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 i n 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 p a r t i a l 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 ii ABSTRACT The theory of conductivity modulation i n semiconductors and the conditions tinder which negative resistance can be obtained are investigated. The ambipolar transport equation i s derived f o r one-dimensional flow i n a two-terminal homogeneous semiconductor with no trapping and no gradients. temperature Charge n e u t r a l i t y 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 c a r r i e r transport with d r i f t and d i f f u s i o n currents i s d i f f i c u l t , the models i n v e s t i gated are mainly concerned with either d r i f t or d i f f u s i o n as the dominant transport mechanism. For a unipolar space-charge l i m i t e d 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 r e s i s t - ance i s found under special conditions f o r the case of no recombination and f o r recombination with a c a r r i e r l i f e t i m e increasing with c a r r i e r density. For recombination with a constant l i f e t i m e , the bipolar d r i f t model gives currentvoltage c h a r a c t e r i s t i c s of the form: l e v e l s , J°cv 2 a t high i n j e c t i o n l e v e l s , and JocV high i n j e c t i o n l e v e l s . law case. Joe V at low i n j e c t i o n y "5 at very Space charge i s important i n the cube Models ignoring d i f f u s i o n are shown to be v a l i d only f o r e x t r i n s i c semiconductors. iii 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 i s found to exhibit negative resistance under certain conditions. The most favourable situation i s 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 i s found that the properties of the contacts are important i n attaining negative resistance. Contacts and their properties are briefly discussed i n relation to the carrier density boundary conditions. Avalanche injection and i t s relation to the conductivity modulation problem i s considered. Several outstanding problems, both theoretical and experimental, arising from this thesis are outlined i n the last chapter. ACKNOWLEDGMENT I wish to thank Professor R. E. Burgess f o r h i s supervision i n carrying out and reporting t h i s research. I also wish to thank the B r i t i s h Columbia Telephone Company and the National Research Council f o r f i n a n c i a l assistance. iv CONTENTS CHAPTER 1. INTRODUCTION 1.1 Purpose 1.2 Review o f P r e v i o u s Work 1 1.3 Scope o f T h e s i s 2 CHAPTER 2. o f the I n v e s t i g a t i o n 1 TRANSPORT OP INJECTED CARRIERS IN SEMICONDUCTORS 2.1 B a s i c Equations 5 2.2 S i m p l i f y i n g Assumptions 7 2.3 The Ambipolar C o n t i n u i t y E q u a t i o n 10 2.4 Steady S t a t e and Small Amplitude A.C. A n a l y s i s 12 CHAPTER 3 . GENERAL CONSIDERATIONS OP CONDUCTIVITY MODULATION AND NEGATIVE RESISTANCE 3.1 C o n d u c t i v i t y M o d u l a t i o n by C a r r i e r I n j e c t i o n 16 3.2 Negative R e s i s t a n c e 17 3.3 Evidence f o r Negative R e s i s t a n c e 20 3.4 Phenomenological 22 CHAPTER 4. Model f o r Negative R e s i s t a n c e CARRIER TRANSPORT BY DRIFT 4.1 Introduction 27 4.2 No Recombination 28 4.21 Neutral Bipolar 28 4.22 Space-Charge-Limited 4.3 Recombination w i t h Constant U n i p o l a r Plow Lifetime— Neutral Bipolar 4.4 Recombination 33 40 w i t h C a r r i e r Dependent Lifetimes—Neutral 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 FIGURE 3.1 Negative Resistance Characteristics 3.2 Semiconductor Filament under Injection 3.3 Current-Voltage Characteristics— Phenomenological Model 4.1 22 23 Current and Voltage—Space-Charge-Limited Unipolar Model 4.2 Facing Page 18 35 Impedance of Space-Charge-Limited Unipolar Model 38 1 CHAPTER 1 INTRODUCTION Purpose 1 .1 of The ant the flow Investigation of injected problem i n semiconductor types exist in a holes of i n the the carriers concentrations conditions valence to drift nation, any an conductivity Review of The culties. under electric Previous equations the under of field, the of of The an carriers The c a n be increased at concentration electrons to Minority carriers conductivity to carrier is two over thermal equilibrium diffusion, thesis of conduction to the carrier these methods this import- band. added to bulk by which are subject and to of the recombisemi- investigate i n j e c t i o n and resistance c a n be to deter- obtained. Work describing the the action of flow may b e of drift, such a complex nature problem presents Solutions certain one conditions negative in solids of is is i n the conduction band. increased. purpose recombination are solution valence semiconductor m o d u l a t i o n due mine under what carriers the and can modulate The Current agency which t r a n s f e r s c a n be under conductor. 1.2 i n the from an electrode carriers physics. which normally exist band and t o injection the by carriers semiconductor—electrons band and p o s i t i v e these current added current diffusion that a rigorous formidable mathematical obtained, simplifying conditions. however, and i n many diffi- cases W. v a n R o o s b r o e c k has 2 presented the theory o f the flow of electrons conductors under charge n e u t r a l i t y ( 1 9 5 0 , charge (1961). tigated problem P a r m e n t e r and R u p p e l state has (1959) and L a m p e r t space-charge l i m i t e d c u r r e n t s insulators. to of transport L a m p e r t a n d Rose semiconductors. studied negative (1956) h a v e (1959) under d r i f t conditions i n (1961) h a v e e x t e n d e d by S h o c k l e y i n j e c t i o n l e v e l s by R i t t n e r has a l s o d e a l t w i t h h i g h qualitatively been developed a space-charge-limited (1962) (1954). one-carrier resistance f o r small The and e x t e n d e d t o Stafeev (1958, currents and resistance A n e v e n more r e s t r i c t i v e (1953) , model has i n which the c u r r e n t current. This model amplitude a.c. i n c e r t a i n state (Shockley 1954). Scope o f T h e s i s This the flow Several on (1949) the appearance o f negative f r e q u e n c y bands b u t n o t i n t h e s t e a d y of analysis i n semi- injection diffusive by S h o c k l e y a n d P r i m exhibits negative 1.3 this can occur under c e r t a i n c o n d i t i o n s . under c e r t a i n c o n d i t i o n s . is steady- t h e same p r o b l e m i n i n s u l a t o r s a n d h a s shown t h a t c o n d u c t o r s has been a n a l y z e d discusses inves- deal with case o f 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 1959) space i n photoconductors. Assuming charge n e u t r a l i t y , Lampert resistance arbitrary i n semi- 1 9 5 3 ) and w i t h Stockman (1956) and R i t t n e r the general and h o l e s t h e s i s i s concerned with a t h e o r e t i c a l a n a l y s i s of injected electrons and h o l e s m a t h e m a t i c a l models a r e s t u d i e d . which the analyses semiconductor filament a r e based or slab. i n semiconductors. The b a s i c structure i s a t w o - t e r m i n a l homogeneous I n j e c t i o n and/or e x t r a c t i o n o f 3 c a r r i e r s may be taking place at both terminals. I n i t i a l l y , the basic equations describing the transport of c a r r i e r s are presented and the ambipolar transport equation formulated using suitable simplifying assumptions. The physical s i g n i f i c a n c e of t h i s equation and the v a l i d i t y 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 d i f f e r e n t i a l equations describing the transport 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 s p e c i f i c models f o r which exact solutions or approximate solutions can be found. A steady state analysis i n one dimension i s carried out and the currentvoltage c h a r a c t e r i s t i c s investigated for negative resistance. Wherever possible, a small amplitude a.c. analysis i s attempted. The analysis i s intended to be as general as possible with no s p e c i f i c reference made as to the mechanism whereby the added c a r r i e r s are injected into the semiconductor. In a l a t e r section, however, contacts and related boundary conditions are discussed i n an attempt to correlate bulk and electrode e f f e c t s . F i n a l l y , since the treatment i s r e s t r i c t e d to s p e c i f i c . models, the fundamental d i f f i c u l t i e s of the general problem are discussed. Reference i s made to further extensions of the 4 analysis and to boundary conditions, tive resistance. other models, p a r t i c u l a r l y concerned w h i c h c o u l d be investigated to with obtain nega. 5 CHAPTER 2 TRANSPORT O F I N J E C T E D C A R R I E R S I N S E M I C O N D U C T O R S Basic 2.1 Equations The blems involving continuity. consists of equations where g-r fundamental the flow for electrons the e l e c t r i c particles, the of current continuity respectively, are (2.1.1) ap/at = g-r - (2.1.2) p current of pair the electron densities and hole concentrations, generation volume, q the and h o l e are given minus the net electronic current densities. by J^ = qM nE + qDngrad n (2.1.3) J = aju piE- qD grad (2.1.4) p the f i r s t term, the electron recombination per unit and J ^ and J the d r i f t (1/q) d i v ? the net rate n p term, p the d i f f u s i o n the gradient of p the conduction of the c a r r i e r s random t h e r m a l m o t i o n to the equation pro- = g - r + ( 1 / q )d i v The second is required' for an/at of pair where charged where and h o l e s , n and p r e p r e s e n t charge, to of which i s of particles I n semiconductors, represents rate the flow equation current i n an e l e c t r i c current density, of the c a r r i e r s the c a r r i e r density, field is and thus densities. due E , and t h e due t o is is the proportional I n thermal equili- 6 brium the e l e c t r o n and h o l e m o b i l i t i e s , , u and.jji^., are r e l a t e d to the d i f f u s i o n c o e f f i c i e n t s , "D and D , by E i n s t e i n ' s n p relations ^ = (q/kT)D , n where k i s Boltzmann's M p = (q/kT)D p constant and T i s the temperature. If the e l e c t r i c f i e l d s are n o t too l a r g e t o "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 c o n s i d e r e d v a l i d . The t o t a l c u r r e n t d e n s i t y i s the sum o f the e l e c t r o n and h o l e c u r r e n t d e n s i t i e s and a displacement c u r r e n t d e n s i t y : J = J* + J* + edE/dt n p (2.1.5) € b e i n g the p e r m i t t i v i t y . One more equation i s r e q u i r e d to s p e c i f y the problem. T h i s i s P o i s s o n ' 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 c e n t r e s i n the semiconductor: d i v grad V = - d i v E = -(q/<f)(p - n + N+ - % ) (2.1.6) * where - stands f o r a l l f i x e d i o n i z e d c e n t r e s . Equations (2.1.1) to (2.1.6) a r e the b a s i c equations d e f i n i n g the t r a n s p o r t problem and are completely g e n e r a l . 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. I f the semiconductor i s homogeneous, then the total electron and hole densities i n the presence of injection, n and p, can be written as n = n where n Q 0 + An, P = P 0 + Ap (2.2.1) and p , the thermal equilibrium densities, are cons0 tants; An and Ap are the injected carrier densities. Thus, the derivatives occurring i n the equations can be rewritten i n terms of the injected carrier densities rather than i n terms of the total carrier densities, e.g. grad n = grad An. The carrier mobilities, be f i e l d independent. and ; U p , are considered to This assumption i s only an approximation and i s valid at low and moderate electric fields only (Shockley 1951). Prom Einstein's relations, the diffusion coefficients w i l l likewise be constants. Temperature gradients are neglected i n the analysis. Any attempt at experimental verification of the analysis must ensure isothermal conditions. The flow i s assumed to be planar, i . e . one-dimensional. This assumption simplifies the mathematics of the problem considerably. However, since every physical problem involves some surfaces, there w i l l be some flow toward the surfaces. Thus the one-dimensional treatment can only be regarded as an approximation. 8 Trapping i s neglected and the impurity centers are assumed to be a l l substantially ionized i n the semiconductor. If neutrality exists i n thermal equilibrium, N^ - Nj may be replaced by n -p . 0 0 Space charge i s then due to the inbalance i n the electron and hole concentration increments An and Ap, and Poisson's equation becomes (5/q.) BE/dx = Ap - A n . (2.2.2) Unless a very strong f i e l d i s 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 w i l l be used i n most of the models considered. For the generation-recombination term, g-r, the most convenient relationship to use i s the one in which there i s no external generation of carriers and where the net rate of recombination r i s assumed to be proportional to the excess carrier density r = Ap/fp = The time constants T n An/r . n and T XT (2.2.4) are the lifetimes of the injected carriers; i n the neutral case, t =tp. n It should be noted that this equality i s valid only i n the case where traps are neglected. I f traps are present, then the neutrality equation must take into account the space charge of the traps and hence An w i l l not be equal to Ap. Thus the lifetimes w i l l be unequal i n 9 in. this case (Shockley and Head 1952). The lifetimes i n general are functions of the excess carrier densities. However, the mathematical d i f f i c u l t i e s which result usually prohibit the use of this relationship i n most analyses. Thus the lifetimes are usually considered constant, independent of the carrier concentrations. Surface recombination i s neglected. This i s primarily an aid for ignoring complicated boundary conditions, and i s related to the assumption of planar flow. The surfaces of the semiconductor w i l l act as a sink for excess carrier pairs and the flow of current w i l l no longer be one-dimensional. However, for a semiconductor filament, i t can be shown (Shockley 1950) that the effect of surface recombination i s to reduce the effective lifetime from the bulk value, 'f^^j» *° some lower u c value T, dependent on the surface recombination velocity and the dimensions of the filament; thus Vt = V r b u l k + V r s u r f ••• < - -5) 2 2 Provided that the appropriate value of the lifetime i s used, then surface recombination can be neglected and one-dimensional flow can be used i n 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 i n the form 9 An/at = - A n / r 3Ap/at = -Ap/r + ju^ndE/ax + ^ E d A n / a x + D 9 A n / a x 2 n p 2 n - MpP9E/6>x - /ipEdAp/ax + D p ^ A p / a x 2 (2.3.1) (2.3.2) M u l t i p l i c a t i o n of the f i r s t equation by ;u\pp and the second by ji^n and adding y i e l d s 9 An . 2 + An Tn t + ^n n dAp + ^p" (2.-3.3) at Mn^p p3An _ n3&p . 9x 3x_ E W p k T Rp9 An n3 /*p 3x2) 2 + 2 + q. _ At t h i s point i n the analysis, with the 9E/^x term eliminated, the assumption of charge n e u t r a l i t y , An=Ap, can be introduced. 