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Theory of conductivity modulation in semiconductors Nishi, Ronald Yutaka 1962

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THEORY OP CONDUCTIVITY MODULATION IN SEMICONDUCTORS RONALD YUTAKA NISHI B.A.Sc, University of British Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics  The University of British Columbia, Vancouver 8, Canada, Date J u n e 13, 1962 i i ABSTRACT The theory of conductivity modulation i n semi-conductors and the conditions tinder which negative resistance can be obtained are investigated. The ambipolar transport equation i s derived for one-dimensional flow in a two-terminal homogeneous semiconductor with no trapping and no temperature gradients. Charge neutrality i s assumed i n the majority of the models studied. A phenomenological model i s considered f i r s t to show how conductivity modulation can lead to negative resistance. Since the general problem of carrier transport with d r i f t and diffusion currents i s d i f f i c u l t , the models investi-gated are mainly concerned with either d r i f t or diffusion as the dominant transport mechanism. For a unipolar space-charge limited d r i f t model, negative resistance i n frequency bands i s found. For bipolar, neutral d r i f t models, negative resi s t -ance i s found under special conditions for the case of no recombination and for recombination with a carrier lifetime increasing with carrier density. For recombination with a constant lifetime, the bipolar d r i f t model gives current-voltage characteristics of the form: Joe V at low injection 2 "5 levels, J°cv a t high injection levels, and JocV y at very high injection levels. Space charge i s important i n the cube law case. Models ignoring diffusion are shown to be valid only for extrinsic semiconductors. i i i Bipolar neutral flow with equal carrier densities leads to diffusion solutions of the ambipolar equation. This case, applies to extrinsic material at high injection levels as well as to intrinsic material and is found to exhibit negative resistance under certain conditions. The most favourable situation is the case where the carrier lifetime increases with carrier density. The dependence of the lifetime with carrier density determines the possibility of defining completely a negative resistance model. It is found that the properties of the contacts are important in attaining negative resistance. Contacts and their properties are briefly discussed in relation to the carrier density boundary conditions. Avalanche injection and its relation to the conductivity modulation problem is considered. Several outstanding problems, both theoretical and experimental, arising from this thesis are outlined in the last chapter. ACKNOWLEDGMENT I wish to thank Professor R. E. Burgess for his supervision i n carrying out and reporting this research. I also wish to thank the B r i t i s h Columbia Telephone Company and the National Research Council for financial assistance. i v CONTENTS CHAPTER 1. INTRODUCTION 1.1 Purpose of the Investigation 1 1.2 Review of Previous Work 1 1 .3 Scope of Thesis 2 CHAPTER 2. TRANSPORT OP INJECTED CARRIERS IN SEMICONDUCTORS 2.1 Basic Equations 5 2.2 Simplifying Assumptions 7 2 . 3 The Ambipolar Continuity Equation 10 2.4 Steady State and Small Amplitude A.C. Analysis 12 CHAPTER 3 . GENERAL CONSIDERATIONS OP CONDUCTIVITY MODULATION  AND NEGATIVE RESISTANCE 3.1 Conductivity Modulation by C a r r i e r I n j e c t i o n 16 3.2 Negative Resistance 17 3 . 3 Evidence f o r Negative Resistance 20 3.4 Phenomenological Model f o r Negative Resistance 22 CHAPTER 4. CARRIER TRANSPORT BY DRIFT 4.1 Introduction 27 4.2 No Recombination 28 4.21 Neutral Bipolar 28 4.22 Space-Charge-Limited Unipolar Plow 33 4 . 3 Recombination with Constant L i f e t i m e — Neutral Bipolar 40 4.4 Recombination with C a r r i e r Dependent L i f e t i m e s — N e u t r a l Bipolar 53 V CHAPTER 5. CARRIER TRANSPORT BY DIFFUSION AND COMBINED DRIFT AND DIFFUSION 5.1 Introduction 56 5.2 Diffusion Model 56 5.3 Combined Drift and Diffusion Models 61 5.4 Validity Ranges for Drift and Diffusion Solutions 66 CHAPTER 6. BIPOLAR FLOW WITH EQUAL'CARRIER DENSITIES 6.1 Introduction 69 6.2 No Recombination 70 6.3 Recombination with Constant Lifetime 72 6.4 Recombination with Carrier Dependent Lifetime 75 6.5 Critique of Stafeev's Model 82 CHAPTER 7. CONTACT PROPERTIES AND RELATED BOUNDARY CONDITIONS 7.1 Boundary Conditions 85 7.2 Contacts 87 7.3 Carrier Multiplication at Contacts 94 CHAPTER 8. CONCLUSIONS AND OUTSTANDING PROBLEMS 97 BIBLIOGRAPHY 99 vi ILLUSTRATIONS Facing Page FIGURE 3.1 Negative Resistance Characteristics 18 3.2 Semiconductor Filament under Injection 22 3.3 Current-Voltage Characteristics— Phenomenological Model 23 4.1 Current and Voltage—Space-Charge-Limited Unipolar Model 35 4.2 Impedance of Space-Charge-Limited Unipolar Model 38 1 C H A P T E R 1  I N T R O D U C T I O N 1 .1 P u r p o s e o f t h e I n v e s t i g a t i o n T h e f l o w o f i n j e c t e d c u r r e n t c a r r i e r s i s a n i m p o r t -a n t p r o b l e m i n s e m i c o n d u c t o r p h y s i c s . C u r r e n t c a r r i e r s o f t w o t y p e s e x i s t i n a s e m i c o n d u c t o r — e l e c t r o n s i n t h e c o n d u c t i o n b a n d a n d p o s i t i v e h o l e s i n t h e v a l e n c e b a n d . T h e c o n c e n t r a t i o n o f t h e s e c a r r i e r s i n t h e s e m i c o n d u c t o r c a n b e i n c r e a s e d o v e r t h e c o n c e n t r a t i o n s w h i c h n o r m a l l y e x i s t a t t h e r m a l e q u i l i b r i u m c o n d i t i o n s b y a n y a g e n c y w h i c h t r a n s f e r s e l e c t r o n s t o t h e v a l e n c e b a n d a n d t o t h e c o n d u c t i o n b a n d . M i n o r i t y c a r r i e r i n j e c t i o n f r o m a n e l e c t r o d e i s o n e o f t h e s e m e t h o d s b y w h i c h t h e c a r r i e r s c a n b e i n c r e a s e d . T h e a d d e d c a r r i e r s a r e s u b j e c t t o d r i f t u n d e r a n e l e c t r i c f i e l d , t o d i f f u s i o n , a n d t o r e c o m b i -n a t i o n , a n d c a n m o d u l a t e t h e b u l k c o n d u c t i v i t y o f t h e s e m i -c o n d u c t o r . T h e p u r p o s e o f t h i s t h e s i s i s t o i n v e s t i g a t e c o n d u c t i v i t y m o d u l a t i o n d u e t o c a r r i e r i n j e c t i o n a n d t o d e t e r -m i n e u n d e r w h a t c o n d i t i o n s n e g a t i v e r e s i s t a n c e c a n b e o b t a i n e d . 1.2 R e v i e w o f P r e v i o u s W o r k T h e e q u a t i o n s d e s c r i b i n g t h e f l o w o f a d d e d c u r r e n t c a r r i e r s i n s o l i d s u n d e r t h e a c t i o n o f d r i f t , d i f f u s i o n a n d r e c o m b i n a t i o n a r e o f s u c h a c o m p l e x n a t u r e t h a t a r i g o r o u s s o l u t i o n o f t h e p r o b l e m p r e s e n t s f o r m i d a b l e m a t h e m a t i c a l d i f f i -c u l t i e s . S o l u t i o n s may b e o b t a i n e d , h o w e v e r , i n m a n y c a s e s u n d e r c e r t a i n s i m p l i f y i n g c o n d i t i o n s . W. v a n R o o s b r o e c k h a s 2 presented the theory o f the fl o w of e l e c t r o n s and h o l e s i n semi-conductors under charge n e u t r a l i t y ( 1 9 5 0 , 1 9 5 3 ) and w i t h space charge (1961). Stockman (1956) and R i t t n e r (1956) have i n v e s -t i g a t e d the g e n e r a l problem of t r a n s p o r t i n photoconductors. Parmenter and Ruppel ( 1 9 5 9 ) and Lampert ( 1 9 5 9 ) d e a l w i t h steady-s t a t e space-charge l i m i t e d c u r r e n t s under d r i f t c o n d i t i o n s i n i n s u l a t o r s . Lampert and Rose (1961) have extended t h i s a n a l y s i s to semiconductors. Assuming charge n e u t r a l i t y , Lampert (1962) has s t u d i e d the same problem i n i n s u l a t o r s and has shown t h a t , n e g a t i v e r e s i s t a n c e can occur under c e r t a i n c o n d i t i o n s . The case of d i f f u s i o n under low i n j e c t i o n c o n d i t i o n s i n semi-conductors has been analyzed by Shockley ( 1 9 4 9 ) and extended t o a r b i t r a r y i n j e c t i o n l e v e l s by R i t t n e r ( 1 9 5 4 ) . S t a f e e v (1958, 1 9 5 9 ) has a l s o d e a l t w i t h h i g h i n j e c t i o n d i f f u s i v e c u r r e n t s and d i s c u s s e s q u a l i t a t i v e l y the appearance o f n e g a t i v e r e s i s t a n c e under c e r t a i n c o n d i t i o n s . An even more r e s t r i c t i v e model has been developed by Shockley and Prim ( 1 9 5 3 ) i n which the c u r r e n t i s a s p a c e - c h a r g e - l i m i t e d o n e - c a r r i e r c u r r e n t . T h i s model e x h i b i t s n e g a t i v e r e s i s t a n c e f o r s m a l l amplitude a.c. i n c e r t a i n frequency bands but not i n the steady s t a t e (Shockley 1 9 5 4 ) . 1 . 3 Scope of T h e s i s T h i s t h e s i s i s concerned w i t h a t h e o r e t i c a l a n a l y s i s of the f l o w of i n j e c t e d e l e c t r o n s and h o l e s i n semiconductors. S e v e r a l mathematical models are s t u d i e d . The b a s i c s t r u c t u r e on which the an a l y s e s are based i s a two-terminal homogeneous semiconductor f i l a m e n t o r s l a b . I n j e c t i o n and/or e x t r a c t i o n o f 3 carriers may be taking place at both terminals. I n i t i a l l y , the basic equations describing the trans-port of carriers are presented and the ambipolar transport equation formulated using suitable simplifying assumptions. The physical significance of this equation and the validity of the underlying assumptions are discussed. Before proceeding to a solution of the transport equation, a general discussion of conductivity modulation and negative resistance i s presented with a simple phenomenological model showing how conductivity modulation can lead to negative resistance. Since the differential equations describing the trans-port problem are non-linear and d i f f i c u l t to solve, no attempt i s made to obtain a rigorous solution. Thus various approxi-mations are made to obtain specific models for which exact solutions or approximate solutions can be found. A steady state analysis in one dimension is carried out and the current-voltage characteristics investigated for negative resistance. Wherever possible, a small amplitude a.c. analysis is attempted. The analysis i s intended to be as general as possible with no specific reference made as to the mechanism whereby the added carriers are injected into the semiconductor. In a later section, however, contacts and related boundary conditions are discussed in an attempt to correlate bulk and electrode effects. Finally, since the treatment i s restricted to specific. models, the fundamental d i f f i c u l t i e s of the general problem are discussed. Reference is made to further extensions of the 4 a n a l y s i s a n d t o o t h e r m o d e l s , p a r t i c u l a r l y c o n c e r n e d w i t h b o u n d a r y c o n d i t i o n s , w h i c h c o u l d be i n v e s t i g a t e d t o o b t a i n n e g a . t i v e r e s i s t a n c e . 5 C H A P T E R 2 TRANSPORT OF I N J E C T E D C A R R I E R S I N SEMICONDUCTORS 2.1 B a s i c E q u a t i o n s T h e f u n d a m e n t a l e q u a t i o n w h i c h i s r e q u i r e d ' f o r p r o -b l e m s i n v o l v i n g t h e f l o w o f p a r t i c l e s i s t h e e q u a t i o n o f c o n t i n u i t y . I n s e m i c o n d u c t o r s , w h e r e t h e e l e c t r i c c u r r e n t c o n s i s t s o f t h e f l o w o f c h a r g e d p a r t i c l e s , t h e c o n t i n u i t y e q u a t i o n s f o r e l e c t r o n s a n d h o l e s , r e s p e c t i v e l y , a r e a n / a t = g - r + ( 1 / q ) d i v ( 2 . 1 . 1 ) a p / a t = g - r - ( 1 / q ) d i v ? ( 2.1 . 2 ) w h e r e n a n d p r e p r e s e n t t h e e l e c t r o n a n d h o l e c o n c e n t r a t i o n s , g - r r e p r e s e n t s t h e n e t r a t e o f p a i r g e n e r a t i o n m i n u s t h e n e t r a t e o f p a i r r e c o m b i n a t i o n p e r u n i t v o l u m e , q t h e e l e c t r o n i c c h a r g e , a n d J ^ a n d J p t h e e l e c t r o n a n d h o l e c u r r e n t d e n s i t i e s . T h e c u r r e n t d e n s i t i e s a r e g i v e n b y J ^ = q M n n E + q D n g r a d n ( 2 . 1 . 3 ) J p = a j u p p i E - q D p g r a d p ( 2 . 1 . 4 ) w h e r e t h e f i r s t t e r m , t h e c o n d u c t i o n c u r r e n t d e n s i t y , i s d u e t o t h e d r i f t o f t h e c a r r i e r s i n a n e l e c t r i c f i e l d E , a n d t h e s e c o n d t e r m , t h e d i f f u s i o n c u r r e n t d e n s i t y , i s d u e t o t h e r a n d o m t h e r m a l m o t i o n o f t h e c a r r i e r s a n d t h u s i s p r o p o r t i o n a l t o t h e g r a d i e n t o f t h e c a r r i e r d e n s i t i e s . I n t h e r m a l e q u i l i -6 brium the electron and hole mobilities,, u and.jji^., are related to the d i f f u s i o n c o e f f i c i e n t s , "Dn and D p, by Einstein's r e l a t i o n s ^ = (q/kT)D n, M p = (q/kT)D p where k i s Boltzmann's constant and T i s the temperature. I f the e l e c t r i c f i e l d s are not too large to "heat" the c a r r i e r s , the E i n s t e i n r e l a t i o n s may s t i l l be considered v a l i d . The t o t a l current density i s the sum of the electron and hole current densities and a displacement current density: J = J* n + J* p + edE/dt (2.1.5) € being the p e r m i t t i v i t y . One more equation i s required to specify the problem. This i s Poisson's equation, which r e l a t e s the e l e c t r o s t a t i c p o t e n t i a l and the space charge due to a l l charged centres i n the semiconductor: div grad V = - d i v E = -(q/<f)(p - n + N+ - %) (2.1.6) * where - stands f o r a l l fixed ionized centres. Equations (2.1.1) to (2.1.6) are the basic equations defining the transport problem and are completely general. 7 2.2 Simplifying Assumptions The equations of the previous section can he reduced to somewhat simpler terms by the use of certain physically reasonable assumptions. If the semiconductor is homogeneous, then the total electron and hole densities in the presence of injection, n and p, can be written as n = n 0 + An, P = P 0 + Ap (2.2 .1) where n Q and p 0, the thermal equilibrium densities, are cons-tants; An and Ap are the injected carrier densities. Thus, the derivatives occurring in the equations can be rewritten in terms of the injected carrier densities rather than in terms of the total carrier densities, e.g. grad n = grad An. The carrier mobilities, and ; U p , are considered to be field independent. This assumption is only an approximation and is valid at low and moderate electric fields only (Shockley 1 9 5 1 ) . Prom Einstein's relations, the diffusion coefficients will likewise be constants. Temperature gradients are neglected in the analysis. Any attempt at experimental verification of the analysis must ensure isothermal conditions. The flow is assumed to be planar, i.e. one-dimension-a l . This assumption simplifies the mathematics of the problem considerably. However, since every physical problem involves some surfaces, there will be some flow toward the surfaces. Thus the one-dimensional treatment can only be regarded as an approximation. 8 Trapping is neglected and the impurity centers are assumed to be a l l substantially ionized in the semiconductor. If neutrality exists in thermal equilibrium, N^ - Nj may be replaced by n 0 - p 0 . Space charge is then due to the inbalance in the electron and hole concentration increments An and Ap, and Poisson's equation becomes (5/q.) BE/dx = Ap - An. (2.2.2) Unless a very strong field is present, Ap must be nearly equal to An. This fact leads to the condition of approximate charge neutrality Ap = An. (2.2.3) Neutrality allows a considerable simplification of the analysis and will be used in most of the models considered. For the generation-recombination term, g-r, the most convenient relationship to use is the one in which there is no external generation of carriers and where the net rate of recombination r is assumed to be proportional to the excess carrier density r = Ap/fp = An/rn . (2.2.4) The time constants T n and T are the lifetimes of the injected XT carriers; in the neutral case, t n=tp. It should be noted that this equality is valid only in the case where traps are neglect-ed. If traps are present, then the neutrality equation must take into account the space charge of the traps and hence An will not be equal to Ap. Thus the lifetimes will be unequal in 9 in. this case (Shockley and Head 1952). The lifetimes in general are functions of the excess carrier densities. However, the mathematical difficulties which result usually prohibit the use of this relationship in most analyses. Thus the lifetimes are usually considered constant, independent of the carrier concentrations. Surface recombination is neglected. This is primarily an aid for ignoring complicated boundary conditions, and is related to the assumption of planar flow. The surfaces of the semiconductor will act as a sink for excess carrier pairs and the flow of current will no longer be one-dimensional. However, for a semiconductor filament, i t can be shown (Shockley 1950) that the effect of surface recombination is to reduce the effective lifetime from the bulk value, 'f^ u^ jc» *° some lower value T, dependent on the surface recombination velocity and the dimensions of the filament; thus Vt = V r b u l k + Vrsurf ••• <2-2-5) Provided that the appropriate value of the lifetime is used, then surface recombination can be neglected and one-dimensional flow can be used in the analysis. No assumptions are made as to the nature of the contacts at the boundaries of the semiconductor at this stage of the analysis. 