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Hall coefficient and resistivity of thin films of antimony and bismuth prepared by distillation Leverton, Walter Frederick 1950

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/fyo fit L-b Hi Csf - 1 HALL COEFFICIENT AND RESISTIVITY OF THIN FILMS OF ANTIMONY AND BISMUTH PREPARED BY DISTILLATION BY WALTER FREDERICK LEVERTON A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS THE UNIVERSITY OF BRITISH COLUMBIA August, 1950 • T H E U N I V E R S I T Y O P BRITISH C O L U M B I A VANCOUVER. CANADA D E P A R T M E N T O F P H Y S I C S September 19, 1950. This w i l l c e r t i f y that the following members on the Committee of Mr. Walter Frederick Leverton have examined his thesis and found that the material f u l f i l s the require-ments of merit and o r i g i n a l i t y prescribed under the regula-tions f o r the Ph.D. degree. Dr. G. yt. Volkoff Dr. K. C. Mann Dr7jV BVWarren Dr. "A-J J . DeK£er The research undertaken by Mr. Leverton was under the d i r e c t i o n of Dr. A. J . Dekker and Dr. A. van der Z i e l . A copy of the l e t t e r of approval by the external examiner, Dr. H. Grayson-Smith, Head, Department of Physics, University of Alberta, i s on f i l e i n the Office of the Regis-t r a r . ABSTRACT An apparatus lias been constructed to measure the H a l l c o e f f i c i e n t i n metals and semi-conductors by an a.c. method. The a.c. method completely eliminates the Ettingshausen effect which i n many cases introduces a con-siderable correction to H a l l voltages measured by d.c. methods. In addition, a small altern a t i n g voltage i s more e a s i l y amplified and measured than a d i r e c t voltage of s i m i l a r magnitude. An apparatus has been devised for the preparation of very pure metal films by evaporation. This apparatus eliminates such sources of contamination as hot filaments. The effect of annealing on both H a l l c o e f f i c i e n t and r e s i s t i v i t y of evaporated films of antimony and bismuth has been examined and a q u a l i t a t i v e explanation based on the electron theory of metals i s presented. The H a l l c o e f f i c i e n t of unannealed antimony films i s +0.216 c.g.s.m. and of annealed films i s +0.244 c.g.s.m. with an accuracy of 1 percent. Previous observers using d.c. methods found values ranging from *0.12 to +0.22 c.g.s.m. The H a l l c o e f f i c i e n t of bismuth films depends on the f i l m thickness. I t i s of the order of 1 c.g.s.m. PREFACE The research described i n t h i s thesis was supported by grants from the Defence Research Board of Canada. I wish to thank the National Research Council of Canada for Studentships granted to me (1948-49 and 1949-50).. I am indebted to Dr. A.J. Dekker and Dr. A van der Z i e l who suggested and directed the problem. Their assistance and encouragement are sincerely appreciated. The University of B r i t i s h Columbia provided the opportunity for me to conduct t h i s research and I consider i t a great honour to be a candidate for the degree of Doctor of Philosophy from t h i s i n s t i t u t i o n . W.F. Leverton August, 1950. TABLE OF CONTENTS Pag© I INTRODUCTION 1 II MEASURING APPARATUS 13 I I I PREPARATION OF SAMPLES 11 IV PURITY OF THE SAMPLES 22 V EXPERIMENTAL RESULTS FOR ANTIMONY (1) H a l l c o e f f i c i e n t s Unannealed samples 24 Annealed samples 24 Impure samples 28 (2) R e s i s t i v i t y measurements 29 VI DISCUSSION OF RESULTS FOR ANTIMONY 33 VII EXPERIMENTAL RESULTS FOR BISMUTH; (1) H a l l c o e f f i c i e n t s Unannealed samples 3& Annealed samples 39 (2) F i e l d - f r e e r e s i s t i v i t y of bismuth 39 (3) R e s i s t i v i t y i n a magnetic f i e l d 45 VIII DISCUSSION OF RESULTS FOR BISMUTH (1) H a l l c o e f f i c i e n t s 47 (2) The f i e l d - f r e e r e s i s t i v i t y 51 (3) R e s i s t i v i t y i n a magnetic f i e l d 52 REFERENCES 55 ILLUSTRATIONS Figure Facing Page 2* C i r c u i t diagram of am p l i f i e r 14 3 . Schematic diagram of movable H a l l contacts 15 4. Schematic diagram of evaporating apparatus 19 5. R e s i s t i v i t y of evaporated deposits of antimony as a function of temperature (1) I . INTRODUCTION I f a conductor carrying a current i s placed i n a mag-netic f i e l d at r i g h t angles to the current, a potential difference develops across the conductor i n a direction" perpendicular to both magnetic f i e l d and current flow. This effect was discovered by is. H. H a l l i n 1879 and the induced transverse potential difference i s known as the H a l l voltage. V o R I H x 1 0-9 (!) t • where I i s the current i n amperes, H i s the magnetic f i e l d i n gauss, t i s the thickness of the conductor i n centimeters i n the d i r e c t i o n of H, and R i s the H a l l c o e f f i c i e n t of the conducting material i n c.g.s. magnetic u n i t s . Suppose the conventional ourrent flows i n the z di r e c t i o n and the impressed magnetic f i e l d acts i n the (2) y d i r e c t i o n . ( F i g . l ) I f conduction i s by electrons, they w i l l be moving i n the negative x d i r e c t i o n and w i l l experience a force tending to move them i n the z d i r e c t i o n . z A F i g . 1 I f points A and B are on an equipotential i n the absence of the magnetic f i e l d , the effe c t of the l a t t e r w i l l be a lower-ing of the potential of A with respect to B. This i s the di r e c t i o n of the observed transverse electromotive force i n materials having a negative H a l l c o e f f i c i e n t . I t was found experimentally that many materials have a po s i t i v e H a l l c o e f f i c i e n t , i . e . the transverse e l e c t r o -motive force i s i n the opposite sense to that predicted om the basis of electronic conduction. A posi t i v e H a l l c o e f f i c i e n t indicates that conduction i s due i n greater part to p o s i t i v e l y charged c a r r i e r s . (3) The existence of p o s i t i v e H a l l c o e f f i c i e n t s could not be explained on the basis of the Sommerfeld theory of e l e c t r i c conductivity. Wilson (1) pointed out, however, that the electrons f i n d themselves i n a periodic rather than i n a constant p o t e n t i a l f i e l d , as assumed by Sommerfeld. The solution of Schrodinger's equation f o r a pote n t i a l with the same p e r i o d i c i t y as the l a t t i c e leads to the existence of energy bands separated by so-called forbidden regions. On the basis of the P a u l i exclusion p r i n c i p l e , the number of possible states i n each band i s just two per unit c e l l corresponding to the two possible orientations of the spin. This "band p i c t u r e 0 enables us to explain why ce r t a i n substances are metals and others insulators or semi-conductors. I t also explains the existence of po s i t i v e H a l l c o e f f i c i e n t s . Consider a s o l i d at the absolute zero of temperature. The energy states, s t a r t i n g from the lowest, are successive-l y f i l l e d by two electrons with opposite spin u n t i l " a l l the valency electrons are accomodated. Suppose there are just s u f f i c i e n t valency electrons per unit c e l l to f i l l an energy band completely. The eff e c t of an external e l e c t r i c f i e l d i s to tend to accelerate the electrons moving i n one d i r e c t i o n and to retard those moving i n the opposite d i r e c t i o n . In the case under consideration t h i s i s impossible because there are no vacant energy l e v e l s to which the electrons may be transferred. The energy l e v e l s of the next higher allowed band are vacant but they l i e s u f f i c i e n t l y high that a f i e l d of the order of lO^volts/om would be required to cause a t r a n s i t i o n . This means that a completely f i l l e d band has an "effective** number