Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Mitchell, Ronald Reid
Master of Applied Science - MASc
University of British Columbia
Germanium space-charge regions have been studied under conditions of double-depletion achieved by applying a reverse bias to both junctions of a transistor structure. The generation and distribution of carriers between the two junctions are described in terms of models for the thermal generation of carrier pairs and for potential distribution in a cylindrical system. Measurements over a wide range of temperature reveal that generation of carriers occurs through the medium of one or more sets of recombination centers and not by direct transition between the valence and conduction bands even at elevated temperatures. The Shockley-Read recombination-generation theory is applied to obtain the activation energies associated with the recombination centers. The impurity density, base width and junction areas are estimated from measurements of punchthrough voltage and junction capacitance. For some specimens the capacitance measurements made with one junction floating confirm the sharp increase in capacitance at punchthrough noted by Barker. The distribution of current between the junctions when both are equally reverse biased is found to be roughly proportional to their areas. It is also shown that control of the reverse current across one junction may be achieved by variation of the reverse-bias on the other junction. The mechanism of this interaction is considered in terms of the diffusion of carriers between the two space-charge regions. The phenomen of slightly non-saturating reverse current is explained in terms of the expansion with reverse bias of the space-charge region within the base. An expression relating the base current and reverse bias in a cylindrical system shown to give good agreement with experiment for one specimen. For the units in which the increase in current with voltage is appreciable, space-charge expansion cannot account for the increase and two other mechanisms are considered: avalanche multiplication in the bulk and along the surface, and a high generation rate on the surface. The surface conduction channels on the base region are investigated in two ways. Maximum floating junction potentials are calculated from measured values of the common emitter amplification factor (using the Shockley 1949 theory) and these are compared with directly measured potentials. A second method involves direct measurement of a.c. surface conductance between collector and emitter when both junctions are reverse-biased to prevent bulk conduction. Both tests reveal very small degrees of surface conductance on all specimens.
A STUDY OF CARRIER GENERATION IN, AND INTERACTION OF, SPACE-CHARGE REGIONS IN GERMANIUM by RONALD REID MITCHELL B.A.Sc, University of B r i t i s h Columbia, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Physics We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1959 ( i i ) ABSTRACT Germanium space-charge regions have been studied under conditions of double-depletion achieved by applying a reverse bias to both junctions of a tran s i s t o r structure. The generation and d i s t r i b u t i o n of c a r r i e r s between the two junctions are described in terms of models for the thermal generation of c a r r i e r pairs and for potential d i s t r i b u t i o n in a c y l i n d r i c a l system. Measurements over a wide range of temperature reveal that generation of c a r r i e r s occurs through the medium of one or more sets of recombination centers and not by dir e c t t r a n s i t i o n between the valence and conduction bands even at elevated temperatures. The Shockley-Read recombination-generation theory i s applied to obtain the a c t i v a t i o n energies associated with the recombination centers. The impurity density, base width and junction areas are estimated from measurements of punchthrough voltage and junction capacitance. For some specimens the capacitance measurements made with one junction f l o a t i n g confirm the sharp increase in capacitance at punchthrough noted by Barker. The d i s t r i b u t i o n of current between the junctions when both are equally reverse biased i s found to be roughly proportional to the i r areas. It i s also shown that control of the reverse current across one junction may be achieved by v a r i a t i o n of the reverse-bias on the ( i i i ) other junction. The mechanism of t h i s interaction i s considered in terms of the d i f f u s i o n of c a r r i e r s between the two space-charge regions. The phenomen of s l i g h t l y non-saturating reverse current i s explained in terms of the expansion with reverse bias of the space-charge region within the base. An expression r e l a t i n g the base current and reverse bias i n a c y l i n d r i c a l system shown to give good agreement with experiment for one specimen. For the units in which the increase in current with voltage i s appreciable, space-charge expansion cannot account for the increase and two other mechanisms are considered: avalanche m u l t i p l i c a t i o n in the bulk and along the surface, and a high generation rate on the surface. The surface conduction channels on the base region are investigated in two ways. Maximum f l o a t i n g junction poten-t i a l s are calculated from measured values of the common emitter amplification factor (using the Shockley 1949 theory J and these are compared with d i r e c t l y measured potentials. A second method involves direct measurement of a.c. surface conductance between c o l l e c t o r and emitter when both junctions are reverse-biased to prevent bulk conduction. Both tests reveal very small degrees of surface conductance on a l l specimens. I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 , Canada. (iv) TABLE OF CONTENTS \ Chapter Page I INTRODUCTION 1 1.1 Double-Depletion in Reverse-Biased Transistors 1 1.2 Potential and Space-Charge Dis t r i b u t i o n s for an Ideal Transistor 2 1.3 Minority-Carrier D i f f u s i o n Current Relations 4 1.4 Mechanisms of Ca r r i e r Generation in Base 6 1.5 Non-Saturating Reverse Currents 7 1.6 Breakdown Mechanisms 9 II CARRIER GENERATION IN THE BASE REGION 12 2.1 Ch a r a c t e r i s t i c s of Transistors Investi- 12 gated 2.2 Tests for Surface Conduction 13 2.3 Capacitance C h a r a c t e r i s t i c s 15 2.4 Activation Energies 16 2.5 Comparison of Space-Charge Generated and Diffu s i o n Currents 20 2.6 Anomalous Behaviour at Low Temperatures 23 III ORIGIN AND DISTRIBUTION OF REVERSE CURRENTS UNDER CONDITIONS OF DOUBLE-DEPLETION 25 3.1 Current D i s t r i b u t i o n under Conditions of Double-Depletion 25 3.2 Non-Saturating Reverse Currents 28 3.3 Operation of Transistors as Amplifiers under Conditions of Double-Depletion 30 (v) Chapter Page IV DISCUSSION OF BREAKDOWN VOLTAGES 35 V SUMMARY 38 APPENDIX 41 BIBLIOGRAPHY 43 ( v i ) ILLUSTRATIONS FIGURE FACING PAGE SPACE-CHARGE AND POTENTIAL DISTRIBU-TION IN THE BASE REGION CURRENT VOLTAGE CONVENTIONS FOR A PNP TRANSISTOR 4 BASIC TRANSITION PROCESSES 6 p-TYPE SURFACE CHANNEL ON AN N-TYPE BASE REGION 5 CIRCUIT FOR MEASUREMENT OF FLOATING EMITTER POTENTIALS 12 6 CIRCUIT FOR MEASUREMENT OF^Q 14 7 CIRCUIT FOR MEASUREMENT OF SURFACE CONDUCTANCE 15 8 CIRCUIT FOR MEASUREMENT OF JUNCTION CAPACITANCE 16 JUNCTION CAPACITANCE CURVES 16 10 THREE-DIMENSIONAL MODEL OF A PNP TRANSISTOR UNDER VARIOUS BIASING CONDITIONS 17 11 LOGARITHM OF BASE CURRENT VS V T 19 12 MODEL FOR COMPARISON OF DIFFUSION AND SPACE-CHARGE GENERATED CURRENTS 21 ( v i i ) FIGURE FACING PAGE 13 LOW TEMPERATURE INCREASE IN BASE CURRENT 23 14 REVERSE CURRENT-VOLTAGE CHARACTERISTIC 27 15 REVERSE CHARACTERISTIC SHOWING INCREASE IN SATURATION CURRENT 3© 16 COMPARISON OF I b / I m VS V-V m WITH THEORETICAL CURVE FOR CYLINDRICAL EXPANSION OF THE SPACE-CHARGE REGION 31 17 WORKING OF A TRANSISTOR UNDER CONDITIONS OF DOUBLE DEPLETION 33 18 MODEL FOR CYLINDRICAL EXPANSION OF THE SPACE-CHARGE REGION 41 TABLE PAGE 2 13 ( v i i i ) ACKNOWLEDGEMENTS The author wishes to thank Professor R.E. Burgess for