: z :_ :5 £3 N915 may. . o... ‘ .. 6 O I» (o. x... . TH ESIS 02., MICHIGAN SSSSSSSSSSSSSSSSSSS IHIIIUIIIHllllillllllil|ll|HlllHl|1|Hl1Lollll|l2|||ll||lll| L m A R y 3129301704 MichIgan State Univcmty MIC‘H'NKN ”KEATS UY‘UVERSZTY DEFAH '3' . .1? {T 0F 23H? 35 C: EASK LAE'VMN’J, MICIVHGAN PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE ABSTRACT RESISTIVITY AND MAGNETORESISTIVITY OF FINE INDIUI WIRES AT LOW TEMPERATURES by Alan Fredrick Burmester . 0 When one of the physical dimensions of a conductor approaches the mean free path length of the charge carriers. ' the observed resistivity of the conductor is increased over i the value found for the bulk material. This increase in,. resistivity is known as the electrical resistance size effect. The mechanism of this effect is the introduction} of surface scattering as a relaxation mechanism which is. comparable to impurity and phonon scattering This size effect in thin films has recently been ’ the object of much experimental and theoretical investi- ’ gation. However, thin wires have not seen so much atten- tion probably due to the more complex geometry and sample. production problems. . This‘investigation has been concerned with the electrical resistance size effect in indium wires ranging in diameter to a lower limit of .0156mm. The temperature dependence of the effect was observed between 4.2'x and l.l'K. lagnetoresistive size effects were observed in' fields up to 20 kilogauss. The increase in resistivity with decreasing size observed in this work agrees well with work done previously _on larger wires. A maximum in the magnetoresistance of some' of the smaller wires was observed. A'eimple calculation Alan I". Burmester based. on this maximum and the size dependence 0f the increased resistivity lead to values of 1.0 x 10""gm-cm/sec for charge carrier momentum and a concentration“ of .4 carriers t per atom . . ' - analmvrrv AND momonnsrsnvnv or rm innwu WIRES AT Low TEMPERATURES By ‘Alan Fredrick Bur-esterfll A THESIS Submitted to lichigan State University . in partial fulfillment for the requirements for the degree of -' name or sermon _ K . ‘ ~Departsent‘of Physics and.Astronosy ' ..i ' _.‘1 , l zfg’continued aid and encouragement throughout the course 61,;55 }_fi . . The guidance of Dr. M. Garber in some of the experié?;lx .If.mental_details and interpretation;of.the data is.alsOQ?f'f. _ jfl_ acknowledged. 1;*3}3'.j-j4,;5; 3.. .1 f, ,u . fl. I55“ ‘y_:.';fliI.amfgratefulfito”thefAtomic:EnergyRCommission”forgj_*.~- ,f:financial"supportrofgthisjwork}5filz : ifflif'idr- 'j“ YZU'I ‘1 . ‘ : ..¢ . f" , -’.""'.’..‘::“‘I-:W;" ...I.--.IA,...‘;: ";.- , - . . . '. - J; .- . 1. g“ l,‘ ' ' I. I . - .3 9g. fi ' . . ..- I. ’ ' . :3- .' E'5L‘l' ' U .. t ‘ .. I: _ j. 5 .‘-n . _' (I, .. A ..- " .fif’VlQ7-- w.. 3.. "‘ ' " j'fird ”U ' pf 1. 'U-ACKNOWLEDGMENTSf t'f' " It is my pleasure to express I n '+ “to Dr. F. J. Blatt who 9 myWsincere'appreciationjv v u 9 suggeSted.theeproblem'and:offered:fm~ Chapter II'. III. Iv. v. TABLE or CONTENTS Page Introduction‘. . . . . . Theoretical o ' e e o e o e o e e e‘ e e. e e 4 , Experimental Arrangement and Procedures. . . . . . . . . . . . . 13 Presentation of Data and‘Discussion of Results. . . . . . . . . . . .'. 25 Conclusions. . i . . . . . . . .'. . . . 55 References 0 e 0' o e e o e e I. e o. e e e e o ' o o o o 58 111 . . it ' ~Figure l. 3. 4. 5. 6.. 80' 9. 10. 11. '12. 13. 14. 15. 16. 17. 18. . 19. ‘2o. 21. The Sample Extrusion Apparatus . . . . . . The Sample Holder. . . . . . . . . . The Measuring Circuit. . . . . . . . . . Resistivity Resistivity Resistivity Resistivity Resistivity' Resistivity Sample “ Resistivity Sample . Resistivity Sample Resistivity Sample Kohler Plot Kohler Plot Kohler Plot Kohler Plot Kohler Plot' Kohler Plot ‘ Resistivity Resistivity Phonon-Surface Resistivity vs; LIST OF FIGURES vs. Magnetic vs. Magnetic vs. Magnetic vs. Magnetic vs. Magnetic vs. Magnetic III-1. vs. Magnetic III-5. vs. Magnetic III-6. . vs. Magnetic IV-6 . . . . I Field; Sample III-l_ Field; Sample III-5 Field; Sample v1-2. Field; Sample Ill-6 Field; Sample IY-B. Field Squared; Field Squared; Field Squared; Field Squared; for sample III-'1 o e e e o o e for Sample IV-2. . . for ’ sample Iv-5 e e o o o o o e for sample Iv-se o o o e e for Various Samples at 4.2'K . for Various Samples at 2.5'K . . _vs. Reciprocal Diameter at 4.2'K vs. Reciprocal Diameter at O'K . .at.4.2'x. Page 13 15 . ' 22 27 28 29 ' .30 31 34 35 36 37 39 4o - g 41 42 46 47 49 .52 53 LIST or TEBLES Table . l . Page 1 Description of Samples . . . a . . . . . . 26 II Tabulation of Resistivities in Zero Magnetic Field at Various Temperatures . 44 \ ’° 3 \ o _. 'z ‘ Q .- ' 1V .2 . - Chapter I Introduction At room temperatures the electrical resistivity of a metallic conductor is independent of the shape or size of the material. However, at low temperatures it has been ob- served that material which has been formed into thin films. or wires has a higher resistivity than the bulk material at the same temperature. This effect, known as the resistance size.effect, is due to diffuse scattering of the conducting charge carrier at the surface. The effect becomes observable only when the mean free path of these carriers is comparable_ to the smallest dimension of the conductor. . This size effect was first observed in thin silver films by Stone in 1899 (1), and was treated theoretically by J. J. Thompson in 1901 (2). With the introduction of.. liquid helium research techniques and the availabilityof ultra pure metals, studies can be conducted in the range where the mean free path is much greater than the sample- dimension. Under these conditions the size dependence of the resistivity is a major effect. A good approximation for the increased resistivity due to this effect in a “phi ' d where pb is the bulk resistivity, L the mean free path in cylindrical wire is given by the expresSion pb + the bulk, d the diameter of the wire, and d a geometrical factor which is close to unity (3). Measurements of resistivity as a function of wire diameter enable a deter- mination of ph and 3 to be made. More importantly, the Q 1 2 product pbt which is directly determined is a measure of 'the momentum of the carriers at the Fermi surface. Closely related to the above effect is a size depend- ence of the magnetoresistance in thin films and wires. _ _ .This effect has been treated theoretically for thin films '(4,5) and for wires in a longitudinal field (6). A successful theory is yet to be developed for thin wires in a transverse field. Size effect experiments of this type may lead to some information regarding the momentum of electrons at the Fermi surface. . Consideration of a temperature dependent size effect .in zero magnetic field may give some information regarding the nature of the resistivity of the bulk metal at very low temperature. The present paper is an extension of the experimental-- work of J. L. Olsen (7) to indium wires smaller in size by about one order of magnitude and in magnetic fields approximately twice as great. Preliminary work on this; problem was done by B. C. LaRoy (8) and a good review of the literature on this subject prior to 1963 appears in his work. ‘ Indium is a good material for these studies because its bulk magnetoresistance saturates at a field well within the available range. 