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Electrodeless techniques for semiconductor measurements and dimorphic phase transformations in compound… Nyberg, Donald Walter 1960

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ELECTRODELESS TECHNIQUES FOR SEMICONDUCTOR MEASUREMENTS and DIMORPHIC PHASE TRANSFORMATIONS I N COMPOUND SEMICONDUCTORS  by DONALD WALTER NYBERG B. A .  So., University  o f B r i t i s h Columbia,  1957  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the  Department of  Physics  We a c c e p t t h i s  thesis  required  as c o n f o r m i n g t o standard  THE UNIVERSITY OF B R I T I S H COLUMBIA April,  I960  the  In the  presenting  this thesis  r e q u i r e m e n t s f o r an  of  B r i t i s h Columbia,  it  freely available  agree that for  Department that  copying or  gain  shall  shall  for reference  and  study.  I  Department o f  for extensive  granted  representatives.  publication  Physics  April 7, I960  by  the  It i s  of t h i s t h e s i s  a l l o w e d w i t h o u t my  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r $, C a n a d a .  Date  be  copying of  Columbia,  of  University  Library  his  be  the  the  p u r p o s e s may  o r by  not  advanced degree a t  fulfilment  I agree that  permission  scholarly  in partial  make  further this  Head o f  thesis my  understood  for financial  written  permission.  ii  ABSTRACT  Eleetrodeleae techniques for semiconductor measurements, baaed on the inductive coupling of the sample to coile, are investigated i n the f i r s t part of the thesis.  The theory and the experimental techniques of  - two main experiments -are developed and applied to several samples.  The  seoond part of the theaia le devoted to diaouaelng dimorphic phase transformations i n compound semiconductors. In the f i r s t experiment the sample i s placed in the core of a solenoid which la exoited by a sine wave generator.  Eddy currents are induced in  the sample and they eet up a secondary magnetic f i e l d which oppoeea the primary f i e l d resulting i n a decrease i n flux through the oore. i s that a complex impedance ie reflected into the ooil.  The result  The increase i n  resistance and decrease in inductance of the ooil is measured by a Q-meter technique and related to the conductivity for long cylindrical and spherical geometry samples.  Design considerations are discussed, and i t i s shown  that the optimum frequency to use, in order to obtain maximum sensitivity with the Q-meter, will depend on the conductivity of the material.  Long-  itudinal and transverse magnat©resistance are observed by applying a static magnetic f i e l d along the appropriate axis of the solenoid. The seoond experiment i s a oroeeed magnetio f i e l d determination of the Hall mobility.  Circulating eddy currents are Induced i n the sample by a  sinueoidally excited coil and a static magnetic f i e l d i s applied along a second axis.  The static f i e l d will rotate the plane of the eddy currents  through the Hall angle, u ^ , the product of the Hall mobility and the  iii  static f i e l d .  The e f f e c t 1B the appearance of an alternating magnetic  f i e l d along the t h i r d axle.  Thie f i e l d , which at low frequencies i e  d i r e c t l y proportional to the H a l l mobility, i e detected by a second c o i l . The technique i e very general and i t i e independent of the conductivity. In p a r t i c u l a r , i t may  be used to determine the H a l l mobility of powders  and l i q u i d s to which i t i s d i f f i c u l t or impossible to attach electrodes. In the second part of the thesis the compound semiconductor s i l v e r selenide 1B investigated.  The  sample could be obtained commercially only  i n powder form and hence the electrodelees techniques were e s s e n t i a l .  The  temperature v a r i a t i o n of conductivity above and below the phase t r a n s i t i o n , a c t i v a t i o n energies, r e l a t i v e oonductivitiee at the phase t r a n s i t i o n and a conductivity-temperature  hysteresis effeot were observed.  Absolute  oonduotivity values were not obtained because of lack of knowledge of the r a d i i of the i n d i v i d u a l powder grains and the H a l l mobility measurements indicated that the sample was very impure, e l e c t r o n i c a l l y speaking, and hence only impurity scattering was being measured.  iv  T A B L E  OP  C O N T E N T S  PART It ELECTR0DELE3S TECHNIQUES FOR SEMICONDUCTOR MEASUREMENTS  Chapter  Page  1  INTRODUCTION  1  2  EXPERIMENTAL TECHNIQUES  7  2.1  Eddy C u r r e n t L o s s e s i n a S o l e n o i d Core  7  (a)  G e n e r a l Techniques  7  (b) (c) (d)  Measurement o f AR and Design C o n s i d e r a t i o n s Frequency Range  2.2  C r o s s e d Magnetic (a) (b) (c)  2.5  5  AL  F i e l d H a l l M o b i l i t y Measurements  Theory Design Considerations D e s c r i p t i o n o f Apparatus  8 11 15 15 15 18 22  E l e c t r o d e l e e e Magnetoreelstance Measurements  24  (a)  T r a n s v e r s e Magnetoreeistance  24  (b)  Mixed M a g n e t o r e e i s t a n c e  25  EXPERIMENTAL RESULTS  27  5.1  C o n d u c t i v i t y Measurements on G r a p h i t e  27  (a)  E l e c t r o d e l e e e Technique  27  (b)  D.C. 4 - E l e c t r o d e Technique  28  5.2  5.5  E l e c t r o n i c P r o p e r t i e s o f N-Type Germanium  29  (a)  C o n d u c t i v i t y Measurements  29  (b)  H a l l M o b i l i t y Measurements  50  E l e c t r o n i c P r o p e r t i e s o f N-Type Indium Antimonide  51  (a)  C o n d u c t i v i t y Measurements  51  (b)  M a g n e t o r e s i s t a n c e Measurements  52  (c)  H a l l M o b i l i t y Measurements  55  V  PART l i s DIMORPHIC PHASE TRANSFORMATIONS IN COMPOUND SEMICONDUCTORS  Chapter  Page  1  INTRODUCTION  2  EXPERIMENTAL TECHNIQUES  4l  2.1  Eddy C u r r e n t Losses i n Powders  41  (a)  Theory  4l  (b)  Design C o n s i d e r a t i o n s  4j  (c)  D e s c r i p t i o n o f Apparatus  44  2.2 5  58  H a l l M o b i l i t y Measurements on Powders  47  EXPERIMENTAL RESULTS  48  J.l  E l e c t r i c a l Conductivity of Silver Selenide  48  5.2  H a l l M o b i l i t y Measurements on S i l v e r S e l e n i d e  55  APPENDIX As Change i n R e s i s t a n c e and Induotance o f a S o l e n o i d due t o a C o n d u c t i n g C y l i n d r i c a l Core  55  APPENDIX Bi Change i n R e a l s t a n c e and Inductance o f a S o l e n o i d due t o a C o n d u c t i n g S p h e r i c a l Core  60  APPENDIX Cs  The I n t e r n a l F i e l d i n an A r r a y o f  Conducting S p h e r i c a l P a r t i c l e s  62  APPENDIX Ds  The C o n d u c t i v i t y Tensor  64  APPENDIX Es  Magnetoreeistance and H a l l M o b i l i t y  67  (1) (2) (5)  T r a n s v e r s e Magnetoreeistanoe Mixed M a g n e t o r e s i s t a n c e Hall Mobility  BIBLIOGRAPHY  68 70 72 74  vi  L I S T  Figure  1  OF  F I G U R E S  Title  Facing Page  T h e o r e t i c a l R e f l e c t e d Impedance v s b r f o r a S o l e n o i d w i t h a C o n d u c t i n g C y l i n d r i c a l Core  8  2  Q-Meter C i r c u i t s  9  5  Q-Meter C i r c u i t Diagram  4  Range of C o n d u c t i v i t i e s Measureable w i t h a  0  C o i l o f Q-Factor Q  5  6  7 8  9  e  as a F u n c t i o n o f Frequency  H a l l M o b i l i t y Measurements (a)  F i e l d and C u r r e n t R e l a t i o n s  (b)  C i r c u i t Diagram  10  12 16  T r a n s v e r s e M a g n e t o r e s i s t a n e e Curves  J2  Mixed M a g n e t o r e e i s t a n c e Curves  55  H a l l M o b i l i t y Curves  55  Temperature V a r i a t i o n o f C o n d u c t i v i t y f o r S i l v e r Selenide  48  vii  A C K N O W L E D G M E N T S  I wish t o thank P r o f e s s o r R. E. Burgess f o r h i s guidance throughout the c o u r s e o f t h i s work, and f o r h i s v a l u a b l e comments and c o n s t r u c t i v e c r i t i c i s m during the preparation of t h i s  thesis.  The a s s i s t a n c e of t h e N a t i o n a l Research C o u n c i l o f Canada t h r o u g h a Postgraduate Studentship,  B u r s a r y and Summer Supplement i s g r a t e f u l l y  acknowledged. I a l s o wish t o acknowledge the a s s i s t a n c e o f t h e Defense  Research  Board i n p r o v i d i n g many o f t h e r e s e a r c h f a c i l i t i e s used i n t h i s work.  1  PART I;  ELECTRODELESS TECHNIQUES FOR SEMICONDUCTOR  MEASUREMENTS  CHAPTER 1 - INTRODUCTION  E l e c t r o d e l e e s t e c h n i q u e s f o r d e t e r m i n i n g the e l e c t r i c a l  conductivity,  m a g n e t o r e s i s t a n e e e f f e c t , and H a l l m o b i l i t y o f a semiconductor w i l l be described.  The t e c h n i q u e s a r e based on i n d u c t i v e c o u p l i n g o f t h e sample  t o c o i l s and have s e v e r a l advantages over t h e standard niques.  4-electrode  tech-  I n p a r t i c u l a r , samples t o which i t would be d i f f i c u l t t o a t t a c h  electrodes,  such as l i q u i d s and c h e m i c a l l y  a c t i v e substances which might  need t o be k e p t i n a vacuum, i n e r t gas atmosphere, or under p r e s s u r e be  studied while i n a s u i t a b l e container,  insulator.  the container  i e an  Another important c l a s s o f m a t e r i a l s t o which e l e c t r o d e s may  not  be a t t a c h e d  the  chemical p r e p a r a t i o n  that,  provided  may  a r e powders.  although absolute  They a r e f r e q u e n t l y t h e i n i t i a l  o f a new m a t e r i a l .  product i n  I t w i l l be ehown i n p a r t I I  c o n d u c t i v i t i e s are not obtainable  by t h e e l e c t r o d e -  l e s s t e c h n i q u e , r e l a t i v e c o n d u c t i v i t i e s a t phase changes, temperature v a r i a t i o n o f c o n d u c t i v i t i e s , a c t i v a t i o n energies, are r e a d i l y measured f o r powders. ohmic c o n t a c t s ,  electrical  techniques.  c o n d u c t i v i t y o f a sample i s determined by p l a c i n g t h e  sample i n t h e core o f a s o l e n o i d . g e n e r a t o r , eddy c u r r e n t s magnetic f i e l d  The B o l e n o i d  i e e x c i t e d by a s i n e wave  a r e induced i n t h e sample, and the secondary  s e t up by the eddy c u r r e n t s opposes t h e e x c i t i n g f i e l d  r e s u l t i n g i n an a t t e n u a t i o n The  I n a d d i t i o n , the problem o f making  a l t h o u g h p a r t l y overcome by 4 - e l e o t r o d e t e c h n i q u e s , i s  c o m p l e t e l y e l i m i n a t e d by these The  and H a l l m o b i l i t i e s  of the f i e l d  as i t p e n e t r a t e s  i n t o t h e sample.  eddy c u r r e n t s f l o w i n g i n the l o s s y m a t e r i a l r e s u l t i n a power l o s s  2  which i s r e f l e c t e d i n t o the solenoid S B an Increase i n resistance.  The  attenuation of the magnetio f i e l d inside the sample r e s u l t s i n a decrease i n f l u x through the solenoid core and hence i s reflected i n t o the solenoid as a decrease i n inductanoe.  These resistance and inductance changes are  measured by a Q-meter technique and are r e a d i l y related to the oonduoti v i t y of the material f o r s p e c i f i c sample geometries. Busch, Wieland and Z o l l e r (1951) have studied the e l e c t r i c a l conducti v i t y of grey t i n as a function of temperature using these techniques. The Q-faotor of a c o i l containing a core of grey t i n powder was measured at frequencies up to JO Mc.  In p a r t i c u l a r ,  where K — a constant dependent on frequency, geometrical shape and dimenaions of the conducting p a r t i c l e s , but independent of temperature, Q,= Q-factor of the c o i l with sample inside, and  Q « Q-factor of the o o i l without the sample. e  The experimenters took care to ensure that during the experiment no change of p a r t i c l e size, due to agglomeration, or due to crumbling of individual grains of the powder, took plaoe.  The constant K was not known due to  uncertainty as to the size of the powder grains, however, since K i s independent of temperature, 0/K  can be measured as a function of tempera-  ture. In general i t i s not possible to obtain absolute conductivity values for powders because K cannot be determined.  However, i n t h i s p a r t i c u l a r  case K was obtained approximately by a comparison method.  The Q-factor,  Q^ of the c o i l was measured f o r grey t i n £«-tin), then the Bample was heated, allowed to revert back to white t i n (0-tin) and the Q-factor,  3  wae measured at the same temperature as that of the grey t i n . The cond u c t i v i t y of white t i n i s known by other methods, hence  Cu  =  I  Qo -  Qoc \  \ Qo - Q  Q_B  J  The reeulte must be corrected f o r a decrease i n volume of 21% i n the 0C.-+P  transition. The H a l l mobility of a semiconductor i s determined by a oroaeed  magnetic f i e l d technique.  Eddy currents are induced i n the sample by an  a l t e r n a t i n g magnetic f i e l d . plane perpendicular  The eddy currents flow solenoidally, i n a  to the d i r e c t i o n of the applied magnetic f i e l d .  