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Electrical properties of thin layers of indium antimonide Parker, Barry Richard 1960

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E L E C T R I C A L P R O P E R T I E S O F T H I N L A Y E R S O F mam A N T I M O W I S E by B A R R Y R. B.A. U N I V E R S I T Y O F PARKER B. C. - 1959 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F THE R E Q U I R E M E N T S F O R T H E DEC-REE O F M A S T E R O F S C I E N C E i n the department of Physics '•ve accept t h i s thesis as confoiminp, to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A MAY - I960 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Tk\js I c $ The U n i v e r s i t y o f B r i t i s h C olumbia, Vancouver 8, Canada. Date Ju/or II Q° I I T A B L E O F C O N T E N T S Chapter Page I INTRODUCTION I I o Object and scope I 2. H a l l E f f e c t 2 3« Magnetoresistance 3 4. D i f f i c u l t i e s 4 5o E l e c t r i c a l p r o p e r t i e s of InSb 5 I I THEORY 9 I o S t a t i s t i c s 9 20 Boltzmann Equation I I I I I EXPERIMENTAL INVESTIGATION 16 I o Apparatus 16 2o R e s u l t s - 18 I V LONGITUDINAL MAGMSTORESISTANCE 19 V APPLICATION OF THEORY TO PROBLEM 22 VI DISCUSSION OF RESULTS 29 I „ D i s c u s s i o n of graphs and Summary 32 I l l LI3T OF ILLUSTRATIONS ANL) GRAPHS ILLUSTRATIONS (FLWG) Page F i g . I , 2 16 F i g . 3 16 F i g . 4 16 GRAPHS Set of Graphs Graphs 1 - 4 Bulk Cominco Sample 5 - 7 Thin Cominco Sample 3 - 1 0 Thin R.R.E. Sample... 20 I Longitudinal Magnetoresistance 21 I I tfM~ « H (Bulk) 30 I I I Log yr / ««7"(Bulk) 30 IV H (Thin) 30 V yoc^ vs -r (Bulk) 30 VI M,, vs T (Thin) 30 V I I ^yi. v* 7- 32 IV A B S T R A C T This Investigation was undertaken i n an eff o r t to compare the ele c t -r i c a l properties of t h i n layers of InSb with those of the bulk material. Layers as t h i n as 20yd. were produced using a squashing device. Laue back-reflection showed that the layers were polycrjrstalline i n nature. Measurements of r e s i s t i v i t y s magnetoresistance and H a l l effect were then taken. From the r e s i s t i v i t y and H a l l effect measurements the Hal l mobility v/as calculated. Trie >;iobility of the t h i n samples was lower than that of the bulk material f o r the same temperature. Considering the increased scattering t h i s , however, i s what was expected. Theoretical calculations were then performed using the experimental r e s u l t s . Since the layers were p o l y c r y s t a l l i n e and were not s u f f i c i e n t l y t h i n the theory of t h i n films could not be applied. Therefore general transport theory was used and the scattering parameters were calculated. From these the dominant scattering mechanisms were then obtained. For the t h i n layers the dominent scattering mechanism vra.s impurity scattering while i n the bulk samples polar scattering by the o p t i c a l modes of vibrati o n proved to be the dominant mode of scattering. V A C K N O W L E D G K E N T 3 I w i s h to< t h a n k D r . R. B a r r i e a n d D r . K. T a y l o r f o r t h e i r g u i d a n c e t h r o u g h o u t t h e c o u r s e o f t h i s w o r k , a n d f o r t h e i r v a l u a b l e comments a n d c o n s t r u c t i v e c r i t i c i s m d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . A c k n o w l e d g e m e n t i s made t o t h e C o n s o l i d a t e d M i n i n g C o . o f Canada f o r p r o v i d i n g t h e m a t e r i a l r e f e r r e d t o i n t h e t h e s i s a s ' 1 C o m i n c o S a m p l e s 1 • a n d t h e R o y a l R a d a r E s t a b l i s h m e n t , E n g l a n d , f o r t h e m a t e r i a l r e f e r r e d t o as " R . R . E . S a m p l e s ' ' . I C H A P T E R I - I N T R O D U C T I O N The object of this thesis was to investigate the electrical properties of thin layers of InSb. Before discussing the investigation a brief summ-ary of the properties of semiconductors vri.ll be given. A semiconductor is defined as an electrical conductor with a conductivity that ranges between that of an insulator and a metal. This conduction can take place as a result of the motion of either electrons or holes0 It is, however, re-stricted to certain bands of allowed energy states. The two bands with which one is usually concerned are called the valence band and the conduction band. At absolute zero a l l electrons will be in the valence bandj however, as the temperature is raised some of the more energetic electrons will be raised into the conduction band. The site in the valence band from which an electron is raised will then be vacant. This vacancy is known as a hole. It has a positive mass and has the properties of a positive electric particle. Separating the valence and conduction bands is a region called the forbidden gap. In the more common semiconductors Ge and 51 this gap is of the order of I ev". Iu3bs however, has an exceptionally small gap of approximately .27 eV at zero temperature, Semiconductors are of two types termed intrinsic and extrinsic. An intrinsic semiconductor has the same number of electrons as i t has holes. The more common type is the extrinsic semiconductor. The carriers in this case arise from impurities. These impurities may be either donors or accept-ors, A donor is an impurity which donates an electron to a band while an acceptor accepts one from a band. The energy levels associated with donors usually l i e just below the bottom of the conduction band] those associated 2 with acceptors usually l i e just above the top of the valence band,, The InSb samples used i n this investigation shoiired i n t r i n s i c behaviour at room temp-erature „ The object of this investigation was to compare the properties of thin layers of InSb with those of the bulk material. Measurements of r e s i s t i v i t y , magnetoresistance and Hall effect