10 2.3 The Ambipolar Continuity Equation With the assumptions of the previous section, equa-tions (2.1.1) and (2.1.2) may be written in the form 9 An/at = -An/r n + ju^ndE/ax + ^ EdAn/ax + D n9 2An/ax 2 (2.3.1) 3Ap/ a t = -Ap/r p - MpP9E/6>x - /ipEdAp/ax + Dp^Ap/ax 2 (2.3.2) Multiplication of the f i r s t equation by ;u\pp and the second by ji^n and adding yields 9 An + An + ^ n n dAp + ^ p" . 2 t Tn at (2 . -3 .3 ) Mn^pE p3An _ n3&p + WpkT R . 9x 3x_ q. _ p9 2An + n32/*p 3x2) At this point i n the analysis, with the 9E/^x term eliminated, the assumption of charge neutrality, An=Ap, can be introduced. The equation then becomes 9 ^ p / a t = -AP/T - ;uEdAp/ax + D ^ A p / a x 2 (2 .3 .4) where p = (n-pJAp/^+n/jOp). D = (n+p)/(p/Dn+n/Dp) (2 .3.5) are known as the ambipolar mobility and the ambipolar diffusion coefficient respectively; X i s the lifetime for both electrons and holes. Equation (2 .3.4) i s known as the ambipolar equation and i s due to van Roosbroeck (1953). For strongly extrinsic n-type material (n>>p), p=p Sr and D=Dp; for strongly p-type material (p»n) and D=Dn. For in t r i n s i c material (n=p), ;u=0 and D=2DpDn/(Dp+Dn). 11 C o m p a r i s o n w i t h t h e o r i g i n a l c o n t i n u i t y e q u a t i o n s g i v e n b y (2.3.1) a n d ( 2 . 3 . 2 ) s h o w s t h a t e x c e p t f o r t h e "9E/9x t e r m , t h e a m b i p o l a r e q u a t i o n f o r s t r o n g l y e x t r i n s i c n - t y p e m a t e r i a l i s i d e n t i c a l w i t h t h e c o n t i n u i t y e q u a t i o n f o r h o l e s , w h i l e f o r s t r o n g l y e x t r i n s i c p - t y p e m a t e r i a l , t h e a m b i p o l a r e q u a t i o n i s i d e n t i c a l w i t h t h e c o n t i n u i t y e q u a t i o n f o r e l e c t r o n s . T h e m o b i l i t y ja d o e s n o t r e p r e s e n t t h e d r i f t v e l o c i t y o f p a r t i c l e s i n a n e l e c t r i c f i e l d b u t r e p r e s e n t s t h e d r i f t v e l o c i t y o f a d i s t u r b -a n c e , i . e . , t h e e x c e s s c a r r i e r p a i r d e n s i t y p a t t e r n . T h u s , t h e d i s t u r b a n c e m o v e s i n a n e l e c t r i c f i e l d i n t h e d i r e c t i o n i n w h i c h t h e m i n o r i t y c a r r i e r s w o u l d m o v e . A t t e n t i o n i s p l a c e d o n t h e e x c e s s m i n o r i t y d e n s i t i e s a n d t h e p r e s e n c e o f t h e n e u t r a l i z -i n g m a j o r i t y d e n s i t i e s i s t a k e n . c a r e o f b y t h e a m b i p o l a r m o b i l i t y a n d d i f f u s i v i t y . I n l e s s s t r o n g l y e x t r i n s i c m a t e r i a l t h e m i n o r i t y d e n s i t y p a t t e r n s h o w s a n e f f e c t i v e m o b i l i t y a n d d i f f u s i v i t y w h i c h a r e l e s s t h a n t h e r e l e v a n t c a r r i e r m o b i l i t y a n d d i f f u s i v i t y , a s i f t h e m a j o r i t y c a r r i e r s w e r e e x e r t i n g a d r a g o n . t h e m i n o r i t y c a r r i e r s . F o r i n t r i n s i c m a t e r i a l , t h e d e n s i t y p a t t e r n o n l y s p r e a d s o u t b y d i f f u s i o n w i t h o u t d r i f t a n d d e c a y s b y r e c o m b i n a t i o n . A m o r e u s e f u l f o r m o f t h e a m b i p o l a r e q u a t i o n f o r t h i s a n a l y s i s i s o b t a i n e d b y r e p l a c i n g n a n d p b y n Q + A p a n d p Q + A p ; t h u s _ AP M p S 3Ap P 0 ( n Q + p 0 + 2 A p ) 3 2 A p at " t " 1+aAp dx + ( n 0 + p 0 ) ( 1 + a A p ) 9x2 (2.3.6) w h e r e 1 2 M o ^ ^ o - P o V C t a o + P o ) , D 0=D pb(n 0+p 0 ) / ( D n 0+p 0) (2 . 3 . 7 ) and a=(b+1 )/.(bn0+p0); \>=}xn/-p.^)f the r a t i o of the m o b i l i t i e s . Prom the current equation (neglecting the displace-ment term) J * c3/0(1+aAp)E + KTjUp(b-1 )dAp/dx = constant, (2 . 3 . 8 ) E may be obtained and substituted i n the ambipolar equation, r e s u l t i n g i n aAp = _.AP _ / i Q 3Ap J - IcT M p(b-1 )3Ap 3t t O^d+a^p) 2 dx|_ 9x t P 0(n 0+p 0+2Ap) 3 2 A p (2 . 3 . 9 ) (n 0+p 0 )0+aAp) 3 x 2 where d'0=qMp(bn0+p0), the conductivity with no i n j e c t i o n . 2.4 Steady State and Small Amplitude A.C. Analysis The steady state a r i s e s under continuous i n j e c t i o n of excess c a r r i e r s into the semiconductor. The analysis f o r the steady state involves only the omission of the time dependent term. In the models concerned, a steady forward bias V i s applied to a slab or filament of semiconducting material. The major part of the i n v e s t i g a t i o n of c a r r i e r transport w i l l involve the steady state current-voltage c h a r a c t e r i s t i c s . A l l variables and quantities written without functional dependence or without subscripts w i l l hereafter r e f e r to the steady state values. 1 3 The potential drop across the semiconductor is V = Edx = IR (2.4.1) t where R is the d.c. resistance. The .differential resistance is given by dV/dl = V/l + I dR/dl (2.4.2) and this is investigated for negative resistance. The a.c. analysis arises, for example, when a sinu-soidally variable signal voltage is superimposed on a steady bias V. The variables consist of a continuous (d.c.) term and a variable (a.c.) term of angular frequency U); thus, I(t) = I + I-e3"* 1 (2.4 .3) V(t) = V + Y ^ e ^ . The coefficients of the a.c. terms, 1^  and , are time independent and are assumed to be small compared to the steady state terms, so that the a.c. equations may be linearized. Substitution of the total variables into the basic equations enables the equations to be separated into a time independent and time dependent part, the time independent part correspond-ing to the steady state equations and the time dependent part ju>t having a time dependence of e . For instance, substitution of Ap(t) = Ap + p-je^, An(t) = An + n ^ ^ , and E(t) = E + E-je 3 U r t into Poisson's equation results in (6/q)OE/3x+3E1/ax e J w t) = Ap - An + (p 1-n 1)e 3 b ) t. For the charge neutral case, p-j =n.p and the a.c. component of the current equation can be written as J 1 = qdup+jA^p-jE + akO+aApjE., + k T(^-u p ) a P l / a x . ( 2 . 4 . 4 ) The basic equations may be combined to obtain an a.c. ambipolar equation; however, the result is fairly complicated and is not derived or written here, since the equation is never used in its entirety in the analyses to follow. Whenever the a.c. analysis is used, the appropriate equations involved are suitably modified to comply with the specific model being consi-dered. The a.c. potential drop is given by V1 = j £ E ^ X = I R 1 + I . , R . ( 2 . 4 . 5 ) The complex impedance is given by Z(w) = V 1 / I 1 = ( R ^ I ^ I .+ R . ( 2 . 4 . 6 ) At zero frequency, the complex impedance corresponds to the slope of the steady state current-voltage characteristics at the d.c. -bias voltage V, that i s , Z (0) = dV/dl. From the a.c. equations, i t can be seen that the frequency u) always appears with the imaginary number j , so that Z (<^) =Z (j<*>). Hence i t can. be easily shown that Z (-*»>) = Z*(w). ( 2 . 4 . 7 ) This result holds true for causal processes. 15 It should he noted that the results of the technique above can also be obtained by analysing the transient response to an impulse of current and finding the "impulsive impedance", and then transforming from the time plane to the frequency plane. This method will be illustrated in Section 4 . 2 1 . , 16 CHAPTER 3 GENERAL CONSIDERATIONS OF CONDUCTIVITY  MODULATION AND NEGATIVE RESISTANCE 3.1 Conductivity Modulation by C a r r i e r Injection The conductivity of a semiconductor containing holes and electrons depends on t h e i r densities and m o b i l i t i e s . The equation f o r the conductivity i s d = q(/i pp + ^ n ) . Under the condition of approximate space charge n e u t r a l i t y , Ap=An, the conductivity takes the form d = d 0 d + a A p ) where o J 0=q(fipP 0 + u Ii no^» t h e conductivity for no i n j e c t i o n and a=q ( jXp+M r l)/(3 0. Thus the r e l a t i v e change i n the conductivity due to the excess c a r r i e r s Ap i s Aq/a'0=aAp. The excess c a r r i e r s can be injected or extracted by suit a b l y biased electrodes and be moved about insi d e the semi-conductor by suitably imposed f i e l d s . Since i n j e c t i o n disturbs the previously e x i s t i n g equilibrium between the holes and electrons, the recombination process attempts to restore the equilibrium condition by reducing the excess p a r t i c l e p a i r s . In an n-type semiconductor, the conductivity i s due to a predominance of mobile electrons; i n a p-type semiconductor, 17 d u e t o a p r e d o m i n a n c e o f m o b i l e h o l e s . T h e d o m i n a n t c a r r i e r i s c a l l e d t h e m a j o r i t y c a r r i e r w h i l e t h e l e s s e r i s c a l l e d t h e m i n o r i t y c a r r i e r . I t s h o u l d b e n o t e d , h o w e v e r , i n i n t r i n s i c ( n Q = p 0 ) o r i n n e a r - i n t r i n s i c m a t e r i a l , n o d i s t i n c t i o n c a n b e made a s t o w h i c h c a r r i e r i s t h e m a j o r i t y o r m i n o r i t y c a r r i e r . I n o r d e r t h a t t h e b u i l d i n g u p o f a l a r g e s p a c e c h a r g e may n o t o c c u r due t o e x c e s s c a r r i e r s o f o n e k i n d b e i n g i n j e c t e d i n t o a s e m i c o n d u c t o r , a c o r r e s p o n d i n g f l o w o f e x t r a c a r r i e r s o f t h e o p p o s i t e s i g n m u s t t a k e p l a c e f r o m t h e o t h e r e l e c t r o d e . I n a s e m i c o n d u c t o r c o n t a i n i n g s u b s t a n t i a l l y o n l y o n e t y p e o f c u r r e n t c a r r i e r , i t i s i m p o s s i b l e t o i n c r e a s e t h e t o t a l c a r r i e r c o n c e n t r a t i o n b y i n j e c t i n g c a r r i e r s o f t h e same t y p e ; h o w e v e r , i n j e c t i o n o f t h e o p p o s i t e t y p e c a n i n c r e a s e t h e t o t a l c a r r i e r c o n c e n t r a t i o n s i n c e t h e s p a c e c h a r g e o f t h e i n j e c t e d c a r r i e r s c a n b e n e u t r a l i z e d b y a n i n c r e a s e d c o n c e n t r a t i o n o f t h e t y p e n o r m a l l y p r e s e n t . T h u s t h e t o t a l i n c r e a s e d c o n c e n t r a t i o n o f b o t h c a r r i e r s c a n m o d u l a t e t h e c o n d u c t i v i t y . 3 .2 N e g a t i v e R e s i s t a n c e N e g a t i v e r e s i s t a n c e o c c u r s a t z e r o f r e q u e n c y w h e n f o r some v o l t a g e V a n d c u r r e n t I , t h e f o l l o w i n g r e l a t i o n s h i p h o l d s d V / d l < 0 . (3 .2 .1) T h e d e r i v a t i v e , d V / d l , i s c a l l e d t h e d i f f e r e n t i a l r e s i s t a n c e a n d i s t h e r e s i s t a n c e r e f e r r e d t o b y t h e t e r m n e g a t i v e r e s i s t a n c e . a ) VOLTAGE - CONTROLLED B> C U R R E N T - CONTROLLED '/// Low Pass d -^>o d i HigK Pass YZZZ Y777 VZ7 ZZZZ -> Band Pass dy di V/V, ///// > '//A '//// Low and Band P a s s s h a d i n g i n d i c a t e s R(<*>)< O c) A.C. C HARACT ERISTICS FIGURE: 3 1 NEGATIVE RES\STANCE CHARACTERISTICS 4 18 Negative resistance may be exhibited in two ways, either voltage-controlled or current-controlled, the differ-ence being in the value of the derivative (eitherooor 0) at the transition point (Figure 3.1a,b). Negative resistance may be manifested by the appear-ance of oscillations under the application of a d.c. voltage. The oscillations would be due to the dominance of the negative resistance over the external circuit resistance at the frequency of oscillation. Some of the possible mechanisms which may combine to produce negative resistance in semiconductors are: carrier multiplication, tunnelling, heating, contact effects, mobility changes, and lifetime changes. For this thesis, the mechanism of importance is that due to contact effects. Negative resis-tance in this case arises from the injection and/or extraction of carriers by a contact. However, the complete effect of the contact is not restricted to injection or extraction of carriers. Other effects can enter, such as carrier multipli-cation or tunnelling, at the contact and these may lead to or be the,primary cause of negative resistance in a semiconducting device. For the purposes of this thesis, only the injection of carriers"into the semiconductor by a contact will be treated in any great detail. If the voltage V=V(I,S)=IR, where S=S(l) is some current-dependent parameter and R=R(S,V), then the differential resistance may be written as 19 O V / B I ) s + iOR/as)TOs/ai)v dV/dl = ±- 1-. (3.2.2) 1 - i(dR/as)y(as/9v)I For current-controlled negative resistance, the numerator must vanish at some current and voltage, while for voltage-controlled negative resistance, the denominator must vanish for some current and voltage. For the a.c. negative resistance characteristics, the relationship R ( C J ) < 0 must hold at some bias and some frequency where R(<o) is the real part of the complex impedance Z( w). The behaviour of the negative resistance in the steady eta'fe and at higher frequencies may be entirely different. If Z(0)=dV/dI >0, then i t is possible that R(w)<0 for some frequency or bands of frequency. Or, i f Z(0)<0, then i t may be possible that negative resistance does not occur at higher frequencies. The various types of frequency dependent negative resistance characteristics are illustrated in Figure 3.1c. Several theoretical models demonstrating the possi-bil i t y of negative resistance in two-terminal semiconducting devices have been proposed. lampert (1962) analyzes double injection in insulators and high-resistivity semiconductors showing negative resistance due to an increasing hole lifetime with increasing injection level. Charge neutrality is assumed and diffusion is neglected. Stafeev (1959) discusses qualita-tively the possibility of negative resistance in semiconductors for diffusive current flow in "long" diodes. The negative resistance is assumed to have its origin in the increase in the free carrier lifetime with injection level resulting in a 2 0 modulation of the d i f f u s i o n l e n g t h . Gartner and S c h u l l e r ( 1 9 6 1 ) d i s c u s s two-terminal t h r e e - l a y e r t r a n s i s t o r - l i k e d e v i c e s which may e x h i b i t n e g a t i v e r e s i s t a n c e . The d.c. and s m a l l s i g n a l a.c. c h a r a c t e r i s t i c s are d i s c u s s e d q u a n t i t a t i v e l y i n terms o f the u s u a l t r a n s i s t o r parameters. Shockley ( 1 9 5 4 ) d i s c u s s e s s e v e r a l models f o r a.c. n e g a t i v e r e s i s t a n c e a r i s i n g from the t r a n s i t time of the c a r r i e r s i n semiconductor d i o d e s . Gunn ( 1 9 5 7 ) d i s c u s s e s the avalanche i n j e c t i o n e f f e c t i n semi-conductors showing the p o s s i b i l i t y of a two-terminal n e g a t i v e r e s i s t a n c e d e v i c e . 3 . 3 Evidence f o r Negative R e s i s t a n c e Experimental evidence f o r n e g a t i v e r e s i s t a n c e has been found i n s e v e r a l two-terminal semiconducting d e v i c e s . Of thes e , three cases w i l l be d e s c r i b e d . Leblond ( 1 9 5 7 ) has found n e g a t i v e r e s i s t a n c e of the c u r r e n t - c o n t r o l l e d type i n i n t r i n s i c and n e a r - i n t r i n s i c germanium a t 8 5°K. T y p i c a l v o l t a g e s and c u r r e n t s a t which the s l o p e of the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s becomes n e g a t i v e 'are around 3 0 v o l t s and 2 mA. O s c i l l a t i o n s have been observed i n the frequency range 2 5 to 3 5 megacycles per second. Two p o s s i b l e e x p l a n a t i o n s are c o n s i d e r e d . In the f i r s t , i n t e r n a l i o n i z a t i o n of the c r y s t a l i s c o n s i d e r e d t o g e t h e r w i t h a l a y e r o f h i g h recombination l o c a t e d immediately i n f r o n t of the e m i t t i n g j u n c t i o n . The n e g a t i v e r e s i s t a n c e i s a consequence o f a p a r t i c u l a r space charge c o n f i g u r a t i o n i n the b u l k . I n the second e x p l a n a t i o n , i n t e r n a l i o n i z a t i o n i s not c o n s i d e r e d but 21 a multiplication factor of minority carriers at the collector junction i s introduced. Gibson and Morgan (1960) have made negative resis-tance diodes using n-type germanium of 20 to 40 ohm cm and n- and p-type s i l i c o n of several hundred ohm cm. The voltage at which the slope dV/dl becomes zero i s around 55 volts for a typical germanium diode and around 65 for s i l i c o n , at currents of about 2 to 3 mA. The slope i s negative up to at least 100 mA or higher. At low currents, the characteristic i s obscured by relaxation oscillations. The sustaining voltage i n this current range i s typically 8 to 11 volts. The theoretical explanation proposed for the negative resistance i s based on the avalanche injection effect described by Gunn (1957). Rediker and McWhorter (1959) have found negative resistance i n compensated p-type germanium at liquid helium temperatures (4.2°K), which they have named a "cryosar". The negative resistance region occurs between a high and low impedance state. They have proposed (1961) a mechanism for this negative resistance involving the inelastic scattering of free carriers by pairs of nearby majority impurities which w i l l form a configuration analogous to a hydrogen molecule ion i f singly ionized by compensation. This extra scattering process disappears after breakdown since the "molecules" become f u l l y