'It was hoped that this would facilitate. . the separation of the various size effects in the magnetic field. From a practical point of view, indium can be extruded to very fine diameters (“.01 mm); furthermore it Q anneals at room temperature. It is also commercially avail- able in a high state of purity. The resistivities referred to herein are a function of both applied magnetic field and temperature and will be symbolized by p(HI,T). A subscript is used to indicate“ whether the symbol refers to the sample resistivity p' (H T), 1 or to the bulk resistivity pbfli, 'D) . Chapter II Theoretical Bulk Resistivity The expression for electrical resistivity of a metal according to the free electron model of metals is m . Wm? - <1) Here'N is the number of free e1ectrons.per unit volume 'and is given by the number of atoms per unit volume times the number of free-electrons per atom, T is a character- ' istic relaxation time which is the average time of travel of the electrons between scattering events, e is the electron charge and.m is its mass.” The free electron model neglects the interaction of the electrons with the lattice potential. As a first approximation, the effect of this potential may be taken into account by replacing m with an effective mass m* which compensates for the effect of the periodic lattice potential on the electron. Since electrons obey the Pauli Exclusion Principle Fermi-Dirac statistics must be applied. ‘In a metal at low temperature these statistics lead to,a highly degenerate distribution. Thus only electrons with energies near the " Fermi level having Fermi velocity v1 participate in the .charge transport.. The Fermi velocity is given by I. '14:}? .' '3 . . . (a) 4 5 where Eris the Fermi Energy. We now define the mean free path by i - vir. Thus ' f I- . p - §:;f-- --§;§2, - (3) where P1 is the Fermi momentum. "The above.discuSSion assumes' one group of charge carrier with isotropic effective mass, I a situation which is rarely encountered in practice. A . better approximation to a real metal is often made by assuming a current which is presumed to arise from the I motion of two types of charge carriers having differing _isotropic effective mass, number density and possibly electric charge. This is known as the two band model. If the scattering of electrons is accomplished by several mechanisms acting independently, a characteristic relaxation time may be associated with each. Then the . effective relaxation time'is 1 ' 1. 1 1 ' 1. 1 -_ - .- - eoeo - - - --- 4 Toff r, + r, + and* set: i, + i, + ( ) In this case resistivities due to each separate mechanism are additive and we arrive at Matthiessens Rule Peff --;§;’< i: + 11-, + ...) :- Pl+Pa+ ---(5). In bulk metals the usual form of Matthiessen's rule is obtained by identifying two scattering mechanisms. Lattice imperfections such as impurities and dislocations 3 r . contribute a temperature independent term while thermally. 6 excited lattice vibrations (phonons).produce a temperature dependent term. Equation (5) is then . p _ f 1 1 ”he“ . W (I? 4' 131D) - ' (6) or pb(T) - ”1 + pp('r) where D1 is the resistivity due to impurity scattering and other temperature independent effects and pp(T) is the resistivity due to phonon scattering. Temperature Independent Size Effect If the conductor has physical dimension comparable - to the mean free.path in the bulk then the charge carriers will be scattered by the surface of the specimen an appreciable portion of the time. If this surface scatter- ing is independent of the other scattering mechanisms, then.' applying Matthiessen's rule: ' . p . f l l 2 Ne (‘b is) ‘where ‘b is the effective mean free path in the bulk umaterial andf8 is the mean free path associated with surface scattering. Identifying ‘s with a physical dimen- sion such as the diameter of a wire, we have the expression derived by Nordheim (3): ~- . " of b . Pr "' ”h (1. + T . (8) o . *where d is a coefficient which expresses the nature of the I Q - i . 7 Surface scattering, ranging from zero, for specular scatter- ing to unity in the case of diffuse scattering. Observations I 'indicate that for most metals the surface scattering is . entirely diffuse (9, 10). More sophisticated treatments based on the solution of the Boltzman equation have been published by Fuchs (11) for thin films and Dingle (12) for thin wires. Calculations by Sondheimer (9) based on Dingle's theory agree to within 5% with results of the Nordheim equation. . . On the basis of equation (8) the temperature dependent part of the resistivity should be independent of the wire size. However, the results of an experiment by Olsen (7) indicated that the temperature dependence of the resistivity of thin indium wires is greater than that of the bulk material. The following explanation suggested by Olsen : has been expanded by Blatt and Satz (13) and has been 'investigated with computer methods by Luthi and Wyder (14). The maximum angle through which an electron can be . deflected by phonon scattering is 1 [momentum of H K 9 .3 phonon _'I_‘_ . _I_)_ max momentum of 9D kf (9) electron where KD is the wave vector of a phonon of energy ken, kf is the wave vectorof an electron at the Fermi surface and 9D is the Debye temperature. .For metals KD/k kf *1 so I . that at low temperatures (T<>d . ' \ b. T<<9D 'c. The differential scattering cross section is a constant for the allowed scattering angles. d. Umklapp scattering is neglected. When compared with the results of Olsen (I) on indium and .Andrew (10) on mercury, the predicted value of pp8 is con- sistently too high. This may be due to the fact that the :measured value of pb which enters in the above equation contains the resistivity due to both Nbrmal and Umklapp . ' p . scattering whereas the derivation of the expression forpp8 ‘9 neglected Umklapp events. By dividing pb by a suitable constant such that predicted values of ppa agree with db- served values, the ratio of Normal to Dmklapp resistivities may be estimated. ' . ' ” . Magnetic Effects If a conductor which is carrying a current is placed in a magnetic field, the Lorentz force on the current carriers- will‘force them into curved orbits. In a metal which has only one type of carrier this curvature deflects all carriers to one side of the conductor which in turn creates a Hall electric field. This field will then just counter- act the Lorentz forces on the carrier producing an undeviated path. To explain magnetoresistance then, one must consider a two band model. In this case the two forces do not cancel a and the carriers are deflected into orbits giving rise to .- -an increase in resistance. It may be expected that the ” ; effect of the applied field on the resistance of the con- ductor will depend upon the radius of the orbit (rc)'com- pared to the mean free path of the carriers. From (H) (p70 T)>- -:<—— j? (N6: ‘ - (Necgté. (13) we see thatH/‘p(o T) is a measure of the dimensionless I parameter L/rc. Consequently, in discussing the effect of the magnetic field, nfip(o T) is used as the independent variable (15).. 