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A theoretical study of magnetic self-pinching in a semiconductor Strome, William Murray 1962

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A THEORETICAL' STUDY SELF-PINCHING  IN A  OF  MAGNETIC  SEMICONDUCTOR  by W I L L I A M MURRAY B.Sc,  A THESIS  University  SUBMITTED  of  in  Alberta,  IN PARTIAL  THE R E Q U I R E M E N T S MASTER  STROME  F U L F I L M E N T OF  FOR T H E • D E G R E E  OF A P P L I E D  the  1960  OF  SCIENCE  Department of  PHYSICS  We a c c e p t required  this  thesis  as  conforming  standard  THE U N I V E R S I T Y  OF B R I T I S H  January,  1962  COLUMBIA  to  the  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study,  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  Physics  The University of British Columbia, Vancouver 8, Canada. Date  J a n u a r y 31, 1962  ABSTRACT  The  subject  of  this  thesis  investigation  of  the  possibility  self-pinching  of  the  hole-electron  semiconductor.  The  derived  for  in  a plasma,  such  generation  and  claim  of  are  reports  of  the  pinching.  It  is  mental that  results  the  above  of  present  pinch equations  the  operated  discussed.  resistance  is  magnetic  be  expected the  in  a  are  to  prevail  effects  of  recombination.  observing  avalanching  theoretical  observing  b o t h w i t h and w i t h o u t  indium antimonide  conditions  of  steady-state  the  plasma  conditions which might  Recent in  of  is  observance  under  sample, indicated  avalanche  A l l these  p i n c h on s l i g h t a purely that  reports  changes  reports  offer  pinch  based  effect  only  compare  Hence  their  overall  a plasma generated  theory.  effect  breakdown  in  secondary  a. p o o r m e d i u m f r o m w h i c h t o with available  of  it  is  circumstantial  of  self-*  by expericoncluded evidence  self-pinching. Finally,  an e x p e r i m e n t a l  w i t h w h i c h one  should  whether  p i n c h can occur  or  not  be a b l e  arrangement  to determine in  is  suggested  unambiguously  semiconductors.  - iii -  TABLE OF CONTENTS  Page 1.  INTRODUCTION  2.  PINCH EQUATIONS 2.1  Model  3  2.2  C y l i n d r i c a l Geometry  6  2.3  S l a b Geometry  8  2.4  Comparison Between the C y l i n d e r and the Slab  2.5  4.  11  E s t i m a t i o n o f Pinch Time i n C y l i n d r i c a l Plasma  5.  1  11  GENERATION AND RECOMBINATION 3.1  Statement of the Problem  14  3.2  Development o f the Equations  14  3.3  C o n d i t i o n s whereby G & R may be Neglected  17  3.4  Numerical S o l u t i o n  18  3.5  Surface Recombination  19  3.6  Recombination Time  20  THERMAL EFFECTS 4.1  Assumptions  22  4.2  C y l i n d r i c a l Geometry  23  4.3  Slab Geometry  24  4.4  Comparison of the Two Geometries  26  -  iv -  Page 5.  6.  PUBLISHED EXPERIMENTAL DATA 5.1  General Review  28  5.2  Critical  31  Summary  EXPERIMENTAL CONSIDERATIONS  7.  of P i n c h  35  6.1  Observable C h a r a c t e r i s t i c  6.£  Radio-Frequency Measurements  36  6.3  Probe Techniques  38  6.4  Infrared  38  6.5  G e n e r a t i o n of a H o l e - E l e c t r o n  6.6  Geometry and M a t e r i a l  41  6.7  Summary  46  Absorption Plasma  40  47  CONCLUSIONS  APPENDIX Some P r o p e r t i e s of Three I n t r i n s i c Semiconducting M a t e r i a l s BIBLIOGRAPHY  49 52  LIST OF FIGURES 2- 1. 3- 1.  O r i e n t a t i o n o f c o o r d i n a t e system w i t h r e s p e c t t o sample C a r r i e r d i s t r i b u t i o n f o r T=0.5 v a r i o u s values of w / u .  (to f o l l o w page 19)-1  C a r r i e r d i s t r i b u t i o n f o r P*1.5 various values o f w / t _ .  (to f o l l o w page 19)-2  0 (  3-2.  9  P|  3-3... C a r r i e r d i s t r i b u t i o n forp--5.0 various values of w / u .  (to f o l l o w  3-4a. Average absolute d e v i a t i o n of *t from u n i t y as a f u n c t i o n of w/lt>jfor three values of r .  (to f o l l o w page 19)-4  P i  page 19)-3  -  LIST  OF F I G U R E S  V  (continued) Page  3-4b.  6-1.  Normalised average absolute d e v i a t i o n o f *t f r o m u n i t y a s a f u n c t i o n o f w/l©i f o r three values of r . E x p e r i m e n t a l arrangement for measuring the carrier d i s t r i b u t i o n by i n f r a r e d absorption.  (to  follow  page  19)-4  - vi -  ACKNOWLEDGEMENTS  I should  like  to express my g r a t i t u d e t o  P r o f e s s o r R. E . Burgess, my t h e s i s s u p e r v i s o r , f o r h i s i n v a l u a b l e guidance and a s s i s t a n c e d u r i n g  the p r e p a r a t i o n  of t h i s work. I am a l s o indebted  t o the N a t i o n a l Research  C o u n c i l of Canada f o r the f i n a n c i a l a s s i s t a n c e g i v e n me through a Bursary and a  Studentship.  1.  INTRODUCTION For a number of y e a r s , s e l f - p i n c h i n g (or magnetic  s e l f - f o c u s i n g ) o f the c u r r e n t c a r r i e r s i n a gaseous plasma has  been d i s c u s s e d i n the l i t e r a t u r e .  During  the past few  years some i n t e r e s t has a r i s e n i n the p o s s i b i l i t y o f o b s e r v i n g a similar effect semiconductor.  i n the h o l e - e l e c t r o n plasma present  T h i s t h e s i s i s concerned w i t h a t h e o r e t i c a l  i n v e s t i g a t i o n of this  possibility.  In chapters 2 and 3 we develop s t a t e magnetic p i n c h .  (G & R) to develop  the pinch  f o r a bounded plasma under c o n d i t i o n s expected i n  a semiconductor. those  the theory o f steady-  We f i r s t n e g l e c t the e f f e c t s of  g e n e r a t i o n and recombination equations  ina  T h i s a n a l y s i s leads to r e s u l t s s i m i l a r to  o r i g i n a l l y d e r i v e d by Bennett  gaseous plasma. cylindrical.  (1934) f o r an unbounded  We c o n s i d e r two sample geometries:  We a l s o g i v e an estimate  f o r p i n c h to develop.  s l a b and  of the time r e q u i r e d  We then c o n s i d e r the e f f e c t s of G & R  i n the slab geometry. The  r e s u l t s of n u m e r i c a l  s o l u t i o n of the r e s u l t i n g  d i f f e r e n t i a l equation show t h a t G & R has a strong e f f e c t on the p i n c h , as would be  inhibitory  expected.  In any pinch experiment one must be c a r e f u l t o e l i m i n a t e , or a t l e a s t account f o r thermal  effects.  chapter 4 i s devoted t o the i n v e s t i g a t i o n of these  Thus effects i n  s l a b and c y l i n d r i c a l shaped semiconductor  samples.  Very r e c e n t l y , a number of workers and S t e e l e , 1959; et a l , 1961;  Glicksman and Powlus, 1961;  and Chynoweth and Murray, 1961)  Ancker-Johnson have r e p o r t e d  o b s e r v a t i o n s of phenomena i n indium antimonide under avalanche  (Glicksman  operated  c o n d i t i o n s which they have a s c r i b e d to p i n c h .  T h e i r measurements i n v o l v e d s l i g h t of the sample d u r i n g " p i n c h " .  changes i n the r e s i s t a n c e  In chapter 5 we  discuss the,  r e s u l t s of the above-mentioned workers and p o i n t out  that  t h e i r evidence f o r p i n c h i s o n l y c i r c u m s t a n t i a l , and  that  avalanching i s a poor method of producing a plasma f o r r i g o r ous  comparison  o f p i n c h theory t o experimental  F i n a l l y , chapter 6 i s devoted  results.  to i n v e s t i g a t i o n  o f p o s s i b l e methods o f d e t e r m i n i n g unambiguously whether or not pinch occurs i n a semiconductor. we  From a l l c o n s i d e r a t i o n s  show that pinch should be d e t e c t a b l e by probe measurements  of the c o n d u c t i v i t y d i s t r i b u t i o n i n a pure germanium s l a b a t about  room  temperature.  3.  ,  2.  PINCH EQUATIONS 2.1  Model F o r the s t a r t i n g p o i n t i n the t h e o r e t i c a l d i s c u s -  s i o n of pinch e f f e c t i n semiconductors s i m p l i f i e d model.  We  we  n=p = n;  assume that  take a r a t h e r i n the u n d i s -  turbed s t a t e where r» und p are the e l e c t r o n and c o n c e n t r a t i o n s r e s p e c t i v e l y , and concentration.  I n i t i a l l y we  n  x  hole  i s the i n i t i a l  w i l l c o n s i d e r only a  carrier distribu-  t i o n of the c a r r i e r s which i s i n dynamic e q u i l i b r i u m without •< any  regard as to how  stability. for  t h i s d i s t r i b u t i o n a r i s e s , or to i t s  ( L a t e r we w i l l t r y to estimate  the time r e q u i r e d  p i n c h to develop.) I t w i l l be assumed that the c a r r i e r m o b i l i t i e s  ( yUp and  p  n  f o r h o l e s and  diffusion coefficients are constants  e l e c t r o n s , r e s p e c t i v e l y ) and  ( D  p  and D  n  f o r h o l e s and  the  electrons)  independent of the c a r r i e r c o n c e n t r a t i o n .  . N e g l e c t i n g g e n e r a t i o n and  recombination  of h o l e - e l e c t r o n  p a i r s , there w i l l be no net c a r r i e r v e l o c i t y i n the t r a n s verse  (or r a d i a l )  directions.  Although  initially  the hole and  t r a t i o n s are both equal to the i n i t i a l we  admit the p o s s i b i l i t y  l o c a l h o l e and other  e l e c t r o n concen-  c a r r i e r concentration,  that under p i n c h c o n d i t i o n s , the  e l e c t r o n c o n c e n t r a t i o n s may  d i f f e r from each  slightly. There are a number of ways i n which the  "steady-  state" pinch equations may be obtained. Bennett (1934) ": originally obtained one form of these equations by considering the r e l a t i v i s t i c inter-particle forces in two oppositely directed streams of oppositely charged particles. In this section, we shall derive the pinch equations through:  J where  = c^ {p  P  (E + v x g ) p  Hp  Jp,Jn Vp,^  n  -  a r e  t h  ©  Q o l e  a n d  -  ,  (Op/^^pJ  f  J  (2- 1)  electron current densities  are the hole and electron velocities i s the electronic charge  B  is the jaagnetic field  and! is the electric f i e l d . We obtain the pinch relations by assuming that the net transverse (or radial) components (denoted by the subscript  j.  *  n  the general case) of current density due  to each type of carrier vanishes:  we form the combination of these vanishing current densities:  and then from (£- 1) we get  (n-p)^ t L ( n ^ - p v - ) . B ] P  1  t V  x  ( f i j n i - £*p) = 0  (2-  2)  5.  shown t h a t u n d e r p i n c h c o n d i t i o n s , £ ^ , ( 0 ^ * 8 ) ^  Now  i t c a n be  and  (0^/f/p)Vx ft (where p may  all  of the  same o r d e r  be r e a d  of magnitude.  may  neglect  the f i r s t  Since  (2-  term o f  v  and  n  2).  p)  n or  either  ^p  are  are  (i.e. n 2 p  ln-pl«n-vp  o p p o s i t e l y d i r e c t e d , then provided we  as  )  Approximately:  3)  (2-  It tion at t h i s  i s customary to i n t r o d u c e  define  tion It  r  the h o l e  must be u s e d w i t h  these d i s t r i b u t i o n s  lattice  temperature,  may  carrier distributions  may  and  be  quite different  from each o t h e r .  that appropriate t o be  perpendicular  analysis,  c o e f f i c i e n t s would  (assumed  be  both  (2-  into  rela-  c o n d i t i o n s where and  the the  "carrier to a  two  temdimen-  approximately  t o the d r i f t  the c a r r i e r m o b i l i t i e s  likely  >  4  Maxwellian.  from  The  ~  This  are  be f a r f r o m M a x w e l l i a n ,  i n the p l a n e  concentration  (2  or e l e c t r o n "temperature".  distribution  I n a more e x a c t  l\  4  be c o n s i d e r e d  sional velocity Maxwellian)  P/ f3  c a u t i o n under h i g h f i e l d  temperatures"  perature" w i l l  sion  k T  i s r i g o r o u s when t h e  "carrier  rela-  point °(3  to  the E i n s t e i n  f i e l d and  and  velocity. the  diffu-  carrier  dependent.  Substituting  <^n [( v - V p ) x S ] B  x  4)  * k(T  n  (2-  3)  we  arrive  -hTp) V ^ n = 0  at  (2- 5)  which we use together with the Maxwell relation:  V x B = p ( J + J , J a /J^r\  (2- 6}  V )  n  n  (where /J is the permeability of the sample) to obtain the pinch equations for the two geometries (cylindrical and slab) in which we are interested. We use the auxiliary condition that  S  carriers are conserved under pinch, i.e. that  n  <JS -  c«»n»*.  *e<*>o« J  2 . 2 Cylindrical Geometry  By assuming cylindrical geometry we obtain the following differential equation for the carrier distribution from equations ( 2 -  (where  v  and  n  vj, are the magnitudes of the respective  carrier d r i f t velocities).  w  h  e  r  e  and n  0  j  5) and ( 2 - 6).  n(r)  =  k> =  n  A solution of this equation i s :  n 0  Q  •  11 + b r J  (2-e)  (v +v,t/*HT +T ) n  n  (2-9)  P  • the carrier density along the axis is found by  invoking conservation of carriers:  ir a n 2  t  r  J n(r)  Znrdr  0  where a  is the radius of the sample.  *  P  where length.  =  Pi  N- TT a* ri£  I " 1  This yields  8*Nk<T i-T l[ n  t>  (2  "  10)  , the number of oarriers of one kind per unit  In Bennett's (1934) original development of the pinch equations, he considered only the case of an unbounded plasma. Ic*  In this situation, one can obtain a c r i t i c a l current,  » which is necessary to maintain the distribution given  by (2-8) appropriate to an unbounded plasma for which n  and  0  b  are undetermined.  I'c.  L 8TT IMk ( T ±Tf»)//j]*'*  =  (2-n)  n  The criterion for self-pinching most often quoted in the literature is that the current, I , must exceed the value of given in (2-11).  However i f we substitute this value into  (2-10) we obtain an infinite value of n  indicating that a l l  0  of the carriers are concentrated on the axis at that current. Since this cannot happen physically, our solution must break d own as  I -* l'  c%  .  For the purpose of our investigation, let us define arbitrarily a c r i t i c a l current, I  t R  , as one whioh w i l l  reduce the number of carriers at the outer surface to one half the i n i t i a l value, m, (or double the concentration on the axis). Putting  r : et  in (2- 8) and substituting the values of r\ and a  k> from (2- 9) and (2-10) we find that from the above definition, Ie* If we define  "k(T +l ,)/sjl"  r  n  T = (I/l «) c  1  t  (2-12)  , a measure of the strength of the  pinch, we may write some of the previous results in terms of it:  (over)  (i-r/ )Ll-  m  t  i r 11  -z -  ('/«)*}]  -1  b "  Li- ir]  ni  r/a  S i n c e c a r r i e r s are conserved  and we assume the  m o b i l i t i e s t o remain constant, the r e s i s t a n c e of the sample i s g i v e n by R  where X  -  /H  X  %  (yu  n  +  yu )  ( 2  p  i s the l e n g t h o f the sample.  From (2-12) and  o b t a i n expressions f o r the l o n g i t u d i n a l e l e c t r i c f i e l d  -  i l 3 )  (2-13) we and the  power d i s s i p a t i o n i n ,the sample:  *  '/2  4  TT k  ( T + Tp) ^ n  (2-14) y  /*> * ( /*« + HP ) 2.3 S l a b  Geometry We w i l l choose a c o o r d i n a t e system o r i e n t e d as  shown i n F i g u r e 2-1  w i t h r e s p e c t to the sample.  (Over)  -dtZ  F i g u r e 2-1. Orientation of coordinate system with r e s p e c t t o sample. I f we  assume that the thicJsness  l e s s than the width  w  , and  ol  i s much  that t r a n s v e r s e components of  current d e n s i t y v a n i s h as b e f o r e , we  need only c o n s i d e r  x -components of the terms i n equations (2- 5) and  (2-  the 6)  s i n c e the j-components w i l l give a n e g l i g i b l e c o n t r i b u t i o n to the p i n c h .  (Note that t h i s i s true only as long as  m a j o r i t y of the c a r r i e r s are X-  i d/t  .)  e n t i a l equation  i .(  As  outside  i n s e c t i o n 2.2,  f o r the c a r r i e r  the  the region bounded by  we  can o b t a i n a d i f f e r -  distribution:  - L . a u )  +  f"f  n • 0  The solution of  is of the form  (2-15)  where  n  c  =  b  L 2 k ( T -«-Tp) /pq  {v^vr^f ]  n  (2-17)  and b is obtained from the conservation of carriers: ni  r  £  d  J  rux)  4x  which leads to the following transcendental equation in b :  /S  tanh (TE w/t)*  (jlt^VfiNkCT^Tp)}  (2-18)  If we define the c r i t i c a l current as in section n(±w/a.\ = ri£ /2. » then we find that  2.2, i.e. that at which I  = [ 8 (<J/w) N K (T + Tp)  C R  //> ]  n  (2-19)  We can also find the current at which our analysis begins to break down —  that at which about half of the carriers are witMn  the region bounded by  x: ± d/2 , This is at a current approx-  [ 0 • S^w/d j"* I  imately equal to  cjl  .  As In section 2.2, with the resistance of the sample as given by  (2-15),  we can obtain the eleotric field*and  power dissipation in the sample: [ ecd/w)k[T +Tp]  c  P  n  s  where f  1  (I/lea)*  eca/ )k(r i-Tp) w  n  ]  '/t  e  p  (2-20)  11.  2.4  C o m p a r i s o n Between L e t us compare  the c u r r e n t ,  power d i s s i p a t i o n r e q u i r e d arbitrary is  definition  assumed  that  to i n i t i a t e  electrio pinching  of the onset o f pinch  f i e l d , and (by o u r  —rue^e)*^).  It  s a m p l e s o f t h e same m a t e r i a l w i t h t h e same  physical properties s e c t i o n a l area  t h e C y l i n d e r and t h e S l a b  and o f t h e same l e n g t h  a r e t o be c o m p a r e d .  At  f  and  cross-  I = lc*  \  :  1 / 1  (2-21)  d/w  Since  current, lower  i s assumed  electric  field,  Estimation In  radial  than u n i t y , the  and power d i s s i p a t i o n c a n a l l be  i n t h e case o f the s l a b  2.5  the  t o be much l e s s  of Pinch  geometry. Time  i n Cylindrical  Plasma  t h e case o f the c y l i n d e r , c o n s i d e r i n g  components,  equations  ( 2 - 1) w i t h  only  ( 2 - 4) may be  written  (2-22)  The  continuity equations are  V- J .  ^ at at  (2-23)  Let us d e f i n e a new  Our  current  density  reason f o r t h i s p a r t i c u l a r combination of  I n - p l  that i f , as u s u a l ,  <<.  n + |0  J„  r  Jp  and  ( i . e . n cr p  r  is  ) ,  approximately:  The  only component o f  From (2-22) and  j ' of i n t e r e s t  i s the r a d i a l  one.  (2-24)  Let us now  define -  3  J  n  <) r  r*  (2-27)  which i s p r o p o r t i o n a l t o the number moment of i n e r t i a of h o l e - e l e c t r o n plasma.  So  plasma i s c o n t r a c t i n g .  long as  i s negative  From (2-25), (2-27) and  the  that c a r r i e r s are conserved at a l l times which l e a d s X(cO  -0  we  get:  at  1 4  which becomes, from (2-6)  and  (2-26)  the  the condition to  13.  "r^T,'  ~1T^  <  R  t2-28)  <-> 22 9  1- id/ic f  ^ I  ni /]  1  n ( « ) i s given by  Notice that when  where  V  \ e i r N k ( T „ + -r»  R  i s defined in equation (2-12), which i s the same  as the expression given by the steady state analysis.  Thus,  let us suppose that the carrier concentration at the outer edge approaches the value given by (2-29) exponentially from , the value before pinching oommences, i.e. assume  where fp is the "pinching time constant" to be determined. Substituting the above into 2-28) -  -  NkCTn^T^/j  f  - t /  V  ( 2  .  3 0 )  Using the definition of 9 given by equation (2-27), the i n i t i a l carrier distribution ( n = n; steady state distribution (at  ) and the final  t =oo ) given by equation  (2- 8) in one form, we can obtain an expression for f : p  Cp =  [ 4 ( - ^ N T ^ ) - H ) ] 8  (-£&•) * ( w - >  (2-31)  14.  (l /l ) -  where f o r compactness we have put The  elt  p o r t i o n o f (2- 31) i n the square bracket i s  a r a t h e r weak i n c r e a s i n g f u n c t i o n of f r =O  3.  to unity at  GENERATION 3.1  Y*.  ranging from 1/3 a t  Y = 2.  AND RECOMBINATION.  Statement of the Problem To t h i s time we have not considered  g e n e r a t i o n and recombination  (henceforth a b b r e v i a t e d G & R)  w i l l have upon the p i n c h d i s t r i b u t i o n . a p p a r e n t l y been overlooked because the e f f e c t  the e f f e c t  T h i s e f f e c t has a l s o  i n the l i t e r a t u r e —  i s negligible  probably  i n a gaseous plasma, and  most o f the work concerned with p i n c h i n semiconductors i s a d i r e c t a d a p t a t i o n o f the gaseous  theory.  In an i n t r i n s i c semiconductor under low f i e l d c o n d i t i o n s , v i r t u a l l y a l l the c a r r i e r s present r e s u l t thermal g e n e r a t i o n balanced  a g a i n s t recombination.  from  Hence,  q u a l i t a t i v e l y , one can see that any e f f o r t t o d i s t u r b the l o c a l c o n c e n t r a t i o n o f the c a r r i e r s w i l l be s t r o n g l y opposed by thermal G & R. 3.2  Development of the Equations We w i l l concern  ourselves i n this discussion  only with t h e case of s l a b geometry.  The c o o r d i n a t e axes w i l l  again be o r i e n t e d as shown i n F i g u r e 2-1.  I f we d e f i n e j'as i n  equation  (2-24) we may combine equations  (2- 1) to get  j  ( t a k i n g only the x -component s i n c e we a r e a g a i n assuming ol/W «  T' J  1 ) :  -  *  a  P» f  [ /,  f  n  * ;^rr^ [  P  * w \  n ( V n + P , B  k  —  ft  n  /~ , \  — — j ; ]  *  J„  At t h i s p o i n t we do not s e t i n the p r e v i o u s c h a p t e r .  1  ^ T • T>) £ n  x  =  (3  1}  s 0 as  I n s t e a d , we make use of the steady-  s t a t e c o n t i n u i t y equation:  •  where g and/t, are the p a i r g e n e r a t i o n and recombination  rates  respectively. Since approximation  n s p , as u s u a l , we may w r i t e to a good  (Smith, 1959, p. 252): V J  P  = - v - J  n  ( H  M  0  -  L l  (n/ ) J 2  n i  or from the d e f i n i t i o n o f J ,  ff =  1.  K CI  -  ]  where (R, i s the thermal g e n e r a t i o n r a t e per u n i t 0  U s u a l l y a t t h i s p o i n t i n any a n a l y s i s  (3- 2) volume. involving  G & R, i t i s assumed' t h a t  (n-n,l<<  may be l i n e a r i z e d  CI ** ( n / n if 3 becomes approximately  2.11 — n/ru]  .  so that  However doing t h i s  appreciably simplify  r\i  , i n which case  i n our case does n o t  the f i n a l e q u a t i o n .  ( 3 - 2)  16.  From e q u a t i o n s  (2- 6 ) ,  ( 3 - 1) and  (3-  2)  (3-  I t was  not p o s s i b l e  above e q u a t i o n . amenable  to f i n d  an  analytical solution  T h u s , t h e e q u a t i o n was  put  to  3)  the  i n t o a f o r m more  t o n u m e r i c a l s o l u t i o n by means o f t h e f o l l o w i n g  sub-  stitutions :  n /ni >  r = /ip  is  k  diffusion  (Tn -f Ti»)  a form  the u n d i s t u r b e d sample.  of ambipolar  If  length is given  n = p=n{  0  l  Lpj  i s the c a r r i e r  into  i the  lifetime.  (3- 3)  length i n  ambipolar  Dn +  7  ; i n t h e s p e c i a l c a s e where  substitutions  diffusion  by  2 <H where  ( 3 - 4)  T h i s i s t h e same a s T  n  l t becomes  =  Tp  .  M a k i n g the  our above  17.  +M%)l -'  i 3.3  of  *°  -  l]AiUin  l3  C o n d i t i o n s whereby G & R may Before  tion  (i  c o n s i d e r i n g the r e s u l t s  ( 3 - 5 ) , we  the e f f e c t s  be  should  o f G & R may  5)  Neglected of a numerical  solu-  look a t the c o n d i t i o n s under which be c o n s i d e r e d  negligible.  The  condition required i s simply:  it-5? A ' - V ) J I  « 21  (3-  6)  0.<f.< 1 .  for  I f we  assume t h a t  ( 3 - 6) h o l d s ,  2  ( 3 - 5)  becomes  r ^= o  whose s o l u t i o n i s  1 where ^  the o u t e r  boundary  8  must be d e t e r m i n e d  d i s c u s s i o n we at  4'-5-<h (M)  neglect  by b o u n d a r y  the e f f e c t s o f s u r f a c e  edge o f t h e s a m p l e , and  condition to apply  current d e n s i t i e s vanish at leads  conditions.  i s that x-  -  hence  an  In  this  recombination  appropriate  t h e h o l e and e l e c t r o n  w/i  t o the f o l l o w i n g t r a n s c e n d e n t a l  .  This condition  equation  i n 4r  (3-  cos  7)  18.  C o n d i t i o n (3- 6) may  now  be w r i t t e n J/t,  w/L  «  D{  ' (3- 8)  +  4' for a l l ^  SnhlAflsech^fKitfii"^]*  -?  i n the range o f i n t e r e s t . The r i g h t - h a n d e x p r e s s i o n of (3- 8) has a m i n i -  mum  at  ^ = 0 and hence we  the r a t i o  VV/L .  «  0  or  on  :  P  W/U {  can o b t a i n the f o l l o w i n g l i m i t  (3- 9)  1  approximately  «  / L O .  It  - f  (3  + ar)"*  (3-10)  i s not the purpose of t h i s s e c t i o n to c o n s i d e r  the p r a c t i c a l i m p l i c a t i o n s o f these r e s u l t s ; however, we note that they would be very d i f f i c u l t available 3.4  will  to s a t i s f y w i t h most  semiconductors. Numerical S o l u t i o n Equation  (3- 5) was  s o l v e d n u m e r i c a l l y w i t h the  a i d of the Alwac I I I - E d i g i t a l computor f o r the values of t ' 0.5, 1  1.5  and 5.0  and  values of K ^ l ,  5, 10, 20 and  30.  While i t would have been d e s i r a b l e to take more values of these parameters, amount of time.  the c a l c u l a t i o n s r e q u i r e d a c o n s i d e r a b l e A l s o as these parameters became l a r g e r ,  machine could no l o n g e r handle  the  the l a r g e range of numbers  which arose i n the i n t e r m e d i a t e stages o f the  calculation.  The  r e s u l t s of the numerical  p l o t t e d i n F i g u r e s 3-1, against  3-2  and  3-3,  where  f o r v a r i o u s r and v*A. . .  $  c a l c u l a t i o n s are  The  p  ^  i s plotted  inhibitory  effect  of the G & R i s immediately obvious i n a l l cases. In Figure 3-4a  we  the three values of V .  0 = We  T7  D against  have p l o t t e d 0  This  i s d e f i n e d as  follows  Ih(f ' > " l l =i/  6  (3-11)  w i l l use t h i s as a measure o f the s t r e n g t h o f the  s i n c e i t i s the d e v i a t i o n of the curve of \ curve which would r e s u l t ^=  w^for  1.  In F i g u r e 3-4b  Here we  from the  i n the unpinched c o n d i t i o n , namely  we have normalized  the curves  r a t e , a weakly d e c r e a s i n g  as would be expected from Surface We  of  4-32.  f u n c t i o n of jjj ,  (3-10).  Recombination  have considered  only recombination,  the body of our semiconductor to t h i s p o i n t . t h i n samples the recombination g e n e r a l l y much higher than Smith (1959, p. 297 the r a t e  f  see that the r e l a t i v e i n h i b i t i o n of the pinch i s ,  i n i t i a l l y at any  3.5  vs  pinch  within  However, i n  r a t e at the s u r f a c e i s  that i n the body of the m a t e r i a l .  f f . ) shows that to a good approximation  at which h o l e e l e c t r o n p a i r s recombine at  s u r f a c e , per u n i t area,  S.  =  B  5  S« i s g i v e n  n ; '[ O v n ; ) '  the  by  - l ]  < - > 3  1 2  Figure.3-1.  