of conduction electrons which i s zero and hence a substance with an electronic structure leading to completely f i l l e d and completely empty bands at T • 0° w i l l be an i n s u l a t o r at that temperature. Next consider a s o l i d i n which there i s just one valency electron per unit c e l l . For most of the electrons an e l e c t r i c f i e l d i s s t i l l i n e f f e c t i v e but the more energetic electrons have plenty of vacant l e v e l s into which they can be transferred by an e l e c t r i c f i e l d and hence a current i s set up. In t h i s case the s o l i d w i l l show met a l l i c behaviour. When a band i s nearly f u l l , the most energetic electrons which alone contribute to the conductivity behave as i f they had a p o s i t i v e charge (or negative mass). I t i s often convenient to say that the vacant spaces or "holes* 1 i n a nearly f u l l band are responsible for the current. (5) In r e a l s o l i d s the s i t u a t i o n i s not as simple as outlined above. A number of materials such as antimony, bismuth and arsenic which would be expected to have a f u l l conduction band and hence to behave as i n s u l a t o r s , a c t u a l l y conduct f a i r l y w e l l . The same i s true of the divalent metals barium, calcium and strontium. This conduction i s explained by assuming that i n the three dimensional p l o t of energy bands the upper f i l l e d band overlaps somewhat with the normally empty band above i t . Some of the electrons leave the f u l l band and go into the empty band. Conduction occurs due to the electrons i n the upper band and the holes i n the lower band. In such a material, i n the pure state, the number of conducting electrons equals the number of holes. I f the mobility of the electrons exceeds that of the holes, electronic conduction predominates and the H a l l c o e f f i c i e n t i s negative. I f the reverse i s true, hole conduction predominates and the H a l l c o e f f i c i e n t i s p o s i t i v e . In addition to metals and insulators there e x i s t s a t h i r d class of s o l i d s known as semi-conductors which show appreciable electronic conductivity at room temperature. They are of great and r a p i d l y increasing technical import-ance. The conductivity of semi-conductors increases with (6) increasing temperature being negligable at very low temperatures. The conductivity of metals increases with decreasing temperature henoe the dependence of conductivity on temperature serves as a convenient c r i t e r i o n f or i d e n t i f y -ing semi-oonductors. Semi-sonductors may be divided into two classes, i n t r i n s i c semi-conductors and impurity semi-sonductors. The former owe t h e i r conductivity to the fact that the f i r s t empty band l i e s very close to the top of the f i l l e d band. At very low temperatures the material acts as an insulator but as the temperature increases, some of the electrons acquire s u f f i c i e n t thermal energy to jump into the empty band thus giving r i s e to conductivity. The number of electrons with s u f f i c i e n t thermal energy to enter the empty band increases r a p i d l y with the temperature hence the conductivity increases with the temperature. Impurity semi-conductors owe t h e i r conductivity to the presence of impurities. The ways i n which impurities can af f e c t the conductivity of materials which are normally insulators may be described by means of the band picture . In an insulator the lower band i s completely f i l l e d and the upper band completely empty. Now suppose an impurity atom with one valency electron i s introduced into the (7) c r y s t a l and further suppose that the energy l e v e l of t h i s electron l i e s between the two bands. This electron cannot take part d i r e c t l y i n conduction i f the number of impurity atoms i s small, since to do so i t would have to jump to a si m i l a r state on another impurity atom. The p r o b a b i l i t y of such a jump decreases very r a p i d l y as the distance between the atoms increases and i s negl i b l e f o r small concentrations of impurities. The electron from the impurity atom can take part i n conduction by being thermally excited into the empty band of the c r y s t a l a f t e r which i t i s free to move through the l a t t i c e . By choosing suitable impurities, the energy difference may be small enough to permit appreciable conductivity at room temperature. The case just described applies to impurity atoms which are el e c t r o p o s i t i v e . The impurity atoms act as ^donors'* of electrons to the empty band. I f the impurity atoms are electronegative and i f they have vacant energy l e v e l s between the f i l l e d and empty bands of the c r y s t a l , they can act as "acceptors™ of electrons from the f i l l e d band. In t h i s case conduction i s by holes In the f i l l e d band. Since the number of free electrons (or holes) increases r a p i d l y with increasing temperature, the conduc-t i v i t y of both kinds of impurity semi-conductors increases (8) i n a s i m i l a r manner. Many metallic oxides and sulphides form impurity semi-conductors. I t i s of in t e r e s t to note that the impurity need not be a foreign atom but may be merely an excess of either the metal or the electronegative atom. The H a l l c o e f f i c i e n t i s related to the number of holes and electrons i n the following manner 0 9 ( n 2 v 2 + n x v x ) 2 where e i s the electronic charge, n-^ , n 2 are the densities of electrons and holes respectively, and v^, v 2 are t h e i r m o b i l i t i e s . I f there are no mobile holes t h i s reduces to, 2 L _ (3) e n x I f there are no free electrons, R = + TT (4) e -n 2 Equation (3) or (4) applies to impurity semi-conductors. (?) I t i s seen at once that a knowledge of the value of the H a l l c o e f f i c i e n t gives us immediately a measure of the density of c a r r i e r s and t e l l s whether electron or hole conduction predominates*. The conductivity i s given by &~ = n e v (j?) where n i s the density of c a r r i e r s and v i s t h e i r m o b i l i t y . By Eq.,5 and E c l . J or 4 v - J L R G~ (6) 3TT . . Thus i f the H a l l c o e f f i c i e n t and conductivity are measured i t i s possible to oalculate the mobility of the c a r r i e r s . Theory shows ( 2) that the mean free path SL of the current c a r r i e r s i s given by jg, * 2 x io-7 Rcr(2 TT n k T ) * ( 7) IT a - . where m i s the mass of the electron, (the mass of a "hole" (10) i s set equal to that of an electron), T i s the absolute temperature, and k i s Boltzmann's constant. To summarize, by measuring the H a l l c o e f f i c i e n t and conductivity i t i s possible to obtain a great deal of valuable information about the conduction processes of metals and more especially of semi-conductors. This i n -formation i s of fundamental importance i n any study of the nature and uses of semi-conductors and i s unobtainable i n any other manner. In the course of t h i s research an apparatus f or accurately determining the H a l l c o e f f i c i e n t s of metals and semi-conductors was designed and a method was developed f o r preparing these ma-serials i n a suitable form for measuring the H a l l c o e f f i c i e n t and conductivity. Examination of the l i t e r a t u r e indicated that most observers used a d i r e c t primary current through the sample i n t h e i r determinations of the H a l l c o e f f i c i e n t . Since such a determination measures the sum of the H a l l and Ettingshausen voltages, the r e s u l t s must be corrected by measuring the l a t t e r separately. (The Ettingshausen effect i s a transverse electromotive force due to the transverse thermal gradient induced i n the sample). Since the voltages are very