A  s t a t i c magnetic f i e l d applied p a r a l l e l to the plane of the eddy currents causes the plane of the eddy currents to rotate about the axis of the s t a t i c f i e l d through an angle which i s proportional t o the product of U the H a l l mobility and to B, the s t a t i o magnetic f i e l d . magnetic f i e l d set up by the eddy currents perpendicular  H  The secondary to t h e i r plane  w i l l also be rotated through t h i s angle, the e f f e c t being an a l t e r n a t i n g magnetic f i e l d along the t h i r d axis, perpendicular  to both the e x c i t i n g  and s t a t i c magnetic f i e l d s , and proportional to the H a l l mobility.  A  c o i l with center l i n e along t h i s axis w i l l detect a voltage whioh i a proportional to the Hall mobility.  In the experimental technique devel-  oped, a pair of Helmholtz c b i l s i s used to provide the e x c i t i n g f i e l d and a solenoid i s the detecting c o i l .  The c o i l e are mounted between the  poles of a do magnet. This eleotrodeless technique f o r making H a l l mobility measurements on semiconductors i s beleived to be new and i t should prove t o be a useful experimental technique because of i t e advantages over conventional niques requiring electrodes.  tech-  In addition to the advantages l i s t e d under  4  the deeeription of the conductivity experiment, this technique has the very important advantage, which will be discussed further i n part II, in that i t 1B the only known general method of obtaining absolute Hall mobility measurements on powders. The reason that the method i s valid for powders is that at low frequencies the results are essentially independent of the conductivity and the shape and size of the sample but are proportional to the rotation of the plane of the eddy our rente which is i t s e l f proportional only to the product u B. H  Bueoh, Wieland and Zoller  (1951) have used dc methods on powders to obtain u , but two H  experimental  difficulties are that the current must flow across particle boundaries, and the current density is not easily defined due to incomplete packing of the granular structure. The eleotrodelees technique overcomes both of these difficulties. Busch, Jaggi and Braunschweig (1955) have described a two electrode technique for making measurements of Hall mobilities.  In particular,  consider a sample of conductivity o~» in the shape of a thin circular disc.  A sinusoldally varying magnetic f i e l d B = B sinat perpendicular 0  to the plane of the disc will set up circulating eddy currents in the disc.  The tangential electric f i e l d and ourrent density are, E  and  j <f  J,  8  C05 uft  -  - CUh ~2~  =  _ ujxTh. B c o s  0  o  tot  2 The magnetic f i e l d will also exert a radial force on the ourrent carriers. The radial force on an electron i s ,  -e£  = -el v  K  Bl  5  If  the charge d e n s i t y  i s n,  then  .•' v;  Substituting for  wcni  =  CT =  , and u s i n g  f = no  c u r r e n t can flow  appear between the o f the  center  C  o 5 col  sin  cut .  t^cosu/t 51n.ut  LUUJIL  i n the r a d i a l d i r e c t i o n , a p o t e n t i a l V  will  and  radius  the rim of the d i s c .  d i s c , then  ^ VI  Hence  The  o  neu,  2  h  Since  6  U  =  II UJk  2  Q  B  I f r , i s the  z  5in  lujt .  __.C^1/JL_—.  =  CU lie'' fiu Sir* 2 W t  major advantage of Busch's method i s t h a t the m o b i l i t y i e  i n d e p e n d e n t l y of the c o n d u c t i v i t y . a t t a c h two little  e l e c t r o d e s to the  However, i t i e e t i l l  sample, and  obtained  n e c e s s a r y to  thus the method eeeme t o o f f e r  advantage over other t e c h n i q u e s employing  electrodes.  M a t e r i a l s w i t h e x c e p t i o n a l l y h i g h m o b i l i t i e s , such as indium o f f e r p o s s i b i l i t i e s f o r obtaining additional information m o b i l i t y as the product u B H  can be g r e a t e r than u n i t y and  magnetoresistanee i s pronounced.  antiraonide,  about the H a l l the phenomena o f  In p a r t i c u l a r , the v o l t a g e  on the  detec-  t i o n c o i l i n the H a l l m o b i l i t y experiment measured as a f u n c t i o n o f B w i l l show a maximum &t a value Hence the v a l u e  of u  H  of u B R  i e obtained  which i e r e a d i l y p r e d i c t e d t h e o r e t i c a l l y . i n termB o f the parameter B, and  e n t l y of f r e q u e n c y , geometry, s i z e and  conductivity.  independ-  6  In the  a d d i t i o n , v a r i o u B m a g n e t o r e s i s t a n e e experiments may be made u s i n g  technique described f o r determining c o n d u c t i v i t i e s .  The r e s i s t a n c e  r e f l e c t e d i n t o t h e s o l e n o i d as a f u n c t i o n o f B may be r e l a t e d ically  t o the product u B . H  theoret-  Two p a r t i c u l a r experiments a r e d e s c r i b e d ,  one a t r a n s v e r s e m a g n e t o r e s i s t a n e e experiment i n which the s t a t i c field  magnetic  i s p e r p e n d i c u l a r t o t h e plane o f t h e eddy c u r r e n t s and a second i n  which the s t a t i c magnetic f i e l d currents.  i s p a r a l l e l t o t h e plane o f t h e eddy  Measurements a r e made on a s i n g l e c r y s t a l of indium antimonide  and t h e r e s u l t s a r e compared w i t h t h e o r y .  7  CHAPTER 2 - EXPERIMENTAL TECHNIQUES  2.1  Eddy Current Losses In a Solenoid Core (a)  General Techniques.  The sample, assumed t o be of uniform  conductivity and permeability, i s placed inside a solenoid.  The solenoid  must be longer than the sample BO that the exciting magnetic f i e l d i s uniform and a x i a l .  A sinusoidal voltage generator i s connected across  the solenoid and the resistance and inductance changes r e f l e c t e d into the c o i l by the sample are measured by a Q-meter technique.  The change  i n resistance i s related to the conductivity of the material f o r two special oases i n appendices A and B.  Conductivities are measured as a  funotion of temperature by placing the solenoid and sample i n a temperature controlled o i l bath. The formulae for the change i n resistance and induotanoe of a solenoid due to a long c y l i n d r i c a l sample of radius r  0  and length l  c  are  given for the two l i m i t i n g cases aat (i)  low frequency l i m i t (u>o"^ ) r 0  a  o  = br (A19) K«(br.r  AL  and  (A15)  where and r  (A18)  c  •  radiuB  and l  c  « length of solenoid,  ( i i ) high frequency l i m i t br * 10,  AR  (A22)  Fa c i  8  n  £t=  o  -Rc('-Y)  8  AL L  and  (A21)  D  The g e n e r a l f o r m u l a e f o r A R / o L br  0  AL/LQ  ( A l 6 ) and  ( A 1 5 ) as a f u n o t i o n o f  are p l o t t e d i n F i g . 1 .  c  The formulae f o r the change i n r e s i s t a n c e and i n d u c t a n c e of a s o l e n o i d due t o a s p h e r i c a l sample of r a d i u s r frequency l i m i t b r  0  - 2 by  0  are g i v e n i n the low  :  (B7)  (B6)  (B8)  (b)  Measurement  of  A R and  AL.  The changes i n r e s i s t a n c e and  i n d u c t a n c e o f the s o l e n o i d a r e measured w i t h a Q-meter.  Let L  0  r e f e r t o the i n d u c t a n c e and a e r i e s r e s i s t a n c e of the s o l e n o i d . Q  O  =  CLJ.L^/R,,.  Similarly l e t L  S  , R  S  and Q  w i t h the sample i n p l a c e i n the c o r e .  No commercial able. the  Consequently  parallel (i)  S  an i m p r o v i s e d Q-meter was  Now  hence  ( F i g . 2a)  I 2 I  Then  Then  Q-meter o f t h e n e c e s s a r y frequency range was  tuned  Q  r e f e r t o t h e same q u a n t i t i e s  designed.  Two  tuned and the s e r i e s tuned, were c o n s i d e r e d :  Parallel  and R  avail-  circuits,  Facing  Po^e <\  FIGURE I Q- METER  (a)  PARALLEL  (b) 5 E.R I ES  TUNED  TilN/ED  CIRCUITS  Q- METER  Q-METER L R -ortnrwp—/www-, e Ea  Re-  -A/WVNA-  L  e 0  e  £  /I <<  fc) SERIES  Z-H  TUNED  Q- METER  R, R  fr  WITH  SHUNT CAPACITANCE  9  If  ^ j ^ j = Q » I, 2  J-j-j  I*  n a s  maximum very close to the  8  uJoLC ** 1.  condition Thus  Q  or  Q  LU C  =  0  i  (2)  ( i i ) Seriee tuned ( F i g . 2b) At resonance, that i e when  £ =-£ C  L  £ =  and  and f o r Q >  1  O  10  say,  e.  R  Q  Hence  OJ L0 »  =  ^  =  -|"  •  (5)  A comparison of the two c i r c u i t s shows that the input impedance of the p a r a l l e l tuned c i r c u i t ,  u) LQ, may be quite high making i t d i f f i c u l t 0  to supply a constant current. must be measured and able.  thiB  I f the current i e not kept constant, I  i s d i f f i c u l t i f no suitable ammeter  1B  avail-  On the other hand, to supply a constant voltage to the series  tuned o i r c u i t requires only that r « R.  The series tuned c i r c u i t  was  chosen and i n practice the constant voltage condition was not e s s e n t i a l since e and  £  were e a s i l y measured with vacuum tube voltmeters. -  The effeot of the stray shunt capacitanoe G lower the apparent Q.  Refer to F i g . 2c.  oircuit,  -j  _  s  of the c o i l i s to  The input impedance of the  i 7 ^  +  JUJ(;  +  R+jiuL _  i - cu L( C + C ) 4 j q j R ( C + 2  s  jujC.[i Now  -cu LC 2  5  + j l u C  s  C ) s  R j  1 Z 1 has a minimum very close to the condition 1 0 * L(C + O j  s  1.  FIGURE Z Q-METER  CIRCUIT  DIAGRAM  O  •2 ft. RG5Sfi COAX.  i  SOLENOID  •2 f+. RG-58A CO/1 X.  j  SINE WAVE G-ENERATOR VACUUM  I it R&58A  TUBE  l/OLTMETE R  COAX.  V A C U U M TUBE VOLT MET E R  10  C + C  -z Hence  t  —  C  (VC R «  But  Hence  The Q-moter was  +jOJC R S  '  L U / L C ,  S  thus  uJolC  -L  =  Q  —  .—!—^  =  uU=  =  |-  c o n e t r u o t e d ae f o l l o w s .  c  .  ^  -  A Shaeta Model 501A  sine  wave generator end two Hewlett Packard Model 400D, vacuum tube v o l t m e t e r s were connected  as Bhown i n F i g . 5»  are as shown.  The  cps t o 1 Mc.  frequency range a v a i l a b l e w i t h t h i s g e n e r a t o r was  R.  10  I n f i x i n g the v a l u e s of the c i r c u i t elements i t i s d e s i r a b l e  t h a t c e r t a i n c o n d i t i o n s be met. of  The v a l u e s of the c i r c u i t elements  Secondly,  a frequency  F i r s t o f a l l , r <.< R makes e  independent  should be chosen so t h a t the t u n i n g c a p a c i -  t a n c e C i s l a r g e compared t o G ,  the s t r a y capaoitanoe o f the c o i l ,  s  and t o  t h e combined c a p a c i t a n c e of the c o n n e c t i n g c a b l e and t h e i n t e r n a l o a p a c i t a n c e o f the vacuum tube v o l t m e t e r measuring eay C  m  = 500  p i c k u p of s t r a y 60 cps  c i r c u i t may  the  radiation. be f i x e d a t a c o n v e n i e n t  and the f r e q u e n c y v a r i e d u n t i l resonance  i s obtained.  value  However, i f  AL  t h e n C must be a c a l i b r a t e d v a r i a b l e c a p a c i t o r so t h a t the be re tuned  and  AL  0,  ThiB p o i n t i e  A l l l e a d s are s h i e l d e d t o minimize  A L i s n e g l i g i b l e , then C may  i s non-zero,  The minimum v a l u e o f  pf l i m i t s the maximum i n d u c t a n c e of the c o i l .  c o n s i d e r e d i n the next s e c t i o n .  If  E .  measured when the sample i s added.  11  (c)  Design Considerations.  The physioal dimensions of the sample  are usually f i x e d by p r a c t i o a l considerations. The solenoid i s then made s l i g h t l y bigger so that the sample may be placed inside.  The next  questions one asks are what frequency should one use and what value of inductance should the c o i l have l n order to obtain the maximum observable e f f e c t . The f i r s t design c r i t e r i o n i s that f o r good s e n s i t i v i t y one wishes Q  3  - Q /2 say, where Q 0  s  = Q of the c o i l with the sample end Q  the c o i l without the sample.  ThiB mesne t h a t A R * R , where R o  0  = Q of  0  i s the  For good quality c o i l s , Q i e approximately  resistance o f the c o l l .  0  independent of frequency.  Applying the c r i t e r i o n that A R - R  D  and assum-  ing that Q i s constant one obtains u s e f u l design formula f o r minimum o  and maximum frequency l i m i t s f o r a material of given conductivity. The c y l i n d r i c a l monocrystal  i s considered. ±  (i)  Low frequenoy l i m i t b r = (wcr/t,,) r 1  D  where r  0  o  - £,  = radius of the c y l i n d r i c a l sample.  The change i n resistance of the solenoid i e ,  (A19)  where  (M5)  hc L z  and r  c  = radius and l  c  = length of solenoid.  Requiring that  gives o  6  o- Pao;e it RflM&E COIL  OF  FIGURE  OF  t  CONDUCTIVITIES  Q-FfiCTOR  Q  K  c  •f  0  fl S  = 0.50  fl  MEflSUREABLE FUNCTION  h„ = 1.0 cr*.  (Mc. s e c . - ' )  OP  WITH  A  FREQUENCY  12  or where f  r ^  _  T L ^  -  8 2  T7^  0  /i  2 0  K  (5)  Q '  c  e  a low frequency l i m i t , 10.  ( i i ) High frequency l i m i t b r ^ 0  The change i n resistance of the solenoid i s ,  Requiring that  AR  ^  M  gives  J _  _Ro_  =  » high frequency  C  0  b/u  R„  where f  V2 K Q  limit.  A second design c r i t e r i o n i s that the Q of the c o i l should be ae large as possible i n order that the lower and upper frequency l i m i t s may be extended.  Thus i t would be possible t o study a wider range of conduct-  i v i t i e s with the same c o i l .  A high Q requires i n general that L, the  inductance of the c o i l , be large.  The minimum value of the tuning capaci-  tance, 0 , of the Q-meter l i m i t s the maximum value of the Inductance t o w  L , assuming that the frequency i s f i x e d at t M  0  m  Q  by other considerations.  i s f i x e d by the requirement that the tuning capaoitance be large  compared to 0 , the stray capacitance of the c o i l , and to the combined S  capaoitance of the connecting cable and the internal capaoitance of the vacuum tube voltmeter measuring 6 .  