SUPERCONDUCTIVITY IN THIN FILMS by RAM DAS CHAUDHARI B.Sc. University of Agra, 1949 M.Sc. University of Agra, 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1963 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia,, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 5 Canada The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RAM DAS CHAUDHARI B.Sc., Agra University, India, 1949 M.Sc, Agra University, India, 1953 MONDAY9 JANUARY 13, 1964, at 3:00 P.M. IN ROOM 302, HENNINGS BUILDING (Physics) COMMITTEE IN CHARGE Chairman: F.H. Soward R. Barrie J.B. Brown R.E. Burgess J. Grindlay J.A.H. Lund L. Young External Examiner: D.C. Baird Royal M i l i t a r y College SUPERCONDUCTIVITY IN THIN FILMS ABSTRACT The c r i t i c a l , currents and magnetic f i e l d s required to destroy superconductivity have been measured for thi n films of indium and t i n i n the thickness range of 585 A to 3600 A, The measurements were made i n the region close to the t r a n s i t i o n temperature, T c . o The c r i t i c a l current measurements on a 585 A thick indium f i l m are the f i r s t reported which com-bine the use of a compensated geometry avoiding the d i f f i c u l t i e s associated with specimen edges, and fast r i s i n g current pulses i n which the t r a n s i t i o n i s not obscured by specimen heating. The fast current pulses used had a r i s e time of 7 nanoseconds." It was found that the. temperature dependence of the c r i t i c a l currents i n the region near the t r a n s i t i o n temperature, 0 ^ ^AT<^ 0,15°K i s in. agreement with the Ginzburg-Landau theory. For a number of films the c r i t i c a l currents were measured using pulses having a r i s e time of 1.2 microseconds.. The c r i t i c a l currents have been found to vary l i n e a r 1 with the f i l m thickness, in agreement with the G-L theory. Measurements with fast pulses seem to indicate the existence of a t r a n s i t i o n delay of about 7 nano-seconds, independent of the current amplitudes. The t r a n s i t i o n from the. superconducting to the., normal state immediately following the t r a n s i t i o n delay appears to be very f a s t . For the c r i t i c a l f i e l d data, the temperature depen dence i s i n accordance with the G-L theory i n the range 0 ^ AT ^ 0. 3°K for both indium and t i n f i l m s , The e f f e c t i v e penetration depth calculated i n the manner of Itt n e r , and Douglass and Blumberg was found to be dependent on the thickness and the mean free path. The c r i t i c a l magnetic f i e l d s were found to be inversely proportional to the f i l m thickness i n agree ment with G-L theory. GRADUATE STUDIES E i e l d - o f Study: Physics Electromagnetic Theory Quantum Mechanics Physics of the S o l i d State Introduction to Low Temperature Physics Low Temperature Physics G.M. Volkoff W„ Opechowski R = E, Burgess J , B o Brown. J . B . Brown. Related Studies: Theory and Applications of D i f f e r e n t i a l Equations Applied E l e c t r o n i c s Computer Programming CA, Swans on M.P, Beddoes JoR.EL Dempster ABSTRACT This t h e s i s describes•experiments on d e s t r u c t i o n of superconductiv-i t y - -by e l e c t r i c currents and magnetic f i e l d s i n t h i n f i l m s of Indium and t i n , o of v a r y i n g t h i c k n e s s . The indium f i l m s i n the th i c k n e s s range of 585 A t o o 3540 A were deposited on sapphire rods a t l i q u i d n i t r o g e n temperature i n r- o o vacuum b e t t e r than 10TJ mm. of H g . Only two t i n f i l m s , 910 A and 3250 A i n thi c k n e s s deposited at room temperature on sapphire rods, were examined. A l l the measurements were made i n the temperature r e g i o n o A T ^ 0-^-°K where A T - T ^ - T • o The c r i t i c a l current measurements on a 585 A t h i c k indium f i l m are the f i r s t measurements ever made by combining the use of a compensated geometry and f a s t r i s i n g c u r r e n t pulses. The current pulses used had a r i s e time of 7 nano seconds. The measurements were made by using T e t r o n i x 661 sampling o s c i l l o s c o p e having a r i s e time of 0.35 nano seconds. The temperature dependence of the c r i t i c a l c urrents i n the r e g i o n A T ^ O I 5 k. i s i n complete agreement w i t h G - L theory. For a number of indium f i l m s , the c r i t i c a l currents were measured by using pulses having a r i s e time of 1.2 micro seconds. The c r i t i c a l c u r r ents have been found to vary l i n e a r l y w i t h the f i l m t h i c k n e s s , i n agreement w i t h G - L theory. Measurements w i t h f a s t pulses seem to I n d i c a t e the ex i s t e n c e of a t r a n s i t i o n delay of £ nano seconds, Independent of current amplitudes. The t r a n s i t i o n from the superconducting t o the normal s t a t e , immediately f o l l o w i n g the t r a n s i t i o n delay, appears to be extremely f a s t . A n a l y s i s of i i the thermal r i s e i n the r e s i s t a n c e a f t e r the t r a n s i t i o n , seems t o give a q u a l i t a t i v e evidence f o r the e x i s t e n c e of thermal propagation mechanism suggested by Bremer and Newhouse. For the c r i t i c a l f i e l d d ata, the temperature dependence i s i n accordance w i t h G - L theory i n the range 0 ^ A T ^ 0.3° K f o r both indium and t i n f i l m s . The e f f e c t i v e p e n e t r a t i o n depth c a l c u l a t e d i n the manner of T t t n e r , and Douglass and Blumberg was found to be dependent on t h i c k n e s s and mean f r e e path. The c r i t i c a l magnetic f i e l d s were found t o be i n v e r s l y p r o p o r t i o n a l t o the f i l m t h i c k n e s s i n agreement w i t h G - L theory. J . B. Brown V ACKNOWLEDGEMENTS I t gives me great pleasure to express my indebtedness to Dr. J.B. Brown, my supervisor, f o r his guidance, help, kindness and for arranging f i n a n c i a l support. I am al s o indebted t o Dr. D.V, Osborne, now at the Uni v e r s i t y of St, Andrews, Scotland, f o r h i s valuable help and kindness during Dr. Brown's absence. Besides, I would l i k e to express my apprecia-t i o n and thankfulness to the following: 1. The Government of Madhya Pradesh, India, f o r granting me leave of absence. 2. Mr. R.W. Weissbach f o r making l i q u i d helium a v a i l a b l e , helping i n constructing the apparatus and doing the drawings. 3. Mr. J, Lees f o r making a l l the glass equipment. 4. Workshop s t a f f f or making many pieces of apparatus. 5. Prof. R.E. Burgess and Dr. J.B. Warren for lending many pieces of t h e i r equipment, 6. Dr. B.L. White and Mrs. A.E. Aldridge for weighings with the micro-balance. 7. Mr. W.R. Irvine f o r his help i n taking p i c t u r e s with the ele c t r o n microscope. 8. Dr. G. Jones and Mr. John Turner f o r t h e i r advice on e l e c t r o n i c s problems. 9. Dr. P.W. Matthews, Mr, J.D. Jones, Mr. S.N. Sharma, Mr. A.R. Kshatriya and Mr, D.S. Sahri for t h e i r assistance i n taking down the data. Mr. J.R. Henderson for h i s help In programming the IBM 1620 computer. Mrs. R.E. B a r t l e y f o r typing the t h e s i s . TABLE OF CONTENTS Page ABSTRACT i LIST OF ILLUSTRATIONS i i i ACKNOWLEDGMENTS v CHAPTER I • I n t r o d u c t i o n and Phenomenological Theories 1 I I M i c r o s c o p i c Theories of Su p e r c o n d u c t i v i t y 15 H I S u p e r c o n d u c t i v i t y i n Thin Films - a Review 30 IV Specimen P r e p a r a t i o n and Mounting 46 V Experimental Set-up and Procedure 56 VI Experimental R e s u l t s and Their D i s c u s s i o n 68 VII Summary and Conclusions 86 BIBLIOGRAPHY 89 i i i . ILLUSTRATIONS Figure Facing Page Photograph showing the cryostat and the magnetic f i e l d coil,-compensating coils mounted on the portable aluminium frame 64 4.1 Vacuum system for the vaporisation chamber 46 4.2 Vaporisation chamber 47 4.3 a,b Specimen holder 52 4.4 a,b E l e c t r i c a l representation of specimen holder 53 5.1 Vacuum system for the cryostat 58 5.2 Circuit diagram of the thyratron pulser 61 5.3 Circuit diagram of the discharge line pulser 61 6.1 Graph showing variation of ^ as a function of d = 1 70 6.2 Plot of ( f d ) " 1 versus log d 70 6.3 Graph ofSTc versus d 72 6.4 (a,b sc) Electron micrographs of the indium and t i n films - 73 6.5 Transition curve for a t i n film 73 6.6 Plot of normalised 1-^ versus /ST 74 6.7 a,b Oscillographs of voltage pulses from the specimen 76 o 6.8 Plot of I versus AT for a 585 A thick film 76 P l o t of the Joule heat Q developed i n the specimen versus the r a t i o of the observed to the c a l c u l a t e d temperature r i s e P l o t of the c r i t i c a l c urrents versus t h i c k n e s s f o r v a r i o u s values of AT A p l o t of normalised H c versus A T f o r v a r i o u s f i l m s of indium A p l o t of H c versus d"-*-A p l o t of \ % versus y f o r the indium f i l m s A p l o t of ^€ versus y f o r the t i n f i l m s A p l o t of K c / ( l - t ) 1 / / 2 versus (1-t) f o r t i n f i l m s A p l o t of H / ( 1 - t ) 1 / 2 versus (1-t) f o r indium f i l m s CHAPTER I INTRODUCTION Superconductivity was discovered i n 1911 by Kamerlingh Onnes while he was investigating the variation of the ele c t r i c a l resistance of mercury at the liquid helium temperatures, available for the f i r s t time as a consequence of his successful liquefaction of helium gas. He found that the resistance of mercury dropped abruptly to zero at about 4° K. This phenomenon of the disappearance the e l e c t r i c a l resistance was called superconductivity. The i n f i n i t e conductivity i n the superconducting state was further demonstrated by Onnes and Tuyn (1924) by inducing 