his guidance of thi s investigation and help in preparation of this thesis. Financial support in the form of a Defence Research Board grant i s g r a t e f u l l y acknowledged. i CHAPTER I INTRODUCTION 1.1 Double-Depletion in Reverse-Biased Transistors I Double-depletion in pnp and npn junction t r a n s i s t o r s occurs when both junctions are reverse-biased so that the space charge regions normally present at thermal equilibrium are extended into the base. In thi s circumstance there i s no i n j e c t i o n of c a r r i e r s into the base region and the current across the junction i s due almost e n t i r e l y to c a r r i e r s thermally generated in the base. If s u f f i c i e n t l y large reverse biases are applied the two space charge regions w i l l meet afte r which a further increase i n the reverse bias of one junction w i l l cause the space-charge region of that junction to extend across the base ultimately reaching the opposite junction - a condition known as "punchthrough". At punchthrough there i s a continuous potential gradient between the two junctions and a current due to i n j e c t i o n of c a r r i e r s from the less biased junction w i l l flow between them. With further increase i n the variable reverse bias t h i s current w i l l increase rapidly to tend to keep the voltage drop across the space-charge regions constant. I f , however, the two reverse biases are kept equal and increased, punchthrough w i l l not occur; there w i l l be only a slow change i n the currents across the junctions u n t i l the occurence of break-down characterized by a rapid increase i n current. SPACE-CHARGE AND POTENTIAL DISTRIBUTION IN THE BASE REGION emitter space charge region collector space charge region w F I G . 1 Ji 2 This work i s concerned with three points of fundamental interest i n the study of the in t e r a c t i o n of the two depletion regions: the c h a r a c t e r i s t i c s of the c o l l e c t o r and emitter currents before punchthrough, the mechanism of hole-electron pair generation in the base, and the mechanism of breakdown. Relevant previous t h e o r e t i c a l and experimental investigations are summarized in the remainder of t h i s chapter. The three succeeding chapters describe experimental work on selected tran s i s t o r s and correlate the r e s u l t s with e x i s t i n g theory. The f i n a l chapter consists of a summary of the work and a description of a simple model which agrees with the experi-mental r e s u l t s . 1.2 Potential and Space-charge D i s t r i b u t i o n s for an Idealized Transistor. Figure 1 shows the potential and space-charge d i s t r i b u -tions under the condition of double depletion for an i d e a l i z e d one-dimensional pnp t r a n s i s t o r . The b u i l t - i n potential steps at the junctions are neglected as small compared to the applied reverse biases. In order to have zero net charge in a junction, the r a t i o of the widths of the space-charge regions i n the p- and n-type materials must be i n the inverse r a t i o of their impurity ion densities. Here the n-type r e s i s t i v i t y i s much greater so the expansion of the space-charge region into the p-type material may be neglected. It 3 i s further assumed that the boundaries of the space-charge regions are abrupt whereas for a r e a l junction they are spread over one Debye length, L d , where ^d = ~y €*T/4 J f q 2 N 1.2.1 and £ = permitivity, k = Boltzmann's constant, q = e l e c t r o n i c charge, and N = impurity ion density i n the base. The potential d i s t r i b u t i o n i s obtained by a simple double integration of Poisson's equation under the boundary condi-tions that the potential be equal to the applied biases at the junctions and zero within the undepleted region of the base. The width of the space-charge region for each junction i s found to be «fre - V e) , d„ - 1 (d>c - V c) 1.2.2 2 P 1 2 P where p i s the space charge density, and (J)e ? (fyc the b u i l t -in biases at the emitter and c o l l e c t o r and V e, V c the magnitudes of the applied bias at the junctions. In the case of a single space-charge layer, punchthrough w i l l occur at the emitter-collector potential drop given by V p - 2Pw 2 * 1.2.3 At punchthrough a s h o r t - c i r c u i t path e x i s t s between emitter and c o l l e c t o r and the c o l l e c t o r and emitter bias voltages w i l l remain locked together, that i s I V c - Vgl = Vp 1.2.4 CURRENT VOLTAGE CONVENTIONS FOR A PNP TRANSISTOR . 1 . EMITTER BASE COLLECTOR P N P F I G . 2 From figure 1 i t may be seen that complete depletion of the base corresponds to a potential d i s t r i b u t i o n in which the two parabolic segments unite to form a single parabola. The voltage r e l a t i o n s h i p i s then V^ e ~ V e + -\J$C - V c - -\/vp" 1.2.5 The presence of the ionic charge Q = NqAd in a depletion layer r e s u l t s i n a dynamic capacitance measured between base and either c o l l e c t o r or emitter of C = dQ = A . 1.2.6 ~dV 2(V + or geometrically C e b = - £ A C c b = C A 1.2.7 d e d c where A i s the cross-sectional area of the ideal planar diode and d e and d c refer to the widths of the corresponding space charge regions. 1.3 Minority-Carrier D i f f u s i o n Current Relations The theory of current flow developed by Shockley (1949) can be applied to the model described above i f we make the assumptions of non-degenerate d i s t r i b u t i o n s for electrons and holes and long bounding p regions. With the current and voltage conventions shown i n figure 2 Shockley's equations are: I e = AqDpPn L p I c =-AqPppn coth ^(e KT - 1) - csch(e - 1) L L P coth w(e * r - 1) - cschCe"^ - 1) 1.3.1 1.3.2 where q = e l e c t r o n i c charge, p Q = thermal equilibrium concentration of holes i n the base and Dp = hole d i f f u s i o n constant. For double depletion both V e and V c are negative so there w i l l be no hole i n j e c t i o n . The two components of each current reach saturation values when the reverse biases exceed a few times kT/q. When w i s within an order of magnitude of L, the f i r s t term i n the expressions for I e and I c w i l l dominate and the reverse currents w i l l be v i r t u a l l y independent of the bias at the other junction. However, as was pointed out by Early (1952) the widening of the space charge layers w i l l reduce the e f f e c t i v e width of the base, and thus some control of the c o l l e c t o r current by v a r i a t i o n of emitter bias becomes possible. Stafayev, Tuchkevich and Yakovchuck (1956) have shown that operation of reverse biased t r a n s i s t o r s as amplifiers i s possible i f the e f f e c t i v e base width can be decreased to the magnitude of L before the currents are f u l l y saturated. The requirements for achieving useful amplifica-tions are: a high operating temperature to provide a large number of thermally generated minority c a r r i e r s i n the base, a narrow base width so that the e f f e c t i v e base width w i l l BASIC TRANSITION PROCESSES E, Energy of an E l e c t r o n EJ (a) (b) (c) (d) Arrows denote e l e c t r o n t r a n s i t i o n . (a) e l e c t r o n capture, (b) e l e c t r o n emission, (c) hole capture, (d) hole emission FIG. 3 be small before the reverse currents have saturated, and low impurity density in the base i n order that the depletion regions w i l l expand into the base rather than into the emitter and c o l l e c t o r . 