9 ^p/at The equation then becomes = -AP/T - ;uEdAp/ax + D^Ap/ax (2.3.4) 2 where p = (n-pJAp/^+n/jOp). D = (n+p)/(p/D +n/D ) n (2.3.5) p are known as the ambipolar mobility and the ambipolar d i f f u s i o n c o e f f i c i e n t respectively; X i s the l i f e t i m e f o r both electrons and holes. Equation (2.3.4) and i s due to van Roosbroeck i s known as the ambipolar equation (1953). For strongly e x t r i n s i c n-type material (n>>p), p=p Sr and D=Dp; f o r strongly p-type material ( p » n ) and D=D . n For i n t r i n s i c material (n=p), ;u=0 and D=2D D /(D +D ). p n p n 11 Comparison w i t h the o r i g i n a l (2.3.1) a n d ( 2 . 3 . 2 ) continuity shows t h a t ambipolar equation identical with the continuity strongly extrinsic identical with equation the continuity electric ance, field i . e . , the excess disturbance which the the minority c a r r i e r s densities minority density drag on.the density decays spreads for is of particles of a i n disturb- Thus, the i n the d i r e c t i o n i n Attention strongly i s placed of the neutraliz- ambipolar extrinsic the relevant material m o b i l i t y and carrier carriers were mobility exerting For intrinsic material, out by d i f f u s i o n on without a the d r i f t and recombination. A more u s e f u l analysis is The pattern. shows a n e f f e c t i v e than the equation velocity o f by t h e as i f the majority only velocity and the presence minority carriers. pattern by pattern which are less diffusivity, field In less while for electrons. density i s taken.care and d i f f u s i v i t y . diffusivity pair w o u l d move. excess m i n o r i t y densities mobility and carrier by material for holes, the d r i f t moves i n a n e l e c t r i c ing majority the the d r i f t but represents n-type the ambipolar equation 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 an extrinsic p-type material, given f o r t h e "9E/9x t e r m , except for strongly equations i s obtained form o f the ambipolar by r e p l a c i n g equation n and p by n +Ap Q for this and p +Ap; Q thus _ at where AP "t M p S 3Ap " 1+aAp dx P (n +p +2Ap) 0 + Q 0 (n +p )(1+aAp) 0 0 3 Ap 2 9x 2 (2.3.6) 12 D =D b(n +p )/(Dn +p ) Mo^^o-PoVCtao+Po), 0 and a=(b+1 )/.(bn +p ); \>=}x /-p.^ 0 n 0 Prom the c u r r e n t )f p 0 0 0 0 (2.3.7) the r a t i o of the m o b i l i t i e s . e q u a t i o n ( n e g l e c t i n g the d i s p l a c e - ment term) (2.3.8) J * c3 (1+aAp)E + KTjUp(b-1 )dAp/dx = c o n s t a n t , / 0 E may be obtained and s u b s t i t u t e d i n the ambipolar e q u a t i o n , resulting i n a Ap 3t = _.AP _ / i 3Ap J - IcT (b-1 )3Ap O ^ d + a ^ p ) dx|_ 9x Q t Mp 2 P (n +p +2Ap) 3 A (n +p )0+aAp) 3 x (2.3.9) 2 t 0 0 0 0 p 2 0 where d' =qMp(bn +p ), 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 . 0 2.4 0 Steady S t a t e 0 and Small Amplitude A.C. Analysis The steady s t a t e 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 i n t o the semiconductor. steady s t a t e i n v o l v e s term. only The a n a l y s i s f o r the the omission of the time dependent I n the models concerned, a steady forward b i a s V i s a p p l i e d to a s l a b or f i l a m e n t of semiconducting m a t e r i a l . major p a r t of the i n v e s t i g a t i o n of c a r r i e r t r a n s p o r t i n v o l v e the steady s t a t e c u r r e n t - v o l t a g e The will characteristics. A l l v a r i a b l e s and q u a n t i t i e s w r i t t e n without f u n c t i o n a l dependence or without s u b s c r i p t s w i l l h e r e a f t e r r e f e r to the steady s t a t e values. 13 The potential drop across the semiconductor i s V = Edx = IR (2.4.1) t where R i s the d.c. resistance. The .differential resistance is given by dV/dl = V/l + I dR/dl (2.4.2) and this i s investigated for negative resistance. The a.c. analysis arises, for example, when a sinusoidally variable signal voltage i s 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-e "* 3 (2.4.3) 1 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 - j e ^ , An(t) = An + n ^ ^ , E(t) = E + E-je 3Urt and into Poisson's equation results i n (6/q)OE/3x+3E /ax e 1 J w t ) = Ap - An + ( p - n ) e 1 1 3b)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 ) a / a x . p P l (2.4.4) The basic equations may be combined to obtain an a.c. ambipolar equation; however, the result i s f a i r l y complicated and i s not derived or written here, since the equation i s never used i n i t s entirety i n the analyses to follow. Whenever the a.c. analysis i s used, the appropriate equations involved are suitably modified to comply with the specific model being considered. The a.c. potential drop i s given by V 1 = j£ = IR E^X 1 + I.,R. (2.4.5) The complex impedance i s given by Z(w) = V /I 1 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). This result holds true for causal processes. (2.4.7) 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 w i l l be illustrated i n 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 I n j e c t i o n The c o n d u c t i v i t y o f a semiconductor c o n t a i n i n g h o l e s and e l e c t r o n s depends on t h e i r d e n s i t i e s and m o b i l i t i e s . The e q u a t i o n f o r the c o n d u c t i v i t y i s d = q(/i p + ^ n ) . p Under the c o n d i t i o n of approximate space charge n e u t r a l i t y , Ap=An, the c o n d u c t i v i t y takes the form d = d d + 0 where o =q(fipP J 0 a=q(jXp+M )/(3 . rl due 0 +u 0 i o^» n I t h e aAp) c o n d u c t i v i t y f o r no i n j e c t i o n and Thus 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 to the excess c a r r i e r s Ap i s Aq/a' =aAp. 0 The excess c a r r i e r s can be i n j e c t e d o r e x t r a c t e d by s u i t a b l y biased electrodes and be moved about i n s i d e the semi- conductor by s u i t a b l y imposed the p r e v i o u s l y fields. Since i n j e c t i o n disturbs e x i s t i n g e q u i l i b r i u m between the h o l e s and e l e c t r o n s , the recombination process attempts to r e s t o r e the e q u i l i b r i u m c o n d i t i o n by r e d u c i n g the excess p a r t i c l e p a i r s . In an n-type semiconductor, the c o n d u c t i v i t y i s due t o a predominance o f mobile e l e c t r o n s ; i n a p-type semiconductor, 17 due is to a predominance called minority the of majority carrier. It mobile holes. carrier should while The the be n o t e d , (n =p ) or i n n e a r - i n t r i n s i c material, made to Q 0 as which c a r r i e r In may n o t into the In a occur due that to sign current carrier, concentration injection of concentration c a n be by the i t take is present. by carriers can modulate 3.2 Negative Resistance Negative for some voltage no of place type resistance V and c u r r e n t a can of the charge injected carriers of electrode. only one the type total of carrier same type; however, the total carrier increase of space extra be carrier. being other increase charge the intrinsic large kind substantially total the flow of called or minority from the to carrier d i s t i n c t i o n can one an increased both in carriers carriers the however, of space Thus is majority impossible the lesser b u i l d i n g up containing opposite since the corresponding injecting neutralized normally a must semiconductor the excess semiconductor, opposite a order is dominant the injected concentration increased of carriers the type concentration of conductivity. occurs I, the at zero frequency following when relationship holds (3.2.1) dV/dl < 0. The derivative, and is the resistance. dV/dl, resistance is called referred to the by differential the term resistance negative a ) VOLTAGE - CONTROLLED > B CURRENT- '/// d CONTROLLED Low Pass HigK Pass -^>o di YZZZ Y777 dy di V/V, VZ7 ZZZZ > Low and Band ///// '//// '//A Pass shading c) A.C. FIGURE: 3 1 4 -> Band Pass i n d i c a t e s R(<*>)< O C HARACT ERISTICS NEGATIVE RES\STANCE CHARACTERISTICS 18 Negative resistance may be exhibited in two ways, either voltage-controlled or current-controlled, the d i f f e r 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 appearance 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 produce negative resistance in semiconductors are: combine to carrier multiplication, tunnelling, heating, contact effects, mobility changes, and lifetime changes. For this thesis, the mechanism of importance i s 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 i s 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 w i l l be treated in any great detail. If the voltage V=V(I,S)=IR, where S=S(l) i s some current-dependent parameter and R=R(S,V), then the differential resistance may be written as 19 dV/dl = OV/BI) s + iOR/as) Os/ai) ±1-. T 1 - i(dR/as)(as/9v) y v (3.2.2) 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(CJ)< 0 must hold at some bias and some frequency where R(<o) i s the real part of the complex impedance Z ( ) . w The behaviour of the negative resistance i n the steady eta'fe and at higher frequencies may be entirely different. I f Z(0)=dV/dI >0, then i t i s 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 i n Figure 3.1c. Several theoretical models demonstrating the possib i l i t y of negative resistance i n two-terminal devices have been proposed. semiconducting lampert (1962) analyzes double injection i n insulators and high-resistivity semiconductors showing negative resistance due to an increasing hole lifetime with increasing injection level. and diffusion i s neglected. Charge neutrality i s assumed Stafeev (1959) discusses qualita- tively the possibility of negative resistance i n semiconductors for diffusive current flow i n "long" diodes. The negative resistance i s assumed to have i t s origin i n the increase i n the free carrier lifetime with injection level resulting i n a 20 modulation (1961) o f the d i f f u s i o n length. discuss two-terminal three-layer w h i c h may exhibit The d . c . and s m a l l parameters. (1954) Shockley 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 the t r a n s i t Gunn ( 1 9 5 7 ) conductors time devices are discussed quantitatively i n terms o f the u s u a l t r a n s i s t o r discusses Schuller transistor-like negative resistance. s i g n a l a.c. c h a r a c t e r i s t i c s from G a r t n e r and o f the c a r r i e r s d i s c u s s e s the avalanche showing the p o s s i b i l i t y arising i n semiconductor injection effect diodes. i n semi- of a two-terminal negative resistance device. 3.3 Evidence f o r Negative Resistance Experimental evidence been found these, i n several germanium a t 8 5 ° K . has found 30 volts type i n i n t r i n s i c a n d 2 mA. emitting of the c r y s t a l junction. of a p a r t i c u l a r o f the near-intrinsic Oscillations 2 5 to 3 5 megacycles have been observed per second. In the f i r s t , immediately i n front The n e g a t i v e r e s i s t a n c e space becomes n e g a t i v e charge explanation, internal layer of the i s a consequence c o n f i g u r a t i o n i n the bulk. ionization Two internal i s considered together with a of high recombination located second and explanations are considered. ionization Of 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 t h e the frequency range possible devices. negative resistance 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 'are a r o u n d in semiconducting be d e s c r i b e d . (1957) current-controlled slope two-terminal three cases w i l l Leblond f o r n e g a t i v e r e s i s t a n c e has In the i s n o t c o n s i d e r e d but 21 a m u l t i p l i c a t i o n f a c t o r of minority c a r r i e r s at the c o l l e c t o r junction i s introduced. Gibson and Morgan (1960) have made negative tance diodes using n-type germanium of 20 to 40 ohm n- and p-type s i l i c o n of several hundred ohm cm. resis- cm and The voltage at which the slope dV/dl becomes zero i s around 55 v o l t s f o r a t y p i c a l germanium diode and around 65 f o r s i l i c o n , at currents of about 2 to 3 mA. 100 mA or higher. The slope i s negative up to at least At low currents, the c h a r a c t e r i s t i c i s obscured by r e l a x a t i o n o s c i l l a t i o n s . The sustaining voltage i n t h i s current range i s t y p i c a l l y 8 to 11 v o l t s . The theoretical explanation proposed f o r the negative resistance i s based on the avalanche i n j e c t i o n e f f e c t described by Gunn (1957). Rediker and McWhorter (1959) have found negative resistance i n compensated p-type germanium at l i q u i d helium temperatures (4.2°K), which they have named a "cryosar". negative resistance region occurs between a high and impedance state. The low They have proposed (1961) a mechanism f o r t h i s negative resistance involving the i n e l a s t i c s c a t t e r i n g of free c a r r i e r s 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 s c a t t e r i n g process disappears a f t e r 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 f o r i t s i n i t i a t i o n . L Injecting Contact "-0 MI FIGURE 3 2 SEMICONDUCTOR UNDER FILAMENT INJECTION 22 3.4 Phenomenological Model f o r Negative Resistance A p n e n o m e n o l o g i c a l m o d e l s h o w i n g how modulation can lead section. derived to negative The s t e a d y resistance state current-voltage and t h e parameters i n v o l v e d negative resistance can occur. m o d e l a n d no c o n s i d e r a t i o n i s developed t o see i f i s a purely i s here given i n this characteristics are are varied This conductivity mathematical to the actual physical mechanisms b e h i n d t h e m o d e l . A across a semiconductor filament s e c t i o n A. the steady bias V i s a p p l i e d C a r r i e r s are being positive electrode i n the forward d i r e c t i o n o f l e n g t h L and u n i f o r m cross- i n j e c t e d into the filament a t 3.2). (Figure The c o n d u c t i v i t y i s assumed t o be o f t h e f o r m <f = d 0 0 Where d 0 lation +^e" x / L i s t h e 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 ; parameter and L the penetration 1 -x/L dependent. the The t e r m ae ' <x t h e modu- l e n g t h may be current f represents the r e l a t i v e change i n c o n d u c t i v i t y due t o t h e 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 from the i n j e c t i n g contact. be (3.4.1) ') high, that If i s assumed t o is, <X»1. the current a = K^I their The i n j e c t i o n l e v e l x logarithmic mi d e p e n d e n c e o f a. and , L 1 mo L' = K g l i s assumed t o be (3.4.2) d e r i v a t i v e s m^ a n d Ek, a r e m-, = ( l / a ) d a / d l , m 2 = ( I / V )dL»/dI. (3.4.3) L' = C o n s t a n t a) MODULATION PARAMETER CURRENT-DEPENDENT, PENETRATION LENGTH CONSTANT o m, . b) MODULATION V PARAMETER AND PENETRATION LENGTH CURRENT FIGURE 3 3 , y DEPENDENT 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 i n L' can change the conductivity considerably whereas small changes i n a. w i l l not. Thus modulation of the penetration length by the current indicates that negative resistance may be possible. The resistance i s found to be • lfl», r * = Xjo(Vc3')dx = R [ l 1' + Y 0 l n UCLe .1 ~1 (3.4.4) " ~ T ^ where R =L/Ad i s the resistance with no carrier injection. 0 0 From V=IR, the differential resistance i s given by dV/dl = V/I + (cvdR/acx)m 1 1) + (I»3R/^1» )m « (3.4.5) 2 If a and L' are current independent (m^,m =0), then 2 the current-voltage characteristic i s ohmic (VoC I ) , 2) If a i s current dependent and L' i s not (m^O,m2=0), then d=d (.1 +K<ji"* e ). As I increases, R varies as'I 1 1 m 0 1 in and the voltage as I "~ 1. From physical considerations, m-j i s found to l i e i n the range 0<m^<1. At the limit m.j=1, the voltage approaches a vertical asymptote. Thus there i s no possibility of the I-V curve bending back to give negative resistance (Figure 3.3a). 