ionized allowing the breakdown to be sustained at a lower f i e l d than that required for i t s i n i t i a t i o n . Injecting Contact " -0 MI L FIGURE 3 2 SEMICONDUCTOR FILAMENT UNDER INJECTION 22 3.4 Phenomenological Model f o r Negative R e s i s t a n c e A pnenomenological model showing how c o n d u c t i v i t y modulation can l e a d to n e g a t i v e r e s i s t a n c e i s developed i n t h i s s e c t i o n . The steady s t a t e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s a re d e r i v e d and the parameters i n v o l v e d a re v a r i e d t o see i f n e g a t i v e r e s i s t a n c e can occur. T h i s i s a p u r e l y mathematical model and no c o n s i d e r a t i o n i s here g i v e n to the a c t u a l p h y s i c a l mechanisms behind the model. A steady b i a s V i s a p p l i e d i n the forward d i r e c t i o n a c r o s s a semiconductor f i l a m e n t o f l e n g t h L and u n i f o r m c r o s s -s e c t i o n A. C a r r i e r s are be i n g i n j e c t e d i n t o the f i l a m e n t a t the p o s i t i v e e l e c t r o d e ( F i g u r e 3.2). The c o n d u c t i v i t y i s assumed to be of the form <f = d 0 0 + ^ e " x / L ' ) (3.4.1) Where d0 i s the c o n d u c t i v i t y w i t h no i n j e c t i o n ; <x the modu-l a t i o n parameter and L 1 the p e n e t r a t i o n l e n g t h may be c u r r e n t -x/L f dependent. The term ae ' r e p r e s e n t s the r e l a t i v e change i n the c o n d u c t i v i t y due to the i n j e c t e d c a r r i e r s a t a d i s t a n c e x from the i n j e c t i n g c o n t a c t . The i n j e c t i o n l e v e l i s assumed to be h i g h , t h a t i s , < X » 1 . I f the c u r r e n t dependence of a. and L 1 i s assumed to be mi mo a = K^I , L' = Kg l (3.4.2) t h e i r l o g a r i t h m i c d e r i v a t i v e s m^  and Ek, are m-, = ( l / a ) d a / d l , m2 = ( I / V )dL»/dI. (3.4.3) L' = Cons tan t a) MODULATION PARAMETER CURRENT-DEPENDENT, PENETRATION LENGTH CONSTANT m , o . , y V b) MODULATION PARAMETER AND PENETRATION LENGTH CURRENT DEPENDENT FIGURE 3 3 CURRENT-VOLTAGE CHARACTERISTICS PHENOMENOGICAL MODEL tfcx) = oU l + ae ) 23 Inspection of the conductivity equation shows that since L' occurs in-the exponential, small changes in L' can change the conductivity considerably whereas small changes in a. will not. Thus modulation of the penetration length by the current indicates that negative resistance may be possible. The resistance is found to be .1 ~1 (3.4.4) • l f l » , r 1' UCLe * = Xjo(Vc3')dx = R 0 [ l + Y l n " ~ T ^ where R0=L/Ad0 is the resistance with no carrier injection. From V=IR, the differential resistance is given by dV/dl = V/I + (cvdR/acx)m1 + (I»3R/^1» )m2« (3.4.5) 1) If a and L' are current independent (m^,m2=0), then the current-voltage characteristic is ohmic (VoC I), 2 ) If a i s current dependent and L' is not (m^ O,m2=0), then d=d0(.1 +K<ji"*1 e ). As I increases, R varies as'I m1 1 in and the voltage as I "~ 1. From physical considerations, m-j is found to li e in the range 0<m^ <1. At the limit m.j=1, the voltage approaches a vertical asymptote. Thus there is no possibility of the I-V curve bending back to give negative resistance (Figure 3.3a). 3) If bothaand L' are current dependent (m^  ^ m^O), then negative resistance is a possibility. If o/ _ -a3R _ R0L' aO-e~L//L') °<1 - T - - - 2 ZrTfT (3.4.6) R d a R L (1+ae L / L )d+a) then equation (3.4.5) can he written as f = (l/V)dV/dI = 1 - o ( i m i - oC2m2. (3.4.8) The limits on o(1 and are found to be For negative resistance, . Hence <(1-o<'1m1)/o(2. (3.4.9) Since CKm-j-O and L/L' is found to be such that e L / / I , ,»1 at this limit, i t can be seen that m2 can be quite small and s t i l l satisfy the inequality. Thus a small change in the penetration length with current is sufficient to cause negative resistance (Figure 3.3b). Since m2>0, L* increases with current. The penetration length can be of two types. For 1/2 diffusion, L' is given by L'=(D"C) , where D is the diffusion coefficient andfis the lifetime; for drift, by L' =Eu.T, where p. is the carrier mobility and E the applied field. The model given above is not entirely self-consistent. If L 1 varies with the current, there is a possibility that i t may also vary with position. Hence the simple exponential law for the conductivity will not be valid. This inconsistency may be seen i f L* is assumed to take the form for drift. If the conductivity arises from a differential equation of the form 25 d(d- d 0 ) / d x = - ( f l ' - ^ A ' , (3.4.10) the o r i g i n a l s o l u t i o n i s not v a l i d when L' i s a f u n c t i o n o f p o s i t i o n . The e l e c t r i c f i e l d E i s r e p l a c e d by J/ d'(x), J b e i n g the constant c u r r e n t d e n s i t y . By s u b s t i t u t i n g E i n L 1 , the d i f f e r e n t i a l e q u a t i o n becomes d (d-Oo)/dx = -(tf-o^oyjjur-. F o r a g i v e n v a l u e o f J , ju and t a r e assumed to be co n s t a n t . The s o l u t i o n i s found to be d= oVO-Ce- x / L") (3.4.11) where L" i s g i v e n by /aXS/d0 and C i s a constant s e t by the boundary c o n d i t i o n s . I f the c o n d u c t i v i t y a t the i n j e c t i n g e l e c t r o d e i s taken to beo y(0), then 0=1-0^/(^(0). F o r v e r y heavy i n j e c t i o n , C i s approximately equal to u n i t y . The s o l u t i o n s f o r R a n d / a r e R = R 0[1 - ( L » A ) C ( 1 - e ~ L / L " ) ] / = (l/V)dV/dI = 1 - c / ^ - .cfyn where m^ = (I/C)dC/dI, m 4 = (I/L")dL"/<H o< 3 = -(C/R)3R/9C = C(R 0L»/RI')( 1-e~ L / / L") = -(L"/H) 9R/9L" = ( C V E ) [ ( I - A ) ( 1 - e - L / l " ) - . - V t " ] . I f C»'(0)=a^( 1+a), where a i s the modulation parameter o f the original model, then C=c/(1 + C0 and m^ =-m1/( 1+a). For very-heavy injection, o \ » 1 , my will be very small. In L " =pXJ/d0, i f ^iT is not current dependent, then m^=1. Therefore, with these two conditions (m^=0,m^=1), i t is found' that]f>0 always and that negative resistance can not occur. If jut is assumed to increase with current, then 1114=1+1115, where m^I/^ut)d(ju'0/dI>0. In this case, i f negative resistance is to occur, then m^  must satisfy the following inequality: ( L / L » ) ( U , ^ ) - 2 ( 1 - e - L / L " ) . M 5 > -LA", , ; ( 3 . 4 . 1 2 ) 5 / 1 - e W h (L/L"+1) If m^  does satisfy the inequality, then negative resistance is possible. Such values of m^  may not be physically realizable, but the purpose of this analysis is to show the various mathematical conditions whereby negative resistance can occur. In the original model, where the penetration-length is a function of current only, negative resistance could be produced quite easily by small changes in the penetration length. On the other hand, when the penetration length is position dependent as well, then negative resistance becomes difficult to obtain unless the penetration length changes considerably with current. 27 CHAPTER 4  CARRIER TRANSPORT BY DRIFT 4.1 Introduction In a d r i f t model, the current i s assumed to he f i e l d driven and d i f f u s i o n currents are neglected. Thus the current density equations are given by J p = qu p(p 0+^p)E ( 4 . 1 . D J n = QMn(n 0 + An)E. The ambipolar equation s i m p l i f i e s to 9Ap/at = -Ap/t - (jW/o^O+aAp)2) 3ap/dx. (4.1.2) These equations are v a l i d when the current produced by the f i e l d E i s much larger than the current due to d i f f u -sion. The assumption of space charge n e u t r a l i t y remains v a l i d except at very high i n j e c t i o n l e v e l s , where n=p. There-fore most of the models treated i n t h i s section w i l l obey t h i s assumption. The exceptions are a unipolar model i n which the current i s space—charge-limited, and a bipolar model at a very high i n j e c t i o n l e v e l . 28 4.2 No Recombination 4.21 Neutral Bipolar A neutral bipolar d r i f t model with no recombination i s f a i r l y simple to analyse and gives r i s e to no unusual e f f e c t s . I f the t r a n s i t time of the injected c a r r i e r s across the semiconductor i s short compared to the l i f e t i m e s of the c a r r i e r s , then recombination can be assumed to be n e g l i g i b l e . Prom the continuity equations, J" and J are found to be P n constants. The ambipolar equation i n t h i s case reduces to ( f o r the steady state) 2Ap/ax = 0. (4.21 .1) The solutions f o r Ap and, from the current density equation, f o r E are found to be constants: Ap = constant (4.21.2) E = constant = J/c£(1+aAp). The conductivity d = c£(1+aAp) therefore does not vary with p o s i t i o n . Due to the nature of the d i f f e r e n t i a l equation (4.21.1), no unique r e l a t i o n s h i p can be obtained r e l a t i n g the c a r r i e r density and the current density. The applied voltage i s given by V = JL/tf 0(1+aAp) (4.21.3) where L i s the length of the semiconductor, and the d i f f e r -e n t i a l resistance by 2 9 dV/dl = [ R o / ( l + a A p ) ] [ l - maAp/(1+aAp)] ( 4 . 2 1 . 4 ) where R Q=L/Ad 0, the d.c. r e s i s t a n c e w i t h no i n j e c t i o n , A being the c r o s s - s e c t i o n a l a r e a of the semiconductor; the c u r r e n t I=JA, and m = (I/Ap) dAp/dl, ( 4 . 2 1 . 5 ) assuming t h a t Ap depends on l y on the v a l u e of the c u r r e n t pass-i n g through the semiconductor. U n l e s s m > 1 , i t i s e v i d e n t t h a t n e g a t i v e r e s i s t a n c e cannot occur. ' I n the a.c. a n a l y s i s , the v a l u e s A p ( t ) = Ap + P l ( x ) e ^ J ( t ) = J + J i e ^ a r e s u b s t i t u t e d i n the time dependent ambipolar e q u a t i o n , r e s u l t i n g i n jwp., = - ( M 0 J / a f 0 ( 1 + a A P ) 2 ) aPl/ax. ( 4 . 2 1 . 6 ) Since Ap i s a constant independent of x, the s o l u t i o n i s simply-P 1 = P 1 0 e x p [ - ( j ( J c T / u o J ) ( 1 + a A p ) 2 x ] ( 4 . 2 1 . 7 ) where P-JQ i s the i n t e g r a t i o n constant s e t by the boundary c o n d i t i o n a t x=0. U n l i k e the steady s t a t e c a r r i e r d e n s i t y , the p e r t u r b a t i o n d e n s i t y v a r i e s w i t h p o s i t i o n . Prom the c u r r e n t d e n s i t y equation 30 J- = <i(1+aAp)E„ + d ap E, l o 1 o 1 the f i e l d E.j i e found and i n t e g r a t e d t o get the a.c. v o l t a g e (4.21.8) V = J 1 L _ ^ o j 2 P l Q D - e x p ( - j ^ ) ]  1 cf 0(l+aAp) 0/0(l+aAp)4juJo0 where the ambipolar t r a n s i t time Z& = L/uE = L c r o ( l + a A p ) % 0 J (4.21.9) has been i n t r o d u c e d to s i m p l i f y the n o t a t i o n . Thus the complex impedance can be w r i t t e n as ' Z((J) = R, (1+aAp) 1 + 3 a I p 1 Q [ l - e x p ( - j U / i ^ ) ] I-, W^O+aAp) (4.21.10) Another assumption made i s t h a t p 1 Q / l 1 = d A p / d I . T h i s assumption i s v a l i d o n l y i f i n e r t i a l e f f e c t s a re n e g l e c t e d a t the plane o f the c o n t a c t . The r e a l p a r t ( a . c . r e s i s t a n c e ) and imaginary p a r t (reactance) of the complex impedance are g i v e n by R(w) = R, (1+aA.p) aApm s i n ^ rti (1+aAp) a J x(u>) = R o m a ^ p ( 1 _ cosUjra). (4.21.11) (4.21.12) I n v e s t i g a t i o n o f R(u3) shows t h a t the second term i n the p a r e n t h e s i s reaches i t s maximum value a t zero frequency. As the frequency i n c r e a s e s , t h i s term o s c i l l a t e s and s u c c e s s i v e peaks become s m a l l e r . Hence i f n e g a t i v e r e s i s t a n c e does not 31 occur i n the steady state, then i t cannot occur at any other frequency. Investigation of the reactance X(a>) shows that i t i s always p o s i t i v e or zero: 0 < I M < 2R 0maAp/(1+aAp) 2t)? a. The reactance i s therefore inductive. Since t h i s model i s f a i r l y simple, i t i s easy to calculate the response AV^-(t) to an impulse of current AIt(t)=Q§(t) carrying a t o t a l charge Q, and to show that the Fourier transform can give r i s e to the complex impedance Z(o>). The time-dependent terms are introduced into the analysis i n a manner s i m i l a r to that of the a.c. perturbation method, and are assumed to be of small amplitude: J ( t ) = J + A J t ( t ) E(t) = E + AE t(x,t) Ap(t) = Ap + Ap t(x,t) where J , E, and Ap are the steady state values. The r e s u l t i n g equations f or the perturbation f i e l d and continuity are A E t = AJ t /o / 0 (1+aAp) - JaAp t /o^(1+aAp) 2 3Apt/at = -v 3Ap t / ax (4 . 2 1.13) where v=jiiE=>i0J/O^(1+aAp)2 i s the ambipolar v e l o c i t y . The continuity equation i s simply a wave equation with a propa-gation v e l o c i t y v; i t s so l u t i o n i s Ap^ .=f ( x - v t ) . The behaviour of Ap-f. and hence the voltage response to an impulse of c u r r e n t Al^.=Q S ( t ) , where <5(t) i s a d e l t a f u n c t i o n , i s d e s i r e d . Since recombination and d i f f u s i o n have been n e g l e c t e d , t h e r e are no d i s p e r s i v e e f f e c t s t o a l t e r Ap-fc. Thus Ap-f. should r e t a i n the same form f o r 0<t<'Ta, where ^ a=L/v i s the ambipolar t r a n s i t time of the i n j e c t e d c a r r i e r s ; thus APt = (QV<lAL) h(t - x / v ) . (4.21.14) The v o l t a g e response A v ^ . = A E t ( x , t ) d x i s g i v e n by = QL 6 ( t ) a l v U ( t ) (QZA < • AV*) = 1 ^ ) " A<ro(1+a^)2l5l)' 0 < t < ^ U - 2 1 ' 1 5 ) . where U(t) i s a u n i t s t e p f u n c t i o n . The F o u r i e r t r a n s f o r m of AV-t(t)/Q, ^ e v o l t a g e response per u n i t charge, l e a d s t o the complex impedance Z(co): Z(fc>) = J ^ a [ A V t ( t ) / Q ] e - ^ d t R o a I v f?a \0-exp(-3a>^)] (i+aap) " A c y i + a A p ) 2 l q A L J U.21.16; where R 0=L/AC^. The term ^/qAL i s the v a l u e of Ap^/AI^ a t the p o s i t i o n x=0. I f i n e r t i a l e f f e c t s are n e g l e c t e d a t the plane of the c o n t a c t , then the assumption <Za/qAL=dAp/dI can be made. With the s u b s t i t u t i o n s v = L / t a and m=(l/ap)dAp/dI, the complex impedance takes the same form as t h a t g i v e n by the a.c. p e r t u r b a t i o n method; thus (1+akp) aApm [ l - e x p ( - j u ) ^ a ^ (1+aAp) j w ^ 33 4.22 No Recombination—Space-Charge-Limited  Unipolar Flow A drift model in which the current is an unipolar space-charge-limited current involves more restrictive assump-tions than bipolar drift models. It is an excellent example showing the existence of band pass negative resistance in a semiconductor with no negative resistance in the steady state. This case has been treated previously by Shockley and Prim (1953) and by Shockley ( 1 9 5 4 ) . The basic assumptions underlying the model are the following! a) Current consists of a flow of minority carriers only which are supplied by some injecting mechanism. b) A potential barrier for minority carriers exists in the region near the injecting electrode, and the position co-ordinate and a l l dependent variables are measured from the potential extremum. c) The region to the right of the potential extremum is swept of both majority and minority carriers with the majority carriers collecting at the potential barrier. d) The space charge to the right of the extremum consists of majority impurity ions and mobile minority carriers. e) There is no recombination, i.e., transit time of the minority carriers across the semiconductor is much smaller than the carrier lifetime. f) Diffusion currents are neglected. 34 In addition to these s p e c i f i c assumptions are the general assumptions of onerdimensional flow, homogeneous semiconductor and field-independent m o b i l i t i e s . Another assumption used i s that the In j e c t i o n l e v e l at the po t e n t i a l extremum i s very high. Assumptions (a) and (d) are the s i g n i -f i c a n t departures from the bipolar models considered, where the current consists of both majority and minority c a r r i e r s and where the space charge consists of majority c a r r i e r s and majority impurity ions as well as minority c a r r i e r s and minority impurity ions. For convenience, an n-type semiconductor i s chosen, although the analysis w i l l apply equally well to a p-type semiconductor. The basic equations of Section 2.1 must be modified s l i g h t l y to account f o r the ad d i t i o n a l assumptions to be used. The steady state current density and Poisson's equation from assumptions (a) and (d) are J = Jp = qUpApE = constant (4 .22 . 1 ) (f/q)dE/dx = n 0 + Ap. ( 4 . 2 2 . 2 ) The two terms on the r i g h t of Poisson's equation are due to the space charge consisting of majority impurity ions K D=n 0-p 0 and mobile minority c a r r i e r s p=p0+Ap. For no current, the two equations can be solved f o r the voltage V f = q n Q L 2 / 2 € . ( 4 . 2 2 . 3 ) This voltage i s referred to as the "punch-through" voltage at which the space charge just supports the applied voltage. b ) VOLTAGE V/S. A p L / n o F IGURE 4.1 C U R R E N T A N D VOLTAGE SPACE-CHARGE-LIMITED UNIPOLAR MODEL I 35 For voltages greater than V^, a space-charge limited flow of holes takes place. A current density can he defined as the current that would flow i f were applied to an intrinsic semiconductor (i.e., space charge consists only of mobile minority carriers);, thus J f = 9£u pV f 2/8lA (4.22.4) The two equations (4.22.1) and (4 .22.2) are solved for space-charge limited flow with the assumption that ApQ»Apj(, where A P Q and Ap^ are the values at the boundaries x=0 and x=L. The solution gives a transcendental equation specifying &Pj/no in terms of J« > i pq 2n 0 L A J = nQ/ApL - ln(1+no/ApL) (4.22.5) The voltage is given by V = (€J 2 Aip 2q 3n 0 5)[u 2 / 2 - u + ln(1+u)j (4.22.6) where the substitution u=n /Ap^ has been made. The current and voltage can be expressed in terms of J^. and V^ . as J/J f = (32/9) [u - ln(1+u)]"1 1/7 £ = [u 2 - 2u + 21n(l+u)][u - ln(1+u)]~2. These two relationships are shown in Figure 4.1. The differ-ential resistance is given by dV 9 V f [2ln(1+uj - 2u + u ln(Uu)] . 