1.4 Mechanisms of Carrier Generation in Base i The saturation currents across the junctions may arise through several mechanisms. Most important of these are bulk and surface generation of c a r r i e r s . There may also be some current due .to hole i n j e c t i o n by the contacts to the base. Creation of hole-electron pairs i n the bulk of the base may occur either by d i r e c t t r a n s i t i o n s of electrons from the valence to the conduction band or through the medium of recombination centres having energy levels i n the forbidden zone. These centers arise from defects of a chemical or physical nature i n the c r y s t a l l a t t i c e . Shockley and Read (1952) have used a s t a t i s t i c a l approach to derive the recombination rates i n the body of a non-degenerate semiconductor having a single trap l e v e l . C a r r i e r recombination and generation are considered to proceed by means of the four basic processes i l l u s t r a t e d i n figure 3, at rates determined by the a v a i l a b i l i t y of holes and electrons and the densities of occupied and unoccupied centers. Under equilibrium conditions the rates of hole and electron emission are equal to the rates of hole and electron capture respectively, and under steady state conditions the net rate of electron capture i s equal to the net rate of hole capture. 7 With these assumptions Shockley and Read f i n d the net re-combination rate per unit volume to be 2 r - g = U = P N - n i 1.4.1 (n - n1)Tp o + (p - Pi)Tno where n = density of electrons i n the conduction band, p = density of holes i n the valence band, (p^} = density of electrons (holes) in the conduction (valence) band when the Fermi l e v e l f a l l s at the trap l e v e l , Tpo = l i f e t i m e for holes in highly n-type material, Tno = l i f e t i m e for electrons in highly p-type material, and n^ = density of holes or electrons i n an i n t r i n s i c specimen. In a space charge region the c a r r i e r s are swept out before they can recombine consequently the generation rate for hole-electron pairs becomes 2 8s.c.^ 1 ^ 1.4.2 %o n i + Tno P X An analysis of surface recombination has been c a r r i e d out by Brattain and Bardeen (1952) along l i n e s s i m i l a r to the above treatment. In their model, surface states replace the recombination centers present in the bulk. Surface recombina-tion i s more d i f f i c u l t to investigate as there i s almost c e r t a i n l y more than one trap l e v e l . 1.5 Non-saturating Reverse Currents For an ideal t r a n s i s t o r under conditions of double deple-tion the base current w i l l be constant over the range between 8 a few tiroes kT/q and the onset of breakdown. However, in most pn junctions the reverse current increases with increasing bias. This was o r i g i n a l l y attributed to surface leakage, but in cases where the space charge generated current predominates an additional voltage dependence may exist due to expansion of the space charge region with increasing reverse bias. The l a t t e r type of voltage dependence has been i n v e s t i -gated by Sah, Noyce and Shockley (1956). They have shown that the space charge generated current w i l l tend to dominate i n t r a n s i s t o r s i n the case of short l i f e t i m e s , low r e s i s t i v i t i e s and large energy gaps. For a step-junction with a reverse bias over a few times kT/q the generation rate w i l l be v i r t u a l l y constant throughout the space charge region so the reverse saturation current w i l l increase as the square root of the applied reverse bias. Brown (1952) has studied surface leakage v i a areas of n-type conductivity on the surface of p-type germanium. In his theory he proposes that previous chemical and e l e c t r i c a l treatment has resulted in a layer of positive charge on the surface of the p-type material. E l e c t r i c a l n e u t r a l i t y then requires a net d e f i c i t of negative charge within the c r y s t a l and thus there arises an inversion layer just below the surface, in which there i s an excess of electrons. These electrons provide a surface conducting path or "channel". Later experiments (Christensen 1954) have revealed the I p-TYPE SURFACE CHANNEL ON n-TYPE BASE REGION type Ge V = V , n-type Ge space charge r e g i o n i - channel length 'V FIG. \ occurrence of p-type channels on n-type germanium. The Brown model has been extended by McWhorter and Kingston (1954). They assert that the channel forms an extension of the normal reverse-biased junction as shown in figure 4. There w i l l be an increase i n reverse current due to bulk generated electrons and surface generated holes being swept into the channel. The current flowing in the channel causes a voltage drop along i t s length; the e f f e c t i v e length of the channel over which i t c o l l e c t s c a r r i e r s extends from the junction to the point at which the voltage in the channel i s equal to the thermal energy of the c a r r i e r s , kT/q. Taking the surface resistance of the channel as proportional to the applied voltage, McWhorter and Kingsley deduced that for the junction of figure 4, the excess reverse saturation current would vary as (lnV/V a)* where V i s the applied bias and V Q the voltage at the extreme of the e f f e c t i v e length of the channel. 1.6 Breakdown Mechanisms Two d i s t i n c t mechanisms have been advanced to account for the sharp r i s e in the current-voltage c h a r a c t e r i s t i c of a reverse biased junction. According to the Zener breakdown i* f theory the rapid increase of current with voltage i s due to the d i r e c t e x c i t a t i o n of electrons from the valence to the conduction band under the influence of a strong e l e c t r i c f i e l d . The avalanche breakdown mechanism i s analogous to Townsend 10 discharge in gases. The electrons or holes gain s u f f i c i e n t energy in the high f i e l d region to cause impact i o n i z a t i o n of the neutral atoms. The electron-hole pairs thus formed in turn produce other c a r r i e r pairs and a m u l t i p l i c a t i v e process ensues. The form of the current voltage c h a r a c t e r i s t i c at break-down serves to determine which mechanism i s responsible. Zener breakdown occurs abruptly when a ce r t a i n c r i t i c a l f i e l d i s reached somewhere i n the semiconductor. Whereas i n avalanche breakdown current m u l t i p l i c a t i o n commences before Zener breakdown. In the case of avalanche breakdown the application of reverse bias w i l l widen the area of the junction and increase the probable number of io n i z i n g c o l l i s i o n s . The extent to which the reverse bias widens the junction area Is dependent on the impurity density; thus extent of avalanche m u l t i p l i c a t i o n i s dependent on impurity density while the f i e l d for Zener breakdown i s not. 'The form of the reverse current c h a r a c t e r i s t i c before breakdown when the avalanche mechanism i s operative i s ob-tained by multiplying the saturation current by an empirical factor due to M i l l e r (1955): M(V) = 1 1.6.1 1 - ( V / V B ) 3 where V Q i s the breakdown voltage which in Ge i s given approximately by , V R - 1 0 1 3 N " 0 - 7 2 8 1.6.2 CHARACTERISTICS OF TRANSISTORS TRANSISTOR 13 21 h6 TYPE 2N137 2N137 2N137 V P 12 v 7 v Ik v BASE WIDTH 1.3x10" 3 cm 1.3x10 3 cm 1.3x10 1 cm 0.0021 cm 2 0.0060 cm z O.OOUx) cm2" A c 0.00M+ cm 2 O.OO30 cm 2 0.0073 cm 2 N i • 14 _> 3.0x10 cm5 l . O x l O1 4 cm"3 l A x l o ' 4 cm" 3 V B 328 v 795 v 608 V TABLE 1 11 Zener breakdown need not be considered i n interpreting the r e s u l t s of this work. M i l l e r has shown that very high f i e l d s are required for Zener breakdown - greater than 320 kv cm"*-, in germanium. M i l l e r has pointed out that surface breakdown may occur at voltages lower than bulk breakdown voltage. Brattain and Garrett have investigated t h i s phenomena. Their r e s u l t s indicate that surface breakdown l i k e bulk breakdown i s m u l t i p l i c a t i v e , and that i t w i l l occur at voltages lower than bulk breakdown voltage i f the surface charge i s the same sign as that of the fixed body impurity on the high r e s i s t i v i t y side of the junction. CIRCUIT FOR MEASUREMENT OF FLOATING EMITTER POTENTIALS EMITTER BASE COLLECTOR P N P VTVM Ve) Res. = 10 XL • W W — i © I FIG. 5 12 CHAPTER II CARRIER GENERATION IN THE BASE REGION » 2.1 C h a r a c t e r i s t i c s of Transistors Investigated The t r a n s i s t o r s used i n the investigation were chosen from commercially available germanium fused junction, high frequency units. The thin base of high frequency types generally r e s u l t s in a punchthrough voltage lower than the breakdown voltage, and a fused junction comes nearer than a diffused junction to providing the step junction of the theory. Several parameters which are useful i n the s e l e c t i o n of units for various measurements and i n the evaluation of r e s u l t s may be obtained by measurement or calculated from the manufacturer's s p e c i f i c a t i o n s . These are l i s t e d for three units i n Table 1. The base widths were obtained from the manufacturer's alpha-cutoff frequency and the formula w The punchthrough voltages were measured i n the c i r c u i t of f i g . 