3) I f bothaand L' are current dependent (m^ ^m^O), then negative resistance i s a possibility. I f o/ _ -a3R _ R L' aO-e~ ') °<1 -T--ZrTfT R d R L (1+ae )d+a) L//L 0 2 a L / L (3.4.6) then equation (3.4.5) can he written as f = (l/V)dV/dI = 1 - o ( The limits on o( and - oC m . i m i 2 (3.4.8) 2 are found to be 1 For negative resistance, . Hence <(1-o<' m )/o( . 1 1 (3.4.9) 2 Since CKm-j-O and L/L' i s found to be such that e this limit, i t can be seen that m 2 L//I,, » 1 at can be quite small and s t i l l satisfy the inequality. Thus a small change i n the penetration length with current i s sufficient to cause negative resistance (Figure 3.3b). Since m >0, L* increases with current. 2 The penetration length can be of two types. For 1/2 diffusion, L' i s given by L'=(D"C) , where D i s the diffusion coefficient a n d f i s the lifetime; for d r i f t , by L' =Eu.T, where p. i s the carrier mobility and E the applied f i e l d . given above i s not entirely self-consistent. I f L The model 1 varies with the current, there i s a possibility that i t may also vary with position. Hence the simple exponential law for the conductivity w i l l not be valid. This inconsistency may be seen i f L* i s assumed to take the form for d r i f t . I f the conductivity arises from a differential equation of the form 25 d(d- d ) / d x = - ( f l ' - ^ A ' , (3.4.10) 0 the original position. the s o l u t i o n i s not v a l i d The e l e c t r i c constant field equation The -(tf-o^oyjjur-. t o be oVO-Ce- x/L ") where L " i s g i v e n b y /aXS/d 0 boundary c o n d i t i o n s . e l e c t r o d e i s taken injection, s e t by t h e I f the c o n d u c t i v i t y a t the injecting t o b e o ( 0 ) , t h e n 0=1-0^/(^(0). For very y equal tounity. solutions for R and/are R = R [1 0 / - (L»A)C(1-e~ L / L = (l/V)dV/dI = 1 - c / ^ m^ = ( I / C ) d C / d I , o< (3.4.11) and C i s a constant C i s approximately The where 1 o f J , ju and t a r e assumed t o be c o n s t a n t . s o l u t i o n i s found d= By s u b s t i t u t i n g E i n L , t h e becomes d(d-Oo)/dx = For a given value i s a functionof E i s r e p l a c e d by J/d'(x), J b e i n g current density. differential when L' m 4 ")] - .cfyn = (I/L")dL"/<H = -(C/R)3R/9C = C ( R L » / R I ' ) ( - e ~ 1 3 0 L / / L ") = -(L"/H) R/9L" 9 = ( C V E ) I f C»'(0)=a^( 1+a), [(I-A) (1-e-L/l") where a i s t h e m o d u l a t i o n - .-Vt"]. parameter o f the heavy original model, then C=c/(1 0 and m^=-m/( 1+a). +C 1 heavy injection, o\»1 , my w i l l be very small. i f ^iT i s not current dependent, then m^=1. these two conditions (m^=0,m^=1), In For veryL" =pXJ/d , 0 Therefore, with i t i s found' that]f>0 always and that negative resistance can not occur. If jut i s 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 i s to occur, then m^ must satisfy the following inequality: (L/L») M 5> 5 / ( U , ^ ) -LA", 1 - e W h - 2(1-e- / "). L , L ; (3.4.12) (L/L"+1) If m^ does satisfy the inequality, then negative resistance i s possible. Such values of m^ may not be physically realizable, but the purpose of this analysis i s to show the various mathematical conditions whereby negative resistance can occur. In the original model, where the penetration-length i s a function of current only, negative resistance could be produced quite easily by small changes i n the penetration length. On the other hand, when the penetration length i s position dependent as well, then negative resistance becomes d i f f i c u l t 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 c u r r e n t i s assumed to he f i e l d d r i v e n and d i f f u s i o n c u r r e n t s a r e n e g l e c t e d . d e n s i t y equations J Thus the c u r r e n t a r e g i v e n by = qu (p +^p)E p p 0 (4.1.D J n = QMn(n The ambipolar equation 9Ap/at +A 0 n)E. s i m p l i f i e s to (jW/o^O+aAp) ) 3ap/dx. 2 = -Ap/t - These equations (4.1.2) a r e v a l i d when the c u r r e n t produced by the f i e l d E i s much l a r g e r than the c u r r e n t due to d i f f u sion. The assumption o f space charge n e u t r a l i t y remains valid except a t very h i g h i n j e c t i o n l e v e l s , where n=p. There- f o r e most o f the models t r e a t e d i n t h i s s e c t i o n w i l l obey t h i s assumption. The exceptions a r e a u n i p o l a r model i n which the current i s space—charge-limited, high i n j e c t i o n level. and a b i p o l a r model a t a very 28 4.2 No Recombination 4.21 Neutral Bipolar A n e u t r a l b i p o l a r d r i f t model w i t h no recombination is fairly simple to a n a l y s e and g i v e s r i s e to no unusual effects. I f the t r a n s i t the 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 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 t o be n e g l i g i b l e . Prom the c o n t i n u i t y e q u a t i o n s , J" and J a r e found to be P n constants. (for The ambipolar e q u a t i o n i n t h i s case reduces to the steady s t a t e ) 2Ap/ax = 0. (4.21 .1) The s o l u t i o n s f o r Ap and, from the c u r r e n t density e q u a t i o n , f o r E are found to be c o n s t a n t s : Ap = constant (4.21.2) E = constant = J/c£(1+aAp). The c o n d u c t i v i t y d = c£(1+aAp) t h e r e f o r e does not vary with p o s i t i o n . Due to the n a t u r e of the d i f f e r e n t i a l e q u a t i o n (4.21.1), no unique r e l a t i o n s h i p can be o b t a i n e d r e l a t i n g the c a r r i e r d e n s i t y and the c u r r e n t density. The a p p l i e d v o l t a g e i s g i v e n by V = JL/tf (1+aAp) 0 (4.21.3) where L i s the l e n g t h of the semiconductor, and the d i f f e r e n t i a l r e s i s t a n c e by 29 dV/dl = o where R = L / A d , t h e d . c . Q r e s i s t a n c e w i t h no 0 the c r o s s - s e c t i o n a l a r e a o f the I=JA, semiconductor; depends o n l y on ing semiconductor. through the Unless m > 1 , cannot being current (4.21.5) the v a l u e occur. current pass- ' Ap(t) substituted analysis, = Ap J(t) resulting of the i t i s evident that negative resistance I n the a.c. = + J + i n the time P l the values ( x ) e ^ J i ^ e dependent a m b i p o l a r equation, in jwp., = - ( M J / a f ( 1 + a A ) ) 2 0 S i n c e Ap 0 P a /ax. (4.21.6) Pl i s a constant independent o f x, the solution simply- P 1 = P exp[-(j(JcT/u J) 1 0 o where P-JQ i s t h e i n t e g r a t i o n c o n d i t i o n a t x=0. the the (I/Ap) dAp/dl, a s s u m i n g t h a t Ap is injection, A and m = are (4.21.4) [ R / ( l + a A p ) ] [ l - maAp/(1+aAp)] U n l i k e the (1+aAp) x] constant steady (4.21.7) 2 s e t by state the carrier perturbation density varies with position. density equation boundary density, Prom t h e current 30 Jl the field V E.j i e f o u n d = 1 = <i(1+aAp)E„ + d a p E, o 1 o 1 J 1 and i n t e g r a t e d t o g e t t h e a . c . v o l t a g e _ ^o L cf (l+aAp) Z D-exp(-j^)] PlQ 0 (l+aAp)4juJo 0 transit 0 time = L/uE = L c r ( l a A p ) % J & o has been i n t r o d u c e d + to simplify R, 3aIp [l-exp(-jU i^)] / 1 + 1 Q This assumption i s v a l i d and only i fi n e r t i a l of the contact. imaginary part (4.21.10) I-, W^O+aAp) (1+aAp) the plane Thus t h e complex ' A n o t h e r a s s u m p t i o n made i s t h a t at (4.21.9) 0 the notation. i m p e d a n c e c a n be w r i t t e n a s Z((J) = (4.21.8) / 0 where t h e a m b i p o l a r j 2 The r e a l (reactance) p 1 Q /l =dAp/dI. 1 effects part are neglected (a.c. resistance) o f t h e complex i m p e d a n c e a r e g i v e n by R(w) = x(u>) = R, aApm s i n ^ t r (1+aA.p) R o m a ^p _ I n v e s t i g a t i o n o f R(u3) the parenthesis As the frequency reaches a J cosU jr ). (4.21.12) a shows t h a t t h e s e c o n d t e r m i n i t s maximum v a l u e increases, this p e a k s become s m a l l e r . (4.21.11) (1+aAp) ( 1 i a t zero term o s c i l l a t e s Hence i f n e g a t i v e frequency. and s u c c e s s i v e r e s i s t a n c e does n o t 31 occur i n the steady s t a t e , then i t cannot occur a t any other frequency. I n v e s t i g a t i o n 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 maAp/(1+aAp) t)? . 2 0 a The reactance i s t h e r e f o r e i n d u c t i v e . Since t h i s model i s f a i r l y simple, i t i s easy t o c a l c u l a t e the response AV^-(t) t o an impulse of current AI (t)=Q§(t) c a r r y i n g a t o t a l charge Q, and t o show t h a t the t F o u r i e r transform can give r i s e to the complex impedance Z(o>). The time-dependent terms are introduced i n t o the a n a l y s i s i n a manner s i m i l a r t o that of the a.c. p e r t u r b a t i o n method, and are assumed to be of small amplitude: J(t) = J + AJ (t) E(t) = E + AE (x,t) t t A p ( t ) = Ap + A p ( x , t ) t where J , E, and Ap are the steady s t a t e v a l u e s . The r e s u l t i n g equations f o r the p e r t u r b a t i o n f i e l d and c o n t i n u i t y are AE = AJ /o (1+aAp) t t / 0 JaAp /o^(1+aAp) 2 t (4.21.13) 3Ap /at = -v 3 A p / a x t t where v=jiiE=>i J/O^(1+aAp) 0 2 i s the ambipolar v e l o c i t y . The c o n t i n u i t y equation i s simply a wave equation w i t h a propag a t i o n v e l o c i t y v; i t s s o 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 o f 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 desired. S i n c e r e c o m b i n a t i o n and t h e r e a r e no retain dispersive t h e same f o r m transit time = t o a l t e r Ap-fc. a carriers; (QV<lAL) h(t - thus x/v). QL 6(t) a l v U(t) V*) = 1 ^ ) where U ( t ) i s a u n i t AV-t(t)/Q, ^ e o < 0 < t < + step function. The per u n i t • ^ t o charge, leads to 1 5 ) . the The term ^/qAL If inertial U.21.16; i s the v a l u e o f Ap^/AI^ a t effects are neglected a t plane of the c o n t a c t , then the assumption With ' f?a \0-exp(-3a>^)] a I v t h e p o s i t i o n x=0. be made. 2 1 Fourier transform of (i+aap) " A c y i + a A p ) 2 l q A L J 0 - e - ^ d t a where R =L/AC^. U Z(co): Z(fc>) = J ^ [ A V ( t ) / Q ] R (QZA A<r (1 a^)2l5l)' " voltage response c o m p l e x impedance (4.21.14) i s g i v e n by t = ambipolar a v o l t a g e response A v ^ . = A E ( x , t ) d x A Thus Ap-f. s h o u l d f o r 0<t<'T , where ^ = L / v i s t h e of the i n j e c t e d APt The effects d i f f u s i o n have been n e g l e c t e d , the s u b s t i t u t i o n s v = L / t a < Z /qAL=dAp/dI a and thus aApm (1+akp) (1+aAp) [l-exp(-ju)^ ^ a jw^ can m=(l/ap)dAp/dI, t h e c o m p l e x impedance t a k e s t h e same f o r m a s t h a t t h e a . c . p e r t u r b a t i o n method; the g i v e n by 33 4.22 No Recombination—Space-Charge-Limited Unipolar Flow A d r i f t model i n which the current i s an unipolar space-charge-limited current involves more restrictive assumptions than bipolar d r i f t models. It i s an excellent example showing the existence of band pass negative resistance i n a semiconductor with no negative resistance i n the steady state. This case has been treated previously by Shockley and Prim ( 1 9 5 3 ) 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 i n 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 i s 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 i s no recombination, i.e., transit time of the minority carriers across the semiconductor i s much smaller than the carrier lifetime. f) Diffusion currents are neglected. 34 I n a d d i t i o n t o these s p e c i f i c assumptions a r e t h e g e n e r a l assumptions o f o n e r d i m e n s i o n a l flow, semiconductor and f i e l d - i n d e p e n d e n t homogeneous mobilities. Another assumption used i s t h a t t h e I n j e c t i o n l e v e l a t t h e p o t e n t i a l extremum i s v e r y h i g h . f i c a n t departures Assumptions (a) and (d) a r e t h e s i g n i - from the b i p o l a r models c o n s i d e r e d , the c u r r e n t c o n s i s t s o f both m a j o r i t y and m i n o r i t y where carriers and where the space charge c o n s i s t s o f m a j o r i t y c a r r i e r s and majority impurity impurity i o n s as w e l l as m i n o r i t y c a r r i e r s and m i n o r i t y ions. F o r convenience, an n-type semiconductor i s chosen, although the a n a l y s i s w i l l a p p l y semiconductor. modified The b a s i c equations The steady s t a t e c u r r e n t d e n s i t y and P o i s s o n ' s from assumptions (a) and (d) a r e J = J p = qUpApE = constant (f/q)dE/dx The o f S e c t i o n 2.1 must be s l i g h t l y t o account f o r t h e a d d i t i o n a l assumptions t o be used. equation e q u a l l y w e l l t o a p-type = n 0 (4.22.1) (4.22.2) + Ap. two terms on the r i g h t o f P o i s s o n ' s equation a r e due t o the space charge c o n s i s t i n g o f m a j o r i t y i m p u r i t y i o n s K = n - p D and mobile m i n o r i t y c a r r i e r s p=p +Ap. 0 two equations 0 0 F o r no c u r r e n t , t h e can be s o l v e d f o r t h e v o l t a g e V = qn L /2€. 2 f Q (4.22.3) T h i s v o l t a g e i s r e f e r r e d t o as the "punch-through" v o l t a g e a t which t h e space charge j u s t supports the a p p l i e d voltage. b ) VOLTAGE V/S. Ap /no L F I G U R E 4.1 C U R R E N T A N D V O L T A G E SPACE-CHARGE-LIMITED UNIPOLAR MODEL 35 I For voltages greater than V^, a space-charge limited flow of holes takes place. A current density current that would flow i f can he defined as the were applied to an intrinsic semiconductor (i.e., space charge consists only of mobile minority carriers);, thus J = 9£u V /8lA (4.22.4) 2 f p f 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/n o i n terms of J« > i q n L A J = n /Ap - ln(1+n /Ap ) (4.22.5) 2 0 p Q L o L The voltage i s given by V = (€J Aip q n 5)[u /2 2 2 3 0 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 i n terms of J^. and V^. as J/J f = (32/9) [u - ln(1+u)]" 1 1/7 = [u - 2u + 21n(l+u)][u - ln(1+u)]~ . 2 2 £ These two relationships are shown i n Figure 4.1. The d i f f e r - ential resistance i s given by dV di = 9 V [2ln(1+uj - 2u + u ln(Uu)] ; z 16 JfA [u - ln(l,+u)j f . (.4.22.7) 0 0 36 and can be does not shown t o be exhibit negative This Q 2 emission The in for i s not values relationship for of o f x, the S h o c k l e y and o f x and For i n a vacuum the exact the a.c. (4.22.8) for diffusion current is negli- p o t e n t i a l extremum s i n c e t h e diffusion. s o l u t i o n s a b o v e do not Thus hold. solutions including diffusion error for large electric fields current worked higher is small. analysis, E(t) = E + Ap.(t).= Q diode. P r i m shows t h a t t h e relatively n : space-charge-limited i s p r e d o m i n a n t l y c a r r i e d by small values by independent f v a l i d near the this region state current- (V>V >0). 3 assumption that the Comparison with out the i n s e m i c o n d u c t o r s i s a n a l o g o u s t o C h i l d ' s Law thermionic gible shown t o be p current-voltage emission be (9/8)6u V A , = steady resistance. c h a r a c t e r i s t i c s can J Thus the large injection values, Ap/n V>1, For voltage always p o s i t i v e . bp + E (x)e^ W t 1 p-jWe^ J(t) = J + J e* & r t 1 are s u b s t i t u t e d i n the continuity resulting equations. in current The density, Poisson's and time dependent terms are the separated, 37 (£/q) ^/dx = p (4.22.9) 1 3WP-J = ->i 3 ( E p + E A p ) / 9 x . 