small, the H a l l voltage determined i n (11) t h i s manner may be i n serious error where the Ettingshausen voltage i s large compared to the H a l l voltage. The Ettingshausen voltage may be eliminated by using H a l l contacts of the same material as the sample, however, t h i s i s practicable only i n a l i m i t e d number of cases. In the measurements described i n t h i s t hesis an al t e r n a t i n g current was used i n the determination of H a l l c o e f f i c i e n t s . Since the temperature gradient which leads to the Ettingshausen effect requires a time of the order of seconds to become e s t a b l i s h e d ^ ) , use of a l t e r n a t i n g current completely eliminates t h i s effect from the observed trans-* verse voltage. A further advantage of the a.c. method Is that i t i s much easier to amplify and measure very small al t e r n a t i n g voltages than to amplify d i r e c t voltages of the same magnitude. Despite these advantages few observers(2-5)have used a.o. methods. Smith(^) found the H a l l c o e f f i c i e n t of bismuth to be constant w i t h i n experimental error for frequencies up to 120,000 cycles per second. Wood^) found s i m i l a r r e s u l t s f o r t e l l u r i u m up to 10,000 cycles per second. Busch and L a b h a r t ^ worked only at l i n e frequency (50 c y c l e s ) . Some e a r l i e r observers(^"^) used a.c. of the same (12) frequency i n both the H a l l sample and the electromagnet producing the magnetic f i e l d . This, of course, resulted i n a d i r e c t H a l l voltage and did not eliminate the Ettingshausen e f f e c t . (13) I I . MEASURING APPARATUS The magnetic f i e l d was supplied by a water-cooled electromagnet with 4 inch diameter f l a t pole faces. This magnet produced f i e l d s up to 14,000 gauss with a £ inch gap. The f i e l d was uniform within 0,5 percent over the volume between the pole,faces. The f i e l d was measured by means of a fluxmeter and two search c o i l s . One c o i l gave f u l l scale d e f l e c t i o n for 5,000 gauss and the other f u l l scale d e f l e c t i o n for 20,000 gauss. These c o i l s were mount> ed on either side of the sample and a separate determina-t i o n of the magnetic f i e l d was made for each measurement of the H a l l voltage. The fluxmeter was guaranteed by the manufacturer to be accurate to better than one percent, however, i t s c a l i b r a t i o n was checked frequently i n accurately known f i e l d s produced i n nuclear magnetic moments experiments i n t h i s department.^) This reduced the error p r a c t i c a l l y to the aotual error i n reading the instrument. This was le s s than 0.5 percent. F i g . 2 C i r c u i t diagram of " l s 160 ohms R2 6b K R 5 1 M % 1 K R 5 220 Z R 6 S3 80 K R7 • 22 K RB 330 K to face page 14 amplifier G1 » 50 mf. C 2 = 16 C 3 - 0.03 C 4 - 0.0033 C c m 0.0066 C 6 = 0.05 C 7 = 0.25 L - #151 Hammond choke C14) Current was supplied to the H a l l sample by a 210 cycle resistance-condenser o s c i l l a t o r of good wave form and frequency s t a b i l i t y . This frequency was chosen to minimize pick up of harmonics of 60 cycles. The o s c i l l a t o r had a transformer output with several taps to permit match-ing to samples with a wide range of impedances. The amplifier f o r the H a l l voltage had a high gain (up to 10?) and was sharply tuned at 210cyoles per second. The input impedance was high so that negligable current flowed through the H a l l contacts. I f current i s drawn through the H a l l contacts the measured voltage i s of course smaller than the H a l l voltage. The sharp frequency char-a c t e r i s t i c , obtained by using a p a r a l l e l T feed-back net-work, reduced the amount of tube noise amplified. The power supply for the H a l l amplifier was c a r e f u l l y f i l t e r e d to reduce random flu c t u a t i o n s . F i g . 2 i s a wiring diagram of the H a l l amplifier. The H a l l voltage could be applied d i r e c t l y to the g r i d of the f i r s t tube or through a step-up transformer. The l a t t e r feature was valuable i n meas-uring very small H a l l voltages because the transformer raised the signal voltage above the noise l e v e l of the f i r s t tube thus improving the signal to noise r a t i o . To measure the H a l l voltage the sample, i n the Fig.3 to face page 15 Schematic diagram of movable H a l l contacts. (1) Lead to amplifier (2; Lucite support (3) Brass sleeve (4) Steel b a l l bearing 1 mm.in diameter Distance between contacts i s 2 cm. (13) form of a t h i n layer on a 2.3 x 7.5 cm. glass microscope s l i d e , was mounted on a piece of i n s u l a t i n g board and held i n place by clamping the ends between layers of t i n f o i l . The clamps served as electrodes for the primary current. The contacts f o r the H a l l voltage were constructed as shorn i n F i g . 3 . The contacts consisted of s t e e l b a l l bearings 1 mm. i n diameter mounted i n brass sleeves. The b a l l bear-ing, free to r o l l i n i t s mounting sleeve, made a point contact which could be moved along the sample without scratching i t . The two H a l l contacts were held at a f i x e d separation somewhat less than the width of the sample by mounting them i n a l u c i t e block. The contacts were held against the sample by spring mounting the l u c i t e block. This ensured good e l e c t r i c a l contact. With current flowing through the sample but no magnetic f i e l d , the H a l l contacts were adjusted so that t h e i r p o t e n t i a l difference was small compared to the H a l l voltage. This residual voltage (as w e l l as the t o t a l voltage with magnetic f i e l d ) was meas-ured by noting the amplifier output and then su b s t i t u t i n g for the unknown voltage a known voltage which could be varied to give the same amplifier output. This c a l i b r a t i o n voltage was obtained by taking a known f r a c t i o n of the voltage developed across a standard resistance and could (16) be calculated by measuring the current flowing through the standard resistance. This c a l i b r a t i o n was repeated for each determination, hence the accuracy of the measurements of H a l l voltage did not depend on the long-time s t a b i l i t y of the am p l i f i e r . The same ammeter (Electronic Instruments Co., Model No.44) was used to measure both the current through the H a l l sample and the c a l i b r a t i o n voltage. Since i n the expression for the H a l l c o e f f i c i e n t , the H a l l voltage appears i n the numerator and the primary current i n the denominator, t h i s arrangement tended to cancel any error i n the ammeter c a l i b r a t i o n . The ammeter had a guaranteed accuracy of 0.3 percent. (17) I I I . PREPARATION OP SAMPLES An examination of the expression f o r the H a l l v o l t -age V » H I E x i o - 9 ( l j indicates that f o r any given material the voltage may be increased by increasing the magnetic f i e l d strength H,and the current I , through the sample and by decreasing the thickness t , of the sample. The magnetic f i e l d strength i s l i m i t e d by the size of the magnet av a i l a b l e . The ©urrent which may be used i s l i m i t e d by the amount of Joule heat, which may be allowed without damaging the sample or i n t e r f e r i n g seriously with temperature control of the sample. Por a given allowable Joule heating, an upper l i m i t i s set upon I 2 r where r i s the resistance of the sample. I f t , the sample thickness, i s decreased by a p factor p, r i s increased by the same factor and Jr must (18) be decreased by tlie same facto r . Thus I i s decreased by py* . I t follows that l / t increases and hence the H a l l voltage also increases. A s i m i l a r argument indicates that a larger H a l l voltage may be obtained by making the sample wider and shorter. However, the r a t i o of length to width must be of the order of 3 or greater to avoid i n t r o -ducing an appreciable error due to s h o r t - c i r c u i t i n g of the H a l l voltage by the current electrodes. The i d e a l sample shape f o r measurement of the H a l l c o e f f i c i e n t appears to be a very t h i n slab with the r a t i o of length to width approximately 3« Layers of antimony were prepared by evaporation i n vacuum using glass microscope s l i d e s 2.5 x 7*5 em. as. targets. At f i r s t , a conventional vacuum evaporating apparatus with removable glass b e l l j ar was used. To ensure uniformity of metal thickness along the sample, multiple sources were used. The sources were 20 m i l molybdenum wire wound i n funnel-shaped s p i r a l s . The nominal thickness of the metal layers was determined by weighing and using the known sample area and the density of antimony i n bulk. Layers from 0.2 to 10 microns i n thickness were prepared with t h i s apparatus. Their e l e c -t r i c a l r e s i s t i v i t i e s ranged from 10 to 20 times greater F i g . 