'persistent' currents i n a superconducting metallic ring when subjected to a changing magnetic f i e l d . Today a large number of metals, compounds and alloys are known to be superconductors. An exhaustive l i s t of these has been piven by Matthias et a l (1963). The temperature at which a metal becomes superconducting is known as the transition temperature, Tc,and i s characteristic of the metal. For pure, homogeneous and strain free bulk specimen, the transition i s usually quite sharp. Below T. the superconductivity can be destroyed by either applying a magnetic f i e l d or passing, an electric current through the superconductor, the value of the minimum- magnetic f i e l d or electric current being a 2 function of the temperature for any bulk superconductor. To a good approximation, the c r i t i c a l magnetic f i e l d , which i s the threshold f i e l d needed to destroy superconductivity i n a wire placed with i t s axis p a r a l l e l to the f i e l d i s given by, where H Q i s the c r i t i c a l f i e l d at the absolute zero and T i s the tempera-ture of the metal. S i m i l a r l y the minimum current needed to destroy the superconductivity i s known as the c r i t i c a l current. According to what i s known as Silsbee's hypothesis, the c r i t i c a l current i s that which produces a magnetic f i e l d equal to the c r i t i c a l f i e l d at the surface of the specimen. This, however, i s not true for a l l types of superconductors. For a number of years following Onnes' discovery, i t was believed that the superconducting state was f u l l y characterised by i n f i n i t e conductivity. However, t h i s i s not so as the following example w i l l show. From Maxwell's equations we know that „ -> dB V x E - - - ^ 1.2 where E and B are the e l e c t r i c and the magnetic f i e l d s . I f we make use of Ohm's law, E = z. where J i s the current densitv and cr i s the e l e c t r i c a l cr conductivity and put tr'aoo f o r a superconductor, we f i n d B = 0 or B i s constant i n time. However, the a p p l i c a t i o n of t h i s r e s u l t leads to the following paradox. Consider, f i r s t l y , cooling a sphere below T c and then applying a magnetic f i e l d to i t ; secondly applying the magnetic f i e l d f i r s t and then cooling the sphere below T , These two processes should lead to two d i f f e r e n t s i t u a t i o n s : i n the f i r s t case there should be no magnetic f i e l d i n s i d e the superconductor whereas i n the second case, a l l 3 the magnetic f i e l d should be frozen i n a f t e r the external magnetic f i e l d has been removed. Thus the magnetic behaviour of a superconductor should depend upon the past h i s t o r y of the superconductor, r e s u l t i n g i n an i n f i n i t y of states. Hence the state of the i n f i n i t e conductivity does not lead to a state of thermal equilibrium because the two external co-ordin-ates do not lead to a unique i n t e r n a l state. Meissner and Ochsenfeld (1933) discovered experimentally that the correct state of a superconductor i s not s p e c i f i e d by the condition that B = constant but that B = 0, This discovery i s known as Meissner - Ochsenfeld e f f e c t . Thus a superconductor i s not only a perfect conductor but also a perfect diamagnet. In f a c t , the diamagnetic aspect of superconductivity i s more basic than the existence of p e r s i s t e n t currents a r i s i n g due to i n f i n i t e conductivity because the former state i s a state of thermal equilibrium as opposed to l a t t e r which i s only metastable, I0NDONS1 THEORY F. London and H„ London (1935) proposed a phenomenological theory to account f o r the electromagnetic properties of a superconductor. E s s e n t i a l l y , i t was an attempt to blend the two c h a r a c t e r i s t i c s of a superconductor - the i n f i n i t e conductivity and the perfect diamagnetism, into a si n g l e theory. They assume that the supercurrent i s always determined by the l o c a l magnetic f i e l d . For a superconductor, they replace Ohm's law by the following two equations: CAAsd (A J ) ZZ. - H C ' 1.3 where A i s a constant c h a r a c t e r i s t i c of the superconductor, J g i s the supercurrent. I f these equations are applied to solve the problem of a 4 superconductor in a magnetic f i e l d . It turns out that the f i e l d penetrates only a thin layer of a bulk superconductor,, The penetration depth, X, in a bulk superconductor i s defined by the relation, > = H T . , ( „ H M X * where H (0) i s the magnetic f i e l d at the surface of the superconductor. According to the London picture, the magnetic f i e l d w i l l decay exponentially within a superconductor. In fact, the experiments generally give integrated flux in the interior of a superconductor and i t is d i f f i c u l t to deduce how the f i e l d changes in the interior. The application of the Londons' equations to the problem of the current flow in a bulk superconductor leads to the similar conclusion that the supercurrent is confined to a thin layer on the surface. The f i r s t direct experimental evidence of penetration effects was obtained by Shoenberg (1939). He measured the temperature variation of the magnetic susceptibility of mercury colloids of particle size between lCT^cm and 10""^ cm. The penetration depth was found to vary with the temperature i n accordance with the law, ^ _ n | ^ MI n IT 1/ -\ e Lock measured the temperature variation of ,\ by measuring the change in the ft - — magnetic susceptibility of thin films, A second method to study the temperature variation o f ^ i s due to Casimir who measured the change i n the mutual inductance of two coils containing closely f i t t i n g cylindrical specimens. This method was also successfully used by Desirant and Shoenberg in their measurement on t i n and mercury. Shalnikov and Sharvin used a variation of Casimir's method. Instead of using oscillating currents i n the primary, they oscillated the temperature. A third method 5 suggested and applied by Pippard i s based on the measurement of surface impedance of a superconductor at microwave frequencies. The penetration depth*X, besides being temperature dependent i s also found to vary with specimen thickness i n the case of t h i n superconductors. This v a r i a t i o n w i l l be discussed l a t e r . THERMODYNAMICS The superconductive t r a n s i t i o n , i n the absence of the magnetic f i e l d i s a t r a n s i t i o n of second order. Keesom and Kok were the f i r s t to show a jump i n the s p e c i f i c heat. In 1934, they also showed that there was no la t e n t heat i n the absence of a magnetic f i e l d . In the i n i t i a l stages the thermodynamics of the t r a n s i t i o n was developed by Keesom, Rutger and Gorter. Gorter's analysis i s based on the assumption that the t r a n s i t i o n i s a r e v e r s i b l e one. That t h i s i s so was vindicated by the discovery of Meissner e f f e c t . I f we assume the normal state to be non-magnetic and the super-conducting state to be diamagnetic and f u r t h e r , neglect the surface e f f e c t s and penetration e f f e c t s , then by the usual procedure of equating the Gibb's energies i n the two phases, we r e a d i l y a r r i v e at the following expressions. > - M T. . A C ~ C s - U v - ^ T " * L T > J V T H c dH<-Here C g, C n; S s, S n are the s p e c i f i c heats and entropies r e s p e c t i v e l y i n the superconductive and normal states and V i s the atomic volume of the 6 material. These equations have been experimentally v e r i f i e d . Gorter's two f l u i d model was the most successful one to account fo r the thermal properties of a superconductor. In the two f l u i d models, the superconductive state i s supposed to be formed as a r e s u l t of the condensation of the metal e l e c t r o n s , the degree of condensation, &>, being a v a r i a b l e parameter, | . The free energy may be w r i t t e n as • F I T , CO) = U o - ^ - i V T " K ( c O , T ) + F , ( T ) where the k e r n e l K (to, T ) ^ , 0* K (0, T) = 1, >^= constant, \j0 i s zero point free energy, "j i s Sommerfeld constant and (T) i s the l a t t i c e f r e e energy. In thermal equilibrium, CO adjusts i t s value so as to minimise the free energy. The various two f l u i d models may be obtained by assuming d i f f e r e n t expressions f o r the kernel K (to). Schafroth i960 has summarised the various models proposed. SURFACE ENERGY AND INTERMEDIATE STATE As remarked e a r l i e r , the above d e s c r i p t i o n does not take i n t o account the surface and the penetration e f f e c t s . The existence of a surface energy must be postulated for the following three reasons. ( i ) Consider an oblong superconductor to be placed i n a long-i t u d i n a l magnetic f i e l d H > H C and l e t the f i e l d be reduced below H c. I t can be shown that the free energy w i l l be lower for the case when the superconductor breaks into a domain structure of a l t e r n a t i n g superconducting and normal layers oriented such that the magnetic f i e l d passes through normal lay e r s . Thus i t w i l l not be necessary f o r the specimen to exclude the magnetic f l u x and thus there w i l l be no Meissner e f f e c t . This i s inconsistent not only with the experimental findings but the very concepts 7 of Gorter •= Casimir theory. However i f we assume the existence of a surface energy necessary f o r the creation of a superconducting •= normal boundary, t h i s w i l l i n h i b i t the breaking of the specimen into normal and superconducting regions and the f l u x w i l l have to be excluded, giving