were made. From the r e s i s t i v i t y and Hall effect measurements the Hall mobility of the sample was obtained where mob-i l i t y i s defined by A plot of temperature vs mobility was made. Graphs of RH vs H were also plotted from which the scattering parameters of the system were calculated by curve f i t t i n g , In this investigation close agreement was obtained between the theoretical and experimental graphs for both thin and bulk Cominco samples, With the above information an assumption could then be made as to the type of s catt ering involved, •'/ith the measurements obtained other properties of InSb were also inves-tigated. These included conductivity, gap width and density of carriers. The longitudinal magnetoresistance of InSb was also measured. Although the class-i c a l theory of solids does not predict this effect i t was found to exist, which i s i n agreement v/ith the results of other investigations. HALL EFFECT The Hall effect i s important because i t provides a direct indication of the carrier type and concentration, and in conjunction with the conductivity yields the Hall mobility. The effect occurs when a substance carrying an 3 electric current i s subjected to a magnetic f i e l d in a direction perpendicular to the direction of current flow. Referring to a Cartesian coordinate system, i f we l e t the current flow in the x-direction, the magnetic f i e l d be i n the y-direction then there i s a potential set up in the a -direction. This potential i s usually referred to as the Hall potential. The equation giving i t s magnitude i s RM •> H A L L c o * r T t SST 6 *- / c M £ S S O F JS)*1P££ (CMS.) (I) The effect i s a result of the carriers experiencing a Lorentz force |= i,irhich tends to direct 'hem towards the side of the sample. Charge builds up on the sides and the resulting space charge gives rise to the Hall potential. The carrier type and concentration can be obtained from the equation. /X>M -= - — (cf*3/couX.o/*io) (2) The plus sign i s used when the carriers are holes and the negative sign is us d when they are electrons. This formula applies only to simple metals and impure semiconductors. For other cases i t must be multiplied by a constant factor. Magnetoresistance It i s well known that there is an increase in the r e s i s t i v i t y of a 4 material when i t i s subjected to a magnetic f i e l d , Magnetoresistance effects, as they are known, are of two types namely transverse and long-i t u d i n a l . Transverse magnetoresistance occurs when the magnetic f i e l d i s perpendicular to the current and longitudinal magnetoresistance occurs when f i e l d and current are p a r a l l e l , A sa t i s f a c t o r y t h e o r e t i c a l explanation of longitudinal magnetoresistance was not given u n t i l quite recently. An elementary explanation of magnetoresistance i s as follows. Since 2 the electrons have different v e l o c i t i e s when t r a v e l l i n g through a substance i n a magnetic f i e l d , the H a l l f i e l d can compensate for the Lorentz force only on the average, to make the net current i n the y d i r e c t i o n equal to zero. Fast and slow electrons w i l l be deviated one way or the other. They w i l l suffer more c o l l i s i o n s while t r a v e l l i n g the length of the sample and the mean free path along the sample w i l l thereby be reduced thus causing a change i n the resistance. D i f f i c u l t i e s i n Measuring these Quantities Numerous d i f f i c u l t i e s are met when the above quantities are measured. When measurements of H a l l voltage and r e s i s t i v i t y are taken the results are found to involve other quantities. These must be compensated f o r . The f i r s t and probably largest of these i s known as the IR drop. When no magnetic f i e l d i s ^ a applied, the equipotentials are l i n e s perpendicular to the di r e c t i o n of flow. I f we do not have our probes A and B on exactly the same equipotential 5 a potential w i l l be measured between them at zero f i e l d . In addition to this there is the Nernst Effect where a potential gradient appears in the y-direction when a thermal current flows i n the x-direction and a magnetic f i e l d i s applied in the ? -direction. Another similar effect called the Righi - Leduc effect must also be compensated for. It is shown in the following i l l u s t r a t i o n . * OH C A e /V 7" Finally we have the troublesome Ettingshausen Effect. 4 r There is no easy method for separating out this effect; however as i t is usually much smaller than the Hall voltage and was not observed in this experiment i t was disregarded. It may be removed using a. c. tecniquesj however, for Hall constant measurements d. c. apparatus is usually more convenient. The above effects may be compensated for by taking a series of four measurements, reversing magnetic f i e l d and current in a l l possible comb-inations. The average of these readings then gives a result in which a l l errors except the Ettingshausen effect are eliminated. Ele c t r i c a l Properties of InSb Several samples of InSb were available which varied in impurity con-1 6 14 -3 centration from 2 x 10 to less than 10 cm . 