5 devised by Schenkel and Statz (1954). v p i s indicated by a sharp increase i n the f l o a t i n g potential of the open c i r c u i t e d junction as the reverse bias on the other junction i s increased. The emitter and c o l l e c t o r areas were obtained from measurements of junction capacity at punchthrough as described FLOATING EMMITER POTENTIALS AND CONDUCTANCES TRANSISTOR TYPE MEASURED fi MEASURED ( v o l t s ) v f ( c a l c . ) = ¥ln(n*) g ( mhos) 13 2N123 ±7 0.075 0 .095 0.0032 21 2N137 27 0.10 0.080 39 2N137 ko 0.135 0 .09^ 0.0011 2N137 31 0.068 0.0$g 0.0056 12 2N123 5V 0.107 0.101 ' 0.0009 TABLE 2 13 * in section 2.3, and the expression for geometrical capacity in equation 1.1.6. The bulk breakdown voltages were calculated from the empirical equation of M i l l e r (1956} 13 -0.72S Vfi = 1.0 x 10 Nj _ 2.1.2 where Nj i s the impurity density in the base region. 2.2 Tests for Surface Conduction Since the theory outlined i n the introduction i s based on bulk properties of the base region, the units used i n the investigation must be chosen to be as free as possible from surface conduction channels between the c o l l e c t o r and emitter. These can be detected ( i ) i n d i r e c t l y by measure-ment of the maximum f l o a t i n g emitter potential when the c o l l e c t o r junction i s reverse biased below punchthrough or ( i i ) d i r e c t l y by measurement of the ac conductance between emitter and c o l l e c t o r when both junctions are reverse biased to prevent bulk conduction. From equation 1.3.2, when I e 0 the maximum f l o a t i n g emitter potential i s V f = kT l n d - O C ) 2.2.1 q where fY = sechw i s the common-base current gain. Brown (1953) found that the presence of surface channels leads to a f l o a t i n g emitter potential several times t h i s value. CIRCUIT FOR MEASUREMENT OF fj 0 . 5 ^ FIG. 6 14 Specimens of four d i f f e r e n t types of tra n s i s t o r were tested i n the c i r c u i t of figure 5, and the measured f l o a t -ing emitter potentials were f i r s t compared with those calculated from the manufacturer's value of alpha. Ia a l l cases the measured values were s l i g h t l y lower than those calculated indicating that the actual alpha's were lower than those s p e c i f i e d . Alpha's for the in d i v i d u a l t r a n s i s t o r s were therefore obtained experimentally using the c i r c u i t of f i g . 6 which measures common emitter current gain as / ? - V 2 / R L 2 . 2 . 2 Vl/Ri i f R L i s small compared with the output impedance of the tran s i s t o r and R large compared with the input impedance, ^ i s related to Of by a = 2 . 2 . 3 Measurements were made at a frequency of 10 kc. The values of thus obtained are necessarily inaccurate because at high frequency the emitter capacity shunts the input current. The purpose of the high impedance c o i l i s to reduce t h i s inaccuracy by tuning the emitter capacity. However, emitter capacity varies from unit of unit, and the f i n i t e r e s i s -tance of the c o i l necessitates readjustment of the supply voltage; hence t h i s i s not a fixed condition test. The values of Vf calculated from the measured alpha's are in most cases s l i g h t l y lower than measured. Measured CIRCUTt JJ-OR MEASUREMENT OF SURFACE CONDUCTANCE Resistance Box - W W 0 210 cps 0.1 v rms Internal Res. -1 10 k 0.1/Af Output c i r c u i t tuned to 210 cps FIG. 7 15 and calculated values are compared in Table 2. A method of measuring surface conductivity which i s more d i r e c t l y quantitative than the Brown test i s that of Statz, de Mars, and Davis (1958). The two junctions are provided with a common reverse bias to prevent bulk conduction as shown i n f i g . 7, and an ac signal i s injected into the emitter-base c i r c u i t . The conducting path provided by the surface channels i s replaced by a resistance box and the resistance adjusted to produce the same voltage at the output of the tuned amplifier. The disadvantages of thi s method are that for very low conductances noise may obscure the s i g n a l , and the reverse biasing may p r e c i p i t a t e surface breakdown with a consequent m u l t i p l i c a t i o n of c a r r i e r s and increase i n conductance as i s probably the case with specimens 46 and 13. 2.3 Capacitance C h a r a c t e r i s t i c s The d i f f e r e n t i a l capacities of reverse biased junctions were'measured in several tran s i s t o r s using the c i r c u i t of figure 8. Not only do these measurements allow an estimate of the junction areas and impurity densities, but also provide c r i t e r i a of uniform impurity density and planar geometry, for, in t h i s case, from equation 1.6.1, a plot of In C vs In V should y i e l d a straight l i n e of slope - when V » § . The curves shown in f i g . 9 were c h a r a c t e r i s t i c of two tr a n s i s t o r s . Others did not show the sharp increase in CIRCUIT FOR MEASUREMENT OF JUNCTION CAPACITANCE SIGNAL GENERATOR SCOPE FIG. 8 JUNCTION CAPACITANCE CURVES x em i t t e r and c o l l e c t o r T . . J 1 1 1 — i — - » 1 1 1 <— 2 3 4 f 6 S 'O \l Zo Jo Vp Reverse B i a s ( v o l t s ) F IG. 9 16 c o l l e c t o r and emitter junction capacities at punchthrough. A l l gave slopes of around -0.35. The sharp increase in junction capacity at punchthrough was f i r s t noted by Barker (1956). His explanation i s that the r e a l geometry of a tran s i s t o r i s spherical rather than plane and when as in f i g . 10b the expanding c o l l e c t o r space charge region just touches the emitter space charge region, the emitter capacitance i s suddenly shunted into the c o l l e c t o r c i r c u i t . Those specimens exhi b i t i n g the Barker e f f e c t were those that showed low breakdown voltages with " s o f t " breakdown c h a r a c t e r i s t i c s . Perhaps in the Barker e f f e c t t r a n s i s t o r s the low conductance associated with i n c i p i e n t breakdown re s u l t s i n a low resistance path between the two space charge regions when their outer boundaries just touch, while in the non-Barker tr a n s i s t o r s a low resistance contact i s not established u n t i l the two regions have overlapped to the extent of a Debye length at which point the emitter space charge w i l l not have much external area. 2.4 Activation Energies The generation rate of c a r r i e r pairs v i a recombination centers in a space charge region i s , from equation 1.4.2 2 g = i 2.4.1 X?° n l + Tno p l . where THREE DIMENSIONAL MODEL OF A PNP TRANSISTOR UNDER VARIOUS BIASING CONDITIONS 17 nl = 2 2 7Tme kT 2 exp - Ec ~ Et Pi = 2 27T*mK kT i 3 / 2 exp -kT Et - E, 2.4.2 kT E = energy l e v e l of the trap, E = l e v e l of the bottom of the conduction band, E = l e v e l of top of valence band, m e f f e c t i v e mass of electron, m^ = e f f e c t i v e mass of hole, and h = Planck's constant. The l i f e t i m e s can be expressed as r = 2.4.3 <vCONt where v = the thermal v e l o c i t y , (f the capture cross sec-tion and N t = the density of recombination centers. If the capture cross section i s assumed independent of temperature, / OC (velocity ) a oc T Z . Generally one of the terms in the denominator of 2.4.2 predominates so that the generation rate i s of the form g = constant x T exp(- C/kT) 2.4.4 where £" = a c t i v a t i o n energy or separation of trap l e v e l and band edge. It i s customary to assume that a c t i v a t i o n energy follows the r e l a t i o n € = 6 0 + rT 2.4.5 £"0 being the value at absolute zero and y some constant. 