The radius r  c  solenoid are also fixed by the size of the sample. 2  TTn2  and length 1 Hence  o f the  15  as the maximum number of turns on the solenoid. A l i m i t a t i o n , applying i n p a r t i c u l a r to low conductivity materials, ie the requirement  that the displacement  currents be n e g l i g i b l e .  Now  conduction currents are proportional to (J and displacement currents are proportional to LU€ .  *  T where € = K  - 1/100  Hence i f one requires that  ioOTfeTe.  =  ^  say.  Then  (9,  Mo ohn m,  r e l a t i v e d i e l e c t r i o constant of the sample and f  frequency l i m i t due to displacement currents.  D  » high  The r e s u l t s of t h i s section  are summarized i n Pig. A. (d)  Frequency Range.  Maximum s e n s i t i v i t y w i l l be aoheived when the  f r a c t i o n a l change i n resistance,AR/R i e a maximum. mmHnmm  i  B  The reason f o r t h i s  seen by considering the two oompeting f a c t o r s .  At low frequen-  cies the e x c i t i n g magnetic f i e l d i s uniform throughout the material and the eddy currents increase l i n e a r l y with radius and frequency.  At higher  frequencies, that i s when the penetration depth l/b = l/^cr/^) becomes small oompared to the physical dimension r  0  of the sample, the e x c i t i n g  f i e l d i s attenuated as i t penetrates into the sample and hence the eddy currents are concentrated i n the surfaoe, again decreasing A R . v a r i a t i o n of Afy/a>L and A V 0  0  The  with frequency are shown l n F i g . 1 f o r the  long monocryetalline c y l i n d r i c a l sample case. The low frequency region appears to be the moBt desirable one from an experimental viewpoint since the eddy currents are distributed more u n i formly than i n the high frequency region.  A disadvantage of the high  frequency region i s that the current i s concentrated i n the surface end  14  henoe any surface i r r e g u l a r i t i e s ,  chemical layers and other inhomogene-  i t i e s w i l l effeot the value of 0". A further purely experimental d i s advantage of using the high frequency region i s that AL must be measured i n addition to A Q , thus increasing the sources of error.  Considering  the f a c t o r s mentioned above i t was decided that the beet frequency to use would be the one which made br 6  Here A R  i s a maximum under  the constraints that A L / L ^ 1% and that the low frequency formula f o r A R be accurate t o better than 1%.  This frequency w i l l be denoted by f , Q  15  2.2  QroBeed Magnetic Plaid H a l l Mobility Measurements (a)  Theory.  magnetic f i e l d B  Consider a spherical sample exoited by an alternating  i 0  .  C i r c u l a t i n g eddy currents J^, are then set up i n the  sample i n the x-y plane, as shown i n Pig. 5 «  A s t a t i c magnetic f i e l d B  a  i e now applied.  x  The force on the current c a r r i e r s i s ,  V  and the current density i n the x-y plane i e ,  where n ° density of ourrent c a r r i e r s .  Hence  n ^ £  e  ff.  The component of current along the z-axie i e ,  =  where z" l e a u n i t veotor. 0  U  Note that  H  Q  %  3"  COS (j)  4  (11)  i s a maximum where i t crosses the  x-axie and zero where i t crosses the y-axie.  Hence the eddy current  planes are i n e f f e c t turned through an angle tan u B > M  between the internal and external magnetic f i e l d s , rotated through the H a l l angle tan u B . H  2, o ,  x  The difference  x  B . - ff ?  lo  w i l l also be  The r e s u l t i e a component of  magnetic f i e l d , B , along the y-axie, where B  3 y  u B (B j - B H  x  2  i 0  ) . Thie  Facing" Pa.^e 16  Fl G-U R £  HALL («J  (t)  FIELD  AND  CIRCUIT  CURRENT  5  MOBILITY  MEASUREMENTS  RELATIONS  OlflG-RflM  cw  1  e  >WAWW  v  •e-  16  field  i s detected  thus  by a c o i l  p l a c e d a l o n g the y - a x i B  and t h e H a l l m o b i l i t y  obtained. A r i g o r O U B d e r i v a t i o n f o r J i n the general  E, p a r t 5 u e i n g t h e c o n d u c t i v i t y t e n s o r  case i s g i v e n i n appendix  d e r i v e d i n appendix D.  The  c o n d u c t i v i t y t e n s o r was d e r i v e d under the assumptions o f a v e l o c i t y endent r e l a x a t i o n time, The is  X  , and t h a t l/m i s a s c a l a r .  Refer  indep-  to F i g .  5« D  r e s i s t a n c e r e f l e c t e d i n t o c o i l 2 when the sample i e e x c i t e d by c o i l 1 c a l c u l a t e d i n t h e low frequency  special (i)  limit br  = (ui(T^o) r  - £ , f o r two  Q  cases: A cylinder of radius r  and l e n g t h 1  L B  and  Q  ( i i ) a sphere o f r a d i u s r  z.  »  r gives o  H  gives  c  Uo  M  fn,  where n, = number o f turns/m on c o i l turns/m on c o i l  o  I +(u B>f  R  10  —  , where l  Q  Ho )  1 and n  z  U h  -  . * S  ^  ,  (E5o)  number o f  2.  With r e f e r e n c e t o F i g . 5b,  £>/i , i f i  s  (  2  - 0.  AIBO  L  and  i f 0^ > > 1, t h e n  R  Hence  and f o r t h e two  cases:  Z I  =  OJL,  (i4)  17  (i)  A cylinder gives  £± _ < e  and  (bftof /U m, m^T tuL " Li  a * B, i t (u B f  (Uof  u  H  (15)  v  ( i i ) a sphere gives £2 _  e,  2 0 '  m, m^%ttio  B  H  L7~  7  (16)  x  ' 1 V(U L\) ' H  Z  A further extension of the theory makes i t possible to make measurements independent of the parameter b and independently o f the s i z e and the shape of the sample at low frequencies. be known. the  In partioular, i f R  |(  Hence the conductivity need not  Is the resistance change r e f l e c t e d into  e x c i t i n g c o i l , ae given by (E22) and (E25), then  (E29) and (E50). Thus on combining (15) end (17),  using  62  _Rn_  =  (U B>Va  fjh  H  ( 1 8  "Mi.--n-Jwhere Q  $|  Finally,  L _ \  (_\ s  ing  ^  without the sample.  0|  fa  H BJ/&  (  1  9  )  Qo, / m, ' I + Cu„BJ>£'  £. ~ V Q , If R  <l)  i s the Q of c o i l 1 with the sample and Q §z -  )  , the resistance r e f l e c t e d into c o i l 2 when used ae the e x c i t -  c o i l , i e measured, rather than R „ , then  R  a  =  R .2L . t t  nil giving  =  £.  wL,  ,  ^ H 6 J / 2  1 + (u £V)>£  ( 2 0 )  rt  . ("wSJ/a  ir(a 6 )7 H  x  . 2  (21)  18  Now  AR  where Q  S I  UJL, ( J .  =  _  ( 1 )  J_ ^ .  i e t h e Q of c o i l 2 with the sample and Q_ without t h e sample. oz  Finally,  , ,  £  £, (b)  A 2  I  /TLLa  A  0>J  \Q  5<i  Design C o n s i d e r a t i o n s .  B,\A l£  (UH  (22)  nvL, i + (a Bv)y2 H  C o n s i d e r t h e d e s i g n o f an experiment t o  measure the H a l l m o b i l i t y , u , o f a c y l i n d r i c a l  sample i n t h e low f i e l d  H  2.  r e g i o n where ( u B ) < < l .  In particular,  H  the most u s e f u l f o r m u l a f o r  e x p e r i m e n t a l work i s ,  §1  I i  -  ]  l  ('in Sv.)/2  inA±  (  where c o i l 1 i s t h e e x c i t i n g c o i l and c o i l 2 i s t h e pickup One  2  2  )  coil.  r e q u i r e s t h a t t h e pickup v o l t a g e , ^ • be a maximum i n o r d e r t o  obtain greatest s e n s i t i v i t y . 1/Qs2 ~ l / Q c i be a maximum.  T h i s r e q u i r e s t h a t B , h /n , x  z  z  E,n,/L, and  The p h y s i c a l s i z e o f the sample w i l l be  assumed t o be f i x e d a t r a d i u s r  o  and l e n g t h 1  Q  by o t h e r c o n s i d e r a t i o n s .  The f o l l o w i n g l i m i t s t o t h e parameters c a n then be s e t t (i)  b r = f i s the l a r g e s t v a l u e a t which t h e low frequency 0  i m a t i o n i s a c c u r a t e t o b e t t e r than 1$.  f.  = 8TTS 'A  O  —  -  (-L  approx-  Thus a t  (10)  "-o  —  \  (U  i s a maximum. ( i i ) B^ i s l i m i t e d t o approximately 0.6 W nf^by t h e d e s i g n o f t h e magnet and by t h e s i z e o f the sample. ( i i i ) l^/ti  z  n ^ V 2 , where V . - volume o f c o i l 2, i s l i m i t e d by t h e C  C2  19  shunt capaoitanoe C uency f Now 0  2  2  2  of c o i l 2.  I f o o i l 2 resonates at f r e q -  due to 0 » then one would require that f 2  2  - 10f  o  say.  includes the internal capacitance of the vaouum tube  voltmeter and of the connecting cable and w i l l be of the order of 50 t o 100 pf.  Henoe i f the minimum volume of c o i l 2 which  w i l l hold the sample i s V . then, c2  i  But  a  n  giving  e Max  =  !  , ,  -  2(Tkc (  Ac  5  (10)  V_  iQVcz/ '  as the maximum number of turnB/m. ( i v ) The maximization of ^ n / L o c  Consider the c i r o u i t i n  (  Pig. 2a.  At resonance, Z = Q UJ L,. (  Hence  £^ £, m,  giving  Cu L, Q ,  _ -  L,  LO T T C Q ,  R  fi  * tuLQ,  which i s maximized when  OJU, corresponding to an impedance matoh of the generator to the turned inductor. Note that n  (  i s defined as the equivalent number of turns/m on a  solenoid producing the  Bame  field.  In p a r t i c u l a r , f o r a solenoid  B,  =  yu m 0  ]  X,  ,  while f o r a Helmholtz pair, the f i e l d at the center i e ,  (2J)  20  .  5  (25)  r  where 21 * length of eide of square c o i l , 2a » separation of the two c o i l s , N - turns per c o i l , and  i « current per c o i l (  m, =  Hence  ±2L  ^  (26)  (f+0^*1*+a  T  2  Now n, w i l l be constant only i n a region near the oenter of the Helmholtz p a i r and hence the sample should be small enough so that i t i s contained i n t h i s region. Large error signals w i l l be detected i f the e x c i t i n g and pickup are not p r e c i s e l y perpendicular.  coils  Hence i t i s p a r t i c u l a r l y important that  the c o i l s be adjustable i n order to achieve t h i s condition.  In p a r t i c u l a r ,  i f the c o i l s are out of perpendicular by an angle 0, then the error signal  OD  =  =  where V  B ,n ?  2  \/  L, c u {jUL m,  c z  sin 0  m  0  2  Vc  2  ) s<n9,  (27)  volume of c o i l 2.  s c 2  Now c" 2 » i, ^ R r i  = t, UJ tor  (b^f(U  H  BMMO  m^TtL  a cylinder at low frequencies, ( b r - £ ) , and f o r low f i e l d s such Q  that (u„B )V< 1. y  Thus,  (E29)  21  Vcz £2  where the volume o f the  V  eample, V  Sin e  16  (28)  u B„(bU* H  s  -  s  2.  I f say 7  ct  = 27  s  and  (br ) o  = ^,  then  b^f s in 0  f  2  o  0/57.5.  (29) 2.  Consider a t y p i c a l material with u =  0.10  -I  1  ~2  -I  V  sec  and  0.5  B,=  W m .  I t Is seen t h a t an angle of 6 = g r e s u l t s i n an e r r o r s i g n a l f = 11 £ and t h a t the s e r i o u s n e s s o f t h i s type of e r r o r i n c r e a s e s w i t h m a t e r i a l s e  of decreasing  mobility.  A source o f e r r o r can be r e s i s t a n c e and i n t o the  c o i l s from the p o l e s  Helmholtz c o i l s ,  s o l e n o i d and  i n d u c t a n c e changes r e f l e c t e d  and frame o f the e l e c t r o m a g n e t . sample are c e n t e r e d  induced eddy c u r r e n t s w i l l  c a n c e l by  symmetry.  c o i l s by  I t should  pieces.  of the Helmholtz p a i r was  I n p a r t i c u l a r , a 1%  the that  permeability  change i n reBonant frequency  noted when the f i e l d was  increased  from 0 to  0.5  2  Another source of e r r o r i s e l e c t r o s t a t i c c o u p l i n g . may  the  be noted  t h e s e changes can be a f u n c t i o n of B because of the v a r i a b l e o f the s t e e l pole  If  a c c u r a t e l y between the  magnet p o l e s the r e s i s t a n c e changes r e f l e c t e d i n t o the  W m  z  be e l e c t r o s t a t i c a l l y  n o n - i n t e r s e c t i n g wires. point  The  pickup  coil  s h i e l d e d by e n c l o s i n g i t i n a g r i d of o r t h o g o n a l They are  connected t o a common ground only  so t h a t eddy c u r r e n t s are not  s e t up  around the  grid.  at  one  22  (c)  D e s c r i p t i o n o f Apparatus.  The s t a t i c magnetic f i e l d s were  produced by a Newport Instruments Type A E l e c t r o m a g n e t .  The e l e c t r o -  magnet was c o n t r o l l e d by a V a r i a n A s s o c i a t e s Model V2500A Power Supply and Model V2J01A Ourrent R e g u l a t o r ,  The m i l d s t e e l p o l e p i e c e s were 10  cm i n diameter and t h e s p a c i n g was c o n t i n u o u s l y a d j u s t a b l e from 0 t o 11.0 cm.  The maximum continuous a l l o w a b l e c u r r e n t per winding f o r the e l e c t r o -  magnet was 4 emp.  These windings were connected i n s e r i e s s i n c e t h e  maximum c u r r e n t a v a i l a b l e from the power supply and r e g u l a t o r was a l s o 4 amp.  Some u s e f u l maximum f i e l d  A i r gap (cm)  Field strength a t maximum r a t e d c u r r e n t o f 4 amp per c o i l (W m-) 2  Approximate r a d i a l d i s t a n c e from c e n t e r a t which f i e l d f a l l s o f f by 1% from c e n t r a l v a l u e (cm)  10 5  O.56  5  0.77  2.5  0.87  2  0.97  and homogeneity  d a t a i s g i v e n below.  Approximate r a d i a l d i s t a n c e from c e n t e r a t which f i e l d f a l l s o f f by 10% from c e n t r a l v a l u e (cm)  1.5  3.5  2.4  4.1 —  4.6  5.6  The s t a t i c magnetic f i e l d s were measured u s i n g a Radio Frequency t o r i e s , Model 1295 G-aussmeter.  I t s range i s 0 . 0 1 t o 2 W m  z  Labora-  i n 9 ranges  and i t has s t a n d a r d magnets f o r c a l i b r a t i o n . A p a i r o f Helmholtz c o i l s were b u i l t  around the p o l e p i e c e s and a  s o l e n o i d was mounted on the frame o f t h e Helmholtz c o i l s , ween the c o i l s and t h e magnet p o l e f a o e s .  