where 4 ..... n + J kT f3 = q' (1.2.3) p n = n 0 + tm and p = p 0 + flp, (ii) the charge con- servation equations, 1 (<rJ•J ) q n =r (1.2.4) 1 ("J•J ) = - r, q p (1.2.5) and (iii) Poisson's equation (1.2.6) Here, the symbols have their usual meanings, in particular, net recombination rate, while n , p 0 0 and 6.n and flp r is the are the thermal equilibrium and net excess carrier densities for the electrons and holes respectively. by - 1/~p· Equation (1.2.4) is now multiplied by By adding, 1/~ n and (1.2.5) the following equation results (b+l) r ~n where b = ~n/~p· Rewriting Eq.(1.2.3) in terms of (1.2.1) and (1.2.2) gives J = ~ p [p0 + bn0 + b.p + bb.n] E + f3(b"Jn-"JP) (1.2.8) 5 These equations, (1. 2 .7) and (1.2.8), are now subject to a set of assumptions called the "high-level approximations" which state that the excess electron and hole carrier densities are approximately equal and are much greater than the thermal equilibrium carrier densities; that is, (a) 6n >> n (b) 6p >> p (c) 6p ~ 6n 0 (1.2.9b) 0 (except that the value of 6p-6n (1.2.9c) is taken from Poisson's equation) • The net recombination rate is defined by 6n 6p T T r =- n which implies that there is a common high-level lifetime T = T n- (1.2 .10) p T where T P (1.2.11) Under the high-level approximations, the individual retention in Eq.(l.2.7) of the first term, second term and third term produces respectively the semiconductor, insulator, and diffusion dominated regimes. Thus, the diffusion dominated regime is described by Eq. (1.2.8) and (1.2.12) 6 which leads to the definit.ion of an ambipolar diffusion length La ' where La ='/2f3T ~ n ~ p /(~ n+~ p ) V (1.2 .13) Retaining only the second term of Eq.(1.2.7) results in (b+l) ~ n T (1.2 .14) p This leads to the Lampert insulator regime in which the one dimensional I-V characteristic is given by , where L is the length of the diode. (1.2.15) Similarly, the retention of only the first term yields (p -n ) 0 0 V•E = - (b+l) ~ p n T (1.2 .16) In this case, the resulting semiconductor regime is described by the current-voltage characteristic (1.2 .17) The initial behavior at low injection levels is, of course, given by 3 = q ~ p (p0 + bn0 ) E (1.2.18) 7 which is just the ohmic regime. The investigation here is totally focused on a double injection diode made from high resistivity rr type silicon. This diode is "sister" to those previously investigated in Reference 1.10 and has parameters such that the semiconductor regime is the principal double injection mode. Therefore, the primary concern with the remainder of this work will be the semiconductor regime described by Eqs.(l.2.8), (1.2.16) and (1.2.17). 1.2.2. Influence of Geometry on Semiconductor Regime I-V Charac- teristics. The theoretical and experimental investigation of the current-voltage characteristics of Lampert's two-carrier semiconductor regime have been entirely concerned with the one and two dimensional planar cases (i.e. rectangular diodes). In this section the effect of geometry on the de properties is examined. regime p + rr n+ Only the semiconductor type structures are investigated. planar case is illustrated in Figure 1.2.1. supplies the holes and the p+ n+ As Here, the contact the electrons. rr (high resistivity p-type) + n I Planar Figure 1.2.1 + + double injection diode p rr n an example, the + p contact 8 From Eqs.(l.2.8) and (1.2.16), the current density can be written as j = - ~ p ~ n T(po -no )(v•E) E (1.2.19) Since the divergence of the current density is zero (i.e. - v•J = o), and if the condition of spatial uniformity for the mobilities, common high-level lifetime and the net thermal equilibrium carrier density (p -n ) 0 0 is maintained, then (1.2.20) Equation (1.2.20) is now solved for the electric field subject to the following conditions: (i) The diode is forward biased as shown in Figure 1.2.1. voltage across the structure is voltage drop across either the (ii) V. There is no appreciable + or n+ regions. The electric field at the electron emitting boundary (n+ -~ (iii) p contact) is zero (i.e. The -E(R 2 ) R 2 = 0). The electric field is a function of only one coordinate direction, that is, E = E(~) ~l • The potential and charge distributions can be determined fran the electric field by (1.2.21) 9 and 1-l T(p -n ) n o o (1.2.22) b + 1 Condition (iii) is fUlfilled by the cylindrical, spherical and planar geometrical configurations. By applying the divergence operator for curvilinear orthogonal coordinate systems, Eq.(l.2.20) is easily integrated twice for the electric field (see Appendix A). electric field along ~ from R1 to R2, Integrating the the expression for the current-voltage characteristic results in 2 -n. I . = o q !-!p !-! n -r(p o -no ) V I_ (1.2.23) o depends only on the geometry and is therefore called the The factor geometrical factor. ship of the I-V This factor does not alter the square-law relation- characteristic but influences only the magnitude. It is interesting to observe that since the spatial dependences of the charge carriers and thus the electric field are functionally the same for pure unipolar space charge limited current, the geometrical factors are identical.(l.l3 ) I * s.I. That is, for single injection* (1.2.24) In this case, however, the appropriate boundary condition is that the electric field be zero at R1 rather than at R2 • Also, the introduction of the notion of space charge curren~ is relevant since for double injection the small difference in excess charge densities (~p-6n) does indeed form a space charge and can therefore be discussed in terms of single injection space charge current. I~ is also noted that R in Figure 1.2.1 is located at the 1 p -rt contact. lO By taking the ratio of Eqs.(l.2.23) and (l.2.24) and assuming where ~R ~ = ~n' is the dielectric (or Ohmic) relaxation time and is defined by ~R = E/q ~ p (p 0 -n0 ) • (l.2.26) Here, the ratio of double injection current to space charge limited current is not only independent of the geometrical factor but is determined solely by the bulk lifetime and dielectric relaxation time. * Equation (l. 2 .25) thus provides a means for determining the dominant mode of injection. For a given situation in which the semiconductor regime can exist, the bulk lifetime will in general be much greater than the relaxation time. Under this condition, the contribution of any space charge current to the overall current density will be negligible (see Section l.2.3). * It should be noted that a similar comparison between the double injecti on "insulator" regime (see Eq.(l.2.l5)) and single injection space charge limited current leads to ~.I(Is.I. = i ~ ~p V where ~ is defined to be the geometrical factor associated with the insulator regime. The quantity 5/6 is, in general, a function of geometrical terms in contrast to expression (l.2.25). In particular, for the planar case where the transit time Tp Tp is defined as =L2 /(~ p V). Here the ratio of the doUble injection insulator regime current to that of space charge current is determined not only by geometrical factors but by the bulk lifetime and a transit time Tp • ll The low-level injection behavior is described by Ohm's Law (Eq. (1.2.18)) and can be expressed as l II"'\ =- q 1-1 ( p u 5 p (1.2.26) + bn ) V 0 0 where (1.2.27) h 3 and 2 h 2 are the diagonal elements of the metric 3 tensor of the coordinate system and A is the cross-sectional area. The terms In this case, the quantity for the ohmic mode. A 1/6 is the appropriate geometrical factor transition voltage VT is now defined to be that voltage which makes the ohmic current equal to the double injection current. voltage Therefore, from Eq.(l.2.25) and (1.2.27) the transition VT is 1 1 (p +bn ) 0 0 (1.2.28) (p -n ) 0 0 When the thermal equilibrium hole carrier concentration larger than the electron concentration l 1 n , 0 p 0 is much Eq.(l.2.28) reduces to (1.2.29) 7)& 1-l ,. n Since the quantity lj6& has dimensions of length squared (i.e. 1j6&: 2 1 for the planar case), Rose has suggested that the onset of the 12 semiconductor regime occurs when the minority carrier transit time is of the order of the bulk high-level lifetime (i.e. Tn = l/(66~ vT) ~ T). n Table 1.2.1 summarizes the analytical expressions for the geometrical factors, electric fields and carrier density profiles.(l.l4,l.l5) convenience of comparison, the planar case is also presented. For Two cases are considered with respect to the cylindrical and spherical structures. In case (a), the total current density is directed radi- ally outward, whereas in case (b), the total current density is directed radially inward. Figure 1.2.2 contains qualitative plots of the electric fields and carrier densities. clearly evident. Here, the influence of geometry is In particular, the electric field for case (b) of the spherical geometry possesses a maximum value when (i.e. pjp 1 equals 41 /3 pjp1 = 1.587). * Figure 1.2.3 illustrates the appropriate nomal- ized geometrical factors as a function of the ratio of the radii. For the cases (a) and (b) in either the cylindrical or spherical geometries, the ratio of the double injection currents is not equal to unity but depends on the ratio of the radii. This current ratio, which is just the ratio of the geometrical factors, for the cylindrical case is given by [ ~] * (1.2.30) CyUndrica1= The symbol p instead of r is used to designate the radial variable to avoid confusion with the notation foF the dynamic impedance. TABIE 1. 2.1 Geometrical factors, electric fields and charge distributions for semiconductor regime double injection diodes in the planar, cylindrical and spherical geometries, GENERAL RElATIONSHIPS: TYPE PlANAR OF STRUCTURE u1 = x DOUBIE-INJECTION GECMETRICAL FACTOR 9A 'S 13 6 OHMIC GECMETRICAL FACTOR 2 ~ , I • = 6 q ~ n~ pT( po-n o ) V E(~) CHARGE- DENSITIES n(~) : L A 3 V ( 1 - r;) X ' 21 t y_L2 Q' (1~L )-t p(u1 ) DEFINITION OF SYMBOLS = p ~ ~ 6 EIECTRIC FIEID CYLINDRICAL CASE (b) ~ ln (z+(z2-l)')- (z2-l) p2 e z = p J-2 [ 2rrH 0 z~-2 0 [ 2] . e -p 1 2rrH t[t,2]'- J fH:1JJf ~2r -r,rw f[~ 2 ~b 1 n T(p o-n o + ~ )J2 = 0 u:rsr{ ~ ~ Tr J 3 02 p -pl ~ do 6rr o 1['2-p] !hi P2P1 1['2-'1] [M ~23_ o~J ~ ~ ~3- 013]] ' 2 p !hi 0201 --2 0 [,r,1']• M = I/(6rrqT~~p(p0 -n0 )) P = Ii.) rrHqT~ ~ (p -n )) 1 \~ npoo Q= = p -jf ~2r,f z = o/o1 (p 2>o1 ) SPHERICAL CASE (a) I SPHERICAL CASE (b) p1 -1l n 01 ~n ~ 2 -1 1 z -1)t-cos 012 1l ne [ - 2] 2;d{ L = Length H = Height A = Area p = inside radius 1 02 =outside radius =!_[~] 7io T CYLINDRICAL CASE (a) ~ 2 vT N=tQ It; PLANAR GEOMETRY CYLINDRICAL GEOMETRY SPHERICAL GEOMETRY CASE (b) CASE (a) CASE (b) CASE (a) ...... +="" xiL PIP2 0 PIP1 10 0 PIP2 0 PIP1 Figure 1.2.2. Qualitative representation of the electric field and charge distributions for the semiconductor regime double injection in planar, cylindrical and spherical configurations (p =radial variable). 10 15 10 2 0 w N _j <( ~ 0:: 0 z ro 1 G() 0:: 0 1-- u ~ _j <( u ro 0 0:: 1-- CYLINDRICAL CASE (a) w ~ 0 w <.9 10- 1 SPHERICAL CASE (a) 10-2 L-~--~~--~--~~--~~~~--~~--~--~~ 0 2 3 4 5 6 7 8 9 I0 II I2 I3 14 P2 /P1 Figure 1.2.3. Normalized geometrical factor as a function of the radii ratio p jp for cylindrical and 2 spherical1 geometries. 16 and for the spherical case by [~]Spherical p2 =- pl 1 J 2 p2 3 1/.2 (1-x ~ pl (x3 -1~ 2 X X 1/.2 dx 2 .(1.2.31) p2 Equation (1.2.30) and the numerical solution of Eq.(1.2.31) are plotted in Figure 1.2.4. Here, in each case, the current ratio smoothly increasing function of the ratio of the radii. ~/Ia is a It is also noted that for a given radii ratio, the spherica1 geometry current ratio is always greater than that for the cy1indrical geometry. Since the same geometrica1 factors app1y to unipo1ar space charge 1imited current (see Eq.(1.2.24)), Figures 1.2.3 and 1.2.4 are also va1id for single injection structures. 1.2.3. 0 Experimental Resu1ts on a Long Si1icon P+ 1f n+ 0 from 1 40 K to 350 K. A p1anar doub1e injection diode is made from high- resistivity f1oat zone grown p-type si1icon. by lithium diffusion whereas the a1uminum and alloying. Structure + p The n+ region is formed region is formed by evaporating Electrica1 contact to the n+ and p+ region is accomp1ished by app1ying a mixture (1:1) of gallium and indium. Further construction detai1s are presented in Ref. 1.10. Figure 1.2.5 contains the physica1 dimensions of the diode and approximate doping as specified by the manufacture. The measured current-vo1tage character- istics as a function of temperature is a1so shown in Figure 1.2.5. * The ambient temperature extends from 140°K to 350°K and all points are taken * Appendix F contains the description of the experimental equipment and procedure for varying the ambient temperature. 8 80 7 70 6 60 5 CYLINDRICAL~ - (LEFT SCALE) 0 ~ 0 .......... 4 ~ ~ 30 2 20 SPHERICAL {RIGHT SCALE) 10 0 - 40'-...._ .0 0 2 3 4 5 6 7 8 9 10 II 12 P2/P1 Figure 1.2.4. Current ratio ~/Ia versus the radii ratio and spherical geometries. p2/p1 for cylindrical 0 .!:5 18 DIMENSIONS, mm [3~ ~~ 101 V (volts) Figure 1.2.5. I-V characteristics of a silicon double injection diode between 140°K and 350°K ambient temperature. The solid dots indicate the operating points at which noise measurements have been made. 19 with the d i ode in the "dark". For the purpose of the ensuing noise measurement, a respectable square law range need exist. This criterion is met by the present diode since a quadratic behavior is observed over approximately a decade of current for each of the considered temperatures. At 140°K, the deviation from the square law (dashed line) at the higher current levels could possibly be due to diffusion 2 . or hea t l.ng e f fee t s as d"1.scussed b y R. Baron. (l.l ) The d evl.a . t•1.on f ran the ohmic behavior at low injection levels is attributed to junction effects. For a given voltage, the square law current is seen to in- crease with decreasing t emperature. This behavior is expected since the electron and hol e mobilities each possess this type of temperature variation. The temperature dependence of the double injection I-V characteristic in the semiconductor regime is treated in Chapter II. Since the host material is concentration p concentration n • ~ type, the thermal equilibrium hole is much greater than the thermal equilibrium electron 0 Therefore, Eq.