r i s e to Meissner e f f e c t . Obviously t h i s energy w i l l have to be p o s i t i v e , i t s / v X Hc magnitude can be shown to be such that the surface energy °^y\s. / ^ TT" ( i i ) Existence of a negative surface energy i s necessary for explaining the unusually high c r i t i c a l magnetic f i e l d s f o r the hard superconducting metals and a l l o y s . S i m i l a r l y even i n the s o f t super-conductors some regions having s t r a i n s or defects must be postulated to be a seat of negative surface energy which i s necessary f o r the growth of superconducting n u c l e i and t h e i r consequent growth throughout the body. Faber ( 1 9 5 2 ) has proposed a mechanism f o r the propagation of the super-conducting phase on the basis of the existence of a negative surface energy. The observed v e l o c i t i e s of propagation are found to be i n agreement with h i s model. Abrikosov ( L 9 5 7 ) and Goodman ( 1 9 6 2 ) have found i t necessary to assume existence of a negative surface energy to explain the magnetic behaviour of a l l o y s . ( i i i ) Consider a superconducting body of non-zero demagnetising c o e f f i c i e n t e.g. a sphere to be placed i n a homogeneous magnetic f i e l d . The sphere, i n order to keep the magnetic f i e l d away from the i n t e r i o r , makes the f i e l d inhomogeneous as a r e s u l t of which the sphere breaks up into a mixture of normal and superconducting domains as the magnetic f i e l d 2 i s increased from a value ^ H c to H c. In order that these domains be of f i n i t e s i z e , a p o s i t i v e surface energy at the i n t e r f a c e of the normal and superconducting boundary must be postulated. 8 The mixed state into which a superconductor of non-zero demagnetising coefficient breaks in a magnetic f i e l d close to H c, is known as the intermediate state. The f i r s t experimental evidence for the existence of the intermediate state was obtained by Shalnikov (1945) by mapping the magnetic f i e l d inside a sphere using a bismuth wire probe. The magnetostatics of the intermediate state has been discussed by Shoenberg (1952) and London (1950). THE PIPPARD THEORY Pippard (1953) suggested a modification of London equations by introducing the concept of coherence length"^. According to him, the current density i n a superconductor does not merely depend upon the vector potential at a given point but on the vector potential i n a region surrounding the point. He replaced London equation by the relation, r* _ _ 3 f g L R • A( 3 e * P C ~ .) P ( " > ' where R = r - r , 1 i s the mean free path of the electrons and ^ , i s a characteristic length dependent upon the metal and independent of impurity content. The coherence length^, is related to "^ eby the equation, where a i s a constant of the order of unity and 1 i s the mean free path, ^depends upon impurity content, for pure metals %*> % D - 10"^ cm. Pippard cited a number of properties i n favour of his concept of the coherence length. These are: (1) The sharpness of the phase transitions i n zero f i e l d . 9 (2) The variation of the penetration depth \ with mean free path. (3) The existence and the size of the surface energy. (4) Small change i n the penetration depth with magnetic f i e l d . can be interpreted as a measure of the size of the wave packets of the superconducting electrons. On the basis of the uncertainty principle, Faber and Pippard (1955) calculate ^ 0and find i t to be given by, So k - r t where a i s an adjustible parameter, v is the Fermi velocity and k is the EoTtemann constant. According to the Pippard theory, the penetration of the magnetic f i e l d inside a superconductor i s not exponential but has a much more complicated form and actually reverses sign at large depths. Sommerhalder (1961) has experimentally verified the direction reversal of the magnetic f i e l d . The Bardeen - Cooper - Shrieffer theory, (B C S) to be described in the following chapter, gives a relation between J and A similar to Pippard's. GINZBURG - LANDAU THEORY The phenomenological theory proposed by Ginzburg and Landau (1950) can be considered to be an extension of the general theory of the second order phase transitions given by Landau and Lifshitz (1958). The superconducting state i n G-L theory i s characterised by an order parameter y such that at T = T c, Y= 0; T < T c, ."Y>0 and I'rl*' = n s where n s is the density of superconducting electrons. ^ i s treated as a kind of wave function of the superconducting electrons. In the absence of a magnetic f i e l d and near T c, the free energy i n the superconducting state, F s o , is written as, 1.9 where F n o i s the free energy in the normal state and o< and p are temp-erature dependent coefficients. Near T c, c< and p are assumed to have the form, * ( T ) = ( T c - T ) ( T T " ) T - T C p a ) ^ pc-^ O - t Remembering that, Fso — F-n.0 — i t is easy to show that ^ 1.10 TT 1.11 This equation has been verified experimentally thereby lending justification for the above form for 0, Let —> the magnetic f i e l d H be direc t e d along the Z axis and the current J and vector p o t e n t i a l A along Y a x i s . I t i s c l e a r that Y= (X), J . n = 0, di v J = = 0 and ^ i s r e a l . Making use of the above equations we f i n d that the equations 1.13 a f t e r some work reduce to the following, 3 Now we make the following s u b s t i t u t i o n s , L - [ ' " ( T J ^ J 1.27 where "Xco i s the w e a k - f i e l d p e n e t r a t i o n depth at 0° K. 14 Wear T , Y SL ( A T ) *\j and i * T lit. Z Hi T U^vfe & T - " T c - T 2.1 where £(k) i s the e l e c t r o n i c energy — , 7 W , b e i n g the e f f e c t i v e mass of the electron,K\^S i s the phonon energy, s being the v e l o c i t y of sound £? —* and iP\j are the wave vectors of the e l e c t r o n and phonon re s p e c t i v e l y . The r e s u l t i n g state K i s given by K = K _ <5j/„ The lowest energy of the system i s obtained when the lowest states are f i l l e d , t h i s can be regarded i n momentum space as f i l l i n g of a Fermi sphere. When the Bloch - Sommerfeld theory outlined above f a i l e d to give even a q u a l i t a t i v e d e s c r i p t i o n of the superconducting state, i t was f e l t that the free e l e c t r o n model does not give correct i n t e r a c t i o n between the electrons. In view of the experimentally established f a c t that i n t r a n s i t i o n from normal to superconducting state the l a t t i c e does not undergo a change and also that the ions i n the l a t t i c e are much more massive than the electrons, i t was considered u n l i k e l y that the l a t t i c e would play an important r o l e i n the establishment of the superconductive state. However, i n such a s i t u a t i o n F r b h l i c h (1950 ) introduced a new idea that the e l e c t r o n phonon i n t e r a c t i o n was responsible for superconduct-i v i t y . F r b h l i c h et a l (1950 ' ) were the f i r s t to introduce f i e l d theory in t o s o l i d state physics. By making use of the f i e I d - t h e o r e t i c approach, F r o h l i c h was able to show that the e l e c t r o n phonon i n t e r a c t i o n contributes a small term to the energy of the system. His Hamiltonian for a system of N free electrons contained i n a volume V, which i n t e r a c t with phonons, treated i n Debye approximation can be written as, H = H 0 + H ' Z - Z where rJ l ^ " 7. •? H p . 2: i i - - ^ ^ Here He r e f e r s to the electrons, Hp^ to the phonons and H to the i n t e r -a c t i o n between electrons and phonons and = 0, 1, 2 ... i s the occupation number of a phonon having wave number q, the corresponding frequency = (q|S. The i n t e r a c t i o n H i s assumed to be, w h e r e i s the p o s i t i o n operator of the vtk. electron, g i s a coupling constant and bq and b^ are absorption and emission operators f o r a phonon, obeying the commutation r e l a t i o n s , Treating the Hamiltonian by second order perturbation theory leads to two consequences. ( i ) E l e c t r o n S e l f Energy The s h i f t i n the energy eigenstate of He which i s analogous to the appearance of the s e l f energy of the electron i n quantum f i e l d theory. This s h i f t Is taken care of by renormalisation procedure and Ho i s con-sidered to describe a 'clothed' e l e c t r o n which i s accompanied by a phonon cloud. ( i i ) E l e c t r o n - E l e c t r o n I n t e r a c t i o n The i n t e r a c t i o n energy £• ( K ^ between two electrons i n states KT and K 0 i s found to be . \\* u ^ a ^ t a D ^ ^ . . . . . v « rr-1 — 2 . 