0 0 = ; z (.4.22.7) di 16 JfA [u - ln(l,+u)j 36 and can be shown to be always p o s i t i v e . Thus the steady s t a t e does not e x h i b i t n e g a t i v e r e s i s t a n c e . F o r l a r g e i n j e c t i o n v a l u e s , A p / n Q V > 1 , the c u r r e n t -v o l t a g e c h a r a c t e r i s t i c s can be shown to be independent of n Q : J = ( 9 / 8 ) 6 u p V 2 A 3 , ( V > V f > 0 ) . (4.22.8) T h i s c u r r e n t - v o l t a g e r e l a t i o n s h i p f o r s p a c e - c h a r g e - l i m i t e d e m i s s i o n i n semiconductors i s analogous to C h i l d ' s Law f o r t h e r m i o n i c emission i n a vacuum diode. The assumption t h a t the d i f f u s i o n c u r r e n t i s n e g l i -g i b l e i s not v a l i d near the p o t e n t i a l extremum s i n c e the c u r r e n t i n t h i s r e g i o n i s predominantly c a r r i e d by d i f f u s i o n . Thus f o r s m a l l v a l u e s of x, the s o l u t i o n s above do not h o l d . Comparison wi t h the exact s o l u t i o n s i n c l u d i n g d i f f u s i o n worked out by Shockley and Prim shows t h a t the e r r o r f o r h i g h e r v a l u e s o f x and r e l a t i v e l y l a r g e e l e c t r i c f i e l d s i s s m a l l . For the a.c. a n a l y s i s , E ( t ) = E + E 1 ( x ) e ^ W t A p . ( t ) . = bp + p - j W e ^ J ( t ) = J + J 1 e * & r t a r e s u b s t i t u t e d i n the c u r r e n t d e n s i t y , P o i s s o n ' s and the c o n t i n u i t y e q u a t i o n s . The time dependent terms are separated, r e s u l t i n g i n 37 (£/q) ^ /dx = p 1 3WP-J = ->i 3(Ep 1 + E 1Ap ) / 9 x . (4.22.9) Thi s set of equations can be solved exactly f o r E^ or . Since i n the steady state, the r e l a t i o n s h i p between the po s i t i o n co-ordinate x and the injected c a r r i e r density Ap i s known, E-j can be found i n terms of p; thus E-,(Ap) = J-jAp (1+n0/Ap) 1 (l+n 0/Ap) 0-30) 300-30) where etO/^UpqnQ. The a.c. v o l t a g e i s g i v e n by JlJ£ V l " 2 J W 2 J*p * n 0 u ln(1+u) (1+u) 1" 3^ - 1 30+ (1-30) " ' 300-30)* (4.22.10) I t i s advantageous at t h i s point to introduce a new va r i a b l e , the t r a n s i t time f o r holes 7^ ., defined as T t = J^O/UpE) dx = (^/u pqn 0) ln(1+u). (4.22.11) A rearrangement of the terms gives u = n Q/Ap L = e ^ - 1 , ^ = ^ p q n 0 t T / ( C = V^R* i n which T R i s the d i e l e c t r i c relaxation time. With t h i s s u b s t i t u t i o n , the complex impedance i s given by J -1 (3 Z(<£) = - 1 30 i-3^ > 300-30)' (4.22.12) 2TT 4TT a) R E A L P A R T OF IMPEDANCE 87T b) IMAGINARY PART OF IMPEDANCE F I G U R E 4 .Z I M P E D A N C E O F S P A C E -C H A R G E - L I M I T E D U N I P O L A R M O D E L Separating the complex impedance Z(4>) into real and imaginary parts yields the a.c. resistance R(<£) and the reactance X(<£) respectively; thus R(4>) = 2 - 2e^cosp^> @ 0 +j6 2)2 • (1+02) 13(1 - ^ ) 6 ^ 8^(3^ (1+02)2|3# l ( e^ -1) (3t X(^) = J 6 ^pVn 0 3A (4.22.13) 0+#2) (1-^) 2)(e ( 3cos^^ -1) + 2e^sin 0(1-^ 2) 2 _ (4.22.14) where @<f)= Cdt^. Investigation of R(^) shows that a bandpass negative resistance occurs. The maximum value of R i s attained at zero frequency, i.e., R(0)=dV/dI. For a given value of the transit time (fixed (3), the value of R(^ >) oscillates. For low values of (3, i.e., short transit times, heavy injection, R(^ >) remains positive for a l l frequencies. However as (3 increases, R(^ >) becomes negative (Figure 4.2a). Investigation of the reactance shows that X((f>) i s always negative and hence capacitive over the whole frequency range (Figure 4.2b). An example i l l u s t r a t i n g typical values to be expected from this model i s given below. A germanium specimen of length L = 10 cm, a r e a A = 10 cm and n Q = 7*10 cm~^ i s con-s i d e r e d . For germanium, the m o b i l i t y p. = 1700 cm 2volt"*^ sec"^ -1? -1 and the p e r m i t t i v i t y 6 = 1.4*10 f a r a d cm . The d i e l e c t r i c r e l a x a t i o n time i s found to be ^ = 7 . 3 , 1 0 ~ 1 0 sec. The "punch-through" v o l t a g e i s found to be V f = 40 v o l t s and the c u r r e n t I f = AJj. = 43mA. Thus u n t i l V becomes 40 v o l t s , no c u r r e n t can f l o w a c r o s s the s t r u c t u r e . F o r v o l t a g e s g r e a t e r than 40 v o l t s , a space-charge l i m i t e d f l o w of h o l e s takes p l a c e . I f the o p e r a t i n g v o l t a g e i s taken to be 50 v o l t s , then from F i g u r e 4.1b, the f o l l o w i n g can be found: ^ P j / n Q = 0.045, 1 = 8 mA, and hence dV/dl = 865 ohms and V / l = 6250 ohms. The t r a n s i t time of the i n j e c t e d c a r r i e r s a c r o s s the semi-conductor i s = 2.3*10""9 sec, g i v i n g |3 =TT. I t can be seen t h a t f o r a l o n g specimen of around 1 cm, the v o l t a g e becomes v e r y l a r g e and p h y s i c a l l y u n r e a l i z a b l e (~10^ v o l t s ) . The c u r r e n t , however, does not change a p p r e c i a b l y . Thus the l e n g t h must be kept v e r y s h o r t to o b t a i n reasonable v a l u e s . Another l i m i t i n g f a c t o r i s the . magnitude of n Q . T h i s has t o be kept f a i r l y s m a l l so t h a t • remains p r a c t i c a l . From the p l o t of R(^) v s . ( F i g u r e 4.2a), the n e g a t i v e r e s i s t a n c e bands are found to l i e i n the frequency bands between 400-575 Mc/s, 880-1000 Mc/s and 1350-1420 Mc/s. The f i r s t n e g a t i v e r e s i s t a n c e peak has a magnitude of 33 ohms, a drop of approximately 95 percent of the d.c. v a l u e , a t a frequency of 450 Mc/s. The maximum v a l u e of the r e a c t a n c e i s 565 ohms ( c a p a c i t i v e ) a t a frequency of 220 Mc/s. The impedance due t o the s t a t i c c a p a c i t a n c e o f t h e s t r u c t u r e a t t h i s f r e q u e n c y i s 515 ohms. I t can be seen t h a t the f r e q u e n c i e s o f i n t e r e s t a r e a p p r o x i m a t e l y o f the o r d e r o f t h e r e c i p r o c a l o f the d i e l e c t r i c r e l a x a t i o n t i m e . As the t r a n s i t time i n c r e a s e s ( d e c r e a s i n g A p j / E ^ ) , the f r e q u e n c y a t w h i c h n e g a t i v e r e s i s t a n c e f i r s t o c c u r s d e c r e a s e s and the number o f n e g a t i v e r e s i s t a n c e bands i n c r e a s e s . F o r v e r y heavy i n j e c t i o n , ^ P i / n 0 l a r g e and f^. s h o r t , n e g a t i v e r e s i s t a n c e does n ot o c c u r . 4.3 R e c o m b i n a t i o n w i t h C o n s t a n t L i f e t i m e — N e u t r a l B i p o l a r I n t h i s s e c t i o n , a n e u t r a l d r i f t model w i t h r e c o m b i -n a t i o n w i t h a c o n s t a n t c a r r i e r l i f e t i m e i s a n a l y z e d . I n t h i s c ase t h e r e c o m b i n a t i o n i s r e p r e s e n t e d by r = A p / t where T i s n e i t h e r dependent on t h e c a r r i e r d e n s i t y Ap o r on t h e p o s i t i o n x. The s t e a d y s t a t e a m b i p o l a r e q u a t i o n f o r t h i s case i s -Ap/ = [^ o J/cT o (1+aAp) 2 ]9Ap/ax. (4.3 .D T h i s e q u a t i o n , s o l v e d f o r x, g i v e s x/L' = -[l / ( 1+aAp) - ln[(1+a&p)/aAp] ' . (4.3.2) where L' = ^ p-^C/c'0» The r e s i s t a n c e i s g i v e n by R = V/I = R Q [1 - ( L ' / L ) ( f L - f 0 ) J (4.3.3) ]Ap(x) JAp(O) where R =1/A0' i s the resistance for no injection and o ' o 0 f = 1/2(1+aAp)2, 0<f<1/2. The terms f^ and f^ represent the values at x=0 and x=L. The corresponding values of Ap w i l l be represented as Ap^ and Ap^. The differential resistance i s (1/RQ)dV/dI = 1 - ( I « A ) [ 2 ( f L - f ) + I d ( f L - f 0 ) / d l ] . ( 4 . 3 . 4 ) In i t s complete form, this equation i s d i f f i c u l t to analyze. Thus this model w i l l be investigated under the conditions of low, moderate, and high injection. The a.c. analysis, carried out as in the previous models, yields the a.c. component of the ambipolar equation: . + j g g l l , 2 * * * . ( 4 . 3 . 5 ) ^ c 5 0 ( l + a A p ) 2 L 3x 9x J f ( i + a A p ) Equation (4.3.1) is used to change the above equation to 3p-|/9Ap = (1+ju*)p1/Ap + 2ap1/(l+aAp) - J.,/J ( 4 . 3 . 6 ) Prom the current density equation, the f i e l d E^ i s given by E 1 = J^tf (1+aAp) - ap^ / c^d+aAp) 2 and the voltage V1 Eidx can be found. Unfortunately, the differential equation for p^  can not be solved exactly. Only the high injection case w i l l be treated. A useful relationship between the i n i t i a l injected carrier density and the carrier density at any point x i n the 4 2 semiconductor can be obtained by i n t r o d u c i n g the concept of the ambipolar t r a n s i t time, s a , up to the p o s i t i o n x* s a = J *0/uE) dx where p3 i s the ambipolar v e l o c i t y . I n the n e u t r a l b i p o l a r case, p. = ^ / ( 1 + a A p ) . The i n t e g r a l g i v e s the simple r e l a t i o n s h i p A p ( x ) = A p Q e ~ B a / ^ . (4.3.7) a) Low I n j e c t i o n For low i n j e c t i o n , the assumptions a^p <^<;1 , aAPj , «1 a r e made so t h a t f £f =1/2, T h e r e f o r e the c u r r e n t - v o l t a g e U L c h a r a c t e r i s t i c departs very l i t t l e from the l i n e a r case and ohmic flow r e s u l t s : J = q.(^i nn o+u pp o)V/L. ( 4 . 3 . 8 ) Since the c o n t r i b u t i o n from the i n j e c t e d c a r r i e r d e n s i t i e s i s n e g l i g i b l e , n e g a t i v e r e s i s t a n c e i s not p o s s i b l e . At low i n j e c t i o n l e v e l s , the c a r r i e r d e n s i t y d i s t r i b u t i o n takes the simple form Ap = A n Q e - x / L ' where L'=Ju T / o l can be c o n s i d e r e d as an e f f e c t i v e " d r i f t " o o l e n g t h . 43 b) Moderate Injection For moderate injection, a A p Q ^ I , and aAp^<1, giving fQ^f^. The condition akp^«1 can be relaxed to include cases where aAPj>1, as long as 1+aAp^<< aAp Q. In this case, equations (4.3.4) and (4.3.2) become (1/RQ)dV/dI = 1 - ( L ' A ) ( 2 f L + Idfj/dl) (4.3.9) (I/L») = -1/(1+aApL) + ln(1+1/aApL). (4.3.10) Equation (4.3.10) differentiated with respect to I and substituted into equation (4.3.9) gives (1/RQ)dV/dI = 1 - L « A(1+aAp L) 2 + aApL/(1+a/^L). (4.3.11) It can be shown by substituting equation (4.3.10) for L A ' in this equation that negative resistance can never occur. In this moderate injection range, the current changes from a linear to a square law dependence on voltage. The square law relation will be shown to be the case for the high injection condition treated next. An interesting feature of this particular case occurs i f the assumption fL=1/2»fQ is used. The following equations result: R A 0 = 1 - L'/2L I»A' = -1 + ln(1+1/aApL) ( l A 0)dV/dI = 1 - L ' A = 1 44 T h e f i n a l e q u a t i o n s h o w s t h a t n e g a t i v e r e s i s t a n c e i s p o s s i b l e i f L1/L 1. H o w e v e r , t h e e x a c t a n a l y s i s s h o w s t h a t n e g a t i v e r e s i s t a n c e c a n n o t o c c u r . A c l o s e r i n s p e c t i o n o f t h e a b o v e s e t o f e q u a t i o n s s h o w s t h a t i f b e t t e r a p p r o x i m a t i o n s a r e m a d e , t h e n e g a t i v e r e s i s t a n c e r e g i o n - b e c o m e s s m a l l e r . W i t h n o a p p r o x i m a t i o n s m a d e , t h i s r e g i o n i s f o u n d t o d i s a p p e a r . T h u s t h i s e x a m p l e s h o w s t h e f o l l y o f m a k i n g a p p r o x i m a t i o n s t o o e a r l y i n t h e a n a l y s i s , r e s u l t i n g i n e r r o n e o u s c o n c l u s i o n s . c ) H i g h I n j e c t i o n F o r h i g h i n j e c t i o n , aAp Q >>1, adp£>> 1. F o r t h i s * c a s e , i t i s m u c h m o r e a d v a n t a g e o u s t o r e w r i t e t h e a m b i p o l a r e q u a t i o n a s - A p / f = ( ^ J / C ^ A P 2 ) d A p / d x (4 .3.12) r a t h e r t h a n t o d r a w c o n c l u s i o n s f r o m e q u a t i o n s (4 .3 .2) a n d (4.3.4). T h e r e s u l t s o b t a i n e d a r e tf0L/J>i0? = 1 / 2 ( a A p L ) 2 V = j 2 ^ o T / 3 c V 2 ( a A p L ) 5 w h e r e t h e a s s u m p t i o n a A p Q » a A p ^ h a s b e e n u s e d a g a i n . T h u s t h e c u r r e n t - v o l t a g e r e l a t i o n s h i p f o r t h i s c a s e i s J = ( 9 / 8 ) q ^ n ( n 0 - p 0 X V 2 / l 3 , 0<V, n Q > p Q (4 .3.13) w h e r e C ^ o ^ p P ^ n ^ ) a n d F 0 ^ n ^ Q ' V Q ) / ( M p P o + ^ n o ) h a v e b e e n s u b s t i t u t e d . I t c a n b e s e e n t h a t n e g a t i v e r e s i s t a n c e c a n n o t o c c u r i n t h i s r a n g e . Lampert and Rose (1961) have obtained the same current-voltage relationship by taking space charge into account. In this region "the ohmic-relaxation" regime, where both the transit time of the injected carriers and the lifetime are greater than the dielectric relaxation time,^=£/C^. The space charge associated with the injected holes are "relaxed", i.e., neutralized by the simultaneous injection of electrons. This is a result of the charge neutrality assumption. It is of interest also to compare the current-voltage characteristic obtained here with that of the unipolar space-charge-limited model. In both cases, J.«*V2/L3. However, in the bipolar model, the multiplicative factor is QJ^Hp^^-Po) whereas in the unipolar model, i t is Cp. , It can be seen that for the lifetime much greater than the dielectric relaxation time, and at the same voltage and specimen length, the bipolar case allows a much greater current to flow than in the unipolar model since there is no space charge to be overcome. Another feature of the bipolar case is that much longer specimen may s t i l l operate under reasonable voltage values for a given current density. For lifetimes of the order of the dielectric relaxation time, the two characteristics are approximately same. An investigation of the current-voltage character-i s t i c , equation (4.3.13)» shows that the solution is not valid when the bias voltage is reversed since the current remains positive. The use of p-type material (p0>nQ) also violates the equation. However, the use of p-type material and a 46 reversed bias i s valid. This w i l l be discussed i n greater detail later. Another anomaly in this equation i s that as n Q-*p 0, J->0. This i s , of course, erroneous, and i s due to the neglect of diffusion which becomes important when the d r i f t term becomes small and comparable with the diffusion term. In this case, there i s no voltage range over which Jo^V^. The transition from the low injection regime to the high injection regime occurs approximately at the voltage where the curves (4 . 3 . 8 ) and (4 . 3 . 1 3 ) intersect: Vc S *2VVpM*o-Po>r U.3.U) The current-voltage characteristic i n the neighbourhood of this voltage i s given by the moderate injection analysis. The a.c. characteristics have a simple form under high injection. The ambipolar equation for this case becomes 9p1/dAp = ( 3+j^^p/Ap - J ^ J (4.3.15) The solution of this equation, to a f i r s t approximation i s P 1 = J1Ap/j(2+jo>t) (4.3.16) The actual solution has an added term which depends on the i n i t i a l injected carrier density, but this term can be assumed to be negligible. It i s indeed found that i n the f i n a l result, this term contributes very l i t t l e to the solution. With this solution, the a.c. component of the f i e l d and voltage may be readily found: 47 J 1 (1+jwr) J i J M n r d+3wr) jg _ 1 . y 1 o 1 (£aap(2+j<*>*) ' 1 3^,2(3^1^)3(2+36^ ' The complex impedance and i t s components are Z(CJ) R(6J) X ( U J ) where L 1 *=Jp.0rt/o~o a*id R0=L/AO^. An equivalent c i r c u i t f o r t h i s 1 /2 two-terminal system i s a resistance R-j=(R0/3) (8L/L') / i n series with a p a r a l l e l combination of a resistance R-j and an inductance R]fC/2. The e f f e c t of the term neglected i n equation (4 .3.16) i s to add a small o s c i l l a t o r y component to the equation f o r Z(iO). As mentioned previously, equation (4 . 3 . 1 3 ) i s not v a l i d f o r a reversed bias or f o r p-type material. This i s an in t e r e s t i n g feature and requires further i n v e s t i g a t i o n . The ambipolar equation (4 . 3 . 1 ) f o r bipolar neutral d r i f t with a constant l i f e t i m e can be rewritten i n the steady state as dp/dx = - ( p - p 0 ) A E ; i where ^= <UpU n(n-p)/(MpP+u nn) i s the ambipolar mobility; f o r n-type material ji>0, f o r p-type p.<0f and f o r i n t r i n s i c u=0. R 0 / 8 L ^ / 2 ( U j L o r ) 3 VL' / (2+jjwt) R O / 8 L \ 1 / 2 ( 2 +U ) V ) 3 \L« j (4+W2'c2) 1/2 3 U ' / U+V2?.2) ( 4 . 3 . 1 7 ) (4.3.18) ( 4 . 3 . 