(l.2.17) describing the semiconductor 0 regime reduces to (1.2.32) The measured room temperature -2 em values are 2 69.5 x 10- 6 _a_ v-ern A = 9. 67 x 10 (T = 298°K) 2 IJ.n = 1280 ~ 1 T = 30.7 X v-s -1 and L = 5.92 x 10 em. From Eq.(l.2.32) the theoretical double injection square law current at is 1.4 rna. 30 volts This theoretical value is approximately 14% below the measured value of 1.6 rna (see Figure 1.2.5). However, R. Baron 20 has shown that this is expected from the effects of diffusion and . therma l genera t 1on. (1.12) From Eq.(l. 2 .25) the ratio of the measured bulk high-level lifetime ( 3 0. ~ sec) to the calculated dielectric relaxation time (15.3 ns) is 2 x 103 • This illustrates the magnitude of two carrier injection current over that of single injection and provides further assurance that the dominant mode is double injection. regime discussed by The negative resistance M. A. Lampert and J. W. Mayer was not observed under any of the present experimental conditions described herein.(l.7, 1.8) 1.3. Differential Step Response of Double Injection. As pointed out by R. Baron et.al., additional information on the properties of a double injection diode can be easily obtained from its differential step response. (l.l 6) Consider the case of an I~ v2 • acteristic as shown in Figure 1.3.1 where I-V char- If the current is indeed entirely due to double injection as prescribed by Eq.(l.2.l7), an incremental voltage step ~V applied at t = 0 should generate the current response shown by the solid line of Figure 1.3.2. ~i(O) instantaneous increase of current second increase of ~i( ro) at t = 0+ There is an followed by a after several time constants The explanation of this behavior rests on the hypothesis that injected charge can not change instantaneously. the application of the voltage pulse Therefore, immediately after ~V, the diode behaves as a series of resistive elements whose values depend on the amount of charge stored in every interval At t = 0+ dx along the length the resistance is accordingly given by L of the diode. 21 I v Figure 1.3.1. Ideal characteristics of a double injection d~ode in the semiconductor regime (I~ V ) (see text far details). 140~~~~~~~~~~~~~~~~~~~~ V=30V 120 ~V= +I.OOV 100 THEORETICAL 80 ~I (t ), ,u.A 60 N N 40 20 0 -20 0 20 40 60 80 100 120 140 160 180 200 t, ,usee Figure 1.3.2. Current response of the diode of Figure 1.2.5 to a differential voltage step of + lV at an operating point of 30V, 1.6 ma. The solid line is a least squares fit of 6i(t) = 6i(O) + 6i(oo) [1-exp(-t/r)] to the data. 23 L r ::: j [Aq(IJ.nn(t=O) + IJ.p p( t:::o)) r L 1 dx = J 1 [Ao(t=o)r dx (1.3 .1) 0 0 Since, in general, the densities n holes vary with the distance the requirement that the current 6i(O) x, and p of the electrons and be free of divergence demands that the additional field generated by 6V will not be constant either. to adjust itself everywhere such that constant. 6E(x) It will, in fact, have 6E(x)•Ao(t=0) = 6i(O) is This is achieved by minute rearrangements of charge in the bulk which takes place within a time span of the order of the dielectric relaxation time e = E/o. Hence, 6E(x)•Ao(t=0) = 6i(O) and L J t-.E(x)dx = 6V. As long as diffusion contributes negligibly to the 0 de current, the de value E(x) of the electric field satisfies L E(x)•Ao(t=O) = I 0 , 1 r 6i(O) nv and I0 E(x)dx = V • It thus follows that 0 Io (1.3. 2) = v-o as indicated in Figure 1. 3 .1. The relationship between the initial step and the second increment follows from the law I ~ v2 and is 6V + V 0 M(oo) = --.,.,,..-Ai(O) (1. 3 . 3 ) v0 where V 0 is the initial bias voltage as shown in Figure 1. 3 .1. incremental step voltage then 6i(oo) ~ 6i(O). 6V If the is much smaller than the bias voltage In the limit as 6V v0 ' approaches zero (differential input), Eq.(l.3.3) reduces to the differential form di(ro) = di(O) If the de characteristic is of the form (l.3.4) I a:.~ , then similarly, di( ro) = (n-1) di(O) (l.3.5) By assuming the presence of a single linear recombination mechanism, this current transition will take place with a single time constant Tl which is equivalent to the previously defined high-level lifetime T (i.e • ,. l = ,. ) ; hence, 6I(t) (l. 3 . 6) The step response of single injection space charge current contrasts with that of double injection in that the characteristic time scale is the transit time (of the order of lifetime. L2 /~V) rather than a It is also noted that the above argument is independent of the geometrical configuration of the double injection structure. It follows thusly, that Eqs.(l. 3 .2) and (l.3.5) are also valid for cylindrical and spherical geometries as well. An experimental incremental pulse response of the double injection diode of Figure 1. 2 .5 biased at 30V and 1. 6ma is shown in Figure l.3. 2 (open circles). The excellent agreement between the theoretical current response (Eq.(l.3. 6 )) and the experimental data indicates that the assumption of negligible diffusion is valid. The differential step response not only provides a convenient means for verifying the 25 double injection semiconductor regime but also provides a direct measurement of the high-level lifetime ential resistance 1.4. T and high frequency differ- r. AC Properties of the Semiconductor Regime. 1.4.1. Small Signal AC Equivalent Circuit. To facilitate the understanding of the noise properties of double injection, it is advantageous to obtain the small signal ac equivalent circuit of the diode. Of course, the transient analysis described in section 1.3 is an equally valid means with which to establish a small signal equivalent circuit. However, since the noise measurement involves a frequency spectrum, a more equivalent approach is to establish the properties of the double injection diode as a function of frequency. The de equations of Section 1.2.1 are generalized to include time dependent quantities (i.e. E(t) and p(t)). With the same assumptions of high-level injection and negligible diffusion, the equations describing the semiconductor regime are given by (1.4.1) and - (p -n ) 'V•E = 0 0 (b+l) (1.4.2) 1-Ln The charge density, electric field and current density are now written in the form 26 + p le jmt (1.4.3a) E = E + Ele jmt (1.4.3b) = J 0 + J le jmt (1.4.3c) p =p 0 0 J where each of the variables is assumed to be a function of only one spatial coordinate u1 as in Section 1.2. By substituting these variables into Eqs.(l.4.1) and (1.4.2), the zero subscript values are easily seen to be the de solutions. neglecting higher order e jmt The ac equations are linearized by terms. The resulting differential equation for the electric field is (1.4.4) where cp = : : (hjmr) ~: jo:£~ 1 , + h (l.4.4a) ov ~ = ~ (l+jaYr) hlh2h3 (1.4.4b) 0 and v0 Ro =I 0 The ac impedance z1 describing the double injection diode is sUbse- 27 quently determined by solving for the electric field and integrating R2 this solution over u (i.e. V1 = +j[ E1 (u ) du1 ) yielding (see 1 1 Rl Appendix B) (1.4.5) For the planar case (h =h 2=h =l), 1 3 u' [ ~~ ] 3 8 = ·-2 R0 c0 where and u' and Eq.(l.4.5) takes the form (l+jm-r) exp[jW8(l+j~)(u-u')] du (1.4.6) is the geometrical capacitance. are dummy integration variables. evaluated for case (i) in which W8 << l Here u Equation (1.4.6) is now and case (ii) where W8 >> l. In case (i), the exponential term (Eq.(l.4.6)) is expanded in powers of W@. Retaining terms of the order of m18 and performing the integration gives Z ,..., R l - o ( l+ jw-r) [l (2+jm-r) - jw83 (1.4.7) --,:r- This expression for the ac impedance is synthesized into the equivalent circuit illustrated in Figure l.4.la and reduces to the circuit shown in Figure l.4.lb when the geometrical capacitance and terms involving 8/-r quencies the conductance is (m << 1/-r) are considered negligible. g = 2/R 0 C 0 At low fre- = oi/oV , or just the de differential conductance. Correspondingly, for high frequencies (m >> 1/-r, but m << l/8) the conductance is given by g = 1/R • 0 28 4 T2 9 8 Ro TRo a9 Co (I+ ~ ~) Ro 4 T 3 9 Ro Ro Figure 1.4.la. Small signal ac equivalent circuit representation of a double injection diode in the semic~nductor regime for frequencies less than l/(2rr®) where @ = - R C • 2 0 0 r=Ro Figure 1.4.lb. First order small signal ac equivalent circuit as derived from Figure 1.4.la when T >> e and the geometrical capacitance C is negligible. 0 29 These results are in agreement with the differential step response analysis of double injection given in Section 1.3. In fact, the current response of the first order ac equivalent circuit is just that of Eq.(l. 3 . 6) and the high frequency behavior is given by r = R0 (see Eq.(l.3.2)). In case (ii), the integral is expanded in powers of 1/w@. The first two terms of the admittance are given by (.l) ' where G 0 = l/R 0 • >> 1/8 (1.4.8) As R. Baron points out, the frequency is sufficiently high that (a) the space charge distribution cannot change thus, the capacitance becomes just the geometrical capacitance and (b) the conductance is given by the motion of the steady state carrier distribution in an excess electric field free of divergence. (l.l7) The first order equivalent circuit shown in Figure 1.4.lb is also valid for cylindrical double injection diodes. * However, the magnitude of the second order elements (components involving are affected by geometrical factors. conductance ratio 8 For frequencies in Figure 1.4.la) m >> 1/8, g/G 0 of a cylindrical double injection structure is a function of the ratio of the radii which contrasts the result g/G 0 = 1 * the for the frequency range 1/T . < m < 1/8. A detailed discussion of the cylindrical case is presented in Appendix B. 30 Extension to the spherical case requires numerical methods; however, it is speculated that the first order equivalent circuit of Figure 1.4.lb is valid for this case also. 1.4.2. Experimental verification of Results. Experimental data demonstrating the validity of the first order equivalent circuit shown in Figure 1.4.lb are illustrated by the current response data of Figure 1.3.2. In order to test the validity of this equivalent circuit independently of the differential response, the real and imaginary parts of the diode admittance have been measured from to lOOMHz 1.4.3. (T = 298°K). 90Hz These results are shown in Figures 1.4.2 and Here, the dependences (solid lines) predicted from the first order equivalent circuit are given using a least squares fit. The agreement is quite good for frequencies between 90Hz and approximately 30MHz. Re(Y) (dashed lines) Above 30MHz, the increasing conductance indicates that the approximation rue << 1 is no longer valid. * This is indeed the case since at the bias point of 30V, condition a@ =1 is satisfied at 33MHz . As a result, the conductance begins to increase towards the value of (4/3 ) G 0 greater than 3Q~z. 1.6ma, the for frequencies In the differential step response data of Figure 1 . 3.2, bandwidth limitations of the measuring equipment precluded the detection of this effect. In table 1.4.2, the values of r, ~l' and r 1 determined from the ac measurements are compared with those obtained from the differential step response at various operating points. * The discrepancies In the frequency range from 30MHz to lOOMHz, the errors in the conductance measurements which are obtained on a H.P. 250 RX bridge may be as large as 2o% . I I - R X 0 V, volts/ 1 30 /'1 o ~-"" x/ I I 25 /I x--x-X-x-X-X-x..x-,"" I Re (Y),40 ,umhos .__ 30 - 0 A 0 ~ 20 D I 0 QO-••J I I I. '11 I I I 6 II 0 n I 15 .II 0 I II otr"" 6 I I w ...... I I - a tl ~ 0 ~ OOK"" 10/ 2 v v vv v 100 103 10 4 10 5 106 107 108 f, Hz Figure 1.4.2. Real part of the diode admittance versus frequency. The solid lines are least-squares fits to the data. The fitting function is that derived from the equivalent circuit of Fig. 1.4.lb. The parameters obtained are given in Table 1.4.2. 10 1 -I m (Y), J.Lmhos w [\) 100 103 104 f, Hz Figure 1.4.3. Imaginary part of the diode admittance versus frequency. The solid lines are least squares fits to the data. The fitting function is that derived from the equivalent circuit of Fig. 1.4.lb after subtraction of aC = Im(Y)f .... CXl • 105 TABLE 1.4.2 Room temperature values of the parameters for the equivalent circuit of Figure 1.4.lb for the double injection diode of Figure 1.2.5. Operating point from step response from frequency response Re(Y) ' V(V) I(mA) r(kD) t 1 (H) r (kD) 1 r(kO) Im(Y) t 1 (Y) r (kO) 1 t 1 (Y) r (kO) 1 0.69 0.86 16.8 21.2 0.63 0.80 17.9 21.4 30.0 1.60 18.1 0.70 18.5 25.0 1.12 21.2 0.83 22.5 17.8 21.4 20.0 0.710 26.7 1.08 29.7 27.0 1.08 28.7 1.01 28.5 15.0 10.0 0.405 34.5 47.1 68 1.54 35.4 48.7 1.52 2.6 43.5 81.4 1.39 2.3 42.5 2.4 43.5 81.9 78 -- -- 73.5 -- -- -- -- 2.0 ! I ' 0.196 0.024 -- --- - - --- --- - -- -- -- lAl lAl nowhere exceed lo%. The large effective inductance (of the order of a henry) which increases as the operating point moves from the quadratic region towards the linear range is an especially interesting feature of double injection. f = l/( 2 rrT), lifetime T Unfortunately, the is less than l/2. all lie between 3~ Q, which is maximum at Above lOV the ratios sec. and 3~ sec. t /r 1 1 for the The simple first order equivalent circuit shown in Figure l.4.lb thus adequately represents the small signal behavior of a double injection diode in the semiconductor regime from low frequencies up to over three decades above the transition point ~ = l. When the ambient temperature of the device is monitor ed f rom l 40°K to 350°K by means of a thermostat (see Appendix F), the transient and ac measurements are extended to six different temperatures. The solid dots displayed in the I-V characteristics of Figure 1 .2 . 