4 to faoe page 19 Schematic diagram of evaporating apparatus. (1) 8 mm outside diameter Pyrex glass tubing (2) Metal to be evaporated (3) Ground glass j o i n t to l i q u i d a i r trap and pumping system (4) Target (2.5 x 7.3 cm.glass microscope s l i d e ) . (19) than that f o r antimony i n bulk. The layers were bright and highly r e f l e c t i n g but had a f i n e l y mottled or banded appearance. Annealing i n vacuum reduced the r e s i s t i v i t y somewhat but was l i m i t e d by the fact that at temperatures above 400°C the antimony re-evaporated from the s l i d e s . Annealing at 500*0 i n an atmosphere of 20-25 cm. Hg of carbon dioxide gas reduced the r e s i s t i v i t y to 5 or 6 times that of antimony i n bulk with only a small l o s s of metal due to re-evaporation. Because of t h e i r high r e s i s t i v i t i e s , these layers wereconsidered unsatisfactory. In an e f f o r t to produce better l a y e r s , the vacuum evaporating apparatus shown i n F i g . 4 was constructed. I t was made e n t i r e l y of Pyrex glass. Heat necessary f o r evaporation of the metal was supplied by a 700 watt furnace which was lowered over the apparatus down to l e v e l AA. This apparatus had several advantages over the conventional vacuum evaporating appara-tus: (1) I t could be completely outgassed permitting more thorough evacuation, (2) Metals other than that being evaporated, rubber gaskets and hot filaments were eliminated thus reducing the p o s s i b i l i t y of contamination, (20) (3) Tjapurities with b o i l i n g points lower than that of the metal to be evaporated tended to evaporate more r a p i d l y and were concentrated i n the f i r s t portion evaporated^ impurities with higher b o i l i n g points tended to remain i n the residual metal. Due to t h i s f r a c t i o n a l d i s t i l l a t i o n , by discarding the i n i t i a l and f i n a l portions, appreciable p u r i f i c a t i o n i s achieved. This apparatus could be used at temperatures up to 600°C. The thickness of layers prepared i n t h i s apparatus was uniform within 1 percent along the length of the sample. This was determined by weighing the metal deposited on several small targets of known area placed at i n t e r v a l s along a 2.5 x 7*5 cm. s l i d e . Layers of antimony from 0.1 to 3*0 microns i n thickness were prepared using t h i s apparatus. A l l were highly r e f l e c t i n g and none showed any structure. Their e l e c t r i c a l r e s i s t i v i t i e s before annealing ranged from 1.8 to 2.2 times greater than that f or bulk antimony (1,2 to 1.4 after annealing i n carbon dioxide). During evaporation, the gas pressure was kept below 10"^ mm. Hg. and the antimony was heated to 580 to 600°0. Antimony was deposited at rates ranging from 2.6 to 33 (21) miorograms/cm2/min. No v a r i a t i o n of r e s i s t i v i t y with t h i c k -ness or rate of deposition was observed. A s i m i l a r apparatus was constructed of vyeor glass, a temperature r e s i s t a n t glass manufactured by the Corning Glass Co. The Vyoor apparatus could be used at temperatures up to 1,100°C. Layers of bismuth from 0.2 to 1.3 microns i n t h i c k -ness were prepared using the Vycor apparatus. Their e l e c t r i c a l r e s i s t i v i t i e s ranged from 1.3 to 2.3 times great-er than that f or bulk bismuth. Annealing at temperatures up to the melting point produced no appreciable effect on the r e s i s t i v i t y . During evaporation the bismuth was heated to 750 to 850°C. Bismuth was evaporated at rates ranging from 30 to 210 micrograms/cm2/min. The r e s i s t i v i t y was found to vary with the thickness, being greater f o r thinner samples. (22) IV. PURITY OF THE SAMPLES The antimony used was 99•8 percent pure. The max-imum l i m i t s of impurities i n percent were l i s t e d as arsenic 0.10, copper 0 .03, i r o n 0 .05, lead 0.04, t i n 0.030, zinc 0.010. Examination of the arc spectra of several of the evaporated layers showed that the concentration of each of these impurities was considerably lower than i n the o r i g i n a l bulk antimony. Arsenic and zinc were concentrated i n the f i r s t f r a c t i o n sublimed while copper, i r o n , lead, and t i n were concentrated i n the resi d u a l melt. The f i r s t f r a c t i o n of each melt (about 5 percent of the t o t a l melt) produced a layer with r e s i s t i v i t y 3 to 11 times that of bulk antimony. This high r e s i s t i v i t y was probably due to the increased concentration of arsenic. The bismuth used was supplied as 99*8 percent pure but impurities were not l i s t e d . The r e s i s t i v i t y of samples prepared from the f i r s t f r a c t i o n of the bismuth melt was (23) not s i g n i f i c a n t l y d i f f e r e n t from the r e s i s t i v i t y of the samples which followed. This indicated that the impurities probably remained behind i n the residual melt. This would be the case i f t h e i r b o i l i n g points were much higher than that of bismuth. (24) 7 . EXPERIMENTAL RESULTS FOR ANTIMONY (1) H a l l Coefficients Uhannealed Samples The H a l l c o e f f i c i e n t R was measured at room temperature for 5 unannealed samples of antimony. Results are given i n Table I . The term 7 m refers to the voltage developed across the H a l l contacts and must be m u l t i p l i e d by the r a t i o of sample width to separation of H a l l contacts i n order to obtain the H a l l voltage. This r a t i o was 1.267* There appeared to be no v a r i a t i o n of R with sample thickness from 0.7 to 1.1 microns, strength of applied magnetic f i e l d from 4,000 to 12,000 gauss, nor with current densities from 180 to 500 amps/cm2. The mean value of H for these samples was •*- 0.215c c.g.s.m. at 20°C. The probable error of the mean was 0.0004 c.g.s.m* Annealed Samples Antimony samples annealed i n carbon dioxide at 20 cm. (25) TABLE I. H a l l Ooeffioients of Unannealed Antimony Layers at 20°C Sample t 54 0.970 55 56 0.703 1.063 63 65 1.047 0.862 89.7 89.7 89.7 89.7 89.7 89*7 89.7 69.2 89.7 89.7 89.7 69.2 69.2 6?.2 4 8 . 6 4 8 . 