1 1 T h i s s e t o f equations can be s o l v e d e x a c t l y f o r E^ o r . S i n c e i n the steady s t a t e , the r e l a t i o n s h i p between the p o s i t i o n c o - o r d i n a t e x and the i n j e c t e d c a r r i e r d e n s i t y Ap i s known, E-j can be found i n terms o f J-jAp E-,(Ap) = 0 " (l+n /Ap) 0 0-30) The a.c. v o l t a g e JlJ£ V l 1 (1+n /Ap) etO/^UpqnQ. where p; thus 2 Jn J*p * W u 2 ln(1+u) 30 (1-30) + 300-30) i s given by (1+u) " ^ - 1 1 " 3 (4.22.10) ' 300-30)* 0 I t i s advantageous a t t h i s p o i n t t o i n t r o d u c e a new v a r i a b l e , the t r a n s i t time f o r h o l e s 7^., d e f i n e d as T t = J ^ O / U p E ) dx = (^/u qn ) ln(1+u). p (4.22.11) 0 A rearrangement o f the terms g i v e s u = n /Ap Q i n which T R L = e^-1, ^=^ qn t /(C p 0 T = i s the d i e l e c t r i c r e l a x a t i o n time. V^R* With t h i s s u b s t i t u t i o n , the complex impedance i s g i v e n by J -1 Z(<£) = 30 (3 i-3^> - 1 300-30)' (4.22.12) 87T 4TT 2TT a) R E A L P A R T OF IMPEDANCE b) IMAGINARY PART O F IMPEDANCE F I G U R E 4.Z IMPEDANCE CHARGE-LIMITED OF SPACE- UNIPOLAR MODEL Separating the complex impedance Z(4>) into r e a l and imaginary parts y i e l d s the a.c. resistance R(<£) and the reactance X(<£) respectively; thus @ 2 - 2e^cosp^> R(4>) = 0 j6 )2 • (1+02) 2 + 13(1-^)6^8^(3^ (4.22.13) (1+0 ) |3# 2 2 X(^) = l ( e ^ -1) J6 (3t ^pVn A 3 0+# ) 2 0 (1-^) )(e cos^^ -1) + 2e^sin 2 (3 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 zero frequency, i . e . , R(0)=dV/dI. i s attained at For a given value of the t r a n s i t time (fixed (3), the value of R(^>) o s c i l l a t e s . For low values of (3, i . e . , short t r a n s i t times, heavy i n j e c t i o n , R(^>) remains positive f o r 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 t y p i c a l values to be expected from t h i s model i s given below. A germanium specimen of length L = 10 cm, area A = 10 cm and n Q = 7*10 F o r germanium, t h e m o b i l i t y p. sidered. = 1700 -1? the p e r m i t t i v i t y 6= and relaxation through" If = AJj. = 43mA. 40 v o l t s , If 1=8 The 4.1b, f mA, and transit conductor time is It 1 cm, hence dV/dl of the unrealizable change a p p r e c i a b l y . = 865 injected ohms and The t o be k e p t T h i s has negative resistance o f R(^) first from = 0.045, a c r o s s the semi- =TT. of around physically limiting factor fairly short to i s the . s m a l l so t h a t of approximately Mc/s. 95 The (Figure 4.2a), to l i e i n the 880-1000 Mc/s negative resistance f r e q u e n c y o f 450 565 |3 vs. bands a r e f o u n d bands b e t w e e n 400-575 Mc/s, is Q then • practical. From t h e p l o t a drop no c u r r e n t Thus t h e l e n g t h must be k e p t v e r y magnitude of n . The the c u r r e n t c u r r e n t , h o w e v e r , does n o t Another remains "punch- V / l = 6250 ohms. carriers obtain reasonable values. Q 50 v o l t s , c a n be f o u n d : ^ P j / n volts). The flow of holes takes place. becomes v e r y l a r g e and (~10^ and seen t h a t f o r a l o n g specimen the v o l t a g e dielectric sec. 1 0 = 40 v o l t s = 2.3*10""9 s e c , g i v i n g c a n be The For voltages g r e a t e r than limited the f o l l o w i n g 2 . , t h e o p e r a t i n g v o l t a g e i s t a k e n t o be Figure cm volt"*^ s e c " ^ V becomes 40 v o l t s , structure. a space-charge cm = 7.3 10~ t o be V Thus u n t i l c a n f l o w a c r o s s the farad t o be ^ v o l t a g e i s found i s con- -1 1.4*10 time i s found cm~^ peak has and the frequency 1350-1420 Mc/s. a m a g n i t u d e o f 33 p e r c e n t o f the d.c. ohms, value, at a maximum v a l u e o f t h e r e a c t a n c e ohms ( c a p a c i t i v e ) a t a f r e q u e n c y o f 220 Mc/s. The i m p e d a n c e due t o t h e 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 this frequency It i s 515 ohms. c a n be s e e n t h a t t h e f r e q u e n c i e s o f i n t e r e s t a r e approximately of the order o f the r e c i p r o c a l of the d i e l e c t r i c r e l a x a t i o n time. Apj/E^), As t h e t r a n s i t the frequency occurs decreases increases. time i n c r e a s e s ( d e c r e a s i n g a t which negative r e s i s t a n c e f i r s t a n d t h e number o f n e g a t i v e r e s i s t a n c e b a n d s F o r v e r y heavy i n j e c t i o n , ^ P i / l a r g e and f^. s h o r t , n 0 n e g a t i v e r e s i s t a n c e does n o t o c c u r . 4.3 Recombination w i t h Constant In this Lifetime—Neutral Bipolar 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 nation with a constant c a r r i e r l i f e t i m e case i s analyzed. recombiIn this 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 = Ap/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 the p o s i t i o n x. The s t e a d y -Ap/ state ambipolar equation f o rt h i s = [^ J/cT (1+aAp) ]9Ap/ax. 2 o o case i s (4.3.D T h i s e q u a t i o n , solved f o r x, g i v e s ]Ap(x) = - [ l / ( 1 + a A p ) - ln[(1+a&p)/aAp] JAp(O) ' . (4.3.2) w h e r e L' = ^p-^C/c'0» The r e s i s t a n c e i s g i v e n b y x/L' R = V / I = R [1 Q - (L'/L)(f -f )J L 0 (4.3.3) where R =1/A0' i s the resistance f o r no i n j e c t i o n 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 d i f f e r e n t i a l resistance i s (1/R )dV/dI = 1 - ( I « A ) [ 2 ( f - f ) + I d ( f - f ) / d l ] . Q L L 0 (4.3.4) In i t s complete form, t h i s 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 i n j e c t i o n . The a.c. analysis, carried out as i n the previous models, y i e l d s the a.c. component of the ambipolar . ^ j g g l l , + c5 (l aAp) L 2 0 + 3x 9x J 2 * * * equation: . (4.3.5) f(i+aAp) Equation (4.3.1) i s used to change the above equation to 3p-|/9Ap = (1+ju*)p /Ap + 2ap /(l+aAp) - J.,/J 1 1 (4.3.6) Prom the current density equation, the f i e l d E^ i s given by E 1 = J ^ t f (1+aAp) - a p ^ / c ^ d + a A p ) and the voltage V 1 2 Eidx can be found. Unfortunately, the d i f f e r e n t i a l equation f o r p^ can not be solved exactly. Only the high i n j e c t i o n case w i l l be treated. A useful relationship between the i n i t i a l injected c a r r i e r density and the c a r r i e r density at any point x i n the 42 s e m i c o n d u c t o r c a n be o b t a i n e d by i n t r o d u c i n g ambipolar t r a n s i t s where p3 time, the concept of the s , up t o t h e p o s i t i o n a x* = J * 0 / u E ) dx a i s the ambipolar v e l o c i t y . c a s e , p. = ^ / ( 1 + a A p ) . The i n t e g r a l I n the n e u t r a l gives bipolar the simple relationship Ap( ) x a) Low Ap B t h e a s s u m p t i o n s a^p^<<;1 , low i n j e c t i o n , made so t h a t f £ f =1/2, U L characteristic ohmic f l o w little from the l i n e a r case and results: negligible, At distribution (4.3.8) = q.(^i n +u p )V/L. n the c o n t r i b u t i o n o p o from the i n j e c t e d negative resistance i s not carrier densities i s possible. low i n j e c t i o n l e v e l s , t h e c a r r i e r takes the simple Ap = A n form e- / ' x Q density L where L'=Ju T / o l c a n be c o n s i d e r e d a s a n e f f e c t i v e o o length. aAPj,«1 Therefore the c u r r e n t - v o l t a g e departs very J Since (4.3.7) e~ a/^. Q Injection For are = "drift" 43 b) Moderate Injection For moderate injection, fQ^f^. and aAp^<1, giving aApQ^I, The condition a k p ^ « 1 can be relaxed to include cases where aAPj>1, as long as 1+aAp^<< a A p . Q In this case, equations (4.3.4) and (4.3.2) become (1/R )dV/dI = 1 - ( L ' A ) ( 2 f + Idfj/dl) Q (4.3.9) L (I/L») = -1/(1+aAp ) + ln(1+1/aAp ). L (4.3.10) L Equation (4.3.10) differentiated with respect to I and substituted into equation (4.3.9) gives (1/R )dV/dI = 1 - L « A ( 1 + a A p ) + aAp /(1+a/^ ). (4.3.11) 2 Q L L L It can be shown by substituting equation (4.3.10) for L A ' i n 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 w i l l 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 f =1/2»fQ i s used. L result: RA 0 = 1 - L'/2L I»A' = -1 + ln(1+1/aAp ) L (lA )dV/dI = 1 - L ' A = 1 0 The following equations 44 The i f final equation 1. L /L 1 shows However, resistance can not set of equations the negative the shows in example the shows analysis, c) High i f the region folly of i t is equation is smaller. found negative the above to With made, no disappear. making approximations r e s u l t i n g i n erroneous Thus too early conclusions. Injection aAp >>1, Q much more a d v a n t a g e o u s to adp£>> 1. rewrite For the this* ambipolar as = (^J/C^AP ) -Ap/f rather that possible approximations are region-becomes this shows is inspection of better For high i n j e c t i o n , case, resistance analysis A closer that resistance negative exact occur. a p p r o x i m a t i o n s made, this that than to (4.3.4). draw c o n c l u s i o n s The r e s u l t s 0 0 (4.3.2) and from equations obtained tf L/J>i ? = (4.3.12) dAp/dx 2 are 1/2(aAp ) L 2 V = j2^ T/3cV (aAp )5 2 L o where the the assumption aApQ» aAp^ has current-voltage been used relationship for J = (9/8)q^ (n -p XV /l3, 2 n where C ^ o ^ p P ^ n ^ ) substituted. occur in this It 0 and F c a n be range. 0 0 seen ^ n ^ Q that ' V Q this case 0<V, ) / ( M negative again. p is n > p Q P o Thus + ^ n (4.3.13) Q o ) resistance h a v e b e cannot e n 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 i s a result of the charge neutrality assumption. It i s of interest also to compare the currentvoltage characteristic obtained here with that of the unipolar space-charge-limited model. In both cases, J.«*V2/L3. However, i n the bipolar model, the multiplicative factor i s QJ^Hp^^-Po) whereas in the unipolar model, i t i s 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 i s no space charge to be overcome. Another feature of the bipolar case i s 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 characteri s t i c , equation (4.3.13)» shows that the solution i s not valid when the bias voltage i s reversed since the current remains positive. The use of p-type material (p >n ) also violates the equation. 0 Q However, the use of p-type material and a 46 reversed bias i s v a l i d . This w i l l be discussed i n greater detail later. Another anomaly i n this equation i s that as n - * p , J->0. This i s , of course, erroneous, Q 0 and i s due to the neglect of d i f f u s i o n which becomes important when the d r i f t term becomes small and comparable with the d i f f u s i o n term. In t h i s case, there i s no voltage range over which Jo^V^. The t r a n s i t i o n from the low i n j e c t i o n regime to the high i n j e c t i o n regime occurs approximately the curves ( 4 . 3 . 8 ) and ( 4 . 3 . 1 3 ) Vc S at the voltage where intersect: * VVpM*o-Po> 2 U.3.U) r The current-voltage c h a r a c t e r i s t i c i n the neighbourhood of t h i s voltage i s given by the moderate i n j e c t i o n The a.c. c h a r a c t e r i s t i c s high i n j e c t i o n . analysis. have a simple form under The ambipolar equation f o r t h i s case becomes 9p /dAp = ( 3 + j ^ ^ p / A p - J ^ J (4.3.15) 1 The solution of t h i s equation, to a f i r s t approximation P 1 = J Ap/j(2+jo>t) is (4.3.16) 1 The actual solution has an added term which depends on the i n i t i a l injected c a r r i e r density, but this term can be assumed to be n e g l i g i b l e . I t i s indeed found that i n the f i n a l t h i s term contributes very l i t t l e to the With t h i s solution, and voltage may result, solution. the a.c. component of the f i e l d be r e a d i l y found: 47 J jg _ 1 (1+jwr) 1 J J n . 1 i M r y (£aap(2+j<*>*) ' 1 d+3wr) o 1 3^,2(3^1^)3(2+36^ ' The complex impedance and i t s components a r e R /8L^/ ) 2 0 ( U j L o r 3 VL' / Z(CJ) R O/8 \ L 1 / 2 (2+U)V) 3 \L« j R(6J) (4.3.17) (2+jjwt) (4.3.18) (4+W 'c ) 2 2 1/2 X(UJ) 2 where L *=Jp. t/o~o * i d R =L/AO^. r 1 a 0 (4.3.19) U+V ?. ) 3 U' / 0 2 An e q u i v a l e n t circuit f o r this 1 /2 two-terminal system i s a r e s i s t a n c e R-j=(R /3) (8L/L') 0 / in s e r i e s w i t h a p a r a l l e l combination o f a r e s i s t a n c e R-j and an i n d u c t a n c e R] C/2. The e f f e c t o f t h e term n e g l e c t e d i n f e q u a t i o n (4.3.16) i s t o add a s m a l l o s c i l l a t o r y component t o the e q u a t i o n f o r Z(iO). As mentioned p r e v i o u s l y , v a l i d f o r a reversed equation ( 4 . 3 . 1 3 ) b i a s o r f o r p-type m a t e r i a l . i s not T h i s i s an i n t e r e s t i n g f e a t u r e and r e q u i r e s f u r t h e r i n v e s t i g a t i o n . ambipolar e q u a t i o n ( 4 . 3 . 1 ) The f o r bipolar neutral d r i f t with a c o n s t a n t l i f e t i m e can be r e w r i t t e n i n the steady s t a t e as dp/dx = -(p-p )AE;i 0 where ^= UpU (n-p)/(MpP+u n) < n n i s the ambipolar m o b i l i t y ; f o r n-type m a t e r i a l ji>0, f o r p-type p.<0 f and f o r i n t r i n s i c u=0. 48 This equation i s 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>P . Q 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 i s 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 i s impossible. This i s curious i n view of the fact that physically, hole injection can not occur i n p-type material, and nowhere i n any of the defining equation i s this fact incorporated. The only assumption made i n the defining equation i s that holes move i n the direction of positive current. A similar situation occurs when the bias voltage i s reversed. In this case, electron injection i s found to be possible i n p-type material and not i n n-type. Perhaps the answer to this seeming paradox may be i n the formulation of the ambipolar continuity equation. In Section 2.3» i t i s shown that the injected carrier pair density pattern moves i n an electric f i e l d i n the direction i n 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 i n the opposite direction. 49 d) Very High Injection—Bipolar, Space Charge For very high injection, n=p, space charge neutrality may no longer be valid. The thermal densities are neglected and the current density, Poisson s and the continuity equations 1 become (£/q)dE/dx = p - n (4.3.20) (M M )p/^- >i ^ = E9(n-p)/3x + (n-p)3E/9x J + n p n p If the assumption p^/qCju^+u^E i s made i n the continuity equation. This set of equations yields the following d i f f e r - 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 w i l l not be too important. The exact analysis can be carried out by replacing n from Poisson's equation but the solution i s not as useful since i t i s d i f f i cult to see the relationship between the position and the f i e l d intensity. The solution of (4.3.21) can be obtained with the d d substitution E ^ = with the solution dE 2 The equation then becomes = -K, E = -Ky /2 + C,y + C . 