8 19 where = 2 ° 9 This energy i s found to represent a true dynamical effect and was regarded by Frohlich to be the cause of superconductivity,, Considering that 0"Q is small compared to KQ, the wave number at the Fermi surface, this inter-action can be considered to be a short range attraction i n K space,, Frohlich's theory predicted isotope effect, T c XfM = constant where M is the isotopic mass of the ion. In fact the isotope effect was experimentally discovered by Maxwell (1950) and Reynolds et a l (1950). The mathematical technique used by Frbhlich is questionable and many of the predictions made by the theory are found to be wrong. The importance of the theory however l i e s i n the isolation of the right interaction. The Frohlich Hamiltonian, however seems to be a lasting contribution to the theories of super-conductivity as i t forms the starting point of the Bogoliubov theory. The mathematical problem of handling the electron-phonon interaction proved to be quite d i f f i c u l t . If one bears i n mind the great qualitative difference i n the two phases, i t is not surprising that the perturbation expansion from the normal state was found to be inadequate. In fact the development of the f i r s t successful microscopic theory, namely B C S theory, had to await two main developments? one was the treatment of electron phonon interaction by Bardeen and Pines (1955) which was based on the collective mode description formulated by Bohm and Pines (1953) and the other was the role of pair correlations giving rise to attractive interaction-between the electrons, given by Cooper (1956). The collective mode treatment of the electron interaction is based on the model of the elementary excitations in the solids. There are three principal excitations in the solids, phonons, plasmons and quasi-particles. An excitation is considered to be well-defined i f i t s l i f e time X* is large enough to make 20 the uncertainty i n the energy small i n comparison with the e x c i t a t i o n energy. A l l the ex c i t a t i o n s are designated by an appropriate wave vector K. The phonon energies do not change much by the t r a n s i t i o n s and can be taken care of by renormalisation procedure. The plasmons, the quanta of the plasma o s c i l l a t i o n s of the electron gas, have too high an energy to be excited at low temperatures and thus the q u a s i - p a r t i c l e s are important f o r the problem of superconductivity. In the normal state at T = 0° K, a l l states below Fermi l e v e l are occupied and a l l above are empty. An e l e c t r o n may be excited above Fermi l e v e l thereby leaving a hole below the Fermi l e v e l . The excited p a r t i c l e s above and the holesbelow the Fermi surface are to be regarded as the elementary q u a s i - p a r t i c l e e x c i t a t i o n s , t h e i r energies c ( K ) being measured r e l a t i v e to the Fermi energy Ep. In the normal state 6- (K) i s a continuous function of K, vanishing at the Fermi surface. The various excited configurations can be described i n terms of occupation numbers i n K space. In a superconductor, the e x c i t a t i o n spectrum d i f f e r s i n that a f i n i t e energy equal to the energy gap i s required to excite a p a r t i c l e from the superconducting ground state. In a super-conductor the energy of a q u a s i - p a r t i c l e may be written as gap parameter. A q u a s i - p a r t i c l e i s to be treated as being 'clothed' by in t e r a c t i o n s with phonons, plasmons and other e x c i t a t i o n s . While discussing the dynamics of q u a s i - p a r t i c l e s , i t i s e s s e n t i a l to include the screening a c t i o n and the backflow of electrons, former a r i s i n g out of high m o b i l i t y of the electrons and l a t t e r being necessary i n order that q u a s i - p a r t i c l e s obey 2.10 where £ (K) i s the Bloch energy i n the normal state and Apr i s the energy 21 the equation of continuity. The a p p l i c a t i o n of f i e l d theory to the i n t e r a c t i o n between electrons and phonons admits of the p o s s i b i l i t y of the emission and absorption of v i r t u a l phonons analogous to the exchange of v i r t u a l photons giving r i s e to Coulomb p o t e n t i a l . In f a c t , i n view of the small phase space available for the e x c i t a t i o n s , a consequence of the extreme smallness of condensation energy, the e f f e c t i v e i n t e r a c t i o n between the electrons, giving r i s e to superconductivity may be pictured as follows. A p a r t i c l e near the Fermi surface i n a state K-j_ emits a v i r t u a l phonon of wave vector q and i s ->/ - _ scattered to a state K^ = - q, the v i r t u a l emission being possible because of the uncertainty r e l a t i o n AE. A"b . ^ s econd e l e c t r o n i n the state Kg absorbs the phonon, thereby going to a state Kg — Kg + q. The net -> -» t / e f f e c t i s to s c a t t e r electrons from t h e i r o r i g i n a l states K^, Kg to KT_, Kg with the.conservation of the wave vector, —*> -» —> KT_ + Kg = KT_ + K g . 2,11 This I n t e r a c t i o n between the p a i r of the p a r t i c l e s comes out to be a t t r a c t i v e i f the energy difference between the i n i t i a l and the f i n a l states i s l e s s than the energy of the v i r t u a l phonons, "t-v * Superconductivity r e s u l t s i f t h i s a t t r a c t i v e i n t e r a c t i o n dominates the repulsive screened Coulomb i n t e r a c t i o n . Cooper (1956) showed that even a very small net a t t r a c t i v e i n t e r -action was s u f f i c i e n t that two q u a s i - p a r t i c l e s may form a bound state. He also showed that f o r a t t r a c t i v e i n t e r a c t i o n s , the Fermi sea i s unstable against the formation of such bound p a i r s . This bound state i s depressed i n energy below the normal continuum of l e v e l s , thereby leading to a. gap i n the energy spectrum. Existence of such a gap was conceived e a r l i e r 22 by many workers to account f o r various superconductive properties. Thus Cooper's discovery supplied the f i n a l missing l i n k f o r the successful s o l u t i o n of the problem. The existence of the Cooper's p a i r s has been vindicated by the experiments on f l u x quantisation In the superconductors„ The p h y s i c a l consequence of the p a i r i n g i s to produce a long range corre-l a t i o n between p a r t i c l e s of opposite spin extending over distances of the order 10"^ cm, i n agreement with Pippard's concept of the coherence length. The coherence e f f e c t s associated with the paired wave functions introduce a di f f e r e n c e between the sc a t t e r i n g i n the normal and the super--* —* —t conductive states. In the normal state, the sca t t e r i n g from K, 0^ , where o~ -v -v i s the spin wave vector, to K ,cr i s e n t i r e l y independent of the scattering from -K, - 2. 2.15 A detailed discussion by Pines (1958) shows that this criterion is i n reasonable agreement with the one empirically established by Matthias (1957) for the appearance of superconductors in the periodic system. The next step is to introduce pair correlations by including only those configurations i n which the states are occupied in pairs, the pairs having vanishing total momentum and spins. Introduction of the pair correlations at this stage amounts to the introduction of selected sub-sets in.the matrix elements so as to give a coherent low state. A pair is designated by a wave vector K independent of the spin. To treat the pair c o r r e l a t i o n s new creation and a n n i h i l a t i o n operators are introduced as defined belows 2.16 These operators s a t i s f y the commutation r e l a t i o n s . In terms of these the Hamiltonian 2.12 may be written as H ^ - l ^ _ K. K K k _ < k p k L , 2.18 Since the i n t e r a c t i o n term i s negative i n sign, VK K1 ^e p o s i t i v e f o r a superconductor. I f to i s the average phonon frequency, the important matrix elements l i e i n the region, Hence % / i s replaced by, V - 0 2.20 , otherwise, assuming that i s i s o t r o p i c and constant over a t h i n s h e l l . Thus the Hamiltonian 2,18 becomes / where ^ extends only i n the region defined by 2 . i f . B C S employ a s e l f consistent method to obtain wave function f o r the Hamiltonian 2.18 and the best of these wave functions i s chosen 25 by a v a r i a t i o n p r i n c i p l e , A wave function f o r N independent pa i r s i s written as ^ , f t n [ jo^Q + ^ t t J t 2 . 2 2 where (J3^ i s the p r o j e c t i o n operator s e l e c t i n g states having exactly W p a i r s , ^5 i s the vacuum operator and the quantities k«_? O ^ Uvc ^ \ 2.23 are the parameters to be so determined as to minimise the expectation value of the energy W(o) given by, V)(o) ( T - 0 > r W 2,24 ¥(o) i s found to be, f J A . The f i r s t term i s the difference i n the k i n e t i c energy of the two phases, the f a c t o r 2 i s due to p a i r i n g . The f i r s t term can be p o s i t i v e or negative. The second term gives the c o r r e l a t i o n energy for t r a n s i t i o n from the state (K, - K) to (K- - K ) . The parameter kv^can be interpreted to be — * _ » the p r o b a b i l i t y that the state (K, - K) i s occupied. Minimising W(o) with respect to k * , r e s u l t s ; 2.26 2.28 Etc - t f ^ H r ^ C O j and t v c - fc 2.29 Sub s t i t u t i n g Q.26 into 2,2% we obtain, 2.30 2 6 I f we replace the summation by i n t e g r a t i o n and remembering that f or the states within the range \d nl ? V = 0 , we have, z (e^-v- 2 . 3 1 where N(o) » density of states at the Fermi surface. From 2 . 3 1 we have, ^° = - ^ T C L l / W o ) 7 3 2 . 3 2 Making use of 2 . 2 9 j , 2 . 2 8 , 2 . 2 4 and 2 . 3 2 , the expression for ¥(o) the energy i n the ground state i s found to be, \AJ(°) — ——vTTuTT 2.33 B C S also calculate the magnitude of the energy gap which i s equal to the energy to break up a p a i r . At 0°K, t h i s i s found to be X € 0 As N(' 0 y v " ^ 3 - 4 , from equation 2 . 2 9 , B C S extended the above treatment to expressions f o r energy gap at temperatures above 0 ° K and also derived an expression f o r T C . For temperature T, 0 ^ ^ T c^ ~ ° Equations 2.41 and 2.34, yield 3'5Zke>"Tc 2 . 4 2 The B C S theory successfully establishes the following: 1. A criterion for the appearance of superconductivity. Pines (1958) has discussed the appearance of superconductivity i n the periodic system and finds that the theory qualitatively explains the va r i -ation of T c from one metal to another. 2 . Existence of an energy gap. Biondi et al. (1958) give a review of experimental work done for the measurement of the energy gap. The temperature dependence of the energy gap predicted by the theory has also been verified. 