5 indicate the operating points at which these measurements are performed. 1.4. 3 summarizes the results. Table Here, along with the measured lifetimes, the high frequency resistances as obtained by ac bridge measurements * are compared to the values determined from the differential step response. Agreement between these values is quite good for temperatures above 220°K, however, at the lower ambient temperatures the difference is as large as 17%. This error at the lower temperatures is caused by an overshoot (~ lo%) in the step response at * (T ~ 220°K) t = 0+ . Each value presented in Table 1.4.3 represents an average of eight measurements taken between lMHz and 22MHz. 35 TABLE 1.4.3 Values of the high frequency resistance and high-level lifetime as a function of temnerature for the double injection diode of Figure 1.2.5. r(kO) from step response T( !-Lsec) from step response 44.9 40.1 39.9 37 . 3 39.5 Temperature v(v) I(ma) r(kO) from frequency response 350°K 40.0 35 .0 25 .0 10.0 2 .0 2 .10 1. 61 O. b6b 0.164 0.023 17 .4 19 . 6 26 . b 53 .9 77 . 6 298°K 30.0 25 .0 20.0 15 .0 10.0 2 .0 1. 60 1.12 0. 710 0.40) 0.196 0.024 17 . 8 21.4 27 .o 3).4 4b . 7 73 .5 17 .4 19. 6 27 .0 55. 3 77 .0 18 .1 21. 2 2o. 7 34.5 47 .1 6b .O 25.0 20.0 15.0 b .O 2 .0 1. 35 O. b45 0.4b0 0. 767 0.02ts 18 .5 22 . 7 30.b 4b .l 62 .4 18 . 5 23 .4 30. b 4b . 2 5b . 3 20.0 17 . 5 15 .0 6 .0 1. 34 1.02 0. 7 46 0.160 0 .037 13 .1 15.1 17 .9 31. 2 37 .4 13 .4 15 .4 l tl .4 26. 2 36 . ts 17 . 3 16 . 9 16 .7 1. 58 1.1) 7 .9 9.4 ll.O 11. 3 10.b 10.9 9. 2 Operating Point 273°K 220°K ?. . 0 l 7 5°K l40°K 17 . 5 15 .0 12 .5 10.0 5.0 . o. csoo .L.L.O 0.5 20 0.170 13 .1 l 7 .b 8. 3 10.1 1 2 .0 14. 6 20.13 10.0 ts .O 6 .0 4.0 0.7 60 0.510 0. 292 0.145 6 .9 7.9 9.2 10.2 7.4 b. 3 9.4 10.4 37 .8 35.9 36 .4 35.4 29.4 ---- 26 . 3 25.1 23 .5 ---- ---- ------- 8. 6 tr.o 6 .4 ---- 1.4. 3 . Nonlinear Effects. In deriving the small signal equivalent circuit for double injection, the nonlinear aspects of the diode are of course neglected. Moreover, the ac measurements used to verify the equivalent circuit are obtained with an impedance bridge which employs a frequency tuned detector, thus the nonlinear properties of the diode are also suppressed in the experimental case. In order to investigate the nonlinearity, consider the double injection diode to be biased at v = v1 cos mt voltage v0 ' I 0 and let the sinusoidal be applied across the diode. If the frequency is sufficiently low, the R.M.S. current will consist of a component at the applied frequency frequency (i.e. 2m). m and a component at twice the fundamental These components are given by 20~ n ~ p (p o -n o ) T V oVl (1.4.9) and (i 2 ( 2mt) / / 2 = {2 o~ ~ (p -n ) T -v 2 n p o o 1 where vl = vl//2. (i 2(mt) )1/2 2 (i (2mt)lf 2 (1.4.10) The ratio of these components, = {2v0 vl ' (1.4.11) is independent of any physical properties of the device (i.e. geometry, doping, mobility, etc.) and the derivation only requires that the current-voltage characteristic be quadratic. From Eq.(1.4.11), the 37 second harmonic com~onent of the current will be small when {2 vo;v1 >> 1. The output of a signal generator operating at a frequency of 100Hz is applied to the double injection diode of Figure 1.2.5 which is biased at V 0 = 30v and I 0 = 1.6ma (T = 298°K). In Figure 1 . 4.4, the measured R.M.S. current at the fundamental and the second harmonic frequencies is given as a function of the voltage -v 2 1 respectively. and Here, the current component at the generator frequency (fundamental) is shown to be directly proportional to the R.M.S. value of the applied signal, whereas the second harmonic is proportional to the square of the applied R.M.S. signal. When the period of the applied ac signal is comparable or greater than the high-level lifetime decrease. T, the magnitude of the second harmonic should The experimental data presented in Figure 1.4.5 demonstrate this attenuation of second harmonic generation of the diode as a function of frequency. Even though the nonlinearity under practical operating conditions is quite small (see Figure 1.4.4) it loses these properties altogether at sufficiently high frequencies. As a conse- quence, the small signal ac equivalent circuit of Figure 1.4.lb is also valid for large signals when the frequency is greater than 1/T as expected from the discussion of the transient response (Section 1.3). 1.5. High Frequency Noise in Double Injection Silicon Diodes. 1.5.1. Generalities. In performing the noise measurements, the unknown noise from the double injection diode is compared to that of a standard calibrator shot noise source. Schottky's theorem states that the shot noise developed across a temperature limited vacuum diode far v~ (volts) 4 0 5 6 7 8 9 20 400 <{ <{ ::!... ::!... I- I- z w 0::: 0::: w z 0::: 0::: 300 15 ~ u (f) SECOND HARMONIC (right upper scales) 2 a (f) 2 ~ u 0::: u 0::: z _J 0 <{ 2 I- z 200 w 10 0::: <{ 2 I 0 0 ~ 0 <{ z z LL u FUNDAMENTAL (left bottom scales) w a 100 (f) 5 oL-~~~~~~-J--~~~--~~~~~~~~--~~o 0 1.0 2.0 3.0 vI ( vo Its) Figure 1.4.4. Fundamental and second harmonic ac current response of the double injection diode given in Figure 1.2.5. The data are corrected to account for the small amount of second harmonic content in the applied signal (H.P. 200CD genera tor) • --... 0~ ......... 100 .. 1z 90 w cr: 0:: 80 ~ 0 0 z 0 ~ 70 60 50 0:: ~ 401 0 30 0 0 20 (f) 10 z w I ' 0 10-l 10° 10 1 10 2 f (kHz) Figure 1.4.5. Percentage of second harmonic current as a function of frequency. 10 3 w \0 40 frequencies less than TT (TT is the carrier transit time) is given by 2q I where I c c ~f (1.5.1) , is the de operating current.(l.lS) The experimental tech- nique of comparing an unknown noise source to a known noise source provides an accurate means with which to obtain the noise spectral density of a two terminal device. * Motivated by experimental evidence, a thermal noise hypothesis is introduced in discussing the high frequency fluctuation phenomena occurring in double injection. Recently, it has been deduced that the thermal noise in a passive resistance R(f) at the ambient temperature 2 (v ) = :JfR(f)[hf/2 + hf/(exp(hf/kT-1))] df (1.5.2) T is represented by where (v2 ) 1 / 2 is the open circuit noise voltage fluctuation.(l.l9) When quantum mechanical effects are negligible (i.e. hf << kT) Eq.(l.5.2) reduces to Nyquist's result (1.5.3) * A detailed description of the experimental arrangement, procedure, and calibration of the entire noise measurement system is contained in Appendix C through G. 41 where the frequency interval resistance R(f) ~f is sufficiently small that the is constant over this bandwidth. "Thermal noise" is adopted here to refer also to quasithermal noise of quasilinear systems in which (v2 ) = 4kT r ~f, where r is a linear element characterizing a (nonlinear) system which is not in true thermal equilibrium. 1.5.2. Model of High Frequency Noise in Double Injection. A model of the diode noise is now developed from the physical properties of the device. t -r 1 1 In the equivalent circuit shown in Figure l.4.lb, the branch has a time constant carriers (i.e. T1 = T). Tl equal to the lifetime of the This means, physically, that the recombina- tion ~rocesses, which play a dominant role in establishing the static characteristic, are too slow to affect the current at frequencies m << 1/T. Thus, at high frequencies, the fluctuations are not affected by the recombination processes either. The total high frequency noise voltage developed across the double injection diode arises from fluctuations associated with the junctions (p+ -rr, rr-n+ ) and the long high resistivity region. * Van der Ziel has shown· that for a p-n junction with the I-V characteris tic (1.5.4) * Any noise associated with surface phenomena is considered negligible at high frequencies. 42 the noise voltage developed across the junction under forward bias . approx1ma . t ,,.(1 . 20) q 1s (V >> kT I) e~ (1.5.5) If the long high resistivity section of the double injection diode is assumed to exhibit only thermal noise at high frequencies, then one may further assume that the noise voltage developed across this region is of the order of (1.5.6) where V /I 0 is the de resistance of the diode. 0 From Eqs.(l.5.5) and (1.5. 6), the noise voltage developed across the injecting contacts will be negligible when the bias voltage kT/q (i.e • V >> 25mv at 0 v0 is much greater than From Figure 1.2.5, this condition is fulfilled for all cases in which the diode is operating in the semiconductor regime (V 0 >lOv). When the diode is forward biased into the quadratic range, it is not in strict thermal equilibrium. If, regardless of conduction, (i) there is a stationary distribution of thermal velocities and (ii) this distribution involves only a small perturbation of the distribution from a Maxwellian, the diode can be considered to be in a state of quasithermal equilibrium. carriers (~ 107 cm/s) drift velocity Since the thermal velocity of the charge is, in the present case, much greater than the (~ 7 x 104 cm/s), this picture of quasithermal equilibrium appears quite justifiable for double injection. With this "thermal hypothis is ", the high frequency noise current of the double injection then becomes simply 4kT g 6f where g (1.5.7) is the conductance of the diode at high frequencies. terms of an equivalent saturated shot noise diode current In (Ieq)' the noise spectral density is given by (see also Appendix C) I 2kT eq q (1.5 .8) g • Using the high frequency conductance of the diode derived in Section 1.4, the equivalent noise current of double injection is thus given by Eq.(l.5.8) where I g 0 v0 ' 1/T < ill < 1/8 (1.5.8a) lj@ < ill < 1/T c (1.5.8b) and g (T c is the mean time between thermal scatterings). A different approach to the explanation of the high frequency noise is developed from the transient response. step and t:,i(O) p Here, the current is determined by the electron and hole concentrations immediately before the voltage step 6v is applied. n Electrons 44 and holes contribute independently to the flow at that instant and without changing their concentration. Van der Ziel and van Vliet proved recently that carriers of a single type injected into a volume element of a device will generate thermal noise. If, however, recom- bination is the only mechanism coupling the holes to the electrons and becomes ineffective at ro >> 1/T, van der Ziel and van Vliet's results applies to both electrons and holes independently. One must then expect 4kT[A cr (ro)/L+ A cr (ro)/L] 6f n p = 4kT g 6f (1.5.9) (1.5.10) 1/T < ro < 1/8 This result is in agreement with Eq.(l.5.8). Figure 1.5.1 summarizes the noise spectral density of a planar double injection diode based on the thermal hypothesis for frequencies greater than 1/T I eq 2kT Io ---q v 0 1/® freq. 1 T Figure 1.5.1. Equivalent noise current versus frequency for a planar double injection diode. c Since the first order equivalent circuit given in Figure 1.4.lb is independent of diode geometry, the equivalent noise current for the cylindrical and spherical geometries in the frequency range 1/T <ill< 1/8 is also given by Eqs . (l.5.8) and (1.5.8a) (see Appendix B). However, for the frequency range 1/8 < ill < 1/Tc , the value of the diode conductance and thus the equivalent noise current will, in general, depend upon geometrical terms. * 1.5.3. Measured Spectral Noise Density. The measured de charac- teristics and ac properties of the diode described in Figure 1.2.5 have been shown in Sections 1.2, 1.3 and 1.4 to closely conform to the predictions of the model for double injection. injection silicon diode provides a sound basis with which to test the thermal noise theory presented in Section 1.5.2. To compare this theoretical model with the actual experimental noise of the device the measurements should cover as wide a frequency range as possible. This not only extends the amount of information but also helps minimize the possibility of systematic errors. From the conductance measurements illustrated in Figure 1.4.2, the ideal frequency range over which the spectral density should be measured extends from 500Hz to over 50 MHz. It is quite difficult to perform noise measurements over five decades of frequency and still optimize * The functional dependence of the conductance g on the geometrical factors for the cylindrical double injection configuration is given in Appendix B. 46 the apparatus (impedance, system noise, etc.) to yield the maximum amount of information about the noise of the device. with the measurement Nevertheless, system described in Appendix C through G, the equivalent noise current of the device can be obtained from 10kHz to 22MHz. Provision in the experimental arrangement allows variation of the ambient temperature of the device and thus extends the noise measurements over a factor of two in absolute temperature. The solid dots displayed in the I-V characteristics of Figure 1.2.5 indicate the operating points at which the noise measurements (and conductance) are performed. There is at least one point at every temperature where the diode is operating in the quadratic range. Figures 1.5.2a, b, c, d, e, f show the spectra in terms of an equivalent noise current for the temperatures 350, 298, 273, 220, 175 and 140°K. It is seen that at all operating points and for all temperatures the noise spectra reach a constant level. The solid lines have been obtained by the least squares fitting of an assumed dependence I eq = c1 + c2 [/~ to the data as described in Appendix G. (1.5.11) At 140°K, the low frequency noise is attenuated for the lOv and 8v spectra to prevent amplifier saturation. This is accomplished by the insertion, at the amplifier input, of an L-C network which is tunable from 5MHz to 22MHz (see Appendix D). V (VOLTS) ., t-i 1o 1 10 3 f (KHz) Figure l.5. 