0f these, two were i n -6 vestigated, the results of which will constitute the thesis. The imp-14 15 -3 urity concentration of the R. R. E. sample was 10 - 10 cm while that 14 -3 from Cominco was less than 10 cm . The bulk material was made into thin layers by a squashing process which will be described in a later chapter. -3 This process produced samples whose thickness vra.s approximately 3 x 10 cms. The object of this thesis was to investigate the electrical properties of these layers. First of a l l , however, a brief summary of the properties of InSb as they are now known will be given. The particular attention that has been shown to InSb lately can be attributed to its extraordinary high electron mobility. Surprisingly enough, even though the mobility is very high, i t is hard to understand why i t is not even higher. This fact is based on a knowledge of the effective mass of electrons in InSb. From cyclotron resonance experiments i t was found to be about .0I3m, where m is the mass of the free electron. Estimates of the effective masses of both electrons and holes have also been obtained in other ways. Agreement of values obtained from the different methods is fairly good. Another property of InSb that has attracted particular attention is the small energy gap. At a temperature T i t can be expressed as t - o.z7 - cs»/o'M7-j J.V (3) This narrow gap width, when taken in conjunction with the small effective mass, has the consequence that the electron distribution in pure InSb is degenerate at room temperature. As a result, the density of carriers rises less rapidly with increasing temperature than would be the case for a non-7 degenerate system. The high electron mobility in InSb can be attributed to the low effective mass. The observed mobilities have increased steadily with improvement in 4 2 o purifying techniques. Hall mobilities of 7.5 x 10 cm /volt-sec at 300 K are now easily obtained „ Mobilities at high temperatures are correspond-ingly lower. At sufficiently low temperatures impurity scattering limits the mobility. At high temperatures polar scattering is the prevalent mode of scattering and here again limits mobility. The onset of degeneracy at high temperatures also causes the mobility to decrease. Cunnell and Saker give a formula for the temperature dependence of electron mobility in InSb. It is o 2 vrhich at 300 K gives,.*, =70,000 cm /volt-sec. This is in fair agreement with the samples used in this experiment. Low temperature measurements have been made on InSb to temperatures as low as that of liquid Helium. No de-ionization of impurities was found for fairly pure n-type materials even at very low temperatures. This implies that the impurity levels overlap the bottom of the conduction band. The amount of this overlap should be a function of impurity concentration. With the ex-istence of an overlap the question will be raised as to the extent the band will be modified by the impurities. InSb is assumed to have spherical energy surfaces. This fact is in-ferred from the angular dependence of magnetoresistance. As far as is known this is the first time thin layers of InSb have been made from a squashing process. The main method of producing thin films is by evaporation. This method, however, tends to separate out the In and Sb 8 and thus give incorrect r e s u l t s , A f a i r amount of work has been done on t h i n semiconductor layers i n general; however l i t t l e attention seems to have been paid to t h i n layers of InSb. Therefore there are no d i r e c t l y comparable r e s u l t s . The theory of t h i n films could not be applied i n t h i s case. This was because the layers used wore not s u f f i c i e n t l y t h i n . One f i n a l point should be mentioned i n regard to uniqueness. Unique curves of magnetoresistance and H a l l effect f o r InSb cannot be drawn. This i s mainly because of the dif f e r e n t impurity concentrations i n the various samples. 9 C H A P T E R 2 - T H E O R Y 3ince there is very l i t t l e fundamental difference between the theoretical treatment appropriate to the transport of electrons in metals and semiconductors we shall consider the general case fi r s t and later spe-cialize i t to semiconductors and InSb in particular 0 The main difference between a metal and a semiconductor is that a metal is highly degenerate at a l l temperatures of interest, whereas in a semiconductor Boltzmann statistics are frequently applicable. The selection of a model is of prime importance. We shall assume a model in which the electrons are quasi-free. That is, they move as i f in a vacuum, statistics are also an important aspect of solid state work. A brief outline of the statistics involved will be .given followed by the development and solution of the Boltzmann transport equation. Statistics of Free Electrons Electrons are elementary particles of spin 1/2. If we consider an assembly of electrons in equilibrium at temperature T, then the probability that a quantum state in a band is occupied is given by (4) where k c = Boltzmann constant The symbol -?i denotes what is called the Fermi energy. It is particularly important because i t is the energy level where the probability of occupation is 1/2. 10 The assumption of a model with quasi-free electrons leads to the following expression f o r the energy. 2. (5) Using t h i s and the fact that the density of states i s given by ^ \ / / d£ \ ( 6 ) we can obtain an expression f o r the density of carriers by substituting the above into W r j / . ( 7 ) Equation (7) then takes the form and i f we l e t x - ^ / X T y - 7 / X T (8) (9) The i n t e g r a l involved i n t h i s expression i s of a special type called Fermi-Dirac i n t e g r a l s ; t h e i r values are tabulated. Special cases of the above analysis may be considered. In a metal the prevailing s i t u a t i o n i s one of extreme degeneracy. This i s because of the large number of conduction electrons, ^ /° / <^ ~~ In t h i s case the expression for the density of carriers i s II 7V = te~S)*'*'*rtt (10) This condition also occurs in the InSb in certain circumstances,, In a semiconductor the number of carriers i s usually quite small and under these conditions classical approximations may be used. The expression for the density of carriers then becomes y 7TJ £ J Boltzmann Equation The Boltzmann equation for the general case w i l l be derived f i r s t . The solution w i l l then be specialized to the case of semiconductors„ An electronic system