3. Isotope effect, "Although i t may be noted that Geballe et a l (1961) found that ruthenium does not exhibit any isotope effect. 28 4. Coherence properties,, The measurement of ultrasonic attenuation on single crystals by many workers e.g. Morse et a l (1959) are in good agreement with the theory. The theory also accounts satisfactorily for; (1) The temperature variation of the specific heat in the supercon-ducting state, (2) The second order phase transition. B C S theory does not yet provide an acceptable explanation of the. Meissner effect and persistence of currents. The theory i s also unable to account for the Knight shift. Bardeen (1962a). Another f a i l i n g of the theory is the discovery of the existence of superconductors amongst the ferromagnetic materials. (Bardeen (1962a.) THE BOGOLIUBOV THEORY Bogoliubov (1958) developed a new approach for treating super-conductivity. As a starting point he assumes the Frohlich Hamiltonian for an electron gas interacting with phonons. In view of the fact that only the electrons i n a thin shell near the Fermi surface are important, one can think of a description of the electronic state in terms of excitations over an appropriate ground state. Out of many ways of defining such excitations and the ground state, Bogoliubov considers the ones involving the pairs of electrons with opposite spins and momenta, with a coefficient which is undetermined to start with. The coefficients must be such as not to produce non-integrable divergencies in the perturbation theory. This condition determines the coefficients. For an attractive interaction one finds a new ground state lower than the Fermi state, Bogoliubov in t e r p r e t s t h i s to be the superconducting ground state, Bogoliubov theory i s supposed to be based on be t t e r mathematical foundations than B C S theory, Yosida (1958) has demonstrated the equi-valence of the two methods i n so f a r as the energy spectrum at the absolute zero i s concerned. The Bogoliubov theory s u c c e s s f u l l y explains a l l the superconducting properties explained by B C S theory. The methods developed by B C S and Bogoliubov are f i n d i n g increasing use i n the theory of elementary p a r t i c l e s . Wambu and Jona - Lasimo (1961) have constructed a dynamical model of the elementary p a r t i c l e s analogous to superconductivity model. They suggest that the nucleon mass arises l a r g e l y through the same mechanism as the appearance of the energy gap i n the theory of superconductivity. The microscopic theories of superconductivity, besides securing a be t t e r p h y s i c a l understanding of the phenomenon, have given r i s e to a great spurt i n the experimental work. CHAPTER III SUPERCONDUCTIVITY IN THIN FILMS ~ A REVIEW There are a large number of factors which govern the structure and the properties of evaporated films. Some of these are; vacuum conditions, rate of deposition, presence of the residual gases, the nature and the geometry of the substrate, stresses in the film and angle of the incidence of the vapour beam. In order that the films with reproducible characteristics may be obtained, a good vacuum is essential. Vacuums of the order of 1 0 " ^ mm of Hg' or better are desirable, A good vacuum has two important effects; f i r s t l y i t prevents the oxidation of the film and secondly i t reduces the amount of absorbed gases in the films, thereby ensuring a smoother structure. The above are also helped by high rates of deposition. Levinstein (1949) found that the tendency for agglomeration formation decreased as the rate of deposition increased. Presumably, with higher rates, more nuclei are formed i n i t i a l l y and these act as nucleation centres to ensure a fine grain size resulting i n a smooth structure. Behrndt et a l (I960) found that the films deposited at the liquid nitrogen temperature possessed smoother surface than the ones deposited at the room temperature. In practice, the rates of deposition are limited by the amount of power the heater can handle and also by the consideration of the homogeneity of the film. Kahan et a l (I960) found that the high rate of deposition results i n the 31 f i l m having low r e s i d u a l r e s i s t i v i t i e s at low temperatures but the grains are found to be coarse, the average grain diameter varying between 5 and 25 microns. On the other hand the f i l m s of higher r e s i d u a l r e s i s t i v i t y , deposited with the substrate at room temperature and lower deposition rate showed f i n e grain structure, almost near the l i m i t of o p t i c a l r e s o l u t i o n . This seems to be i n disagreement with the findings of L e v i n s t e i n stated above. Perhaps i n Kalian's f i l m s the higher evaporation rates r e s u l t e d i n the temperature r i s e of the substrate, thereby making the n u c l e i more mobile which resulted i n the formation of bigger grain s i z e . Presence of gases excercises considerable influence on the structure and properties of t h i n f i l m s . Of a l l the gases present, oxygen i s most undesirable, i t s presence r e s u l t s i n the oxidation of the f i l m . This can be minimised by having good vacuum and employing some getters to get r i d of oxygen. Some metals notably tantalum and tungsten have a great a f f i n i t y f o r oxygen r e s u l t i n g i n the formation of v o l a t i l e oxides which could be c o l l e c t e d on a s h i e l d before commencing deposit on the substrate. High deposition rates also are h e l p f u l i n diminishing the oxidation of the f i l m because of the reduced t r a n s i t time and the condensation time - the periods when the atoms seem to be i n an active state of being oxidised. Water vapour, another common impurity i n the vacuum system, can be s u b s t a n t i a l l y removed by a suitable l i q u i d nitrogen t r a p . The nature of the substrate i s an important f a c t o r governing the f i l m q u a l i t y . The substrate must be cleaned thoroughly. Various methods have been used f o r cleaning, ranging from use of detergents, acids, followed by degreasing baths to u l t r a - s o n i c cleaning. In order to free the sub-strate from the occluded gases, the substrates have been baked i n vacuum. 32 The thermal conductivity of the substrate and the coefficient of expansion relative to the condensate are important factors governing the grain size and the stresses i n the film. The crystalline state of the substrate i s a factor which affects the directional growth of the nuclei. Evans and Wilman (1952), on the basis of the electron diffraction studies show that the amorphous or inactive crystalline substrates have tendency to cause the films to grow in the preferred directions. The temperature of the sub-strate, which is an important factor governing the mobility of the con-densate nuclei, may influence this process. Rhodin (1949) found with different substrate materials that there was a characteristic base temper-ature leading to maximum orientation. On this basis, he concludes that the condensed atoms must possess a certain minimum kinetic energy to enable them to move into preferred positions. In the single crystal substrates, the deposit crystals are often found oriented with respect to the substrates. This phenomenon i s known 'epitaxy'. No satisfactory theory of epitaxy exists, the model which has received more support is that of Franck and Van der Merwe (1949). According to them, the f i r s t step i n the growth of the oriented layer is the formation of a monolayer of the deposit which has the same spacing as the substrate. This monolayer is in a state of constraint and i f the constraint exceeds a certain c r i t i c a l value, disloca-tion w i l l occur which w i l l hinder the growth of the nuclei i n the i n i t i a l orientation. The creation of a dislocation needs some activation energy, hence the deposition at the low temperatures w i l l tend to reduce and prevent the formation of dislocation. However, the deposition at very low tempera-tures, such as liquid helium temperatures, may lead to many faults or defects being frozen in. The condensation i n such situations corresponds to quenching. ELECTRICAL CONDUCTION IN THIN FILMS The e l e c t r i c a l r e s i s t i v i t y £ of the metals can be written as; i ? ^ ^ * t o t a l - l a t t i c e + ' r e s i d u a l + ' thickness The t h i r d component a r i s e s from the boundary s c a t t e r i n g of the electrons, the f i r s t two are the same as f o r the bulk metal. The r e s i s t i v i t y of t h i n films depends upon thickness. For a f i r s t few angstrom t h i c k l a y e r s , r e s i s t i v i t y may be i n f i n i t e . For a d e t a i l e d examination of the dependence of r e s i s t i v i t y on f i l m thickness, the f i l m may be divided into four domains i n thickness: (1) When f i l m thickness i s too small f o r the conduction to begin. The thickness of t h i s domain w i l l depend upon a large number of factors such as the deposit material, substrate and conditions of conden-sation, (2) The conduction just sets i n , the e l e c t r i c a l r e s i s t i v i t y i s s t i l l very high, o (3) Extending from about 200 A up to a c e r t a i n value t i l l the r e s i s t -i v i t y continues to decrease with thickness, (4) The r e s i s t i v i t y becomes independent of thickness or decreases very slowly with thickness. The theory of the mean free path e f f e c t s (see Sondheimer 1952 f o r a review) i s applicable to the f i l m thickness i n the t h i r d and the fourth domains. The r e s i s t i v i t y of the f i l m i s given by; - ~ - \+ ^ ~ ~ T 3.2 where d i s the f i l m thickness, 1 i s the mean free path and "j» i s the bulk r e s i s t i v i t y . 