1 9 ) 48 This equation is completely general and can apply to either n- or p- type material. No assumptions have been made as to the nature of the contacts at either end. For a positively "biased filament, the above equation gives a negative density gradient for n-type material and P>PQ. This corresponds to carrier injection at the positive contact. Since the electrode -is- positive, holes are being injected and neutralized by electrons from the negative electrode. For a p-type semi-conductor, however, the gradient is positive. This obviously can not correspond to carrier injection since the density grows with increasing distance. Thus the ambipolar equation states that hole injection into p-type material is impossible. This is curious in view of the fact that physically, hole injection can not occur in p-type material, and nowhere in any of the defining equation is this fact incorporated. The only assumption made in the defining equation is that holes move in the direction of positive current. A similar situation occurs when the bias voltage is reversed. In this case, electron injection is found to be possible in p-type material and not in n-type. Perhaps the answer to this seeming paradox may be in the formulation of the ambipolar continuity equation. In Section 2.3» i t is shown that the injected carrier pair density pattern moves in an electric field in the direction in which the minority carriers would move. Thus the injection of majority carriers into a semiconductor would be opposed by the tendency of the injected excess pairs, i f they exist, to move in the opposite direction. d) Very High Injection—Bipolar, Space Charge 49 For very high injection, n=p, space charge neutrality may no longer be valid. The thermal densities are neglected and the current density, Poisson1s and the continuity equations become (£/q)dE/dx = p - n (Mn+Mp)p/^-n>ip^ = E9(n-p)/3x + (n-p)3E/9x J (4.3.20) If the assumption p^/qCju^+u^E is made in the continuity equation. This set of equations yields the following differ-ential equation This assumption can be justified by noting that in the current density equation, p and n are additive so that the small difference between n and p will not be too important. The exact analysis can be carried out by replacing n from Poisson's equation but the solution is not as useful since i t is d i f f i -cult to see the relationship between the position and the field intensity. The solution of (4.3.21) can be obtained with the d d d 2E substitution E ^ = The equation then becomes = -K, with the solution E = -Ky2/2 + C,y + C0. The field distribution with .respect to y is parabolic. It can readily be seen that the absolute value of y is not important. Therefore, to simplify the analysis, y=0 can be set at the field maximum. If holes are being injected at the anode, then the space charge due to the injected holes is largely neutralized by the injection of electrons from the cathode. The boundary conditions for this case can be taken as the vanishing of the electric field intensity at both anode and cathode. Rather than the usual procedure for setting the position co-ordinate as 0<x<L, i t is found to be advisable to take -L/2<x<L/2".It is assumed that the field intensity at the contacts is negli-gible compared to elsewhere in the semiconductor. Since the field distribution is symmetric with respect to y, the boundary condition E=0 at x=-L/2 and x=L/2 is analogous to E=0 at y=-yQ and y=Yj=+y^. Thus the solution is E = (Ky Q 2 / 2)fl - (y/y Q) 2]. ( 4 . 3 . 2 2 ) Prom E=dx/dy, the relationship between x and y is x = (Ky 0V6 ) [ ( 3 y/y 0) - (y/y 0) 3], y Q 3 = 3 L / 2 K . ( 4 . 3 . 2 3 ) This equation shows that x is an odd function of y and also that E is symmetrical about x=0. The voltage is found to be V = fao E2dy = ( 4 / 1 5)K 2y Q 5 J -y 0 which gives the current-voltage relationship J = (125/18) erp^^/L3. (4.3.24) This current-voltage relationship has also been derived by Lampert and Rose (1961) from the general differential equations for drift with space charge. The carrier densities are approximately inversely proportional to the field intensity so that the density distributions have a minimum near the center of the semi-conductor. The actual distribution of p and n can be found from the current density and Poisson1s equations as p = J + Mn € dE ^ ^ + u p ) E (Mn +%)^ dx n = J u 6 dE ^  Thus the hole and electron distributions are not symmetrically distributed with respect to x. The difference between p and n is (£/q)Ky/E and is quite small. It should be noted that the net space charge over the crystal is zero. At the anode, holes are being injected and electrons being removed; at the cathode, electrons are being injected and holes being removed. The words majority and minority carriers no longer have any meaning since the solution does not depend on which carrier is the greater in number. Inter-changing n and p has no effect on the solution. 52 A n o t h e r i n t e r e s t i n g f e a t u r e o f t h e s o l u t i o n i s t h a t i f t h e b i a s i s r e v e r s e d , t h e s o l u t i o n i s e q u a l l y v a l i d . B o t h t h e c u r r e n t a n d f i e l d b e c o m e n e g a t i v e . T h u s t h e c o n t a c t s m u s t n o t o n l y b e a b l e t o i n j e c t h o l e s a n d r e m o v e e l e c t r o n s w h e n b i a s e d p o s i t i v e l y b u t a l s o b e a b l e t o i n j e c t e l e c t r o n s a n d r e m o v e h o l e s w h e n b i a s e d n e g a t i v e l y . S u c h c o n t a c t s a r e h i g h l y i d e a l i z e d a n d m u s t i m p o s e n o c o n s t r a i n t s o n t h e e n t e r -i n g o r e x i t i n g c u r r e n t s . P h y s i c a l l y , s u c h c o n t a c t s a r e p r o -b a b l y n o t r e a l i z a b l e . A c t u a l c o n t a c t s may n o t b e a b l e t o i n j e c t o n e c a r r i e r a n d r e m o v e t h e o t h e r s i m u l t a n e o u s l y . I t s h o u l d b e n o t e d t h a t t h e a n a l y s i s g i v e s i n f i n i t e c a r r i e r d e n s i t i e s a t t h e c o n t a c t s . T h i s i s a r e s u l t o f n e g l e c t i n g t h e d i f f u s i o n c u r r e n t s . When t h e f i e l d v a n i s h e s , t h e d i f f u s i o n c u r r e n t s b e c o m e i m p o r t a n t . H o w e v e r , a s l o n g a s t h e d i f f u s i o n c u r r e n t s a r e c o n f i n e d t o a n a r r o w r a n g e i n t h e r e g i o n o f t h e c o n t a c t s , t h e d r i f t a n a l y s i s may b e a s s u m e d t o b e v a l i d . T h e r e g i o n b e t w e e n t h e c o n t a c t s s h o u l d b e s e v e r a l d i f f u s i o n l e n g t h s l o n g . T h e r a t i o J d i f f / J d r i f t i s g i v e n b y J d i f f / J d r i f t = [ k T ( u n - ^ p ) / E ] d E / d x w h i c h i s q u i t e s m a l l e x c e p t a t t h e c o n t a c t s w h e r e E=0 a n d d E / d x i s v e r y l a r g e . T h e t r a n s i t i o n f r o m t h e h i g h i n j e c t i o n r e g i m e o f S e c t i o n ' 4.3c t o v e r y h i g h i n j e c t i o n r e g i m e ' I n c u r s a p p r o x i -m a t e l y a t t h e v o l t a g e w h e r e t h e s q u a r e l a w a n d t h e c u b e l a w c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s i n t e r s e c t , n a m e l y vc*d = (81/500) L 2q(n 0-p 0)/6. (4.3.25) If V a - > c, the transition voltage from the low injection to high injection case, equation (4.3.H), is set equal to Vc_>d, then i t can be seen that i f nl /g q(n0~p0) < [(500/81)^e/^ pu ntj (4.3.26) then the high injection range JoCV2 will not exist. This will be the case i f n0=pQ. For a sufficiently extrinsic n-type semiconductor, i t can be seen that Va_ 0^ reaches a minimum value L2//ipT while Vc+& is limited only by the value of n 0-p Q and may be several orders of magnitude larger than Vg^. As no~ >Po' ^a+c i n c r e a s e s a n d Vc-*d decreases. 4.4 Recombination with Carrier Dependent Lifetimes—Neutral Bipolar For the case of carrier dependent lifetimes, the recombination term can be written as r = Ap/tr = rQAp . (4.4.1) In this case, the lifetime is also a function of position. Since the general drift model can not be solved exactly, only the high injection range is treated. The steady state ambi-polar equation is written as -r 0a 2Ap 2 + i > = ()i0J/d0) dAp/dx (4.4.2) and the solutions for Ap^ and the applied voltage V are 1 = J>i0/C5/0a2(1+V)r0ApL(1 + i ; ) (4.4.3) v = J 2u 0/0 J 0 2a3(2+i;)r 0Ap L ( 2 + ) ; ). (4.4.4) The current-voltage relationship becomes v ( u v ) / % o i / V o ( 2 ^ ( i ^ J ~ L(2+ ^ / V P o1 / V a ( l - ^ ( l + j ; ) ( 2 + ^ ) / y U ' 4 - 5 ) (l/V)dV/dI = V/(1+V). (4.4.6) For V=1, i t can be easily shown that the results are identical with the case for constant lifetime of Section 4.3c i f r 0 is replaced by Negative resistance can occur for -1<V<0. It is difficult, however, to see what type of mechanism can give recombination of this form, where the lifetime increases with the injection level in the form fcCAp , 1<y<2. The analysis is not valid for V=0, when the carrier lifetime increases directly with the carrier density. In this case, no approximations need to be made in the ambipolar equation, thus dx/^^J = -adAp/c^O+aAp)2 (4.4.7) where t ^ r ^ a . The solution obtained is O^x/ty^ = 1/(1+aAp(x)) - 1/O+aAp0) and the voltage becomes with the use of this equation 2 " JPo^i v = _ £ 1 + O o L ( 1 + a A p 0 ) (4.4.8) The solution is exact and gives a threshold voltage before 55 current can flow* The d i f f e r e n t i a l resistance i s dV R, di 2(1+'aAp0) 1 - maA' 1+aApQ ( 4 . 4 . 9 ) where R0=L/AC^ and m=(l/A.p 0)dAp Q/dI. Thus i t can be seen that unless m i s greater than unity, negative resistance can not occur. 56 CHAPTER 5  CARRIER TRANSPORT BY DIFFUSION  AND COMBINED DRIFT AND DIFFUSION 5.1 Introduction In the previous chapter, only drift currents were considered as the mechanism for carrier transport. In this chapter, the analysis is extended to include the effect pf diffusive currents. First a purely diffusive flow model, neglecting drift currents, is investigated; then the general problem including both drift and diffusion under specific conditions is discussed. Since the general problem including drift and diffusion is highly complicated, the results of this chapter are not conclusive, but are presented as a transition from the drift analyses of the previous chapter to the high injection level diffusion analyses of the next chapter. The assumption of space charge neutrality is used throughout this chapter. In order to get simplified expressions, boundary conditions of the form A -P=APQ at x=0 and Ap=0 at x=L are taken in most of the models considered. 5.2 Diffusion Model A purely, diffusive flow model arises when excess carriers are injected into a semiconductor under a field-free condition or under fields such that the drift current is negli-gible compared to the diffusion current. The injection of 57 minority carriers is always accompanied by a compensating majority carrier density since the least deviation from charge neutrality is sufficient to set up an electric field. However, this field has very l i t t l e effect on the diffusion of the minority carriers and hence can be neglected to a fi r s t approximation. The presence of majority carriers in the semi-conductor can be ignored except through the use of the ambi-polar diffusivity. The ambipolar equation in the steady state under the above conditions becomes 0 = -Ap/t + D8 2Ap/dx 2. (5.2.1) For convenience, a sufficiently extrinsic n-type semiconductor is assumed, so that JD=Dp. With this substitution, and the boundary conditions Ap=ApQ at x=0, Ap=0 at x=L, the solution of the ambipolar equation is Ap = ApQ sinhjL-xj/L p (5.2.2) sinhL/Lp where Lp=(Dp'r) 1/2 i S the diffusion length of the minority holes. When the length of the specimen is several times the diffusion length so that exp(L/Lp)»1, the solution takes on the simple exponential form Ap = Ap0 e"x/Lp . The current is given by J = -qD^p/ax. (5.2.3) 58 I f t h i s condition i s to hold, then the i n j e c t i o n l e v e l must he low, aAp « 1 . The v a l i d i t y of t h i s equation may be checked by considering the complete hole current density equation j = qPpPpJ + qMpJAP - qD 0(n 0+p 0 +2Ap) Bop  P <*0 cT0(1+aAp) (n 0+p 0)(1+aA P) a x Since e x t r i n s i c n-type n 0 » p 0 and low i n j e c t i o n n Q » A p i s assumed, the hole current density equation reduces to equation (5.2.3). However, under the same conditions, the ambipolar equation (2.3.9) reduces to 0 = -AP _ ^ P J &P + kTjap 2(b-1)/dAp\ 2 + D p 9 2 A p (5.2.4) X d0 3x dQ \dx) dx2 and not to equation (5 .2 .1 ) . Therefore i f the solution i s to be v a l i d , the terms containing the f i r s t d erivative of the hole density must be n e g l i g i b l e compared to the other two terms. The derivative calculated at x=0 i s substituted i n equation (5 .2 .4) , r e s u l t i n g i n the conditions (b -1)Ap/bn 0 «1 J « q b n Q D p / L p , equivalent to E«kT/qL p, to be s a t i s f i e d . The f i r s t i n e q u a l i t y i s s a t i s f i e d by the assumption of low i n j e c t i o n ; the second, however, gives an ad d i t i o n a l condition which must be s a t i s f i e d i f the solutions are to be v a l i d . Since the i n j e c t i o n l e v e l i n t h i s purely d i f f u s i v e flow model i s so low, the conductivity i s not modulated 59 appreciably and negative resistance cannot occur. For the diffusion of sinusoidal disturbances of injected carrier densities, the a.c. ambipolar equation becomes (1+jtJt) P l/t= D p3 2 P l/3x 2. (5.2.5) The solution is similar to that of the d.c. case: sinh (1+jto?)1//2(L-x)/L„ Pi = Pm 775 (5.2.6) 1 1 0 sinh (1 + ju)t) 1 / 2L/L T P which for exp(L/Lp)»1 reduces to ? 1 = p 1 0 expt ( 1 + ju)t) 1/2 x / L 1 . The diffusion length has been replaced by an effective a.c. diffusion length L p / ( 1 + jwt:)1/2> I t c a n b e s e e n t h a t t h e alternating carrier and current densities are attenuated more rapidly than the d.c. ones and suffer a phase shift in transit. The d.c. attenuation is due exclusively to carrier recombi-nation while the a.c. attenuation is due to recombination and a certain "inertia" of the diffusion process—the need to establish a certain excess carrier density before the current can flow. The propogation of the disturbance can be considered to proceed with a complex wave number (l + jw'c) 1/ 2/^ whose real part gives an effective wavelength and whose imaginary part yields the spatial attenuation. At high frequencies such that , this complex wave number takes the simple form . ( ^ / 2 D p ) 1 / , 2 ( l + 3 ) . I t can be seen that the l i f e t i m e has disappeared, so that recombination i s not important at high frequencies. The a.c. f i e l d f o r lov; i n j e c t i o n l e v e l s i s given by = J-jO-aop)^, - a P l J / t f 0 - (kT/d' 0)(ju n- (u p )9p 1 /ax which when integrated over the length gives the a.c. voltage as (with the substitution s=( 1 + j<*>r) 1 / 2 ) _ J<jL J^aApQL p(coth L/L p - cosech I i / L p ) ( 5 . 2 . 7 ) do dQ - J a p 1 Q L p ( c o t h s L / l p - cosech sL/L p) + kT(^ n-M T ?)p 1 Q . do3 do The complex impedance Z(k>) can be found by d i v i d i n g by J 1 . Here the assumption i s made that no i n e r t i a l e f f e c t s occur at the plane of the contact, giving PTQ/J^dAp^/dJ^Ap^y/j.. Thus Z(w)A = £ - I» paap Q(coth I»Ap - cosech L/L p) ( 3 . 2 . 8 ) ~So <*o - maAp QL p(coth sL/L p - cosech s L / l r ) + kT(fx^-pi^)m£pQ. <*>B V I t can be seen that since the frequency dependent term i s small, the a.c. resistance R(i*>) i s approximately equal to the d.c. d i f f e r e n t i a l resistance. For a long specimen, exp(L/L ) » 1 , the a.c. term becomes (JaL p/so / 0)dAp 0/dJ. For low frequencies C O T ^ O , the r e a l and imaginary parte are respectively (-JaLp/tf0)dApQ/dJ 61 and (JaL pwr / 2 d o)dAp o/dJ. For high frequencies 6 J f ^ » 1, they are (-JaL p/d 0 ( 2 c j t) 1/ 2)dAp ( )/dJ and ( J a L p / t f ^ u j ^ V 2 ) d A p ^ d J # 5 . 3 Combined D r i f t and D i f f u s i o n Models The transport problem f o r bipolar neutral flow includ i n g both d r i f t and d i f f u s i o n requires the complete ambi-polar equation f o r i t s solution. Hov/ever, the solu t i o n can be c a r r i e d through only under spe c i a l circumstances due to the complex nature of the equations involved. The ambipolar equation ( 2 . 3 . 9 ) i n the steady state i s 0 = -AP M p U - kTu p(b-1) 3flp~l X 00(1+aAp)2 dx L 3 x J D 0(n Q+p +2Ap) 3 2Ap ( 5 . 3 . D (n 0+p 0)(1+aAp) 3 x 2 where M 0 = > y * n ( n o - P oV i n ^ P ^ n ^ 8 3 1 ( 1 I )o = I )p ] )n (VPo )/ (¥o +Vo )-For the case of low i n j e c t i o n , a A p « 1 , the ambipolar equation can be made l i n e a r . The r e s u l t i n g d i f f e r e n t i a l equation i s 0 = -_^ P - **oJ 9 A P + Do^^P (5.3.2) x (SQ 3 X 3X2 and i t s solution can be e a s i l y obtained. At ultra-high i n j e c t i o n l e v e l s , the thermal c a r r i e r densities can be neglected so that n-p. As a r e s u l t , the ambipolar mobility ju=M0/(1+aAp) becomes very small and the ambipolar d i f f u s i v i t y D=D Q(n 0+p 0+2Ap)/(n 0+p 0)(1+aAp) tends to the constant D i=2D pD n/(D p+D n). Thus the f i e l d dependent terms i n the ambipolar equation can be neglected compared to the d i f f u s i o n term. The r e s u l t i n g equation i s 0 = -Ap/r + D ^ A p / a x 2 . (5.3.3) This equation w i l l not be solved i n t h i s section but w i l l be investigated i n Chapter 6 since the form i s the same as that f o r bipolar flow with equation c a r r i e r d e n s i t i e s . At moderate or high i n j e c t i o n l e v e l s , i t i s d i f f i c u l t to simplify the ambipolar equation to any form which can be solved. The only exception i s the case f o r no recombination when the analysis can be car r i e d out f o r a r b i t r a r y i n j e c t i o n l e v e l s . The solution of the ambipolar equation at low i n j e c t i o n l e v e l s (5.3.2) f o r no recombination i s Ap = * P o 1 - exp(-(l~*)W* nD ) ( 5 > 3 < 4 ) 1 - exp(-LJ;u 0/tf 0 D 0 ) i f Ap=0 at x=L. The soluti o n f o r recombination with a charac-t e r i s t i c l i f e t i m e independent of c a r r i e r density or po s i t i o n i s ( e r - L e r + x - e r+ Le r- x) A p = Ap L ( 5 ^ 5 ) where The boundary conditions Ap=ApQ at x=0 and Ap=0 at x=L have been taken. The current-voltage c h a r a c t e r i s t i c i n both cases 6 3 is approximately linear, being given by the ohmic relation J = q f o p P o + M ^ ) V / L . The departure from linearity is of the order of aApQ and hence very small. The work of Rittner ( 1 9 5 6 ) and Stockman ( 1 9 5 6 ) on low level drift and diffusion models with recombination is of interest here. Rittner obtains the same general form as equation ( 5 . 