18 If i n t r i n s i c generation occurs within the space-charge region the generation rate w i l l be (Shockley 1949) 2 g = (constant) p i 2.4.6 r From Conwell (1952), n i = 9.3 x 10 3 tT 3exp(-87OO/T)cm~ 3 2.4.7 so, for i n t r i n s i c generation g = (constant JT^ exp(-CQ^AT) 2.4.8 where << i s an exponent depending upon the temperature dependence of *f . Now for a pnp t r a n s i s t o r under conditions of double-depletion as in f i g . 9c we make the assumption that a l l the holes generated within the depleted region are car r i e d either to the emitter or c o l l e c t o r and that this space charge generated current predominates over current which arises from holes generated within the undepleted region adjacent to the space-charge region. Then lb = I e - I c = g(volume of space charge) 2.4.9 If generation i s through recombination centers the slope of a In I b vs 1/T plot w i l l be from 2.4.4 s * - So/k - 2T 2.4.10a and i f i n t r i n s i c generation predominates the slope w i l l be s = ~^>&A - f T 2.4.10b LOGARITHM OF BASE CURRENT VS 1/T (Transistor no. h6) 19 Base currents were measured in double depleted tr a n s i s t o r s over a range from 90°C to -60°C. The emitter and c o l l e c t o r leads of the specimens were joined and the junctions reverse biased s u f f i c i e n t l y to cause a large overlap of the two space charge regions as i n f i g . 9c. This condition i s assured by making the reverse bias equal to or larger than Vp. For measurements above room temperature the t r a n s i s t o r was placed in a heated o i l bath and readings were taken as the bath cooled. For measurements below room tempera-ture the o i l bath was frozen in a mixture of dry ice and acetone, and readings taken as the bath warmed. A mercury thermometer was used above room temperature and an alcohol thermometer belowJ in both cases a copper s t r i p provided a thermal connection between the thermometer and the base lead. Temperature r i s e in the base due to power d i s s i p a -tion i s not serious. A t y p i c a l d i s s i p a t i o n constant in a i r i s 0.6°C/mwj at an extreme of 200fia and 10 vo l t s the temperature r i s e would be appreciably less than 1.2°C due to the cooling provided by the base f i n and o i l bath. The voltage drop between the base and collector-emitter was kept constant by adjustment of the power supply. A t y p i c a l plot of log I b vs 1/T i s shown in f i g . 11. The a c t i v a t i o n energies Indicated by the upper slopes of the curves ranged from 0.70 ev. to 0.61 ev. The value of 20 6 o G . i s 0.785 ev. (Conwell 1958), hence i t i s concluded that the currents at these temperatures do not arise from i n t r i n s i c generation but from generation through recombina-tion centers. One wonders i f i n t r i n s i c generation might begin to predominate at a temperature above 90°C. A the o r e t i c a l estimate of "onset temperature", the temperature at which i n t r i n s i c generation dominates may be made i f i t i s assumed that t h i s occurs when the majority c a r r i e r concentration n = 2Nn where Nj) i s the donor density. Using the mass action law n^ = pn 2.4.11 and the charge balance n - p = N D 2.4.12 one obtains nf = 2Np 2.4.13 when n = 2Np. Using equation 2.4.7 for n the onset tempera-ture can be obtained for a given Np. For specimens studied these temperatures ranged between 60 and 80°C. It i s therefore concluded that i n t r i n s i c generatidn does not occur at a l l in the units used in this investigation. 2.5 Comparison of Space charge Generated and Diffusi o n Currents. In making the analysis of ac t i v a t i o n energies in the MODEL FOR COMPARISON OF DIFFUSION AND SPACE CHARGE GENERATED CURRENTS L a I collector FIG. 12 emitter 21 preceeding section i t was assumed that space charge-generated current predominated over d i f f u s i o n current. This assumption w i l l now be tested a n a l y t i c a l l y follow-ing a method of Sah, Noyce and Shockley (1952). Several assumptions regarding physical and geometrical quanti-t i e s must be made, and to be consistent a l l w i l l be chosen to make the space-charge-generated current a minimum. The model in f i g . 12 represents the same physical conditions as that of f i g . 9c only i t i s assumed that the overlapping space-charge regions form a c y l i n d r i c a l column between the emitter and c o l l e c t o r . The cross-section of the column i s chosen as the c i r c u l a r area of the c o l l e c t o r . As discussed i n section 1.4, the d i f f u s i o n current consists of, on the average, a l l the minority c a r r i e r s generated within one d i f f u s i o n length L of the i space-charge region. The electron and hole l i f e t i m e s Tpo and Tn.o are assumed equal and the trap l e v e l i s assumed to f a l l at the i n t r i n s i c Fermi l e v e l . From equation 1.4.4 the genera-tion rate In the space charge region then has i t s minimum value of ' _ ni g s . c . ~ 2.5.1 and from 1.4.2 the generation rate in the undepleted n-type I 22 region i s g d = Po 2.5.2 T Multiplying the generation rates by the appropriate volumes calculated from f i g . 10 gives the r a t i o of the space charge-generated and d i f f u s i o n currents: I s . c . » no . a 2 2.5.3 I d 2ni L 2 + 2aL At temperatures above 300°K, n Q i s of the order of N D or 10 l 4cm~ 3, nt i s 2.5 x 10 1 3cm 3at 300°K, and a i s about 0.05 cm for the tr a n s i s t o r s used. The d i f f u s i o n constant D may be obtained from the mobility of holes in germanium, /jp = 1800 cm volt sec and the Ei n s t e i n r e l a t i o n D p = z / p * T 2.5.4 q The d i f f u s i o n length i s then given by L = -\JDT 2.5.5 It i s necessary to make an assumption regarding the l i f e t i m e . If following Sah, Noyce and Shockley, we assume T = 10" 6sec, then L = 7 x 10 _ 3cm « a and I s . c . / I d - 7 -T= I0~ 5sec, I 8 . c . / I d - 2 - 2 i a n d f o r T - 10" 7sec, I g c / I d ^ 2 2 It i s concluded that the assumption that space charge-generated current predominates i s a reasonable one. LOW TEMPERATURE INCREASE IN BASE CURRENT 0.1 -* 1 • f 7-100 200 250 Temperature (deg. K) FIG. 13 23 2.6 Anomalous Behaviour at Low Temperatures An anomaly was revealed when the measurements of section 2.4 were extended to 190°C by freezing the o i l bath in l i q u i d nitrogen. Three specimens exhibited an increase in current with decreasing temperature below a certain temperature. The curve of f i g . 13 i l l u s t r a t e s t h i s for tra n s i s t o r no. 20, a 2N137 unit. A l l the units showing this anomaly possessed very low breakdown voltages and " s o f t " breakdown c h a r a c t e r i s t i c s . This phenomena cannot be accounted for by the Shockley-Read theory of recombination and generation v i a recombination centers. The net generation rate given by equation 1.4.1 must always decrease with temperature; t h i s i s shown for the case of generation within the space charge region i f we take 7^ ,Tp°cT z • Then equation 1.4.2 i s of the form _ Constant T exp(- ^/kT) 2.6 g s • c • = constTexp-(E c - E t/kT)+const. T exp-(E t - E v/kT| This i s c l e a r l y a monotonlcally Increasing function at a l l temperatures; the same i s true for the more general case given by eqn. 1.4.1. These r e s u l t s are si m i l a r to an anomalous increase i n c a r r i e r concentration at low temperatures observed in heavily doped s i l i c o n p-type specimens by Pearson and 24 Bardeen. These r e s u l t s were associated with degeneracy of electrons in the conduction band. It i s worth considering whether the anomalous r e s u l t s in the present case may not be caused by degeneracy r e s u l t i n g from avalanche m u l t i p l i c a t i o n Of c a r r i e r s in the base. The required density of electrons in the conduction band for degeneracy to occur at a given temperature i s , from Shockley (1949) ndeg 3.7 x 10 15 ,1 £ mnJ*T2 2.6.3 m where mn = e f f e c t i v e mass of electron, about 0.5 m for i n t r i n s i c germanium (Conwell 1958). For no. 20 the current increase occurs at 120°K for which n d e g = 2 x lO^cm, i s required. However, the concentration of electrons that would occur in the absence of breakdown i s n = 4f(Np -VNg - 4nf)- = ND 2.6.4 as NjjX/'n^ Thus at 120°K, n = I0 l 4cm. so a m u l t i p l i c a t i o n factor of 10,000 i s required. The avalanche m u l t i p l i c a t i o n factor as defined by 1 - " s a t u r a t i o n 2 ' 6 ' 5 i s only about 10 at 15 v o l t s reverse bias, room temperature. Thus i t i s concluded that degeneracy i s not responsible for the anomaly. 