6 100.0 100.0 100.0 100.0 100.0 100.0 H 3.99 4.01 4.01 12.30 12.31 12.31 4.03 4.03 12.40 12.40 12.43 12.40 12.40 12.43 12.40 12.40 12.17 12.15 12.15 12.07 12.07 12.07 ¥m 0.625 0.628 0.636 1.968 1.940 1.949 0.875 0.672 1.783 1.759 1.772 1.377 1.359 1.346 0.978 0.965 1.942 1.983 1.987 2.449 2.415 2.363 0.792 0.796 0.806 2.494 2.459 2.471 1.109 0.851 2.259 2.230 2.246 1.744 1.723 1.705 1.240 1.223 2.461 2.513 2.518 3.103 3.061 2.995 R 2.146 2.147 2.174 tm 2.160 2.170  2.174 2.156 2.146  2.151 2.158 2.130 2.140 2.160 2.134 2.106 2.185 2*157 2.146 2.117 2.166 2.170  2.151 2.215 2.186 2.139 2.180 D i f f . from mean O.013 0.012 0.015 0.034 0.001 0.011 0.003 0.013 0.001 0.029 0.019 0.001 0.025 0.053 0.026 0.002 0.042 0.007 0.011 0.056 0.027 0.020 Mean / 2.159 0.019 / mean, giving equal weight to each sample, t i s i n microns, I i n milliamperes, H i n kilogauss, V i n units of 10""2volts, and R i n c.g.s.m. units.x 10~^# V m refers to the measured voltage between the H a l l contacts, (26) TABLE I I . B a l l Coefficient of Annealed Antimony Layers at 20°Q Sample t I m V *J» 1.020 100.0 100.0 12.11 12.11 2.289 2.287 2.900 2.898 65* 0.838 100.0 100.0 2.85 2.85 0.655 0.665 0.830 0.842 50.0 50.0 100.0 3.75 3.75 3.76 0.437 0.435 0.858 0.554 0.552 1.088 100.0 100.0 4.83 4.83 1.115 1.121 1.412 1.420 70.0 70.0 8.24 8.24 1.312 1.312 1.663 1.663 75.0 75.0 75.0 50.0 50.0 12.09 12.09 12.09 12.11 12.11 2.091 2.102 2.087 1.382 1.395 2.650 2.664 2.645 1.751 1.768 67* 0.757 75.0 75.0 12.15 12.13 2.322 2.307 2.943 2.924 70* 1.886 100.0 100.0 12.15 12.15 1.233 1.236 1.562 1.567 rjQ&j$L 1.843 100.0 100.0 100.0 12.13 12.13 12.13 1.260 1.265 1.260 1.597 1.603 1.597 D i f f . R from mean 2.443 0.002 2.441 0.000 2.442 2.440 0.001 2.476 0.033 2.476 0.035 2.467 0.026 2.425 0.016 2.456 2.450 0.009 2^464 0.023 nm 2.416 0.025 2.416 0.025 2.416 2.449 0.008 2.462 0.021 2.444 0.003 2.423 0.018 2.447 0.006 2.445 2.445 0.004 2.433 0.008 2.43? 2.425 0.016 2.432 0.009 2.428 2.426 0.015 2.436 0.005 2.426 0.015 2.429 (27) TABLE I I (continued) D i f f . Sample t I H V m V H from mean 71* 1.468 7 0 . 0 12.11 1.107 1.403 2.430 0.011 70.0 12.11 1.126 1.427 2.471 0 .030 2.450 7 1 4 * 1.441 70 .0 12.11 1.128 1.429 2.429 0.012 70.0 12.11 1.135 1.439 2.446 0.005 2.438 71*** 1.414 100.0 3.75 0.517 0.655 2.470 0.029 100.0 3.75 0.516 0.653 2.462 0.021 2.466 50.0 8.14 0 .560 0.709 2.463 0.022 50.0 8.14 0.553 0.701 2.435 0.006 50.0 8.14 0.560 0.709 2.463 0.022 2.454 70.0 12.11 1.152 1.460 2.435 0.006 70.0 12.11 1.153 1.462 2.439 0.002 70.0 12.11 1.148 1.455 2.427 0.014 2.434 Mean / 2.441 0.014 / mean, giving equal weight to each sample. JL a f t e r one annealing && a f t e r two annealings Afck a f t e r three annealings (28) Hg and 510°C for 45 minutes had a larger H a l l c o e f f i c i e n t than before annealing. Results for 5 annealed samples are given i n Table I I . Repeated annealing caused no further change i n R. The H a l l c o e f f i c i e n t did not vary with t h i c k -ness (0.8 to 1.9 microns), applied magnetic f i e l d (3000 to 12,000 gauss) or current .density (140 to 460 amps/cm ). The mean value of R for the annealed samples was + 0.244]_ e.g.s.m. at 20°C. The probable error of the mean was 0.0002 c.g.s.m. The increase i n R due to annealing was 13 percent. The accuracy of the measurement of the H a l l c o e f f i c i e n t i s estimated to be w i t h i n ± 1 percent, the -chief sources of error being the determination of the u n i -formity of thickness of the layers (0.5 percent) and the measurement of the magnetic f i e l d (0.5 percent). Impure Sample As noted previously, the f i r s t sample prepared from each melt contained considerably more arsenic and zinc than did the o r i g i n a l bulk antimony. The H a l l c o e f f i c i e n t of one of these impure samples was measured as follows^at 20°0. (29) Magnetic f i e l d (1CK gauss) H a l l c o e f f i c i e n t (c.g.s.m.) 3.96 12.24 +0.1888 +.0019 +0.1956 ±.0006 Mean +0.1922 The H a l l c o e f f i c i e n t of t h i s sample was 11 percent lower than that of more pure samples (unannealed) and appeared to vary somewhat with magnetic f i e l d strength. (2) R e s i s t i v i t y Measurements The r e s i s t i v i t y »p of a number of samples was measured at temperatures of 77 and 293°K. Results are given i n Table I I I . Electron d i f f r a c t i o n studies 11) have shown that evaporated antimony deposits are oriented with the p r i n -c i p a l axis perpendicular to the substrate, hence the r e s i s t i v i t y measurements were a c t u a l l y made perpendicular to the p r i n c i p a l a x i s . The r e s i s t i v i t y of bulk antimony perpendicular to the p r i n c i p a l axis i s ^ 2 ^ 4 2 . 6 x 10""^ ohm-cm at 293°E and W) 8.2 x 10"*^ ohm-cm at 77°K,hence unannealed samples was 0.154 x 10~ ohm-om/°K and for the unannealed samples 0.168 x 10"^ohm-cm/°K. In E i g . 5 r e s i s t i v i t y i s plotted against temperature (30) TABLE I H . Temperature Dependence of R e s i s t i v i t y of Antimony Layers Sample ?293 f77 ^P/AT f o r 293 Unannealed 54 88 51.5 0.169 38.5 49.5 55 81 51 0.139 40.5 40.5 56 94 59.5 0.160 47 47 63 85 52 0.153 40 45 65 94 59 0.162 46.5 47.5 67 85 54 0.144 43 42 71 76 43 0.153 31 45 Mean / 86 53 0.154 41 45 Annealed 63* 53 18.5 0.160 6 47 65* 54.5 18 0.169 5 49.5 67* 58.5 19 0.183 5 53.5 70* 55 19 0.167 6 49 70** 53 18.5 0.160 6 47 71* 53.5 18.5 0.162 6 47.5 71** 54 17.5 O.I69 4.5 49.5 A A A -71*** 50.5 16.5 0.157 4.5 46 Mean / 54.5 18 0.168 5.5 49 /mean, giving equal weight to each sample A l l r e s i s t i v i t i e s are i n units 10 ^ ohm-cm. 0 50 100 150 200 250 300 TEMPERATURE *K F i g . 5 to face page 3>1 R e s i s t i v i t y of evaporated deposits of antimony as a function of temperature. A - unannealed deposits} B - annealed deposits; C - single c r y s t a l s , r e s i s t i v i t y perpendicular to p r i n c i p a l axis* (3D for unannealed samples, annealed samples and bulk antimony. The r e s i s t i v i t y of the evaporated layers can be represented conveniently as the sum of a temperature depen-dent r e s i s t i v i t y , f> T , which varies approximately l i n e a r l y with the absolute temperature, and a temperature independent r e s i s t i v i t y p Q . P • f o * P T p o was obtained by extrapolating the r e s i s t i v i t y -temperature plot to 0°K. R e s i s t i v i t y measurements are summarized below? ( R e s i s t i v i t i e s i n units of 10**^ ohm-cm). f 7 7 f 2 Q ? *p/A T p293> f0 ?/?s293 Uhannealed samples 33 86 0.154 4-5 41 2.02 Annealed samples 18 54.5 0.168 49 5.5 1.28 Bulk antimony 8.2 42.6 0.159 46.6 -4.0 1.00 f%93 i s the value of at 293°K. f /fSb i s the r a t i o of r e s i s t i v i t y to that of bulk antimony. Annealing greatly reduced p o , hut s l i g h t l y increased The mean room temperature r e s i s t i v i t y of the annealed samples was 1.28 times that of bulk antimony. The lowest (32) r e s i s t i v i t y previously reported f or e„vaporated layers was 1,35 times that of bulk antimony by Harris and S h a f f e r ^ 1 4 The r e s i s t i v i t y of t h e i r samples was considerably more temperature dependent than that of bulk antimony, the smallest value of A J ) / A T they reported was 0.185 x 10~ ohm-cm/°K. (33) VT. DISCUSSION OF RESULTS FOR ANTIMONY Antimony c r y s t a l l i z e s i n a t r i g o n a l structure. This structure d i f f e r s from that of most other metals i n that there are two atoms per elementary c e l l . According to the modern electron theory of s o l i d s , each energy band provides two possible energy states per elementary c e l l as mentioned i n the introduction. Hence, a contribution of one electron per atom w u l d completely f i l l a B r i l l o u i n zone. Antimony has f i v e valence electrons per atom and i t i s generally assumed that these f i l l four bands completely while the f i f t h band i s nearly f i l l e d and the s i x t h nearly empty. The amount of overlapping of the f i f t h and s i x t h bands has been estimated by Brown and Lane (^5) f r 0m meas-urements of the magnetic anisotropy of tin-antimony a l l o y s . They conclude that there i s about 1 free electron per 100 antimony atoms or 3 .3 x 1 0 2 0 electrons and an equal number of holes per cm^. (34) Apart from the r e l a t i v e l y large number of free electrons and holes (even at T =0) which leads to a posi t i v e temperature c o e f f i c i e n t of resistance, antimony may, i n certain respects, be considered as a semi-conductor. I t s H a l l c o e f f i c i e n t should be given by the same formula which holds f o r a semi-conductor with combined electron and hole conduction: rr # ngvgg - wi2 (2) e (n2V2 + n i v i )'£ where e i s the electronic charge, n^, n,, are the densities of electrons and holes respectively and v.