'i . ._ ,’ ;1 In particular itcangbe shown that ' 10 _Q_ _ - H pb(o.T) ’ 7p our) <13) In this relation, known as Kohler's rule, pb(O,T) is the zero field resistivity, Ap is the change in the resistivity in the applied field H and f is a function of __TD—T) which if is characteristic of the particular metal and independent of temperature. A specific example.of this general rule is an expression derived by Sondheimer and Wilson (16) for. the two band model i I ' | I “56%) l' TBHTfiHZI. (14) Here A and B are constant characteristics of the metal. In”. practice it has been observed that although this expression. correctly predicts the occurrence of saturation in indium the transition from the quadratic to the saturation region. is actually much broader than predicted by equation (14). Moreover the dependence of the magnetoresistance on applied" field at low fields is observed to be somewhat less than quadratic. O The magnetoresistive size effect has been calculated for a thin wire in a longitudinal field by Chambers (6). Olsen has applied his arguments in a qualitative way to the case of thin wires in a transverse magnetic field. He has pointed out that in applying Kohler's rule to thin 'wires, one should replace the pb(O,T) value by pw(O,T) as measured for that particular wire if the same function f —- is used independent of wire sina. With this modifi- cation one would expect Kohler's rule to be obeyed under t ‘t 11 the weak magnetic field condition where rc>d. . . Olsen (7) and LaRoy (8) have observed this to be true I at moderate fields. At fields such that rc>d). That is, px is itself a function ... .. .I of magnetic fields. ' I Chambers in his calculation of the magnetoresistance of a thin wire in a longitudinal field computes pfoor ... u.¥l various L/d and ‘(rc values. No such tabulation exists for the transverse case. One may still construct a Kohler plot based on the measured.p'(O,T) and from observa- tion of departure of this Kohler plot from that of the bulk material one can eatimate rc(H) and hence the Fermi * b t ‘ momentum using . P ":1. sure (16) A maximum in magnetoresistance with a subsequent decrease with increasing transverse fields in thin sodium .‘ .o' ‘1' \ r . _ _ \ - . .fi ~ r .. ‘ .-. _._ _...-........_._ ._-._...._.._.-—. n“. .HW-.._.__..-- -.. 4 ____...._-~__.._.-._ wires has been observed by Mononald and Sarginson (5). This effect was explained theoretically (5) for thin films with a magnetic field in the plane of the film and for a thin wire of square cross section in a transverse field. The origin of this size dependent effect is the reduction of the surface scattering resistivity which occurs when the cyclotron orbits of the conduction electrons become smaller than the dimensions of the conductor. The electrons .traveling in these.small orbits are confined to trajectories which meet the surface less often than the trajectories associated with the Tc) g' condition. The position of , this maximum occurs at a field strength such that rem %5 _From the expression for the cyclotron radius Pf may be evaluated.. Sondheimer (4) has predicted that the magneto- resistance of a thin film in a magnetic field normal to I the plane of the film would be an oscillatory function of the applied field. Both of these effects have recently .3 been observed in thin aluminum and indium sheets by Forsvoll and Holwech (17,18). Previously Babiskin and Siebenmann (19) had observed an oscillatory magnetoresistance in thin sodium wires.~ ; While the theory of magnetoresistance in thin films has been well developed and supported by experimental results, the same can not be said for the case of thin 'wires. Nevertheless, qualitative remarks which have been" 'made here are probably correct and will provide a back- ground and basis of comparison for the present work. ~ Q Chapter III Experimental Arrangement and Procedures Sample Preparation Wire samples were extruded from k inch indium rods supplied by American Smelting and Refining Company. The purity of this material was given as 99.999% by the processor. The extrusion press consisted of a steel cylinder 1" OD x %" ID x 3" long to one end of which a diamond die was bolted. A short length of the indium, as supplied, 'was placed into this cylinder. Pressure was applied from i" steel piston. a bench press and hydraulic jack by a Figure l The Sample Extrusion Apparatus 13 14 A heater of manganin wire was wound around the extrusion cylinder. It was found, that for die sizes smaller than .003 inches, smoothest wire surfaces were formed when the indium was heated to a temperature of about 120°C. Higher tempera- tures produced wire which had a beaded appearance while lower temperatures required pressures which placed a 1 severe stress on the steel punch. For the larger size dies heating was not necessary. . ' The speed of extrusion was found to be important in the formation of wire with smooth surfaces.’ A formation .speed of about 10 centimeters per minute seemed to be the best for the smaller dies. Faster speeds produced twist- ing and jumping in the formation. 'A light weight was 1 attached to the end of the smaller wires during the extru- .sion process to prevent twisting and excessive vibration -in the wire. The surfaces of the wires were examined with a micro- scope and were found to be of generally regular shape and uniform diameter (variations less than 10%) with occasional scratches or nicks in the surface. The wires were allowed to anneal at room temperature for at least 48 hours after :mounting and before being cooled in the resistivity cryostat.. The smallest wire.produced by theSe methods came from ' a die with a-bore of .0005 inch (.013mm). However, due to difficulties encountered in handling and mounting, the smallest wire for which resistivity was successfully measured was from the .0007 (. 018mm) inch die. R 15 The diameter of the wire was determined by measuring, the resistance of the samples R, in a kerosene bath'at ‘ room temperature. At this temperature the mean free paths of the electrons due to phonon scattering are small (~10.°cm) compared to the diameter of the smallest specimen;’hence° the resistivity of the material in the wire should not be. a function of its size, and every wire is considered a bulk specimen. The diameter is thengiven simply by d - 3% . The resistivity, p of indium was taken as 4 1' 8.8 x 108 ohms-cm._at 296°K from White and Woods (20). The length, L, of the wire between the pair of potential con- . tacts in the sample holder was measured with a.traveling microscope. In the case of the larger samples, sections which had been taken from the same wire as the experimental sample were also examined and measured by microscope. The, two values for the diameter usually agreed to within 2 or A 3 percent. The optical measurement was not possible for . wires smaller than about .04mm due to diffraction fringes: which obscured the edge of the wire.. ' The Sample Holder . For the resistance measurements thefsamples were. ' placed in a sample holder which consisted of a one inch diameter glass filled epoxy (glastic) rod 10 inches long. 