c e n t r a l l y bet-  I n p a r t i c u l a r , t h e Helmholtz  p a i r were r i g i d l y mounted on a wooden frame, t h e geometry  o f which ensured  o r t h o g o n a l i t y w i t h r e s p e c t t o t h e s t a t i c magnetic f i e l d .  The s o l e n o i d  23  was suspended from t h e t o p Helmholtz c o i l  by 2 a d j u s t a b l e b r a s s b o l t s .  The a x i s o f t h e s o l e n o i d was p e r p e n d i c u l a r t o t h e axes formed by t h e l i n e s j o i n i n g t h e c e n t e r s o f t h e magnet p o l e s and t h e c e n t e r s o f t h e Helmholtz coils.  The a d j u s t a b l e b o l t s enabled one t o o b t a i n a n u l l  pickup c o i l with the s t a t i c f i e l d  zero.  were used throughout t o minimize f i e l d  s i g n a l on t h e  Brass r a t h e r t h a n B t e e l  fasteners  distortions.  The Sheets s i n e wove g e n e r a t o r and t h e 2 Hewlett-Packard vacuum tube v o l t m e t e r s mentioned  i n 2.1b were u s e d .  Two 2 f t l e n g t h s o f RG58A-U  c o a x i a l cable j o i n e d t h e Helmholtz p a i r , connected i n p a r a l l e l , vacuum tube v o l t m e t e r and t h e s i n e wave g e n e r a t o r .  to a  A oondensor box was  connected a c r o s s t h e g e n e r a t o r and t h e o i r c u i t was tuned f o r p a r a l l e l resonance t o a c h e i v e maximum power i n p u t i n t o t h e c o i l s a B i n d i c a t e d by a maximum i n the  £ , the voltage aoross the c o i l s . f  The c o u p l i n g between  Helmholtz p a i r was n e g l i g i b l e . A t h i r d 2 f t l e n g t h o f RG58A-U connected t h e p i c k u p c o i l  second vacuum tube v o l t m e t e r .  t o the  The Helmholtz p a i r , t h e s o l e n o i d and t h e  c o a x i a l l e a d s were connected t o a common ground near t h e s o l e n o i d . S h i e l d e d l e a d s were such as 60 c p s .  UBed  wherever p o s s i b l e t o minimize s t r a y p i c k u p ,  24  2.J  Eloctrodoloss Magnetoreslstance Measurements  When the product u B N  becomes comparable v i t h or greater than unity,  the phenomena of magnetoreslBtenoe i e pronounced. be  Two  special cases w i l l  considered.  (a) fields, B  Transverse Magnetoreaistance. z o  and B , 2  The e x c i t i n g and s t a t i c magnetic  are p a r a l l e l , r e e u l t i n g l n the s t a t i c f i e l d  perpendicular to the eddy current planes.  B  being  w i l l exert a r a d i a l foroe  2  on the current causing i t to oonoentrate i n the outer or inner layers of the sample, depending on the signs of B  and B^.  2o  Henoe i f the resistance  r e f l e c t e d i n t o the solenoid by the sample i s measured i t w i l l decrease with increasing B . 2  In p a r t i c u l a r , two oases have been calculated f o r  the low frequency region, br  -  Q  assuming constant  X and that l/m i s  a scalart (i)  A cylinder of length 1  and radius r  Q  wU where and  K  ( l i ) a sphere of radius r  -  D  8  ^ n  c  gives,  l+ (u B,) H  a  '  ° ^'  (  E  U  )  (E15)  gives,  . 3  where  K  =  4  .  (El6y  The design c r i t e r i a f o r a suitable c o i l and a Q-meter to measure A R ( B ) , are the same as i n 2.1. 2  However oertain precautions must be  taken because of the following d i f f i c u l t i e s .  F i r s t of a l l , i t i s  25  p o s s i b l e t h a t eddy c u r r e n t s w i l l be induced hence r e f l e c t a r e s i s t a n c e i n t o the c o i l . f u n c t i o n of B and  ?  s i n c e the p e r m e a b i l i t y of the s t e e l i s a f u n c t i o n o f B , ?  both w i t h and without  suitable non-metallic of the two magnet  (b) 0  T h i s e f f e c t would a l s o be a  i t 1B best e l i m i n a t e d by measuring t h e Q o f the c o i l  of  B^  i n the magnet pole p i e c e s and  t h e sample.  The s o l e n o i d i s s e t up i n a  stand w i t h a x i s a l o n g the l i n e  j o i n i n g the center  poles.  Mixed M a g n e t o r e s i s t a n c e .  and B , are p e r p e n d i c u l a r ,  of the eddy c u r r e n t .  eddy c u r r e n t p l a n e s  The e x c i t i n g and s t a t i c magnetic  r e s u l t i n g i n the s t a t i c f i e l d b e i n g  y  t o t h e plane  BB a f u n c t i o n  as e x p l a i n e d  Hence B*. w i l l i n 2.2.  and a z-component appears.  appendix E, p a r t 5 » since there solenoid two cases  The x-component of the eddy  (i)  2  i s measured i t w i l l  Hence i f the r e s i s t a n c e r e f l e c t e d i n t o t h e  decrease w i t h  A c y l i n d e r of length 1  where  x  and r a d i u s r  k  _  6  ( i i ) a sphere o f r a d i u s r  V  _ 5  ~  i  ri° °  In p a r t i c u l a r ,  limit, br  o  ^  c  gives  (E22) F  (  U  N  B  X  )  1  (£15)  L  /u  where  increasing B .  and t h a t 1/m i e a s c a l a r :  c  and  TheBe r e s u l t s are d e r i v e d i n  have been c a l c u l a t e d f o r the low frequency T  currents  The z-component has no e f f e c t on the e x c i t i n g s o l e n o i d  i s no e f f e c t i v e E .  assuming c o n s t a n t  parallel  cause a r o t a t i o n o f the  c u r r e n t s w i l l be u n a f f e c t e d , whereas the y-component o f the eddy i s decreased  fields,  L  2  gives  .  4- ^  3  (El6) '  26  The remarkB c o n c e r n i n g e x p e r i m e n t a l again here. pieces w i l l line  However,  difficulties  i n (a) a p p l y  t h e e f f e c t of t h e eddy c u r r e n t s induced  i n the p o l e  be much s m a l l e r s i n c e t h e B o l e n o i d a x i s i s o r t h o g o n a l t o the  j o i n i n g t h e c e n t e r s o f the pole p i e c e s .  symmetrically  I f the s o l e n o i d i s s e t up  between the p o l e p i e c e s t h e s e e f f e c t s ehould c a n c e l .  27  CHAPTER 5 - EXPERIMENTAL RESULTS  5.1  C o n d u c t i v i t y Measurements on  Graphite  C o n d u c t i v i t y measurements were made on commercial g r a p h i t e eddy c u r r e n t t e c h n i q u e of 2.1  and by a 4-electrode t e c h n i q u e .  by the Thus i t  was p o s s i b l e t o check the accuracy o f the e l e c t r o d e l e s s t e c h n i q u e . Graphite  was chosen because i t was e a s i l y  obtained  i n the form o f a  l o n g s o l i d homogeneous c y l i n d e r and secondly because i t e c o n d u c t i v i t y i s o f the same order discussed  i n part  o f magnitude as t h a t of the s i l v e r  I I . The sample of r a d i u s r  cm W 8 8 prepared from the c y l i n d r i c a l  6.1  (a) (A19)  E l e c t r o d e l e s s Technique.  where  T  K,  and  The l  c  r a d i u s o f the s o l e n o i d r =  10.5  i n 2.1  cm.  and P i g . 2b and 5  Experimental  w  a  128 197  =  1.5  s  c  1  J_  K  ho  I  (50)  (5D  '  c  ic  (A15)  cm and the l e n g t h of the s o l e n o i d mho m  Bee  .  The Q-meter  described  used t o measure the Q ' B .  (J-  l/Q,  5.11*10"* 5.ii*l0" ' 2  o.i5»io  _1  =  core o f a 6-volt dry c e l l .  results:  f (kc)  22.7  c  cm and l e n g t h l  - i>, as  jUc  ho  2.9*10  Hence K, -  o  M  -  = 1.0  salt  The c o n d u c t i v i t y i s g i v e n by (1) and  i n the low frequency l i m i t b r AT  Q  selenide  6.58-10'12.02X10"15.55X10"2  2  2  (mho m"')  1.9*10 2.0*10* 1.75*10* 4  0.25 0.56 0.66  28  (b)  4-Eleotrode Technique.  e l e c t r o d e s are a t t a c h e d  t o c o n t a c t r e s i s t a n c e and  method. silver  The  ends and  standard method i n which the  t o the ends of the  are p l a c e d a l o n g the sample was due  The  two  used.  narrow bands on the sample were p a i n t e d  with  p a i n t i n order t o o b t a i n good c o n t a c t w i t h the e l e c t r o d e s .  Now  where r  = 1.0  (52) cr  cm,  and the d i s t a n c e between the p o t e n t i a l probes 1,  5*2  results:  (kc)  £  (mv)  I  The  discrepancy  o f 2h%  methods i s c o n s i d e r e d  e l e c t r o d e l e s s method. (i)  (J-  («/mp)  (mho  m") 1  2.4*10* 2.7 * 10^ 2.6-10" 2.6-10" 2.4>10'  1.00 5.27 5.6o 1.57 1.00  6.90 20.2 25.5 10.0 6.85  0 0.1 1 10 20  4  f  between the average r e s u l t s o b t a i n e d  from the  s a t i s f a c t o r y because o f e r r o r s i n h e r e n t  i n the  In p a r t i c u l a r ,  F r i n g i n g of the e x c i t i n g f i e l d ,  due  to the l o n g s o l e n o i d approx-  i m a t i o n not b e i n g s a t i s f i e d , r e s u l t s i n too low and  =  Hence  Experimental  two  p o t e n t i a l probes  r e c t i f y i n g c o n t a c t s i n h e r e n t i n a 2-electrode  Tike  f  two  T h i s method e l i m i n a t e s the e r r o r s  4= *= cm.  sample and  current  a v a l u e of observed  ( i i ) D i s r u p t i o n of the eddy c u r r e n t p a t t e r n s a t the ends of  the  sample, an e r r o r which would decrease as the l e n g t h to r a d i u s r a t i o the c y l i n d e r was observed  CT.  i n c r e a s e d , would a l s o r e s u l t i n too low  a value  cr,  of  of  29  5*2  Electronic Properties  Conductivity  o f N-Type  Germanium  and H a l l m o b i l i t y measurements were made on a s i n g l e  c r y s t a l o f n-type germanium t o i l l u s t r a t e t h e t e c h n i q u e s d e s c r i b e d and  2.2.  The c r y s t a l , w h i l e somewhat i r r e g u l a r i n shape, was assumed t o  be s p h e r i c a l f o r purposes o f c a l c u l a t i o n .  In a d d i t i o n , t h e c r y s t a l was  p u r e r a t one end t h a n the o t h e r and hence o o n t a i n e d i m p u r i t y t i o n gradients. position.  l n 2.1  concentra-  Thus the c o n d u c t i v i t y and m o b i l i t y would be a f u n c t i o n o f  However these o f f e o t s a r e n o t c a l c u l a b l e and i t was  assumed  t h a t the t h e o r y gave average v a l u e s f o r t h e c o n d u c t i v i t y and m o b i l i t y . A l l measurements a r e made i n the low f r e q u e n c y r e g i o n b r where  Q  = (toOyU.)^r  -  o  i s t h e r a d i u s o f t h e sphere. (a)  Conductivity  Measurements.  The c o n d u c t i v i t y i s g i v e n  by  ^= r ( i - i> where  K,  L°_^  =  K  0  /,  K =  and  5  (55)  ,  £TTyU ho h  (50)  s  .  (B8)  i n 2.1b and P i g . 2b and 5 was used t o measure the  The Q-meter d e s c r i b e d Q's o f t h e c o i l . Experimental data:  r  0  and  - 1.2  cm and K  ft-  _  °  5  - 0.47, g i v i n g K, » 1.9*10  1.1 * io' (  "  f  \  I  ^  ( ~Q, " Q j  - i n  h  o  m  -  The Q-meter r e a d i n g s a t f - 690 ko a r e 1/0^ = and  1/Q  0  = 0.0124.  Henoe CT  s  5.1»10  3  mho m" s e c ,  mho  m~'.  0.126  50  (b)  Hall Mobility Measurements.  U  =  Z  The H a l l mobility i s given by  (22)  Qo2. Q s z  Vz  f o r ( u B f « 1, the quantities being defined i n d e t a i l i n 2.2a H  x  The apparatus and procedure was as described i n Experimental datat  and Pig. 5b.  2.2c.  L, = 7.5/^h  n, * 40 turns  h  = 15yuh  n^ * 7.7*10 turns m  f  * 500 kc  * 0.10  W nf  V, a 10.5 v  » 0.51  mv  z  Q 02  111  Q =  m"  1  1  10.9  S2  2. -' Substituting the above data i n (22), one obtains u  - 0.12  H  m  v  see.  Also of i n t e r e s t i e the donor density 2.3 N  D  ~  A~ u e  =  1.6  * 10  3 w~.  H  The experimentally determined value f o r u  H  may be compared with the 2.3  value given by Prince (1955). \x « 0.20 n  m  2.  v  -1sec-•and  In p a r t i c u l a r , f o r N »  UH/U* - 0.9,  0  giving u  H  - 0.18  -3  10 m  m , he quotes  2 - 1  v  -1  sec .  The  agreement between the results i s s a t i s f a c t o r y because of the large errors possible due to the irregular shape of the sample and the of N  Q  through the Bample.  non-uniformity  These errors are most serious i n determining  the conductivity and hence i n d i r e c t l y i n obtaining N . D  51  J.J  Electronic Properties of N-Type Indium Antimonide Conductivity, magnetoresietance and Hall mobility measurements were  made on a single c r y s t a l of n-type indium antimonide using the electrodel e s s techniques of 2.1,  2.2 and 2.5 and by dc 4-electrode techniques. The  exceptionally high electron mobility of indium antimonide made i t possible to aoheive experimentally, values of ^ B ^ l , magnetoresistanee was pronounced.  and thus the phenomena of  The sample was irregular i n shape and  was approximated by a cylinder f o r purposes of calculation. (a) (i)  Conductivity Measurements.  Electrodeless Technique.  f u l l y i n 2.1.  In p a r t i c u l a r the conductivity  K, =  where Now r  c  * 1.6 cm, l  K, s  4.6«lo"  Q, •  27.9 and Q  (ii)  The method and apparatus are described  c  '°  ** k  .  (51)  = 5«0 em, r « 0.50 cm and 1  0  e  = 2.7 cm, giving  mho nf sec"'. The Q-meter readings at f = 1  0  »  59.6.  Hence  DC 4-Electrode Technique.  was used.  7  kc are  mho m ', and b r * 0  The same technique as described i n 5«lb  and the distance between the potential probes 1, = 1.7 (T =  Thus  l . l * I 0 _ " L i^no rn\ Z  Experimental r e s u l t s gave £ * 7.8 mv f o r I = 1.0 amp, giving  Q  O.96.  S T  Hence  where A - 1.0 cm  0~ = 1.8*10^  264  - 2.2*10 mho m .  om.  Facing  Pa £e  32  T R f l M5 VER5E  F I G- U R E  (,  MA G N E TO RE 51 S Tfi WC E  CURVES  32  (iii)  DJBouBBion o f R e s u l t s .  s a t i s f a c t o r y because i n $.1,  as mentioned In  The agreement between t h e r e s u l t s i s  o f t h e e r r o r s i n h e r e n t i n the e l e c t r o d e l e s s method, and because  o f the i r r e g u l a r shape o f t h e sample.  p a r t i c u l a r , r , the r a d i u s o f t h e c y l i n d e r , can o n l y be e s t i m a t e d t o Q  ± 25$.  Another  such t h a t b r  G  e r r o r i s due t o making t h e measurements a t a f r e q u e n c y  - 1,  r e s u l t i n g i n an e r r o r o f h% i n the low f r e q u e n c y  formula.  (b)  M a g n e t o r e s i s t a n c e Measurements.  Two magnetoresistance  experi-  ments a r e made. (i)  T r a n s v e r s e MagnetoreBJstance.  The s t a t i c magnetic  p e r p e n d i c u l a r t o t h e plane o f t h e eddy c u r r e n t s . r a t u s were d e s c r i b e d i n 2.1  and 2.J.  AR(B ) _  mentally  whereas t h e o r e t i c a l l y ,  A  The procedure and appa-  In p a r t i c u l a r , one measures e x p e r i -  J  g  f i e l d B^, i s  |_  m  02.  n a  where In F i g . 6,  (b/te) ti io 1  a  the e x p e r i m e n t a l d a t a l / C ^ - 1/Q  p ^ — ^ — g - y i s plotted against u B . z  H  ?  Q  i s p l o t t e d a g a i n s t B^, and  I t i s seen t h a t the e x p e r i m e n t a l  r e s u l t s do n o t f i t t h e t h e o r e t i c a l curve which was d e r i v e d on the b a s i s o f a c o n s t a n t mean f r e e time, f The of  Lorentz-Sommerfeld  , between c o l l i s i o n s .  t h e o r y o f c o n d u c t i o n based  on the assumption  a c o n s t a n t mean f r e e path w i l l be c o n s i d e r e d f o r comparison.  (1954) g i v e s ance e f f e c t .  a curve f o r  Wilson  CT v e r s u s u B f o r the t r a n s v e r s e m a g n e t o r e s i s t -  The assumptions  H  a r e f r e e e l e c t r o n s i n one energy  band,  a c o u s t i c l a t t i c e s c a t t e r i n g , c o n s t a n t mean f r e e p a t h and c l a s s i c a l s t a t i s t i c a l  33  These r e s u l t s are plotted i n Pig. 6 f o r comparison with the experimental r e s u l t s and with the constant  T  theory.  The theory i a extended to a  two band model f o r indium antimonide by Fiaoher and MaoDonald (195®)» but the r e s u l t s are not required here since the sample i s n-type and hence the conductivity i s predominantly due t o electrons since u ^ ^ u ^ , (ii)  Mixed Magnetoresjetence.  The s t a t i c f i e l d B , i s p a r a l l e l to the x  plane of the eddy currents, henoe the resistance i s a mixture of onehalf transverse magnetoresistance  and one-half longitudinal magnetoresist-  anoe as shown i n appendix E, part 2. described i n 2.1 and 2.5.  The prooedure and apparatus were  In p a r t i c u l a r , one measures experimentally  aa i n ( i ) , while t h e o r e t i c a l l y  where  CL  — 6  Ac / 2  e  In F i g . 7, the experimental data, 1/Q - l/Q, i s plotted against B^, and (  l+(tU3»%£  a  i  a  p i t t e d against u a .  0  I t i s noted that the t h e o r e t i c a l AR(B^)  saturates at AR(0)/2 while the experimental values appear t o keep on decreasing. The constant mean free path theory w i l l also be considered f o r compariaon with the experimental r e s u l t s .  Now the t o t a l resistance measured  w i l l have equal contributions from the longitudinal and transverse components.  Hence the curve given by Wilaon (195*0 f o r the tranaverae e f f e c t  may be uaed.  Olickeman (1957) states that there i s no H a l l e f f e c t and no  magnetoresistance  i n the longitudinal case as long as mi* and " f are i s o t r o p i c .  The  curve showing t h e combination o f t h e s e two r e s u l t s i s p l o t t e d i n P i g . 7«  (iii)  Discuesion of Results.  The e x p e r i m e n t a l t r a n s v e r s e m a g n e t o r e s i s t -  anee r e s u l t s agree v e r y w e l l w i t h t h e t h e o r e t i c a l c o n s t a n t mean f r e e curve i n d i c a t i n g t h a t the mean f r e e path h y p o t h e s i s i s s a t i s f i e d . t h a t t h e o r e t i c a l l y as u B - * ° ^ , A/t^,-» H  and c o n s t a n t mean f r e e  0.868  path.  mean f r e e path ourve t h e o r y a t v a l u e s o f u B < H  path curve s a t u r a t e s e t a v a l u e o f However, f o r u^B  fields.  (1957)  I n the l i m i t  Cr/cr  o  r e s u l t s agree w i t h t h e c o n s t a n t 1.  The c o n s t a n t mean f r e e  » 0.95^  assuming s p h e r i c a l  1 t h e e x p e r i m e n t a l curve decreases  >  and f a l l s below t h e c o n s t a n t Glicksman  T  curve which s a t u r a t e s a t  oommente on t h e magnetoresistanee of i n f i n i t e f i e l d s ,  r e l a x a t i o n time and t h e energy  surfaces.  i s independent  The  s u r f a c e s K * 1,  The s a t u r a t i o n  9TT/5  2  (  on t h e energy dependence o f T.  magnetoresistanee  97T/96.  24,000Mo. to  longitudinal and can  For spherical i s zero.  may be much l a r g e r s u r f a c e s and does depend  F o r example, I f T^(energy) , t h e v a l u e  I t i s a l s o noted t h a t t h e s a t u r a t i o n v a l u e s o f  a r e independent  Dresselhaus, Kip, K i t t e l resonance  0.50.  a t s t r o n g magnetic  and t h e l o n g i t u d i n a l magnetoresistanee  energy"^) f o r s p h e r i c a l energy  of 97T/J2 becomes  *  o f t h e energy dependence o f X  s a t u r a t i o n t r a n s v e r s e magnetoresistanee,  than  rapidly  on t h e form o f t h e  d i r e c t l y p r o v i d e i n f o r m a t i o n on K, t h e i s o t r o p y f a c t o r . energy  energy  t h e magnetoresistanee i s  e x p e c t e d t o s a t u r a t e , t h e s a t u r a t i o n v a l u e depending  magnetoreeistance  Note  f o r s p h e r i c a l energy s u r f a c e s  The e x p e r i m e n t a l mixed magnetoresistanee  surfaces.  path  o f the H a l l  and Wagoner  mobility.  (1955) nave  observed  cyclotron  i n a s i n g l e c r y s t a l o f indium antimonide a t a f r e q u e n c y o f about One resonance  an e f f e c t i v e mass m* »  line  observed a t low f i e l d  (0.015  i  0.001)m,  s t r e n g t h s corresponded  and t h e resonance was i s o t r o p i c  Facing  .Pag-e ? 5  FIGURE  HALL  0.50  0-60  0.70  MOBILITY  0.80  oao i.o u  H  B  8  CURVES  1.5  2.0  2.5  55  under r o t a t i o n i n a (100) plane, to within experimental error.  It ie  i n f e r r e d from the i s o t r o p i c maee that the resonance i s associated with conduction electrons.  Preliminary observations on two resonance l i n e s  with heavier masses m* » 0.18m behaviour have been made. with holes.  and m* > 1.2m,  and which show anisotropic  I t i s suggested that they may be associated  Hence f o r n-type Indium antimonide i t appears to be just-  i f i a b l e to take K * 1. The experimental mixed magnetoresistanee curve at high f i e l d s cannot be explained because i t does not appear to saturate.  The long-  i t u d i n a l magnetoresistanee component should be zero since the assumption K = 1 i s s a t i s f i e d according to Dresselhaue et a l the  (1955)*  I  n  addition  transverse magnetoresistanoe curve of F i g . 6 appears to f i t the K = 1, j.  X oc(energy)  (c) the  assumptions.  Hall Mobility Measurements.  H a l l m o b i l i t i e s were measured by  electrodeless technique and by a do 4-electrode technique,  (i)  Crossed Magnetic F i e l d Technique.  to the plane of^ the eddy currents. described i n 2.2.  x  The procedure and apparatus were  In p a r t i c u l a r ,  £, 6  Hence  The s t a t i c f i e l d B , i s p a r a l l e l  \^Uz  ?  =  d  Q.  U  h  2/'  0  nn L ' | +(UHB»)/2 z  ^*  (  t  ,  I +(U B.)72 H  where  A  —  L  (\  ^  In Fig. 8, the experimental pickup voltage '*  i s plotted against u B . M  K  ?•  r h  '  L  ^ .  £ i s plotted against B , L  Now the t h e o r e t i c a l  x  and  £ ( B ) w i l l be a Z  X  *  2 )  maximum when u B M  x  a T2 and I f the t h e o r e t i c a l and experimental curves l n  Pig. 8 are compared to determine B„ at C~ The maximum occurs at B  2Miir  . then u f  0.J15 W m\ hence u  s x  H  H  may be determined.  - 4.5 m  and (22)  At low f i e l d s , one may neglect terms i n ( u B ) H  v~' sec '.  x  becomes  The experimental data gavet L.  s  7«5/^a  n, » 4o turns m"'  h - 15 yuh  n « 7.7*\<r turns ar'  £, s 10.8 v  £ * 10.0 mv  Q = 66.8  Q^*  t  2  2  o2  B Hence u (ii)  » A.6 m  H  2  18.5  s 0.10 W m"2  y  v"' sec"'.  4-Electrode H a l l Oonstant Measurements.  The standard method i n  which current electrodes are attached to the ends of the sample and 2 potential probes are placed along the sample i s used.  where\E A  -  = §/l,, 1, * distance between potential probes, J  = cross-sectional  K  Now  x  * l/A , x  area of sample perpendicular to d i r e c t i o n of I,  u = R cr = (Tfl.E i.B, I  4  B  H  The experimental data are:  1 s 0.93 cm  An - 1.0 cm  B  e  Hence u  » 0.14 W n f H  a 4.1 m  £ * 1.2 mv  (  1  v" sec .  I » 0.50 amp  and  C T = 2.2*10^ mho nf'  37  (iii)  Diacueaion of Reeulte.  There i e an error of approximately 10%  between the results of ( i ) and the results of ( i i ) . However experimental errors w i l l account for this discrepancy.  In particular, the 4-electrode method  involves errors i n measuring the dimensions because of the irregular shape of the sample, and end effects w i l l tend to decrease the observed value of u because the equipotential surfaces are disturbed. B H  also vary a few percent over the sample.  z  will  There are undetermined errors  i n the crossed magnetic f i e l d technique when using the constant c o l l i s i o n relaxation time assumption.  The transverse magnetoresistance results i n  (b) indicated that a constant mean free path assumption better f i t t e d the experimental results.  This assumption would lead to a different conduct-  i v i t y tensor (E17) and would modify ( 2 2 ) so as to give different results for u . H  58  PART II»  DIMORPHIC PHASE TRANSFORMATIONS IN COMPOUND SEMICONDUCTORS  CHAPTER 1 - INTRODUCTION An i n t e r e s t i n g group of compound semiconductors are the s i l v e r salts s i l v e r selenide, s i l v e r t e l l u r i d e and s i l v e r s u l f i d e .  They undergo changes  i n c r y s t a l l i n e structure i n the temperature range 100 - 200° C. e l e c t r i c a l conductivity and charge c a r r i e r density undergo changes at the t r a n s i t i o n due to changes i n band structure.  The  discontinuous A major  obstacle to applying conventional dc 4-eleotrode techniques to determine conductivities and H a l l constants i s that these oompounds are at available commercially only i n the form of powders.  present  I t i s impossible  to  attach electrodes to a powder sample and i n addition the powder w i l l e x h i b i t an indeterminate bulk conductivity due to intergranular resistance. Consequently an eleotrodeless technique would appear to be e s s e n t i a l . I t i s of h i s t o r i c a l i n t e r e s t to note that s i l v e r s u l f i d e i e probably the f i r s t semiconductor to be discovered. i n Faraday's Diary on February 21, 1833*  The o r i g i n a l reference appears Faraday noticed that on  passing  a current through a tube of fuzed s i l v e r s u l f i d e that "the heat rose as the conducting power increased", i n d i c a t i n g that the material had a positive temperature c o e f f i c i e n t of conductivity.  He also observed the abrupt  increase of conductivity at the phase t r a n s i t i o n . Experimental work was done on s i l v e r selenide powder. e x i s t s i n two c r y s t a l l i n e phases, a low temperature c r y s t a l i e face-centered  This compound  fl phase i n which the  tetragonal, and a high temperature o£ phase i n  which the c r y s t a l i s body-centered cubic.  T r a n s i t i o n temperatures of  and 155°0 have been reported i n Hansen (1958)»  122  59  The electrodeless techniques of part I were employed measure the e l e c t r o n i c properties. i v i t y , a c t i v a t i o n energies,  in attempts to  The temperature v a r i a t i o n of conduct-  the approximate r e l a t i v e change i n conductivity  at the phase t r a n s i t i o n , and a conductivity versus temperature hysteresis e f f e c t were observed.  Absolute conductivity values could not be obtained  because of lack of knowledge of r ^ , the r a d i i of the i n d i v i d u a l p a r t i c l e s . The H a l l mobility appeared to be much lower than values quoted i n the l i t e r a t u r e i n d i c a t i n g that the powder was impure and only impurity scattering was being measured. Busch and Junod (1957) have prepared s i l v e r selenide of known stochiometric r a t i o by fusion of the components i n a quartz c r u c i b l e . was then p u r i f i e d by zone r e f i n i n g .  The compound  Measurements of the H a l l constant and  the e l e c t r i c a l conductivity between the temperatures of l i q u i d a i r and 500°0 showed that the type of conductivity i n the two c r y s t a l l i n e phases was d i f f e r e n t .  They found that i n the low temperature phase s i l v e r  i s a semiconductor with a c t i v a t i o n energy conductivity was 10  mho m  AE  =  0.075  A  and the H a l l constant was 2'10  t 20°C the m  t r a n s i t i o n between the two c r y s t a l l i n e phases occured at 1 5 5 ° C high temperature phase the elementary c e l l contracted,  selenide  coul.  The  In the  the e l e c t r i c a l  conductivity suddenly increased and i t s v a r i a t i o n with temperature became comparable with that of a metal.  The charge c a r r i e r density increased by  a factor of 5 and was constant to temperatures i n excess of 500°0. 2. charge c a r r i e r mobility was approximately 0 . 2 0 m  The  -I  v sec  at the t r a n s i -  t i o n and decreased i n proportion to l/T with increasing temperature. Magnetoreeistance measurements showed that the conductivity was e l e c t r o n i c , there being n e g l i g i b l e measureable i o n i c conductivity.  