C a r r i e r d i s t r i b u t i o n f o r V-  0.5;  various values of  W/L .. P  Figure  3-3.  Carrier  distribution for v a l u e s o f vw/«- .. 0  T=  5.0;  various  (to  f o l l o w page  19)-4  Average a b s o l u t e d e v i a t i o n of *\ from u n i t y a s a f u n c t i o n o f ^/ - f o r t h r e e v a l u e s o f r . UOi  Figure  3-4b.  N o r m a l i s e d a v e r a g e a b s o l u t e d e v i a t i o n o f 'h f r o m u n i t y a s a f u n c t i o n o f M//U. f o r t h r e e values of n . p  4  in  intrinsic  case  of  the  m a t e r i a l , where slab,  recombination  we  nor  does for  not  the  surface  (or the  obvious  0t.  account  the generation edge  term,  =  A./ This  that  can take  i n t o account  recombination rate  rate  surface the  ZB  will  increased  of  a of  expect  ^ to  3.6  rise  time,  we a s s u m e  would  take  whose  bulk  (3-13) the  edge  of  the  sample.  considerably at  be r a i s e d  be,  for  that  towards  expect  the  most  near  f=l  towards the  Since the is unity.  distribution  part, one  slab,  as  given  should  unity.  R e c o m b i n a t i o n Time For  to u n i t y .  sharply  at  one w o u l d  except  surface  to  cylindrical  will  effects,  and -3  it  the  the  cylinder), qualitatively it  i n the x - d i r e c t i o n t o -2  p o r t i o n of  the  z  in a  be  In  n ; /«|  S  effects  of  3-1,  major  generations  with  Figures  a constant.  , changing  0  Thus, ^  the  (R.  for  is  s  m e r e l y by m o d i f y i n g  +  value  surface  B  the that  i n the  purpose  of  initially  absence  of  e s t i m a t i n g the \  is  equal  G & R (^)  to  and  N e g l e c t i n g d i f f u s i o n and magnetic  solution  recombination the  value  eventually  it falls  effects:  is  0  lg  cosh  ( 2ft,t/ni)  +  7\, s i n h ( * & * t / n ; ) +  *inn U * . t /  n <  -)  cosh U f c * t / n i )  by  21  T h i s y i e l d s an e x p o n e n t i a l decay only near the c r o s s - o v e r p o i n t where fall  to  However, the time r e q u i r e d f o r I V l  ^=1.  J ^ . - l I /e ranges from 0.45 tT; f o r  to  5 to 1.53  for  \ - 0, where *«  ~  i s the c a r r i e r l i f e t i m e i n the undisturbed  sample.  £;  i s a reasonable  s  m  Thus,  estimate  of the recombina-  t i o n time, * r . I f we  should f i n d  that  of magnitude o r s m a l l e r than  t,  t  r  i s of the same order  the p i n c h time, we  p  may-  assume t h a t the pinch b u i l d s up to the degree suggested  by  the above n u m e r i c a l a n a l y s i s somewhat more slowly than as i n d i c a t e d by ?  ?  (d  U Q  to the r e t a r d i n g e f f e c t o f the G &  On the o t h e r hand, i f  t « V p  ,  r  w e  could expect  b u i l d up to about the extent suggested n e g l e c t i n g G & R i n a time Then a f t e r a t o t a l time  f  r  f  p  R).  the p i n c h to  by the a n a l y s i s  , overshooting i t s f i n a l  l t would f a l l  value.  to the f i n a l s t a t e  i n d i c a t e d by the numerical a n a l y s i s of t h i s  chapter.  4.  THERMAL EFFECTS 4.1  Assumptions The  fold:  Thermal  (or r a t h e r  purpose  i n discussing  pinching gives  rise  t h e r m a l e f f e c t s i s two-  to a carrier  c o n d u c t i v i t y d i s t r i b u t i o n ) o f p r e c i s e l y t h e same  f o r m a s m a g n e t i c s e l f - p i n c h i n g ; and n e g a t i v e to thermal e f f e c t s could which  could  conceivably  6.  =  semiconductors, the conductivity i s  (4_ i )  £ i s the c o n d u c t i v i t y a t temperature t e m p e r a t u r e X,  of the semiconductor. lei  we may  where  G =  £  3  ft  and Eg i s t h e e n e r g y  I n t h e s p e c i a l case where  = I(T-TO)I  «  To  6  a  e  ( 4 - 2)  /ZWT/  For  t h e m a j o r p o r t i o n o f t h i s d i s c u s s i o n , we  be c o n c e r n e d w i t h heat t r a n s p o r t Vd Z  where K  T , £ t h e conduc-  write  6 -  the  pinch.  e  t i v i t y a t the ambient gap  to i n s t a b i l i t i e s  t o a good a p p r o x i m a t i o n , b y : 6  where  r e s i s t a n c e due  give r i s e  mask t h e p r e s e n c e o f m a g n e t i c In i n t r i n s i c  given,  distribution  the steady state  conditions,  i n which  will case  equation i s : +  6  E. /K Z  = O  i s t h e t h e r m a l c o n d u c t i v i t y o f t h e sample  ( 4 - 3) and  Bis  83.  the a p p l i e d e l e c t r i c (4-  3) we  Substituting  ( 4 - 2)  into  obtain Vb  + (*.E /K)  l  As the o u t e r ambient  field.  e  2  G e  = 0  ( - 4) 4  a boundary c o n d i t i o n , i t w i l l  be assumed  s u r f a c e o f t h e sample i s m a i n t a i n e d  temperature  T<>>  suggesting  that  at the  vigorous cooling  o f the  sample. 4.2  Cylindrical  Geometry  In  of a c y l i n d e r , e q u a t i o n  t h e case  ( 4 - 4) may be  written  assuming p u r e l y r a d i a l heat With the boundary the d o u b l e - v a l u e d  c o n d i t i o n quoted  i n s e c t i o n 4.1,  t o ( 4 - 5) i s  - I In ^([l * (l - */Z )*'*] + [l ± (l-JytV*] [r/«f)j  GM where The  solution  flow.  ^ -z  upper sign  that  4  F T  4*E*G/K  i s taken  at the turn-over  lower  sign f o r values  Thus, from  ( 4 - 6)  (4-7)  i n ( 4 - 6) f o r v a l u e s o f c u r r e n t above,  field  (where  A = N  M ( M  o f c u r r e n t below t h i s  . = Z.O ) a n d t h e value.  ( 4 - 6 ) and ( 4 - 2 ) , -Z  6(r) =  4^{[l7  0-^)' J [l±(lW ]L^f} /Z  +  w h i c h , as c a n be s e e n by c o m p a r i s o n w i t h t h e same f o r m pinch.  l  equation  o f c o n d u c t i v i t y d i s t r i b u t i o n a s does  ( 4  _  8 )  ( 2 - 8) g i v e s magnetic  24.  Negative the  turn-over  which  y  t h e power  the e l e c t r i c  =  L  2«a  r  , y  ''  is at  also  This  condition w i l l  give  condition  field  reach  (4-10)  o r exceed  C 4.3  Slab  l K  in Figure  section  2-1.  a slab  takes  ness  of  under  which  to a d i s t r i b u t i o n of  expected  i n magnetic  the current,  power  pinch.  and  electric  j" * 4  11  oriented  as  * 0-55 P / *  0-91  P/''  place  i n the  the sample).  i n which  the axes  the inequality  2 . 3 , we m a y s a f e l y  flow  the conditions  values  (fhcrrn-l)  If  )  value  Geometry Consider  shown  rise  i s met i f  (*K."«lJ  E'/  9  (4-10)  to that  the  4  ) " *  consider Z 60  comparable  V"  i t s maximum p o s s i b l e  =  6(0)  conductivity  exceeds  * « i K / 6  field  We s h o u l d  this  the current  ( 2 6. K / S ) " '  £.'/'• - d ( 2 K  since  i f  is  P and  occurs  value, i ;  at  resistance  assume  that  5-direction  In this  case,  d « nearly  are w  holds a l l of  ( i . e . through ( 4 - 4 ) may b e  as the  the  in heat  thick-  written  25.  The s o l u t i o n o f the  (4-12)  temperature  9C3) where  T  with  the surfaces  at  ^~ * ~z  held at  i s g i v e n by  =  f i n  is given  '  (4-13)  X  s e c h (T )  by the s o l u t i o n o f t h e t r a n s c e n d e n t a l  equation  T Thus  from  sech(D  ( 4 - 2) a n d  as  again  gives  does magnetic  6.  t h e same pinch.  Negative the  turn-over  as  A F E  T  the e l e c t r i c  In ing most  takes  place  pronounced  conductivity are  e  c  h  *  /  (4-14)  2  (  { 4  -  1 5 )  form of conductivity d i s t r i b u t i o n  occurs  (2-21)) i f the current  exceeds  before:  5-*2w(2 6 K/G ) '  Z  0  field  the case  s  8  X K/G-  i s a t i t s maximum  of the slab,  i n the thickness,  (4-16)  value,  since  the thermal  and magnetic  pinch-  pinching  is  i n t h e w i d t h d i r e c t i o n , as f a r as the measured  d i s t r i b u t i o n i s concerned,  orthogonal  other.  G / 8 K ) '  is P "  and  s  resistance  value,  w h i c h t h e power  2  ( c . f . equation  * at  E C d  (4-13),  e ^ ) which  =  t h e two forms  and should n o t a p p r e c i a b l y  interfere  of pinch  with  each  26.  4.4  Comparison In o r d e r  be  compatible  o f t h e Two  that  with  t h e comparison  pinch  ( s e c t i o n 2.4),  cross  s e c t i o n a l area  between t h e t u r n - o v e r  P  ck  4.5  and t h e r m a l  effect tion  f o r the s l a b  (+h«rm.l)  T  '-35  =  (a/w)  = 0-°>l  (+h^m«|)  ( <s|/w)  (<A/a)  =  i n a C y l i n d r i c a l Magnetic consider  the interaction  pinch i n d e t a i l ,  the equations  i f the temperature  \ (4-17)  ,  Pinch between  t h e mag-  b u t t o do s o e x a c t l y  involved.  We c a n l o o k a t t h e  m a g n e t i c p i n c h w o u l d have on t h e t e m p e r a t u r e  negligible. the  (*.«H.O  W r ^ l )  Heating  complicates  u w  , *Ub  %  We s h o u l d netic  The a p p r o p r i a t e comparisons a r e  c u r r e n t , e t c . f o r the c y l i n d e r :  (the*-**)) / P  C +  a g a i n assume t h a t t h e  c u r r e n t , power and f i e l d  /I;  n<V-  we w i l l  o f mag-  o f t h e s a m p l e , and i t s l e n g t h a r e k e p t  t h e same i n e a c h c a s e .  