'Six separate grooves were out along the length of the rod. The grooves were .05 inch deep and .02 inch wide. The . . . r g - . glastic material was selected for the mounting form because Figure 2. The Sample Holder Thermal lsolotlon Chamber Current Contact Radiation -7 ‘ Potential \ 1 fl Contact Helium Level Sample Groove Detectors / BA Carbon \ Thermometer Dn- IJ Heater 16 17 its thermal expansion coefficient closely matched that of the indium samples. Another advantage was that the material was easily machined. Flat surfaces were milled in the rod at each end of the six grooves. At each of these flat surfaces two i" x 33" x g?" copper blocks were attached by means of epoxy cement. The blocks at the extreme ends of. :the rod served as current contacts. rThepotentialocontact'" .was made by a narrow strip of platinum foil which was‘ soldered to the block nearest to the center of the rod. The platinum was placed in such a manner that the area of contact to the sample was a minimum. This was done to localize the point of contact and to minimize rectifying contacts and contact potentials. The sample length be- .tween the “ontacts was about 20mm. The six sample wires were connected in series. This connection permitted monitoring the measuring current in all of the samples at once. However, a common measuring ‘ current proved inconvenient since the resistances of a set of samples usually differed by about two orders of magnitude making different measuring currents desirable. Moreover, with this type of current connection, when individual samples broke or became disconnected during a : measurement run, the entire operation had to beadiscontinued. For the last set of samples these disadvantages were eliminated by mounting only two samples and rewiring the sample holder with separate current leads for each sample. The electrical leads for the measdring circuits were 18 placed in the inside of the sample holder support tube, passed through a wax seal in the dewar cover plate, and terminated in an 18 pin connector. This connector was placed inside of a braSs thermal shield for the purpose of minimizing thermal emf in the potentialileads. Sample Mounting . After annealing, the wire sample was placed into the groove in the glastic sample holder in physical contact ' with the four electrical contact blocks. These contacts .were previously tinned with an indium 33 wt. bismuth alloy . which melts at 70°C. A small drop of Superior #30 flux was placed at the four connections.‘ Besides aiding in the . - soldering process, the surface tension of the'flux helped Ito maintain close physical contact between the wire and the contact'blocks. A small soldering iron heated to approxi- mately lOO°C-was placed on the contact block as far away from the sample as possible. The iron was removed just _ as the solder under the sample melted.‘ In this way con- tamination of the sample through alloying was reduced to a minimum. The entire length of the groove_c0ntaining the .sample was then filled with a warm glycerin.soap solution which jelled at room temperature thus immobilizing the ‘ specimen. It was found that if.this procedure was not' used the Lorentz force due to the magnetic field was sufficient to cause deformation of the sample with an . . - t , accompanying rise in resistance. In fact some of the Q l|-"‘ 19 .‘rlr'la | smaller wires were actually broken by Lorentz‘forces.when ~they had not been Supported by this means." ' N The Cryostat ‘ Temperatures in the 4.2'K to l.l5°K.range were obtained‘w by placing the sample holder with liquid helium in a con- ventional glass double dewar with a'narrow tail section. The sample holder was suspended from the dewar cover plate 'by'means of a g inch stainless steel tube. Attached to the 'support tube were two liquid helium level detectors, one just above the sample holder and the other just below the level of the top of the inner vacuum jacket. A copper radiation shield was positioned about 4 inches above the upper level detector. The level detectors consisted of. 10 feet of .005 inch tantalum wire wound on a one inch glastic form. The coils had a normal resistance at the superconducting transition temperature of 0.3 ohms; The olevel of the liquid helium was detected by observing the. superconducting reSistance transition in the tantalum. In' order to prevent the ceil from being cooled below the transition temperature (4.3'K) by the cold helium gas. during transfer a 30 ohm manganin heatertdiSSipating 100 milliwatts was wound over the tantalum coil and connected in series with it. The general arrangement is-shown in ’ figure 2. . . Temperatures below 4.2’E were obtained by reducing . _the pressure at the surface of the helium.bath. 'In order 0 20 to insure that the bath was in equilibrium.with its vapor at temperatures above the lambda point, 100 milliwatts were dissipated in a 600 ohm heater which was attached to the bottom end of the sample holder. . 1 For the first two runs (III and IV) the vapor pressure was regulated in the 4.2'K to 2.5'K range by use of a cartesian manostat. For temperatures lower than 2.5'K '.the manostat had insufficient pumping capacity and it was necessary to use a one inchneedle valve in parallel with the manostat to achieve lower temperatures. For the last two runs (V and VI) a Walker regulator (21) with 1% inch pumping lines was installed. This regulator held the I ‘pressure constant to within one part in 10"r in the_range “ from 760mm to 6mm (1.2'K). Below 6mm it was necessary to - pump directly through a two inch valve. The vapor pressure was measured by‘a mercury manometer, ' an oil manometer, or a Macleod gauge depending upon the ‘pressure involved. From this, the temperature in the .cryostat was determined by referring to the 1958 Helium Temperature Scale. I - An 87 ohmfiwa'tt carbon resistor wasattached to ' the sample holder in order to monitor the temperature of the samples. The resistance of this thermometer which had a value of 1,285 ohms at 4.2'K, was measured with a simple;' ID.C. bridge using a Leeds & Northrup type 9834 electronic . null detector.. With this combination it was possible to . , . 1 determine the temperature to one part in 10‘. _ -~ \ 21 During the course of an experimental run after the temperature had stabilized at the desired value, the carbon- resistance thermOmeter was balanced and the null detector observed throughout the course of the measurements. These observations indicated that the temperature was maintained constant to within .0005'K. Transverse magnetic fields were applied to the samples by means of a Harvey-Wells 15 inch iron core electromagnet.