^, v g are t h e i r m o b i l i t i e s . For pure antimony n^ » n 2 and since the H a l l c o e f f i c i e n t i s p o s i t i v e , v 2 > v-^  • The r e s u l t obtained f o r the H a l l c o e f f i c i e n t of unannealed antimony, +0.21^ c.g.s.m, at 20°C, i s i n agree-ment with the value +0.219 c.g.s.m. reported by Barlow^ and by Zanu^1*^ but i s i n poor agreement with the value +0.192 c.g.s.m. by Ettingshausen and N e r n s t ^ 1 ^ , and +0.123 c.g.s.m. by Altertnum^ 1?K For annealed antimony samples the H a l l c o e f f i c i e n t was found to be +0.244.^  c.g.s.m. at 20°C, t h i s i s 13 per-cent higher than the value f or the unannealed samples. Since (35) the increase i n the H a l l c o e f f i c i e n t due to annealing does not vary with the thickness of the antimony la y e r , the increase must be due to a bulk effect and cannot be a t t r i b -uted to a surface effect such as p a r t i a l oxidation of the antimony. From equation (2), assuming the r a t i o v^/vj* remains constant, i t may be shown that i n the neighbourhood of n l = n2» 8 1 1 increase i n the H a l l c o e f f i c i e n t can be obtain-ed only by reducing n-^  and/or n 2 . I t i s very u n l i k e l y that annealing would have the effect of trapping either electrons or holes. On the contrary, one would expect the small c r y s t a l l i t e s to form larger c r y s t a l s upon annealing thereby reducing the number of possible surface traps. I t i s i n t e r e s t i n g to refer to an observation made by Stephens^ 2 0) on antimony i n bulk. Stephens measured the H a l l c o e f f i c i e n t of two antimony plates 0.5 cm. i n thickness, one consisting of large c r y s t a l s , the other of smaller c r y s t a l s . He found that the H a l l c o e f f i c i e n t of the f i r s t plate was 9 percent greater than that of the second. On the basis of the foregoing discussion, i t seems reasonable to exclude an explanation based on trapping electrons or holes at the c r y s t a l boundaries. The follow-ing i s suggested as a plausible explanation of the increase (36) i n R. The r a t i o v^Ag i s assumed to remain p r a c t i c a l l y constant during the annealing process. These antimony layers as w e l l as the plates used by Stephens were of very high p u r i t y , i t w i l l therefore be assumed that trap-ping by impurities may be neglected. (In any case we should not expect such trapping to be increased by anneal-ing the samples). Since trapping at c r y s t a l boundaries i s also excluded, » n 2. I t i s clear that the amount of overlapping of the f i f t h and s i x t h bands depends on the c r y s t a l structure. Since the behaviour of the p o t e n t i a l at the c r y s t a l boundaries w i l l be d i f f e r e n t from that with-i n the c r y s t a l s , one might expect the extent of "average overlapping" to depend on the size of the c r y s t a l s compos-ing the sample. I f we assume that c r y s t a l boundaries tend to increase the average overlapping, annealing would r e s u l t i n the recombination of a f r a c t i o n of the free electrons and holes. Annealing would then have the ef f e c t of reduc-ing the number of both electrons and holes and hence would increase the H a l l c o e f f i c i e n t . Any in t e r p r e t a t i o n of the effect of annealing on R must be i n accordance with the observed decrease i n r e s i s t i v i t y . The l a t t e r i s an e f f e c t which i s generally observed i n evaporated layers. I t i s commonly assumed that (37) by increasing the size of the c r y s t a l l i t e s , annealing reduces scattering by c r y s t a l boundaries. This would lead to a decrease i n p Q , the temperature independent part of the r e s i s t i v i t y . However, i n addition to the decrease i n p Q , an increase of about 9 percent i n p"*", the temperature dependent part of the r e s i s t i v i t y , w a s observed. This increase i s consistent with the tentative explanation given above for the behaviour of the H a l l c o e f f i c i e n t . I f a number of free electrons and holes reoombine upon annealing f T w i l l increase. I t would be expected that the percentage increase i n R and p T on annealing to be the same. In view of the sim p l i f y i n g assumptions made, the observed increases of 13 percent and 9 percent respectively, seem to be i n s a t i s f a c -tory agreement. ( 3 8 ) VTI. EXPERIMENTAL RESULTS POR BISMUTH (1) H a l l Coefficients Unannealed samples The H a l l c o e f f i c i e n t R was measured at room temperature for 4 unannealed samples of bismuth. Results are given i n Table IT. The H a l l c o e f f i c i e n t did not depend on current density but varied with the magnetic f i e l d according to an equation of the form R(H) = R Q - bH 2 ( 8 ) where R Q and b are constants. The values of R Q and b d i f f e r e d from sample to sample but for each sample the experimental points f i t t e d a curve of t h i s type w e l l w i t h i n the experimental error. The values of Rcalc. l i s t e d i n Table IV were obtained from the r e s u l t i n g equation. The mean value of the difference ( E c a l c - H) was 0.002 c.g.s.m. There appeared to be no d e f i n i t e dependende of R 0 or b on sample thickness. (39) Annealed Samples Annealing the evaporated bismuth layers i n an atmosphere of 20 cm. Hg of carbon dioxide f o r 40 minutes at 260°0 reduced both R q and b by about 20 percent. Results are given i n Table V. The experimental points f i t t e d a curve of the form given by Eg..(8). The value of R Q and b varied from sample to sample. Table VI l i s t s the values of RQ and b for both annealed and unannealed samples. The mean value of RQ for the unannealed samples was +0.895 c.g.s.m., and for the annealed samples +0.708 c.g.s.m. The mean values of b were +10.1 x 1 0 " 1 0 c.g.s.m./(gauss) 2 and +8.4 x 1 0 " 1 0 2 c.g.s.m./(gauss) respectively. In order to obtain the sign of the H a l l c o e f f i c i e n t , a rough measurement was made using a d i r e c t current method. (2) The Fie l d - f r e e R e s i s t i v i t y of Bismuth The r e s i s t i v i t y i n the absence of a magnetic f i e l d was measured at room temperature (293*K) and at the temperature of l i q u i d nitrogen (77°K) for 6 unannealed and 3 annealed samples. Results are given i n Table V I I . Although the main purpose was to measure the r a t i o P77/P293 n 0 e l a b o r a , b e Precautions were taken to obtain absolute values, a p l o t of room temperature (40) TABLE IV B a l l Coefficients of Unannealed Bismuth Layers at 20°C R c a l c . Sample t I H V 111 0.427 40.0 40.0 3.68 3.68 2.594 2.57? 3.287 3.268 20.0 20.0 8.12 8.12 2.684 2.695 3.401 3.415 14.0 14.0 12.14 12.14 2.617 2.624 3.316 3.325 112 1.206 12.0 12.0 8.13 8.13 0.504 0.507 0.639 0.642 9.00 9.GD 12.14 12.14 0.504 0.499 0.63? 0.632 115 1.010 30.0 30.0 1.496 1.498 0.294 0.294 0.372 0.372 45.0 40.0 3.68 3.77 1.085 0.978 1.375 1.23? 30.0 30.0 5.33 5.33 1.025 1.025 1.29? 1.2?? 25.0 25.0 8.16 8.16 1.238 1.238 1.56? 1.56? 20.0 20.0 10.26 10.26 1.189 1.182 1.506 1.498 40.0 35.0 12.16 12.16 2.684 2.335 3.401 2.958 0.953 0.947 0.950 0.946 0.894 0.898 0.896 0.901 0.833 0.835 0.834 0.831 0.790 0.794 0.792 0.792 0.704 0.698 0.701 0.701 0 . 8 3 7 0.837 0.837 0.841 0.838 0.831 0.834 0.830 0.820 0.820 0.820 0.816 0.777 0.777 0.777 0.780 0.742 0.737 0.740 0.743 0.706 0.702 0.704 0.702 (41) TABLE IV continued Sample t I H "m V R Rcalc. 