2 0 The f i e l d distribution with .respect to y i s parabolic. It can readily be seen that the absolute value of y i s not important. Therefore, to simplify the analysis, y=0 can be set at the f i e l d maximum. If holes are being injected at the anode, then the space charge due to the injected holes i s 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 f i e l d intensity at both anode and cathode. Rather than the usual procedure for setting the position co-ordinate as 0<x<L, i t i s found to be advisable to take -L/2<x<L/2".It is assumed that the f i e l d intensity at the contacts i s negligible compared to elsewhere i n the semiconductor. Since the f i e l d distribution i s symmetric with respect to y, the boundary condition E=0 at x=-L/2 and x=L/2 i s analogous to E=0 at y=-y Q and y=Yj=+y^. Thus the solution i s E = (Ky /2)fl - (y/y ) ]. Q (4.3.22) 2 2 Q Prom E=dx/dy, the relationship between x and y i s x = (Ky V6)[(3y/y ) - (y/y ) ], 3 0 0 0 y = 3L/2K. 3 Q (4.3.23) This equation shows that x i s an odd function of y and also that E i s symmetrical about x=0. The voltage i s found to be V = fao E dy = ( 4 / 1 5 ) K y J -y 2 0 2 5 Q which gives the current-voltage relationship J = (125/18) erp^^/L . (4.3.24) 3 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 f i e l d intensity so that the density distributions have a minimum near the center of the semiconductor. The actual distribution of p and n can be found from the current density and Poisson s equations as 1 p = ^^ n = + J + u p) E Mn € dE (Mn %)^ dx + J u 6 dE ^ Thus the hole and electron distributions are not symmetrically distributed with respect to x. The difference between p and n i s (£/q)Ky/E and i s quite small. It should be noted that the net space charge over the crystal i s 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 i s the greater i n number. Interchanging n and p has no effect on the solution. 52 Another i n t e r e s t i n g i f the the bias i s reversed, current and f i e l d when b i a s e d and remove highly ing holes bably but also currents. not realizable. inject carrier Physically, Actual s h o u l d be n o t e d densities neglecting become the diffusion currents are confined be v a l i d . diffusion The r e g i o n between lengths J which i s quite S e c t i o n ' 4.3c mately diff/ drift J = are gives a result However, analysis J proto vanishes, as long s h o u l d be i s as i n the may b e a s s u m e d diff/ drift J infinite of to a narrow range the contacts The r a t i o small except dE/dx i s very The long. is important. the d r i f t enter- simultaneously. When t h e f i e l d diffusion currents are on the may n o t b e a b l e This the diffusion currents. electrons contacts the analysis at the contacts. of the contacts, electrons Such contacts the other that Both contacts to inject such contacts the region the a n d remove be a b l e that valid. impose no c o n s t r a i n t s one c a r r i e r a n d remove It holes Thus when b i a s e d n e g a t i v e l y . i d e a l i z e d a n d must or exiting negative. to inject positively of the solution i s the solution i s equally become must n o t o n l y be a b l e feature to several given by [kT(u -^ )/E]dE/dx n p at the contacts where E=0 a n d large. t r a n s i t i o n from the h i g h i n j e c t i o n regime to very at the voltage current-voltage h i g h i n j e c t i o n regime'Incurs where the square characteristics of approxi- l a w and t h e cube l a w intersect, namely v c*d = (81/500) L q(n -p )/6. 0 If V a->c (4.3.25) 2 0 , the transition voltage from the low injection to high injection case, equation (4.3.H), i s set equal to V _> , c d then i t can be seen that i f nl /g q(n ~p ) < [(500/81)^e/^ u tj 0 0 p (4.3.26) n then the high injection range JoCV w i l l not exist. 2 be the case i f n =p . 0 This w i l l For a sufficiently extrinsic n-type Q semiconductor, i t can be seen that V _^ a 0 reaches a minimum value L //ipT while V +& i s limited only by the value of n -p 2 c 0 and may be several orders of magnitude larger than Vg^. n o~ Po' ^a+c i 4.4 > n c r e a s e s Q As Vc-*d decreases. a n d Recombination with Carrier Dependent Lifetimes—Neutral Bipolar For the case of carrier dependent lifetimes, the recombination term can be written as (4.4.1) r = Ap/tr = r Ap . Q In this case, the lifetime i s also a function of position. Since the general d r i f t model can not be solved exactly, only the high injection range i s treated. The steady state ambi- polar equation i s written as -r a Ap 2 2+i> 0 = ()i J/d ) 0 dAp/dx 0 (4.4.2) and the solutions for Ap^ and the applied voltage V are 1 = J>i /C5 a (1+V)r Ap / 0 2 0 0 (1 L + i;) (4.4.3) v = J u /0 a3(2+i;)r Ap 2 J 0 2 0 0 (2+);) L . (4.4.4) The current-voltage relationship becomes v J ( u v ) / % i / o ~ (2+^/V 1/V L Po a V o (l-^ ( 2 ( l + ^ ( i ^ j; (2+^)/y ) (l/V)dV/dI = V/(1+V). U ' - 4 5 ) (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 replaced by 0 is Negative resistance can occur for -1<V<0. It i s d i f f i c u l t , however, to see what type of mechanism can give recombination of this form, where the lifetime increases with the injection level i n the form fcCAp , 1<y<2. The analysis i s not valid for V = 0 , when the carrier lifetime increases directly with the carrier density. In this case, no approximations need to be made i n the ambipolar equation, thus d x / ^ ^ J = -adAp/c^O+aAp) (4.4.7) 2 where t ^ r ^ a . The solution obtained i s O ^ x / t y ^ = 1/(1+aAp(x)) - 1/O+aAp ) 0 and the voltage becomes with the use of this equation v=_£ 2 " 1 + J Po^i OoL(1+aAp ) (4.4.8) 0 The solution i s exact and gives a threshold voltage before 55 current can flow* The d i f f e r e n t i a l r e s i s t a n c e i s dV di maA' R, 1 2(1+'aAp ) 1+aApQ where R =L/AC^ and m=(l/A.p )dAp /dI. 0 that unless not occur. (4.4.9) 0 0 m i s greater Q Thus i t can be seen than u n i t y , n e g a t i v e r e s i s t a n c e can 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 i s extended to include the effect pf diffusive currents. First a purely diffusive flow model, neglecting drift currents, i s investigated; then the general problem including both d r i f t and diffusion under specific conditions i s discussed. Since the general problem including d r i f t and diffusion i s 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 i s 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 i s negligible compared to the diffusion current. The injection of 57 minority carriers i s always accompanied by a compensating majority carrier density since the least deviation from charge neutrality i s sufficient to set up an electric f i e l d . However, this f i e l d has very l i t t l e effect on the diffusion of the minority carriers and hence can be neglected to a f i r s t approximation. The presence of majority carriers i n the semi- conductor can be ignored except through the use of the ambipolar d i f f u s i v i t y . The ambipolar equation i n the steady state under the above conditions becomes 0 = -Ap/t + D8 Ap/dx . 2 2 (5.2.1) For convenience, a sufficiently extrinsic n-type semiconductor i s assumed, so that JD=D. With this substitution, and the p boundary conditions p=ApQ at x=0, Ap=0 at x=L, the solution A of the ambipolar equation i s Ap = Ap s i n h j L - x j / L sinhL/Lp Q where L =(D 'r)/2 i 1 p holes. p S p (5.2.2) the diffusion length of the minority When the length of the specimen i s several times the diffusion length so that exp(L/L )»1, the solution takes on p the simple exponential form Ap = Ap e" / p . x L 0 The current i s given by J = -qD^p/ax. (5.2.3) 58 If t h i s c o n d i t i o n i s to h o l d , then the i n j e c t i o n l e v e l must he low, a A p « 1 . The v a l i d i t y o f t h i s equation may by c o n s i d e r i n g the complete h o l e c u r r e n t d e n s i t y j = <* 0 cT (1+aAp) 0 equation - qD (n +p +2Ap) Bop qPpPpJ + qMpJAP P be checked (n +p )(1+aA ) a x 0 Since e x t r i n s i c n-type n » p 0 0 0 0 0 P and low i n j e c t i o n n » A p i s 0 Q assumed, the h o l e c u r r e n t d e n s i t y equation reduces t o (5.2.3). equation However, under the same c o n d i t i o n s , the ambipolar (2.3.9) reduces t o 0 = -AP _ ^ X J &P + kTjap (b-1)/dAp\ 2 P d 3x 0 and not t o equation d Q (5.2.1). 2 + D 9 A p dx Therefore (5.2.4) 2 p \dx) 2 i f the s o l u t i o n i s to be v a l i d , the terms c o n t a i n i n g the f i r s t d e r i v a t i v e o f the h o l e d e n s i t y must be n e g l i g i b l e compared t o the other terms. equation equation The d e r i v a t i v e c a l c u l a t e d a t x=0 two i s substituted i n (5.2.4), r e s u l t i n g i n the c o n d i t i o n s (b-1)Ap/bn «1 0 J«qbn D /L , Q to be s a t i s f i e d . p p equivalent to The f i r s t E«kT/qL , p i n e q u a l i t y i s s a t i s f i e d by the assumption o f low i n j e c t i o n ; the second, however, g i v e s an a d d i t i o n a l c o n d i t i o n which must be s a t i s f i e d i f the s o l u t i o n s are t o 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 p u r e l y diffusive f l o w model i s so low, the c o n d u c t i v i t y 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) /t= D 3 Pl p 2 /3x . (5.2.5) 2 Pl The solution i s similar to that of the d.c. case: sinh (1+jto?) (L-x)/L„ Pi = Pm 775 sinh (1 + ju)t) L/L 1//2 1 1 0 (5.2.6) 1/2 T P which for exp(L/L )»1 reduces to p ? 1 = p expt(1 + j u ) t ) 1 / 2 x / L 1 . 1 0 The diffusion length has been replaced by an effective a.c. diffusion length L / ( 1 + jwt:) /2 1 p > 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 i n transit. The d.c. attenuation i s due exclusively to carrier recombination while the a.c. attenuation i s 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) / /^ whose real 1 2 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 . (^/2D ) p 1 / , 2 (l+3). I t can be seen t h a t the l i f e t i m e has d i s a p p e a r e d , so t h a t recombination i s n o t important a t h i g h 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 g i v e n by = J-jO-aop)^, - a J/tf P l 0 - (kT/d' )(ju - u )9p /ax 0 n p ( 1 which when i n t e g r a t e d over the l e n g t h g i v e s the a . c . v o l t a g e as ( w i t h the s u b s t i t u t i o n s=( 1 + j<*>r) / 2 ) 1 _ J<jL p do p p d Q - Jap 1 Q do The (5.2.7) L / L - cosech I i / L ) J^aApQL (coth L (coth sL/l p p - cosech s L / L ) p kT(^ -M )p + n 3 d T? i s made t h a t no i n e r t i a l by J = £ - I» aap (coth p ~So Q I»A p . Thus (3.2.8) - cosech L / L ) p <*o - maAp L (coth s L / L Q p p - cosech s L / l r ) + kT(fx^-pi^)m£pQ. V <*>B It 1 e f f e c t s occur a t the plane o f the c o n t a c t , g i v i n g PTQ/J^dAp^/dJ^Ap^y/j.. Z(w)A . o complex impedance Z(k>) can be found by d i v i d i n g Here the assumption 1 Q can be seen t h a t s i n c e the frequency dependent term i s s m a l l , the a.c. r e s i s t a n c e R(i*>) i s a p p r o x i m a t e l y e q u a l t o the d.c. d i f f e r e n t i a l resistance. F o r a l o n g specimen, becomes ( J a L / s o ) d A p / d J . / p 0 0 exp(L/L ) » 1 , the a . c . term F o r low f r e q u e n c i e s r e a l and imaginary p a r t e a r e r e s p e c t i v e l y C O T ^ O , the (-JaLp/tf )dApQ/dJ 0 61 and (JaL wr/2d )dAp /dJ. p o For high frequencies 6 J f ^ » 1 , o are (-JaL /d (2cjt) / )dAp /dJ 1 5.3 and 2 0 p () (JaLp/tf^uj^ V 2 ) they d A p ^ d J # Combined D r i f t and D i f f u s i o n Models The t r a n s p o r t problem f o r b i p o l a r n e u t r a l flow i n c l u d i n g both d r i f t and d i f f u s i o n r e q u i r e s the complete p o l a r equation f o r i t s s o l u t i o n . ambi- Hov/ever, the s o l u t i o n can be c a r r i e d through only under s p e c i a l circumstances due t o the complex nature of the equations i n v o l v e d . The ambipolar equation ( 2 . 3 . 9 ) i n the steady s t a t e i s 0 = -AP X U Mp - k T u ( b - 1 ) 3flp~l p 3xJ L 0 (1+aAp)2 dx 0 D ( n + p +2Ap) 3 A p (n +p )(1+aAp) 3 x 2 (5.3.D 2 0 Q 0 where M = > y * n ( n 0 0 o-PoVin^P^n^ 831(1 I) o p n VPo / ¥o Vo =I) ]) ( ) ( F o r the case of low i n j e c t i o n , a A p « 1 , the ambipolar can be made l i n e a r . ) equation 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 - **o J 9 A (S x + Q (5.3.2) P + Do^^P 3X 3X2 and i t s s o l u t i o n can be e a s i l y o b t a i n e d . At u l t r a - h i g h i n j e c t i o n l e v e l s , the thermal d e n s i t i e s can be n e g l e c t e d so t h a t n-p. carrier As a r e s u l t , the ambipolar m o b i l i t y ju=M /(1+aAp) becomes v e r y s m a l l and 0 the ambipolar d i f f u s i v i t y D=D (n +p +2Ap)/(n +p )(1+aAp) tends to Q 0 the constant D =2D D /(D +D ). i p n p n 0 0 0 Thus the f i e l d dependent terms i n the ambipolar e q u a t i o n can be n e g l e c t e d compared t o t h e d i f f u s i o n term. The r e s u l t i n g e q u a t i o n i s 0 = -Ap/r + (5.3.3) D^Ap/ax . 2 T h i s equation w i l l n o t be s o l v e d i n t h i s s e c t i o n but w i l l be i n v e s t i g a t e d i n Chapter for 6 s i n c e the form i s t h e same as t h a t b i p o l a r flow with equation c a r r i e r densities. At moderate o r h i g h 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 s i m p l i f y the ambipolar solved. e q u a t i o n t o any form which can be The o n l y e x c e p t i o n i s the case f o r no recombination when the a n a l y s i s can be c a r r i e d out f o r a r b i t r a r y injection levels. The s o l u t i o n o f the ambipolar injection levels Ap = * e q u a t i o n a t low (5.3.2) f o r no recombination i s P o exp(-(l~*)W* D 1 - ) n ( 5 > 3 < 4 ) 1 - exp(-LJ;u /tf D ) 0 0 i f Ap=0 a t x=L. teristic The s o l u t i o n f o r recombination w i t h a c h a r a c - l i f e t i m e independent (e - e + r A p 0 = Ap L r x of c a r r i e r density or position i s - e + e - ) r L r x L (5^5) where The boundary c o n d i t i o n s Ap=ApQ a t x=0 and Ap=0 a t x=L have been taken. 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 i n both cases 63 i s approximately linear, being given by the ohmic relation J = qfopPo+M^) V / L . The departure from linearity i s 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 i s 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 i s a superposition of four exponential decay processes with four generally different decay ranges. I t 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 i n the steady state give J„ = constant N Jp «= constant. (5.3.6) Therefore the electric f i e l d E can be obtained from the current 64 d e n s i t y equations and s o l v e d f o r dap/dx: _ J p + gP aAp/3x _ J p n d&p = gD aAp/ax n qu^i^+ap) QM (P +Ap) p - 0 JpKbno-PpJp/Jp) + ( b - J / J ) A p ] n p (5.3.7) qD b(n +p +2Ap) p The 0 0 s o l u t i o n o f e q u a t i o n ( 5 . 3 . 