is completely defined when we know the distribution function J-CJiA). In equilibrium this w i l l be the well known Fermi-Dirac distribution function. By definition we can say that the number of electrons whose wave vector l i e s in the interval dk i s ^, f(J, x) A . ' (I2) V 7 7 - J J A free electron under the influence of an electric f i e l d accelerates according to a% - -J^£* (I3) This acceleration continues indefinitely and i s thus not i n accord with Ohm's law. We must therefore assume some kind of f r i c t i o n process„ It i s well 12 known that there i s no scattering i n a perfect l a t t i c e , therefore we must assume that i t comes from the defects i n the l a t t i c e . The two most common scattering mechanisms are l a t t i c e scattering and impurity scattering. One of the main problems i n t h i s investigation i s to f i n d which type of scatter-ing i s prevalent. To derive Boltzmann's equation l e t t *j ^  ^ ^ be the number of electrons i n a volume ilxJ^Ji i n phase space with wave vector i n also i n phase space. I f a small change i n time occurs the d i s t r i b u t i o n function / w i l l change s l i g h t l y . Denote the new function by X <tfj - • j-%* t. ^St/^, - v t + S t ) ^ ^ ^ , / ^ ^ (15) where X, Y, Z, are proportional to the external forces. The volumes e/y. <^ c£i and <u% ^4 however, do not change with time. Therefore the d i s t r i b -ution function (15) must be equal to the o r i g i n a l d i s t r i b u t i o n function (14). We f i n d upon equating them that This i s one form of Boltzmann's equation. Two d i s t i n c t factors contribute to changes of t h i s d i s t r i b u t i o n function. The f i r s t i s the f i e l d , the second i s the scattering or c o l l i s i o n process. Since steady state conditions require o(t (I7) our equation i s then = O (IS) 13 The f i r s t operator has been derived above. The second one, which i s due to collisions, i s given by (HI " f f s ( J ' , J ) f(J')f'-*<*~>1 (I9) The f i r s t term gives the number of electrons scattered into the state k per unit time. The second term i s just the number scattered out of k in the same time. The factor J(Z)£/ -JfA)]] is the probability that the i n i t i a l state i s occupied and the f i n a l state unoccupied. Remembering that f = ~-*ff + J»Z ] (20) we f i n a l l y get the Boltzmann equation in i t s usual general form. It i s + j £•* Z 7. _L V, / * AX. V / - - £z-£> (21) The right side of this equation comes from the fact that under certain cond-itions the operator f'Lf) may be simplified to f ~ jf° . Under these conditions the system is said to have a relaxation time ti . Calculation of this relaxation time i s of fundamental importance. The solution of the Boltzmann equation can be obtained by assuming / has the form £ y « fff,") (22) From the general equation for the current density i n terms of the number of electrons we obtain the equation and using equation (22) above this becomes J = - fx Xr (/Z- • ) <JX* JLJL (24) 1 4 Therefore we get f o r the conductivity ^ (25) For pure semiconductors we can assume Boltzmann s t a t i s t i c s . This means that i n equilibrium the d i s t r i b u t i o n function may be written as £(£) ^ JU^ -f(_V-£)/J.-rJ (26) Using t h i s and the s o l u t i o n of the Boltzmann equation we get the current components f o r the p a r t i c u l a r case of semiconductors. Jx. ~ and 2, (28) where <^> i s defined as <;> * ( 2 9 ) and _ / / . I f we l e t N o i n equation (27) above ive get and from t h i s we obtain the conductivity with no magnetic f i e l d applied a-- = ^  ^  < ^  > , v — (30) inhere the symbols have t h e i r usual notation. D r i f t m o b i l i t y i s the v e l o c i t y imparted to an ele c t r o n per unit applied e l e c t r i c f i e l d . I t i s defined by the r e l a t i o n ^ = ^ /STIJZ, ( 3 D Applying equation (30) above we get 15 I f we assume the relaxation time can be written as -V -r^c* (33) then using equation (28) and (30) we can write equation (32) above as We are, however, more interested i n the H a l l mobility which i s given by where /?a and <ro are the H a l l constant and conductivity at zero magnetic f i e l d . An expression f o r P6 cr, w i l l be derived i n a l a t e r section and from i t the type of scattering w i l l be determined,, 16 C H A P T E R 3 - E X P E R I M E N T A L IN V E J T I G A T I O N APPARATUS As was stated previously the InSb was received in the bulk form and had to be squashed into thin films. The method involved wj.ll be described. First of all two optical flats of quartz vrere thoroughly cleaned with deterg-ent and nitric acid. Quartz was selected because it had an expansion co-efficient very close to that of InSb. A small piece of InSb weighing about I/IO gm was cut from the bulk sample. This was then washed in carbon tet-rachloride and etched in CP4A to get rid of all surface impurities. Finally it was dried under an infrared lamp. The cleaned sample was then places on one of the flats which lay on a heater coil in the cylindrical air tight chamber shown in figure (4)» With the raising of the squashing device the apparatus was ready for the argon. This was used to produce an inert atmosphere and was usually passed through the apparatus for about two hours before the sample was heated to liquefaction. The moment the InSb liquified the squashing device was dropped. It was left this way until the InSb had cooled. Upon raising it was found that the thin layer invariably attached itself to the bottom flat. It was then removed by means of a razor blade. The results using quartz flats were quite good. Other flats such as plate glass, window glass and molybdenum were tried to see i f better results could be obtained, however they did not prove too successful. The squashed layers usually ranged in thickness from 20 to 30 microns, but some were as thin as 15 microns. The measurements of thickness were performed with a gauge F/ tt/R £ 2 C IRC UI T FOfi HALL /r/r peer A A/0 A/I E GA/E TO RE SliTAA/C E~ V777777X SAM ft £ (,-V 8ETTEAY F7e v # E Z. ClOS/tV P OE HOJLDEft r J." TOP View 1 • f y y y y J t s J v ^ I 1 -—(ft ' f I SIDE VIEW - THE* mocoueiE £ C HA Af ISM A MS s j 3A/*r>*.f S7AA/D C i L I A f O f i IC 4/. 