2a. Equivalent noise current (I ) versus frequency at five operating points foreq the double injection silicon diode given in Figure 1.2.5. (T = 350°K}. 48 V (VOLTS) 10 2 T= 2 98 (°K) <{ :1.. .... H 10' 10 3 f (KHz) Figure 1.5.2b. Equivalent noise current versus frequency at six operating points the double injection silicon diode given in Figure 1.2.5. (T = 298°K). V (VOLTS) D ~~~n_a__~~~-o--~:--~~c-~~~~--~~~L-- f (KHz) Figure 1.5.2c. Equivalent noise current (Ieq ) versus frequency at five operating points for the double injection silicon diode given in Figure 1.2.5 (T = 273°K). 50 t0 2 T=220(°K) <( :1. ... CT H to 1 f (KHz) Figure 1.5. 2d. Equivalent noise current (I ) versus frequency at six operating points foreq the double injection silicon diode given Figure 1.2.5 (T = 220°K). 51 .. H 0' f (KHz) Figure 1.5.2e. Equivalent noise current (Ieq ) versus frequency at five operating points for the double injection silicon diode given in Figure 1.2.5 (T = 175°K). 52 V (VOLTS) 0" &I H f (KHz) Figure 1.5.2f. Equivalent noise current (I ) versus frequency at four operating points foreq the double injection silicon diode given in Figure 1.2.5 (T = 140°K). 53 1.6. Evaluation of Results and Conclusion. c1 From the constant of Eq.(l.5.ll), the measured equivalent noise resistance of the diode is given by r 2kT eq (l.6.l) qCl Table 1.6.1 compares the values of r eq determined from the noise measurement with the high frequency resistance r of the diode for each operating point and over a temperature range from l40°K to 350°K. Here, the high frequency resistance r is measured throughout the frequency range in which the equivalent noise current is constant and corresponds to the values "from the frequency response" in Table 1.4.3. The very close agreement between r eq and the real part r of the diode impedance at high frequencies indicates that (1. 6 . 2 ) g = 1/r. with a close to unity and for a is obtained from a plot of the experimental values of A better estimate of the value I eq versus g as a function of ambient temperature as shown in Figure 1.6.1. Also shown there (dashed lines) are the dependencies predicted from the equation I 2kT eq q g • The ratios of the experimental values of I eq (1.6. 3 ) to this predicted value 54 TABIE l.6.l Comparison of the real part r of the diode impedance with the equivalent noise resistance r at high frequencies. eq Operating Point r V(v) I(ma) (kO) 350°K 4o.o 35 .0 25 .0 10.0 2 .0 298°K 30 .0 25 .0 20 .0 15.0 10.0 2 .0 2 .10 1.61 o.e6e 0.164 0.023 1.60 1.12 0.710 0.405 0.196 0.024 273°K 25.0 20 .0 15.0 e.o 2 .0 1.35 0.845 o.4eo 0.767 0.028 17.4 19.6 26 .e 53 . 9 77.6 17.8 21.4 27.0 35.4 4e.7 73.5 18.5 22 .7 30.e 48.1 62.4 220°K 20 .0 17.5 15.0 6 .0 2 .0 l75°K 17.5 15.0 12.5 10.0 5.0 10.0 e.o 6.0 4.0 1.34 1.02 0.746 0.160 0.037 1.58 1.15 0.800 .0. 520 0.170 7.4 9.4 ll.O 13 .l 17.8 10.9 13.3 18.1 0.760 0.510 0.292 0.145 6.9 7.9 9.2 10.2 6.9 7·9 9.3 10.3 Temperature l40°K 13 .l 15.1 17.9 31.2 37.4 r eq (kO) 16 .6 19.9 27 .7 51.6 74.3 18.0 21.6 26 .1 32 .6 45.1 70.5 18.3 23.5 30 .4 4-7 .T; 59.o 13 .1 15.1 19.1 30.4 38 .0 8.2 8.9 I I I I I I T (o I I I o I )' I / ' I //I I / I K) = 3501 1 / / I 298 1 / // /"I / / / 273 220 1 ~ I 3 .0 - / I 0 1/ / / <( 1('/1'0 / :L 2.0 .........., I I/ I I I // ;o./rf / 0" Q) ~ I // :r// /r'. I / 4/ / / / /-o // ;;; // 'j/ 175 /I ,// ,../ / ,/ // ? /}f ,. . . . I I I ...'(' :r/ // / I il JY I 1 T~~/;;;r-11---rl---r1--,// J //1-' I I ,....... ,... .... " /' // )';r' 9/ ,, 140 .... , ,J:J/ /' / \.11 \.11 /' I I I r" / // / ............ / ,.....____Th eoreticol 7b~~ a_~fl /rf//"............ <9';J-//// //' /1/ / / / //' 1#///'/ 1,;/:-"/ /' ~~ 100 110 120 130 140 150 Figure 1.6.1. Equivalent noise current (I ) versus the high frequency conductance of the silicon double injectionegiode at the operating points indicated in Figure 1.2.5. 56 of 2kT g (i.e. q a) have been evaluated by least squares fits as l.oo, l.oo, 1.01, 1.01, l.o4, and 1.03 ± 0.05 for 140, 175, 220, 273, 298 and 350°K respectively. This established that a =l to within 5% and thus the high frequency noise is given by 4kT g b.f 0 0 where l 40 K ~ T ~ 350 K and 1/T < (1) < l/8 g (1.6.4) is the high frequency conductance of the device at the lattice temperature T. The quoted error of 5% is attributable to experimental causes, as indicated by the errors of individual values of r eq •* It is also noted that for a given tempera- ture the thermal noise expression (Eq.(l.6.4)) is valid for every operating point, even where the square law does not hold. agrees with the idea of thermal noise. This again The overall conclusion is therefore drawn that Eq.(l.6.4) describes the noise of double injection accurately as long as recombination effects and diffusion are negligible. The thermal noise current source (i 2 ) ; 4kT g b.f in the first order equivalent circuit shown in Figure 1.6.2 expresses these ideas formally. At lower frequencies, the equivalent noise currents as illustrated in the spectra of Figures 1.5.2, etc., have a "l/f" frequency dependence. Prompted by these results, Bilger, et al, have investigated the low * The error associated with the determination of c at each of the 1 six ambient temperatures is an R.M.S. deviation of the fitted curve from the data (see Appendix G) and is generally less than 1%. Since the error in the lattice temperature of the diode is of the order of 3%, the stated error of 5% is a conservative estimate. -------., '\ C')( <>I L2 C>\, ..e, I c...( I ) c _ _i_ -T- r (i2) r +------, I 1 l I I I I I <"' r2< ;> r, ' ) <"' 1 I I • • • Figure 1. 6. 2. Equivalent circuit containing internal noise sources of a double injection diode operating in the semiconductor regime. --...:] .J.. ~·'22)'<_'l ! I ( ' I ' I / T I _ _ _ _ _ JI • _ _ _ _ _ ., I \J1 I ') J \ 58 p+ v n + frequency noise of a 1 22 structure down to a few hertz. ( • ) The spectra reveal qualitatively that the noise can be explained for frequencies effects. m << 1/T in terms of generation-recombination Therefore, in the equivalent circuit of Figure 1.6.2, this . t'~ng e ff ec t can b e f orma lly expresse d b y assoc~a with noise. * g-r current (g-r) I eq, g-r Figure 1.6.3 shows the measured at injection diode current I. ** 100kHz g-r (;12) • and (;22) • equivalent noise as a function of the double It is noted, however, that the noise measurements presented in Figure 1.6.3 are not obtained with currentless probe contacts which eliminate unwanted contact noise; therefore, the absolute magnitude of the g-r 1.7. The conclusion that at Discussion and outlook. 1/T < m < l/8 noise may be in error. 2 (i ) = 4kT g 6f necessitates the rejection of alternative theories 1 2 advanced to explain high frequency noise in double injection.C • 3, 1.24,1.25) According to van der Ziel * A careful experimental analysis revealed traces of a second time constant T of about l/4 Tl' but the effect was judged to be 2 sufficiently small to be neglected. ** The low-frequency double injection current noise theory of A. 2 FazakaS and A. Friedman predicts ( l. 23 ) I ex I • Reference 3 /2 eq,g-r 1.22, however, predicts I ex I for diodes operating in eq, g -r the semiconductor regime. 59 FREQ. 100kHz lo- 3 ~----~--~~~~~~~----~--~--~~~~~ 10-1 10° I, mA Figure l.6. 3 . Observed g-r noise at 100kHz versus diode current at four operating points (T = 298°K). 10 1 60 where (l.7.la) a= From the measured temperature dependencies of the electron mobility and the hole mobility 2 1 " ex T- • 8) ,....p ( ,....p 11 of the silicon double injection diode presented in Chapter II, the representation a = 4~-.~.p~-.~.n/(IJ.p+IJ.n) 2 must be rejected because it is inconsistent with the result a= l.OO ±.05 throughout the temperature range from l40°K to 3500 K. Other reported noise measurements, although difficult to cast into the light of an ac equivalent circuit owing to the lack of experimental data, are in agreement with the thermal noise model as expressed by Eq.(l.6.4.)*(l.l,l. 2, 1 · 3 ) results of J. H. Liao on a germanium However, the recent experimental p+ v n + structure seem to indi- cate that van der Ziel's result (Eqs.(l.7.l), (l.7.la)) is correct at 1 T = 298°K.C • 25 ) Unfortunately, insufficient data on the de charac- teristics and high frequency properties of the device preclude the verification of semiconductor regime double injection. ** * Since the correct relationship between the low frequency conductance G and the high frequency conductance g is g = ~ G , the term "Roise suppression" of Ref. l.3 is not appropriate in aouble injection as long as it refers to thermal noise. ** The germanium diode reported on by Liao exhibits an I cxv 2 from approximately O.lv to 0.5v. The observed noise, therefore, may be largely influenced by diffusion and contact effects. 61 There is no physical reason why the noise should not be thermal for frequencies above l/8 since this characteristic time pertains to the macroscopic phenomenon of dielectric relaxation. One thus expects that Eq.(l.6.4) is valid at least up to frequencies of the order of l/T • c Moreover, there is no physical reason why the noise should not be thermal for temperatures above 350°K and below l40°K as long as the double injection diode can be characterized by the Lampert model. Therefore, when recombination and diffusion effects are negligible, it is predicted that the thermal noise model for double injection given by (i 2 ) = 4kT g ~f holds for any double injection device at any temperature and for all operating points over the frequency range Extension to various diode geometries is accomplished through the relationship between the geometrical factors (i.e. frequency conductance of the diode. l/T < ru < l/8, o) and the high Thus, over the frequency range the equivalent noise current for the cylindrical and spherical geometries as a function of the radii ratio is given by Figure 1.2. 3 to within the normalized scale factor. range l/8 < ru < l/T c , For the frequency the conductance and thus the noise of the cylindrical configuration is determined by ge ometrical terms as discussed in Appendix B, whereas in the spherical case, the conductance must be resolved by numerical methods. At frequencies below l/T, the noise can be explained in terms of generation-recombination effects . (l. 22 ) Therefore, the conclusion is drawn that the observed low frequency equivalent noise current is due primarily to g-r noise. It is also noted from Figures 1.5.2 a-f 62 that the frequency at which the g-r noise and thermal noise levels are equal increases with decreasing temperature. A more qualitative investigation of the low frequency noise in double injection is presently being pursued to establish the form of I eq,g-r With the high frequency noise in double injection reduced to a thermal causes, an interesting question now arises as to when ceases to be unity. Since times of the order of the mean free time between collisions is needed to "the rmalize" the carrier motion, one would expect therefore noise which is quite different from Nyquist noise at frequencies w ,..., 1/'r • c c is usually in the Unfortunately, microwave range and complicates the experimental situation. violation of quasithermal for a f. l. The equilibrium provides another possibility Therefore, double injection structures operating under large current densities may offer unique opportunities to study fluetuations of tepid and hot charge carriers in a solid. As discussed in Section 1.5.2 and illustrated in the experimental data of Figures 1.5. 2 a-f, the high frequency noise associated with the + P -rr, rr-n+ contacts is indeed negligible for the long silicon double injection diode of Figure 1. 2 .5. &van der Ziel has demonstrated both theoretic- ally and experimentally that the noise developed in a p-n junction is shot like noise due to the diffusing of carriers. gressively shortening the physical length L Therefore, by pro- of a double injection diode, a transition in the spectral noise density from thermal noise to semiconductor shot noise is expected. Presumably the transition range can be expressed in terms of the overall physical length the device and the ambipolar diffusion length L of La • An investigation of 63 this nature may yield valuable information in estimating the influence of diffusion on thermal noise. 