3 4 SUPERCONDUCTING TRANSITION WITH ZERO FIELD AND CURRENT Unlike the transition i n single crystal and strain-free super-conductors, the superconducting transition in thin films is rather broad 0 The transition temperature when measured by resistance measurements i s defined to be the temperature at which the resistance becomes half of i t s value in the normal state. Similarly i f the transition is observed by measuring the magnetic induction, T c i s defined to be the temperature when the induction has half the value in the normal state. Likewise for sub-critical temperatures, the c r i t i c a l currents and fields are defined to be the ones required to restore the resistance to half the values i n the normal state. De Haas and Voogd (1931) showed that the resistance of a single crystal of t i n vanishes discontinuously at T c„ At T c, the transition i s of the second order. The discontinuous drop i n the resistance to zero value i s cited by Pippard as an evidence for the existence of a coherence length i n the superconductors. The presence of physical or chemical impurity is known to lead to broader transitions. Aziz and Baird (1959) investigated the effect of grain size on the width of the transitions. They find, i n agreement with the original findings of de Haas and Voogd, that the existence of grain boundaries broadens the transition i n the direction of increasing temperature. Their experiments with samples of differing grain size showed that smaller grain size leads to wider transitions and that any crystal mis-orientation w i l l also result in wider transitions. According to Faber (1957) any misoriented inclusion imbedded in a single crystal w i l l lead to the estab-lishment of s t r a i n s r e s u l t i n g i n the appearance of c e r t a i n p a r ts of the c r y s t a l having T c greater than t h a t f o r an i d e a l c r y s t a l . Thin f i l m s are f a r from a t h i n s l i c e of an i d e a l superconductor, which they are supposed to be f o r most t h e o r e t i c a l treatments. The t r a n s i t i o n s i n f i l m s are broad and t h i s i s a consequence of s t r a i n s , inhomogeneity, edges and weak spots associated with the f i l m s . The t r a n s -i t i o n temperature f o r f i l m s i s o f t e n d i f f e r e n t from i t s value i n b u l k . Lock (1951) was f i r s t to p o i n t out the connection between the s h i f t i n T c and the s t r a i n s i n the f i l m r e s u l t i n g from the d i f f e r e n t i a l c o n t r a c t i o n between the f i l m and the s u b s t r a t e . He could show exp e r i m e n t a l l y t h a t the t r a n s i t i o n temperature f o r the f i l m s could be above or below the bulk value depending upon whether the substrate contracted l e s s or more than the f i l m . By matching the thermal expansions, i t was p o s s i b l e to make f i l m s having t h e i r t r a n s i t i o n temperature the same as that i n the b u l k . Toxen (1961) and Jennings and Swenson (1958) discussed q u a n t i t a t i v e l y the e f f e c t of s t r e s s e s on T c and found t h a t the change i n T c can be represented by the equation, where d i s f i l m t h i c k n e s s of t h e i r f i l m s . The b u l k of the work done on t h i n f i l m s has been done on f i l m s deposited on plane surfaces. Presence of the edges i s an important cause of broad t r a n s i t i o n s , Kahan et a l (I960) f i n d t h a t the t r a n s i t i o n s can be made much sharper by removing the edges. Annealing and etching a l s o improves the sharpness of the t r a n s i t i o n s , " De Lano, J r . (I960) a l s o d i s -cussed the edge e f f e c t s and t h e i r removal by annealing, e t c h i n g and trimming. f T CURRENT TRANSITIONS I N FILMS 36 The c r i t i c a l c u r r e n t s i n t h e b u l k s p e c i m e n , a r e g o v e r n e d b y S i l s b e e ' s h y p o t h e s i s . I n t h i n f i l m s t h e s i t u a t i o n i s d i f f e r e n t . H e r e , e s p e c i a l l y i n t h e n e i g h b o u r h o o d o f T c , t h e m a g n e t i c f i e l d p e n e t r a t e s t h e w h o l e t h i c k n e s s . The f o l l o w i n g c o n s i d e r a t i o n b a s e d on L o n d o n t h e o r y shows t h a t I c s h o u l d v a r y a s ( T c - T ) ' / ^ . I n c a s e o f c u r r e n t f l o w o f c u r r e n t d e n s i t y J g , t h e i n c r e a s e i n t h e f r e e e n e r g y i s g i v e n b y , . .. AF - ^M1") j / 3.4 where A ( T ) i s assumed t o b e i n d e p e n d e n t o f J _ , w h i c h may n o t be v a l i d f o r l a r g e c u r r e n t s . To o b t a i n t h e c r i t i c a l c u r r e n t we e q u a t e t h e i n c r e a s e i n t h e f r e e e n e r g y t o t h e d i f f e r e n c e o f e n e r g y b e t w e e n n o r m a l a n d s u p e r c o n -d u c t i n g p h a s e s w h i c h i s ^ / Thus we h a v e , 1 / 2 A (T) J s c 2 = Es! 3.5 8 TT N e a r T c , b o t h H c a n d J/V ( T ) ] " 1 v a r y as ( T c - T ) , l e a v i n g J g c v a r y i n g as ( T c - ' T ) 3 / * . ' R o g e r s (I960) c a l c u l a t e s t h e c r i t i c a l c u r r e n t s on t h e b a s i s o f m i c r o s c o p i c t h e o r y b y t a k i n g I n t o a c c o u n t t h e change i n t h e d i s t r i b u t i o n o f q u a s i - p a r t i c l e s a n d e n e r g y gap w i t h c u r r e n t . H i s v a l u e s a r e f o u n d t o b e 25% l o w e r t h a n t h e ones c a l c u l a t e d o n s e m i - p h e n o m i n o l o g i c a l t h e o r i e s . H o w e v e r , t h e t e m p e r a t u r e d e p e n d e n c e i s f o u n d t o r e m a i n u n c h a n g e d . E x p e r i m e n t a l l y t h e s i t u a t i o n i s c o m p l i c a t e d b y t h e f a c t t h a t t h e a p p e a r a n c e o f t h e f i r s t t r a c e o f r e s i s t a n c e r e s u l t s i n t h e p r o d u c t i o n o f J o u l e h e a t i n g w h i c h r a i s e s t h e t e m p e r a t u r e o f t h e f i l m , t h e r e b y l o w e r i n g t h e a m b i e n t c r i t i c a l c u r r e n t . A s s t a t e d e a r l i e r , i n t h e f i l m s , t h e 37 t r a n s i t i o n i n the absence of Joule heating, i s broad, occupying several m i l l i - d e g r e e s , a consequence of inhomogeneity and presence of weak spots i n the f i l m . Quite misleading r e s u l t s may be obtained i f proper precautions are not taken to eliminate or account f o r the Joule heating. I t i s w e l l known that the c r i t i c a l currents are found to be considerably lower when continuous currents are used instead of pulses. Most of the work has been done on f l a t f i l m s . The presence of sloping edges may be another f a c t o r f o r lower c r i t i c a l currents as superconductivity at t h i n edges would be destroyed f i r s t r e s u l t i n g i n the heating of the f i l m before the currents c r i t i c a l t o the bulk p o r t i o n of the f i l m have been reached. Heating of the f i l m w i l l depend upon the current being passed through the f i l m , the r e s i s t -ance of the f i l m and also the thermal c o n d u c t i v i t y and capacity of the substrate. These f a c t o r s w i l l be important f o r the thermal h y s t e r i s i s observed i n the f i l m s . Wo s a t i s f a c t o r y model to account f o r heating e f f e c t s i n t h i n f i l m s e x i s t s . Bremer and Wewhouse (1958) were the f i r s t to show experimentally the thermal propagation e f f e c t s i n the f i l m s . In t h e i r experiments, they passed a maintaining current I-^ through the f i l m , 1^ being l e s s than the c r i t i c a l current„ A current pulse known as the nucleating current was passed through a wire held very close to the f i l m and perpendicular to i t . The magnetic f i e l d associated with the pulse together with that created by Ijyj was s u f f i c i e n t to restore resistance j u s t below the nucleating wirej t h i s r e s u l t e d i n the production of joule heating due to current L^. The normal region was found t o propagate along the f i l m r e s u l t i n g i n the r e s t o r a t i o n of normal state throughout the f i l m . The v e l o c i t i e s of prop-agation of these thermal wave fr o n t s were found to be low, of the order of 20 cm per second. Evidence f o r the existence of t h i s mechanism of thermal 38 propagation has been found by Broom and Rhoderick (1959, I960) Cherry and Gittleman (I960) and Kolchin et al. '1961), Broom and Rhoderick find that the velocities of thermal propagation i n their experiment were many times larger than observed by Bremer and Newhouse. This difference they attribute to larger currents and thinner films used i n their experiment. Indeed these authors (I960) also report the successful construction of a new type of bistable element capable of being switched from one state to another i n an interval of 10 nyis. The bistable element works on the principle that at a given temperature below T for any film there exists a unique current c which maintains the interphase boundary stationary. If the current is increased, the normal region spreads and i f i t is decreased, the supercon-ducting region grows. Thus a current pulse in either direction riding over the biasing current w i l l switch the element into the superconducting or the normal state. This device can be taken to be an evidence for the existence of the movement of the interphase boundary due to the thermal effects. Cherry and Gittleman also found an evidence for thermal propagation of the interphase boundary on