17 comparator which was graduated i n steps of 2^, . Checks were made using a t r a v e l l i n g microscope. The r e s u l t s of measurements using the two d i f f e r e n t methods were reasonably consistent. Laue b a c k - r e f l e c t i o n photographs were taken of the layers and confirmed t h e i r p o l y c r y s t a l l i n e nature. When the sample was held up to the l i g h t small 8 pinholes could be seen through i t and when examined under a microscope the surface was found to be very uneven. Thus our estimate of 3o/* f o r the t h i c k -ness must be considered as an average value. This also indicates that theory can only be applied to a f i r s t approximation, as a rigorous a p p l i c a t i o n of the theory could only be applied to t h i n , even, s i n g l e c r y s t a l l a y e r s . As the experiments were performed at temperatures varying from l i q u i d a i r temperature up to room temperature a dewar was needed as a container f o r the l i q u i d a i r . The assembly f o r holding i t i s shown i n f i g u r e (3). Since the holder was to be mounted on the magnet i t was constructed of aluminum. The dewar i t s e l f was s p e c i a l l y constructed to f i t i n s i d e the magnet. The e l e c t r i c a l apparatus consisted of a Rubicon potentiometer, a 6 v o l t d.c. battery and standard resistances along with associated switches. The switch boxes were constructed so that both the current through the sample and the f i e l d could be reversed. The setup i s as shown i n f i g u r e ( i ) . The electromsgnet was of the standard type with poles shaped as shown i n the diagram. The f i e l d was checked f o r homogeneity and was found to be s a t i s -f a c t o r y . Although 10,000 gauss could be obtained quite e a s i l y the magnet was u s u a l l y only used up to 7,000 gauss. This was p a r t l y due to the great amount of heat, generated at the l a r g e r f i e l d s . The gap i n the magnet was about 3/4 of an inch which was a convenient s i z e . 18 The specimen holder i s shown i n f i g u r e (2) a l o n ~ w i t h i t s dimensions. I t was set up so t h a t f i v e contacts could be f i x e d to the specimen. The current leads were at e i t h e r end and the H a l l and r e s i s t i v i t y probes were on e i t h e r s i d e . With t h i s setup both M a l l e f f e c t and magnetoresistance could e a s i l v be measured. Set i n a groove around the o u t s i d e of the holder was the heater w i r e . This was held i n place by Sauereisen cement. To the center of the s i d e opposite the specimen came the leads of the thermocouple. A chromel-alumel thermocouple was found most s u i t a b l e f o r the temperature ranges i n v o l v e d i n these experiments. Good contact w i t h the specimen was hard t o o b t a i n , s i l v e r p a i n t leads proved t o be the best although at low temperatures contact was f r e q u e n t l y l o s t . The a c t u a l j o i n between the InSb and the s i l v e r p a i n t always seemed to have a f a i r l y high r e s i s t a n c e . S i n c e a potentiometer was used to measure a l l the voltages t h i s should have made l i t t l e d i f f e r e n c e t o the r e s u l t s . For the l o n g i t u d i n a l magnetoresistance l a r g e r f i e l d s were needed, there-f o r e a d i f f e r e n t magnet was used i n which f i e l d s of up t o 25,000 gauss could be obtained. Measurements of l o n g i t u d i n a l magnetoresistance were taken. Results The r e s u l t s obtained f o r the both bulk and t h i n InSb are shown i n the graphs. A thorough d i s c u s s i o n of these w i l l be given i n the l a s t chapter. Numerous r e s u l t s were taken. The r e s u l t s shown i n the graphs are t y p i c a l of those obtained. Accuracy of p o i n t s i s shown wherever p o s s i b l e . >f S V -075 B e /. // 46- C Co /VST. Vs. H. 3. A/f 4 C <V £ T Qfi £ S/S TAW c * Vs 7~£ M Mo -Zoo -'Ca _•«.« -ro -«e -/60 - / t o -e -S U S A M R ' fV / / V s 8. HA<.<- £O*>ST. VS. Tf^p T s M p UfJ Vo 19 C H A P T E R 4 - LONG! Tl 1DINAL MAGI'' E^ORES I STA?! CE A s p e c i a l study has been made of l o n g i t u d i n a l magnetoresistance. This p a r t i c u l a r property of semiconductors i s i n t e r e s t i n g because i t cannot be predicted from c l a s s i c a l theory of s o l i d s . I t s p r e d i c t i o n can only come about as a r e s u l t of quantum mechanical methods. Because l o n g i t u d i n a l magnetoresistance i s much smaller than the s i m i l a r transverse e f f e c t higher f i e l d s were needed. Therefore, as mentioned e a r l i e r , a. l a r g e r magnet was used. To show that the e f f e c t i s not predicted by c l a s s i c a l methods l e t us consider Boltzmann's equation - J~ Z * £ 7 • _L V,J -r ^ .V j- - - Lzl- . ( 2 1 ) 1 C J A. * * We again assume a s o l u t i o n of the form j - / - z J (22) Substituting t h i s into the l e f t side of equation (21) we f i n d a f t e r some manipulation If v,e lat ^ _ ^ then the component equations can be written as ^ - - t ^ V 7±1^ Z * ±JL ^ ^ (33) (37) 20 From the second equation i t can be seen that a magnetic f i e l d has no effect on the current i f the electric f i e l d i s parallel to i t . "hus we can see from this that using classical analysis longitudinal magnetoresistance cannot be predicted. Experimentally, however, longitudinal magnetoresistance has been observ-ed by many workers. In this work longitudinal magnetoresistance