“ The dewar required a 3 inch pole tips separation. For runs' III, IV, and V the magnet was equipped with 12 inch pole : tips.with which the maximum field was 17.7 kilogauss. For run VI, 6 inch pole tips were used with a maximum field 21.8 kilogauss. Sample lengths were less than 80% of the, . diameter )f the pole tips and were carefully centered in the field. Measurements with a nuclear magnetic resonance probe indicated the field to be uniform to 0.5% over the sample length. During the course of the measurements the 1 field was measured with a Rawson type 720 rotating coil . 57. flux meter. The field strength.was determined to within 1%.. 3 -Electrical Measurements The current and potential leads were brought from the thermal shielding chamber on top of the cryostat through A inch copper tubing to a switch box on a shielded measuring 3 4 . table. The six pairs of potential leads were connected by means of a Leeds & Northrup rotary switch to a Rubicon six- dial potentiometer. -'Nu11 was detected with a Guildlinetype I '- . ‘1'- ' O. . O' ’C. at; ....- Figure 3. The Measuring Circuit 4 ‘fi fi‘ a K- 3 Six - Dial r-- - - - -- - - - - ----------- '1 PO'ODNOH‘IOIB' I ' Cuffs", l Cryostat I I l Supply 1 l l l I | l I Samples l | l l l l ' I l I l l L ................ _l 22 23 5214 photocell galvanometer amplifier operated in the series feedback mode with a sensitivity of 50 centimeters per micro- volt. Noise, stability and drift were such that the resolution was approximately 2 x 10‘”8 volts. The measuring current was supplied by a series parallel bank of four large lead-acid batteries (300 amp-hrs). Uhder constant load the current from these batteries remained stable to within on part in 105. In order to vary the current a constant impedance (n - network) attenuator was employed. The current was monitored continuously by ' measuring the potential across a 10 ohm series resistor with a separate potentiometer (type K-3) with electronic null detector. With this arrangement a variation in the current of one part in 10p produced a deflection of one cm on the null detector. Experimental Procedure At liquid helium temperatures the first concern was to determine a proper measuring current. This was done by starting with very low currents, about 1 ma, and measuring the resistance of each sample. The current was then increased until the resistance of the smallest wire in the set showed an increase.- This increase was due to the :magnetoresistance produced by the self-field of the- :measuring current. By keeping the self-field of each sample below one gauss the resistance pf the wires could ‘be considered independent of the measuring current. .A Q . 24 current of 10 to 50 ma fulfilled this requirement and still produced an emf of about 5 microvolts in the larger wires. The initial decrease of resistances with increasing measur- iing current observed by Cochran and‘Yaqub (22) in single, crystal gallium wires was not observed. Calculations indicated that, assuming a Hall coefficient of 0.4 x 10"“ - volt-cm/amp-Oe (15), the Hall voltage across the potential .contacts under the worst possible conditions could not exceed 2 x 16" volts. Since this is Just the resolution of the experiment no corrections were made. The magneto- resistance and zero field data were obtained by fixing the temperature and the magnetic field and measuring the potential drop across each sample, first with the current in one direction and then with the current reversed to eliminate thermal emf's. The magnetic field was then changed to the next desired value and the above procedure repeated. .The temperature was changed after data had been obtained for all of the desired values of magnetic field. Recent work by Bate, Martin and Hills (23) has indicated‘ a decrease in the resistivity of a wire as its age increases. They have attributed this to a slow recrystallization of the material after extrusion, into small crystallites whose sizes are comparable ta.that of the wire. To investigate this effect several samples were allowed to age at room temperature for a period of several months.. Then measurements were made simultaneously on the aged wires and on wires which had Ibeen recently extruded through the same die. \ . Q Chapter IV Presentation of Data and Discussion of Hesults Four experimental runs (III through VI) were made. Runs III and IV each consisted of six samples measured in magnetic field strengths ranging from zero to 17.7 Kila- gauss at eight temperatures between A.2°K and 1.15'K. In run V four samples were measured at 4.2'K only. In 'run VI three samples were measured at fields ranging from zero to 21.2 Kilogauss at four temperatures between 4.2'K i ' and 1.20'K. ‘A description of these samples is contained in Table 1. ‘ This discussion is based on the results of all of these measurements. The graphs presented show the general trend of the results as well as the deviations from those trends. High Field Magnetoresistance Plots of the resistivity vs. magnetic field at several.. 'temperatures are shown for 5 samples in Figures 4 to 8. These plots all display a saturation-of magnetoresistance which is characteristic of indium. ‘The degree of satura- tion is seen to be generally temperature dependent. There is a sharper break in the curve and a flatter plateau for lower temperature. In addition the smaller samplesddisplay a definite decrease in resistance in moderate magnetic fields. ' This effect is likewise temperature dependent, that is, the 25 1 ' 1 TABLE I Description of Samples Die Size ' j.d‘ l/d Age Designation (mm) (mm) ..(mm31) (days) III 1 .254 “ .248 4.03' ‘ 21 III 2 .0889 .0847 11.8 III 3. .0889 .0795 12.6 III 4- .0635 .0603 . 16.6 III 5 \.0399 ‘.O366 27.3 *III 6 '.0229 .0189 ‘ 53.0 IV 1 .0635 3.0597 16.7 Iv , 2 .0635 .0544 18.4 * 1v 3 .0315 .0362 27.6 * IV 4 .0330 .0362 30.3 * IV 5 .0256 .0229 39.1 Iv. 6 .0173 .0156 64.1 v 1 .635 .642 1.56 ** V' 2 .254 .249 4.02 * v 3 .0315 .0352 . 28.4 t v 6 .0229 .0185 54.4 1 v1 2 . .0284 .0273 36.7 4 v1 3 .157 ‘ .159 6.29 4 v1 4 4 o 4 .0508 .0416. 2 . *Damaged dies' A . . **This sample was taken from the same wire as III-l 26 ‘ Q Pllo. (SI-cm) 3.5 L figure 4. Reeieiiviiy vs. Magnetic Field; Sample III-I “*2.“ 3.0 -— " T23.“ e A A H 2.5 r- I T-2.5'K I. 2 G U H , . ‘NJI'K V v 5' ¥ ? 2.0 - " . ., m-I H d-.248mm I'I” I.5 ,., i L l J J l l l l I 2 4 6 8 IO I2 I4 I6 I8 HiKG) Figure 5. Resistivity v3. Magnetic Field; ,Sarn I2 -— ‘ p x i0. ; (SI-cm) . Temperature (' K) o 4.2 A 3.8 D 2.5 v LII d - .0366 In In 2 4 6 8 IO l2 l4 I6 l8 23 H (er '2 __ F'Oure 6. Resistivity vs. Magnetic Field; Sample Iv-z px IO. (Q'cm) / I0 Temperature (’ K) 0 4.2 a 2.5 08.0273mm N .p O a) 5 K) I 3 3 29 H (KG) [- Figure 7. Resistivity vs. Magnetic Field; Sample III-6 C C H px IO' .‘ (SI-cm) ‘ ‘ _ I3 *- i .’ I2 i. Temperature (‘Kl O 4.2 A 3.6 D 2.5 V L! ds.0i887mm 1 L J 1 J I l I 2 4 6 8 I0 l2 I4 I6 I8 30 I5 I2 Figure 8. Resistivity vs. Magnetic Field; Sample IV- 6 ’ t p x i0. . ‘7 4 (9.- cm) ; 1 ‘ . 0 '/ V - Q " 'I' 7 I Temperature PM I- . .. O 4.2 A 3.8 a 2. 5 V I.I d 8 .0l56 m m l 1 l I A l l l 2 4 6 8 IO I2 I l6 l8 H (KG) 32 decrease is sharper at lower temperatures. This decrease. was observed in all samples having a diameter less than .0273mm and in a few samples with diameters as large as . .085mm (III-2, III-3, VI-4). This effect was not observed in any of the four samples having d.) .085mm. In all cases. when it was observed the maximum in the p vs. H curves was more pronounced at l.l5°K than at 4.2'K. In some cases the decrease of p with respect to field was observed only at the lower temperatures (Fig. 6).