25 CHAPTER III ORIGIN AND DISTRIBUTION OF REVERSE CURRENTS UNDER CONDITIONS OF DOUBLE-DEPLETION 3.1 Current D i s t r i b u t i o n under Conditions of Double-Depletion The d i s t r i b u t i o n of current i n an ideal t r a n s i s t o r under conditions of double-depletion may be obtained d i r e c t l y from the Shockley equations (1.3.1 and 1.3.2). However, there are several factors which w i l l cause the current d i s t r i b u t i o n in a r e a l t r a n s i s t o r to deviate from that predicted by theory: design practice i s to make the c o l l e c t o r area several times that of the emitter i n order to improve; there may be unequal impurity density through-out the base material; and avalanche m u l t i p l i c a t i o n may occur at one or both junctions. The simplest modification of the Shockley equations which takes into account unequal electrode areas i s the replacement of the common area A for the c o l l e c t o r and emitter by the d i f f e r i n g areas A 0 and A . The equations © c then become, for a pnp tr a n s i s t o r , 3.1.1 P JAcCschfCe* 1* --D - Aecoth£(e < r - 1) 3.1.2 26 For the si t u a t i o n in which the c o l l e c t o r i s f l o a t i n g and the emitter i s reverse biased to saturation, ie I c = 0, we have l e o = - Pnq DP A^tanh e 3.1.3 and, s i m i l a r l y , when the o o l l e c t o r i s biased to saturation and the emitter i s f l o a t i n g I c o . + En^A ctanh£ 3.1.4 the r a t i o of the reverse saturation currents i s then eo _ _ A 0 CO 3.1.5 as for an ideal t r a n s i s t o r . Another s i t u a t i o n of interest i s that for which both junctions are reverse biased to saturation but not to so high a voltage that the e f f e c t i v e base width i s s i g n i f i c a n t l y decreased by expansion of the space charge regions. For this case we have Ie = w w - Accsch-r; + A e c o t h — and Ic " f A e c s c h ^ - A ^ c o t h ^ The r a t i o of these currents i s q PnDp 3.1.6 3.1.7 Ie = - A p c s c h 4- A e c o t h % A c s c h ? - A c o t h * e L c L 3.1 .8 ( A c - A e ) - ^ r ( 2 A e + A „ ) f o r w « L ( A c " V • £ i T ( 2 Ac + V REVERSE CURRENT-VOLTAGE CHARACTERISTIC Reverse B i a s ( v o l t s ) F I G. Ht 27 Experimental measurements of the currents described above were made for two tr a n s i s t o r s and a t y p i c a l curve i s i l l u s t r a t e d i n f i g . 14. The r a t i o Ieo/^co i s only -1.07 implying, contrary to fact, that A e ^ A c . The r a t i o of the saturation currents when both junctions are reverse biased i s I e / I c = -0.28. The largest value of A c/A e for which the r a t i o of equation 3.1.8 i s possible i s about 3.6 and this happens for the case that w y L, whereas i n fact L i s larger than w for the units tested. The simple modification of the Shockley equation introduced i s thus unsatisfactory since i t leads to incon-s i s t e n c i e s in the choice of the parameters w/L and A c/A e to disagreement with the values of A c and A e obtained by capacity measurements. We therefore conclude that the other factors mentioned at the beginning of the section are important i n determining the d i s t r i b u t i o n of reverse currents in the units tested. As the regulation of the reverse current at one junction by v a r i a t i o n of the reverse current at the other w i l l be discussed i n section 3.3, i t i s of interest to note that for the t y p i c a l unit of f i g . 14, the c o l l e c t o r saturation current can be varied over a range of 13//.& by varying the emitter bias, while the emitter current can be varied 4*7// a by varying the c o l l e c t o r reverse bias. 28 3.2 Non-saturating Reverse Currents It has been shown in Section 2 that the reverse current at a junction arises mainly from c a r r i e r s generated in the space-charge region. To determine whether the non-saturating of the reverse current can be ascribed to expansion of the space charge region an experimental s i t u a t i o n must be chosen for which the volume of the space charge region can be related to the applied bias. For a transistor i n which the c o l l e c t o r and emitter are hemi-spheres of p-type material extending into the base, ,this may be arranged by applying an equal reverse bias to the emitter and c o l l e c t o r of such a value that the two space charge regions overlap to form a c y l i n d r i c a l column of space charge extending outwards into the base wafer. The volume of t h i s space-charge region may then be obtained by an integration of Poisson's equation in c y l i n d r i c a l co-ordinates with suitable boundary conditions. This i s c a r r i e d out in the Appendix with the r e s u l t , i n mks units, where V i s the reverse bias, V m the minimum reverse bias at which c y l i n d r i c a l geometry can be applied ( i . e . at which the region d i r e c t l y between the junctions i s j u s t depleted), R the radius of the cylinder at the l a t t e r voltage, and a the radius of the cylinder for V>V m. Now, the space-charge generated current i s d i r e c t l y proportional to the volume of the space charge region which 3.2.1 29 o i s in turn d i r e c t l y proportional to , so equation 3.2.1 may be written V - V = /^R 2 [ 1 - I_ + I , 1 j 3.2.2 where I i s the base current, that i s , the t o t a l reverse ions, and 1, t h i s may current across the emitter and c o l l e c t o r junct s, i I m i s the base current at V •» V m. For be approximated as ^ I__ - 1 Im V - V- = P R 2 ' m 8 ^ I - I jm 3.2.3 Lm which i s the equation of a parabola. To f i t experimental values of V and I to the curve of equation 3.2.3 one can estimate V m and take I m as the experi-mental value of I at t h i s point. The constant P R 2 may be used as a parameter and chosen to make the t h e o r e t i c a l curve coincide with one experimental point. The other experimental points w i l l l i e along the curve i f the excess current does a r i s e from the expansion of the space charge region. Base current as a function of bias voltage was measured for several t r a n s i s t o r s at 50°C and at room temperature. The r e s u l t s for a l l except one unit proved unsuitable for comparison with theory because of early breakdown or high punchthrough voltage. In the l a t t e r case i t was thought that the formation of a c y l i n d r i c a l space charge region did RWERSE CHARACTERISTIC SHOWING INCREASE IK SATURATION CURRENT FIG. 15 30 not occur u n t i l the reverse-bias was in the region of break-down . The current-voltage c h a r a c t e r i s t i c for tran s i s t o r No. 39, a 2N137 unit, i s i l l u s t r a t e d i n f i g . 15. The value of V m was taken to be Vp/4, the value of V at which the two space charge regions would meet i n an ideal planar t r a n s i s t o r . Because of the very small increase in current over the range between Vra and breakdown, i t was p r a c t i c a l to take only-three points from the experimental curve. The constant was chosen such that the theoreti c a l curve passed through the experi-mental point having the lowest value of V. It w i l l be seen from f i g . 16 that the agreement of the two other experimental points i s very good. From th i s f i t the following numerical information i s obtained: yOR2 = 8.9 x 10 4volt -3 14 which i s consistent with p = 50 coul.m (N D = 3 x 10 cm" 3), R = 0.5 x 10"3m., £ = 1.4 x lO'^F.m." 1 which are reasonable values for the structure examined. 