occurs at 9 o o room temperature (20 c) and at l i q u i d air temperature (-196 c) for both thin and bulk Cominco InSb. The ^eneral results in this case seem to agree with those for the transverse case in that the bulk is much more temperature dependent than the thin films. It w i l l be noticed that the effect in the bulk material becomes quite appreciable at li q u i d a i r temperatures. This is in disagreement with the new quantum mechanical theory of longitudinal magnetoresistance. "his theory predicts a f a i r l y small effect at these temp-eratures „ A formula for longitudinal magnetoresistance using quantum mechanics was given by Argyres. It is as follows. g f tC-jt/'J cf (39) where '*/*• A relation i s needed between -*JU and ^ 0 . Because of the smaller energy gap in InSb i t must be assumed that the number of electrons i n the band is altered by the magnetic f i e l d ; using s t a t i s t i c a l mechanics we can then arrive at the following formula which gives this relation. * - 'f'Jiry £•>/.• (-"W'J (40) y rr - o 21 This formula, when used i n conjunction with the one above predicts a longitudinal magnetoresistance; however, the values are much lower than those which are obtained i n t h i s experiment 0 There are, however, some encouraging r e s u l t s , "he theory predicts that the effect should not appear u n t i l f a i r l y large f i e l d s were reached ( 8 , 0 0 0 -IC, 0 0 0 gauss). In the results obtained i t w i l l be noticed that at room temperature there was no longitudinal magnetoresistance u n t i l a f i e l d of 8 , 0 0 0 gauss was reached. At lower temperatures, however, the effect was observed at f i e l d s below t h i s . For the th i n Cominco the eff -ect was observed at f i e l d s as low as 4 , 0 0 0 gauss. Summarizing, we can say that although c l a s s i c a l theory does not pred-i c t l o n g i t u d i n a l magnetoresistance, the effect has been observed and i s v e r i f i e d i n t h i s experiment. Using a new quantum mechanical s o l i d state theory longitudinal magnetoresistance can be predicted. The results obt-ained i n t h i s experiment, however, are not i n good agreement with t h i s theory. They show an effect which i s much too large. Longitudinal magnet-oresistance i s usually much less than one. As can be seen from the graph values of up to 3.5 are obtained at temperatures as high as l i q u i d a i r f o r the bulk case, "'his i s d e f i n i t e l y too large. I / . 6 J T U O / fit A 1. 22 C H A P T E R 5- APPLICATION OF THEORY TO THE PROBLEM I f we assume there i s a relaxation time ii and i t i s given by V--*.T*t («) then oxw main problem w i l l be to solve for r, s and 7? . From these we can obtain <n> the number of carriers and ^ the relaxation time. There are four basic assumptions which are as follows: 1) Consider a one band model only. 2) This band i s a standard band where energy i s given by t - ^L^? . 3) Cl a s s i c a l s t a t i s t i c s apply. 4) We are considering the effects due to the holes to be negli g i b l e i n comparison to the effects due to the electrons. Our f i r s t problem w i l l be to obtain an expression for the H a l l mobil-i t y . Previously we obtained the following expressions f o r the currents i n the x and y directions. r £ / * - \ / £ / ^LM-^-— r . snJi r r y v _ v _ £ y <=c //12' Setting J * o i n (20) we obtain and therefore (27) (28) (42) ± £ 9 / < * > - C+fff<^//+(*"*f>- <xs/,+wJkO) Now we know that the H a l l constant i s given by 2 3 Substituting the above values i n we e a s i l y obtain 7/e also have f o r the conductivity -2> „S Jrf Jg ° ( 4 6 ) For calculation purposes the above notation i s a l i t t l e cumbersome, therefore equivalent but more concise expressions w i l l be used. I f we denote A and B by the following and A = ^ ^ - i - A ^ — , a n y ~ j then the H a l l constant and conductivity can be written as follows: X."' * •>• and^/c^is defined by the relaxation (48 ( 4 9 ) and ( 5 0 ) w h e r e Ur ^ u * +s)J-z <*> X(>-)= / ^ * ( 5 2 ) ' , r ° " 3 /z ~% / ( 5 3 ) ( 5 4 ) From t h i s we can obtain an expression f o r the H a l l mobility. F i r s t of a l l solving equations ( 4 9 ) and ( 5 0 ) for A and B terms of R, H and 24 We get ^ J r ^ — i — (55) and cw)2y* . (56) D i v i d i n g B by -A we o b t a i n _ 0_ = ^ (57) Therefore from previous expressions (47) and (48) f o r A and B we have y = -<ri= Z'M/xfrJ (58) S u b s t i t u t i n g the valu e f o r 7" i n we o b t a i n ' XH<r - " Cl*> ^ ^ • (59) We a r e , however, more i n t e r e s t e d i n the expressions as the magnetic f i e l d H approaches zero „ In t h i s expression as //-*o we f i n d t h a t a l s o r - 9 o . This gives us an expression f o r the H a l l m o b i l t y as f o l l o w s J Now and we know th a t Therefore we o b t a i n f o r ^y^-^ ^ = ± T * " r ' * / j L * s ) / r C K ) • ( 6 2 ) Using t h i s i n expression (60) above we get Z<r - ^ r „ i'r**' rfo^/r*'***) • (63) From t h i s expression we can o b t a i n (r-* s) by t a k i n g t h e n a t u r a l l o g a r i t h m of both s i d e s and p l o t t i n g 2 5 where rf s _f_ r J r + Zs)/r s) . I t c a n e a s i l y b e s e e n f r o m t h i s e q u a t i o n t h a t ( r ^ s ) w i l l b e t h e s l o p e o f a p l o t o f ^ c r j g T „ Once we h a v e o b t a i n e d ( r ^ - s ) we c a n t h e n e a s i l y o b t a i n r a n d s a l o n e . T h i s i s done b y o b t a i n i n g a n e x p r e s s i o n i n v ; h i c h s i s t h e o n l y unknown . C o n s i d e r t h e e q u a t i o n * ' IJ • (49) S u b s t i t u t i n g i n f o r A a n d B we o b t a i n *„ • - ^ ' ' r " } z (64) w h e r e r ^ s ) J W s i S?~ - - ? ^ / y *c^// /re*?