7 ) i s , assuming c o n d i t i o n Ap=Ap a t x=0, Q qD b(n -p ) p J P X = t h e boundary 0 0 (b-J^Jp) ^bn -p J /Jp + (b-J /J )Ap ^bn -p J /J + ( b - J / J )Ap 0 lnl 0 0 n 0 n n p N 0 n 2(Ap -Ap) + (5.3.8) 0 n o" o p I f t h e i n j e c t e d h o l e d e n s i t y a t x=L i s taken t o be z e r o , then the h o l e c u r r e n t d e n s i t y can be expressed as = P y u - )r / _H^J Ji" p + L ( b - J / J ) |_ n \ p + bn -p Jn/J 0 0 / p n -p , Q 0 (5.3.9) The a p p l i e d v o l t a g e i s found t o be y = kT(b+J /J ) n p l q(b-J /J ) n p n / + V b-J /J n \ p bn ^p J /J 0 0 n Because o f the constant r a t i o 7 p J n /J p (5.3.10) which may change w i t h the c u r r e n t p a s s i n g through the semiconductor, i t i s difficult to o b t a i n an unique current-voltage c h a r a c t e r i s t i c . I n the case where the c u r r e n t a t the i n j e c t i n g c o n t a c t x=0 i s composed s o l e l y o f h o l e c u r r e n t , i . e . , the e l e c t r o n c u r r e n t i s zero at x=0 and hence everywhere i n the semiconductor, the analysis becomes much simpler. (5.3.8) Since J / J p 0 » equation = n can be written as Ty J Yl+Ap /n V \1 + Ap/n / 2(Ap -.Ap) n -p o 0 Q 0 0 Q At the limiting cases of low injection and high injection, Ap varies linearly with position. I t i s unlikely that the departure from linearity i n the intermediate injection range i s appreciable. The voltage and differential resistance take on the simple forms V = (kT/q)lnO+Ap /n ) 0 V dV 1 d = 0 2 A / l n ( U A p / n ) - (n -p ) Po I 0 0 Q 0 (n +p +2Ap ) 0 0 0 It can be noted here that the total current density, J=I/A, i s equal to the hole current density J . p Since J = 0 , the N electron conduction current and diffusion current must be equal i n magnitude. Thus diffusion plays an important part i n this analysis and can not be neglected. Another approach to the combined d r i f t and diffusion problem i s to assume that the d r i f t solution i s 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& ±f-t T and comparing i t s magnitude along the filament to make sure 66 that the ratio i s less than unity. This method can he used i n the analysis of Section 4.3 for instance, hut leads to a complication i n that the ratio of Ap^/Ap^ i s required. Since the analysis of Section 4.3 has conveniently disregarded ApQ, the method i n this case i s not too helpful. 5.4 Validity Ranges for Drift and Diffusion Solutions In the preceding section, the problem of considering both d r i f t and diffusion was briefly examined. The d i f f i c u l t i e s involved i n the general problem can be readily seen. I f 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 i n 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 i s valid for fields such that >«H/,l where I ^ J J i s d l „ "the diffusion length. For the d r i f t 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 where I) =J dr M f/o' 0 0 range for d r i f t . dr i s the drift length, gives the validity In the intermediate region kTAL d i f f <E<kT/0LL d r both d r i f t 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. intrinsic case n =p . 0 0 Diffusion i s also important i n the However, the intrinsic case i s straight- forward since the drift solutions break down and the problem of defining the validity ranges does not arise. On the other hand, there i s 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 i n finding the validity ranges i s similar to that for the low injection ranges—to compare the neglected and retained terms i n the ambipolar equation by substituting "a posteriori? either the d r i f t solution or the diffusion solution. For diffusion, the solution Ap=ApQe"~ '' i s substiX,/ tuted i n equation ( 5 . 3 . 1 ) • L a 2 1 /2 where L =Jji T/^ the d r i f t length and L =(D f) ' dr o i Thus the condition dr/ i^<( *P) L I 0 i i the diffu- sion length, must hold for the diffusion solution to be valid. For d r i f t , the solution Ap=Ap e~ a/^ , equation ( 4 . 3 . 7 ) , where s 0 s a i s the ambipolar transit time, i s substituted i n 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 It solution. should be remembered t h a t i n both cases aAp » 1 injection level The contacts approximation. c o n t a c t s a t which Ap=An/0)• can be almost p u r e l y d i f f u s i v e . There the i f diffusion diffusive c u r r e n t s are l a r g e o n l y over narrow r e g i o n s c o n f i n e d to the v i c i n i t y of- the c o n t a c t s , then analysis w i l l flow However, i f the l e n g t h of the semiconductor i s taken-.. to be much g r e a t e r than the drift high d r i f t a n a l y s i s i s not v a l i d a t or n e a r non-ohmic (i.e., l e n g t h , and the g i v e a f a i r l y good approximation. the 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 D /(Dp+D ) • n ambipolar equation T n e n for this case becomes quite simple: 3Ap/9t = - A p / r + D 3 Ap/9x . 2 (6.1.1) 2 I Due to the absence of the f i e l d dependent term, i t can be seen that the spatial distribution of the carrier density i s not subject to d r i f t under the f i e l d , but only to diffusion and recombination. Only the neutral case i s studied. This can be considered as only a rough approximation, since for high injection or for high r e s i s t i v i t y material, space charge becomes important. Boundary conditions of two types are used at x=0. The f i r s t i s simply Ap=Ap at x=0, and the second i s J ^ O H X j J , Q where Y i s 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 i s easily found. The hole current can be written as 70 Jp = qu (p +Ap)E - qD 9Ap/3x p 0 =V J p qD^Ap/ax - po where L =1/(b+1) for both the intrinsic and the high level extrinsic cases. J At x=0, p = W ( 0 ) 3Ap/9x| lx=0 ~ <* i D a A p / 5 x x = 0 = V (6.1.2) = -(yP -LP°)J/qD . T) n (6.1.3) 1i The expression for Ap obtained from the ambipolar equation using the boundary condition Ap=Ap^ at x=0 substituted i n equation (6.1.3) gives the relationship between Ap 6.2 Ho Q and' }f . Recombination For no recombination, the ambipolar equation i n the steady state becomes 3 Ap/9x = 0 2 (6.2.1) 2 If the boundary conditions Ap = Ap^ at x=0 and Ap = 0 at x=L. are assumed, the solution of the ambipolar equation i s Ap = Ap (1- /L) 0 (6.2.2) X where L i s the length of the semiconductor. When this solution i s substituted i n the current density equation, the electric f i e l d E may be found and integrated to get the voltage: E . - " V " " - * ^ ) J tf [l ± + AP (1-XA)/P ] 0 0 ( 6 . . 2 3 ) kT(b-l) JL V = 1 R (b+l)A dV di 0 m (1+Ap / ) 0 (m-O Ap /p 0 l n ( 1 + A /p ) 1 kT (b-1)Ap IL + Mp Po (6.2.4) P(> q(b+1)__ P() n Ind+ApQ/Po) (6.2.5) 0 where m=(l/Ap )dApQ/dI, 0 For heavy injection, A Q > > P , and negligible d i f f u P 0 sion current, the differential resistance i s (1/R )dV/dI = ( P / A P ) [ 1 - ( m - 1 ) l n ( A / p ) ] . 0 0 0 Po o For negative resistance, under this condition, m must be greater than unity. However, this i s unlikely, unless the injection ratio Vp, increases with increasing current. equation (6.1.3), the relationship between APQ and A *0 - ( V ^ i ' D From i s found: (6.2.6) From this equation, the relationship between m and m* where m' = [ l / ( V - ^ ) ] d ( y - ^ ) / d I i s found to be p 0 p m = 0 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 Pl ( x ) = C xp( jw/Di) ie 1//2 x + C exp-( jW/L^) /^. 1 2 The constants, C-j and C , are determined by the boundary 2 conditions p^(L)=0 and P^(0)=P Q, giving 1 sinh ( j ( j / D ) / 2 ( , sinh (jCo/Di)'/ ! 1 i P^x) ' = p 1 Q 1 U L-x ) . (6.2.9) 2 For (^/2D ) / L»1, the solution simplifies to 1 2 1 P-l(x) = p exp[-(^/2D ) / (l + j)x] 1 1 0 (6.2.10) 2 i and i s identical to the purely diffusive flow of injected carriers at high frequencies which was worked out i n Section 5.2. The voltage and a.c. impedance cannot be worked out exactly for this model and i s not attempted. 6.3 Recombination with Constant Lifetime If the lifetime X i s independent of the carrier density and of the position, the ambipolar equation in the steady state for bipolar flow with equal carrier densities becomes B Ap/2x 2 where L. =(D.f) 1y/2 2 = ApA (6.3.D 2 ± , the effective diffusion length. high injection level, n=p, For ultra- the generalized ambipolar equation also reduces to equation (6.3.1) as shown in Section The boundary conditions Ap=Ap 5.3. at x=0 and &p=0 at x=L 73 are used to obtain the solution Ap =Ap sinh(L-x)/L. Q sinh L/L, u If the substitutions u=e^^~ ^ ^ x // Ji (6.3.2) ±. and du/dx—u/I^ are used i n the equation for the f i e l d E = j/cY (1+aAp) - [kTUpfb-D/Cod+aApjjaAp/ax, 0 then the f i e l d can be readily integrated to obtain the applied voltage and the differential resistances V = dV dl- LjJ kT(b-1) l n ( 1 + a A p ) ln Q 0 C^(1+4M ) /2 2 1 q kT(b-1) maAp ( b + R L./L 0 Iq(b+1)(1+aAp ) 0 (6.3.3) l) 1+4M (1-m) 2 (1+4M ) L(l+.4M ) /2 2 2 4M(2Mcoth l / 2 I i 1 ln Q - 1)m (6.3.4) (coth L/21^) + 4Mcoth l/2L - 1 2 ± where M aA /(e / i-e- / i) L L L L P o coth L/2L + 2M + (1+4M ) / 2 coth I/2L. + 2M - (1+4M ) / 2 2 Q= R Q 1 ± 2 = 1/ACf,, 1 m = (l/Ap )dA /dI. 0 Po The case where the length of the semiconductor i s several times the diffusion length giving rise to e ^ A l ^ l ^ taken. (i) There are two situations to be investigated: high injection i n short filament aAp e~ 0 I,//l 'i»1. s 74 In this case, the voltage and differential resistance become V = dV dl IR e L/Li 0 2aAp 0 R e L / L 0 2aAp, i « + —kT(b-1) i L. i n aAp q(b+1) Z l l n 4aAp L A (1-m)^i l n 4aAp - 1^ + m IJL Q Q 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 i s sufficiently larger than unity. (ii) high injection i n long filament aAp e~ 0 Ii//l 'i«1, aApQ» 1 . The voltage and differential resistance are V = IR fl - ^1 In 2 a A p + L L J | j = R |1 - ^ l n 2aAp 0 0 M(b-1) l n aAp q(b+1) m^ + 0 kT(b-l)m Iq(b+1) Since (Lj./L)ln aAp i s much smaller than unity, i t i s unlikely Q that negative resistance w i l l 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 L For e> 'Ai cosh ( L - x ) A i cosh I/L^ (6.3.5) both solutions reduce to Ap = Ap 0 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 1 - " IR q(b+D/ lnl (6.3.7) \l+aAp e- > ^ L 0 /L 0 M +aAp Q 1 +aAp ' - L / L j 0 RL m (l+aAp e- /Li) L 0 kT(b-l) 1 L Iq(b+1) 0 For very heavy injection such that aApQ (6.3.8) 1, negative resistance can occur. For the a.c. analysis, the solution i s for e Pl (x) = p exp[-(1 + j w r ) / 2 / ] x i»1 (6.3.9) 1 1 Q L//l, Li where the diffusion length has been replaced by an effective a.c. diffusion length L j / O + j u t ) / . 1 comparison to P l At frequencies high i n 2 the solution reduces to (x) = p exp[-(W/2D ) / (1+j)x]. 1 1 0 2 i It can be noted that this limiting solution i s identical with that for no recombination equation (6.2.10). Thus at high frequencies, recombination i s not important i n the a.c. solution. 6.4 Recombination with Carrier Dependent Lifetime If the carrier lifetime i s a function of the carrier density, then the recombination term can be written, as i n the drift case, as r The ambipolar densities = Ap/t equation i n the 2 solution using the of this The b o u n d a r y limits be V to The n e g a t i v e carrier becomes differential (6.4.1) equation root increasing is d Ap/dx 2 vanish. greater is can 2 be obtained by the of used boundary to make condition' to the also thus (6.4.2) 1+I taken value u9u/3Ap. 2BAp V(1+V). = = ApQ(1 = 2 Ap=0 i s This than -1; position x; Ap x=0 equal = Ap/DjT = BAp^. 2 (dAp/dx) Q state flow with c o n d i t i o n dAp/dx=0 a t constant Ap bipolar = dAp/dx, integration where V 0 substitution u with r &p . for steady d Ap/dx The = obtain a solution which decays thus + K x ) " the 2 / ( ^ 1 injected (6.4.3) 5 carrier density at and K F o r V=1, the situation becomes i d e n t i c a l section. = (^-1)(BAp Thus, 0 l '- /2(1+V)) 1 1 / 2 (6.4.4) . for constant lifetime, w i t h the solution found equation i n the (6.4.3) preceding 77 +J>=li~ 2 L = l i m A p „ [ * 11 V-*1 L Ap U ± which i s the same as equation The equations (6.3.6). f o r the v o l t a g e and d i f f e r e n t i a l r e s i s t a n c e , however, w i l l n o t be i d e n t i c a l with and ( 6 . 3 . 8 ) (6.3.7) be made i n the f i e l d equation due t o the m o d i f i c a t i o n t o as d e s c r i b e d i n the next paragraph. Due t o the d i f f i c u l t i e s i n v o l v e d , the a n a l y s i s w i l l be c o n f i n e d to the h i g h i n j e c t i o n case ( a A p » 1 ) . The f i e l d and v o l t a g e i s then g i v e n by E _ k T u ( b - 1 ) dAp = p OiaAp O^aAp (6.4.5) dx J(V-1)/(^+D 0>+1)/(y-O L 4)^20 k T ( b - l ) 2 ln(1+KL) (6.4.6) q(b+D(y-1) The d i f f e r e n t i a l r e s i s t a n c e can be w r i t t e n as dV = m(l^+D [ ( 1 + K L ) "drift di 2 |J1 KL) A A + 1 + - l] - 1] (b -D(Y -)( )KLm(V l)" 2 + p po 4(1+KL)[(1+KL) where m = (l/Ap )dAp /dI 0 ^ = 2/0/-1) 0 + A + 1 - 1 ] (6.4.7) qUp(b+1)KAp The equation g i v i n g KAp was o b t a i n e d f r o m 0 0 Y i n terms o f t h e i n j e c t i o n r a t i o equations (6.1.3) and ( 6 . 4 . 3 ) . T he condition f o r negative resistance i s x m(y+1)/2 2 (1+KL) (1+KL) A + 1 - 1 the range written; K'=-K. -1<V<1. f o r V a r e taken: F o r V>1, t h e e q u a t i o n s >1. (6.4.8) t h e range V>1 c a n be u s e d a s f o r -1<V<1, t h e e q u a t i o n s a r e m o d i f i e d b y s u b s t i t u t i n g Equation (6.4.3)» Ap In P—22 2(1+KL) - 1 Two s e p a r a t e r a n g e s and (b -l)(y-V)KL = f o r example, Ap (1-K«x) 2 / ( 1 0 - U ) becomes . o r d e r t o keep t h e 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 , t h e constraint required K'L<1 i s imposed. f o r KL i n t h e range For K L « 1 , However, no s u c h constraint i s V>1. the condition f o r negative resistance becomes m [l In a semiconductor (b -l)U -Y 2 it - (b -1)(Y -Y )(V-1)/4]>1. 2 p p o s u c h a s germanium, t h e c o e f f i c i e n t )/4 i s a p p r o x i m a t e l y i s highly u n l i k e l y that 1 / 2 f o r *=1. negative resistance T h u s f o r V>1, will 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 i s For the range -1<V<1, the condition for K'L«1 m>1. gives m[l + (b -1)(Y -^ )(1-V)/4>1 2 p o which i s possible for at least some values of m. values of K'L, disappears. At higher that i s , as K' L-*1, negative resistance From the equations for the two ranges, i t can be seen that for negative resistance to occur, the term due to diffusion (b -1) ()f 2 (1-^)/4 i s detrimental i n the case of V>1 and advantageous i n the case of -1<V<1. If the injection ratio V also changes with current, p then the possibility of negative resistance i s increased. In this case m = 2(1+m')/(V+1) (6.4.9) where m' = [l/( V - K^)] d( )f -^ )/dI. p p There are two p o s s i b i l i t i e s : 0 the injection ratio increasing with increasing current, m'X), and the injection ratio decreasing with increasing current, m'<0. In any event, i t i s unlikely that m* w i l l be very large. It can be noted that a changing Y m>1. P admits the possibility of The differential resistance can be rewritten i n terms of m' as 80 dV « V drift " iftl+KLpT-l] KLd+KL)^ - m'[(l+KL) - f) (b -D(y -^ )KL(Um') 2 p 0 (6.4.10) 2(1+KL) For V>1 a n d m ' < 0 , n e g a t i v e occur. is F o r m*>0, sufficiently there large enough. t h e range any m*; f o r m'<0, dV/dl this m o d e l may be s e e n V drift definitely not For larger values resistance -1<^<1, n e g a t i v e b o t h m'>0 a n d m ' < 0 . The r e l a t i v e will may be a p o s s i b i l i t y a t K L « 1 i s no p o s s i b i l i t y of negative In for there resistance i s negative importance + f however, occurring. is i s always possible negative for when - m ' < K ' L . o f .drift and d i f f u s i o n i n by t a k i n g the r a t i o 2b(V-1)[(1 KLV of KL, resistance F o r m'>0, dV/dl i f m A + 1 - V ( j ; r ift / dif fusion* y V 1] (6.4.11) (^-OUp-lfpoW+l) •diff For K L « 1, the r a t i o reduces mO+KL) to drift v Unless (b^-l)(!