3.3 Operation of Transistors as Amplifiers under Conditions of Double-Depletion The findings of Stafayev et a l (1956) mentioned i n the introduction indicate that a tr a n s i s t o r may be operated p r a c t i c a l l y as an amplifier under conditions of double-depletion. It i s worth while to review here the simple COMPARISON OF lj^Am VS V — V M tflTH THEORETICAL CURVE FOR CYLINDRICAL EXPANSION OF THE SPACE CHARGE RSGION 0 I 2 3 f 5 6 7 8 9 10 II 12 13 / f V - V or c o n s t ( l - x + x l n x ) 31 p r i n c i p l e s of operation of such a device, e s p e c i a l l y as the r e s u l t s of section 3.1 indicate that such operation i s possible for the units investigated. A germanium pnp tr a n s i s t o r conventionally biased as an amplifier acts on an i n j e c t i o n system described as follows where I c o i s the saturation current across the c o l l e c t o r Junction when the emitter i s unbiased and i s the grounded base current amplification factor. In a system of double depletion the r e l a t i o n becomes Ic - I Co " (XW 3.3.2 This r e l a t i o n obtains only when the base width i s of the order of L; then the two space-charge regions are to some extent i n t e r a c t i n g for the boundaries of the regions are not sharp in a r e a l t r a n s i s t o r . Under th i s condition one electrode may draw c a r r i e r s not only from the undepleted region of the base but also by a "robbing" action from the space charge region of the other electrode. The two electrodes w i l l be referred to as the control electrode and the c o l l e c t o r . Amplification becomes possible with asymmetry of the working of the control electrode and the c o l l e c t o r which can be achieved by a d i s s i m i l a r geometry or by application of d i f f e r e n t bias values. The bias value on the c o l l e c t o r must be great enough to assure operation in the c o l l e c t o r saturation region. Amplification w i l l 32 occur u n t i l the control electrode current becomes saturated. Since I c o i s small i n Ge tr a n s i s t o r s except at high tem-peratures i t i s only p r a c t i c a l to use them as reverse-biased amplifiers at high temperatures. The Russian investigators made measurements on pnp germanium tr a n s i s t o r s using the electrode with the larger area as the control electrode. Measurements of current and power amplification with grounded control electrode as i n f i g . 17(a) were made at various temperatures with both forward and reverse biases on the control electrode. Small signals were used at a frequency of 1000 cps. Some of the re s u l t s from the Russian paper are shown i n f i g . 17. When the control electrode Is forward biased, the amplifier c i r c u i t i s equivalent to the grounded emitter configuration so the current amplification factor w i l l be referred to as J3 . For the tra n s i s t o r s used by the Russians, I c o was about 2 ma at 100°C. and 1 ma at 81°C. as shown in f i g . 17 (b), (c). Under double-depletion conditions, they achieve a maximum current amplication of 13 db and a power amplification, Cj , of 26 db at an operating temperature of 81°C. The maximum c o l l e c t o r current i s of course l i m i t e d An important feature of these curves i s the displacement of the maximums of J2 and g into the region of negative currents at higher temperatures. As pointed out by the Russians t h i s appears to contradict the generally accepted WORKING OF A TRANSISTOR UNDER CONDITIONS OF DOUBLE-DEPLETION 33 explanation for the maximum in J$ due to Webster (1954) which asserts that the f a l l - o f f of J3 at low reverse biases i s owing to the increased e f f e c t of surface recombination at low f i e l d s , while the decline at higher values of I e i s caused by decreased emitter e f f i c i e n c y . The r e s u l t s of our own investigation can be u t i l i z e d to explain t h i s anomaly. As deduced from the curve of f i g . 11 surface recombination and generation are n e g l i g i b l e at higher temperatures; therefore there would be no apparent decrease in J$ as I e approaches zero. The decline which occurs with increasing reverse bias i s caused by saturation of the control electrode current. This reasoning i s supported by the fact that the sharp decreases of to zero i n the Russian r e s u l t s occurs at c o l l e c t o r currents of about the value of I c o . The Russian investigators note that, for specimens tested with the smaller electrode as control electrode there was a displacement of the maximum of J3 i n the di r e c t i o n of lower I e but i t remained in the i n j e c t i o n region. In t h i s case saturation of the control electrode occurs at a lower current and does not a t t a i n i t s maximum u n t i l I e i s p o s i t i v e . F i g . 17 (d) shows the r e l a t i o n of the input resistance J to the current of the control electrode. When the control electrode current i s far from saturation, the input r e s i s -tance i s much smaller than the output resistance (about 50 ohms) so that voltage amplification i s also possible. 3 4 The t r a n s i s t o r s used in the present investigation would not be suitable for use as reverse biased amplifiers because of their r e l a t i v e l y low saturation currents up to 100°C. With appropriate geometry and doping, however, tra n s i s t o r s might be made which u t i l i z e the advantages of the double-depletion system; higher power amplification at high temperature and stable power amplification with v a r i a t i o n in temperature. 35 CHAPTER IV DISCUSSION OF BREAKDOWN VOLTAGES In the course of t h i s investigation breakdown voltages were measured for the purpose of selecting suitable specimens for the main experimental work. These measure-ments w i l l now be discussed in the l i g h t of e x i s t i n g theory for breakdown at p-n junctions. Breakdown voltage was defined for purposes of measure-ment as that value of the reverse bias across a junction at which the reverse current has increased to 5 times i t s saturation value. This c r i t e r i o n i s unsatisfactory i n that i t does not take into account the increase of current with voltage, and i t i s possible that a r e l a t i v e l y slow increase i n current due to the non-saturating of reverse current rather than a sharp increase due to avalanche breakdown may cause an increase to 5 times the value of the saturation current. One such factor leading to non-saturation of the reverse current has been discussed in Chapter I I I , another of importance i s a large recombination-generation rate on the surface of the base in the v i c i n i t y of the emitter and c o l l e c t o r regions. As the space-charge region around the emitter or c o l l e c t o r expands, more of the surface contributes to the minority c a r r i e r flow into the base. For a l l specimens tested, the breakdown voltage as defined above was much lower than the bulk breakdown voltage 36 calculated using M i l l e r ' s formula (equation 2.1.2) and t estimates of impurity density obtained as i n section 2.1. For instance, for the tr a n s i s t o r s described in Table 1, the measured breakdown voltages were 12, 17 and 30 v compared to calculated bulk values of 328, 795 and 608 respectively. As pointed out in the introduction, breakdown may occur between the surface and the space-charge region of the bulk, therefore the r e s u l t s obtained were examined to fi n d i f there was any c o r r e l a t i o n between the measured breakdown voltages and the f l o a t i n g emitter and surface conductance measurements obtained as described i n section 2.2. No regular increase or decrease of breakdown voltage with surface conductance or f l o a t i n g emitter voltages was found. The only theory available for surface breakdown i s that of Garrett and B r i t a i n (1956) which i s based on a one-dimensional model. They obtain a r e l a t i o n between surface breakdown voltage, bulk breakdown voltage and impurity density as follows V u ,, - v * = 27fe d A _ vbulk v s u r f . — — 4.1 where i s the surface charge density, 6" the semiconductor d i e l e c t r i c constant, Nj the impurity density and KT( the minority c a r r i e r density. This r e l a t i o n cannot r e a d i l y be applied to the experimental 37 data because of the uncertainty in (f . Not only i s i t s value uncertain, but i t i s very l i k e l y that i t i s dependent on such factors as the reverse bias and the impurity density. 