64 CHAPI'ER II TEMPERATURE DEPENDENCE OF THE COMMON HIGH-LEVEL LIFETJME AND CONDUCTIVITY MOBILITY OF CARRIERS IN SILICON FROM DOUBLE INJECTION 2.l. Introduction. The purpose of this chapter is, in part, to explain the temperature dependence of the I-V characteristic from l40°K to 350°K for a double injection silicon diode operating in the semiconductor regime. In Lampert's model for two-carrier injection into a semiconductor, the magnitude of the current is given by I= o ~n~pT(p -n )v2 • 0 0 Therefore, the explanation presented in Section 2.2 of the measured I-V characteristics involves investigating the temperature variation of the charge carrier lifetime and mobility. Section 2.3 thus gives the measured temperature dependence of the common high-level lifetime as determined by differential step response techniques. In Section 2.4, the temperature dependence of the conductivity mobility majority carrier is determined. * and ~p ~n (for ~p) (for ~n ) • In addition, ~ p of the the absolute values of are measured at room temperature with the Hall-Effect and with the large signal transient method of R. H. Dean (2 .l) These measurements in conjunction with the I-V charac- teristic provide a means with which to determine the temperature dependence of the minority carrier mobility ~n from l40°K to 350°K. The excellent agreement with other independent values found in the * No distinction is made between conductivity and drift mobility since they are considered equivalent throughout this discussion. 65 literature for the variation of the electron mobility with temperature provides further support for the model of double injection and validates the approximations of the Lampert representation.( 2 • 2 ) This shows that trap-free double injection current can be used to study the common high-level carrier lifetime and conductivity mobility in high-resistivity materials. 2.2. Temperature Dependence of the Current-Voltage Characteristic for Double Injection. In Figure l. 2 .5, the measured I-V characteristic for the silicon double injection diode C-l27 3, 3, 2.6. B is given at six ambient temperatures from l40°K to 350°K. The long high-resistivity rr region is boron doped with an impurity concentration of approximately l.l x lol2 cm-3 Therefore, with the acceptor level (0.045ev) fully ionized over the considered temperature range, the number of thermal equilibrium majority carriers acceptor impurities (p ) 0 Na. is constant and equal to the number of From the discussion of Section l.2.3, the I-V characteristic of the diode is properly represented by v2 I = A[9/8]q ~ ~ TN -p n a L3 Over the range f'rom l40°K to 350°K, the temperature variation of and ~n (2.2.l) T, can (approxi.nia.tely) be described by the power-law dependences T (2.2.2) (o) J.lp = J.lp [~orp (o) [Tarn T J.ln = J.ln 66 (2.2.3) ' (2.2.4) ' (T ) where the room temperature 0 values are designated by the zero and a superscript and the terms n are constants.< 2 • 2, 2 ·3) Such dependences seem characteristic of drift mobilities dominated by 2 4 lattice scattering. ( • ) Rewriting Eq.(2.2.l) in terms of the applied current and substituting the expressions (2.2.2) to (2.2.4) gives (2.2.5) Figure 2.2.1 gives the values of the square of the voltage from l40°K to 350°K for currents between 2ma and o.6ma. The measurements are performed under constant current conditions to eliminate the deviation from the I ~ v2 law at l40°K for current levels greater than 2ma. The average of these four slopes is 2.0; therefore, = 2.0 2.3. (2.2.6) Common High-Level Lifetime Versus Temperature in Si. The measured common high-level lifetimes tabulated in Section 1.4 0 0 4 3) for temperatures from l 40 K to 350 K are obtained by the (Table l •• differential step response method described in Section 1.3. These measurements indicate that when the operating points of the diode are I ( ma) 2.0 1.5 1.0 - - C\J (/) C\J > Figure 2.2.1. Square of the de voltage from 140°K to 350°K for the double injection constant current levels 2.0, 1.5, 1.0 and o.6ma. 68 in the square law range, the common high-level lifetime for all temperatures considered does not vary appreciably with injection level. This proves that it is indeed valid to introduce a common high-level lifetime which is independent of the magnitude and spatial distribution of the charge carriers as is done in the high-level approximations of Section 1.2. In Figure 2.3.1, the measured lifetime values are given as a function of temperature. It is seen that Eq.(2.2.2) properly represents the temperature dependence of the common high-level lifetime. Thus, equals y 1.93 and the room temperature value ,. (o) is equal to 30.7~ sec, so that (2.3.1) The nearly square law behavior of the lifetime versus temperature is in excellent agreement with previous measurements by D. M. Evans.( 2 .3) 2 .4. Majority and Minority Carrier Mobility Versus Temperature in High Resistivity Silicon. From Section 2 . 2 and 2.3 the temperature dependence of the product of the mobilities ap + an = 3. 93. ~p~n The procedure now is to determine individually the room t emperat ure magn1•t udes dependences ~(o) ~(o) [Tolap+an where p n TJ is given by (ap' an) ( ~p(o) , (o)) ~n and the temperature of the carrier conductivity mobilities. In the ohmic range (low level injection, Eq.(l.2.18)), a I = ~~o) [~a] p [A (2.4.1) I 100 200 400 500 700 1000 Figure 2 . 3 .l. Measured commgn high-level lifetime of charge carriers in Si from 140 K to 350°K. 70 Equation (2 .4.1) thus provides a means with which to determine and (o) J..l.n ap Unfortunately, the ohmic regime, as illustrated in • Figure 1.2.5, is not well defined throughout the complete temperature range (l40°K- 350°K). This is due to the onset of the semiconductor regime and junction effects . Figure 2.4 .1 shows potential probe measurements (open dots) with the double injection diode biased at 0. 3V. These measurements are made by placing the diode in a fixture containing a tungsten probe which is mounted in a micromanipulator and traversing along a lapped (3200 mesh alumina) side of the structure. The potential does not decrease uniformly across the entire diode but is greatly influenced by the rc-n+ junction. Similar probe measure- ments with the diode biased into the semiconductor regime are also shown in Figure 2 .4.1 (solid dots). These measurements are in accord- 2 ance with potential distributions obtained by J. w. Mayer et al.< • 5 ) The difficulty encountered with the rc-n+ junction in the ohmic range precludes the use of the double injection diode in establishing the majority carrier mobility. * a p+ rc p+ This problem is circumvented by fabricating structure ( semiconductor resistor) from the same high- resistivity rc type silicon. Here, the p+ contacts are made by evaporating aluminum onto chemically etched end surfaces (3.17 x 3 .14mm) and heating to the alloy temperature. * Potential probe measurements The resistivity and thus the majority carrier mobility (J..J.(o)) can be determined from the potential probe measurements of thepdouble injection diode given in Figure 2.4.1 (0. 3V) . However, similar probe measurements proved to be intractable at low temperatures (T < 29(3°K) . SAMPLE C-1273, 3, 2.68 L=5.92mm r=30.7f-Lsec • 30V APPLIED J= 1.65 x I(J" 2 A/cm2 o .3V APPLIED J= 2 .04 xlo-5 A/cm2 > 20 _j . <! 1-- z • w b Q_ •• w OJ 0 0:: 0.5 0 0 • • • • Q_ ••• • Figure 2 .4.1. Potential distribution along a p+ rr n+ diode forward bias ed into the ohmic regime, open data points, and into the semiconductor regime, solid points. Insert illustrates diode orientation. SAMPLE C-1273, 3, 2.6 AM L= 4 .8 mm A= 9 .8 (mm)2 J= 4 .26 x 10-5 A/cm2 0 o .3V APPLIED > <i 2.0 1-- zw b Q_ w OJ 0 0:: Q_ 0 .5 Figure 2 .4. 2 . Potential distribution along a p+ rr n+ structure. Insert illustrates the orientation of the experimental arrangement. 72 shown in Figure 2 .4. 2 illustrate the nearly ohmic behavior of the p+ rr p+ structure. The resistivity of the determined by the ohmic device is ture of the + p rr p + rr type silicon as 14. 2 kO-cm. By varying the tempera- structure under constant voltage bias, the current according to Eq.( 2 .4.1) will exhibit the temperature dependence (~p). of the majority carrier conductivity mobility Figure 2.4.3 shows the measured current of this semiconductor resistor over a 0 0 temperature range from 1 40 K to 350 K. equal to 2.18 (i.e. ap = 2 .18). The slope of this curve is Since a p +an = 3.93, the tempera- ture dependence of the electron mobility (i.e. an= 1.75). (~n) is equal to 1.75 These values are in good agreement with measurements reported in the literature which give 2 •3 ~ a ~ p 2 •7 1. 5 ~ a and n ~ 2.6. (2.2) The absolute values of the conductivity mobilities ~~o) are now considered. ( o) ~p and R. H. Dean has shown that when a step voltage from zero bias to a point in the semiconductor regime is applied to a double injection diode, the derivative of the resulting current response contains a "cusp". The time t a occurs is related to the minority carrier mobility (o) ~n at which this cusp ~(o) by n (2.4.2) = which reduces to (o) iln = 5 L 2 6 ta v (2.4.3) 73 <( ::L 10 I H 2 3 4 1000/T (oK- 5 6 1 ) Figure 2 .4.3. Measured current of thS p+ 1C p+ (C-l273, 3.2.6 AM) from l40°K to 350 K. structure for p 0 >> n • 0 A large signal step voltage is therefore applied across the double injection silicon diode and the derivative of the current response is measured. The result of such a measurement is shown in Figure 2 .4.4. The cusp occurs at 3 . 2 5~ sec. From Eq.(2.4.3), the minority carrier mobility is calculated to be ~(o) = 1 280 ~ n v-s 2 which is within 6% of 2 the average value obtained by Ludwig and Watters The majority carrier mobility ( o) ~p (~~o) = 1350 ~s). ( 2 • 6 ) is now calculated from the measured resistivity and the doping concentration quoted by the manufacturer (l.l x 1012 cm-3) 2 to be 2 ~(o) = 405 em p v:s . ( .7) As an alternative check, a Hall mobility measurement is performed on the structure using a 3900 gauss magnetic field. at a de current of 3~, With the device operating the measured Hall mobility is 2 em v-s (o) ~pHall = 348 Using the value for the ratio of the Hall hole mobility to the drift mobility of 0.84 as determined by J. Messier and M. Flores in 2 8k~cm silicon, the calculated hole mobility is (o) = 415 ~ ~P v-s (2.8) This value is in good agreement with the hole mobility obtained from the resistivity and doping level. Therefore, the average hole 2 mobility of 410 ~ v-s is taken as the absolute roam temperature (T = 298°K) value. In summary, the high-level lifetime and majority and minority conductivity mobilities in high-resistivity silicon as measured from double injection are T = I. 6 T ] 1.93 sec, (30.7 X 10- ) L298 (2.4.4) -u 7 Q) (/) ~6 0 -z E ~ 5 w 0:: 4 0:: ::> u ~ 31 I> I w > 2 ~ :::> 0:: w 0 o--~--~~~~~~~~~~~~~~~~~~~~~~ 0 5 10 15 t (fLsec) Figure 2.4.4. Derivative of the current response for an applied step voltage (70v) to the double injection silicon diode of Figure 1. 2.5. A passive RC network is used to perform the differentiation. 20 ~ \Jl 76 t298] 2 .18 IJ.p = 410 - T 2 em v-s t298Jl.75 2 em v-s IJ.n = 1280 T 0 0 where 1 40 K ~ T ~ 350 K. (2.4.6) An estimate of the error in all determined ( T' 11,....p' 11,....n' y' a p' an ) values (2.4.5) ' is ~ ~- uyG Figure 2 .4.5. illustrates the electron and hole conductivity mobility given by Eqs.( 2 .4.5) and ( 2 .4. 6 ). The excellent agreement between these measured quantities and the values quoted in the literature indicate that the temperature dependence of the I-V characteristic from 140°K to 350°K of double injection semiconductor regime diodes is determined entirely by the temperature variation of the common high-level lifetime and the majority and minority carrier conductivity mobilities. 2 . 5. Conclusion. It is established that from 140°K to 350°K the dependence of the I-V characteristic of a silicon double injection diode on the lattice temperature is consistent with Eq.( 2 . 2 .1) and the variation of IJ.n(T) and T(T) quoted in the literature. IJ.P(T) , Thus, the validity of Lampert's representation for two-carrier injection in high resistivity p-type silicon is demonstrated throughout the temperature range from 140°K to 350°K. The excellent agreement found between theory and experiment suggests that double injection, once its presence has been established, offers access to the values of IJ.p (T), by relatively simple de and pulse measurements. 1J. n (T) and T(T) To assure the presence of trap-free double injection current may, however, not be an easy 77 104 300 500 100 200 I / v I ELECTRON MOBILITY~ (SLOPE= 1.75) --Ejf U> -..__,.- , 1- l/ v v v I/ (\J >- 103 v _j I CD 0 I I :1 ~ I /._ v' '-HOLE MOBILITY (SLOPE = 2.18) 3 4 I 2 5 1 6 7 8 9 10 1000/T (°K- ) Figure 2.4.5 Electron and hole conductivity mobility from l40°K to 3506K in high-resistivity silicon (> 3~cm) as determined from double injection. task, in general. Nevertheless, from the results presented herein, the properties of two carrier injection may possibly be used in the future as a primary tool for investigating the transport behavior of carriers in high-resistivity materials. 79 APPENDIX A SOLUTION FOR GEOMETRICAL FACTOR Equation (l.2.20) is written as = 0 (A.l) which is easily integrated once resulting in where Cl is the integration constant. 2 w(ul) = E (ul)' With the substitution of Eq.(A.2) is reduced to dw 2w dul+ h h- (A.3) = 2 3 Solving (A.3) for w(ul)' and subsequently the electric field, results in (A.4) where (A.5) Since as J = - ClJ.l p n _j.l ,- ( p -n ) ( '\7 • E) E o o , the constant cl is evaluated So (A .6) Substituting this value of Cl into Eq.(A.4) gives (A.7) where I/A = J. The current-voltage characteristic is determined from Eq.(l.2.2l) and the boundary condition (ii). That is, (A.8) and finally 2 I = o~ p ~ n T(po -no ) v (A.9) ~2 J ~ :~ (A.lO) where 0 = _ Rl A'J • Fe l l/2 duj -2 81 APPENDIX B SOLUTION OF AC SMALL SIGNAL IMPEDANCE The linearized ac equations which result from the substitution of Eqs.(l.4. 3 a,b,c) into Eqs.(l.4.l) and (1.4.2) are given by (B.l) and (B.2) where pT = p 0 - n • Solving Eq.