40  Miyatani  (1959)  has observed r e s u l t s which tend to confirm Busch's  r e s u l t s with regard to the mode of conductivity.  His r e s u l t s f o r the  e l e c t r o n i c conductivity, <T , and the i o n i c conductivity, 0~; , of s i l v e r t  selenide at the t r a n s i t i o n temperature aret p> phase -  0~ = (2 - 5)*10  mho m~'  6"; = (5 -20)  mho m"'  e  oc phase -  Q£ =  (2 - 5)* 10  mho m"  0^- 4*10 mho m" Ranges of conductivity are quoted since according to Miyatani the value i s dependent on composition.  Miyatani also states that the material i e  n-type, with an electron mobility u ^ * 0.15 m ~ v~'seo~' and an e f f e c t i v e 2  eleotron mass of 0.11 as approximately Miyatani  m at 124°C.  The t r a n s i t i o n temperature i e given  l4o°0.  (1958)  has discussed the e f f e c t of the stoichiometric r a t i o  on the t r a n s i t i o n temperature of s i l v e r t e l l u r i d e .  The r e s u l t s are given  here as they may explain the conductivity versus temperature hysteresis observed f o r s i l v e r selenide. p  i  s  l°  w o r e a  In p a r t i c u l a r , the t r a n s i t i o n temperature  with increasing deficiency of s i l v e r , while the p to oC  t r a n s i t i o n temperature  changes very l i t t l e .  The e l e c t r o n i c conduct-  i v i t y also decreases rapidly with small d e f i c i e n c i e s of s i l v e r and the change of (J^ at the oc to ^  t r a n s i t i o n i s very Blow.  The slowness of the  to £3 t r a n s i t i o n may be due to the p r e c i p i t a t i o n of tellurium, while at the p  to oC t r a n s i t i o n i t i s necessary to heat the specimen t o higher  temperatures f o r the precipitated tellurium to reenter the matrix.  41  CHAPTER 2 - EXPERIMENTAL TECHNIQUES  2  «1  Eddy C u r r e n t Losses I n Powders  (a.)  Theory.  The  t h e o r y f o r eddy c u r r e n t l o s s e s  i n powders d i f f e r s  somewhat from t h a t f o r a homogeneous m a t e r i a l .  In p a r t i c u l a r ,  important to know whether the  c o n f i n e d t o the  eddy c u r r e n t s are  p a r t i c l e s by h i g h i n t e r g r a n u l a r f l o w a c r o s s the  resistance,  individual  or whether they are  free  to  p a r t i c l e boundaries.  C o n s i d e r a semiconducting powder made up packed, s p h e r i c a l  p a r t i c l e s of r a d i u s r - . 6  l e s s medium of p e r m i t t i v i t y electrically  i t is  insulated  0  and  of Z i d e n t i c a l ,  They are  permeability  from each o t h e r .  The  close-  imbedded i n a  JLL  loss-  SO t h s t they  0  are  d e n s i t y of p a r t i c l e s , N,  is  assumed t o be u n i f o r m .  A uniform s i n u s o i d a l l y  v a r y i n g magnetic  field,  H ,  f r e q u e n c y i s low,  i s br  eddy  i s applied  2o  and  c u r r e n t s i n the  the  individual  f r e q u e n c y f o r m u l a and  the  induoed magnetic d i p o l e seen by due  to  a p a r t i c l e due a l l the  p a r t i c l e s may  t h e n be  ;  -  The  the  low  i n t e r a c t i o n between p a r t i c l e s i n terms of  the  moment i n i n d i v i d u a l t o the  exciting f i e l d  r e p r e s e n t e d by  particles. and  t o the  The  total  secondary  field field  o t h e r p a r t i c l e s i s then,  H  =  where  1/r  _  "To  k TT  3 v. N = K  s  u  ^7T  r  i  Now  that  -  the  ( ) c11  -  k  :  '  » volume occupied by the volume of c o n t a i n e r  e v i d e n t t h a t f o r br-  T "  particles  secondary magnetic f i e l d  < 1,  hence i t i s  seen by  a particle  42  due to a l l the other p a r t i c l e s i a n e g l i g i b l e . Thus there i a no i n t e r action between the p a r t i c l e B and the t o t a l power loss w i l l be the sum of the individual p a r t i c l e power losses.  Hence  AR = J where, from ( B 7 ) and (B8),  A RL _  and 2 a number of p a r t i c l e s .  Finally  w h e r e  K  and r  c  ^  H Z O * ^  "  c  = radius and l  c  1/  -  3>/ <^>~  AR  ,  ;  (i) (2)  (b/t;) .  4-?  hi  ...  W 7 7 '  ( 4 )  a length of solenoid.  The determination of the conductivity of a powder from (5) i s seen to involve two serious experimental d i f f i c u l t i e s .  F i r s t of a l l , one  would require that the inter-granular conductivity be n e g l i g i b l e compared to the conductivity of the i n d i v i d u a l grains.  Secondly the size of the  p a r t i c l e s must be known, which p r a c t i c a l l y requires that e l l the grains be of uniform s i z e . The temperature  since  v a r i a t i o n of conductivity may however be studied  ARD A  =  K_ i, fI Ksu.ujtiL  i,, h  z  )(HT)  where the constant of proportionality i s independent  of temperature.  O)  An  experimental d i f f i c u l t y i s that agglomeration or crumbling of p a r t i c l e s  4?  may occur, thus changing the constant.  Relative conductivities at solid  state phase transitions may also be determined i f the change i n r. , due to shrinking or expanding of the crystal structure, ie small. (b)  Design Gonslderations.  Consider an aggregate of Z identical  conducting spherical particles, of radius r., which are insulated from each other.  The same design formula as were established i n I 2.1 apply  again for powders with the exception that the c r i t i c a l radius i s now r^ , the radius of the individual particles, rather than r^, the overall radius of the sample. ±  Thus the optimum frequency defined by br  -  (^> (5jU ) 0  o  r - ^ i s given l  and the low frequency limit from (5) as  (6) 3  K, = * k  where  £  (7)  - total volume of particles, volume of solenoid Note that the volume of the particles i s not the volume of the container since there will be a certain percentage of voids. The optimum value of inductance for the solenoid is given ae i n I 2.1c by  L  H  =  '  f  *  (8)  and the maximum number of turns by  (9)  44  where C f  w  i s the minimum v a l u e  i s given by The  (5),  and r  displacement  = r a d i u s and  c  6^=  (c)  l  c  ^  i.SO* i o  D e s c r i p t i o n of Apparatus.  i s unchanged at  o / i w rm s e c " '  8  r e l a t i v e d i e l e c t r i c constant  of the Q-meter,  = l e n g t h of s o l e n o i d .  c u r r e n t l i m i t to the frequency  _£  where  of the t u n i n g capaoitance  ( ) 1 0  o f the m a t e r i a l .  The  c y l i n d r i c a l g l a s s Bample c o n t a i n -  e r s f i t c o a x i a l l y i n s i d e the s o l e n o i d s which are a l s o wound on g l a s s forms. The  s i z e s of these and  the p r e v i o u s  section.  because of i t s low up t o 600  the optimum v a l u e s Glass was  of inductance  used f o r these  were d i s c u s s e d i n  c o n t a i n e r s and  forms  d i e l e c t r i c l o s s , mechanical s t a b i l i t y a t temperatures  C, r e l a t i v e chemical  i n a c t i v i t y and  a l s o because of the t r a n s -  parency a l l o w i n g v i s u a l o b s e r v a t i o n of the sample t o be made. A s p e c i a l f e a t u r e of the sample c o n t a i n e r , which i s c l o s e d a t end  and has  one  a removeable top f o r adding the sample a t the o t h e r end,  a s m a l l c a p i l l a r y extending  from the c l o s e d end  c e n t e r to the midpoint of the c o n t a i n e r .  One  of the c o n t a i n e r up  end  is the  of a thermocouple i s  p l a c e d i n t h i s c a p i l l a r y where i t comes i n good thermal c o n t a c t w i t h  the  sample. The  s o l e n o i d s are wound u s i n g T e f l o n i n s u l a t e d w i r e .  T e f l o n i s very  u s e f u l as i n s u l a t i o n s i n c e i t r e t a i n s a l l ' i t s i n s u l a t i n g p r o p e r t i e s a t temperatures i n excess of the 200°C temperatures achieved  i n this experi-  ment. The  Q-meter used was  temperature was  d e s c r i b e d i n I 2.1b  c o n t r o l l e d by p l a c i n g the  and F i g . 2 and  5-  The  s o l e n o i d w i t h sample l n an o i l  4  bath.  Heating and s t i r r i n g of the o i l acheived the necessary degree of  temperature 1254,  5  control.  The o i l used was a special i n s u l a t i n g o i l , Aroclor  which can be used at temperatures up to J00°0.  Aroclor 1254 i s  one of a group of o i l s composed of chlorinated biphenyl and chlorinated polyphenyls and i t should be noted that at temperatures i n excess of 100°C the fumes from the o i l are very i r r i t a t i n g to the skin and arrangements should be made to remove the fumes. 1000 ml beaker.  The o i l bath container was a  The heater was a 100 ohm 5-watt power r e s i s t o r supplied  through a General Radio 5 amp  115 v Variao.  The s t i r r i n g motor, an  Eastern Engineering Co., Variable Speed S t i r r e r , Model 4, 110 v do or ac, was also supplied through a similar variao.  Heating and cooling rates  were varied by manipulation of the heating power and s t i r r i n g rate. Radiation and convection heat losses were lowered by wrapping the beaker with £ i n . of corrugated cardboard and by placing a wooden cover on the beaker.  The cover also prevented the o i l vapors from esoaping and  i t was used to anchor the top of the solenoid, the s t i r r e r shaft, a thermometer, a thermocouple  wire and the heater lead-in wires.  I t would have  been possible to obtain much better temperature regulation i f a Dewar f l a s k and a better method of temperature regulation had been used.  How-  ever, the Dewar f l a s k has a serious f a u l t l n that eddy currents are induced i n the s i l v e r layers on i t s walls, thus greatly reducing the Q of the solenoid.  Special Dewar flaeke i n which the B i l v e r layers are  interrupted by v e r t i c a l gaps to prevent the flow of the eddy currents would minimize t h i s e f f e o t .  These experimental sophistications were not  employed however, since very sharp t r a n s i t i o n s were observed i n s i l v e r selenide using the experimental techniques described i n the previous paragraph.  46  In making a r u n the f o l l o w i n g g e n e r a l procedure was used. h e a t i n g power was g r a d u a l l y i n c r e a s e d so as t o j u s t o f f s e t losses.  The  radiation  I n t h i s way h e a t i n g r a t e s o f 1°C per minute o r lower were a t t a i n e d .  The power r e q u i r e d by t h e s t i r r e r d e c r e a s e d w i t h i n c r e a s i n g ae t h e v i s c o s i t y o f t h e o i l d e c r e a s e d r a p i d l y .  temperature  Two temperatures were  r e c o r d e d , the thermometer r e a d i n g t h e o i l b a t h temperature and t h e thermoc o u p l e r e a d i n g t h e temperature i n the sample core, ( t h e r e b e i n g o n l y a s m a l l temperature g r a d i e n t through t h e w a l l a o f the c a p i l l a r y ) . way  In t h i s  i t was p o s s i b l e t o see more e a s i l y t h e c o o l i n g or h e a t i n g r a t e s by  n o t i n g t h e temperature g r a d i e n t between sample and o i l b a t h . a d d i t i o n a time l o g o f temperature and Q r e a d i n g s was k e p t .  In When t h e  sample was w e l l above the t r a n s i t i o n temperature t h e h e a t i n g power was g r a d u a l l y decreased, so as t o keep i t s l i g h t l y below t h e r a d i a t i o n l o s s e s , and a c o o l i n g r u n made.  A c a l i b r a t i o n r u n was a l s o made w i t h t h e c o i l  alone t o determine t h e v a r i a t i o n o f Q  D  w i t h temperature.  47  2.2  H a l l M o b i l i t y Meaauremente on Powders The H a l l m o b i l i t y , u , i s g i v e n i n t h e low f r e q u e n c y br. - t , low H  ( U B ) < < 1 , l i m i t by  field  H  UH  =  1_L  WiL,  6 £,  <n, k '  K  where  Q  o  2  I-(22)  .  Q - Qsz oz  £, * v o l t a g e on e x c i t i n g c o i l of i n d u c t a n c e L, and n, turns/m, £  = v o l t a g e on d e t e c t i o n c o i l o f i n d u c t a n c e L  z  B„ * magnetic  z  f i e l d a p p l i e d orthogonally t o both  and n  2  turna/m,  coils,  Q - Q o f d e t e c t i o n c o i l w i t h sample, 5 2  Q ^ - Q, o f d e t e c t i o n c o i l without and  r  ;  * radius of individual  sample,  particles.  T h i a r e s u l t i s thought t o be independent  o f t h e shape and t h e s i z e o f t h e  i n d i v i d u a l p a r t i c l e s because i t depends on t h e r o t a t i o n o f t h e p l a n e s o f the eddy c u r r e n t which i s p r o p o r t i o n a l t o u B . H  independent  o f c o n d u c t i v i t y and f r e q u e n c y .  The r e s u l t i s a l s o  The t e c h n i q u e should prove t o  be v e r y v a l u a b l e s i n c e t h e r e a r e no other e x p e r i m e n t a l t e c h n i q u e s which can be a p p l i e d i n g e n e r a l t o powders. e l e c t r o d e technique w i l l  F o r i n s t a n c e , t h e s t a n d a r d dc 4-  n o t work i f t h e powder e x h i b i t s no dc conduct-  ivity. The  d e s i g n c o n s i d e r a t i o n s a r e much t h e same as t h o s e f o r a monocryatal  aa g i v e n i n I 2.2b  except t h a t , i f t h e eddy c u r r e n t s a r e c o n f i n e d t o  i n d i v i d u a l p a r t i c l e s , then t h e optimum f r e q u e n c y i s g i v e n by  The  apparatus i s d e s c r i b e d i n I  2.2o.  FIGURE  <?  