1 38 CHAPTER V SUMMARY i This work has been concerned with an investigation of space-charge regions in germanium by u t i l i z a t i o n of ( the double-depletion condition. Experimental data have been discussed in terms of s p e c i f i c geometrical and physical models for c a r r i e r generation processes. ' In Chapter II methods of obtaining i n d i r e c t l y the ele c t r o n i c and geometrical parameters of junction t r a n s i s t o r structures are described. A c y l i n d r i c a l three-dimensional model under the condition of double-depletion i s described and i t i s shown that, adopting the Shockley-Read recombination-generation theory, the minority c a r r i e r current flowing i n the base arises predominantly in the space charge region and not by d i f f u s i o n from the neutral region of the bulk. Experimental data between 100°C and -190°C have been analysed and a c t i v a t i o n energies of c a r r i e r s derived by application of the Shockley-Read theory. It' was further deduced that two sets of recombination-generation centers are active; the set having the higher a c t i v a t i o n energy was ascribed to bulk impurities and the lower to surface states. It was deduced that i n t r i n s i c generation of c a r r i e r s does not contribute to the hole flow i n specimens investigated, even at very high temperatures. 39 Some specimens showed an anomalous increase i n current at low temperatures which could not be accounted for by the Shockley-Read theory or by degeneracy of electrons i n the conduction band at low temperatures. It i s conjectured that some unknown surface generation mechanism may account for t h i s unusual behaviour. In Chapter III the current d i s t r i b u t i o n between the two junctions when both are reverse-biased i s discussed. It i s deduced from experimental data that the current d i s t r i b u -tion i s roughly i n proportion to the area of the junctions when both are equally reverse-biased. The generation rate of c a r r i e r pairs per unit volume w i l l be the same i n the two depletion regions and the current flow to the junctions w i l l therefore be proportional to the volumes of the regions and hence approximately proportional to the junction areas. It was further found that interaction between reverse-biased junctions separated by less than a d i f f u s i o n length becomes s i g n i f i c a n t at elevated temperatures. This i s related to the a b i l i t y of each junction to receive (by di f f u s i o n ) c a r r i e r s from the space-charge region around the other junction and consequently ( d l e / d l c ) i s not n e g l i g i b l e compared with unity under these conditions. A discussion i s included of the proposal of Stafayev, Tuchkevich and Yokovchuk for the use of junction t r a n s i s t o r s as amplifiers at high temperatures under conditions of double-depletion. It i s shown that the phenomena of non-saturating reverse 40 current can arise from expansion of the space charge region into the base. A r e l a t i o n r e l a t i n g the base current and the reverse bias at the junction i s developed for a c y l i n d r i -c a l model assuming that the rate at which c a r r i e r s are t generated in the space charge region greatly exceeds that at which c a r r i e r s d i f f u s e into i t from the undepleted region. The th e o r e t i c a l current-voltage r e l a t i o n i s shown to give a very good agreement with the experimental data for one specimen. MODEL FOR CYLINDRICAL EXPANSION OF THE SPACE-CHARGE REGION ~T~ a R R= r a d i u s o f space-charge r e g i o n a t V - V, ZL_ V l b FIG. 18 APPENDIX DEPENDENCE OF THE VOLUME OF A CYLINDRICAL SPACE CHARGE REGION ON REVERSE BIAS When a tr a n s i s t o r consisting of hemi-spherical p-type emitter and c o l l e c t o r extending into an n-type base wafer i s biased as shown i n f i g . 17, then at some value of reverse bias, V - V m, the space charge regions of the emitter and c o l l e c t o r w i l l overlap to form a c y l i n d r i c a l column of space-charge of length w and radius R. For V ^ V m the extent of the space charge region may be determined by the solution of Poisson's equation in c y l i n d r i c a l co-ordinates. For the case of r a d i a l symmetry and constant length as i n th i s s i t u a t i o n Poisson's equation becomes, - P v x r dr / £. A . l where ^ i s the pote n t i a l , r the r a d i a l distance from the axis, and p and £ the space charge density and p e r m i t t i v i t y i n mks units. If a i s the value of r at the boundary of the space charge region, then the boundary conditions are: \ 3 r /a ( i ) ( i i , ¥(a> - ¥ ( R ) - V - V m The general solution of equation A . l , v a l i d for V > V m and r> R i s A.2 «' ( r ) ^ B l n g ^ C 42 where B and C are constants of integration. Using boundary condition ( i ) B = 2T A.3 and from boundary condition ( i i ) v-vm -£R> A.4 1 - * + R 1 R 2 R 43 BIBLIOGRAPHY Brown, W.L., "N-Type Surface Conductivity on P-Type Germanium", Phys. Rev., 91 (1953), pp. 518-527. Barker, A.S., "A Study of Space-Charge and Avalanche M u l t i -p l i c a t i o n Processes in Germanium", Thesis for Master of Science Degree, University of ' B r i t i s h Columbia, September, 1957 Christensen, H., "Surface Conduction Channel Phenomena in Germanium", Proc. IRE 42 (1954) pp. 1371-1376. Conwell, E.M., "Properties of S i l i c o n and Germanium", Proc. IRE, 40 (1952), pp. 1327-1337. Conwell, E.M., "Properties of S i l i c o n and Germanium; II", i Proc. IRE, 46 (1958), pp 1281-1300. Early, J.M., "Effec t s of Space-Charge Layer Widening i n Junction Transistors", Proc, IRE, 40 (1952), pp. 1401-1406. i Garrett, C.G.B., Brattain, W.H., "Some Experiments on, and a Theory of, Surface Breakdown", J. App. Phys., 27 (1956), pp. 299-306. M i l l e r , S.L., "Avalanche Breakdown i n Germanium", Phys. Rev., 99 (1955), pp. 1234-1241. 44 McWhorter, A.L., Kingston, R.H., "Channels and Excess Reverse Current in Grown Germanium P-N Junction Diodes", Proc. IRE, 42 (1954), pp. 1376-1380. Nichol, D.W., "Space-Charge-Limited Currents in Germanium", Thesis for Master of Science degree, University of B r i t i s h Columbia, September 1958. Sah, C.T., Noyce, R.N., Shockley, W., "Carrier Generation and Recombination in P-N Junction Characteristics,' Proc. IRE, 45 (1957), pp. 1228-1243. Schenkel, H., Statz, H., "Voltage Punch-Through and Avalanche Breakdown and their E f f e c t on the Maximum Operating Voltages for Junction Transistors", Proc. Nat'l Electronics Conf., 10 (1954), pp. 614-625. Shockley, W., "The Theory of P-N Junctions in Semiconductors and P-N Junction Transistors", B e l l Sys. Tech. » J r . , 28 July 1949, pp. 435-489. Shockley, W., "Electrons and Holes i n Semiconductors", Van Nostrand, 1950. Shockley, W., Sparks, M., Teal, G.K., "P-N Junction Transis-tors", Phys. Rev., 83 (1951), pp. 151-162. Shockley, W., Read, W.T., " S t a t i s t i c s of Recombination of Holes and Electrons", Phys. Rev., 87 (1952), pp. 835-842. 45 Shockley, W., "Electrons, Holes and Traps", Proc. IRE, 46 (1958), pp. 973-990. Stafeyev, V.I., Tuchkevich, V.M., Yakovchuk, N.S., "Operation of a Transistor in Conditions of i Depletion", Zh. Tekh. F i z . , 26 (1956), pp. 15-21. Statz, H., deMars, G., Davis, L., Adams, A., "Surface States on S i l i c o n and Germanium Surfaces", Phys. Rev., 106 (1957), pp. 455-464. Statz, H., de Mars, G.A., " E l e c t r i c a l Conduction v i a Slow Surface States on Semiconductors", Phys. Rev., I l l (1958) pp. 169-182. Webster, W.M., "On the Variation of Junction Transistor Current-Amplification Factor with Emitter Current", Proc. IRE, 42 (1954), pp. 914-920.
Thesis/Dissertation
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Physics
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A study of carrier generation in, and interaction of, space-charge regions in germanium
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