* The size dependent reduc- tion of resistivity with increasing magnetic field has previously been observed for sodium.wires in a transverse field by McDonald and Sarginson (5) and for aluminum and .. indium films by Forsvoll and Holwech (17, 18). This effect '” has been ascribed by McDonald to the reduction of surface scattering of the conduction electrons when their cyclotron . diameter equals the wire size that is when rc - $3, The fact. that this effect was not observed in larger wires is not surprising since on the basis of McDonald's argument the - maxima for these wires should occur in a region where the bulk magnetoresistance has not yet saturated.. The effect would then be obscured by the rapid increase of the bulk ‘ resistance with magnetic field. ~*In contrast to this behavior which is considered to be a . real effect, several of the samples in the d - .027mm range - namely samples III-5, IV-3 and IV-4 (see Fig. 5) - 'showed no decrease in resistance at any temperature. These samples usually displayed incomplete saturation -- that is, the resistance continued to increase'slowly with increasing . field beyond the 10 Kilogauss region.w ‘ 33 . If we use equation (16) and assume that the maximum in the magnetoresistance occurs at r - g5 we obtain the c average value of P - 1.0 i .1 x 10“9gm.cm/sec in fair f ' vagreement with values reported by Forsvoll from his size effect experiment (1.3 x 10"°gm.cm/sec) and in good agree- fr. ment with Rayne's (24) magnetoacoustic measurements (.83 x 10““gm.cm/sec). Similar results have recently been obtained for thin indium sheets by Cotti and Olsen (25). Effects of this type wereiobserved by Olsen (7) for wires. I in a longitudinal field; however, he did not identify his ' maximum with the above orbital condition. Low Field magnetoresistance . Figures 9 through 12 display the resistivity of a" given wire as a function of the square of the applied magnetic field in the region below 1000 gauss. These 'plots indicate the general deviation from the H2 rule. Moreover, it was noted that no single exponent could be chosen which would characterize the magnetic field depend- - ence of all samples. These changes in the field dependence 1. do not seem to be correlated with size or age of the sample or the temperature. , ; Since indium becomes a superconductor below 3.39'K and since the construction of a.K0hler plot requires p'(o,T);: it is necessary to obtain values of the zero field resistivity for temperatures less than 3.39'K by indirect methods. Olsen suggested extrppolation of the:p vs. I!” plot to 3-0. With - '2-1‘...‘ 4- . v—‘afi.-- - - .. n a 34 use: Nz 00.. Ngv 3... 9.20..an0... mm. on. 0N. 2.6.? 62.1 ... Z. 2955 mooeoaum 32... 05:00:. .2. 3.33.31 .m 953.... 0N 35 .85 «I 00.. A D I. _._ ON I m.m N.¢ .. C. L oeaaoeaneh m I... 5238. u e Baeow ..u 953.5 22... 2.2302 .2, 2.3533... .0. 2:3... :5 9 025. V 13: NI 8.. as. on. on EE ago. a u 36 min. m.» C. o. oeaaoeeanh N.¢ w I :_ 2.25m ..eoeoaum 30E 23:00: .2, £333: .__ 852... aoxvux 37 9N 9n 3. o. 3:3;qu... N6 86 00.0. m->_ oEEow "noeoacm 32a 2.2.3: .2, 33:23”. .m. 2.6.... 0.0 38 this method of extrapolation in mind a high density of points was taken in the low field region in the present experiment. However, due to the complicated low field behavior described above, a reliable extrapolation was' not possible. An extreme example of the difficulty involved here is shown in Figure 11. ‘Another difficulty which beset this method of obtaining pw(0,T) was that the magnetic superconducting transition was about 50 gauss wide in.some cases.‘ This broadening made it im- possible to distinguish between points which represented normal magnetoresistance and points which represented a .partial superconducting state. This was an especially ~serious defect since the points in question were the closest ones to Hi0 and thus should have received the greatest weighting in the extrapolation to H=0. The inconsistent dependence of the magnetoresiStance with respect to wire size and the broadened superconducting transition are no doubt associated with the conditions of sample preparation and the amount of impurities (especially ferromagnetic) present in each sample. As a result of these observations it was concluded that extrapolations based on this plot would be uncertain to such an extent as to invalidate the further analysis of this data. The procedure finally used to obtain pw (0,T) for all samples was based on the use of Kohler diagrams (Figures 13 to 16). For T > 3. 4' K the :7???» vs m points were computed directly from the measured values p" (0, T) and t' 1 .5-c<:39 N..o. x .a: 0~ 0. o 0 t N _ o. o. v. N. _. 00. _1 _ _ _ _ _ _ a _ _ _ esoew ... ... D as a a.» a 9 3 we 0 3... 2282.63 .-.: 225m .3 SE :23. .2252... No. 00. 00. AEoIwaaaaoo. «..o. x 60v: N . w. .. 00. N0. — # — q q EEwwno. u u _ _ D n.N D Qn 0 Ne 0 0.5 226.35.» NI). flqeow .2 3.5 323. .3 2:3“. N0. 00. 41 .EquxangQ «to. x ix: “XL u _ m. N. _ oo. «o. d 3 q A — u a _ u D a 56003.. e a ... D D ON a 9n d «é o 0.20.368. oEEom 3. SE .623. .m. 2.6... .1 -1 N0. 00. 42 .Eo..d<::oo. «..o. x ix... N _ o. N _ 00. «0 q _ 1 n _ a _ _ fl . E E $0.0. .. .0 ... D rtu a o o.n d «6 O 0.60 22939:.» 0 I). gasom .2 .Sa 323. .m. 23E .1-1 N0. 43 pw(HfiT). The plots for these temperatures were used to L pwcoxr) was assumed to be independent of temperature in the low define the function f< > graphically. The function field range. Here"low fieldVis defined to be fields corre- sponding to the condition rc > £5 The data for T.S 3.4'K was fitted to this function in the low field region by adjusting pw(0,T). Superconducting points were easily dis-o ltinguished by the fact that they could not be made to fit the function at low fields. pw(0,T) values obtained in this manner are presented in Table 2 and are probably not in error by more than 1% for T > 2.0'K and 5% for T.g 2.0'x. In the saturation region of these Kohler diagrams a temperature dependent deviation from Kohler's rule is observed. VF0r wires with d > .08mm decreasing temperature produces a lower saturation value of 9%3 This effect de- creases as the wire size becomes smaller, vanishes for d 3 .06mm (Ivel and IV-2), and in fact, reverses for smaller diameters. The results for d.2 .08mm are in qualitative.' agreement with those of Olsen whose smallest reported sample had d - .10mm. The reversal of the effect has not been previously noted. However, LaRoy's "II-7" with d 3 .06mm 0 I seemed to show a weaker temperature dependence than did his larger samples. This deviation.at high magnetic fields for the larger wires probably has its origin in the effect proposed by Olsen.. That is, in the Kohler relation the quantity pw . should be replaced by pb since under the condition r§ < d Q "— '- _Iw—I-v-r r7 '7 'rvww‘ 44 0*.H \ mm.s_.eos x «sensse see. 055 9H6 a. 8.0 .2 x 50-5.0230... mofips>wpmfimom Ufiofim caposmss oaoN mo sowpmasnae HH mumdfi Amev assuaaonaoa me.s mv.a . me~.0 mmn.0 m0«.0 muH.0 ems.0 Hm.0 mm.0 em.0 we.0 «0.0 08.0 mm.0 peas 0\H .m> 0.50s“ eeeeaeeeeexm ses>spmemem seem mo.m . us.n . . HH.0 . mm.e .uexu mm.H emm.H ma0.