* *s) a n d a l l o t h e r s a r e as p r e v i o s l y d e f i n e d „ T h i s r e d u c e s down t o We a l s o h a v e t h e f o l l o w i n g e x p r e s s i o n f o r o~ . cr = 8* ( 5 0 ) S u b s t i t u t i n g i n a p p r o p r i a t e v a l u e s we o b t a i n r^^y*^ *ry 7 , f ^_ey^ /TA ( 6 6 ) w h i c h r e d u c e s t o S o l v i n g f o r n we o b t a i n S u b s t i t u t i n g t h i s i n t o e q u a t i o n ( 6 5 ) t h e f o l l o w i n g e x p r e s s i o n i s a r r i v e d « ' ~ cr- / y % +s) (69) S i n c e we r e q u i r e H a l l m o b i l i t y r a t h e r t h a n d r i f t m o b i l i t y a n e x p r e s s i o n f o r i t i n t e r m s o f d r i f t m o b i l i t y must b e d e r i v e d . H a l l m o b i l i t y i s d e f i n e d b y ? — ^ <12 ^ < r y (70) 26 e>£> where < t> - f f x "*,v* J/7r 7 X X ^ (71) Therefore < t > - / * & ^% (72) ' o and -o 2 which with the use of (73) /?y * -e'Vx - / i r ^ + ') (74) reduces to ^ * ^ r 0 4 0 s rr%^s)/rt^s) ( 7 5 ) Now we know that Therefore we get {j^jj}] * ^  (76) Substituting t h i s i n expression (69) the following i s obtained ^ ' " y cr- KCT) rtf/i. +ZJ) Since we know everything i n t h i s expression except s we can solve i t fo r s by curve f i t t i n g „ Once t h i s i s done $ since we know r»* s , we w i l l know r , s, and te separately. In regard to our experimental results we notice that no use has been made of the magnetoresistance curves , They should be used to check the consistency of our results given by the H a l l constant and r e s i s t i v i t y curves . Using the values of r and s obtained from these i n the theo r e t i c a l expression for the magnetoresistance we should get values very close to those obtained experimentally. We s h a l l see however^ that with the assumptions used these curves are of l i t t l e use to us i n 27 checking our scattering parameters r and s . The expression for transverse magnetoresistance is as follows: 4* * +X hr*/.* 7 - 7 (78) yz> L <t/1+r) > J where <// r""0 V -? , To simplify the calculation let s - 0 and r= - 1 ; this will be shown in the next section to be true for the bulk case , the above equation then becomes: i f ~o Vf f ¥ e * x i t ' tZ. fe'lf* J^rrT* J /+ fr^T")* **s. f°J^L . Thus with the assumptions used the magnetoresistance is equal to zero. This is the result of assuming only one band. If we consider both electrons and holes we would find that the magnetoresistance would give a non-zero result . This , however, is a more difficult task as we must know both the hole mobility and density of carriers 0 In this case these are not known , therefore we must be satisfied with the one band approximation . This leads us to consider how accurate our results are when we neglect the holes „ The formula for conductivity considering both electrons and holes for the intrinsic case is as follows: CT = * (79) This can be rewritten as a~0 ^ y c . ( /-fy*/>/+t„ Mi jy<„ (/ + '/6) (30) where b ^  ^ and . * ^ It is known that is approximately equal to 80 for the case of Inob . Therefore we can say that in neglecting the holes we introduce an error of 1% in the value of the conductivity . The exp-ression for the Hall constant , considering both electrons and holes where r ' = constant depending on the material This easily reduces to /?„ - - ^  r / - j •71. S ^ ^ ' > and from a,bove In neglecting the holes we introduce an error of about 2», Now from our expression for /?B we can get an expression the density of carriers „ For the bulk material with s* 0 we get and for the thin material with s = 1/2 Graphs of these wil l be given in the next section. 29 C H A P T E R 6 - DICU33I0N OF RESULTS We s h a l l f i r s t use the graphs obtained to calculate the types of scattering involved and the times of relaxation . Once these are knovm we are i n a better position to explain the results .The following formula was used to plot H a l l mobility against temperature < o~0 (85) The results are shown i n graphs (5) and (6). I f we assume that the relaxation time can be written as r = T . a s ( 4 D then the H a l l mobility /?„ cr0 can be written as This formula was derived i n the l a s t section . As previously stated we can obtain ( r + s ) by p l o t t i n g This i s done i n graph (IV) and from the slope we obtain ( r v s ) equal to -.90. As a f i r s t approximation we can say t h i s i s equal to - I . This i s j u s t i f i e d when one looks at the graph , as the points have a s l i g h t scatter around the straight l i n e drawn through them .To obtain s alone an expression v/as obtained i n v/hich s was the only unknown . Using t h i s i t v/as determined that f o r the bulk material s = 0. The graph of t h i s i s shown i n ( I I ) . The room temperature curve was selected because i t probably v/as the most accurate .Other curves were also plotted corresponding to other temperatures . The f i t i n a l l cases v/as quite good except at l i q u i d nitrogen temperature (77°K) . This i s probably the result of impurity scattering taking over at t h i s temperature. For the t h i n material 30 we found s = 1/2 . The f i t i n t h i s case , however , was not quite as good as f o r the bulk material,, Let us refer f i r s t of a l l to the graph giving the bulk material r e s u l t s . A value of s equal to zero means that the relaxation time i s energy independent and that polar scattering i s the dominant mode of scattering . This means that the scattering i s by the longitudinal o p t i c a l branch of the l a t t i c e vibrations which polarize the c r y s t a l when the two atoms i n the unit c e l l are not a l i k e . Scattering of t h i s type i s known to occur i n ]nSb but i s absent i n crystals l i k e germanium and s i l i c o n o Other investigators tend to agree that t h i s type of scattering i s dominant i n InSb ( 2 . Summarizing we can say that for bulk InSb the scattering i s due to the interactions of the electrons with the o p t i c a l modes of v i b r a t i o n . For the t h i n Cominco layers we get a value for s of 1/2. This indicates that impurity scattering predominates „ The correspondence between the experimental and theo r e t i c a l curves i s not as good as for the bulk case. Referring again to formula (63) , f o r the bulk case we have s = 0 and r - - I , therefore and tt(-fjc) = ^ 7 (86) Substituting appropriate values i n (86) we obtain For the t h i n material we have s = 1/2 „ Therefore 1 (A s) - *° ^  ^* ^ f & ) (87) Conoco g (,Q" Q 0ot> cr 20'c Cons T . 