UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Theory of conductivity modulation in semiconductors Nishi, Ronald Yutaka 1962

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1962_A7 N4 T4.pdf [ 5.42MB ]
Metadata
JSON: 831-1.0085897.json
JSON-LD: 831-1.0085897-ld.json
RDF/XML (Pretty): 831-1.0085897-rdf.xml
RDF/JSON: 831-1.0085897-rdf.json
Turtle: 831-1.0085897-turtle.txt
N-Triples: 831-1.0085897-rdf-ntriples.txt
Original Record: 831-1.0085897-source.json
Full Text
831-1.0085897-fulltext.txt
Citation
831-1.0085897.ris

Full Text

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.  Radioelect.  I . and R e d i k e r , R.H.  121.  26.  12,, 95.  1962.  J . A p p l . P h y s . j53_, 1892. P a r m e n t e r , R.H.  1959.  and R u p p e l , W.  J . A p p l . P h y s . ;50,  R e d i k e r , R.H.  and McWhorter,  A.L.  1959.  R e d i k e r , R.H,  and McWhorter,  A.L.  1961.  Solid Rittner,  E.S.  1954.  P r o c . I.R.E.  State Electron.  P h y s . Rev.  2,  41,  100.  1161.  Rittner,  E.S. ' 1956. Photoconductivity (Atlantic City C o n f e r e n c e 1 9 5 4 ) , W i l e y , New Y o r k , p.215. v a n R o o s b r o e c k , W. 1950. B e l l S y s t e m T e c h . J . 29_, 560. v a n R o o s b r o e c k , W.  1953.  P h y s . Rev.  91,  v a n R o o s b r o e c k , W.  1961.  P h y s . Rev.  123,  282. 474.  S h o c k l e y , W.  1949.  S h o c k l e y , W.  1950. Electronsand Holes i n Semiconductors, D. Van N o s t r a n d Co. I n c . , P r i n c e t o n , N . J .  Bell  S y s t e m T e c h . J . 28,  1548.  435.  1207.  Shockley, W.  1 9 5 1 . B e l l System Tech. J . 29_» 9 9 0 .  Shockley, W.  1 9 5 4 . B e l l System Tech. J . 22* 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.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085897/manifest

Comment

Related Items