116 0.971 40.0 40.0 3.68 3.68 1.070 1.074 1.356 1.361 0.894 0.897 o.m. 0.897 20.0 20.0 8.12 8.12 1.113 1.101 1.413 1.395 0.845 0.g?5 0.840 0.839 18.0 18.0 12.18 12.18 1.327 1.333 1.681 1.689 0.745 0.748 0.746 0.747 R i s i n c.g.s.m. uni t s . (42) TABLE 7 H a l l Coefficients of Annealed Bismuth Layers at 20°0 Sample t I H V m 7 R R c a l c • 112 1.203 30.0 25.0 3.76 3.67 0.552 0.448 0.699 0.568 0.746 0.745 0.746 0.748 10.0 8.14 0.378 0.479 0.707 0.704 9.00 12.12 0.453 0.574 O.63? 0.635 114 0.795 50.0 50.0 25.0 25.0 3.66 3.66 8.08 8.08 1.182 1.181 1.206 1.205 1.498 1.496 I.528 1.527 0.651 0.650 0.650 0.602 0.602 0.602 0.651 0.602 20.0 12.10 1.264 1.601 0.^ 27 0.527 116 0.964 45.0 45.0 3.75 3.75 0.956 0.962 1.211 1.219 0.692 o.6?7 0.694 0.692 20.0 20.0 8.12 8.12 0.865 0.863 1.096 1.093 0.651 0.649 0.6^6 0.654 16.0 16.0 12.16 12.16 0.951 0.950 1.205 1.204 0.597 0.<>?6 O.596 0.594 ( 4 3 ) TABLE VI H a l l Coefficients of Bismuth Layers at 20°C. Snmma-ry Sample t RQ b 1 1 1 0 . 4 2 7 0 . 9 5 8 8 . 7 1 1 2 1.206 0 . 8 6 6 1 1 . 2 1 1 3 1 . 0 1 0 0 . 8 4 3 9 . 5 116 0 . 9 7 1 0.912 1 1 . 1 Mean 0 . 8 9 5 1 0 . 1 Annealed 1 1 2 1 . 2 0 3 0 . 7 6 0 8 . 5 1 1 4 0 . 7 9 5 , 0 . 6 6 3 9 . 3 1 1 6 0 . 9 6 4 0 . 7 0 2 7 . 3 Mean 0 . 7 0 8 • 8 . 4 - 1 0 »o b i s i n units 1 0 c.g.s.m./(gauss) c (44) TABLE V I I . R e s i s t i v i t y of Bismuth Layers t f293 ?77 P77/P293 ftl2 ) fil2)/f 25 0.235 240 748 3.12 257 1.07/ 0.306 203 599 3.06 221 1.09 0.461 215 600 2.79 236 1.10 0.770 164 455 2.78 187 1.14 0.864 157 m» - -1.049 178 mm - 203 1.14 1.091 172 475 2.77 198 1.15 1.302 143 243 1.70 174 1.22 Annealed 0.859 154 368 2.38 182 1.18 1.041 166 411 2.48 198 1.19 1.299 148 252 1.70 181 1.22 f*293 and P77 are the f i e l d - f r e e r e s i s t i v i t i e s at 293°K and 77°K, P(12) i s the r e s i s t i v i t y i n a magnetic f i e l d of 12 kilogauss at 293°IC with the magnetic f i e l d p a r a l l e l to the p r i n c i p a l c r y s t a l a x i s . A l l r e s i s t i v i t i e s are i n units 10"*^  ohm-cm. (45) r e s i s t i v i t y versus thickness revealed a d e f i n i t e decrease of r e s i s t i v i t y with increasing thickness. The highest r e s i s t i v i t y at room temperature, 240 x 10 ohm-om, was measured for the thinnest sample, 0.24 microns i n thickness. The lowest room temperature r e s i s t i v i t y , 143 x 10~^ohm-cm corresponded to the thickest sample, 1.3 microns i n th i c k -ness. These values may be compared with measurements by Kaye(21) # He found the r e s i s t i v i t y of bismuth single -6 c r y s t a l s perpendicular to the t r i g o n a l axis to be 109 x 10 ohm-cm. at 293 °K. For a l l samples measured, the r a t i o f^q /P293 Y f & a larger than unity, i . e . the temperature c o e f f i c i e n t was negative. P^/p 2oj decreases from 3*1 for t =0.24 microns to 1.7 for t » 1.3 microns. I t i s to be noted that the temperature c o e f f i c i e n t of the r e s i s t i v i t y of bulk bismuth i s p o s i t i v e . Annealing did not have any appreciable e f f e c t upon the r e s i s t i v i t y . (3) R e s i s t i v i t y i n a Magnetic F i e l d The influence of a magnetic f i e l d upon the r e s i s t i v i t y of bismuth layers was measured at room temperature with the magnetic f i e l d p a r a l l e l to>the p r i n c i p a l c r y s t a l a x i s , i . e . perpendicular to the sample baoking. A t o t a l of 7 (46) unannealed and 2 annealed samples were investigated. For each sample the r e s i s t i v i t y increased with increasing f i e l d strength according to the formula P(H) = f w * a H 2 ( 9) where P293 i s t h @ f l e l d - f r e e r e s i s t i v i t y at 293°Z and a i a a constant. The r a t i o of the r e s i s t i v i t y at 12,000 gauss to the f i e l d - f r e e r e s i s t i v i t y i s l i s t e d i n Table VII. The increase i n r e s i s t i v i t y due to a f i e l d of 12,000 gauss varied from 9 percent for the thinnest sample up to 22 percent for the thic k e s t . Annealing had no p a r t i c u l a r effeot upon the constant a. A few measurements with the magnetic f i e l d perpendic-ular to the t r i g o n a l a x i s , i . e . p a r a l l e l to the sample back-ing, showed that the change i n r e s i s t i v i t y was about twice as large as for the case where the magnetic f i e l d was p a r a l l e l to the t r i g o n a l a xis. This r e s u l t i s i n agreement with measurements of the magneto-resistance of bismuth single c r y s t a l s by K a y e ( 2 1 ) , assuming that our layers are p o l y c r y s t a l l i n e with the axis perpendicular to the backing as found by Lane(22) # (47) VTII. DISCUSSION OF RESULTS FOR BISMUTH (1) H a l l Coefficients Although there ex i s t s a great amount of l i t e r a t u r e on the e l e c t r i c and magnetic properties of single c r y s t a l s of bismuth and bismuth a l l o y s ( 2 2 ) , r e l a t i v e l y few reports on t h i n layers can be found. Grosz^ 2*) prepared bismuth layers i n d i f f e r e n t ways and investigated the change of r e s i s t i v i t y i n a magnetic f i e l d as a function of the layer thickness. His work as w e l l as that of K i r e h n e r ^ 2 ^ and i n p a r t i c u l a r that of Lane( 2 2) shows that evaporated bismuth layers are p o l y c r y s t a l l i n e , the t r i g o n a l axis being perpendicular to the substrate. No reports were found of measurements of the H a l l c o e f f i c i e n t of bismuth films prepared by a technique s i m i l a r to that employed i n t h i s research. In the following discussion, d i s t i n c t i o n w i l l be made between Rn (H p a r a l l e l to t r i g o n a l c r y s t a l axis) (48) and R x (H perpendicular to t r i g o n a l a x i s ) . I t w i l l also be assumed that there i s c i r c u l a r symmetry around the t r i g o n a l axis so that the symbols (TN and o x only w i l l be used. The l i t e r a t u r e on r e s u l t s obtained with bismuth single c r y s t a l s Indicates that R may be p o s i t i v e or negative depending on the angle between H and the p r i n -c i p a l axis and also depending on the temperature. Heaps( 2^ found from experiments on plates cut from single c r y s t a l s that R x at room temperature i s negative and Ry/ i s po s i t i v e (about 1 c.g.s.m.). He also observed a s l i g h t increase i n R ff With increasing f i e l d strength. Onnes and Beckman ( 27) reported R/( = +0.06 c.g.s.m. I t should be remarked that many of the discrepancies found i n the reported values of the H a l l c o e f f i c i e n t are probably due to the fact that the number of free electrons i n bismuth i s very small (about 10"^ per atom according to Thompson^28^ and others). For t h i s reason the H a l l c o e f f i c i e n t and conductivity jDf bismuth are extremely sensitive to small amounts of impurities. The value of R // reported here agrees f a i r l y w e l l with Heaps1 values referred to above. However, i t was found i n this research that R// decreases with increasing (49) f i e l d strength contrary to Heaps1 observations. According to Jones^ 2?), the H a l l c o e f f i c i e n t of pure bismuth (density of electrons = density of holes = n) i s given by R'« " eTn * 2 - 1 ) and Rj_ = - -2- ( X x + X„ - 1) where X i s the r a t i o of the conductivity due to electrons to the t o t a l conductivity and C i s a po s i t i v e constant. I f Heaps* results are v a l i d f o r pure bismuth at least as fa r as the sign of the H a l l c o e f f i c i e n t i s concerned i t follows that Xj_ 0.5 and X„> 0.5, i . e . v^ ^  v g i n a di r e c t i o n perpendicular to the c r y s t a l axis and v-^  > V g p a r a l l e l to the axis. (v - p V g are electron and hole mobilities r e s p e c t i v e l y ) . The r a t i o of the m o b i l i t i e s i s therefore anisotropic. At a temperature of 87°K, Heaps found R to be positive i n a l l d i r e c t i o n s . This means that at s u f f i c i e n t l y low temperatures v^ £ v 2 both perpendicular to and p a r a l l e l to the p r i n c i p a l a x i s . Jones 1 theory predicts that i n the case n^ .