3 . 5 ) » his analysis being similar to the present analysis of this section. Stockman, on the other hand, takes a different approach by including Poisson's equation and obtaining a linear differential equation of the fourth order, which reduces to equation ( 5 . 3 . 2 ) for no space charge. The solution obtained is a superposition of four exponential decay processes with four generally different decay ranges. It should be noted, however, that both Stockman and Rittner are concerned with photoconductors with the boundary condition p^=An=0 at both electrodes. Thus the solutions obtained are not exactly similar to that of this analysis. For the case of no recombination, a different approach can lead to an exact analysis valid at any arbitrary injection level. The continuity equations for holes and electrons with no recombination in the steady state give J„ = constant N ( 5 . 3 . 6 ) Jp «= constant. Therefore the electric field E can be obtained from the current 64 density equations and solved f o r dap/dx: _ J p + gP paAp/3x _ J n - g D naAp / a x qu^i^+ap) QMp(P0+Ap) d&p = JpKbno-PpJp/Jp) + (b-J n/J p)Ap] qD pb(n 0+p 0 + 2 A p ) (5.3.7) The solution of equation (5.3.7) i s , assuming the boundary condition Ap=ApQ at x=0, qD pb(n 0-p 0)  J P X = (b-J^Jp) lnl ^bn 0-p 0J n/Jp + ( b - J n / J p ) A p 0 N ^bn 0-p 0J n/J + ( b - J n / J )Ap + 2(Ap 0-Ap)  n o " p o (5.3.8) I f the injected hole density at x=L i s taken to be zero, then the hole current density can be expressed as = y u - p ) r / + _ H ^ J + J i " P L ( b - J n / J p ) |_ \ bn 0 - p 0 J n / J p / n Q-p 0 , (5.3.9) The applied voltage i s found to be y = kT(b+J n/J p) l n / + b - J n / J p \ q(b-J n/J p) V b n 0 ^ p 0 J n / J p 7 (5.3.10) Because of the constant r a t i o J n / J p which may change with the current passing through the semiconductor, i t i s d i f f i c u l t to obtain an unique current-voltage c h a r a c t e r i s t i c . In the case where the current at the i n j e c t i n g contact x=0 i s composed s o l e l y of hole current, i . e . , the electron current i s T yYl+Ap0/n oV 2(ApQ-.Ap) \1 + Ap/n0 / n 0-p Q zero at x=0 and hence everywhere in the semiconductor, the analysis becomes much simpler. Since J n/Jp =0» equation ( 5 . 3 . 8 ) can be written as J At the limiting cases of low injection and high injection, Ap varies linearly with position. It is unlikely that the departure from linearity in the intermediate injection range is appreciable. The voltage and differential resistance take on the simple forms V = (kT/q)lnO+Ap0/n0) V dV = 2 A P o/ln(UAp 0/n 0) - (n Q-p 0)  1 d I (n 0+p 0 +2Ap 0 ) It can be noted here that the total current density, J=I/A, is equal to the hole current density J p . Since J N=0, the electron conduction current and diffusion current must be equal in magnitude. Thus diffusion plays an important part in this analysis and can not be neglected. Another approach to the combined drift and diffusion problem is to assume that the drift solution is the dominant solution and then substitute this solution into the diffusion term to obtain a small correction term. The validity of such an analysis can be checked by taking the ratio &±ff/3&T±f-t and comparing its magnitude along the filament to make sure 66 that the ratio is less than unity. This method can he used in the analysis of Section 4.3 for instance, hut leads to a complication in that the ratio of Ap^/Ap^ is required. Since the analysis of Section 4.3 has conveniently disregarded ApQ, the method in this case is not too helpful. 5.4 Validity Ranges for Drift and Diffusion Solutions In the preceding section, the problem of considering both drift and diffusion was briefly examined. The difficulties involved in the general problem can be readily seen. If the analyses are restricted only to diffusion or only to drifts' the problem generally becomes tractable. However, i f these solutions are to be used, the range in which the solutions are valid must be specified. For low injection levels, the validity ranges for diffusion only, drift and diffusion, and drift only, can be easily obtained. In Section 5.2, i t was shown that diffusion alone is valid for fields such that > « H / , l d l „ where I ^ J J i s "the diffusion length. For the drift solution, i t can be easily shown from the results of Section 4.3 and equation (5.3.2) that the condition E » kT/qL dr where I ) d r = JM 0f/o' 0 is the drift length, gives the validity range for drift. In the intermediate region kTAL d i f f<E<kT / 0 L L d r both drift and diffusion components have to be considered. For moderate and high injection levels, the corresponding ranges can not be as easily defined. In Section 5 .3 , i t was shown that the diffusion solution becomes important again at high injection levels. In Chapter 4 , however, i t was shown that drift solutions are also valid at high injection levels. Diffusion is also important in the intrinsic case n0=p0. However, the intrinsic case is straight-forward since the drift solutions break down and the problem of defining the validity ranges does not arise. On the other hand, there is a problem in defining the validity ranges for the non-intrinsic high injection diffusion-dominated solutions and for the high injection drift-dominated solutions. The procedure in finding the validity ranges is similar to that for the low injection ranges—to compare the neglected and retained terms in the ambipolar equation by substituting "a posteriori? either the drift solution or the diffusion solution. For diffusion, the solution Ap=ApQe"~X,/'I'i is substi-tuted in equation ( 5 . 3 . 1)• Thus the condition L d r / L i ^ < ( a * P ) 2 1 / 2 where L d r=Jji oT/^ 0 the drift length and Li=(Dif) ' the diffu-sion length, must hold for the diffusion solution to be valid. For drift, the solution Ap=Ap0e~sa/^ , equation ( 4 .3 .7 ) , where s a is the ambipolar transit time, is substituted in equation ( 5 . 3 . 1 ) . The condition L d r / L i » (a*p) 2 i s obtained f o r the v a l i d i t y range f o r the d r i f t s o l u t i o n . I t should be remembered that i n both cases aAp » 1 the high i n j e c t i o n l e v e l approximation. The d r i f t analysis i s not v a l i d at or near non-ohmic contacts ( i . e . , contacts at which Ap=An/0)• There the flow can be almost purely d i f f u s i v e . However, i f the length of the semiconductor i s taken-.. to be much greater than the d i f f u s i v e length, and i f d i f f u s i o n currents are large only over narrow regions confined to the v i c i n i t y of- the contacts, then the d r i f t analysis w i l l give a f a i r l y good approximation. 69 CHAPTER 6 BIPOLAR PLOW WITH EQUAL CARRIER DENSITIES 6.1 Introduction Bipolar flow with equal carrier densities, n=p, applies not only to intrinsic material but also to extrinsic material under conditions of high injection, as mentioned in Section 5.3. In this case, the ambipolar mobility vanishes due to the equality of n and p, and the ambipolar diffusivity becomes a constant, ^=2D Dn/(Dp+Dn) • T n e ambipolar equation for this case becomes quite simple: 3Ap/9t = - A p / r + D I3 2Ap/9x2. (6.1.1) Due to the absence of the field dependent term, i t can be seen that the spatial distribution of the carrier density is not subject to drift under the field, but only to diffusion and recombination. Only the neutral case is studied. This can be considered as only a rough approximation, since for high injection or for high resistivity material, space charge becomes important. Boundary conditions of two types are used at x=0. The fi r s t is simply Ap=ApQ at x=0, and the second is J ^ O H X j J , where Y is known as the hole injection ratio and gives the fraction of the current carried by the holes at x=0. The relationship between the two boundary conditions is easily found. The hole current can be written as 70 Jp = qup(p0+Ap)E - qDp9Ap/3x = VpoJ - q D ^ A p / a x where L =1/(b+1) for both the intrinsic and the high level extrinsic cases. At x=0, J p ( 0 ) = W ~ < * D i a A p / 5 x x = 0 = V (6.1.2) 3Ap/9x| = -(yT)-Ln)J/qDi. (6.1.3) lx=0 P P° 1 The expression for Ap obtained from the ambipolar equation using the boundary condition Ap=Ap^  at x=0 substituted in equation (6.1.3) gives the relationship between ApQ and' }f . 6 .2 Ho Recombination For no recombination, the ambipolar equation in the steady state becomes 32Ap/9x2 = 0 (6.2.1) If the boundary conditions Ap = Ap^ at x=0 and Ap = 0 at x=L. are assumed, the solution of the ambipolar equation is Ap = Ap0(1-X/L) (6 . 2 . 2 ) where L is the length of the semiconductor. When this solution is substituted in the current density equation, the electric field E may be found and integrated to get the voltage: E . J - " V " " - * ^ ) ( 6 . 2 . 3 ) tf±[l + A P 0 ( 1 - X A ) / P 0 ] V = J L kT(b-l) (b+l)AP() q(b+1)__ ln ( 1 + AP(>/p ) (6.2.4) 1 dV m R0 di (1+Ap 0/ P o) 1 + kT M p(b-1)Ap n IL ( m - O Ap0/p0 Ind+ApQ/Po) (6.2.5) where m=(l/Ap0)dApQ/dI, For heavy injection, A PQ>>P 0, and negligible diffu-sion current, the differential resistance is (1/R0)dV/dI = (P 0/AP 0 )[1 - (m-1)ln(A P o/p o)] . For negative resistance, under this condition, m must be greater than unity. However, this is unlikely, unless the injection ratio Vp, increases with increasing current. From equation (6.1.3), the relationship between APQ and is found: A*0 - ( V ^ D i ' (6.2.6) From this equation, the relationship between m and m* where m' = [ l / ( V p - ^ 0 ) ] d(y p-^ 0)/dI is found to be m = 1 + m' . (6.2 .7) Thus, i f )L increases with increasing current, m'>0, then m>1. The ambipolar equation for the a.c. analysis becomes JWp., = D ^ p ^ x 2 , (6.2.8) with the solution P l(x) = C i exp( jw/Di) 1 / / 2x + C2exp-( jW/L^)1/^. The constants, C-j and C 2 , are determined by the boundary conditions p^(L)=0 and P^(0)=P 1Q, giving sinh (j(j/D i) 1/2( L - x) P^x) = p 1 Q , . (6.2.9) ' 1 U sinh (jCo/Di)'/2! For (^/2D1)1/2L»1, the solution simplifies to P-l(x) = p 1 0 exp[-(^/2D i) 1/ 2(l + j)x] (6.2.10) and is identical to the purely diffusive flow of injected carriers at high frequencies which was worked out in Section 5.2. The voltage and a.c. impedance cannot be worked out exactly for this model and is not attempted. 6.3 Recombination with Constant Lifetime If the lifetime X is independent of the carrier density and of the position, the ambipolar equation in the steady state for bipolar flow with equal carrier densities becomes B2Ap/2x2 = A p A ± 2 (6.3 .D where L. =(D.f) 1 y / 2, the effective diffusion length. For ultra-high injection level, n=p, the generalized ambipolar equation also reduces to equation (6.3.1) as shown in Section 5.3. The boundary conditions Ap=Ap at x=0 and &p=0 at x=L 7 3 are used to obtain the solution sinh(L-x)/L. A p =Ap Q ±. u sinh L/L, ( 6 . 3 . 2 ) If the substitutions u=e^^~x^//^Ji and du/dx—u/I^ are used in the equation for the field E = j/cY0(1+aAp) - [kTUpfb-D/Cod+aApjjaAp/ax, then the field can be readily integrated to obtain the applied voltage and the differential resistances V = dV dl-LjJ ln Q kT(b-1) ln(1+aAp 0) where C^(1+4M2)1/2 q ( b + l ) kT(b-1) maAp0 R L./L Iq(b+1)(1+aAp0) (1+4M2) 1+4M2(1-m) L(l+.4M2)1/2 ln Q 4M(2Mcoth l/2I i - 1)m M Q = (coth L/21^)2 + 4Mcoth l/2L ± aA P o/(e L/ Li-e- L/ Li) coth L/2L± + 2M + (1+4M2)1/2 coth I/2L. + 2M - (1+4M2)1/2 - 1 ( 6 . 3 . 3 ) ( 6 . 3 . 4 ) RQ = 1/ACf,, m = (l/Ap 0)dA P o/dI. The case where the length of the semiconductor is several times the diffusion length giving rise to e ^ A l ^ l ^ s taken. There are two situations to be investigated: (i) high injection in short filament aAp0e~I,//l'i»1. 74 In this case, the voltage and differential resistance become V = IR0e L/Li 2aAp 0 Z l ln 4aAp L A kT(b-1) . « + — i L in aApQ q(b+1) dV R 0 e L / L i dl 2aAp, (1-m)^i ln 4aApQ - 1^  + m IJL kT(b-1)m Iq(b+1) For m<1, then dV/dI>0; for m>1, negative resistance may be possible i f m is sufficiently larger than unity. (ii) high injection in long filament aAp0e~Ii//l'i«1, aApQ» 1 . The voltage and differential resistance are V = IR fl - ^ 1 In 2 a A p + M(b-1) ln aAp0 L L J q(b+1) | j = R0|1 - ^ l n 2aAp0 - m ^ kT(b-l)m  + Iq(b+1) Since (Lj./L)ln aApQ is much smaller than unity, i t is unlikely that negative resistance will occur. An alternative boundary condition at x=L can also give rise to a similar solution as equation (6.3.2). The boundary condition 9Ap/9x=0 at x=L yields the solution = Ap. 0 cosh (L-x)Ai cosh I/L^ (6.3.5) > LAi For e ' both solutions reduce to Ap = Ap0 e ~ x / / 1 i . (6.3.6) With this new boundary condition the voltage and differential 75 resistance can be written as V = IR, dV di " R ° L " IR0q(b+D/ \l+aAp0e-L>/L^ (6.3.7) 1 - l n l M +aApQ 1 +aAp0' - L / L j m (l+aAp0e-L/Li) R 0L 1 kT(b-l) L Iq(b+1) (6.3.8) For very heavy injection such that aApQ 1, negative resistance can occur. For the a.c. analysis, the solution is for e L / / l ,i»1 P l(x) = p 1 Q exp[-(1 + jwr) 1/2 x/ L i] (6.3.9) where the diffusion length has been replaced by an effective a.c. diffusion length Lj/O+jut) 1/ 2. At frequencies high in comparison to the solution reduces to P l(x) = p 1 0 exp[-(W/2Di)1/2(1+j)x]. It can be noted that this limiting solution is identical with that for no recombination equation (6.2.10). Thus at high frequencies, recombination is not important in the a.c. solution. 6.4 Recombination with Carrier Dependent Lifetime If the carrier lifetime is a function of the carrier density, then the recombination term can be written, as in the d r i f t c a s e , a s r = A p / t = r 0 & p V . T h e a m b i p o l a r e q u a t i o n f o r b i p o l a r f l o w w i t h e q u a l c a r r i e r d e n s i t i e s i n t h e s t e a d y s t a t e b e c o m e s d 2 A p / d x 2 = A p / D j T = B A p ^ . (6.4.1) T h e s o l u t i o n o f t h i s d i f f e r e n t i a l e q u a t i o n c a n b e o b t a i n e d b y u s i n g t h e s u b s t i t u t i o n u = d A p / d x , d 2 A p / d x 2 = u9u/3Ap. T h e b o u n d a r y c o n d i t i o n dAp/dx=0 a t Ap=0 i s u s e d t o make t h e i n t e g r a t i o n c o n s t a n t v a n i s h . T h i s b o u n d a r y c o n d i t i o n ' a l s o l i m i t s V t o b e g r e a t e r t h a n - 1 ; t h u s ( d A p / d x ) 2 = 2 B A p 1 + IV (1 + V ) . ( 6 . 4 . 2 ) T h e n e g a t i v e r o o t i s t a k e n t o o b t a i n a s o l u t i o n w h i c h d e c a y s w i t h i n c r e a s i n g p o s i t i o n x ; t h u s Ap = A p Q (1 + K x ) " 2 / ( ^ 1 5 (6.4.3) w h e r e ApQ i s t h e v a l u e o f t h e i n j e c t e d c a r r i e r d e n s i t y a t x=0 a n d K = ( ^ - 1 ) ( B A p 0 l ' - 1 / 2 ( 1 + V ) ) 1 / 2 . (6.4.4) F o r V=1 , t h e s i t u a t i o n f o r c o n s t a n t l i f e t i m e , e q u a t i o n (6.4.3) b e c o m e s i d e n t i c a l w i t h t h e s o l u t i o n f o u n d i n t h e p r e c e d i n g s e c t i o n . T h u s , 7 7 Ap = lim A p „ [ * 1 V - * 1 U L  + J > = l i ~ 2 L ± which i s the same as equation ( 6 . 3 . 6 ) . The equations f o r the voltage and d i f f e r e n t i a l resistance, however, w i l l not be i d e n t i c a l with ( 6 . 3 . 7 ) and ( 6 . 3 . 8 ) due to the modification to be made i n the f i e l d equation as described i n the next paragraph. Due to the d i f f i c u l t i e s involved, the analysis w i l l be confined to the high i n j e c t i o n case ( a A p » 1 ) . The f i e l d and voltage i s then given by E = OiaAp _ kTu p(b-1) dAp O^aAp dx ( 6 . 4 . 5 ) J(V-1)/(^+D 0>+1)/(y-O 4 ) ^ 2 0 L kT(b-l)2 ln(1+KL) q(b+D(y-1) ( 6 . 4 . 6 ) The d i f f e r e n t i a l resistance can be written as dV di = " d r i f t m(l^+D [(1+KL) A - l ] 2 | J 1 + K L ) A + 1 - 1 ] + (b 2-D (Y p-)( p o)KLm(V +l)" 4 ( 1 + K L ) [ ( 1 + K L ) A + 1 - 1 ] where m = (l/Ap 0)dAp 0/dI ^ = 2 / 0 / - 1 ) ( 6 . 4 . 7 ) qUp(b+1)KAp 0 The e q u a t i o n g i v i n g KAp 0 i n terms of the i n j e c t i o n r a t i o Y was obtained from equations (6.1.3) and (6.4.3). T h e c o n d i t i o n f o r n e g a t i v e r e s i s t a n c e i s m(y+1)/2 ( 1 + K L ) A + 1 - 1 x (b2-l)(y-V)KL (1+KL) - 1 P — 2 2 2(1+KL) >1. (6.4.8) Two separate ranges f o r V are taken: the range V>1 and the range -1<V<1. F o r V>1, the equations can be used as w r i t t e n ; f o r -1<V<1, the equations are m o d i f i e d by s u b s t i t u t i n g K'=-K. Equa t i o n (6.4.3)» f o r example, becomes Ap = A p 0 ( 1 - K « x ) 2 / ( 1 - U ) . I n o r d e r t o keep the s o l u t i o n p h y s i c a l l y r e a s o n a b l e , the c o n s t r a i n t K'L<1 i s imposed. However, no such c o n s t r a i n t i s r e q u i r e d f o r KL i n the range V>1. For K L « 1 , the c o n d i t i o n f o r n e g a t i v e r e s i s t a n c e becomes m [ l - ( b 2 - 1 ) ( Y p - Y p o ) ( V - 1 ) / 4 ] > 1 . I n a semiconductor such as germanium, the c o e f f i c i e n t ( b 2 - l ) U - Y )/4 i s approximately 1/2 f o r *=1. Thus f o r V>1, i t i s h i g h l y u n l i k e l y t h a t n e g a t i v e r e s i s t a n c e w i l l occur since m must be large. Investigation of the inequality (6.4.8) shows that i t cannot be satisfied as KL becomes larger, the upper limit being m>2KL/(y+1). ForV=1, the condition for negative resistance is m>1. For the range -1<V<1, the condition for K'L«1 gives m[l + (b 2 -1)(Y p-^ o)(1-V)/4>1 which is possible for at least some values of m. At higher values of K'L, that i s , as K' L-*1, negative resistance disappears. From the equations for the two ranges, i t can be seen that for negative resistance to occur, the term due to diffusion (b2-1) ()f - (1-^)/4 is detrimental in the case of V>1 and advantageous in the case of -1<V<1. If the injection ratio V palso changes with current, then the possibility of negative resistance is increased. In this case m = 2(1+m')/(V+1) (6.4.9) where m' = [l/( Vp- K^)] d( )f p-^ 0)/dI. There are two possibilities: the injection ratio increasing with increasing current, m'X), and the injection ratio decreasing with increasing current, m'<0. In any event, i t is unlikely that m* will be very large. It can be noted that a changing YP admits the possibility of m>1. The differential resistance can be rewritten in terms of m' as 8 0 dV V d r i f t « " i f t l + K L p T - l ] K L d + K L ) ^ - m ' [ ( l + K L ) - f ) ( b 2 - D ( y p - ^ 0 ) K L ( U m ' ) 2 ( 1 + K L ) ( 6 . 