1/j ^ _ / 6 O "77 Mo 3//T r ytr,, vs r Sr. COM /A/C O T MS AS & J M f=>4, £ 31 Substituting appropriate values i n we obtain f o r t h i s case Ta (-/jZ)= f-2 */o sfc - f*z es From these values we can obtain the relaxation time since r r.(«,s) J0J T A " (88) For the bulk case t h i s i s and f o r the t h i n material Referring to the graphs i t w i l l be noticed that r e s i s t i v i t y -i s much more temperature dependent i n the case of the bulk material than i n the corresponding t h i n f i l m case„ This effect can also be noticed i n regard to magnetoresistance . With the use of the acquired data we can calculate the gap width f o r both the t h i n and bulk materials . From the formula /> * />o ^ (89) where yoo depends on T, we get f o r the io n i z a t i o n energy A£ - f 1 (90) Substituting i n our values f o r the bulk case we get .25e?0 For the th i n f i l m we get .15 eV. These values are i n f a i r agreement with most other quoted values . Cunnell and Saker give the following formula f o r the gat> width of InSb. * O.Z7 ~(3*/o''i'7-)sz.\/ . (91) Other authors give values that range from .2/*.V to . 17 e V a* NOXTT l e t us turn our attention to the calculation of the 32 number of carriers . The formulas needed were derived i n the theory section. They are as follows: f o r the bulk material and ' ^ f o r the t h i n material „ Using these formulas vie obtained the graphs shown. Discussion of the graphs and summary One of the surprising things i n regard to the graphs i s that the results from the two t h i n f i l m samples do not seem to agree . In fact they are quite different . This can be p a r t i c u l a r l y seen i n the graph of ca r r i e r s vs temperature „ There are two things that tend to make us view the results f o r the R.R.E. sample with scepticism. These are the negative magnetoresistance encountered and the fact that the r e s i s t i v i t y i s so large. Actually , i n very t h i n films , because of additional surface scattering the r e s i s t i v i t y should be higher than i n the bulk case. This result can be predicted t h e o r e t i c a l l y . The r e s i s t i v i t y i n the R.R.E. sample i s d e f i n i t e l y higher than that i n the Cominco bulk sample j however, the corresponding values i n the t h i n Cominco are lower than those f o r the bulk case .It i s not understood why the t h i n R.R.E. sample does not get into the i n t r i n s i c region . Otherwise the graphs are as expected and tend to agree with the results of other investigators .The extreme values at the lower temperatures i s another property that i s expected and i s confirmed i n t h i s experimental investigation. A summary of the results obtained i n t h i s investigation i s now i n order „ F i r s t of a l l , the scattering mechanism i n both bulk and t h i n Cominco InSb were determined . Polar scattering by the o p t i c a l modes i IX. 33 of v i b r a t i o n predominated i n the bulk material at both room temperature and at lower temperatures « For the t h i n material impurity scattering occur; at both low and high temperatures „ The energy gap for these materials was then calculated „ For the bulk material (Cominco) a value of .25eV was obtained „ This i s i n general agreement with the values obtained by other investigators „ The t h i n Cominco sample gave a gap width of „I5 eV o This tends to be a l i t t l e small„ The accuracy of t h i s calculation , however, i s not as great as that f o r the bulk case since not as many points were obtained i n the i n t r i n s i c range of the material. A calcu-l a t i o n of the density of carriers was then completed . The results are 3hown i n graph (VTI) . F i n a l l y an investigation into longitudinal magnetoresistance i n InSb was made „ The results are shown i n graph (I) o The effect was observed 0 In fact , the values obtained proved to be very much larger than expected . One f i n a l remark i n regard to dis l o c a t i o n scattering . Dislocation counts have been made on both strained and unstrained Cominco InSb . The values obtained are approx-imately 1 0 /cm for the unstrained material and 1 0 dislocations / cm for the strained material . Since the mobility i s usually not effected with under IO 8dislocations /cm1 there must be l i t t l e d i s l o c a t i o n scattering i n the t h i n samples. B I B L I O G R A P H Y I . Dekker A.J. , S o l i d S t a t e P h y s i c s , P r e n t i c e - H a l l , (1957) 2. Dunlap W.C, I n t r o d u c t i o n t o Semiconductors, Wiley, (1957) 3. Linbergh 0 . Proc. I.R.E., 4 0 , 1414 (1952) 4. Ehrenreich H., J . Phys. Chem. S o l i d s 2 131, (1957) 5. Welker and Weiss, S o l i d S t a t e P h y s i c s V o l . I l l , Academic Press (1957) 6. Cunnell and Saker, Progress i n Semiconductors, V o l . I I Pergamon Press 7. B l a t t F . J 0 , S o l i d S t a t e P h y s i c s , V o l . IV, Academic Press (1957) 8. Bate G. and Taylor K,, J . A. P. - 31, 991 (i960) 9. Broom R.F., Proc. Phy. Soc. 71 470 (1958) 10. Argyres P.N., J . Phy. Chem. S o l i d s 4, 19 (1958) I I . Wilson A.M., The Theory of K e t a l s , Cambridge (1952) 12. Hilsum C. & B a r r i e R., Proc. Phy. Soc. LXXI, 676 (1958) 13. K i t t e l C., I n t r o d u c t i o n t o S o l i d State P h y s i c s , Wiley (1953) 

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