(B.l) for and J 0 P1 (u1 ) and substituting = AqP0 ~ p E0 (b+l). Since 2 I o = o~ p ~ n TpTVo into (B. 2 ) yields where J 1 = I 1 /A 0 (see Eq.(A.9)). Eq.(B. 3) can be rewritten as where R 0 = V /I • 0 0 This equation is now placed in the form (B. 5) &V0 cp = AE 0 (l+i(.l)'T") (I 0 /E 0 +jm=:A) h1 , and 82 6V w = AEo (l+j mT) h1 h 2h • The solution of Eq.(B.5) and therefore the 3 electric field is easily found to be 0 (B.6) By integrating the electric field along voltage u 1 from R 1 v1 and subsequently the ac impedance z1 = to vl 11 R , 2 the is determined. The result is which is Eq.(l.4.5). Consider the cylindrical geometry (case (a)) discussed in Section 1.2.2. From Eq.(B.7), the ac impedance z1 after some manipulation is brought into the form ~[~·] (1+jmT) 0 exp[jill8(l+jmT)(u-u')] where (B.8) H is the height of the diode and 8 ER0 (2~H6) / 2 . W8 to O(ruB) the integral (Eq.(B.8)) in powers of 1 2 Expanding and integrating, the resultant expression for the ac impedance is given by (B.9) where ij = I, 2 Llne(z + (z -1) l/2 (B.lO) 2 l/2]2 ) - (1-z- ) Case (a) and z = p jp • 2 1 In this case, the impedance (B.9) is formally identical to the expression for the planar geometry impedance (Eq.(l.4.7)). The difference arises in the magnitude of the second order terms which involve the factor "8". In the planar configuration the ratio 8/(R 0 C0 ) is constant and equal to 3/2 8/(R 0 C0 ) = 3/2). How- ever, for the cylindrical configuration the ratio ®/(R 0 C0 ) where c0 (i.e. is the geometrical capacitance of the cylindric~l diode, is not constant but depends upon the value of the radii ratio in Figure B.l. As toward the value pzfp 3/2 p jp 2 1 as shown approaches unity the ratio ~,{R c ) tends 0 0 1 which is anticipated from the planar result. The form of the ac impedance for case (b) is identical with Eq.(B.9) except 8 is now given by @ = ItL:z 2 -1) l/ 2 1 l (B.ll) 2 - cos- (1/z)J Case (b) From Figure B.l, the ratio ®;(R 0 C0 ) in e~ch case (a and b) is a rather insensitive function of the radii ratio and can thus be a~proximated by the constant value 3/2 when p2 jp1 < 10. 3~--~--~--~----~--~--~--~----~--~--~----~~ CYLINDRICAL STRUCTURE CASE (b) 2 0 u 0 0::: "-.. I® I CASE(a)7 00 2 3 4 5 6 7 I 8 9 10 P2/P, Figure B.l. Ratio R®C versus the radii ratio 0 0 configuration. p2 pl for the cylindrical II 12 (X) + 85 When the second order terms involving ~ are negligible, the first order small signal equivalent circuit shown in Figure 1.4.lb represents both the planar and cylindrical geometries. This is in agreement with the step response analysis of Section 1.3 which is independent of any arguments based on geometrical properties of the double injection diode. The ac impedance for the spherical geometry must be determined from Eq.(B.7) by numerical techniques. Nevertheless, by analogy to the planar and cylindrical cases, it is assumed that the first order equivalent circuit illustrated in Figure 1.4.lb is applicable to the spherical geometry also. For frequencies 1/(WS) through w >> 1/S, o(ae)- 2 • Eq.(B.8) is expanded in powers of The small signal admittance f~r the cylindri- cal configuration is thus given by = g + jo.:C 0 ' w >> 1/S (B.l2) where g = Case (a) 1 1/2 1 2 1/2 2 1/2 ) - (1-z-) ][tanh- (1-z-) ] [ln (z+ (z -1) e G 2 0 (lne z) (B.l3) 1/2 2 [(z -1) g = Case (b) and C 0 - cos -1 (1I z)][tan -l (z 2-1) (ln z) e (the geometrical capacitance) is given by 1/2 J G 0 (B.l4) 86 In the planar geometry, the ratio is constant and equal to 4/3 g/G 0 (i.e. g/G 0 = 4/3 ). ratio g/G 0 is not constant but depends upon the value of the radii ratio P2/Pl as shown in Figure B.2. ratio g/G 0 tends toward the value However, for the cylindrical configuration the For both cases (a and b), the 4/3 as p jp ~ The conductance g in the frequency range 2 1 approaches unity. 1/® < rn < 1/T c now depends upon geometrical terms and contrasts the result for the first order equivalent circuit given in Figure 1.4.lb. Numerical means must be used to determine the ac admittance from Eq.(B.7) when the geometrical configuration is spherical. 3r---~--,----.---.----.---.----.--~----~--~--~--~ CYLINDRICAL STRUCTURE 2 0 ~ 0"1 ~~------------------~----- ~ CASE (o)J O0 I 2 3 4 5 6 7 8 9 10 II p2j p1 where g P2/P1 Figure B. 2. The conductance ratio g/G 0 versus the radii ratio is determined over the frequency r ange 1/8 < m < 1/1 c • 12 88 APPENDIX C THEORY OF NOISE MEASUREMENT AND EXPERIMENTAL PROCEDURE C.l. Noise Spectral Density. For a two-terminal network, the noise in a frequency interval D.f can be represented by an equivalent noise 2 l/2 voltage (v ) in series with the network or alternatively by an 2 l/2 equivalent noise current generator (i ) in parallel with the network~C.l) In general, the network will consist of both passive and active elements. Another representation of the noise current is 2 (i ) obtained by introducting s.l. (f) where s.l. (f) per unit bandwidth ' (C.l.l) is the "noise spectral density". In the measurement procedure, the unknown noise source of the device is compared to a known shot noise diode calibrator source. Thus, an alternative representation of the noise current is 2 q I where I eq eq D.f (C.l.2) is an "equivalent" saturated noise diode current giving the same mean square value of the short-circuit noise current in the frequency interval D.f. From equations (C.l.l) and (C.l.2), the noise spectral density is given by 2 q I eq (C.l.3) I eq will be used throughout the remainder of this discussion as a measure of the noise current spectral density. C. 2 . Measurement System. A block diagram of the measurement system is outlined in Figure c. 2.1. In this case, noise signals from the device are passed through an amplifier and wave analyzer. The wave analyzer consists of a variable narrow bandpass filter and a linear half-wave rectifier. The rectified output voltage is integrated and displayed on a digital voltmeter. By activating a shot noise calibrator source, the equivalent noise current of the device is subsequently compared to that of a known noise source~C- 2 ) This technique eliminates the need for accurate amplifier gain and filter bandpass measurements. The temperature of the device is varied by the temperature control unit. This allows measurements to be made as a function of ambient temperature. C. 3 . Small Signal AC Equivalent Circuit and Notation. The small s ig- nal ac equivalent circuit along with the pertinent noise sources of the amplifier input are shown in Figure C.3.1. represented by a noise generator 2 1/2 (i ) , The device noise is where (1.•2) is given by Eq. (C.l.2). The following notation and symbols are defined as: q = lql, magnitude of the electronic charge, 1.6 x 10-l9 coulombs. T = Temperature, °K k = Boltzmann's constant, ~f 1.38 x 10- 23 joule/°K = Frequency interval of narrow bandpass filter .---------, . I DEVICE I -~ CURRENT I SUPPLY L _____ _jI I .-------, I I AMPLIFIER I SUPPLY I I L __ , _ _ .J I I I r---L---, DEVICE WAVE I I 1--f - -1 AMPLIFIER f-- f - ANALYZER I 10KHz-22MHz IL _ _ T _ _ _j ' DIGITAL INTEGRATOR I - VOLTMETER I I r--L--, TEMPERATURE CONTROL I I I SHOT NOISE DIODE I I I L---r--.J I r __ .LI _ _ , 1 I NOISE DIODE I I ~-SUPPLY _j Figure C.2.l. Block diagram of' experimental noise measurement apparatus. (va 2) T 0 w A v Z L..+.-.J (i2) :? rd )<b((ic2):? rb )<!:>( 4b2) "'T' Cb ~ IDEAL AMPLIFIER _j_ S2 I I E A N A L y z Sl I I I I I Figure C.3.1. Small signal ac equivalent noise circuit of amplifier input and shot noise source. I E R \0 I-' 92 Shot noise current generator of the temperature 2 ) = 2 q I ~f c c 3) diode(C. is the de plate current of the I c where (i limited diode (5722 vacuum tube), rd = Real part of the dynamic impedance of the device rb = Real part of amplifier input impedance ~ = Total input capacitance of amplifier (includes device capacitance) Thermal noise current generator due to 4kT(l/rb) ~f(c. 4 rb; (~ 2 ) = ) (v 2 ) = Equivalent noise voltage source of amplifying system. a location of <va 2> The . in F1gure C.3.1 expresses the fact that this noise source is found experimentally to be insensitive Z and the plate current to the amplifier input impedance I Sl c (see Section D.5) Switch which activates shot noise source. This is illus- trated symbolically in Figure C.3.1 by a switch in series with the noise source (i 2 c ) S2 = Shorting switch at amplifier input Z = Total input impedance (includes Ga =Voltage gain of system with rd) 2 (va ) as the source. This gain is experimentally found to be insensitive to the input impedance Z and plate current I c (see Section D.5) Gl =Voltage gain of system with Sl open, S2 open 2 (i ) and (~ 2 ) as sources, 93 sources, sl ~=Gain ratio, closed, s2 2 (ic ) 2 (i ), G =Voltage gain of system with 2 and (~ 2 ) as open ~ = ja1 j 2 jja 2 j 2 • ~ is The introduction of motivated by experimental evidence indicating that in C.4. Measurement Procedure. The variable narrow bandpass filter f (wave analyzer) is set at a center frequency 2 l/2 2 l/2 2 l/2 voltages (va ) , (v1 ) and (v ) 2 and the output noise 0 are measured by employing the following procedure (see Figure C.3.l): ( i) With Sl open, S2 closed, (ii) With Sl open, S2 open, measure (iii) With closed, Sl S2 2 l/2 (va ) measure 2 l/2 <vl ) 2 l/2 <v2 ) open, measure and the noise diode plate current I (i.e. I c c 2 l/2 adjusted such that <v2 ) >> (vl2)l/2, see is Appendix D.l.l) (iv) The device is removed from the input. and (iii) are now repeated yielding ~ 2 l/2 and I . (V2 ) c By repeating this procedure for various 's f quantities in conjunction with the value Since ' these eight measured 0 ' ~ are sufficient to obtain a measure of the device noise spectral density C• 5 • Analysis • Steps (i), (ii) ~ 2 l/2 ~ 2 l/2 (va ) , <vl ) I eq • are smoothly Ga ' varying functions of frequency, they can be considered to be constant over a sufficiently small frequency interval 6f. The voltage gain G a of the amplifier equivalent noise source is assumed to be independent of the input impedance z. * voltages and With these assumptions, the noise (v 2 ) 2 are expressed as (C.5.1) (C.5.2) 2 2 2 2 2 2 <v2 ) = (va >1Gal + (i )lz\ IG 2 1 + 2 (~ >lzi 2 IG 2 1 2 + (ic >lzi 2 \G 2 12 2 (C.5.3) Here, for example, the terms in Eq.(C.5.3) represent the amplifier noise, device noise, amplifier input resistance noise and the shot noise sources respectively. Solving Eqs.(C.5.1), (C.5.2), and (C.5.3) for the equivalent noise current eq of the device gives (1-11) <va 2) <v 2) - <v 2) 1 a I I I 1 - eq 2 11 <v22)- (Vl ) + (1-11) (Va 2) 11 <v 2) - (v 2> 2 1 2kT qrb c (C.5.4) The second term of Eq.(C.5.4) is recognized as the equivalent noise current (I I * eq, b eq,b ) of rb' where (C.5.5) Any change in the system gain due to "Miller effect" is found experimentally to be negligible (see Appendix D). 95 This equivalent noise current is experimentally determined by invoking step (iv) of the measurement procedure (c.4). That is, I eq = 0 and Thus, Eq.(C.5.4) becomes I c - I eq,b (C.5.7) At the center frequency f , 0 the equivalent noise current is now totally determined by experimentally measured quantities. value of ~ When the (see Appendix D) is equal to l.O, Eq.(C.5.7) simplifies to (C.5.8) In this case, system gain. I eq does not depend upon the characteristics of the APPENDIX D REALIZATION OF NOISE MEASUREMENT APPARATUS D.l. Description of Amplifier and Input Circuitry. The amplifier, shot noise calibrator diode and associated power supplies are illustrated in Figures D.l.l and D.l.2. Here, combination of Rl, R3 S2, and Sl, R2 and Sl, Ql S2, and the parallel represent respectively the elements (to first order) in the ac equivalent circuit shown in Figure D.3.l. The resistors Rl, R2 and R3 constitute the input biasing networks and are considered as part of the input impedance of the amplifier. The vacuum tube Ql (type 5722), which operates as a temperature limited diode, is used as the shot noise calibrator source.(D.l) This noise source is activated by the switch in the filament supply. Sl located A 200vdc supply in conjunction with Rl provides the device current, whereas a similar voltage supply with provides the plate current I the filament current via R62, for the shot noise diode. c Metering circuits Ml and By adjusting the equivalent noise current the shot noise diode can be varied from M2 0 (I c ) of to 500~A (see table D.l.l). continuously monitor the device current and noise diode plate current respectively. The first stage of ampli- Q2 and Q3, arrangement and the output stage Ql2 is an emitter follower. fication, which is denoted by R2 is a standard cascade somewhat unusual mid-stage gain section is differential. to provide an option for having a differential input. The This is done However, this possibility was not exploited since the single ended cascade arrangement proved adequate. <D ® rio - ® - Cl2 Cll @ R28 R29 L5 ~R5 ~R6 R4 L3 ~ C3 1( R14 I I t I Rll$ . .. •u r Rl3 - ~CI6 R1s;> I u .. . I I (oa \ I -· I R23> \010/ ... I ,-.1~ 1D I I EI I yI I c I LEI .,_ 9 - 11"0:_\ ?R3 I I ?R7 ?R8 TJ I L Figure D.l.l. Input amplifier and shot noise source. I I I R27 T [ 012 I ~ ~ R37 06 R33 R46 07 R47 + R45 _C23 R39 PLATE SUPPLY FOR 5722 (0 1) NOISE DIODE v~ ~ ---------::c~:u~=:---------, R4~5 O (SAME AS ABOVE) r...--------------------------..1~----4--+--~----------~ R51 R50 R52 015 019 R57 24V D.C. '-~~v-~--r---------------------~3 013 R5B R53 R56 Tl R55 R59 014 R60 AMPLIFIER SUPPLY DRESSEN BARNES MOOEL 40-12 12V DC SOLID STATE SUPPLY FILAMENT SUPPLY FOR 5722 (0 1) NOISE DIODE Figure D.1.2. Power supplies for amplifier, shot noise calibrator source and device. 