the c r i t i c a l  between t h e c y l i n d e r  t h e comparison g i v e n i n the case  netic  and  Geometries  distribu-  dependence o f the conductance i s  L e t u s assume t h a t , i n t h e c a s e  of the c y l i n d e r ,  c o n d u c t i v i t y i s g i v e n by 6(f)  ^  6;  (l +  baM  /(l +  f r o m s e c t i o n 2.2 where 6i i s t h e i n i t i a l  br ) 1  1  c o n d u c t i v i t y and  6.  is  as  defined  in section  which y i e l d s  upon  w h i c h has section  same  4.2.  pinching, be  (^i  the  a.  I + too* (  TiT  e  Thus,  by  \  2  radial  we w o u l d  enhanced  Equation  (4-  2)  now  becomes  integration  /  0(r) =  2.2.  I * ba*  as  considering  the  j  [T+b?J  dependence  without expect  f  l n  that  found  increased  conductivity  in magnetic  distribution  to  factor  r i±^1 1  for  moderate  thermal  or  large  and m a g n e t i c  b . pinch  Thus to  for  be  the  interaction  of  negligible,  g k V K i.e.  E  «  Of us  a warning  pinch  E  T  of  course, to  look  investigation,  enhance  each  other.  (4-16) equation  the  above  carefully since  (4-  9)  analysis for  magnetic  only  thermal and  serves  effects  thermal  in  pinch  to  give  any  5.  PUBLISHED EXPERIMENTAL DATA 5.1  G e n e r a l Review The  f i r s t r e p o r t c l a i m i n g observance  of magnetic  s e l f - p i n c h i n g i n a semiconductor was by Glicksman and S t e e l e (1959).  T h i s was based upon a comparison  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 o f an n-type indium antlmonide sample i n the presence of a l o n g i t u d i n a l magnetic varying strength. magnetic smaller  field,  of a  slightly  ( i n the avalanche breakdown region) than w i t h a  longitudinal f i e l d present. which  f i e l d of  I t was observed that i n the absence  the s l o p e of the I-V curve was  (InSb)  Since the v a l u e of c u r r e n t a t  the curve without a magnetic  w i t h the f i e l d  f i e l d departs from that  Is a not unreasonable estimate o f I  CR  (given  i n s e c t i o n 2.2) and s i n c e i t i s w e l l known that a l o n g i t u d i n a l magnetic the  field  tends t o i n h i b i t  p i n c h i n a gaseous  observed phenomenon i s a s c r i b e d  to pinch.  plasma,  The reason  g i v e n f o r the i n c r e a s e d r e s i s t a n c e under p i n c h c o n d i t i o n s i s t h a t the c u r r e n t c a r r i e r s are compressed  into a smaller  c r o s s - s e c t i o n than the g e o m e t r i c a l one, "...hence the  apparent  increasing  resistivity".  In a more recent paper  (Glicksman,.and Powlus, 1961)  f u r t h e r attempts are made t o c o r r o b o r a t e the observance of p i n c h i n n-type InSb. of  In t h i s c a s e , the temporal behaviour  the v o l t a g e a c r o s s the sample i s observed when a constant  c u r r e n t pulse i s passed through i t . electric field  I t i s assumed that the  throughout the e n t i r e l e n g t h o f the sample  29.  remains across  homogeneous, the  sample  is  divided the  rise  very  to  electric  field  to  a  greater  (after  about  to  about  denoted  by  t ,  at  which  until  it  the  of  The The  initial  down d i d  voltage  not  to  occur.  time  was  ascribed  plasma the  of  compared  to  The  1  rose  from very  to  a  an  interval  very  of  slightly  slight  to  the  were  interpreted result  to  the  value  and  the  slight  establishment  of  as  rise  a  ( w h i c h was the  defined  electrons  theoretical  =  ( I  a l ,  1961)  of  the  plasma",  which would  reasonable  the  total  originally  a  breakdown. the  current  present)  less  and  was  ( 5 - 1)  / A where  velocity  f o r m as  as  after  relation:  drift  same  was  to  magnetic  against  2  break-  required  plotted  of  follows:  i f  t  the  value  fluctuations,  reciprocal  a  time, to a  The  to  an  corres-  by  due  to  voltage  value  which would  condition,  electron  son y i e l d e d  m." ).  After  d r o p was  the  of  .  value  is  is  (corresponding  created  Uncker-Johnson et  R  was  plasma  0  C  sample  hole-electron  '/V, ,K  I  the  the  current  current  a  The  breakdown  pinching  V.  4  observations  was  the  g  voltage  pulse.  sustain t  across  again  aside  above  rise  the  -1 V.m.  the  to  length.  0.04 microsecond)  remained,  end  equal  high value  than 3 x l 0  2x10  the  2  its  voltage  4 ponding  simply  by  Initially observed  dropped  and  be  and  p,  is  /v^ p  1  "the  in section  for p  p  and  r  p  2.2.  k (r  p  where  radial  approximatelyZp  given  values  A = lira  n  in  mobility  this The  ^  case.  compari-  F u r t h e r work done at R.C.A. L a b o r a t o r i e s Johnson, et a l , 1961)  involved v i r t u a l l y  as quoted above, except the to  the same experiment  that p-type InSb was  used.  Also,  i n i t i a l drop i n v o l t a g e as d e s c r i b e d above was  ascribed  i n j e c t i o n of c a r r i e r s r a t h e r than a v a l a n c h i n g .  o s c i l l o g r a p h s shown i n t h i s work are those shown by Glicksman and increased  geometrical  c a r r y i n g the  Here a g a i n ,  explained  c u r r e n t was  d i f f e r e n t methods.  Murray  The  used by Glicksman and that Chynoweth and current  The  critical  a l s o derived  (1961) a l s o put f o r t h  f i r s t was  Steele  They used  (1959) d e s c r i b e d  Murray were c a r e f u l to see  occurred  three  above, except that the  criti-  w e l l i n t o the avalanche breakdown r e g i o n .  c u r r e n t was  observed  the c r i t i c a l  the w e l l known theory  to be about 4 amperes.  I<  critical  R  i s as g i v e n  =  of gaseous p i n c h  Ic.  +  ( B > / I  i n s e c t i o n 2.2.  c u r r e n t of 4.4  e x c e l l e n t agreement w i t h  They  c u r r e n t by p l o t t i n g the c u r r e n t at This  i s compared  (e.g. L i n h a r t ,  ): W  where  evidence  almost i d e n t i c a l to that  which " p i n c h " set i n a g a i n s t magnetic f i e l d .  p. 221  "...the c r o s s -  reduced from the  observed p i n c h i n n-type InSb.  cal  by:  the  c r o s s - s e c t i o n to the p i n c h c r o s s - s e c t i o n . "  Chynoweth and of having  (The  i d e n t i c a l i n form to  Powlus.)  observed r e s i s t a n c e was  s e c t i o n a l area  to  (Ancker-  amperes and  P  The  , ^  Mi)*  plot yielded a  gave a value of  the a c t u a l r a d i u s of the  a  in  sample.  1961,  The tal  third  method u s e d i n v o l v e d a s i m i l a r  a r r a n g e m e n t as t h a t o f G l i c k s m a n  d e s c r i b e d above. ent.  e R  (1961)  However t h e o b s e r v a t i o n s were q u i t e  The s a m p l e was s u b j e c t e d  Below I  and P o w l u s  experimen-  to constant  t h e v o l t a g e remained constant  current  throughout  differ-  pulses. the pulse,  and w e l l a b o v e I< t h e v o l t a g e r e m a i n e d c o n s t a n t a t a s l i g h t l y R  higher value.  I n the neighborhood of I  of current a t which these  In I„ c  t o be  I>  .  fluctuThe v a l u e  f l u c t u a t i o n s w e r e t h e most I  (  A  .  [ I  C  R  - 4 t o 6 amp.)  t h i s work, t h r e e s e p a r a t e methods o f o b t a i n i n g  g a v e good a g r e e m e n t .  C h y n o w e t h and M u r r a y a s c r i b e t h e  increased r e s i s t a n c e o f the pinched effects  the voltage  I< let. and t h a t when  a t e d b e t w e e n t h a t when  p r o n o u n c e d was t a k e n  ( R  plasma t o the combined  o f h o l e - e l e c t r o n s c a t t e r i n g w h i c h i s i n c r e a s e d due t o  the h i g h e r  c o n c e n t r a t i o n of c a r r i e r s near the center o f the  sample, and t o a l e s s e r e x t e n t They s t a t e t h a t these  increased  magneto-resistance.  changes cannot a c c o u n t f o r a change i n  r e s i s t a n c e a s l a r g e a s was o b s e r v e d ,  at l e a s t not according  to  observed  present  theory.  The f l u c t u a t i o n s  e x p l a i n e d by a p i n c h i n g - u n p i n c h i n g in the "pinched" 5.2  c o n d i t i o n was  when I - I » were  instability.  f  No  instability  observed.  C r i t i c a l Summary Probably  t h e most d i s t u r b i n g f e a t u r e o f t h e a b o v e  r e p o r t s of observance o f magnetic s e l f - p i n c h i n g i s that a l l t h e a r g u m e n t s a r e b a s e d on an o b s e r v e d  a p p a r e n t change o f  r e s i s t a n c e o f t h e sample between t h e " p i n c h e d "  and " u n p i n c h e d "  states.  I t i s p a r t i c u l a r l y d i s t u r b i n g t o see  in resistance and  Steele  dismissed  and  Because the  so g l i b l y  resistance  of  the  confined  sample  pinching,  way  (see  resistance  section  5.1). smaller  r e a s o n f o r the  overall  w h a t s o e v e r , i f t h e number  t h e i r m o b i l i t i e s r e m a i n the  any  change  Glicksman  to t r a v e l i n a  i s no  t o c h a n g e i n any  c a r r i e r s and  i s done by  Ancker-Johnson et a l  c a r r i e r s are  cross-section  as  this  change w i l l  be  same.  