(-lf) d i f f )L=}Lo» p o p t h e d i f f u s i o n component When K * L a p p r o a c h e s u n i t y o r K L becomes become thus quite neglected large, cannot large, be neglected. the r a t i o can e n a b l i n g t h e d i f f u s i o n term t o be and s i m p l i f y i n g the a n a l y s i s The p h y s i c a l s i g n i f i c a n c e considerably. of the constant K c a n be 81 s e e n by d e f i n i n g an effective diffusion length L ^ f r o m r i g h t hand s i d e o f e q u a t i o n BAp = Ap/Di? = V B = Ap - /I 1 Hence In K = the level V 1 bulk. d e c r e a s e s , and very high KL APQ heavy necessary f o r negative the injecting t h i s range, the i s p r o p o r t i o n a l to Ap . Q injection the lower L^(0) increases, range, increases. injection. give as w e l l as N e g a t i v e r e s i s t a n c e i s f o u n d t o be In In rise to be seen r e s i s t a n c e may contact the becomes l a r g e r . injection. t h a t as very t h a t as p h y s i c a l s i t u a t i o n w h i c h may -1<y<1 r a n g e . Ly(0) so 2 v would i m p l y considering constant. (6.4.12)' L (0)oCl/Ap^"" ^ , -1<V<1 , Ly(0)oTApQ^"'^, so behavior = 2 y would imply A 2 v = Ap^-^/L^O) 2 y increases, Ly(0) Thus K ' L « 1 Ap/L thus (V-1 )//2(y+1) L ( 0 ) . r a n g e V>1, Thus K L » 1 (6.4.1); the the the by semiconductor attainable i n effective diffusion Hence as the the length injection level 1 V increases, carriers the lifetime i n j e c t e d by time i n the the also increases contact semiconductor bulk, mentary i n c r e a s e d penetration t o a f u r t h e r d e c r e a s e i n the leads and bulk leading bulk, the which i n t u r n i n c r e a s e s the lead . the to a resistivity. This in p o t e n t i a l between the increase lifetime the i n the i n the contact carrier bulk. The life- compli- injected carriers p o t e n t i a l drop across to a corresponding ToCp ~ then increase which w i l l o f the to a r e d i s t r i b u t i o n of the the will since leading turn contact rising and injection Such a "feedback" process behaviour necessary 6.5 Critique levels. (1959) discusses q u a l i t a t i v e l y In a long diode, consists the diffusion mechanism. ohmic c o n t a c t s . The obtains i s similar levels, the he At l e n g t h of the diode e a c h end to equation j u n c t i o n and exponential, the diode j u n c t i o n as o f the diode are At high the attached that he injection and the diode (including c=2(b+coshL/L . ) / ( b + 1 ) ; I diffusion length s m a l l changes i n can example, i f L/I^=6, 1=10^1^, a 20 t o change I by a f a c t o r of appears i n contacts, is a J f u n c t i o n of the parameters o f the diode Since sufficient The C p-n For w i t h a p-n injection f o r m I = I ( e x p ( q V / c k T ) - i j where V i s t o t a l voltage applied across bulk) possibility i s much carriers. (6.3.4). the material. diode voltage-current characteristic o b t a i n s the complicated the at ultra-high l e n g t h o f the of an unsymmetrical injecting the Model r e s i s t a n c e i n "long diodes" l o n g e r than to f o r negative resistance. of Stafeev's Stafeev of negative o f self-enhancement would g i v e r i s e and of the the cause l a r g e changes i n I . percent change i n is 50. 1 /2 B e c a u s e L^oC X ' L^. The o f the lifetime injected c e n t r e s i n the begin of the carriers. , changes i n the carriers I n the semiconductor, the d e p e n d s on presence levels. p r o c e e d s a c c o r d i n g t o t h e b i m o l e c u l a r law The value of the of the will vary concentration recombination occupancy of these t o change a t c e r t a i n i n j e c t i o n conditions. lifetime centres Recombination under these current at which t h i s occurs depends on the c o n c e n t r a t i o n and a c t i v a t i o n recombination occupied centres. energy of the At l a r g e c u r r e n t s ( c o n c e n t r a t i o n o f c e n t r e s much l a r g e r than c o n c e n t r a t i o n of unoccupied c e n t r e s ) , the l i f e t i m e a g a i n become independent of injection level. With t h i s background, S t a f e e v proceeds w i t h a qualitative argument. He assumes a c u r r e n t - v o l t a g e i s t i c a p p r o p r i a t e to a low current and a 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 c u r r e n t L^. character- (independent o f a p p r o p r i a t e to a l a r g e Since the l i f e t i m e i s a f u n c t i o n of l e v e l over a c e r t a i n narrow c u r r e n t i n t e r v a l , the injection transition from the low c u r r e n t to h i g h c u r r e n t c h a r a c t e r i s t i c ; occur i n t h i s i n t e r v a l . The current) will l i f e t i m e can i n c r e a s e or decrease w i t h the i n j e c t i o n l e v e l depending on the p r o p e r t i e s of the recombination centres. For an i n c r e a s i n g l i f e t i m e w i t h forward c u r r e n t i n a diode w i t h 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 S e c t i o n T h i s l e a d s to a t r a n s i t i o n from the low current c h a r a c t e r i s t i c high and r e s u l t s i n a n e g a t i v e r e s i s t a n c e . Although S t a f e e v s 1 c o r r e c t , he makes one c u r r e n t t o the 6.4. argument may be approximately assumption t h a t i s i n v a l i d . He assumes t h a t the l i f e t i m e changes w i t h i n j e c t i o n l e v e l ( t h a t i s , c u r r e n t ) and t h a t the l i f e t i m e a t t h a t g i v e n i n j e c t i o n i s same throughout the whole semiconductor. case, however. The level T h i s i s not the l i f e t i m e i s a f u n c t i o n of the l o c a l c a r r i e r d e n s i t y and hence of p o s i t i o n a v a r i a b l e l i f e t i m e occurs. a t i n j e c t i o n l e v e l s where At h i g h c u r r e n t s where the occupancy of is a At low the the constant l i f e t i m e may be This not change, of c a r r i e r I n the the occupancy o f the by m e a s u r i n g t h e could intermediate range, carrier recombination probably lifetime lifetime density. a f u n c t i o n o f b o t h p o s i t i o n and change i n l i f e t i m e experimentally along be checked the length of semiconductor. Stafeev s 1 utilizing the regions range, value can the to the be be obtained density of the i n the range of values lifetime the voltage be and a f i r m e r base be could 1 carrier d e t e r m i n e d by found The appropriate on found referring by d e n s i t y and (Shockley the and 6.4 the life- 1952). Read favourable that i s , f o r increasing Thus t h e i n c l u d i n g the the u s e transition lifetime i n Section -KV<1, density. be The lifetime the used. dependence o f the w h i c h was characteristic In by 6.4. constant Shockley-Read model carrier Section 6.3. can f o r m fo&p ""^ r e s i s t a n c e was with 6.3 6.4 r e l a t i o n s h i p between t h e f o r negative on from S e c t i o n r e s u l t s of Section t i m e s u c h as put c h a r a c t e r i s t i c s i n the of V which gives carrier could model can r e s u l t s of Section current-voltage The same i s t r u e . d e p e n d i n g on centres. do i n d e p e n d e n t o f p o s i t i o n and currents,.the density the recombination centres complete negative of the current- resistance models p r e s e n t e d range in this chapter. Recent and Rediker negative 1962) e x p e r i m e n t s on long d i o d e s of InSb (Kelngailis seem t o 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 resistance occurs c a r r i e r l i f e t i m e with from an carrier increase density. of the injected 85 CHAPTER 7. CONTACT PROPERTIES AND RELATED BOUNDARY CONDITIONS 7.1 Boundary In the contacts and ^p=^p Tj Conditions the previous chapters, have b e e n e x p r e s s e d a t x=L. the boundary c o n d i t i o n s a t i n t h e f o r m a s Ap=^p^ a t x=0 No r e f e r e n c e h a s b e e n made t o s p e c i f i c contact p r o p e r t i e s o r i n j e c t i o n mechanisms. setting boundary c o n d i t i o n s l e a d s usually the behaviour o f the i n j e c t e d c a r r i e r current get i s not e x p l i c i t l y an e x p l i c i t condition the hole a i s , J (0) = ^ J . p )( i s i n g e n e r a l voltage p the transport # p K p at contact be case, a r e much s m a l l e r c a n be u s e d i s t o s o l v e field drift than elsewhere, c a r r i e r densities at the contacts. E, as i s at very i t i s assumed t h a t b o u n d a r y c o n d i t i o n E-0 i s u s e d a t b o t h c o n t a c t s . infinite current- unobtainable. problem i n terms o f the e l e c t r i c the contacts current a f u n c t i o n o f J , an e x p l i c i t In this i s that and J , b u t s i n c e t h e i n j e c t i o n a l t e r n a t i v e approach that levels. boundary i t i s possible to get 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 injection Another o f the t o t a l With t h i s , p r e l a t i o n s h i p may s t i l l An since d e n s i t i e s with relationship. be some f r a c t i o n r e l a t i o n s h i p between Ap@, ratio t o many d i f f i c u l t i e s t h a t has been used a t t h e i n j e c t i n g that problem o f known, and t h e n i t i s i m p o s s i b l e t o current-voltage current density, This high the f i e l d s so t h a t t h e This leads to As mentioned 86 previously, this i s a result w h i c h become i m p o r t a n t of n e g l e c t i n g a t low fields. In general, s o l u t i o n i n t e r m s o f E i s more d i f f i c u l t systems, the the voltage field E=0 or the d e n s i t i e s and i n advance When d i f f u s i o n it electric be an the currents are the been found advantageous to analyses t o v/ork w i t h and and the the the injected c a n n o t be problem. The taken i n t o account, contacts. the system i s Therefore current s o l u t i o n o f the however, the a p p r o x i m a t i o n w h i c h must be n e g l i g i b l e at has on current. on i t s boundary c o n d i t i o n s of the i s thus only not total currents, since i n a l l physical parameter impressed i s a v a r i a b l e dependent carrier fied independent diffusion Because of speci- use of used with care. the may this field difficulty, e l i m i n a t e E i n most o f excess c a r r i e r density as the the dependent v a r i a b l e . In r e l a t i n g there are contact This effect contact the can will blocks, two questions meet t h e concern injects on the boundary c o n d i t i o n s t o be considered. requirements of such m a t t e r s as or flow through the the to a c t u a l First, boundary to whether the contact o f the S e c o n d , how the injection level injection changes? rate v a r i e s with semiconductor. of conditions? o r whether i t has carriers. contacts, what t y p e extracts carriers b e h a v e when the p r o b l e m o f how passing the does This the no the involves current 87 7.2 Contacts In this s e c t i o n , t h e r e l a t i o n s h i p between t h e boundary c o n d i t i o n s and c o n t a c t s discussion i s not r e s t r i c t e d contacts tures, but a l s o i n c l u d e s will The to metal to semiconductor other inhomogeneous j u n c t i o n s t r u c - f o r example, a p - n j u n c t i o n . describe be d e v e l o p e d . No a t t e m p t t h e numerous t h e o r i e s p r o p o s e d metal to semiconductor contacts, i s made t o 1957) (Henisch for none o f w h i c h h a s p r o v e d t o be s a t i s f a c t o r y . The properties of contacts made t o t h e s u r f a c e s e m i c o n d u c t o r d e p e n d s l a r g e l y on t h e p r o p e r t i e s conductor surface with factors the a which t h e contact are important—the surface of the semi- i s made. Three p o t e n t i a l b a r r i e r a t the surface, recombination-generation r a t e and t h e presence o f f o r e i g n i n s u l a t i n g l a y e r on t h e s u r f a c e . The of the abrupt of p o t e n t i a l b a r r i e r a t the surface This a t the surface leads presence and those i n t h e b u l k o f the compensating bulk to a p o t e n t i a l b a r r i e r , the height the excess surface The material surface charge. space charge g i v e s rise d e p e n d i n g on t h e amount o f charge. e l e c t r o n i c surface the formation structure t o a d i f f e r e n c e between t h e w h i c h may l e a d t o a n e t p o s i t i v e o r n e g a t i v e The i s the r e s u l t 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 the semiconductor. energy l e v e l s to of a o f the surface recombination-generation s t a t e s w h i c h may g i v e rise charge can a l s o a c t as centers f o r c a r r i e r s . The r e s u l t i n g 88 recombination and rate defines d e p e n d s on layer w i l l be the necessary and potential condition is the again The equilibrium of the are being infinite V carrier J and Q the the h o l e p = p , Q The of i s made. The a first transport i n either field ohmic c o n t a c t a t any direction point with The the the no second condi- since contact t r a n s p o r t i n the maintains semicon- thermal of both c a r r i e r s . current i s characterized The ratio by >"n o^p o n irrespective p holds The boundary c o n d i t i o n f o r the of the c o n d i t i o n s i n the ohmic c o n t a c t semiconductor. i s t h u s An=Ap=0 a l l times. D e p a r t u r e s from the finite p o t e n t i a l b a r r i e r and at surface. the c o n d i t i o n i s no by absence removed o r s u p p l i e d by and at the absence of space charge e f f e c t s densities, n e l e c t r o n to insulating recombination-generation contact r a t e demanded by ideal of the considered. to space charge f o r m a t i o n . e n s u r e s the at the ductor. an be conditions are p r o p o r t i o n a l t o the carriers exactly sufficient due effect "ohmic" c o n t a c t w i l l ensures that the complications The velocity later. s u r f a c e where the strictly tion an b a r r i e r and r a t e a t the surface recombination barrier height. discussed First, the the nature ohmic c a s e c a n a finite Thus i n a non-ohmic longer o f the recombination contact, c h a r a c t e r i z e d by contact and the result the rate exist boundary Ap=An=0 b u t current i f a passing i s set through i t . First, contact into the will the defined. flows current will through the in be contact Another o r x = L may n o t but may b e chosen some of applied the v/hen J > 0 . electron where the the the At other, to charge at or For instance, a eliminate gradients. charge plane x=0 was There are the is that the in the i f the current will will differ of the boundary of the the to the eliminate regions which solution of condition J a limited flow model at the is negative high illustrated of be n from very example may and J > 0 such as Another be p at contact c r i t e r i o n which could conditions to the direction plane w h i c h may r e s u l t taken contact, currents diffusion used be positive; the away i n o r d e r assumptions could contact semiconductor since noted be from the contact, two the contact. distance densities space the to termed into i n the the regions x=0 be one only be relative flows will current to small would current unipolar current semiconductor necessarily plane This concentration current current relative the model. at at a space particular the and feature x=0 violate the i n terminology defined complicated the of termed n e g a t i v e . d i r e c t i o n but is If the from in direction; current direction semiconductor, current agree be the Section potential in 4.22 barrier extremum. If can the current supply contact is through excess said semiconductor, four to types the carriers be holes of contact into injecting. will contacts be is the to be positive discussed. and semiconductor, For instance, injected with a the contact then i n an the n-type corresponding increase i nthe electron density to maintain charge n e u t r a l i t y . can The boundary, c o n d i t i o n f o r s u c h a c o n t a c t b e c h a r a c t e r i z e d AP=APQ>0 w i t h J>0. p-type m a t e r i a l , e l e c t r o n s w i l l contact. be i n j e c t e d direction will I n the to follow, the current the current through the other J P the t o t a l due t o t h e f l o w o f e l e c t r o n s N o f the' e l e c t r o n o r t h e h o l e c u r r e n t c a n be d e f i n e d . with holes being injected, F o r a n ohmic c o n t a c t , ratio current F o r n-type m a t e r i a l , the injection ratio i s i s n o t zero but equal T h u s i t c a n be s e e n t h a t i n j e c t i o n If the contact c o n s i s t si n due t o t h e f l o w o f h o l e s , a n i n j e c t i o n ft c o n s i s t i n g o f t h e r a t i o t h e current through occurs for t o Yp =p /(bn +Pg). 