~ 0m.» 00.». 0mm.» . mmm.m . _ 0mH.0 n0m.s. -- 800N.~ Hm.0 em.e He.e mm.m 08.5 sm.m sm0.0s www.0s we.» as.» mm.s me.s be.» mm.s mm0.m 0vm.m Hm.m em.m 00.m me.m _me.m ms.0 mm.0 eme.o me.e 00.m 0H.m em.m me.m H0.m mem.m me.o om.m Hm.m mm.m so.m none as.» 0am.m msm.m m0.m . w0.m seem 0H.m Hm.m me.m men.m 0Hw.m so we.» 0w.s H0.s mH.m me.m 05.w .me.m .0He.m em mn.v mm.v sm.¢ .mm.v. ma.m Hm.e 0m.m Hom.e we. be." 0m.m em.m mm.m e0.m as.” new.” emm.m m0. mm0.m m0.~ ms.u ea.m mm.u em.m eee.u 000.m me m0.m ,ms.n Hm.« e0.m HH.m mam.m «H0.» 300 em0.~ m0.H 50H.H. meH.H oesws sem.s «He.s «mm.s . A .3 a 2.6.53 333633 n.~ 0.H 0.x m.m 0.m v.” m.m N.e bawo. bud. ammo. ‘I”‘ H > mmao. ammo. een.. ue0.. TV”° b' emso. memo. «m0. memo. eemc. sumo. HNMV‘IDCD l > H meO. 0000. m000. m050. bvwo. mfim. A250 NMV'IDCD HIHHH ofimadm 45 the charge carriers generally are confined to trajectories which do not strike the surface. By using pw we decrease - the %3 value for wires since pw > pb' 'This deviation p w should be more pronounced at lower'temperatures where b is larger. Regarding the apparent reversal of a trend which we expect to be maintained, we note that here the low field- magnetoresistance also appears to be a function of wire size itself. Our results indicate that £3 at low fields. increases as d is reduced. The reason f0: this will be described subsequently. However, it is clear that the K0hler function derived from low field data then is anomalously large, particularly for the smallest wires and lower temperatures for which this effect is greatest.. It is possible that the additional magnetoresistivity mechanism operative at low fields persists to fields in excess of saturation and is responsible for the positive‘ flf discrepancy. ’I Size Dependence of Kohler Diagrams Figures 17 and 18 are Kohler diagrams for several samples at 4.2°K and 2.5°K reSpectively.? In the 4.2°K plot the general features of Olsen‘s high field results are confirmed. The quantitative differences are probably ' due to differences in sample purity and preparation. The apparent independence of Kohler's rule with . _’, respect to wire size at the lower fields suggested ~o“- “ . I . ~ 46 .Eoudxuasow. 8.03.6.2. O o. o N . o. N . mo. . _ . _ a J 1 q T q A I a b 00.0. 0I>. > mooo. e-... meN. N-> ocN. .-.: New. .-> .66. 3.0503 0.955 e u - - c v..N.¢ .0 «0.955 30...; .0. 3.... .0200. .... 05...“. N0. 00. 47 25-0.3360. N..0. x as. O. m N . m N. _. @O. s _ . . e a . a a J a a .D a L N00 0 D a D l d. a 00. 00.0. 0-... e my 4 wnNo. 0.2 D I . I _ onno. 0.2 n. D nmso. m..: G .. n. Q¢N. .-.: O D .02... 3.0605 0.060m n. I N .. D D 46 D 4 DD 0% \s L 0. D . - .. . D ....-e. .aua .. .0”.-. . Aw . L . xomN .0 00.0E0m m:0..0> .0. .0... .023. .m. 0.00.... 48 by Olsen's data, is net true for the finer wires of this study. This has also been noted by LaRoy who ascribes the. 'increase in gB-at low fields and decreasing wire size to an enhancement of surface scatteringadue to the curvature. of the carrier trajectory. That is, even weak magnetic fields cause electrons which are traveling axially in the wire to follow a curved orbit to the surface, where they suffer diffuse scattering. . In both the high and low field magnetic effects the results of this experiment were not whally self consistent. Exceptions to the high field trend are samples III-5, IV-l,‘ '29 V-3, while some anomalous behavior occures at low field." in samples 111-2, '5, Iv-2, v-1, VI-2 and V154. Size Effect in Zero Field -- %-Plot' According to the Nordheim equation a plot of p'(0,T) " vs. l/d for the various samples at 8 given temperature should yield a straight line with a slope of pb(O,T) x t and intercept at l/d-O of pb(0,T). pb(0,T) is the zero field bulk resistivity and z is the bulk mean free path . at the temperature T. “The data for T - 4.2'K is presented in this way in Figure 19. The results of several other workers are also shown for comparison. The scatter of the data points in this figuregreatly exceed the 1% experi- mental error which can reasonably be assigned.) The cause . for this inconsistency is not understood at present. It should be noted, however, that subsequent to these measure- ' .2 , ‘5 . I. ’ . Figure I9. Resistivity vs. Reciprocal Diameter at 4 p 1 IO' (SI-cm) AQ a‘ “A e .2°K 0 indicates damaged die 7 1 1 1 1 L 1 1 IO 20 3O 40 50 60 7O 49 —'— (mm)- 5O ments the dies with which the wires had been formed were examined by the manufacturer. This examination indicated that several of the dies had been damaged. The wires which had been produced by dies which were found to be damaged are marked by an asterisk in Table I. The damage to the dies probably occurred sometime during the extrusion process wherein large pressures were exerted. unfortunately it is impossible to tell when this damage occurred and hence all data resulting from these wires is subject to some question.. The data in figure 19 in the l/d < 20mm."I range indi-, _cates a linear relationship with an intercept pb (bulk, I . resistivity) - .931: .01 x 10'9 ohm-cm and a slope of 1.63 i .07 x 10"1ohm-cm’. TheSe results are in fair agreement with pb - .75 x 10"“ ohm-cm and pbz - 2.1 x .10-“ ohm-cma (Olsen) and to ob - .70 x 10" ohm-cm and pbz - 1}4 x 10“3ohm-cm (Forsvoll and Holwech). The differences can probably again be attributed to differences' in purity and sample preparation. This contention is - supported by the fact that LaRoy using the same raw materialand the same preparation techniques obtained results which agree closely with those found here. The . lower bulk resistivity obtained by otherTworkers indicates that either the initial purity of their material was better. or that impurities and unannealed dislocations were introduced into the samples at the time of preparation.- . Measurements have not been made previously in the region . . A 9 . l/d > 20nn"'(d < .10mm). Our results suggest that the resistivity increases less rapidly in this region, a departure ."t“i 51 from the theory of Dingle and.Nordhemn. This observation. is, however, in disagreement with the model of Blatt and ‘ .Satz according to which there should be an increase in Ap for the finer wires. As will be seen below the effect de- . pends upon temperature since for T - 0 the p vs l/d graph is linear to the smaller sizes. . Values of pw(0,0) are tabulated in table II. The l/d plot for T - 0 (Figure 20) is of special interest since here . ' the Blatt and Satz phonon-surface resistivity pps’ should not be present, leaving the slope phi dependent upon the. _l'bulk material only. (For T > 0, t is affected by the pps term.) We note that I ' p , p (”bur-o ' (116:7) 7%; From previous information, Pf - 10“’gm~cm/sec and from the slope of the curve in figure 20, phi - 1.4 x 10"“ohm cm”. We obtain a value of 0.4 carriers per atom. The p'(o,o) vs 1/d plot also yields pb(0,0) - .50 x 10" ohm-cm from x. which the bulk mean free path may be calculated to be £(0,0) - .276mm.