In  a purely  magnetic  secondary  e f f e c t , s u c h as  t h a t s u g g e s t e d by C h y n o w e t h and  Murray.  can  e f f e c t o f m a g n e t o - r e s i s t a n c e on  the  dismiss  the  in resistance  as  follows.  tion f o r transverse R/R*  A  =  U s i n g the  f a c t o r 10  -  A  *  6/6.  with  m.  derived  We  will,  10  weber  has  and  f o r the  a uniform conductivity  field.  change  approxima-  B*  ( 5  been t a k e n f r o m  6  %)--  and  B  are  /ni  Hosier  (1957).  The  change of r e s i s t a n c e and  without further  as g i v e n  applied  transverse  above in a  magnetic  j u s t i f i c a t i o n , assume  whereas i f the have  i n s e c t i o n 2.2,  i n terms  sample  that (5-  of  , 4*  r  current  2 )  the  6 B* rc\r If  _  -2  measurements of F r e d e r i k s e r e l a t i o n was  field  We  magneto-resistance  4  where the  small  of  10 d e n s i t y w e r e homogeneous we i  would  3)  W i t h no G & R a t and  sample  (5-  4)  and  (5-3)  Thus of  the  radius  2.5x10  -4  (  $ ( ) 2)  / (Chynoweth  m.  of  :  0  4  amperes  and M u r r a y ,  1961)  gives  the  the  gives  =r  AR/R  effect  of  is  negligible,  It  is  rather  their  equation  of  theory  (2-28)  we  5 * /a" " 5  magneto-resistance  sample  time dependence since  of  c r i t i c a l current  the is  even  on the  during  difficult  unpublished. that  resistance  pinch.  t o comment  pinch obtained  see  total  on  the  form  by G l i c k s m a n a n d  However,  if  we  of  Powlus  look  at  initially  (81 If  the  would  plasma were get  the  contract  following  t -..u  ir„»or  the  factor  we  the  same  that these  so  p  f o r m as  occurs  conditions,  Johnson  (5et  if  given  7),  a l .  In  the  the  as  Mr ?  until  (J 0 , =  pinch  ).  1^  given  This  I  £  R  to  ,  time:  is  the  above,  effects  if 3  I by  not he  to  quite  assumes  ^/dt ^ 0 (I  f o r m quoted of  previous  identical  s t i l l  for  replace  then gives the  is  i n the  However, then  we  ( 5 - 7)  Mr*, I/A  by G l i c k s m a n . at  rate  lit**  he w o u l d h a v e which  for  -  have m u l t i p l y i n g I  no p i n c h i n g  equation  this  l/(MrVkp/ln<**")  Ancker-Johnson*s  section;  at  expression  ' T^Tf  fl  ( c f .  to  under in  by  Ancker-  diffusion  have  34.  been  completely ignored.  o f InSb a t 7 7 ° K . q u o t e d 1/A  *  3xl0 rV  Our  6  coul." ,  .  3xl0  e s t i m a t e o f 1/  ranging  tfp  f r o m 2.5x10  by G l i c k s m a n and P o w l u s  properties  (1961),  giving  1  *  U s i n g t h e v a l u e s f o r the  6  (I - I»  ) secf  c  1  f r o m e q u a t i o n (2-31) g i v e s v a l u e s  sec."  f o r v e r y weak p i n c h i n g  to  6 — 1 0.85x10 about  sec."  f o r very strong pinching.  t h e same s i z e  current  of pinch  dependence.  a l l o f i t was  under which still  dons u n d e r  o f t h e above work i s  a v a l a n c h e breakdown  These  produce non-uniform plasma  conditions w i l l  densities.  The  t h e o b s e r v a t i o n s made by t h e a b o v e w o r k e r s unexplained  out.  or t o i n j e c t i o n  t h e samples  space charge  t e l l whether  were o f e f f e c t s and  that  (not  an o b j e c t i o n i f the  the r e s u l t s  p i n c h were p r e s e n t o r n o t .  responses which  could  involving  mechanisms. to operat-  experiments  could  unambigu-  Measurements  o f s e c o n d a r y i m p o r t a n c e : change o f  time dependent  con-  completely ruled  as p o s s i b l e  i n the a v a l a n c h e r e g i o n  that  t o as y e t  avalanche  instabilities  n o t be s o g r e a t  were d e s i g n e d i n s u c h a way ously  a r e due  c a n n o t be  effects) merit attention There would  ing  effects,  tend to  possibility  b e h a v i o u r of semiconductors under  In p a r t i c u l a r ,  magnetic  conditions,  the t h e o r y of the b e h a v i o u r o f s e m i c o n d u c t o r s i s  rather uncertain.  ditions,  yield  time, but c o m p l e t e l y d i f f e r e n t  Another d i s t u r b i n g feature that  B o t h methods  resistance  conceivably  be  caused  by some o t h e r m e c h a n i s m t h a n p i n c h . process, the l o n g i t u d i n a l e l e c t r i c /Jp  are  H e n c e we  During field  the  and  the r a t i o  inhomogeneous i n the l o n g i t u d i n a l  have e f f e c t i v e l y  a l o n g i t s l e n g t h and  avalanche  direction.  a plasma whose p r o p e r t i e s v a r y  i n time.  A l l d e r i v a t i o n s of  pinch  t h e o r y assume t h a t t h e p r o p e r t i e s o f t h e p l a s m a a r e longitudinally equal to  Mn /MP  and •  of  r e q u i r e (Jn^ / Jp^  uniform  ) t o be c o n s t a n t  Hence a plasma g e n e r a t e d  by  and  avalanche  g i v e s an u n s a t i s f a c t o r y medium f o r a r i g o r o u s c o m p a r i s o n theory to experimental r e s u l t s . likely  to occur under these  s t a n t i a l and  6.  of  Anomalous e f f e c t s a r e a l s o  c o n d i t i o n s and  i n c o n c l u s i v e evidence  hence o n l y  circum-  of pinch i s a v a i l a b l e .  EXPERIMENTAL CONSIDERATIONS 6.1  Observable We  Characteristic  of  Pinch  have seen t h a t the o v e r a l l r e s i s t a n c e changes  which might occur d u r i n g the s e l f - p i n c h i n g are o n l y secondary  importance.  The  main c h a r a c t e r i s t i c  of  of p i n c h i s  a change i n t h e d i s t r i b u t i o n o f the c a r r i e r s , accompanied  by  a  The  corresponding  only type  change i n the c o n d u c t i v i t y d i s t r i b u t i o n .  of experiment  whether or not  w h i c h can u n a m b i g u o u s l y  pinching i s present  i s one  i n d i c a t i o n o f the c a r r i e r d i s t r i b u t i o n will  w h i c h g i v e s us  i n the sample.  c o n s i d e r a few p o s s i b l e schemes f o r o b s e r v i n g  r e d i s t r i b u t i o n o f c o n d u c t i v i t y and sible  determine  experimental approach to  then  follow.  an We  this  t r y to suggest  a  pos-  6.2  magnetic We w i l l  Radio-Frequency  Measurements  Let  the  us  field make  consider  to  the a)  the  sample  following  In  the  field  a p p l i c a t i o n of by means  simplifying  absence is  of  of  the  homogeneous  a  a  high-frequency  coaxial  solenoid.  assumptions:  sample, and  of  the  the  magnetic  same  phase  throughout. b)  Between  c)  the  solenoid  (radius  the  same  as  The  skin  depth,  angular d)  sample  € to «  if  the  so  the  test  displacement  is  present.  a .  «  r.f.  field  not  (2./JJ6U>)  of  that  magnetic  sample were  8-  the  <x ) a n d  r„ ) , t h e  frequency 6  (radius  co i s  the  signal,  currents  are  negligible. When t h e and  the  within  where the  the  L  0  skin  is  the  sample.  small  changes  carrier of  boundary  the  above  conditions,  for  from Maxwell's  small  changes  in  equations  conductivity  depth  inductance  While the  of  above  (as  is  the  the  apparent  inductance  the  strength  of  the  the is  in conductivity  distribution is radius  h o l d , we g e t  at  solenoid  in  the  only  valid  for  the  outer  edge,  absence  relatively i f  a monotonically decreasing case will  pinch.  i f be  there an  is  For greatest  the function  no s u r f a c e  increasing  of  G & R)  function  sensitivity,  of the  37.  radius of the sample should be as close to the radius of the solenoid as possible. An alternative method of using r . f . techniques to obtain an estimate of carrier redistribution is to measure the impedance seen by a longitudinal r . f . electric f i e l d . We retain assumptions c) and d) above.  for small changes in conductivity within the skin depth. One of the most important assumptions used in the above analysis is that  . This condition must be met  in order that changes in the distribution w i l l be measureable. We can find the size of co required to meet this condition for two semiconductors whose properties are given in the appendix. InSb at 160°K. : Ge  «>>7  8xl0 /  sec."  1  at 300°K. : u>>? 5xl0 /  sec."  1  where 4 is measured in meters.  5  3  For other considerations we  w i l l see that we should not allow a to be much greater than -3 10  m.  Hence the frequency which would be required is well  into the kilomegacycle region. There are two important drawbacks to the use of microwave measurements. The f i r s t i s that any changes in overall resistance would have to be carefully considered. The other is that this method does not actually measure the conductivity distribution, but only gives an Indication of  38.  