their experiment on a t i n film of 1 micron thickness and 0.5 mm.width. They postulate an early formation of a normal region which could have been brought about due to a variety of causes e.g. local heating, local magnetic f i e l d , any metallurgical inhomogeneity, or by eddy current heating during the rise time of a fast current pulse (Pippard process). The velocities of the thermal propagation observed by them ranged from 10 to 10,000 cm, per second depending upon the current magnitudes and the bath temperature. Kolchin et al. also assume two different mechanisms for the thermal propagation,for slow rates of rise Joule heating i s predominant, for a fast rate, eddy current heating becomes quite 39 e f f e c t i v e . The f i r s t attempt to measure the c r i t i c a l currents i n evaporated t h i n f i l m s was made by Shalnikov (1940) by using continuous currents. The observed values were found to be considerably lower than those expected on the basis of the Silsbee's hypothesis, Alekseevski-r and Mikheeva (1957) used a compensated geometry to avoid any edge e f f e c t s . Their f i l m s were deposited on a substrate i n the form of a d i s c , the current being introduced through a lead perpendicular to the plane of the d i s c and taken out from the periphery. The fil m s were prepared by sputtering t i n on a substrate held at the l i q u i d nitrogen temperature. The films.were kept under vacuum u n t i l they were immersed i n l i q u i d helium. The c r i t i c a l currents were measured by using pulses of 0,1 sec duration and the temperature region over which the measurements were made was 0.5° below T c. The H c j was found to be proportional t o l ^ X ) where n = 0.6. In t h e i r l a t e r measurements (i960) these authors i n v e s t i -gated the e f f e c t of r i s e time of the current pulse on the value of the c r i t i c a l current and found that the c r i t i c a l current continued to increase with the decrease i n the r i s e time a f t e r which i t did not show any change. By extrapolating the I c t h i s way they f i n d I c oC AT. However, i n case of the fi l m s deposited at room temperature, near T c, the dependence was found to be i n agreement with the p r e d i c t i o n of Ginzburg - Landau theory. Glover I I I (1957) measured the c r i t i c a l currents i n fil m s of t i n and lead to quite low temperatures by using continuous currents. His o fil m s were 50 A i n thickness. The c r i t i c a l current was found to vary as f l - (jjj-) 2J over e n t i r e temperature range but near T c B C S function f o r c c r i t i c a l currents was found to be a bett e r f i t . 40 Schmidlin and Crittenden (1958) with indium f i l m s f i n d I oC AT close to T 0 However, i n some fil m s they f i n d that the p r o p o r t i o n a l i t y departs from l i n e a r i t y thereby causing I c - A T curve bending towards the A T a x i s . Crittenden, Cooper and Schmidlin (i960) determine c r i t i c a l current as a function of temperature f o r a t i n f i l m , 0 6 / ^ t h i c k , 5 mm long and 60^-wide deposited on o p t i c a l l y polished glass. The c r i t i c a l currents were measured by employing a c i r c u i t which increased the current slowly u n t i l the specimen switches when the current i s cut o f f to zero w i t h i n a micro second. The D.C, c r i t i c a l current below ^ point i s approximately I T A. f i t t e d to the function _£ = 1 - (ijr-) , where I c o i s the extrapolated ico c c r i t i c a l current at the absolute zero. They observe a discontinuous jump i n the c r i t i c a l current at the A point. This could be a t t r i b u t e d to the f a c t that the t r a n s i t i o n s observed were not free from heating e f f e c t s . Near the t r a n s i t i o n temperature, the threshold c r i t i c a l currents C J-fc) were found to be lower than I c but the two became one at lower temperatures. These workers also f i n d evidence f o r the thermal propagation of the i n t e r -phase boundary s i m i l a r to one observed by Bremer and Newhouse (1958). Smallman et a l . (I960) determine c r i t i c a l currents f o r t i n f i l m s of various width and thickness by using pulses of r i s e time 60^s.and decay time 40/A-»S„ The substrates used by them were quartz and glass plates. The f i l m s of the same thickness on quartz showed greater c r i t i c a l current than the ones on glass, both on D.C. as w e l l as on pulse t e s t s . This difference a r i s e s due to b e t t e r thermal conductivity of quartz and has been observed by several other workers. The f i l m on glass under D.C, test showed a convex curvature with respect to temperature axis. They also noted thermal h y s t e r i s i s and estimated the temperature r i s e of approximately 0.20°K, Bremer and Newhouse (1959) on their t i n films deposited on single crystal sapphire and glass substrates seem to verify Ginzburg - Landau theory for the temperature region close to T (within 30 milli-degrees). Their measurements were made i n the temperature region A T ^ 0.3°K, i n this range the calculated penetration depth for the bulk t i n b e i n g ^ 0.1 micron which was comparable to their film thickness of 0.30 microns. They also find that the c r i t i c a l current varies linearly with the film width, interpreting this to mean that the current was uniformly distributed over the film width, for the films whose thickness was comparable with X . They also find l c for the films on sapphire to be greater than the ones on glass. Until now with the exception of the films of Alekseevskiiand Mikheeva, most of the films used were deposited on f l a t substrates, using uncompensated geometry. The data, on such films is hard to interpret because of the extreme distortion of the magnetic f i e l d of the current at the edges of the films. The f i r s t experiment on films evaporated on cylindrical substrates was done by Ginzburg and Shalnikov (i960). Their measurements were limited to the temperature region close to T . The t i n films used by them ranged i n thickness from 3.3 x 10"^ cm to 6.5 x 10"^ cm. The films were deposited at room temperature in a vacuum of the order of 10 mm. of Hg. For observing transitions they used d.c. The temperature and size dependence of their c r i t i c a l currents and c r i t i c a l magnetic fields i s i n agreement with the Ginzburg - Landau theory. The transition from the superconducting to the normal state is an extremely fast one occuring i n times of the order of afew nano-seconds. Kolchin et al.(1961) find that for sufficiently large currents the trans-i t i o n takes place i n about 5 nano-seconds, the transition time depending upon the current amplitude. Schmidlin et a l (1960 ) find the transition 42 times dependent upon the current amplitude and the f i l m thickness and could vary within wide l i m i t s . DESTRUCTION OF SUPERCONDUCTIVITY BY MAGNETIC FIELD The measurement of the c r i t i c a l f i e l d s provide a t e s t f o r the v a l i d i t y of the various theories of superconductivity. London theory i s unable to account f o r the high c r i t i c a l magnetic f i e l d s observed f o r t h i n f i l m s . The f a i l u r e a r i s e s due to two reasons, f i r s t l y London theory does not properly take i n t o account the surface energy between superconducting and normal regions and secondly i t i s a weak f i e l d theory and i t s a p p l i c a -b i l i t y i s l i m i t e d to the magnetic f i e l d s ^< H . Ginzburg - Landau theory i s free from above shortcomings but l i k e London theory i t i s a l o c a l theory. For the explanation of the c r i t i c a l f i e l d data, i t has been found e s s e n t i a l to incorporate the non-local aspects i n the G.L. theory. The c r i t i c a l f i e l d s are calculated mainly f o r two l i m i t i n g cases* the Pippard l i m i t (non-local) when the coherence length X ^ > \ j London l i m i t ( l o c a l ) when "\ y?^r The second condition i s easier to s a t i s f y when working with t h i n specimen near the t r a n s i t i o n temperature. and the temperature dependence. THE SIZE DEPENDENCE: According to London theory the c r i t i c a l magnetic f i e l d of a t h i n f i l m of the thickness d i s given by, . where H ^ i s the c r i t i c a l f i e l d f o r the bulk material, AL i s London penetration depth and H c i s the c r i t i c a l f i e l d for the f i l m . In the above Two types of dependence have been investigated, the size dependence 3.6 expression, the v a r i a t i o n of magnetic f i e l d i n s i d e a f i l m of t h i c k n e s s d i s assumed t o be, Ittner ( I960), assuming the f i e l d v a r i a t i o n to be according t o the c a l c u -l a t i o n s of S c h r i e f f e r (1957) where k n = ^ n + 1),^, a n d K(k) i s given by being the current d e n s i t y and vect o r p o t e n t i a l r e s p e c t i v e l y , shows th a t t h equation (3»fe) represents the c o r r e c t behaviour i f \ ; L i s replaced by Xe where V i s found t o vary w i t h f i l m t h i c k n e s s and p u r i t y . .. The p e n e t r a t i o n depth has been found to vary w i t h t h i c k n e s s i n the experiments of Lutes (1957), Schawlow (1958) and Glover and Tinkka/w (1956, 1957). A l l microscopic t h e o r i e s p r e d i c t v a r i a t i o n of \ w i t h t h i c k n e s s and mean f r e e path. Ittner c a l c u l a t e d the value of ^ V d ^y s u b s t i t u t i n g h i s data on- c r i t i c a l f i e l d s i n t i n f i l m s i n equation 3'6 He found t h a t i f the mean f r e e path and f i l m t h i c k n e s s was greater than % the p e n e t r a t i o n depth d i d not vary a p p r e c i a b l y . I n case e i t h e r mean f r e e path or f i l m t h i c k n e s s was l e s s than >^ , the p e n e t r a t i o n depth increased s i g n i f i c a n t l y w i t h t h i c k n e s s . I n h i s experiment the v a r i a t i o n of \ e C T ) / X f (•) as a f u n c t i o n of j j L -(•£-) 4 J i s found to be d i f f e r e n t from the p r e d i c t i o n on the b a s i s of B C S theory but i s i n agree ment w i t h the experiment of Schawlaw and D e v l i n . Toxen (1961) on h i s measurements w i t h indium f i l m s found the v a r i a t i o n of Xwith t h i c k n e s s i n agreement w i t h Ittner's r e s u l t s . Toxen (1962) obtained expressions for the c r i t i c a l fields of thin films. By making use of G - L theory he finds a relationship between c r i t i c a l f i e l d and magnetic susceptibility in the weak f i e l d limit. Then by combining this with Schrieffer% (1957) calculation of weak f i e l d sus-ce p t i b i l i t y i n terms of non-local parameters, he obtains expressions for the c r i t i c a l fields i n terms of non-local parameters. His results can be stated as follows: (i) Assuming specular reflection at the film surfaces - (whereas a i s half t h i c t o s s ) , K_ „ X.0[U%X)/«?]'"