99 TABLE D.1.1 Parts lis t for amplifier and power supplies. RESISTORS * Rl R2 R3 R4, 5, 16, 23 R6 R7 , 17, 21, 22, 24 R8 R9, 13 RlO,l2 Rll Rl4, 15 Rl 8, 19 R20 R25, 26 R27,54 R28, 29, 30, 31 R32 R33 R34 R35 R36 R37 R38 R3 9 R40 R41 R42 R43 R44 R45 R46, 47 R48 R49 R50 R51, 52 R53 R55, 60 R5 6 R57 R58 R59 R61 R62 R63 RMl RM2 lOOk, l/2w, MF lOOk or 3 OOk, 1/ 2w, MF lOMO, l/2w, MF l.5k, l/2w, MF 3 .48k, l/4w, MF 750, l/4w 10. 7k, l/4w, MF lOk, lj4w, MF 620, l/4w 39, l/4w 40. 2k, l/2w, MF 221, l/8w, MF 24, l/4w 267, l/8w, MF 2k, l/2w 8 . 2k, l/2w, MF lk, 1/lw 4.7k, 1/lw lMO, lw 68ok, 1j2w 2. 2k, l/2w 4. 7k, l/2w llOk, l/2w lOOk, l/2w lOk, trim.pot. 47k, l/2w 68k, l/2w 2 .4k, l/2w l 20k, l/2w lOk, lOw 5k, lw lOk, Helipot lOk, lw 3 . 3k, l/2w 470, l/2w 27k, l/2w lk, l/2w 2.8k, l/2w 3, l/2w 1. 82k, l/2w 510 trim.pot. l.lk, lw 500 Helipot 2 .15, 25w {selected meter shunts tl/2w, MF 100 Table D.l.l (Continued) CAPACITORS ** Cl, 2, 22 C3,7 c4, 8 C5, 28 c 6, 11, 12, 13 C9 ClO, 14, 32,33 Cl5 Cl6 Cl7 Cl8,19 C20, 21, 25 C23, 24 C26 C27, 30 C29 C3l C34 0.02 0.47 l.O 0.1 0.47 2 .0 0.01 5-80 pfd 2 . 2 pfd 3 .9 pfd 100 pfd 80 4o 0.5 100 0.001 4ooo 50 INDUCTORS Ll 12 13 L4 15 l~h, 50 turns #22 wire on Micrometal T80-7 toroid core 2 .~h, 20 turns #16 wire on Micrometal T80-6 toroid core 4.7mh &.th 8~h TRANSISTORS Q2 Q3, 6, 7 Q4,5 Q8, 9,12 QlO,ll Ql3 Ql4, 15,16 Ql7 Ql8, 20, 21, 22 Ql9 Q23 Q24 SF-5868 or 2N3819 (selected for low noise) 2N250l 2N3572 2N426l MM 999 RCA 40424 RCA 40327 2N3906 2N3904 Tektronix 151-140 2N442 2N3715 DIODES & ZENERS Dl D2,3,4,5 IN 962 IN 4005 101 Table D.l.l (Continued) D9, 10, ll, 12, 13, 15 D6 D7 1 8 Dl4 Tektronix l52-0o40 IN 752 IN 227 IN 960B NOISE DIODE 5722 (Sylvania) METERS Ml, M2 * ** 0-50t-J.A TAUT BAND All resistor values are in ohms; MF =metal film. All capacitor values are in microfarads unless otherwise specified; capacitors associated with the amplifier are ~ 50 VWDC, whereas the capacitors associated with the power supply are ~ 250 VWDC. 102 D.2. Amplifier and Wave Analyzer Characteristics. In order to perform the necessary noise measurements, the amplifier must be linear, low noise and have sufficient voltage gain over a wide frequency range to . enable the wave analyzer to detect the small noise signals. The wave analyzer is a Hewlett-Packard HOR-312A with a frequency range from ldcHz to 22MHz. A filter bandpass of 3kHz is used for all measurements. Since the device dynamic resistance is of the order of 1 kO to 10 kO the real part of the amplifier input impedance should be greater than 10 kO in order to reduce attenuation of the device noise signals. How- ever, the output impedance of the amplifier should be about 50 0 or less since this is the value of the input impedance of the wave analyzer. A junction Field-Effect-Transistor (FET), type SF-5868, is utilized in the first stage of amplification since it fulfills the criterion of low noise and high input impedance (the real part being greater than lOOMeg 0). Figure D.2.1 shows the amplifier gain as a function of frequency for an input ac generator source impedance of 50 0. For this case the amplifier is also terminated in 50 0. The gain is greater than 40db over the frequency range from lO~z to 22MHz. This gain is more than adequate for the sensitivity of the wave analyzer. 300 kO, R2 = 100 kO and R3 = lOMeg 0, impedance is approximately 75 kO. Since Rl = the real part of the input The measured output impedance is 8.4 o. Van der Ziel has shown that for a junction FET the theoretical equivalent noise resistance is given by 70.--,,-,,rn~---.-.-.~~n---~-r-rTT.n.---~~~~~~--~~~ --- 1-' 0 w o~--~~~~~~--~~~~~~--~~~~~~--~~~~~UL--~--~~ 100 101 10 2 103 f (KHz) Figure D.2.1. Voltage gain of the input amplifier (Figure D.l.l) from 10kHz to 40MHz. 104 104 (D.2.1) where is the transconductance of the FET. (D. 2 ) gml of 4.5 ma/v A measured for the SF-5868 corresponds therefore to a VGS=O theoretical equivalent noise resistance of 148 O. The actual measured short-circuit equivalent noise resistance of the amplifier is shown in Figure D.2.2. A high frequency limiting value for mately 275 0 indicates that possibly other noise sources besides that Req of approxi- of the FET are contributing to the overall short-circuit noise of the amplifier. For device dynamic resistances greater than 275 O, the noise of the amplifier is sufficiently low to insure accurate measurements of D.3. I eq . High Frequency Effects. the amplifier is 15.4 pfd. The total measured input capacitance of Approximately 6pfd, 7pfd and 2.4pfd are associated with the FET, noise diode, and distributed wiring capacitance respectively. Depending on the dynamic resistance of the device, this input capacitance may appreciably attenuate the noise signals at high frequencies. As a result, the signal noise approaches that of the amplifier, and the error in the measured spectral noise density of the device may become very large (see Eq.(C.5.7)). a tuned circuit consisting of a high To overcome this effect, Q inductor and variable capacitor is added to the input impedance of the amplifier. consists of the elements This circuitry Ll, L2, Cl5 and 83 as shown in Figure D.l.l. The combination of the inductor Ll, input capacitance (""' ~) and 800 700 600 s CT 300 Q) a:: I-' 0 \Jl 200 1 IOO'10 I ' I 1 I 1111 10 2 I I I I II II I 103 I I I I II II I . /1 f{KHzl Figure D.2.2. Equivalent noise resistance of amplifier (Figure D.l.l) from 10kHz to 22 MHz. I I 106 variable capacitor 5MHz to 9MHz, 22 MHz. Cl5 while L2 is tunable over a f'requency range f'rom in a similar arrangement covers 13 MHz to With the wave analyzer set, for example, in the f'requency range where Ll is applicable, the variable capacitor adjusted for maximum noise output signal. Cl5 is Under this resonance condi- tion, the reactive part of the input impedance is eliminated, and the parallel combination of Ll Rl, R2, R3 plus the losses associated with determines the magnitude of the real part of the amplifier input (> lOkO ) • impedance The theory and procedure of the noise measure- ment described in Appendix C is not altered by the introduction of the tuned circuitry at high frequencies. Since the Q of the wave analyzer is much greater than that of the tuned circuit, the f'requency bandpass 6f is still determined by the wave analyzer. D.4. Low Frequency Effects. If the dynamic resistance of the device is sufficiently small, the increasing impedance of the coupling capacitor Cl at low frequencies causes the measured noise spectral density to decrease. Under these conditions, the correct equivalent noise current is calculated f'rom I eq = s(f0 ) I eq measured , (D.4.1) where s(f ) 0 (D.4.2) 1~ Since the value of the capacitor Cl is 0.02~fd, which in this case is the maximum value attainable within the given space restrictions, Eq.(D.4.1) is utilized when rd is less than lkO over the appropriate low frequency range. D.5. Measurement of Gain Ratio The deviation of the gain ratio ~. from 1.0 is found experimentally to depend only on the magnitude of and the noise diode plate current I c• (i.e. ~ ~ 1) R3 This probably arises from a de leakage current through C2 developing a bias voltage across subsequently alters the ~ R3 which of the FET Q2. This effect is negligible when the high frequency L-C network is switched into ~ the amplifier input since Ll and de voltage appearing across R3. various values of R3 I and c L2 effectively short circuit any The measured values of ~ for are given in Table D.5.1. TABLE D.5.1 Measured values of the gain ratio ~ I c ~5~ L-C out I c ~ 10~ I ~. c = 20~ I ' ~~~ c 1 0.953 0.883 0.828 1 1 1 0.950 l l 1 1 R3~10Meg0 L-C out R3~500KO L-Cin It is noted that for I c less than 5~A, the gain ratio to 1 and does not depend on the amplifier input impedance. ~ is equal 108 D. 6 . Comments on Construction. The amplifier, input circuitry, noise diode and associated power supplies form an integral system. The utilization of solid state devices permits all of these units to be placed in a single metal enclosure which acts as a Faraday shield. This construction also facilitates the location of one signal ground which is labeled ~ in Figure D.l.l. The general construction techniques of high frequency equipment are employed in laying out the amplifier and input circuitry. Metal film resistors which exhibit nearly ideal thermal noise over a wide range of frequency and current are used extensively. low capacity. The shorting switch Inductors Ll typically greater than 175. and 12 82 is a point contact of are toroids and have Q's All power supplies are of the series regulator type and have peak to peak ripple of less than l millivolt. D.7. Active Integrator. With a noise signal at the input, the output voltage of the narrow bandpass wave analyzer fluctuates around a mean value v (v ~ lV). To improve on the expected error of this reading, the output voltage is integrated for T seconds (see Appendix H). The integrator consists basically of an operational amplifier with an RC feedback network. integrating scheme. with C2 and R6 Figure D.7.1 illustrates such an Here, the operational amplifier through Ql in conjunction Rll make up the integrator for the fluctu- ating input voltage. The total integration time a similar arrangement (Q2, C3, Rl2-Rl7) which integrates a constant input voltage of lV. When switches 82 Q4 and Q5 are on; by resistors Rl8 and R20 T is determined by is in the RESET position, the FET capacitors respe.ctively. C2 and C3 are discharged Placing 82 in the INT. 109 PIN CONNECTIONS FOR SP65U (01,02,03) 54 SPAR ELECTROSTATICS POWER SUPPLY MODEL 500 -15 +15 Rl Rl8 R6 -=- RESET Sl INPUT 1-= OUTPUT 1 - 15V 7 Bl -15V -= Rl2 0 .5 R20 Rl7 - 15V Figure D.7.1. Wiring diagram for active integrator. llO TABLE D. 7 .l Parts list for active integrator. RESISTORS * Rl,29 R2,3 R4 R5, l8, 20,30 R6, l2 R7, l3 R8,l4 R9,l5 RlO, l6, 22 Rll, l7, 24 Rl9, 2l R23 R25 R26 R27 R28 R3l R32 JMO, l/2W lOOMO, l/4w l4k, l/2W lk, l/2W l.6MO, l/2W 800k, l/2w 4ook, l/2W 200k, l/2w lOOk, l/2W 50k, l/2W 500k, l/2w l50k, l/2W 23lk, l/2w lOk, l/2W 30, l/2W lOOk, l/2W 470, l/2W lOk, lOw CAPACITORS ** Cl, C2, C3 lj.lfd DEVICES Ql, 2, 3 Q.4,5 * ** Philbrick SP65Au operational amplifier 2N38l9 All resistors values are in ohms; All capacitor values are :<! MF = metal film. 50 VWDC. lll position starts the integration. Q3 activates the READY indicator simultaneously. Ql After B2 T seconds, the level detector and turns off Q4 and Q5 The integrated input voltage is read at the output of with a digital voltmeter. One of the main advantages of such an integrator is that the relative standard error is smaller than with a passive RC integrator of the same time constant. (D.3) 112 APPENDIX E CALIBRATION MEASUREMENTS E .l. Wave Analyzer Detector. An investigation of the wave analyzer shows that the detector is a linear half-wave rectifier followed by a passive RC time averaging circuit. tude For an input sine wave with ampli- the half wave-rectified output voltage v0 ' <vh.w.r )S V - is 0 1'( The output of the wave analyzer, however, is calibrated in R.M.S. (root-mean-square) voltage and reads v0 2 l/2 (E.l.l) (V )S INDICATED ' Therefore, for a sine wave 2 l/2 (V )S INDICATED rc ----~'--------- = --<vh.w.r. )S (E.l.2) Now consider an input noise voltage being applied to the same halfwave rectifier. The noise voltage is assumed to have a normal (Gaussian) distribution which is defined as p(v) = where p(v) dv between v and J ~." ~ff v exp [ ~ [v.;i]j (E.l.3) is the probability that the noise voltage has a value v + dv and veff is the standard deviation. For a half-wave rectifier, the expected value of the output noise voltage is 113 Cvh.w.r. )N (-v-) 2] dv = veff (E.l.4) and the R.M.S. value is given by (E.l.5) Therefore, for a noise voltage 2 1/2 <v >N > = h.w.r. N (v, F . (E.l.6) From Eqs.(E.l.2) and (E.l.6), the actual R.M.S. noise is given by 2 2 1/2 -(V )N INDICATED {rf ' Since the system is used as a comparator (see Eq.(C.5.7)) this correction does not appear explicitly in the evaluation of the result. It does appear, however, in the testing procedure discussed in Section E.2. E.2. Verification of Nyquist 's Theorem. According to Nyquist's theorem, the mean square value of the voltage developed across a resistance at the ambient temperature T °K in a narrow frequency interval R ~f is(c.4) (E.2.l) 114 A lOkO metal film resistor in lieu of the device is placed at the input of the amplifier. The measured value of this resistor in parallel with the input biasing networks of the amplifier is 8.2 kO at a temperature of 298°K. With the wave analyzer set at a center frequency of 50kHz and a bandwidth 6f of 3kHz, the voltage gain of the system as measured by the wave analyzer is 173. From Eq.(E.2.1) the theoretica1 noise voltage is <v2> theoretical The output reading is 97~V (RMS at wave analyzer). Therefore the actual noise measured with respect to the input is (0.401 ± .Ol)(~v) 2 The experimental resu1t agrees very well with the theoretical expression of Nyquist and demonstrates that meta1 film resistors exhibit nearly idea1 thermal noise. E.3. Verification of Schottky's Theorem. In accordance with Schottky's theorem, the mean square noise current from a temperature limited vacuum tube diode in a frequency interval 2q I where I c c 6f 6f is given by ' is the de plate current of the diode.(C.3) (E.3.1) ll5 Under experimental conditions identical with those described in E. 2 the output noise voltage with the shot noise diode activated is (see Eq.(C.5.3)) (E.3.2) where Figure E. 3 .l shows a plot of the output noise the shot noise diode plate current I . as a function of The result demonstrates the and also establishes the linearity linear dependence of on of the amplifying system. From the slope of this curve and Eq.