4 . 1 0 ) F o r V>1 a n d m ' < 0 , n e g a t i v e r e s i s t a n c e w i l l d e f i n i t e l y n o t o c c u r . F o r m * > 0 , t h e r e may b e a p o s s i b i l i t y a t K L « 1 i f m f i s s u f f i c i e n t l y l a r g e e n o u g h . F o r l a r g e r v a l u e s o f K L , h o w e v e r , t h e r e i s n o p o s s i b i l i t y o f n e g a t i v e r e s i s t a n c e o c c u r r i n g . I n t h e r a n g e - 1 < ^ < 1 , n e g a t i v e r e s i s t a n c e i s p o s s i b l e f o r b o t h m ' > 0 a n d m ' < 0 . F o r m ' > 0 , d V / d l i s a l w a y s n e g a t i v e f o r a n y m * ; f o r m ' < 0 , d V / d l i s n e g a t i v e w h e n - m ' < K ' L . T h e r e l a t i v e i m p o r t a n c e o f . d r i f t a n d d i f f u s i o n i n t h i s m o d e l may be s e e n b y t a k i n g t h e r a t i o V ( j ; r i f t y / V d i f f u s i o n * V d r i f t 2 b ( V - 1 ) [ ( 1 + K L V A + 1 - 1 ] • d i f f ( ^ - O U p - l f p o W + l ) m O + K L ) ( 6 . 4 . 1 1 ) F o r K L « 1 , t h e r a t i o r e d u c e s t o d r i f t v d i f f (b^ -l)(!(p-lfpo) U n l e s s )L=}Lo» t h e d i f f u s i o n c o m p o n e n t c a n n o t b e n e g l e c t e d . When K * L a p p r o a c h e s u n i t y o r K L b e c o m e s l a r g e , t h e r a t i o c a n b e c o m e q u i t e l a r g e , t h u s e n a b l i n g t h e d i f f u s i o n t e r m t o b e n e g l e c t e d a n d s i m p l i f y i n g t h e a n a l y s i s c o n s i d e r a b l y . T h e p h y s i c a l s i g n i f i c a n c e o f t h e c o n s t a n t K c a n b e 81 seen by d e f i n i n g an e f f e c t i v e d i f f u s i o n l e n g t h L ^ f r o m the r i g h t hand s i d e o f equation (6.4.1); thus BAp V = Ap/Di? = A p / L v 2 B = A p 1 - V / I y 2 = A p ^ - ^ / L ^ O ) 2 = c o n s t a n t . Hence K = (V-1 )//2(y+1) L y ( 0 ) . (6 . 4.12)' I n the range V>1, L v ( 0)oCl/Ap^"" 1 ^ 2 , so t h a t as the i n j e c t i o n l e v e l i n c r e a s e s , Ly ( 0 ) decreases, and KL becomes l a r g e r . Thus K L » 1 would imply v e r y h i g h i n j e c t i o n . I n the lower range, -1<V<1 , Ly(0)oTApQ^"'^, so t h a t as APQ i n c r e a s e s , L^ ( 0 ) i n c r e a s e s . Thus K ' L « 1 would imply v e r y heavy i n j e c t i o n . A p h y s i c a l s i t u a t i o n which may g i v e r i s e to the b e h a v i o r n e c e s s a r y f o r n e g a t i v e r e s i s t a n c e may be seen by c o n s i d e r i n g the i n j e c t i n g c o n t a c t as w e l l as the semiconductor bulk. Negative r e s i s t a n c e i s found to be a t t a i n a b l e i n the -1<y<1 range. I n t h i s range, the e f f e c t i v e d i f f u s i o n l e n g t h Ly(0) i s p r o p o r t i o n a l to A p Q . Hence as the i n j e c t i o n l e v e l 1 V i n c r e a s e s , the l i f e t i m e a l s o i n c r e a s e s s i n c e ToCp ~ . The c a r r i e r s i n j e c t e d by the c o n t a c t w i l l then i n c r e a s e the l i f e -time i n the semiconductor bulk, which w i l l l e a d to a c o m p l i -mentary i n c r e a s e d p e n e t r a t i o n of the i n j e c t e d c a r r i e r s l e a d i n g to a f u r t h e r decrease i n the bulk r e s i s t i v i t y . T h i s i n t u r n l e a d s to a r e d i s t r i b u t i o n of the p o t e n t i a l between the contact and the bulk, the p o t e n t i a l drop a c r o s s the c o n t a c t r i s i n g and l e a d i n g to a c o r r e s p o n d i n g i n c r e a s e i n the c a r r i e r i n j e c t i o n which i n t u r n i n c r e a s e s the l i f e t i m e i n the b u l k . Such a "feedback" process o f self-enhancement would g i v e r i s e t o the behaviour necessary f o r n e g a t i v e r e s i s t a n c e . 6.5 C r i t i q u e of S t a f e e v ' s Model S t a f e e v ( 1 9 5 9 ) d i s c u s s e s q u a l i t a t i v e l y the p o s s i b i l i t y o f n e g a t i v e r e s i s t a n c e i n " l o n g diodes" a t u l t r a - h i g h i n j e c t i o n l e v e l s . I n a long diode, the l e n g t h of the diode i s much l o n g e r than the d i f f u s i o n l e n g t h of the c a r r i e r s . The diode c o n s i s t s of an unsymmetrical diode w i t h a p-n j u n c t i o n as the i n j e c t i n g mechanism. At each end of the diode are a t t a c h e d ohmic c o n t a c t s . The v o l t a g e - c u r r e n t c h a r a c t e r i s t i c t h a t he o b t a i n s i s s i m i l a r to e q u a t i o n (6.3.4). At h i g h i n j e c t i o n l e v e l s , he o b t a i n s the form I=I C(exp(qV/ckT) - i j where V i s the t o t a l v o l t a g e a p p l i e d a c r o s s the diode ( i n c l u d i n g c o n t a c t s , p-n j u n c t i o n and bulk) and c=2(b+coshL/L J. )/(b+1); I i s a c o m p l i c a t e d f u n c t i o n of the parameters o f the diode and of the m a t e r i a l . S i n c e the d i f f u s i o n l e n g t h appears i n the e x p o n e n t i a l , s m a l l changes i n can cause l a r g e changes i n I . F o r example, i f L/I^=6, 1=10^1^, a 20 percent change i n i s s u f f i c i e n t to change I by a f a c t o r of 50. 1 /2 Because L^oC X ' , changes i n the l i f e t i m e w i l l v a r y L^. The l i f e t i m e of the c a r r i e r s depends on the c o n c e n t r a t i o n o f the i n j e c t e d c a r r i e r s . I n the presence of recombination c e n t r e s i n the semiconductor, the occupancy of these c e n t r e s b e g i n to change at c e r t a i n i n j e c t i o n l e v e l s . Recombination proceeds a c c o r d i n g to the b i m o l e c u l a r law under these c o n d i t i o n s . The value of the c u r r e n t a t which t h i s occurs depends on the concentration and a c t i v a t i o n energy of the recombination centres. At large currents (concentration of occupied centres much lar g e r than concentration of unoccupied centres), the l i f e t i m e again become independent of i n j e c t i o n l e v e l . With t h i s background, Stafeev proceeds with a q u a l i t a t i v e argument. He assumes a current-voltage character-i s t i c appropriate to a low current (independent of current) and a current-voltage c h a r a c t e r i s t i c appropriate to a large current L^. Since the l i f e t i m e i s a function of i n j e c t i o n l e v e l over a c e r t a i n narrow current i n t e r v a l , the t r a n s i t i o n from the low current to high current cha r a c t e r i s t i c ; w i l l occur i n t h i s i n t e r v a l . The l i f e t i m e can increase or decrease with the i n j e c t i o n l e v e l depending on the properties of the recombination centres. For an increasing l i f e t i m e with forward current i n a diode with e^/^i»1, the i n j e c t i o n of c a r r i e r s i s enhanced by the "feedback" process mentioned i n Section 6.4. This leads to a t r a n s i t i o n from the low current to the high current c h a r a c t e r i s t i c and r e s u l t s i n a negative resistance. Although Stafeev 1s argument may be approximately correct, he makes one assumption that i s i n v a l i d . He assumes that the l i f e t i m e changes with i n j e c t i o n l e v e l (that i s , current) and that the l i f e t i m e at that given i n j e c t i o n l e v e l i s same throughout the whole semiconductor. This i s not the case, however. The l i f e t i m e i s a function of the l o c a l c a r r i e r density and hence of p o s i t i o n at i n j e c t i o n l e v e l s where a variable l i f e t i m e occurs. At high currents where the occupancy of the recombination c e n t r e s do not change, l i f e t i m e i s a constant independent o f p o s i t i o n and of c a r r i e r d e n s i t y . A t low c u r r e n t s , . t h e same i s t r u e . I n the i n t e r m e d i a t e range, the l i f e t i m e may be a f u n c t i o n of both p o s i t i o n and c a r r i e r d e n s i t y depending on the occupancy of the recombination c e n t r e s . T h i s change i n l i f e t i m e c o u l d probably be checked e x p e r i m e n t a l l y by measuring the l i f e t i m e a l o n g the l e n g t h of the semiconductor. S t a f e e v 1 s model can be put on a f i r m e r base by u t i l i z i n g the r e s u l t s of S e c t i o n 6.3 and S e c t i o n 6.4. The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s i n the constant l i f e t i m e r e g i o n s can be obtained from S e c t i o n 6.3. I n the t r a n s i t i o n range, the r e s u l t s of S e c t i o n 6.4 can be used. The a p p r o p r i a t e v a l u e of V which g i v e s the dependence of the l i f e t i m e on the c a r r i e r d e n s i t y of the form fo&p 1""^ c o u l d be found by r e f e r r i n g t o the r e l a t i o n s h i p between the c a r r i e r d e n s i t y and the l i f e -time such as i n the Shockley-Read model (Shockley and Read 1952). The range of v a l u e s which was found i n S e c t i o n 6.4 f a v o u r a b l e f o r n e g a t i v e r e s i s t a n c e was -KV<1, t h a t i s , f o r i n c r e a s i n g l i f e t i m e with c a r r i e r d e n s i t y . Thus the complete c u r r e n t -v o l t a g e c h a r a c t e r i s t i c i n c l u d i n g the n e g a t i v e r e s i s t a n c e range c o u l d be determined by the use of the models presented i n t h i s c h a p t e r . Recent experiments on l o n g diodes of InSb ( K e l n g a i l i s and R e d i k e r 1962) seem to i n d i c a t e t h a t c u r r e n t - c o n t r o l l e d n e g a t i v e r e s i s t a n c e occurs from an i n c r e a s e of the i n j e c t e d c a r r i e r l i f e t i m e w i t h c a r r i e r d e n s i t y . 85 CHAPTER 7.  CONTACT PROPERTIES AND  RELATED BOUNDARY CONDITIONS 7.1 Boundary C o n d i t i o n s I n the p r e v i o u s c h a p t e r s , the boundary c o n d i t i o n s at the c o n t a c t s have been expressed i n the form as Ap=^p^ a t x=0 and ^p=^pTj a t x=L. No r e f e r e n c e has been made to s p e c i f i c c o n t a c t p r o p e r t i e s or i n j e c t i o n mechanisms. T h i s problem of s e t t i n g boundary c o n d i t i o n s l e a d s to many d i f f i c u l t i e s s i n c e u s u a l l y the behaviour o f the i n j e c t e d c a r r i e r d e n s i t i e s w i t h c u r r e n t i s not e x p l i c i t l y known, and then i t i s i m p o s s i b l e to get an e x p l i c i t c u r r e n t - v o l t a g e r e l a t i o n s h i p . Another boundary c o n d i t i o n t hat has been used a t the i n j e c t i n g c o n t a c t i s t h a t the h o l e c u r r e n t be some f r a c t i o n #p o f the t o t a l c u r r e n t d e n s i t y , that i s , J p ( 0 ) = ^ p J . With t h i s , i t i s p o s s i b l e to get a r e l a t i o n s h i p between Ap@, Kp and J , but s i n c e the i n j e c t i o n r a t i o )( p i s i n g e n e r a l a f u n c t i o n o f J , an e x p l i c i t c u r r e n t -v o l t a g e r e l a t i o n s h i p may s t i l l be u n o b t a i n a b l e . An a l t e r n a t i v e approach t h a t can be used i s to s o l v e the t r a n s p o r t problem i n terms of the e l e c t r i c f i e l d E, as i s done i n S e c t i o n 4.3d f o r b i p o l a r n o n - n e u t r a l d r i f t a t ve r y h i g h i n j e c t i o n l e v e l s . I n t h i s case, i t i s assumed t h a t the f i e l d s a t the c o n t a c t s are much s m a l l e r than elsewhere, so that the boundary c o n d i t i o n E-0 i s used a t both c o n t a c t s . T h i s l e a d s to i n f i n i t e c a r r i e r d e n s i t i e s a t the c o n t a c t s . As mentioned 86 p r e v i o u s l y , t h i s i s a r e s u l t of n e g l e c t i n g d i f f u s i o n c u r r e n t s , which become important a t low f i e l d s . I n g e n e r a l , however, the s o l u t i o n i n terms of E i s more d i f f i c u l t s i n c e i n a l l p h y s i c a l systems, the independent parameter impressed on the system i s the v o l t a g e or the t o t a l e l e c t r i c c u r r e n t . T h e r e f o r e the f i e l d i s a v a r i a b l e dependent on the c u r r e n t and the i n j e c t e d c a r r i e r d e n s i t i e s and i t s boundary c o n d i t i o n s cannot be s p e c i -f i e d i n advance of the s o l u t i o n o f the problem. The use of E=0 i s thus o n l y an approximation which must be used w i t h c a r e . When d i f f u s i o n c u r r e n t s are taken i n t o account, the f i e l d may not be n e g l i g i b l e a t the c o n t a c t s . Because of t h i s d i f f i c u l t y , i t has been found advantageous to e l i m i n a t e E i n most o f the a n a l y s e s and to v/ork w i t h the excess c a r r i e r d e n s i t y as the dependent v a r i a b l e . I n r e l a t i n g the boundary c o n d i t i o n s to a c t u a l c o n t a c t s , t h e r e are two q u e s t i o n s t o be c o n s i d e r e d . F i r s t , what type of c o n t a c t can meet the requirements of the boundary c o n d i t i o n s ? T h i s w i l l concern such matters as to whether the c o n t a c t b l o c k s , i n j e c t s or e x t r a c t s c a r r i e r s or whether i t has no e f f e c t on the flow of the c a r r i e r s . Second, how does the c o n t a c t behave when the i n j e c t i o n l e v e l changes? T h i s i n v o l v e s the problem o f how the i n j e c t i o n r a t e v a r i e s with the c u r r e n t p a s s i n g through the semiconductor. 87 7 . 2 Contacts In t h i s s e c t i o n , the r e l a t i o n s h i p between the boundary c o n d i t i o n s and c o n t a c t s w i l l be developed. The d i s c u s s i o n i s not r e s t r i c t e d to metal to semiconductor c o n t a c t s but a l s o i n c l u d e s other inhomogeneous j u n c t i o n s t r u c -t u r e s , f o r example, a p-n j u n c t i o n . No attempt i s made to d e s c r i b e the numerous t h e o r i e s proposed (Henisch 1 9 5 7 ) f o r metal to semiconductor c o n t a c t s , none of which has proved t o be s a t i s f a c t o r y . The p r o p e r t i e s o f c o n t a c t s made to the s u r f a c e of a semiconductor depends l a r g e l y on the p r o p e r t i e s of the semi-conductor s u r f a c e w i t h which the co n t a c t i s made. Three f a c t o r s are i m p o r t a n t — t h e p o t e n t i a l b a r r i e r a t the s u r f a c e , the s u r f a c e r e c o m b i n a t i o n - g e n e r a t i o n r a t e and the presence of a f o r e i g n i n s u l a t i n g l a y e r on the s u r f a c e . The p o t e n t i a l b a r r i e r a t the s u r f a c e i s the r e s u l t of the abrupt d i s c o n t i n u i t y i n the p e r i o d i c c r y s t a l s t r u c t u r e o f the semiconductor. T h i s l e a d s t o a d i f f e r e n c e between the energy l e v e l s a t the s u r f a c e and those i n the bulk m a t e r i a l which may l e a d to a net p o s i t i v e or n e g a t i v e s u r f a c e charge. The presence o f the compensating bulk space charge g i v e s r i s e to a p o t e n t i a l b a r r i e r , the h e i g h t depending on the amount of the excess s u r f a c e charge. The e l e c t r o n i c s u r f a c e s t a t e s which may g i v e r i s e to the fo r m a t i o n o f the s u r f a c e charge can a l s o a c t as rec o m b i n a t i o n - g e n e r a t i o n centers f o r c a r r i e r s . The r e s u l t i n g 88 recombination r a t e d e f i n e s the s u r f a c e recombination v e l o c i t y and depends on the b a r r i e r h e i g h t . The e f f e c t of the i n s u l a t i n g l a y e r w i l l be d i s c u s s e d l a t e r . F i r s t , an "ohmic" c o n t a c t w i l l be c o n s i d e r e d . The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s are the absence of a p o t e n t i a l b a r r i e r and an i n f i n i t e r e c o m b i n a t i o n - g e n e r a t i o n r a t e a t the s u r f a c e where the c o n t a c t i s made. The f i r s t c o n d i t i o n ensures t h a t the c a r r i e r t r a n s p o r t i n e i t h e r d i r e c t i o n i s s t r i c t l y p r o p o r t i o n a l to the f i e l d a t any p o i n t w i t h no c o m p l i c a t i o n s due to space charge f o r m a t i o n . The second c o n d i -t i o n a g a i n ensures the absence of space charge e f f e c t s s i n c e the c a r r i e r s are being removed or s u p p l i e d by the c o n t a c t e x a c t l y a t the r a t e demanded by the t r a n s p o r t i n the semicon-d u c t o r . The i d e a l ohmic c o n t a c t maintains the thermal e q u i l i b r i u m d e n s i t i e s , n Q and p Q , of both c a r r i e r s . The r a t i o of the e l e c t r o n to the h o l e c u r r e n t i s c h a r a c t e r i z e d by V J p = > " n n o ^ p p o and h o l d s i r r e s p e c t i v e of the c o n d i t i o n s i n the semiconductor. The boundary c o n d i t i o n f o r the ohmic c o n t a c t i s thus An=Ap=0 a t a l l times. Departures from the ohmic case can r e s u l t i f a f i n i t e p o t e n t i a l b a r r i e r and a f i n i t e recombination r a t e e x i s t a t the s u r f a c e . Thus i n a non-ohmic c o n t a c t , the boundary c o n d i t i o n i s no l o n g e r c h a r a c t e r i z e d by Ap=An=0 but i s s e t by the nature of the c o n t a c t and the c u r r e n t p a s s i n g through