the changes which occur near the outer s u r f a c e . s u r f a c e 0 & R w i l l tend  Hence any  to mask the changes which take  p l a c e f u r t h e r i n t o the sample. 6.3  Probe Techniques Probes to measure  the change i n c o n d u c t i v i t y d.I.s . r  t r i b u t i o n could be a p p l i e d to e i t h e r the s l a b or c y l i n d e r . In the case o f the c y l i n d e r , however, only an i n d i c a t i o n of the changes i n the d i s t r i b u t i o n would the r . f . methods,  be o b t a i n e d .  the e f f e c t s of s u r f a c e G & R would  to mask any i n t e r n a l changes. would a c t u a l l y measure  As w i t h tend  In the case o f the s l a b one  the c o n d u c t i v i t y d i s t r i b u t i o n by  means o f an a . c . t e s t s i g n a l a p p l i e d to p a i r s of probes attached  to opposite  f a c e s o f the s l a b at various  from the l o n g i t u d i n a l a x i s . effects w i l l  i n no way  distances  In t h i s arrangement the edge  hide the i n t e r n a l c a r r i e r  redistri-  bution. 6.4  Infrared Harrick  Absorption (1956) has d e s c r i b e d a method by which the  a b s o r p t i o n of i n f r a r e d by the f r e e c a r r i e r s i n a semiconduct o r may be used to determine t h e i r d i s t r i b u t i o n . experimental 6-1.  The  arrangement i s shown s c h e m a t i c a l l y i n F i g u r e  For each p o s i t i o n of the s l i t and d e t e c t o r on the  sample, the t r a n s m i t t e d  intensity I would  out any c u r r e n t passing  through the sample.  be balanced  T  out by a b r i d g e c i r c u i t .  be measured  with-  T h i s would  then  The current would  then  39.  0.5 * 3 S  SLIT „ mn t II  INFRARED  |j  THERMOPILE  B R I D G E  F i g u r e 6-1.  D E T E C T O R  Experimental arrangement f o r measuring the c a r r i e r d i s t r i b u t i o n by i n f r a r e d a b s o r p t i o n .  be a p p l i e d to the sample, and A l , the change i n t r a n s m i t t e d i n t e n s i t y would be measured.  The c a r r i e r d i s t r i b u t i o n  could  then be found through the r e l a t i o n g i v e n by H a r r i c k : A I / I  T  d  where c i s a constant  - 1 )  ( e  and d i s the t h i c k n e s s of the sample.  On the b a s i s of H a r r i c k ' s measurements on a sample about 1 cm. thick,  one should  be able to d e t e c t changes i n c a r r i e r con19  c e n t r a t i o n of about 10 would adequately  3 m.~  i n a s l a b 0.5 mm. t h i c k .  This  demonstrate pinch i f i t were to o c c u r .  I t should be noted that i t takes about one second to make a measurement, the time constant  of the thermopile.  Thus i f p i n c h In semiconductors turns out to be a s h o r t - l i v e d phenomenon as i t i s i n gaseous plasmas, t h i s method would n o t detect i t .  40.  6.5  Generation We  will  hole-electron generation,  consider  and  photo-ionization.  major  o b j e c t i o n t o u s i n g avalanche  which i s hot  p i n c h i n g might occur c a n n o t be  certain  e f f e c t s ) would  fully  interfere with  distribution,  In a d d i t i o n , the  on  o f c o n t a c t and  type  f u l l y understood. a specific  to give r i s e the  best  e l e c t r o n p l a s m a i s by  method, t h e  sample u s e d  c e n t r a t i o n as may estimate  be  true  density  the r a d i a t i o n l n order  penetration.  I f we  understood  low  even w i t h  i s predominantly  to pass only  by  i s not  an  For  this  impurity  We  can  G & R  con-  readily  ln this  case,  situations.  a surface  photons w i t h  the e f f e c t  for  of c r e a t i n g a  a s l a b of the  those  yet  to search  generation.  practically.  is filtered  injection  effects.  p r e v i o u s l y mentioned  consider  semi-  by a method  method  have as  of pinch  i n the  charge  longitudinal  intrinsic  seem a d v i s a b l e  should  Photo-ionization  conductor  i n an  thermal  one  observations.  t o unknown  obtained  the magnitude  w h i c h i s not  unless  (e.g. space  carrier  I t does n o t  Probably  though  dependence o f c a r r i e r  the  cas-  i n t h i s way,  phenomenon i n a p l a s m a g e n e r a t e d  which i s l i k e l y  hole  the  Even  to a non-uniform  especially  conductor.. the  understood.  t h a t o t h e r phenomena  not  breakdown  It i s a stochastic  i n a plasma g e n e r a t e d  I n j e c t i o n would l e a d carrier  a  breakdown, i n j e c t i o n , t h e r m a l  been o u t l i n e d i n s e c t i o n 5.2.  cade p r o c e s s  Plasma  f o u r methods o f g e n e r a t i n g  plasma: avalanche  The has  of a Hole-Electron  effect  same s e m i good  of s h i n i n g p r o p e r l y  41.  f i l t e r e d l i g h t u n i f o r m l y on both faces o f a s l a b , we can approximate the p h o t o - i o n i z a t i o n by a constant p a i r t i o n r a t e throughout  the sample, &  x-component, equation  where  m.~  s  3  sec." .  +  | j ]''*  1  generaTaking the  (3- 3) becomes:  and  ^ '  £. [ l  The net e f f e c t of t h i s i s to r a i s e the values o f n  «  >  (  w  <  /  i - P i ^  (3- 5).  a n (  * T by a f a c t o r  of ( I  T*& /fl )k t  t>  i n equation  The s i t u a t i o n i s then e s s e n t i a l l y unchanged  p u r e l y thermal g e n e r a t i o n  ,  except t h a t the e f f e c t i v e  from n,- i s  increased. From the above c o n s i d e r a t i o n s , i t would seem that the best method of producing a plasma f o r the study of s e l f p i n c h i n g would be thermal g e n e r a t i o n . s p u r i o u s e f f e c t s are l e s s l i k e l y  With t h i s method  t o i n t e r f e r e w i t h the obser-  v a t i o n s of p i n c h . 6.6  Geometry and M a t e r i a l Having decided upon the means o f g e n e r a t i n g the  plasma,' we should now c o n s i d e r the geometry of the sample. In the case of the s l a b , we can measure the c a r r i e r  distri-  bution whereas with the c y l i n d e r we can o b t a i n only a rough estimate of the way i n which t h i s d i s t r i b u t i o n changes. From t h i s c o n s i d e r a t i o n , i f a l l e l s e were e q u a l , i t would seem that the s l a b geometry i s s u p e r i o r .  42.  We  should  now  compare the s l a b and  the c o n s i d e r a t i o n of minimizing t h i s , we  c y l i n d e r from  thermal e f f e c t s .  To  do  compare the r a t i o s of minimum power r e q u i r e d to  i n i t i a t e s e l f - p i n c h i n g to the maximum power permitted thermal e f f e c t s becoming important From equations  Since  d«W  (2-20) and  geometries.  (4-17)  i t i s apparent that the s l a b i s s u p e r i o r from  this consideration. field  f o r the two  without  Since we  ( f o r magnetic pinching)  r e q u i r e the c r i t i c a l  electric  t o be below the breakdown f i e l d ,  the s l a b i s a g a i n s u p e r i o r . R e c a l l equation We We  do not  (2-20):  should  now  E<« CsLb)/6^(<yi->  -  ( Z - ^ / T T W ;  c o n s i d e r the problem of s u r f a c e G &  expect the bulk G & R to give a p p r e c i a b l y  r e s u l t s i n the two geometries.  R.  different  Because the s l a b has a much  g r e a t e r s u r f a c e area f o r i t s volume than the c y l i n d e r , we might expect s u r f a c e G & R to present the s l a b .  a g r e a t e r problem i n  However, here the main e f f e c t of s u r f a c e G & R i s  to i n c r e a s e the e f f e c t i v e bulk recombination  rate.  t i o n a s m a l l i n c r e a s e of c a r r i e r c o n c e n t r a t i o n w i l l at the edges.  The  In a d d i result  primary e f f e c t i n the c y l i n d e r w i l l be  increase the c a r r i e r c o n c e n t r a t i o n at the outer s u r f a c e .  to This  43.  i n c r e a s e could mask the measurement of the i n t e r n a l i n c a r r i e r d e n s i t y , as we have pointed  out i n the  changes  preceding  sections. Thus, i t appears  that the s l a b geometry  i s superior  from a l l c o n s i d e r a t i o n s . We must now  c o n s i d e r the m a t e r i a l to be used i n  any experiment designed to observe s e l f - p i n c h i n g . appendix, we have l i s t e d  a number of p r o p e r t i e s of three  r e p r e s e n t a t i v e semiconductors: and Indium Antimonide  I n the  Silicon  ( S i ) , Germanium (Ge),  (InSb).  From equations (2-19) and  (4-16) we get  (6- 2)  where we have assumed that temperature. perature. operating  T  h  ? T  - T  p  the  ambient  T h i s i s a s l i g h t l y i n c r e a s i n g f u n c t i o n o f tern-  The minimum temperature at which we the sample would  could  be the one at which the  consider  carrier  d e n s i t y i s about one order of magnitude g r e a t e r than the impurity  concentration.  