0 0 0 ^Vp^VpQ- the contact i s negative, i t 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 neighbourhood of t h e contact. can n o t accept h o l e s r a p i d l y up case. will be d e s c r i b e d ) . g e n e r a l o f two c o m p o n e n t s , one J may negative be o p p o s i t e t o t h a t f o r t h e n - t y p e m a t e r i a l Since to from the f o r an i n j e c t i n g contact i n t h i s other three contact c l a s s i f i c a t i o n s and ( I n t h e case o f 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 be n e g a t i v e which w i l l approximate This can occur i f t h e contact enough a n d t h e h o l e s a t the contact with a corresponding tend increase i ntheelectron d e n s i t y t o m a i n t a i n approximate charge n e u t r a l i t y . c o n t a c t i s s a i d t o be a n a c c u m u l a t i n g to build contact. Such a I n the high l e v e l d i f f u s i o n m o d e l 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 U p and Ap i s given as ApQoc(^pQ )p)J, 0 Thus i f the hole density- increases Ap>0 with a negative current, then #p>Vp.. 0 If the current through the contact i s negative and carriers are being depleted below their equilibrium value because the semiconductor bulk can not supply carriers fast enough, then the contact i s said to be an extracting contact. In this case Ap<0,|Ap|<p and o Vp*>Vp. 0 I t should be noted that there i s a limit on the size of Ap. A similar situation can occur i f the current through the contact i s 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 i n the semiconductor bulk. exclusion. This phenomenom i s referred to as carrier In this case Ap<0, |Ap|<p and 0 ...VpX^po* Other definitions can be applied to contacts. A blocking contact for say electrons i s one which can neither accept nor inject electrons. A saturated contact i s one which i s 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 o f the c o n t a c t w i t h changing c u r r e n t l e v e l i s a l s o an important f e a t u r e of the c o n t a c t . I f the impressed c a r r i e r d e n s i t y a t the plane x=0 i s A p , the e a s i e s t 0 method of i n v 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 c u r r e n t 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 thus m=(l/Ap )dA.po/dI. T h i s form has been used throughout the 0 thesis. power law, In general, m i s d i f f i c u l t to determine and can not be i n f e r r e d from 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 . Although i t i s u s u a l l y necessary i n certain to determine m e x p e r i m e n t a l l y , cases, i t may be p o s s i b l e to o b t a i n m a n a l y t i c a l l y . For i n s t a n c e , i n a p-n j u n c t i o n a t 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 o f S e c t i o n 5.2 i s v a l i d . I n t h i s case, the t o t a l c u r r e n t a t the j u n c t i o n (x=0) i s predominantly a h o l e c u r r e n t so t h a t Vp=1. Since the t o t a l c u r r e n t d e n s i t y through the semiconductor i s a c o n s t a n t , J can be r e p r e s e n t e d J=Jp(0)=qDpApQ/Lp. Hence m=1 f o r t h i s case. by 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 i n c r e a s i n g l y important, the e l e c t r o n c u r r e n t may become s u f f i c i e n t l y l a r g e so t h a t i t can not be n e g l e c t e d . Thus Kp w i l l decreasing hole current decrease on account of the fraction. In the p u r e l y d r i f t a n a l y s i s of Chapter 4 w i t h d i f f u s i v e terms n e g l e c t e d , the r e l a t i o n s h i p between Kp and ApQ is H V °P"°PO where Ifp^qjipPo/tfo, M &Pfi (j^+jap) (1+a4p ) a = n 0 the f r a c t i o n of the c u r r e n t c a r r i e d by h o l e s w i t h no added c a r r i e r s . The l i m i t s s e t on lL by the 93 conditions of low and high i n j e c t i o n l e v e l s are Y p o <V - 1 / ( b + 1 ) I t can be seen that A P Q does not depend e x p l i c i t l y on the current and that the i n j e c t i o n r a t i o Yp increases with current. The relationship between m and m = fl/( Vp-^p0)"] d( Y p - / p 0 ) / d l i s 1 m' m = 1 - <V P W + V ^ o For the i n t r i n s i c and high l e v e l d i f f u s i v e models of Chapter 6 , the relationship where K = 1 / ( b + 1 ) . P O between Vp and A P Q i s In the i n t r i n s i c case, K p0 represents the f r a c t i o n of the current carried by holes with no excess carriers, I t can be seen that m^'+l where m' = [ l / ( tf -^ )] p 0 d( Vp-Y )/dI. po Negative resistance was found to occur f o r both m'>0 and m'<0. Thus a l l that i s required i s that the i n j e c t i o n r a t i o changes s l i g h t l y with current. In the case of an extracting or excluding contact where Ap i s l i m i t e d by the condition |Ap\<p Q , the i n j e c t i o n 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 < t the i n j e c t i o n r a t i o w i l l decrease while f o r an 1 X* ^ XT excluding contact where 0<)| <)f , the i n j e c t i o n r a t i o w i l l p po increase with increasing current. In practice, the behaviour of the contacts with current may be quite unusual. Harrick ( 1 9 5 9 ) has observed injection regardless of the direction of current flow and also extraction regardless of the direction of current flow i n metal to semiconductor contacts. The explanation given for this behaviour u t i l i z e s an insulating film between the metal and the semiconductor surface. I t 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 i n the ambient or the surface etching process. 7.3 Carrier Multiplication at Contacts The injection of excess carriers by a suitably biased contact i s not the only means of increasing the carrier density i n a semiconductor. Avalanche multiplication i s an alternative means of generating hole-electron pairs and can inject carriers into the semiconductor bulk. Avalanche multi- plication may occur i n a narrow space charge region of'high r e s i s t i v i t y where a high f i e l d i s set up. Such a region of high r e s i s t i v i t y may be due to a foreign insulating layer such as an oxide layer between the metal contact and the surface of the semiconductor. 95 A carrier moving i n a high electric f i e l d may acquire sufficient energy to ionize the lattice and create a new holeelectron pair. The primary and the generated secondary carriers travel independently i n the electric f i e l d and may cause further ionization. The probability of i t s doing so i n a distance dx i s expressed asc^dx for an electron and o(pdx for a hole. The ionization coefficients o( are functions of the f i e l d and are defined as the number of ionizing collisions per centimeter path length made by a single particle. For example, i n an n-type germanium filament through which a steadily increasing current i s passed, the electric f i e l d increases steadily until a c r i t i c a l f i e l d i s reached at which the avalanche starts. The holes produced by the avalanche d r i f t towards the negative terminal and increase the conductivity and thus reduce the f i e l d at this end. If the externally applied voltage remains constant, the f i e l d i s enhanced at the positive end where the avalanche i s occurring and increases the avalanche generation rate. Injection into the bulk may be so heavy that only a relatively small f i e l d exists i n this region. G-unn (1957) has carried out a theoretical analysis i n 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 i s f a i r l y large with the carrier space charge resulting from the avalanche process being much greater than the Impurity density. 96 The i o n i z a t i o n c o e f f i c i e n t i s assumed to depend with the f i e l d . exponentially With these assumptions, the voltage across -1 /2 the avalanche tends at large current densities to V °Cj a Thus i f the current increases, the voltage ultimately decreases and such an avalanche i s capable of showing a negative resistance. Avalanche i n j e c t i o n can cause negative resistance i n d i r e c t l y as w e l l . Due to the heavy i n j e c t i o n l e v e l s possible, the injected c a r r i e r 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 f o r negative resistance i n several of the models investigated i n previous chapters. 97 CHAPTER 8 CONCLUSIONS AND OUTSTANDING PROBLEMS The models o f t h i s t h e s i s have shown how d u c t i v i t y of a semiconductor or the can be modulated by the con- injection e x t r a c t i o n o f o a r r i e r s and the' c o n d i t i o n s under which n e g a t i v e r e s i s t a n c e c o u l d be o b t a i n e d . both d r i f t The g e n e r a l case f o r and d i f f u s i o n c u r r e n t s i n the t r a n s p o r t equations has been shown t o be complicated and d i f f i c u l t i t was t o do. However, shown t h a t w i t h the proper c h o i c e o f o p e r a t i n g l e v e l s , e i t h e r the d r i f t the ambipolar o r the d i f f u s i o n term c o u l d be omitted i n equation. Por the d r i f t model with u n i p o l a r space-charge- l i m i t e d f l o w , a band pass a.c. n e g a t i v e r e s i s t a n c e was obtained. The b i p o l a r n e u t r a l models i n g e n e r a l d i d not e x h i b i t n e g a t i v e resistance. Two cases seemed t o show a p o s s i b l e d.c. n e g a t i v e r e s i s t a n c e — t h e case f o r d r i f t w i t h no recombination and the case where the l i f e t i m e i n c r e a s e d w i t h the c a r r i e r d e n s i t y . Por the d i f f u s i o n f l o w a n a l y s e s which d e s c r i b e d the s i t u a t i o n f o r i n t r i n s i c and h i g h l e v e l e x t r i n s i c n e g a t i v e r e s i s t a n c e was The semiconductors, found t o be p o s s i b l e i n s p e c i a l cases. case where the c a r r i e r l i f e t i m e i n c r e a s e d w i t h i n c r e a s i n g c a r r i e r d e n s i t y was I t was the most 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 . found p o s s i b l e to d e f i n e completely a n e g a t i v e resist- ance model when the l i f e t i m e v a r i e d w i t h c a r r i e r d e n s i t y . T he case f o r p u r e l y d i f f u s i v e f l o w w i t h no d r i f t c u r r e n t s was 98 f o u n d t o be v a l i d low only a t very low i n j e c t i o n levels and v e r y fields. Several thesis. problems r e s u l t from the analyses of this Among t h e t h e o r e t i c a l p r o b l e m s a r e t h e many where t h e e q u a t i o n s are intractable. instances 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 t h e impedance f o r c e r t a i n r a n g e s of frequencies. should Space charge be i n v e s t i g a t e d . e f f e c t s i n t h e presence o f t r a p s The e x t e n s i o n of avalanche f o r a one-dimensional c o n f i g u r a t i o n should r e l a t i o n between t h e c u r r e n t in determining investigated negative by assuming s p e c i f i c passing lifetime ing the l i f e t i m e should along should be models f o r c o n t a c t s . t h e r e l a t i o n between J and A through i t . on t h e c a r r i e r important This relation m e a s u r e d t o d e t e r m i n e how t h e c o n t a c t current The d e n s i t y J and t h e impressed resistance. Experimentally, be be p u r s u e d . d e n s i t y a t t h e c o n t a c t A P Q was f o u n d t o b e carrier injection P Q should behaves w i t h t h e The d e p e n d e n c e o f t h e c a r r i e r density should the filament. be done u n d e r i s o t h e r m a l be c h e c k e d by measurHowever, s u c h a n d known s u r f a c e experiments conditions. 99 BIBLIOGRAPHY G a r t n e r , W.W. 1961. and S c h u l l e r , M. G i b s o n , A.P. and Morgan, J.R. 1957. Gunn, J . B . P r o c . I.R.E. 754. 1960. S o l i d S t a t e E l e c t r o n . 1, 54. P r o g r e s s i n S e m i c o n d u c t o r s . V o l . 2, p. 2 1 3 » Heywood and Company, L t d . , L o n d o n . Harrick, K.J. 1959. P h y s . Rev. 11£, H e n i s c h , H.K. 1957. Rectifying Semiconductor Contacts, 876. Oxford, Clarendon Press. Lampert, M.A. 1959. R.C.A. Rev. Lampert, M.A. 1962. P h y s . Rev. Lampert, M.A. L e b l o n d , A. Melngailis, and R o s e , A. 1957. Ann. 20, 682. 12£, 126. 1961. P h y s . Rev. 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B e l l System Tech. J . 22* 9 9 . 7 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. 87, 835. Shockley, W. S o v i e t Phys.-JETP 2» Stafeev, V.I. 1958. Stafeev, V . I . 1 9 5 9 . S o v i e t P h y s . - S o l i d S t a t e 1, Stockman, P. 1502. 763, 769. 1 9 5 6 . Photoconductivity (Atlantic City Conference 1 9 5 4 ) , Wiley, New York, p. 269.
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Theory of conductivity modulation in semiconductors Nishi, Ronald Yutaka 1962
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Title | Theory of conductivity modulation in semiconductors |
Creator |
Nishi, Ronald Yutaka |
Publisher | University of British Columbia |
Date Issued | 1962 |
Description | The theory of conductivity modulation in semiconductors and the conditions under which negative resistance can be obtained are investigated. The ambipolar transport equation is derived for one-dimensional flow in a two-terminal homogeneous semiconductor with no trapping and no temperature gradients. Charge neutrality is assumed in the majority of the models studied. A phenomenological model is considered first to show how conductivity modulation can lead to negative resistance. Since the general problem of carrier transport with drift and diffusion currents is difficult, the models investigated are mainly concerned with either drift or diffusion as the dominant transport mechanism. For a unipolar space-charge limited drift model, negative resistance in frequency bands is found. For bipolar, neutral drift models, negative resistance is 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 drift model gives current-voltage characteristics of the form: J α V at low injection levels, J α V² at high injection levels, and J α V³ at very high injection levels. Space charge is important in the cube law case. Models ignoring diffusion are shown to be valid only for extrinsic semiconductors. 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. |
Subject |
Semiconductors |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085897 |
URI | http://hdl.handle.net/2429/39626 |
Degree |
Master of Applied Science - MASc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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