(E.3.2), the value of jqj I , c c (v2 ) is determined as jqj = (1. 65 ± .06) x l0-l9 coulombs This measured value for the magnitude of the electronic charge is in good agreement with the accepted value of 1. 602 x l0-l9 coulombs. It also verifies t he pure shot noise behavior, as expressed by Eq.(E.3.l), of the temperature limited diode E.4. Resistor Calibration. Ql. In order to fully check the performance of the noise measuring apparatus and verify the theory and procedure described in Appendix c, the noise spectral density for a series of resistors is measured at room temperature (T; 298°K). The theoretical 1.0.-----r----r-----,r----r----r------.---r---.------.---,.---~-----.---,.------, SLOPE 2 x 10-3 fl.-V ~ 0\ o~--~--~L--~----~--~----~--~~---L---~--~----~--~---~--~ -100 -50 0 Figure F.3.1. 50 100 150 200 250 300 Ic (fLA) 350 400 450 500 550 Mean square amplifier output voltage versus noise diode de plate current for 5722 vacuum tube at 50kHz. 600 117 e quivalent nois e current of a resistance I e~heoretical = R is given by 2kT (E.4.1) Figure E.4.1 shows the measured equivalent noise current for a lOOkO, lOkO, lkO and 3010 metal film resistor over a frequency range from 10kHz to 22MHz. The results are tabulated in Table E.4.1. In all cases, the average measured equivalent noise current deviates by no more than 5% from the theoretical equivalent noise current calculated from Eq.(E.4.1). Since the real part of the amplifier input impedance at 22MHz is approximately 17k0, the experimental results for the lOOkO r esistor illustrate the rather remarkable accuracy that can be obtained by this technique. The experimental data presented in Figure E.4.1 demonstrate that the apparatus has a capability of making noise spectral density measurements on two terminal devices which may have an equivalent noise current from 0.5~ to over 17~. The long term stability, reliability and absence of temperature drift of the system, is attributed to well designed solid state circuitry and is reflected in the measurements shown in Figure E.4.1. TABLE E.4.1 Comparison between the theoretical and measured equivalent noise current for resistors. Resistance (0 ) lOOk lOk lk 301 Ie ~easured 0.50 5.2 51 164 (I-LA) I (~-tA) eqtheoretical 0.52 5.2 52 171 ··a;; Error 3 . 9'k -(J!fo 1.% 4.1% 0 "'" " 0 ~ "' R~30~il . 0o "'0 0 0 164 "' p.A~ 0 0 ~ nl'l00 ~ <! ::1... -0" 10 1 5.2 p.A 11) R=IO Kil 1--1 0 I-' I-' CP ---o--o-'"l!o,..,o~:r· •o....-----<oo R=~000Kil o e 0 0 -o"'" · 0 ~ 0.50p.A~ 0 .·- - 0 0 °0-00 0 d·k~ "'0 101 10-11 1 I I 1 1 I I II 2 10 1 I I I I I I 10 I I3 I I I I I I I I I• f (KHz I Figure E.4.1. Equivalent noise current for a lOOkO, lOkO, 1kO and 3010 metal film resistor fran 20kHz to 22MHz. 1 I I ll9 APPENDIX F TEMPERATURE ENVIRONMENT AND THERMOSTAT F.l. General Considerations. The principle applied to vary the temperature of the device under test is to heat _precooled nitrogen gas to the desired temperature and expose the device to this gas. A thermally insulated fixture houses the device and heating element. Feedback is utilized to maintain a constant temperature within the chamber. Temperatures from l00°K to over 400°K have been achieved with this system. F.2. Description of Thermostat . arrangement. Figure F.2.l shows the experimental Nitrogen gas is passed through a coiled copper tube submersed in liquid nitrogen . A reduction valve and flowmeter, which is calibrated from 0 up to 40 SCFH (Standard Cubic Feet per Hour), regulates the gas flow. The precooled gas flows to a heater coil where it is heated and subsequently enters the temperature chamber which contains the device. The ambient temperature is raised in the same manner but without precooling the gas (i.e. no liquid nitrogen). The feedback configuration used for control of the temperature consists of a thermocouple sensor, temperature reference source, comparison voltage source, and amplification system (dark lines in Fig. F.2.l). An iron-constantan thermocouple located in the temperature chamber near the device is the sensor. The iron-constantan wires are carried through the amplifier enclosure to a copper block located in a crystal oven (Monitor Model 5l2, T = 349.7°K) . temperature of the reference junction. This establishes the The output voltage between the REFERENCE TEMP (T=349 .7 °K) MONITOR # 512 COMPARISON VOLTAGE SUPPLY COPPER BLOCK AMPLIFIER INTERFACE _____..,r----- - - - , I 1 I I I I I I I r-- SENSOR (Fe-Co THERMOCOUPLE) I 0-15(mV) + I D.C. VOLTMETER (TEMP) I REDUCT ION VALVE I Co Fe I N2 GAS Fe INPUT OF AMPLIFIER I TEFLON : D.C. AMPLIFIER GAIN 103 Fe I I I 1--' (\) 0 :L ______ jI D.C. VOLTMETER (HEATER) 6V ERROR SIGNAL DEWAR + 2av. D.c. SUPPLY r , ._ Figure F .2 .1. --·-.......~ ~ . TO RECORDER · .. .. D.c. AMP 1 GAIN 106 Experimental arrangement for thermostat. 1 21 sensor and reference thermocouple junctions is compared to a variable (0-15mv) voltage source. An error voltage 6V resulting from this comparison is amplified by two cascaded chopper stabilized amplifiers with a total voltage gain of 106 . This amplified error voltage is the input signal for a power transistor (RCA 40312) with the heater coil as its collector load. If 6V is greater than coil will draw current from its 28v de supply. Ov, the heater The heater, which is composed of Nichrome wire wound around four ceramic tubes and assembled in a "bird cage" fashion, has a small heat capacity and a resistance of approximately 200. Therefore, the heater is capable of delivering 39 watts into the gas stream. Indication of the chamber temperature is provided by a XlOOO chopper stabilized amplifier and a de voltmeter. A recorder output is also supplied. F.3. Operation and Procedure. The chamber temperature is determined by setting the comparison voltage. Depending upon the desired tempera- tureJ the voltage across the heater coil will be Ov (off) or 28v (on). Successful operation of the system is achieved when the measured heater voltage is somewhere between 0 and 28v. This is accomplished by adjusting the flowmeter and or the gain of the error signal amplifier. When the system is properly adjusted, a change in temperature (typically 2Q%) is accomplished within a few minutes. The temperature will generally overshoot and then undershoot once before accurate regulation is achieved. In order that the room temperature characteristics of the amplifier remain constant, a thermal insulating wall which consists of a Teflon disc is used to couple the temperature chamber to the amplifier enclosure. 122 Stainless steel wires, which have a relatively low ~hermal conductivity, are used to carry the noise signals and biasing current of the device through the temperature wall to the amplifier input. Furthermore, the temperature chamber being made of copper provides an extension the Faraday shield. F.4. Calibration. With ice-water ( 273°K) as a reference temperature, the iron-constantan thermocouple is calibrated at liquid nitrogen (77°K) and found to be less than 1% from the values quoted in standard calibration tables~F.l) Using this calibrated thermocouple, the measured temperature of the crystal oven is 349.7°K. Calibration of the comparison voltage supply indicates an error of l~V (less than l% error for voltages greater than lmV). For noise considerations, the sensor thermocouple is not directly attached to the device. Therefore, in order to insure that the device temperature is that of the temperature chamber, the following measurements are performed. A lkO metal film resistor is placed in the temperature chamber and the spectral noise density is measured at l00°K and 200°K. Figure F.4.l shows the results in which the measured tern- perature of the resistor R is determined from T = qRI eq /2k. Table F.4.l summarizes the results. TABLE F.4.l Calibration values for temperature chamber. T (Thermocouple) T (Resistor) % Error l00°K l03°K 3% 200°K 205°K 2.5% 0 A T = 200 °K T = 100 °K R= I K.O. <l: ::L 0'" Q) I-I A 10 1 • • a a • • • • • • • • a 7 •• 17.8 f-LA Figure F.4.1. Equivalent noise current for a 1kO resistor at 100°K and 200°K ambient temperature. a •o-----A I-' [\) VJ 1 24 There is excellent agreement between the temperature as measured by the thermocouple and that of the resistor which is determined from the noise measurement. This illustrates the fact that the lattice temperature of the device (under normal operating conditions) is equal to the ambient temperature of the chamber to within 3%. 125 APPENDIX G CURVE FITTING AND ERROR ANALYSIS G.l. Introduction. In the present case, the experimentally measured equivalent noise current of the double injection diode can be represented by the equation I where f eq = c1 + (G .1.1) is between 10kHz and 22MHz. The first term c1 represents the "white" or thermal noise level, whereas the second term represents the "excess" noise (due to generation-recombination). c1 , improve upon the estimate of In order to Eq.(G.l.l) is fitted to the data by a least-squares procedure. G.2. Least-Squares Curve Fitting of a Function I --------------------------------~~----------------~q Consider the function coefficients C • have no errors. N parameters v y(f,C ) v The data consist v = 1, - ~ which need not be linear in the The independent variable C (i.e. v = y(f,C ). (G.l) of • • • , N) is represented by the number pair f is real and assumed to L points which shall exceed the to be determined. Each point , f. • The I 's are eqi ~ eqi assumed to be uncorrelated, however, the absolute errors in the are not, in general, equal. ated. I Therefore, the weights w.~ are incorpor- A given initial set of estimated coefficients is represented by c~ 0 )(v = 1, ••• , N). The function is now linearized in the coefficients by means of a Taylor expansion through Thus, 126 y(f.C ) - y(f.,C ) l,V V l C C C = C 1 = N ( ) 0 = 6y.l (G.2.1) , 1 (o) N where N 6y. = l 2 ~d 6C + 0 ( 6C ) ~ v v v , (G.2.2) V=l and d v = oy (f.,C) l v oc v (G.2.3) The quantity - 6y.) 2 (G.2.4) l where (G.2.5) is now minimized. the N After differentiation versus the linear equations for the 6C v are given by N coefficients Cv' 127 L L L widld2t.C2 L + ••• + widlycN i=l i=l L + • •• + L wid2~t.CN i=l L L L widl~t.cl L + i=l wid2<\fC2 + •• • i=l (G.2.6) By solving these equations for the t.c v ' an improved estimate (cv ) for the coefficients is given by = C ( o) + t.C v (G.2.7) v The analysis is now repeated using the improved coefficients cv as the initial estimates. G.3. Determination of the Weights wi. A good approximation for the spectral noise density at a center frequency I eq f is (see Appendix C) (G.3 .1) l 28 where the time average symbols are dropped for convenience. The weight, which is proportional to the reciprocal of the variance 2 is given by w = lja . 2 () ' By considering the propagation of errors, the variance is calculated accordingly as a 2 -2 ""2 =a + a (G.3.2) where (G.3.3) and In this case, I quantities 2 >1 / 2 (V l c and ' "' Ic are assumed to have no errors. <v22>1j2 , <va 2>1j2, Also, the e t c. are assumed to be Gaussian distributed with equal relative standard errors. In order to demonstrate that the relative standard errors are approximately equal, 500 measurements are made on each of the noise readings and The readings are taken with the active integrator set for a 2 second total integration time. Figure G.3.l shows the experimental data in which the relative standard errors for and are 0.80% and 0.84% respectively. Gaussian with the same mean and variance is also fitted to the A 130r-----------------------~~---------------------------. BEST FITTED GAUSSIAN • I EXPERIMENTAL DATA • I BEST FITTED GAUSSIAN ta l EXPERIMENTAL DATA----~~>~ r-' N \0 ol I I.Y I ~~.5 ~~.7 ~~.9 2~ 2~.3'V 2 112 (va ) 2~ 2~6 2~8 2:0 2:2 NOISE READING(,uV) I I ">-I 2 112 (vl ) Figure G.3.1. Experimental data for 500 noise readings per histogram; a Gaussian with the same mean and variance is fitted to the experimental data. I 130 appropriate data. The theoretical standard error ex1/2 is given by (G.3.5) where T = total integration time. From Eq.(G.3.5), the theoretical relative standard error is 0.91% which is in good agreement with the measured results. G.4. A computer program is written to perform the Computer Program. necessary arithmetical manipulations involved in calculating I eq , f.) eqi ~ are determined throughout the frequency range from 10kHz to 22 MHz. from Eq.(C.5.7). In general, more than 25 data points (I Equation (G.l.l) is then fitted to the data using the least-squares procedure of G.2. In this procedure, the iterations are continued 2 until the sum of the squares, ~6y , successive iterations m and m+l of the residuals, in two satisfies the condition < E (G.4.l) ~6 2 LY m+l An error matrix is also calculated in which the diagonal terms are the R.M.S. errors given by A ~-'v of the coefficients Cv. Estimates of l3v are 131 [L-N vvJ ls UMS Q, A where A l/2 (G.4.2) is the inverted matrix of the set of linear equations, and SUMSQ, is the magnitude of the sum squares of the residuals after the last iteration. also provided. A graphical plot of the data plus the fitted curve is 1 32 REFERENCES . . . . l.l. S T Ll. u ' S Yamamoto' and A van der Ziel, "Noise in Double Injection Space-Charge-Limited Diodes", Appl. Phys. Letters ~ 3 08; 1967. l.2. F. Driedonks, R. J. J. Zijlstra, and C. Th. J. Alkemad.e, "Double Injection and High Frequency Noise in Germanium Diodes", Appl. Phys. Letters _g, 318; 1967. l. 3 . M-A . Nicolet, H. R. Bilger, and E. R. 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