i t . F i r s t , t h e d i r e c t i o n o f c u r r e n t r e l a t i v e t o t h e c o n t a c t w i l l b e d e f i n e d . I f t h e c u r r e n t f l o w s f r o m t h e c o n t a c t i n t o t h e s e m i c o n d u c t o r , t h e c u r r e n t w i l l b e t e r m e d p o s i t i v e ; i f t h e c u r r e n t f l o w s f r o m t h e s e m i c o n d u c t o r i n t o t h e c o n t a c t , t h e c u r r e n t w i l l b e t e r m e d n e g a t i v e . A t o n e c o n t a c t , t h e c u r r e n t t h r o u g h t h e c o n t a c t a n d t h e c u r r e n t i n t h e s e m i c o n d u c t o r w i l l a g r e e i n d i r e c t i o n ; a t t h e o t h e r , t h e t w o c u r r e n t s w i l l d i f f e r i n d i r e c t i o n b u t i n t e r m i n o l o g y o n l y s i n c e t h e d i r e c t i o n o f t h e c u r r e n t i s d e f i n e d r e l a t i v e t o t h e c o n t a c t . A n o t h e r f e a t u r e t o b e n o t e d i s t h a t t h e b o u n d a r y a t x = 0 o r x = L may n o t n e c e s s a r i l y b e a t t h e p l a n e o f t h e c o n t a c t b u t may be c h o s e n a s m a l l d i s t a n c e a w a y i n o r d e r t o e l i m i n a t e c o m p l i c a t e d s p a c e c h a r g e r e g i o n s o r d i f f u s i o n r e g i o n s w h i c h may v i o l a t e some o f t h e a s s u m p t i o n s u s e d i n t h e s o l u t i o n o f a p a r t i c u l a r m o d e l . F o r i n s t a n c e , a c r i t e r i o n w h i c h c o u l d b e a p p l i e d a t t h e p l a n e x=0 c o u l d b e t h e c o n d i t i o n J p a n d J n > 0 v/hen J > 0 . T h i s w o u l d e l i m i n a t e c o n d i t i o n s s u c h a s n e g a t i v e e l e c t r o n c u r r e n t d e n s i t i e s w h i c h may r e s u l t f r o m v e r y h i g h c o n c e n t r a t i o n g r a d i e n t s . A n o t h e r e x a m p l e i s i l l u s t r a t e d i n t h e u n i p o l a r s p a c e c h a r g e l i m i t e d f l o w m o d e l o f S e c t i o n 4.22 w h e r e t h e p l a n e x=0 was t a k e n t o b e a t t h e p o t e n t i a l b a r r i e r e x t r e m u m . T h e r e a r e f o u r t y p e s o f c o n t a c t s t o b e d i s c u s s e d . I f t h e c u r r e n t t h r o u g h t h e c o n t a c t i s p o s i t i v e a n d t h e c o n t a c t c a n s u p p l y e x c e s s c a r r i e r s i n t o t h e s e m i c o n d u c t o r , t h e n t h e c o n t a c t i s s a i d t o b e i n j e c t i n g . F o r i n s t a n c e , i n a n n - t y p e s e m i c o n d u c t o r , h o l e s w i l l b e i n j e c t e d w i t h a c o r r e s p o n d i n g i n c r e a s e i n t h e e l e c t r o n d e n s i t y t o m a i n t a i n a p p r o x i m a t e charge n e u t r a l i t y . The boundary, c o n d i t i o n f o r such a c o n t a c t c a n be c h a r a c t e r i z e d AP=APQ>0 w i t h J>0. ( I n the case o f p-ty p e m a t e r i a l , e l e c t r o n s w i l l be i n j e c t e d from t h e n e g a t i v e c o n t a c t . Due t o t h e c u r r e n t c o n v e n t i o n u s e d , t h e c u r r e n t w i l l be n e g a t i v e f o r an i n j e c t i n g c o n t a c t i n t h i s c a s e . I n t h e o t h e r t h r e e c o n t a c t c l a s s i f i c a t i o n s t o f o l l o w , t he c u r r e n t d i r e c t i o n w i l l be o p p o s i t e t o t h a t f o r the n-type m a t e r i a l w h i c h w i l l be d e s c r i b e d ) . S i n c e the c u r r e n t t h r o u g h the c o n t a c t c o n s i s t s i n g e n e r a l o f two components, one J N due t o the f l o w o f e l e c t r o n s and t h e o t h e r J P due t o the f l o w o f h o l e s , an i n j e c t i o n r a t i o ft c o n s i s t i n g o f the r a t i o of the' e l e c t r o n o r the h o l e c u r r e n t t o t he t o t a l c u r r e n t can be d e f i n e d . F o r n-type m a t e r i a l , w i t h h o l e s b e i n g i n j e c t e d , t he i n j e c t i o n r a t i o i s F o r an ohmic c o n t a c t , i s n o t z e r o but e q u a l t o Yp0=p0/(bn0+Pg). Thus i t can be seen t h a t i n j e c t i o n o c c u r s f o r ^Vp^VpQ-I f t h e c u r r e n t t h r o u g h t h e c o n t a c t i s n e g a t i v e , i t may be p o s s i b l e t h a t t h e c a r r i e r d e n s i t y i n c r e a s e s i n t h e neighb o u r h o o d o f t h e c o n t a c t . T h i s can o c c u r i f t h e c o n t a c t c a n not a c c e p t h o l e s r a p i d l y enough and the h o l e s t e n d t o b u i l d up a t the c o n t a c t w i t h a c o r r e s p o n d i n g i n c r e a s e i n the e l e c t r o n d e n s i t y t o m a i n t a i n a p p r o x i m a t e charge n e u t r a l i t y . Such a c o n t a c t i s s a i d t o be an a c c u m u l a t i n g c o n t a c t . I n the h i g h l e v e l d i f f u s i o n model o f C h a p t e r 6, t h e r e l a t i o n s h i p between Up and ApQ is given as ApQoc(^ p- )p0)J, Thus i f the hole density-increases Ap>0 with a negative current, then #p>Vp0.. If the current through the contact is negative and carriers are being depleted below their equilibrium value because the semiconductor bulk can not supply carriers fast enough, then the contact is said to be an extracting contact. In this case Ap<0,|Ap|<po and Vp*>Vp0. It should be noted that there is a limit on the size of Ap. A similar situation can occur i f the current through the contact is positive but carriers are being swept away from the neighbourhood of the contact because the contact can not supply sufficient carriers necessary for the transport in the semiconductor bulk. This phenomenom is referred to as carrier  exclusion. In this case Ap<0, |Ap|<p0 and ...VpX^po* Other definitions can be applied to contacts. A blocking contact for say electrons is one which can neither accept nor inject electrons. A saturated contact is one which is incapable of supplying a current density greater than a given saturation value. Thus i f the contact reaches a satu-ration value for the electron current density, then the contact becomes partially blocking for electrons. o It can be seen that the blocking or partially blocking contact would meet the requirements necessary for accumulation or exclusion. 92 The behaviour of the contact with changing current l e v e l i s also an important feature of the contact. I f the impressed c a r r i e r density at the plane x=0 i s A p 0 , the easiest method of inv e s t i g a t i n g the r e l a t i o n s h i p between the current I and the i n j e c t i o n l e v e l A P Q i s to assume a simple power law, thus m=(l/Ap0)dA.po/dI. This form has been used throughout the the s i s . In general, m i s d i f f i c u l t to determine and can not be in f e r r e d from the current-voltage c h a r a c t e r i s t i c s . Although i t i s usually necessary to determine m experimentally, i n c e r t a i n cases, i t may be possible to obtain m a n a l y t i c a l l y . For instance, i n a p-n junction at low i n j e c t i o n l e v e l s , the d i f f u s i o n model of Section 5.2 i s v a l i d . In t h i s case, the t o t a l current at the junction (x=0) i s predominantly a hole current so that Vp=1. Since the t o t a l current density through the semiconductor i s a constant, J can be represented by J=Jp(0)=qDpApQ/Lp. Hence m=1 f o r t h i s case. As the i n j e c t i o n l e v e l r i s e s and d r i f t becomes increasingly important, the electron current may become s u f f i c i e n t l y large so that i t can not be neglected. Thus Kp w i l l decrease on account of the decreasing hole current f r a c t i o n . In the purely d r i f t analysis of Chapter 4 with d i f f u s i v e terms neglected, the r e l a t i o n s h i p between Kp and ApQ i s H V = Mn a&Pfi °P"°PO (j^+jap) (1+a4p0) where Ifp^qjipPo/tfo, the f r a c t i o n of the current ca r r i e d by holes with no added c a r r i e r s . The l i m i t s set on lL by the 93 conditions of low and high injection levels are Y p o < V 1 / ( b + 1 ) -It can be seen that APQ does not depend exp l i c i t l y on the current and that the injection ratio Yp increases with current. The relationship between m and m1 = f l/( Vp-^p0)"] d( Yp- / p 0 ) /d l i s m' m = 1 - < V P W + V ^ o For the i n t r i n s i c and high level diffusive models of Chapter 6 , the relationship between Vp and A P Q i s where K P O = 1/(b+ 1 ) . In the in t r i n s i c case, K p 0 represents the fraction of the current carried by holes with no excess carriers, It can be seen that m^'+l where m' = [l/( tfp-^0)] d( Vp-Ypo)/dI. Negative resistance was found to occur for both m'>0 and m'<0. Thus a l l that i s required i s that the injection ratio changes slightly with current. In the case of an extracting or excluding contact where Ap i s limited by the condition |Ap\<pQ, the injection ratio ^ changes whenever the current becomes such that|Ap\ tends to become greater than p Q . For an extracting contact where l L Q < ) L< 1t the injection ratio w i l l decrease while for an X* ^ XT excluding contact where 0<)|p<)fpo, the injection ratio w i l l increase with increasing current. In practice, the behaviour of the contacts with current may be quite unusual. Harrick (1959) has observed injection regardless of the direction of current flow and also extraction regardless of the direction of current flow in metal to semiconductor contacts. The explanation given for this behaviour utilizes an insulating film between the metal and the semiconductor surface. It was found that often i f the semiconductor surface were clean when the contact was made, no extraction or injection was observed. However, when an insulating oxide layer (^10"*-* cm) was known to be present, the unusual effects were observed. Some other factors are mentioned by Harrick which can change the characteristics of a contact, such as the passage of large currents, a change in the ambient or the surface etching process. 7.3 Carrier Multiplication at Contacts The injection of excess carriers by a suitably biased contact is not the only means of increasing the carrier density in a semiconductor. Avalanche multiplication is an alternative means of generating hole-electron pairs and can inject carriers into the semiconductor bulk. Avalanche multi-plication may occur in a narrow space charge region of'high resistivity where a high field is set up. Such a region of high resistivity may be due to a foreign insulating layer such as an oxide layer between the metal contact and the surface of the semiconductor. 9 5 A carrier moving in a high electric field may acquire sufficient energy to ionize the lattice and create a new hole-electron pair. The primary and the generated secondary carriers travel independently in the electric field and may cause further ionization. The probability of its doing so in a distance dx is expressed asc^dx for an electron and o(pdx for a hole. The ionization coefficients o( are functions of the field and are defined as the number of ionizing collisions per centimeter path length made by a single particle. For example, in an n-type germanium filament through which a steadily increasing current is passed, the electric field increases steadily until a critical field is reached at which the avalanche starts. The holes produced by the avalanche drift towards the negative terminal and increase the conducti-vity and thus reduce the field at this end. If the externally applied voltage remains constant, the field is enhanced at the positive end where the avalanche is occurring and increases the avalanche generation rate. Injection into the bulk may be so heavy that only a relatively small field exists in this region. G-unn ( 1 9 5 7 ) has carried out a theoretical analysis in which he assumes a planar region with the hole and electron ionization coefficients equal. He also assumes that the drift velocities are constants and equal. The current density is fairly large with the carrier space charge resulting from the avalanche process being much greater than the Impurity density. 96 The ionization coefficient i s assumed to depend exponentially with the f i e l d . With these assumptions, the voltage across -1 /2 the avalanche tends at large current densities to V a°Cj Thus i f the current increases, the voltage ultimately decreases and such an avalanche i s capable of showing a negative resistance. Avalanche injection can cause negative resistance indirectly as well. Due to the heavy injection levels possible, the injected carrier density may increase with the current i n such a manner that m=(l/APo)d£pQ/dI>1. This i s the necessary condition for negative resistance in several of the models investigated i n previous chapters. 97 CHAPTER 8 CONCLUSIONS AND OUTSTANDING PROBLEMS The models of t h i s thesis have shown how the con-d u c t i v i t y of a semiconductor can be modulated by the i n j e c t i o n or extraction of o a r r i e r s and the' conditions under which negative resistance could be obtained. The general case f o r both d r i f t and d i f f u s i o n currents i n the transport equations has been shown to be complicated and d i f f i c u l t to do. However, i t was shown that with the proper choice of operating l e v e l s , e i t h e r the d r i f t or the d i f f u s i o n term could be omitted i n the ambipolar equation. Por the d r i f t model with unipolar space-charge-l i m i t e d flow, a band pass a.c. negative resistance was obtained. The bi p o l a r neutral models i n general did not exhibit negative resistance. Two cases seemed to show a possible d.c. negative r e s i s t a n c e — t h e case f o r d r i f t with no recombination and the case where the l i f e t i m e increased with the c a r r i e r density. Por the d i f f u s i o n flow analyses which described the s i t u a t i o n f o r i n t r i n s i c and high l e v e l e x t r i n s i c semiconductors, negative resistance was found to be possible i n s p e c i a l cases. The case where the c a r r i e r l i f e t i m e increased with increasing c a r r i e r density was the most favourable f o r negative resistance. I t was found possible to define completely a negative r e s i s t -ance model when the l i f e t i m e varied with c a r r i e r density. The case f o r purely d i f f u s i v e flow with no d r i f t currents was 98 found t o be v a l i d o n l y a t v e r y low i n j e c t i o n l e v e l s and v e r y low f i e l d s . S e v e r a l problems r e s u l t from the a n a l y s e s o f t h i s t h e s i s . Among the t h e o r e t i c a l problems are the many i n s t a n c e s where the equations are i n t r a c t a b l e . I n the a.c. cases, i t may be p o s s i b l e t o c a l c u l a t e the impedance f o r c e r t a i n ranges o f f r e q u e n c i e s . Space charge e f f e c t s i n the presence o f t r a p s should be i n v e s t i g a t e d . The e x t e n s i o n o f avalanche i n j e c t i o n f o r a one-dimensional c o n f i g u r a t i o n should be pursued. The r e l a t i o n between the c u r r e n t d e n s i t y J and the impressed c a r r i e r d e n s i t y a t the c o n t a c t A P Q was found t o be important i n d e t e r m i n i n g n e g a t i v e r e s i s t a n c e . T h i s r e l a t i o n should be i n v e s t i g a t e d by assuming s p e c i f i c models f o r c o n t a c t s . E x p e r i m e n t a l l y , the r e l a t i o n between J and A P Q should be measured t o determine how the c o n t a c t behaves w i t h the c u r r e n t p a s s i n g through i t . The dependence o f the c a r r i e r l i f e t i m e on the c a r r i e r d e n s i t y should be checked by measur-i n g the l i f e t i m e a l o n g the f i l a m e n t . However, such experiments should be done under i s o t h e r m a l and known s u r f a c e c o n d i t i o n s . 99 BIBLIOGRAPHY Gartne r , W.W. and S c h u l l e r , M. 1 9 6 1 . Proc. I.R.E. 754. Gibson, A.P. and Morgan, J.R. 1960. S o l i d S t a t e E l e c t r o n . 1, 5 4 . Gunn, J.B. 1 9 5 7 . Progress i n Semiconductors. V o l . 2, p. 2 1 3 » Heywood and Company, L t d . , London. H a r r i c k , K.J. 1 9 5 9 . Phys. Rev. 11£, 876. Henisch, H.K. 1 9 5 7 . R e c t i f y i n g Semiconductor C o n t a c t s , Oxford, Clarendon P r e s s . Lampert, M.A. 1 9 5 9 . R.C.A. Rev. 20, 682. Lampert, M.A. 1962. Phys. Rev. 12£, 126. Lampert, M.A. and Rose, A. 1961. Phys. Rev. 121. 26. Leblond, A. 1957. Ann. R a d i o e l e c t . 12,, 95. M e l n g a i l i s , I . and Rediker, R.H. 1962. J . A p p l . Phys. j53_, 1892. Parmenter, R.H. and Ruppel, W. 1 9 5 9 . J . A p p l . Phys. ;50, 1548. R e d i k e r , R.H. and McWhorter, A.L. 1 9 5 9 . Proc. I.R.E. 41, 1207. R e d i k e r , R.H, and McWhorter, A.L. 1961. S o l i d S t a t e E l e c t r o n . 2, 100. R i t t n e r , E.S. 1 9 5 4 . Phys. Rev. 1161. R i t t n e r , E.S. ' 1956. P h o t o c o n d u c t i v i t y ( A t l a n t i c C i t y Conference 1 9 5 4 ) , Wiley, New York, p.215. van Roosbroeck, W. 1950. B e l l System Tech. J . 29_, 560. van Roosbroeck, W. 1 9 5 3 . Phys. Rev. 91, 282. van Roosbroeck, W. 1961. Phys. Rev. 123, 474. Shockley, W. 1 9 4 9 . B e l l System Tech. J . 28, 435. Shockley, W. 1950. E l e c t r o n s a n d Holes i n Semiconductors, D. Van Nostrand Co. Inc., P r i n c e t o n , N.J. Shockley, W. 1 9 5 1 . B e l l System Tech. J . 29_» 9 9 0 . Shockley, W. 1 9 5 4 . B e l l System Tech. J . 22* 7 9 9 . Shockley, W. and Prim, R.O. 1 9 5 3 . Phys. Rev. <K), 7 5 3 . Shockley, W. and Read, W.T. 1 9 5 2 . Phys. Rev. 8 7 , 8 3 5 . Stafeev, V.I. 1958. Soviet Phys.-JETP 2» 1502. Stafeev, V . I . 1 9 5 9 . Soviet Phys.-Solid State 1, 7 6 3 , 7 6 9 . Stockman, P. 1 9 5 6 . Photoconductivity ( A t l a n t i c C i t y Conference 1 9 5 4 ) , Wiley, New York, p. 269. 

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