=* hg, the H a l l c o e f f i c i e n t should be independent of the magnetic f i e l d strength. Small amounts of foreign atoms may change (50) and n 2 and for a l l o y s , his theory leads to the following expression, R(H) * A - (C 2 V Z/e) H 2 ( 1 0 ) 1 + Z 2 C H 2 where A i s a constant, 7 i s the atomic volume of the impurity and Z i s the r a t i o of the number of impurity atoms to the t o t a l number of atoms. I f the impurity atoms accept or donate one free electron each, n 2 - n x = Z/7 In the case of tetravalent impurities Z has to be taken p o s i t i v e , f o r hexavent impurities Z must be taken negative, The term Z 2 C H 2 i n the denominator of Eg.. (10) i s probably 1 i n the layers described here. I f t h i s i s true, the observed r e s u l t s s a t i s f y an equation of t h i s form with Z p o s i t i v e (Eq.(8)). This does not necessarily mean that the observed behaviour of these bismuth layers, was due to impurities. I t i s reasonable to assume that because the c r y s t a l l i t e s are very small an appreciable f r a c t i o n of the free electrons are trapped i n surface states. This would have the same effect as adding t e t r a -valent impurities such as t i n or lead to bulk bismuth. Annealing would increase the size of the c r y s t a l l i t e s , t h u s (51) reducing the number of surface traps and hence reducing the apparent Z. Upon annealing, a decrease i n b of about 20 percent was observed. Further confirmation f o r the idea of surface traps i s afforded by the observed decrease i n RQ upon annealing. Examination of Eq..(2) indicates that an increase i n the number of electrons (for example,by releasing electrons from surface traps) would, indeed, decrease R Q . (2) The F i e l d - f r e e . R e s i s t i v i t y The fact that the observed room temperature r e s i s t i v i t y was larger than has been reported for single c r y s t a l s may be due i n part to surface trapping and i n part to a decrease i n the mean free path of the electrons. The mean free path i s estimated to be about 1 micron. As may be seen i n Table T i l f^f f ^ was found to decrease with increasing sample thickness (from 3.1 at 0.2 microns to 1.7 at 1.3 microns). The observed negative temperature co e f f i c i e n t of r e s i s t i v i t y confirms the findings of Lane(22) on evaporated layers. He found, for samples from 0.1 to 4.0 microns i n thickness, a mean value of f ^ j / /3293Cff> a D 0 U b 2.8. Kaye^ 2 1) found a positive temperature c o e f f i c i e n t for both fh and i n single c r y s t a l s . The negative temperature c o e f f i c i e n t i n t h i n layers (52) may be explained on the assumption that the density of free electrons i s reduced by surface traps as compared to that of larger single c r y s t a l s . The density of free electrons and holes i n "pure" bismuth i s , according to r e s u l t s obtained by Thompson, about 1 0 " p e r atom. An increase i n temperature w i l l lead to two competing processes: i t w i l l increase the p r o b a b i l i t y of l a t t i c e scattering and increase the number of free c a r r i e r s by thermal e x c i t a t i o n . The l a t t e r becomes pronounced at small c a r r i e r densities and w i l l cause a decrease i n r e s i s t i v i t y with increasing temperature i f s u f f i c i e n t electrons are released from traps. The dependence of ^77/ P 293 o n thickness would, on t h i s basis be expected to be i n the d i r e c t i o n observed, so that here again the observed behaviour can be accounted f or q u a l i t a t i v e l y by the same assumption as used i n the previous section. The effect of annealing on f was not noticeable whereas on R Q and b i t was. This may be due to the fact that Ro and b are determined by the difference between n^ and n 2 rather than by t h e i r sum. (3) R e s i s t i v i t y i n a Magnetic F i e l d According to the theory developed by Jones the r e l a t i v e change i n r e s i s t i v i t y In a magnetic f i e l d i s (53) given by ^ f / p - B H 2 1 + Z 2 C H 2 -Provided Z 2 C H 2 ^ 1 , which i s l i k e l y at room temperature and f o r the range of f i e l d strengths used, t h i s i s i d e n t i c a l with Eg,.(9) as derived from the observed r e s u l t s . The increase of RH)/ f> with increasing sample thickness has also been observed by G r o s z ^ 2 ^ who i n v e s t i -gated layers i n the same range of thickness. Gn the basis of the assumption that increases with increasing thickness due to a reduction of the influence of surface trapping, these observations can be explained. The factor B i n Jones* expression (2q.. (11)) i s proportional to °l • , ,2 — — — ~ ( V ! + v 2 ) ( Ol + )2 where (F£t CT^  a r e electron and hole conductivities perpendicular to the t r i g o n a l a x i s . A p o s i t i v e H a l l constant indicates that CT[ ^ CTg. I f i t i s assumed that v^, v 2 at any given temperature do not vary with sample thickness, then an increase i n cr^,leads to an increase i n B. Therefore thicker layers should correspond to larger values of B. This i s i n agreement with the observed v a r i a t i o n * ( 5 4 ) The observation that f> increases more rap i d l y with the magnetic f i e l d for H perpendicular to the t r i g o n a l axis than f or H p a r a l l e l to the axis i s i n agreement with (24. i the r e s u l t s of Grosz* 'on evaporated layers as w e l l as with those of Kaye^ 2^ on single c r y s t a l s . On the basis of the foregoing discussion, i t appears possible to explain the observed r e s u l t s i n a consistent and s a t i s f a c t o r y manner using the assumption that surface trapping of electrons plays an important part i n the e l e c t r i c and galvano-magnetic properties of evaporated bismuth l a y e r s . (55) REFERENCES (1) A.H. Wilson, "Semi-conductors and Metals" (Cambridge 1939) (2) see F. S e i t z , "The Modern Theory of S o l i d s " (McGraw-Hill 1940) page 190 (3) A.Smith, Phys. Rev. ^ 5_, 81 (1912) (4) L.A .Wood, Phys. Rev. 41, 231 (1932) (5) C Busch and H. Labhart, Helv. Phys. Acta. 19, 463 (1946) (6) T. Des'Coudres, Phys. Z i e t s . 2, 586 (1901) (7) Von; Traubenberg, Ann. d. Physik. 12, 78 (1905) (8) H.Zahn, Ann. d. Physik. 2.6, 553 (1911.) (9) see T.L. C o l l i n s , "Measurement of Nuclear Magnetic Moments" a Ph.D. thesis, the University of B r i t i s h Columbia (10) J.A. Pri n s , Nature. 131. 760 (1933) (11) G. Hass, K o l l o i d Z e i t s . 100. 230 (194-6) (12) P.W .Bridgman, Proc. Am. Acad. 6£, 351 (1929) (13) W.J. DeHaas, Proc.Royal Acad.Sc. Amsterdam 16, 1110 (1914) (14) L.Harris and L.H. Shaffer, Phys. Rev. 26, 943 (1949) (13) S.H. Brown and C.T. Lane, Phys. Rev. 60, 895 (1942) (16) G. Barlow, Ann. d. Physik 12, 897 (1903) (17) H. Zahn, Ann. d. Physik 14, 886 (1904) (18) A. Ettingshausen and W. Nernst, Sitzungsberichte Akademie der Wissenschaften i n Wien 94 11.360 (1886) (56) (19) Alterthum, Ann.a. Physik 22, 93? (1912) (20) E. Stephens, P h i l Mag. 347 (1930) (21) G.W. Kaye, Proc. Royal Soc. 170A, 561 (1939) (22) C.T. Lane, Phys. Rev. 48 193 (1935) (23) see for example, N.F. Mott and H. Jones "The Theory of the Properties of Metals and Alloys'' (Oxford 1936) (24) F. Grosz, Z e i t s . f. Physik 64, 520 (1939) (25) F. Kirchner, Z e i t . f. Physik 36, 576 (1932) (26) C.W. Heaps, Phys. Rev. £0, 6 l (1927) (27) H.K. Onnes and B. Beckman Com. Phys. Lab. Leiden No.129. No. 1£2 (1912) (28) N. Thompson Proc. Royal Soc. 155A. I l l (1936) (29) H. Jones Proc. Royal Soc. 155A, 653 (1936) 

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