TEMPERATURE VARIATION OF CONDUCTIVITY T 200  p  IgO i  160 i  /40  i  FOR  SILVER  SELENIDE  C O  tZO  IOQ  1  1  80 i  W i  ^0 1  10  r  48  CHAPTER 5 - EXPERIMENTAL RESULTS  5.1  E l e c t r i c a l Conductivity  The The (a) and  experimental  Selenide  r e s u l t s l / Q - l / Q versus (  l / T a r e p l o t t e d i n F i g . 9.  Q  r e s u l t s may be i n t e r p r e t e d i n two ways: I f the i n d i v i d u a l g r a i n s o f powder a r e i n s u l a t e d from each assumed t o be o f u n i f o r m  _ where K  The  of Silver  (i)  r a d i u s r- , t h e n  r\ ___ _ _ _ _ s  ( 5 )  • t o t a l volume of p a r t i c l e s , and i f b r volume o f s o l e n o i d  experimental  data  ;  r a d i u s of s o l e n o i d , r  c  « 10.2  cm,  = 1.55  c  o v e r a l l l e n g t h o f sample, 1 o v e r a l l r a d i u s of sample, r  0  #  om  » 9*3  cm,  0.72  cm,  a 0  f i l l i n g f a c t o r o f sample c o n t a i n e r ,  g i v i n g K,  =  -  gave:  length of solenoid, l  and  other  k  °  F  a  0.47,  0.12..  —  ( i i ) Q-meter _2_  5  T = 50°C, f = 5.9*10 cps and l / Q - l / Q = 4.0*10 . (  o  S u b s t i t u t i n g t h e d a t a i n (5), one o b t a i n s br. » 1.8,  and hence (5) i s n o t  valid.  C,  On t h e b a s i s o f e s t i m a t e d  t h a t br- - \ would be s a t i s f i e d p a r t i c u l a r using o b t a i n s br.  CT 2*10 B  =0.59.  values  f o r r^ and  i t was expected  i f t h i s i n t e r p r e t a t i o n was t r u e .  5 - 1 mho m , r . =  -1  0.5 10 m  and f =  5.9*10  In  •s ops, one  49  (b)  I f the i n t e r g r a n u l a r r e s i s t a n c e i s assumed t o be low so t h a t t h e  c u r r e n t can f l o w a c r o s s p a r t i c l e boundaries, c y l i n d e r formula a p p l i e s approximately, The  experimental  data g i v e as i n ( a ) :  t h e n the m o n o c r y s t a l l i n e  i f allowance  i s made f o r v o i d s .  K, = t o t a l volume o f p a r t i c l e s volume o f s o l e n o i d  = 0.12, 1/Q - 1/Q » 4.0x10", and br > £ . |  0  Q  Hence u s i n g t h e g e n e r a l f o r m u l a  (Al6),  =  - K I<* Y * c  - Ln Y — £_ f befc,(bh„) b e ^ b ^ J  w h e r e  bft  0  I  + b e i j b / j k c i (b/J 0  + b e i ' ( b^to)  \Dii,(bh ) 0  a  Z  s u b s t i t u t i n g i n (A16), one o b t a i n s , Im Y = 0 . J 1 , and from F i g . 1, b r = 1. o  Now f o r f = 5.9 10  cps and r = 0.72 cm, 0 - 2.2"10 o  mho m .  T h i s v a l u e f o r CT can o n l y be c o n s i d e r e d an e s t i m a t e o f t h e o r d e r of magnitude because o f t h e assumptions made i n the c a l c u l a t i o n .  It i s  i n f a c t about one o r d e r o f magnitude s m a l l e r t h a n the v a l u e s of (.1 - 5)*10 mho m"' g i v e n by Buech (1957) and M i y a t a n i  (1959).  Among t h e many sources o f e r r o r , t h e r e a r e : (a)  F r i n g i n g o f the e x c i t i n g magnetic f i e l d ,  approximation  not being s a t i s f i e d ,  due t o t h e l o n g B o l e n o i d  r e s u l t s i n too low a v a l u e o f observed  conductivity. (b)  D i s r u p t i o n o f t h e eddy c u r r e n t p a t t e r n s a t the ends o f the sample,  an e r r o r which would decrease  as the l e n g t h t o r a d i u s r a t i o o f t h e sample  c o n t a i n e r was i n c r e a s e d , would r e s u l t i n too low a v a l u e o f observed conductivity. (c)  E r r o r s i n measuring t h e dimensions o f the s o l e n o i d and the sample  container w i l l  l e a d t o e r r o r s i n the c o n d u c t i v i t y .  ?  50  (d)  There may be pockets o f powder i n s u l a t e d from t h e r e s t of t h e sample,  thus l o w e r i n g t h e observed (e)  There  is difficulty  conductivity.  l n d e f i n i n g the c u r r e n t d e n s i t y due t o i r r e g u l a r  p a c k i n g o f the p a r t i c l e s . (f)  The percentage o f v o i d s may change and crumbling or a g g l o m e r a t i o n o f  p a r t i c l e s may take p l a c e d u r i n g the experiment. (g)  Small d e v i a t i o n s from s t o i c h i o m e t r i c r a t i o l e a d t o a l a r g e decrease  in conductivity  i f M i y a t a n i ' s r e s u l t s on s i l v e r t e l l u r i d e , mentioned i n  the i n t r o d u c t i o n , a p p l y t o s i l v e r  selenide.  On o b s e r v i n g the e x p e r i m e n t a l CTvereus l / T curve i n P i g . 9, I t i s seen t h a t i n the /S phase, a semiconductor.  dO/dT i s p o s i t i v e and hence s i l v e r  L e t t i n g CT=C^6.  selenide i s  ^ d e f i n e t h e a c t i v a t i o n energy  A E which  may be e s t i m a t e d from the e x p e r i m e n t a l data, one o b t a i n s : (a)  Heating run  CT, ('2o°Z) _  (2o°0  2.06  1-85 '  A E = UoVgl)  giving  = 0.012 (b)  Cooling run  0", (liO°c) _  2.12  G" (26°C)  2.00  Z  giving  &E  = 0.0057  ev.  ev.  The v a l u e o f the a c t i v a t i o n energy and o f the c o n d u c t i v i t y a t a g i v e n temperature  were observed t o change as experiments  progressed.  T h i s c o u l d i n d i c a t e t h a t on h e a t i n g the sample some o f the v o l a t i l e s e l e n i u m evaporated, changing the s t o i c h i o m e t r i c r a t i o ,  s i n c e the work  51  o f M i y a t a n i (1958) on s i l v e r t e l l u r i d e i n d i c a t e s t h a t the  electronic  c o n d u c t i v i t y i s v e r y s e n s i t i v e t o small d e v i a t i o n s i n composition. i s a l s o observed  t h a t t h i s v a l u e o f a c t i v a t i o n energy  It  i s much s m a l l e r  t h a n the v a l u e of 0.075 ev r e p o r t e d by Busch (1957). I n the oc phase, dO/dT i s n e g a t i v e , thus s i l v e r s e l e n i d e shows  f m e t a l l i c behaviour. for  In p a r t i c u l a r , i f (ToC T , one  T , from the e x p e r i m e n t a l  data.  Q~, ( I 2 2 ° C )  f-  giving  Two and T ^ ^  experimental  =r  h  COT/Ok)  -0.65.  -1.  d e f i n i t e t r a n s i t i o n temperatures = 155°C'  data g i v e :  1.48  =  For m e t a l l i c behaviour  The  can c a l c u l a t e a v a l u e  were observed,  T^^  - 121  Busch (1957) r e p o r t e d the t r a n s i t i o n temperature  C as  155°C and M i y a t a n i (1959) as approximately l 4 0 C , i n d i c a t i n g t h a t they C>  performed  o n l y h e a t i n g experiments  temperatures  have been r e p o r t e d i n Hansen.  h e a t i n g r a t e s were as slow as 1°0 experiments  on s i l v e r s e l e n i d e .  The e x p e r i m e n t a l  per minute.  on s i l v e r t e l l u r i d e , has found  Both t r a n s i t i o n cooling  M i y a t a n i (1958), i n  t h a t T^_»^  i s lowered  by  s m a l l d e f i c i e n c i e s o f the s i l v e r component w h i l e T^.^ i s r e l a t i v e l y unaffected.  The  s i m i l a r i t y of s i l v e r s e l e n i d e and  I n d i c a t e s t h a t t h i s phenomena may  silver  telluride  a l s o take p l a c e i n s i l v e r s e l e n i d e ,  thus e x p l a i n i n g the h y s t e r e s i s e f f e c t .  and  52  At the  toe* t r a n s i t i o n a d i s c o n t i n u o u s  takes place.  In p a r t i c u l a r , n e g l e c t i n g volume changes,  _ 0> a  505? decrease  inoreases  i n conductivity.  1.46  2-  Busch  _ 0  0.70,  8  (1957)  suddenly a t the ft t o °c t r a n s i t i o n .  of M i y a t a n i  decrease i n c o n d u c t i v i t y  (1959)>  s t a t e s t h a t the  oonduotivity  Again, appealing  t o the work  the disagreement i n r e s u l t s may  ences l n c o m p o s i t i o n .  be due  to  differ-  55  5.2  H a l l M o b i l i t y Measurements on S l i v e r  Selenide  Attempts were made t o measure the H a l l m o b i l i t y o f s i l v e r u s i n g the e l e c t r o d e l e s s t e c h n i q u e d e s c r i b e d i n 2.2. consistent.  selenide,  The r e s u l t s were n o t  In p a r t i c u l a r , two t y p i c a l experiments gave: (i)  (ii)  Or)  10.5  11.7  n,  ( t u r n s m"')  80  80  L,  ( h)  50  50  (mv)  0.40  O.O55  ( t u r n s m")  1.6-10^  10^  Li  (yuh)  27  105  B»  (W nf )  0.45  0.54  Qsz  5.96  54.0  Q.oZ  57.5  74.5  n  A  1  4  4  f  (Mc)  1.0  0.50  T  Co)  20  20  0.025  0.0062  (m  l  v"' s e c " )  Busch (1957) gave u  m  - 0.20 m" v~* sec"' a t 20°0 and M i y a t a n i 2  (1959) gave  u ^ s 0.15 rn~ v"' sec"' a t 1 2 5 ° 0 f o r t h e e l e c t r o n m o b i l i t y o f s i l v e r s e l e n i d e . The most l i k e l y e x p l a n a t i o n f o r t h e low v a l u e s experiment i s t h a t the sample was r e l a t i v e l y impurity  s c a t t e r i n g was b e i n g measured.  s c a t t e r i n g Erginsoy  of u  H  obtained  i n this  impure and hence o n l y  In p a r t i c u l a r , f o r n e u t r a l impurity  (1950) g i v e s , f o r t h e i m p u r i t y  concentration,  5*  20  and  C • 5C  t r a t i o n of  =  m~  problem o f e r r o r s i g n a l s i s v e r y  where  Ee = e r r o r v o l t a g e due  and  £  = v o l t a g e due  pickup  • 0.015 2  t o an i m p u r i t y  concen-  not  « is  73*  «ec  deviation,  9  , from o r t h o by  1  t o pickup from e x c i t i n g  coil,  effect.  -i  -I  •  severe on low m o b i l i t y m a t e r i a l s .  c o i l s i s g i v e n approximately  'j e°  =  to H a l l  Z  ^e/6  Miystani)  c h e m i c a l l y pure but  to small a n g u l a r  £e  alignment  (from r e s u l t s of  pure.  g o n a l i t y , of the e x c i t i n g and  for u  t T"  , which corresponds  which would be c o n s i d e r e d  From I - (JO), the e r r o r due  Now  e  m  a 0.11m  v" sec , m 25 _3  K  D  0.16$,  m  2.6 10  ; N -  0  electronically The  m  0.015  u  —I  —|  2,  Hence, u s i n g u  3  , B  -2. y  » O.50  Wm  1  we  see t h a t f o r a %  0  mis-  55  APPENDIX A  Change l n R e s i s t a n c e  and  Inductance o f a S o l e n o i d due  t o a Oonduotlng  C y l i n d r i c a l Core  L e t the  sample of c o n d u c t i v i t y  JJ~ be homogeneous and  isotropic.  Q~, MKS  permittivity  s o l e n o i d and  resulting be  c  the magnetic f i e l d  i n a decrease i n inductance  >  1  and  length l B  of  i0  a r e s i s t a n c e a c c o r d i n g l y r e f l e c t e d back i n t o the  calculated specifically.  where l  Q  i n the sample by the magnetic f i e l d  At h i g h f r e q u e n c i e s  Q  and  l  c  solenoid i s uniform  The  i s unable t o p e n e t r a t e o f the  axial.  solenoid.  The  solenoid i s of radius r  i s l a r g e compared t o r  and  c  c  and  The e  .  Eddy  the  solenoid. the  sample  effects will  and  so t h a t the f i e l d  The magnetic f i e l d  i n s i d e the sample are c a l c u l a t e d by  permeability  u n i t s are used throughout.  sample i s i n the form of a l o n g c y l i n d e r o f r a d i u s r c u r r e n t s are induced  £ , and  length l  c  ,  i n s i d e the  current  s o l v i n g Maxwell's e q u a t i o n s .  density A l l time  v a r y i n g q u a n t i t i e s are s i n u s o i d a l . —*  - »  The  relation J  3  (J E and  i n Maxwell's e q u a t i o n s  the t r a n s f o r m a t i o n  d/dt •  substituted  gives (Al)  (A2)  and  S u b s t i t u t i o n o f (A2)  i n (Al) gives  V*(V*rT)= - j 10G> J ( I 4 - j w £ But  V  x r?) =  -S/^W.  56  V' H =  Hence  i  (  I +  j ^  )  H ,  (A3)  z  ~  b  where  t  j  UJ CTyU .  Aeeume t h a t : (a)  displacement c u r r e n t s a r e n e g l i g i b l e , i . e .  (b)  c i r c u l a r symmetry  (c) H(r) r Thus  giving  and i n f i n i t e  H(r)z^, where 7 H  f  2  =  o  ^t-  length, i . e .  z  • u n i t veotor - j b H i  < < —  I , o  =  -  •  i n the z - d i r e c t i o n .  HM)=-J - » t>H  h.  Substitute x = '  br i n (A4)  and o b t a i n ,  ^  T 4  -+ H = o .  ( A 5 )  The s o l u t i o n s o f (A5) a r e t h e B e s s e l f u n c t i o n s , J ( x ) and Y ( x ) . 0  Hence  The boundary c o n d i t i o n s a r e : 0 ) is finite,  (a)  H(r  (b)  H(r - r ) = H ,  s  o  ? 0  by Stokes theorem.  H ( A ) = H * , i ^ ,  Thus  2.  sinoe Y ( 0 ) i e i n f i n i t e ,  and where X  q  = j ' br .  The complex f u n c t i o n J ( x ) may be s e p a r a t e d 0  using, Thus  JM  -  H(b/i)  =  0  ben (U) 0  H  Z 0  i n t o r e a l and imaginary  + j bei" (b-h).  beUU)  (A6)  0  4  j be,- (b;i) p  parts  ^  (  a  ?  )  57  and (A8) bei»lbO + j  bei (b/0 o  Oaloulation of the total complex flux, ^  , through the sample  determines the Impedance reflected into the solenoid.  Thus (A9)  where N s total turns of solenoid linking the  Bample,  s nl^ , where n = tume/m on solenoid. If (A7) i s substituted i n (A9),  = where  L  and  .2.T  (A10)  J K L. Y, t  =  Y  then  (All)  f  be/io ( b i J  + j bd'  0  (  b.U.  = JJL fYl TTA/ IC S induotance of solenoid without the sample, Z  0  0  K  =  ho f-o  - f i l l i n g factor of solenoid.  (A12)  (A15)  He lc Hence the complex change i n inductance of the solenoid  =  $ - $°  =  K ( Y - i j L.  =  AL  c  -4-  (Al*)  56  Thus  AL  =  _  K  c  R  e  ( , _ Y J  bq'.'(UJ ^ and  C  0  0  beCib/jJ  bft  1  b e / t ( U ) - befro'lkftJkeio  0  + be</(bnj  (k/t.  , (A15)  AR « _2_ fbe/Ta(U>)befc>'(b/roJ + U i 0 ( b / J b e ' J ( b/u)  —  beft/"( b O  +  (A16)  bei0 (t>O 2  The l i m i t i n g forms f o r small and large arguments of the Bessel functions are now e a s i l y determinedt (a) br 6 £ The Bessel functions are given by the series expansions,  U^t)  -  I - ill,  d.'f  =  ^ 1(2!  and  °  oil!  --r i l l  -  (sir  r5o  till 4-!  5! 6 '  3!  3.'  211  z  ar  «H  +  51  (A17)  Substituting (A17) In (A15) and ( A l 6 ) one obtains,  and  AL  _  AR UJL  _  v [\>ho) 96  +  0  as the f i r s t approximation f o r the refleoted impedance.  (A18) (A19)  59  (b) br *  10  The Beeeel functions are given to the f i r s t order of approximation by,  and  ^ 1  TT n  x  =  bei  _  e ^  Si'n/jL.  =  e ^  cos  (x)  ke>'>) -  e_!L  cos  (  be*>)  sin  /"A  (JL  JL) ,  +  .  n i ,  TT  y  (A20)  Substituting (A20) i n (AV?) and (A16) one obtains,  Ti-  - - M  JL)  and as the f i r s t approximation f o r the r e f l e c t e d impedance.  (A21)  (A22)  60  APPENDIX B  Change In Inductance and Realstance of a Solenoid due t o a Conducting Spherical Core The sample, a sphere of radius r , conductivity CT , permeability ^4, o  and p e r m i t t i v i t y  £ , i e homogeneous and i s o t r o p i c .  i n a long solenoid of radius r , length l , c  resistance and inductance  c  The sample i s placed  and of n turns/m, and the  changes r e f l e c t e d into the solenoid by the  sample are calculated. Hence one must solve equation ( A 4 ) , i n spherical coordinates. Assume thati (a)  displacement  (b)  the magnetic f i e l d i s not a function of , ». H(/i,e) a H(i,& )zf where z* i s a u n i t vector i n the z - d i r e c t i o n , and  (c)  currents are n e g l i g i b l e , i . e .  < < •> 1  0  i (d)  the frequency Since b r « 0  i s low so that (lu(T^4) r  a  s b r « 1. 0  1, the secondary magnetic f i e l d s , outside the sphere are  expressible i n terms of the equivalent magnetic d i p o l e induced i n the sphere. This magnetization~mj i s In the z d i r e c t i o n , and i s i n general a complex quantity.  (1955),  Wait where  hae given the s o l u t i o n to the more general problem  ^jU-c and displacement  currents are not n e g l i g i b l e .  and assuming that the displacement  Putting  ~  currents are n e g l i g i b l e one obtains,  —>  nm  where  =  -  h  i0  (p +  j ^) V  a volume o f sample, 5  5  s  ,  (Bl)  61  and p and q are r e a l quantities given by  -f  3 ( 5 inlnot - oC C05I10C }  =  3_  =  -\TJ~ (UJ GyU ) k  2  1  and where  cxT  How since br <<  1,  —  1  0  Q  ij  and  —  4-  cosh oC —  -f  4-  '!  3!  1  _2L  (B3) 0  5  3  sirih  bh  (B2)  5!  2. -+  ^  '  '  _2<L  2!  4!  and one may s i m p l i f y (B2) to obtain as the f i r s t order terms:  P  <B4)  105  and Now  (B5)  -(p + jq) s complex magnetization per u n i t f i e l d per unit volume. A AL L  Hence  L  _  K S  P  =  _  K S  a  10 5  AR  and  where  =  UJL  K s =  s C  L  4" ftp s volume of sample . 3h.£ l volume of solenoid c  10  (B6) (B7) (B8)  62  APPENDIX 0  The  Internal F i e l d  Consider  l n an A r r a y  a uniform  o f Conducting S p h e r i c a l P a r t i c l e s  array of uniform  spherical p a r t i c l e s of radius  r- imbedded i n a l o s s l e s s medium o f p e r m e a b i l i t y L  and p e r m i t t i v i t y  The  p a r t i c l e s a r e c l o s e packed and l e t t h e d e n s i t y o f p a r t i c l e s be N.  uniform  magnetic f i e l d  YL  i e a p p l i e d a l o n g t h e a x i s o f t h e c y l i n d e r and  io  1 and hence t h e secondary  the f r e q u e n c y i n low so t h a t br. = (cufly^>) r. < < field  A  o u t s i d e t h e p a r t i c l e due t o i t s e l f  i s e x p r e s s i b l e i n terms o f t h e  e q u i v a l e n t magnetic d i p o l e t h a t i s induced i n t h e s p h e r i c a l p a r t i c l e . Consider the  the magnetization  o f one par t i d e .  The d i p o l e moment o f  c u r r e n t d i s t r i b u t i o n with respect t o the center  w where r = v e o t o r  =  J^-j(h  of the p a r t i c l e i s  * J ) dX  d i r e c t e d from t h e c e n t e r  (ci)  o f t h e sphere t o the c u r r e n t  element,  J  and where H  • local field  K *$  K o w  ~ j - | rn k5i7?0  - K co5  = ji  0  H'J •  e 1? T  h. SinQ z  E v a l u a t i n g the i n t e g r a l s ,  m  =  - j  where  V.  =  ^ '  and  z  s  (j> ,  (02)  o  seen by p a r t i o l e s .  o  m = -j  Hence  =  unit vector i n z-direction.  7r  0  4  hSin k^inB  -  e  =t* .  COS B  IS  -~- JJ.oH flK i  D }  ^  J  in0 oW<|>  d&  (cj)  63  M  The t o t a l magnetization i e then,  I t may  —  ^ nm .  (c4)  be noted that i f the p a r t i c l e s had a permeability JU> J J( , 0  then there  would be a s t a t i c magnetization. To evaluate the l o o a l f i e l d , B', seen by a p a r t i c l e , consider a small spherical cavity containing one p a r t i c l e .  The p a r t i c l e s are assumed to  be close paoked and hence the magnetization of the material surrounding the cavity may be aeaumed to be uniform. looal f i e l d , H , 1  at the center of euch a cavity i s ,  On substituting f o r I t from (03) approximation at low frequenoy, where  Stratton (1941), shows that the  Nv^ =  K  g  s  and (C4), one obtains, as the f i r s t H  —  ^—77—rj  t o t a l volume of p a r t i c l e s . volume of solenoid  *  (06) (07)  64  APPENDIX D  The C o n d u c t i v i t y Tensor  of m a s B m and oharge q i n combined  Consider a p a r t i c l e magnetic f i e l d s .  The f i e l d s  arbitrary directions. field  electric  and  e r e c o n s t a n t throughout space and a r e i n  The magnetic f i e l d  varies sinueoidally.  is static,  while the  Furthermore assume a c o l l i s i o n  electric  relaxation  time , T,whioh i s independent o f v e l o c i t y v and t h a t l/m i s a s c a l a r . The f o r c e s on t h e p a r t i c l e  due t o t h e f i e l d s a r e :  F. =  q.KB -  i/ Pj  ^  i ( % B, -  v B  B  =  <jf (  =  and  ?  V  y  E J,  z  B  x  Y  EJ,  +  x  B-  V,  v  +  J  E  .  (Dl)  S u b s t i t u t e ( D l ) i n the e q u a t i o n s o f motion o f the p a r t i c l e  m v  y  - j cu w V  x  1 u; m v and  jCU >w V  where nv^/X  8  X  f  6  -  ^ (  E*  =  <? (  E  =  rjf ( E  v  4  V  B  Y  +  V  +  V,  e  -  V,  B«  -  V  B  -  4  r  B  ) -  i  -  J22_^* ,  V B )  -  TnV  Y  4  v  "the energy l o s s terms due t o  Introduoe t h e n o t a t i o n  m  obtain:  ,  )  x  B  rnv>  and  ?  collisions.  .  (D2)  65 K =  -  i  T  On c o l l e c t i n g terms (D2) become  ^  + X  Y  -  Y  -+ G. V  - C \J  and  v  c„ V  -t  Y  %  =  KE ,  =  KE,.  Y  (D5)  The solution of (D5) give  5^= D  ^  =  K  Dv,  and  (  C  x  Q  _  c  ?  )  E  y  D =  where  A  2  -t-  14  Y  4 C  2  + (c c, + c J E Y  f  € f  + ( i+c,)E„  y  d*)  a  iw T  J  = n ^ N/^  The general relations  y  y  -KCyC, - c j E  I 4" C , 4- C I  also hold, where n  +c/JE  x  = (ccv+c )E  ^  + (CXQ- c )E ,  d + c / J E * -r ( G c C ^ c j e ,  and  = (J"^  ,  number of p a r t i c l e s per unit volume.  s  Multiply (D4) by nq and note that  §~ - •  2  on  , to obtain  D D X  and  D  ^  0T( i + j u J t j  _  = ( C C +C )E, + ( C Q - C)f ?  Y  y  y  4 Cl 4 C^J E . f  (B5)  66  On comparing  coefficients,  1  cr. =  + c/ ^ C  F  •  where  _  (To  J  D A t low f r e q u e n c i e s  F = and  such t h a t  T).  t  -t C,  +  C  3  (D6)  (D7)  (JJ t <C <C I ,  (T  (D8)  (D9)  67  APPENDIX E  Magnetoreslstanoe and H a l l M o b i l i t y  Coneidor a long c y l i n d r i c a l or spherical sample of conductivity permittivity A or B.  E  (J",  JU^ inside a solenoid, as l n appendices  , end permeability  C i r c u l a t i n g eddy currents are then induced i n the sample.  A  s t a t l o magnetic f i e l d B i e applied and the change i n resistance of the solenoid, measured ae a function of B, i s calculated for two orientations of the f i e l d s .  In a t h i r d oase, the change i n resistance  of a solenoid placed orthogonally calculated.  The  specific  to the e x c i t i n g and s t a t i c f i e l d s i s  c o l l i s i o n relaxation time, T ,  i s assumed to be constant,  and 1/ra i s assumed to be a scalar and hence the conductivity tensor derived i n appendix D may  be used.  In order that the r e s u l t s of appendix D may  be applied, the equiv-  alent e x c i t i n g e l e c t r i c f i e l d must be determined. states that  -r*  ^ B  -k-  (AS-'" &  1  The s o l u t i o n of ( E l ) i s ,  Thus  E  y  -  and  E  Y  -  Now  assume (v?«  at  -  V X D  Hence  -EySin(f Ef  E?)  —  L<p  C  u H  —  ^ f  - jU>  i  _  j  (  UJ  But k  ^  £ ~  j UJ 6 ?  1 and T O ^ r T q ^ ^ / m .  of the p a r t i c l e s , hence ?UJ ^ a  =  —  =  cos cp  Maxwell's equation  0  8  io  -  '  y  , x  (El) (E2)  (EJ) .  (E4)  fq/m a u^,the H a l l mobility  • x,y,z).  (E5)  68  (1)  Transverse  Magnetoresistanee  The e t a t i o f i e l d  Bg i s p e r p e n d i c u l a r  The c o n d u c t i v i t y t e n s o r ,  t o the eddy c u r r e n t  ( D 6 ) , beoomee:  H  6".  0  -u B* H  -(u«BJ7 .  0  o  Hence t h e c u r r e n t  \  o  U 6  (T =  planes.  (E6)  density,  (*7) (To-  and  (E8)  The power, W, d i s s i p a t e d i n t h e sample i s  - J IF f dT.  (To  •* . n  f  I n the low f r e q u e n c y r e g i o n ( b r ^ £ ) , s u b s t i t u t e (E2) and B ^ a  (E9)  «^ n i  i n (E9) t o o b t a i n :  h sin 9 dT . 2  (E10)  Vol. Consider (a)  two s p e c i a l c a s e s :  A c y l i n d e r of radius r  J Hence  r  rL  •ho  J  J  0  and l e n g t h 1  0  gives  n  If tf d*'d<f) -  TT C l (Ell)  69  where  u K = yUo rn t  z  IT  L  (E12)  ft/,  (815)  end  A sphere of radius r  Q  gives  (A sin Bf(h*sin  Hence  i -i  flty) =  -I A /  (  ^  )•  (814)  (815)  where  and  &oM0  *f  /to  (E16)  70  (2)  Mixed  MagnetoreelBtance  The s t a t i c f i e l d B  x  i s p a r a l l e l to the eddy current planes.  The  conductivity tensor, ( D 6 ) , becomesJ  (UHBJ " 2  cr =  Co  0  o  o  \  -u„B„  i  -+-  o  u„  B  I  •  (E17)  Hence the current density,  J* =  and  7  f  =  (T-  E,  (T.  E/  E, r ^ u g, (T. C jp  tr,  i +  M  -  ^  (815)  y  E/.  (E19)  The power dissipated i n the sample i s ,  (E20)  Vol. In the low frequency region ( b r B  fC)  r ^ o n i i n (E20)  0  - £ ) , substitute (EJ), (E4)  and  to obtains (E21) Vol.  The f i r s t term i n the i n t e g r a l represents the longitudinal component and the second term represents the transverse component of resistance. that there i s no longitudinal  magnetoresistance.  Note  71  Consider two special casest (a)  A cylinder o f radius r . h. to r ™  and length l  c  o  and rein© = P, gives  l  J  r  Hence  }  and  (E12)  since  b  and  (b)  (E22)  =  A sphere of radius r  gives  c  f J _  /Co 5  Hence  Ai?  and  AR  since  U o 1  K  z  ±ir 3  ho . 5  (E15)  72  (5)  Hall  Mobility  The s t a t i c  Hence Now,  field B  ~T* -  tT  f  J  (J  U  ~  0  x  i s p a r a l l e l t o t h e eddy c u r r e n t ~  _  +  L  i f a second s o l e n o i d  field  B , r 2  equivalent  C^q  L  — , „  to electric  x  i  Now  R  2|  s  V /l 2  (  f  2  cr  =  0  F„ F ^  a  J  V o )  n i  i  and B ^  - j  +  "  ^  2  x  CT  C  Ev  7~*  /„,«%  0*1  * •  (E18)  (E24)  * ,  '  s e t up by c o i l  F , F Y  4 i  (br  c  •  (E26)  1 t o c o i l 2, i s g i v e n byt  - £ ) , substitute  (E25)  1.  "^T^gji  I + (U„DJ  * Ji "o^\z i» (E27) t o o b t a i n i 0  B  »  L  In the low frequency r e g i o n  0  H  *  t h e t r a n s f e r impedance from c o i l  L, i  r ^  +  =  i n t e r a c t with the currents  J-E  Thus  p -a (  U  fields  £»a =  and  (E2), w i l l  ,  i s s e t up w i t h a x i s a l o n g t h e y - d i r e c t i o n , i t s  E  by  Ey  (2).  p l a n e s as i n  J ( E J ) , (E4),  B^,  73  Consider two special casest (ft)  A cylinder of radius r Y*  D  and length 1 , gives Q  dt —  0 ,  'Vol.  and  c*x <*Y  x  Henoe  (b)  A sphere of radius r , gives o  Yi ,1.  dr=  0,  1 3  and  Henoe  It should be noted that the e x c i t i n g c o l l and pickup c o i l may be interchanged  or  since Onsager's r e c i p r o c i t y theorem states thatt  RJ6)=  R  ~  I  1 2  UB),  I•  (E31)  This theorem i s used to advantage l n experiments where the geometry of the sample makes i t easy to calculate the eddy current losses f o r e x c i t a t i o n by one of the c o i l s , and d i f f l o u l t t o calculate f o r the other c o l l .  7 4  BIBLIOGRAPHY  BuBch  G., WIeland J. and Zoller H., E l e c t r i c a l Properties of Grey Tin, Helv. Phys. Acta., 2k, k9 (1951)-  Busch G., Jaggi R. and Braunschweig P., B a l l i s t i c Method for Measuring the Hall Effect, Helv. Phys. Acta., 26, 592 (1955). Busch G. and Junod P., The E l e c t r i c a l Properties of Silver Selenide, Helv. Phys. Acta., 5_0, 470 (1957). Dresselhaus G., Kip A. P., K i t t e l 0. and Wagoner G., Cyclotron and Spin Resonance l n Indium Antimonide, Phys. Rev., g8, 556 (1955). Erginsoy C , Neutral Impurity Spattering l n Semiconductors, Phys. Rev., J£, 1015 (1950). Faraday M., Faraday's Diary 1820-1862, B e l l and Sons, London, v o l . I I , p. 49. Fischer G. and MacDonald D. K. C , Magnetoresistance and Field Dependence of the Hall Effect i n Indium Antimonide, Can. J . Phys., $6, 527 (1953). Glicksman M., The Magnetoreeistlvity of Germanium and S i l i c o n . PROGRESS IN SEMICONDUCTORS, Heywood and Company, London, v o l . 5,  p. 1-25 (1958).  Hansen M., Constitution of Binary Alloys, McGraw-Hill, New York, 2nd Ed., p. 50 (1958). Miyatani S., Eleotrloal Properties of AgoTe, J. Phys. Soc. Japan, lj)., 541 (1958). Miyatani S., Ionic Conduction In P -AggTe and @ -Ag^Se, J. Phys. Soc. Japan, lk, 996 (1959). Prinoe M. B., D r i f t Mobilities i n Semiconductors! I Germanium, Phys. Rev. 92, 681 (1955). Stratton J . A., Electromagnetic Theory, McGraw-Hill, New York, p. 244-245 (1941). Wait J . R., Complex Magnetic Permeability of Spherical Particles, Proc. Inst. Radio Engrs., 1664(1955). Wilson A. H., Theory of Metals, Cambridge University Press, 2nd Ed., Sections 8.6 - 8.64 (1954).  

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