‘ ' Temperature Dependent Size Effect In order to examine the phonon-surface contribution to the total wire resistivity equation (10) can be rewritten in terms of the experimentally available values pps -rp'(o,T) + 'Pbi0,0) - P'(0.0) - Pb(0,'l‘).' ~ The accumulated probable experimental error here is Figure 20. Resistivity vs. Reciprocal Diameter at 0°K p x lo. (SI-cm) 0 indicates damaged die IO 20 3O 4O 5O 60 7O -1 .1. him) 52 d 39:. ~_. nzcaoaimclooo 3:52.: .3. L .nxu “EN. 3 ®.~ex 41 x“. x .Auo 5.3.103. fi fi _ _ _ u _o _u -~\u .u\« .38 can: No 89 54 25% due, in large part, to the two fold extrapolation required to obtain pb(0,0) and in lesser degree to the single extra-=- polation to obtain both pw(0,0) and pb(0,T). Figure 21 I presents pps (T - 4.2) vs. (33%. There is here no evidence of the linear correlation predicted by Blatt and Satz; however, the large scatter in the data and lack of any systematic relationship leads one to suspect thatTthed actual error is larger than predicted."' f ' Chapter V . Conclusion In this investigation the effects of transverse magnetic }° fields on the resistivity of thin indium.wires has been. studied. Previous work of this nature has produced informa- tion on wires larger than .06mm in diameter; This work extends these efforts to wires having a diameter of .0157mm. Maxima in the magnetoresistance as predicted by MacDonald and Sarginson were observed in several of the smaller wires. From the location of these maxima, a reasonable value for the momentum of electrons at the Fermi surface has been obtained. ‘ A number of size dependent departures from Kohler's Rule were observed. For high fields (”e/d < 1) the saturation" value of Ap/pw(0,T) was found to decrease with decreasing I wire size at 4.2'K as seen in figure 17. This observation , is in good agreement with the results of Olsen within his range of sizes. The saturation values of Ap/po have also been observed to be dependent upon temperature. For the larger wires the saturation value decreases as T is.lowered as shown in Figure 13. This was also observed by Olsen. How- ever for finer wires d < .05mm, the saturation value of Ap/bo (0, T) was found to increase as temperature was lowered as ' illustrated in Figure 16. For wires of diameter approximately equal to .05mm the effect was small or zero (Fig. 18). Turning to the low.magnetic field region, departures from Kohler's rule ' similar to those observed by LaRoy in one case, were observed . . . 65 Z , A - ' 56 to be generally true through the entire range of samples. This effect was that at a given temperature smaller wires. ' exhibited a larger value of ap/p(0,T) than did the large wires for the same value of HZp(0,T).. The size dependence; of the resistivity in zero magnetic field at T - 0 was found to be in good agreement with the theory of Dingle. The slope of pw(0,0) vs. l/d Ploti.pbl ' pig/Ne2 - 1.4 x 10'1Iohmécma. This result together with the value of Pf - 1.0 x 10"°gmscm/sec'gives “the value of n -(L4 for the number of conducting electrons‘ ‘ Per atom."fln. A temperature dependent size effect (departure from MathiessenS'Rule) was studied. The data shows a general qualitative trend in accord with the theory of Blatt and Satz. However, since this effect is observed only indirecté ly, the accumulated experimental error in this case is so large (20-50%) that no quantitative support for the theory can be claimed here. None of the above conclusions have been found to ’depend upon the ages of the samples, that is the length of time that the wire annealed at room temperature. This ' result may indicate that the method of preparation employed' produced large crystallites in the wire since other workers have observed such an age effect. It should be emphasized that many of these results de- pend upon a delicate series of extrapolations. .The'method of analysis employed in this thesis shOuld be verified by I 9 .I' til! ('1 'Ilull 4" I! in!- ... 57 f measurements on a metal which has a low superconducting .transition temperature, for example gallium. The method .could then be applied to metals with higher transition temperatures such as lead and tin. . In addition to the method of analysis the importance of sample preparation has also been indicated by the inconé sistent behavior of a number of samples. Some of this behavior may be the result of impurities introduced during the extrusion process and some by thermal strain produced during cooling in the cryostat. For these reasons it - would seem desirable to produce wire by using a nonferrous die_holder for the extrusion process or by casting the sample in a plastic form. The latter would allow pro- duction of single crystal samples by seeding. The thermal '. strain problem may be eliminated by mounting the samples . _on an indium substrate. Further information about the suspected role of. impurities and crystal imperfections-should be obtained by thermoelectric and thermal conductivity measurements. Extensions of the present work in the directions of single crystals and higher magnetic fields are obvious. Single crystal studies should lead to‘determination of ' Fermi momentum and the number of effective'current carriers' in particular directions of k space. High field measure--i ments would determine the extent of the maxima observed here and may reveal oscillations in the magnetoresistances; . . . (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) <20). <21) '(22) . E. H. Sondheimer, Phys. References J. J. Thomson, Proc. Camb. Phil. Soc. 113 120 (1901) L. Nordheim, Act. Sci. et Ind}, No. 131, 1934 (Hermann Paris) Rev. fig 401, (1950) D. K. C. MacDonald and K. Sarginson, Proc. Roy. Soc. (London) A203, 223 (1950) R. 6. Chambers, Proc. Roy. Soc. A202, 378 (1950) J. L. Olsen, Helv. Physica Acta 31, 713 (1958). a: . C. Laroy, Thesis for M.S. Michigan State University (1963) Unpublished , . H. Sondheimer, Adv. in Physics 1, 1 (1952) . R. Andrew, Proc. Phys. Soc. A62, (1949) E E K. Fuchs, Camb. Phil. Soc. 35, 100 (1938) R. B. Dingle, Proc. Roy. Soc. 5291, 545 (1950) F . J. Blatt and H. G. Satz, Helv. Phys. Acta 33, 1007 (1960) Luthi and P. Wyder, Helv. Physica Acta 33, 667 (1960) J. M. Ziman, Electrons and 2Phonons, Clerandon Press, Oxford (1966),490m E. H. Sondheimer and A. H. Wilson, Proc. Roy. Soc. A190, 435 (1947) K. Forsvoll and I. Holwech, Phil Mag 2, 435 (1964) K. Forsvoll and I. Holwech Phil Mag 19, 181 (1964) J. Babiskin and P. G. Siebenmann, Phys. Rev. 107, 1249 (1957) G. K. White and S. B. Woods, R.S.I. 2g, 638 (1957) E. J. Walker, R. S. I. -30, 834-(1959) 'J. F. Cochran and u. Yagub, phys'ics Letters 5, 307 (1963) .-5a . _ ' , 59 (23) R. T. Bate, Bryon Martin, and P. F.,Hille, Phys. Rev. 131, 1482 (1963) (24) J. A. Rayne, Phys. Rev. 129, 652 (25) P. Cotti and J. .L. Olsen, Cryogenics g, 45 (1964) ., M.‘ (‘35: f . Uii‘ ‘12. .‘2 D! :‘EAT'E‘ ’.‘,V;\/(-!-'Q,T‘" .‘ ,‘V -_ .-r '1', .,_ ' .110-.,‘l U} t 1"a'1‘1'Q's'11f I. Liv-QQHNQ' “Heilii'GAN 11111111111111'1‘11“ 1040027 "111111111 1111111111111 1”