The p r a c t i c a l minimum impurity  con-  19 c e n t r a t i o n o b t a i n a b l e at the present  time i s about 10  Thus we w i l l denote the temperature which gives us a concentration of 1 0  2 0  m."  3  as  temperature f o r a g i v e n sample.  -3 m.  carrier  ~ T j . > our minimum o p e r a t i n g m  n  We w i l l take the temperature  at which the e l e c t r o n gas becomes degenerate as an upper  44.  " T V ^ . . From n u m e r i c a l values quoted i n the appendix  limit,  we have estimated and t a b u l a t e d tc*. / £  b  <W.)  ,  i n t a b l e 6-1. t  f b  p  (T  w i  Tnm., T  m a i (  „.),  .  f  ^Jl^h__y  [Tm-.n.^  ^w,^l  TKT)  and  i s the breakdown e l e c t r i c  r  field.  TABLE 6-1  T  indium antimonide  germanium  Silicon 400  310  160  620  555  285  360(«/»")  60 (d/W)  0.93  0.8xl0~ /w  0.32xl0' /w  1.3xl0"7w  f p (»".)  100 w/  13 w*  4.5 wv*-  f  10  M ;  „  CK)  3  1  Cs«<.)  r  -3  10  5.8xl0 d 3  Note:  Since  i  3  l  -3  10  7.3xl0 «| 3  t  (d/")*  -7  2.5xl0  3  and w are measured i n meters  W , the width o f the s l a b can be made g r e a t e r than  2xl0"" meters without any d i f f i c u l t y , 3  there w i l l be no problem  i n keeping below the breakdown f i e l d i n any o f these materials. From  Pr*(pi««u)/ftt(  + h  *'•""'J  which must be kept l e s s than  45.  u n i t y t o reduce thermal e f f e c t s , we o b t a i n a l i m i t on d/W :  4/w < .05  Si : Ge :  d/w <  InSb:  d/*" < 1.04  .13  The above does not c o n s i d e r the i n t e r a c t i o n between thermal and magnetic e f f e c t s nor the e f f e c t s of G & R.  Hence we  should c o n s i d e r these l i m i t s on olA* as a b s o l u t e maxima. p r a c t i c e i t would be d i f f i c u l t  to o b t a i n a r a t i o  In  of  much l e s s than .05 s o we can r u l e out the use o f s i l i c o n at once. We s h a l l now look at the e f f e c t of G & R i n g e r manium and indium antimonide.  From the d e f i n i t i o n of L p  ;  g i v e n i n equation ( 3 - 4 ) and the n u m e r i c a l values g i v e n i n the appendix: 1-D-, (&e) Lot  *  (InSb)  1.8xl0" *  2  -  l  x  1  0  "  3  5  m. m  '  The sample would l i k e l y have a width of about 1 cm., i n which  case Ge:  w  InSb:  /Lo,-  -  5.5  W/uoj ar 480  From chapter 3 we see that the G & R w i l l  completely dominate  i n the case of InSb without even c o n s i d e r i n g the e f f e c t o f surface recombination.  Thus germanium looks as i f i t would  be the most s u i t a b l e m a t e r i a l .  46.  6.7  Summary In  t h i s s e c t i o n we s h a l l t r y to o u t l i n e a pos-  s i b l e experiment  t o be performed  t o observe  self-pinching.  F o r the sake of a n u m e r i c a l example, l e t us assume that we have a w e l l - e t c h e d s l a b of h i g h l y pure germanium of c r o s s s e c t i o n 0.5 mm.  x 1.0 cm.  1 m./sec. , we f i n d  I f the recombination v e l o c i t y i s  that:  w/uo/  -  12.5  T h i s value i s not too h i g h : from the curves shown i n chapter 3, we estimate that w i t h o b t a i n a measurable p i n o h .  I ?  P~ Z-*i one should  T h i s means a c u r r e n t of. about  / ( 3TC 16 k T m^/ju  )  \  llt  1.5 amp.  From t a b l e 6-1 we have 1.3x10'-3 sec. 1.0x10 -3 sec. 1.8x10'-3 sec. Thus a l l the times tude.  T h i s means that we w i l l have t o take s p e c i a l  to reduce  p  y t  r  oare  thermal e f f e c t s , s i n c e any o f these which occur  w i l l do so about f  i n v o l v e d a r e o f the same order of magni-  we w i l l  the same time as p i n c h i n g takes p l a c e . Since expect the p i n c h t o develop to the value  c a l c u l a t e d w i t h G & R without any a p p r e c i a b l e overshoot.  47.  Some means of c o o l i n g the s u r f a c e would have t o be employed, s i n c e about 600 watts p e r cm. l e n g t h of the sample would be d i s s i p a t e d .  I f p u l s e s o f c u r r e n t a r e used,  these must be at l e a s t an order o f magnitude longer c h a r a c t e r i s t i c times i n v o l v e d , i . e . at l e a s t 10 I t i s proposed that i n i t i a l l y  than the  sec.  the probe method  (see s e c . 6.3) should be employed to measure the c a r r i e r redistribution.  I f i t should  t u r n out t h a t the pinch i s  s t a b l e f o r up to a second, the i n f r a r e d a b s o r p t i o n described  7.  technique  i n s e c t i o n 6.4 might be used.  CONCLUSIONS In the above a n a l y s i s we have shown that i n order  to make a r i g o r o u s comparison between any experiment designed present  t o observe s e l f - p i n c h i n g and the theory a t i t s stage  o f development, we must use a t h e r m a l l y gener-  ated plasma, p o s s i b l y enhanced by i l l u m i n a t i o n w i t h filtered  light.  properly  The only unambiguous i n d i c a t i o n of s e l f -  p i n c h i n g i s the t r a n s v e r s e r e d i s t r i b u t i o n of the c a r r i e r s and  hence the c o n d u c t i v i t y .  Even t h i s i n d i c a t i o n i s s u b j e c t  to i n t e r f e r e n c e from thermal e f f e c t s , which must be c a r e f u l l y accounted f o r i n any experimental  arrangement.  F i n a l l y , we have shown t h a t i f s e l f - p i n c h i n g can  occur in semiconductors, one should be able to observe i t in a thermally generated plasma in a pure germanium slab at room temperature.  Hence i t i s completely unnecessary  to consider using the highly unsatisfactory avalanchegenerated plasma for pinch studies, as has been done to date.  APPENDIX Some P r o p e r t i e s o f Three I n t r i n s i c  Semiconducting M a t e r i a l s :  S i l i c o n , Germanium and Indium Antimonide a)  Silicon  Property  ni  3.88x10^ T  e  m  Smith (1959)  (KT i n ev)  pn  0.15(300/t) ' 2  6  m /V.sec.  Smith  (1959)  M?  0.05(300/r)  2 # 3  m /V.sec.  Smith  (1959)  K (300*K)  2  2  Smith (1959)  84 watts/m. deg.  Cp (Soo*K)  1.64X10  tE  Source o f Information  Value  J . /m.  6  Smith  deg.  3  10~ sec.  Smith (1959)  1.21 ev.  Smith  620° K.  Smith (1959)  3  S  T  2xl0  Jonscher  (1960)  Germanium Source o f Information  Value  Property 22-~3/i  m  1.76x10^ T * e  r»  0.38(300/T )  Mt  0.16  K (SM'K)  (1959)  Chynoweth (1958),  V./m.  7  10 m/sec. b)  (1959)  1  -MZS/kT  ,  6  (300/T) ' 2  m~  m /V.sec  6  2  3 3  m /V.sec. 2  63 watts /m. deg. 1.85xl0  6  3  J/m. deg. 3  Smith (1959) Smith  (1959)  Smith  (1959)  Smith  (1959)  Smith (1959) (over)  50.  Germanium (continued) Property  Value 10~  E  b  Source of Information  sec.  Smith  (1959)  0.785 ev.  Smith  (1959)  555° K.  Smith  (1959)  8xl0  Smith  (1959)  3  V./m.  6  1 m/sec. c)  Jonscher (I960!  Indium Antimonide  Property  Value  ni  Source of Information  1.2 9x10 (T/2 90 r e " ' ^ 22  7.0  (300/ T )  0.09(300/ T ) K(I6«*K)  ( l  1 , 6 8  "^  r? m  m /V.sec. 2  m /V.sec.  2 , 1  2  67 watts /m.deg.  Cp (|*0*K)  0.665xl0  6  J/m  10"  deg.  3  sec.  •  Smith  (1959)  Smith  (1959)  Smith  (1959)  Busch and Schneider,(1954] Gul'tyaev and P e t r o v (1959) Smith  (1959)  .255 ev.  Smith  (1959)  285° K.  Smith  (1959)  2xl0  7  4  V./m.  Kanai,(1959)  In the above, the symbols are defined as follows: fl{ - undisturbed intrinsic carrier concentration  MniMp - electron and hole mobilities respectively K  - thermal conductivity  Cp  - specific heat per unit volume  51.  tJ  M # <  -  maximum p r a c t i c a l l y in  E.^  -  the  undisturbed  e n e r g y gap temperature  ^ley*.-  obtainable  lifetime  sample.  (extrapolated at  carrier  which  to  the  0°  K.)  electron  gas  becomes  degenerate. E  b  -  electric  field  required  to  initiate  avalanche  breakdown. Smin. ~ m i n i m u m p r a c t i c a l l y recombination References Busch,  G.  for  the  A.  G.,  Gul'tyaev,  P.  V.  Jonscher,  A. K . ,  surface  velocity.  Appendix:  and S c h n e i d e r ,  Chynoweth,  obtainable  Phys.  M., Physica, Rev.  and P e t r o v , Principles  109, A. of  20,  1537,  1084,  (1958)  V . , Sov. P h y s . S o l i d 1, 3 3 0 , (1959)  State,  Semiconductor  Operation  Device  (John W i l e y & Sons Kanai,  Y.  J . Phys.  Smith,  R.  A.,  Soc.  Japan,  Semiconductors,  (1954)  14,  1302,  (Cambridge 1959)  Inc.,  1960)  (1959) University  Press,  52.  BIBLIOGRAPHY  Ancker-Johnson, Betsy; Cohen, R. W.;  and Glicksman, M.,  Phys. Rev. 124, 1745 (1961) Bennett, W. H., Phys. Rev. 45, 891 (1934) Chynoweth, A. G. and Murray, A . A . , Phys. Rev. 123, 515, (1961) F r e d e r i k s e , H. P. R. and H o s i e r , W. R., Phys. Rev. 108, 1136 "^"1957) Glicksman, M. and S t e e l e , M.C., Phys. Rev. L e t t e r s , 2, 461 (1959) Glicksman, M. and Powlus, R. A . , Phys. Rev. 121, 1659,(1961) H a r r i c k , N. J . Phys. Rev. 101, 491, (1956) L i n h a r t , J". G. Plasma P h y s i c s ,  (North-Holland P u b l i s h i n g Co., 1961)  Smith, R. A., Semiconductors, (Cambridge U n i v e r s i t y P r e s s , 1959)  

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