~ 3.9 ( i i ) , For random scattering -A comparison of these equations shows that the non-local calculation i s not sensitive to the type of reflection assumed. Toxen's model i s i n agreement with his experimental data on indium films. TEMPERATURE DEPENDENCE OF THE CRITICAL FIELDS Ginzburg (1958) on the basis of G - L theory found that near T c the c r i t i c a l f i e l d i s given by the following expression: (equation 1.28) when \oo i s the penetration depth at 0°K. Both the thickness dependence and temperature dependence have been experimentally verified by Ginzburg and Shalnikov (i960). Douglass and Blumberg (1962) calculated the c r i t i c a l fields i n the London limit. Near T , the c r i t i c a l f i e l d is given by He IT) = 1510 [ A ( M ) / ^ ( A t ) \ l + ^ f ) 3.11 where ^ i s a number independent of film thickness but depends upon the form of expression assumed for the free energy i n the superconducting state. For Ginzburg - Landau form (1950), (? = 0.31 and for Bardeen (1954) £ » - 0.19. They also find the penetration depth to be size dependent. Bardeen (1962) calculated the c r i t i c a l f i e l d near T c by minimising free energy and calculating the equilibrium value of the energy gap parameter. His expression for H c i s given by, r i c =• — • • A 3.12 A rt ° ^ where ^K.t> i s the energy gap at 0UK, AT = T - T and a = • — — p j ; A Ho 2d ^ film thickness g is a dimensionless geometrical factor depending upon the shape of the sample and the type of scattering assumed. P h i l i p s Gauge & Air W W Pump Nitrogen 7y>ap Nitrogen Dewar > Vaporiser Fie* . 4 - 1 Vacuum System for* l/apnrisen CHAPTER TV SPECIMEN PREPARATION AND MOUNTING Most of the work done on s u p e r c o n d u c t i v i t y i n t h i n f i l m s has been done on f i l m s deposited on f l a t s u b s t r a t e s . Such f i l m s though e a s i e r to make s u f f e r from the disadvantage of having s l o p i n g edges, which i f not removed may l e a d t o misleading r e s u l t s . Besides even f o r f i l m s w i t h edges trimmed the problem of f i n d i n g the d i s t r i b u t i o n of current i s a d i f f i c u l t one. I n order to escape these d i f f i c u l t i e s , i t was decided to deposit f i l m s on c y l i n d r i c a l rods. I n the design of the apparatus the f o l l o w i n g requirements were kept i n view. ( i ) A b i l i t y to produce vacuums of the order of 10"^ mm or b e t t e r , ( i i ) P r o v i s i o n f o r c o o l i n g the substrate t o l i q u i d n i t r o g e n temperatures, ( i i i ) Arrangement f o r r o t a t i n g the specimen at s u i t a b l e speeds during d e p o s i t i o n to ensure u n i f o r m i t y of the d e p o s i t . ( i v ) P r o v i s i o n of high current leads f o r r a p i d l y v a p o r i s i n g the metal to be deposited, (v) A s u i t a b l e s h i e l d f o r p r o t e c t i n g the substrate when d e p o s i t i o n i s not d e s i r e d , as i n the i n i t i a l outgassing. The vacuum system as shown i n the f i g u r e 4 , l c o n s i s t s of the f o l l o w i n g p a r t s , ( i ) An o i l d i f f u s i o n pump, backed by a mechanical pump, capable of 7 producing vacuum of the order of 10 mm,of Hg. The f i r s t n i t r o g e n t r a p i s provided to t r a p any o i l vapours escaping i n t o the v a p o r i s a t i o n chamber. 1„ Nitrogen dewar 2, Aluminium flange 3. 0-ring 4„ Evaporation b.ej.1 jar 5. Nitrogen dewars 6. O-ring seal flange 7. Drive shaft 8. Spacer 9. O-rings 10, Drive clutch 11, Brass plate 12, Heavy duty kovar seals 13, Cable 14, E l e c t r i c a l lead and boat support 15, Boat 16, Shield 17, Specimen 18, Specimen holder 19, Brass bar 20, "' Copper flange 21, Kovar glass seal 22, Bolt 23, Nut 24, Water cooled shield 25, 0-ring 26, 0-ring-seal 27, Water outlet 28, Water inlet 47 A Philip's vacuum gauge was used for measuring the vacuum. On the high side of the vacuum the pumping resistance was kept small by using larger size glass tubes and stop cocks. Apiezon N type grease was used at the stop cocks. Kel-F O-rings and gaskets supplied by Vinylloyd Company, Los Angeles, used in the vaporisation chamber and driving assembly helped in obtaining better vacuum. Thus vacuum of the order of 10"^ mm. was not d i f f i c u l t to attain. (2) The second li q u i d nitrogen trap is for maintaining the substrate at liquid nitrogen temperature. Tt is closed at the bottom by soldering a copper block to the kovar piece at the end of the trap. In order to secure a leak free joint, a special type of stainless steel solder, Eutec Rod 157, manufac-tured by the Electric Welding Co. of Canada, was used. On cooling, copper contracts more than kovar, hence the copper piece was made to f i t the outside of the Kovar piece. The thick copper block provided good thermal capacity and conductivity. Attached to this block was another block of brass and a mask for obtaining films at the ends. (3) The vaporisation chamber shown in the figure 4.2 was a cylindrical Corning glass pipe 30 cm. long and 11 cm. i n diameter. The top end of this was tied to the lower part of the system carrying the second nitrogen trap and the bottom end was closed with a brass disc by using standard seals. Instead of using the conventional O-rings, Kel-F O-rings were used. The brass disc carried two heavy current leads through the kovar seals capable of carrying 50 amperes of current. Tt also carried a movable shield through an 0-ring seal. The position of the shield could be adjusted from outside. The vaporisation chamber also had a side tube for the drive shaft for rotating the substrate. The drive shaft consisted of a stainless steel rod passing through a double 0-ring seal. The outer end of the shaft 48 was connected to a d.c. motor whose speed could be varied by adjusting the current through the motor. The end i n s i d e the chamber had an L shaped piece joined to i t to rotate the specimen by a clutch drive. E a r l i e r , instead of using the mechanical drive, a magnetic drive was t r i e d . This consisted of a magnet attached to the c e n t r a l axis of the gear box insi d e the vacuum and made to rotate by a powerful magnetron magnet mounted on an e l e c t r i c motor outside. This drive was not found s a t i s f a c t o r y as the bar magnet moved i n jerks instead of moving uniformly. In the e a r l i e r stages of the pro j e c t , the f i l m s were deposited on glass rods. Seven c a r e f u l l y cleaned glass rods were mounted i n a v e r t i c a l array. A l l the rods could be rotated simultaneously by employing a gear box having seven pinions which had holes for i n s e r t i n g brass plugs to which the rods were attached by n a i l varnish. In order to estimate the f i l m thickness, a deposit was c o l l e c t e d on a glass plate which was weighed and the thickness of f i l m s on the rods was estimated by assuming the v a l i d i t y of the inverse square law. For vaporising indium and t i n , c o n i c a l baskets of tungsten wire were made and painted with alumina i n water. A f t e r drying, these were baked i n vacuum to e x p e l l absorbed gases. A good portion of the charge was vaporised before allowing deposition on the substrates. A water cooled s h i e l d made t h i s p o s s i b l e . However, fi l m s thinner than 0 . 2 ^*-were not found to be conducting. A systematic search was made to look f o r the cause for lack of continuity i n the f i l m s . An examination under high powered microscope revealed the formation of agglomerates. This could have a r i s e n due to a number of reasons. A search was car r i e d out i n v e s t i g a t i n g the following p o s s i b i l i t i e s . 49 (i) Inadequate cleanliness of the surface of the substrate, ( i i ) Formation of agglomerates due to non-normal incidence of the vapour beam, ( i i i ) Temperature rise due to poor thermal contact with the nitrogen cooled metal blocks. (i) It was suspected that nai l varnish due to i t s high vapour pressure might form a thin film on the substrate. Substitution of glyptal in place of n a i l varnish did not improve matters. Next i t was tried to hold the substrates by mechanical grip. Several schemes of making chucks which would grip the substrate were tried but the substrate broke in many cases. Finally the most practical way found was to d r i l l a hole i n the brass holder perpendicular to the substrate, tap i t and hold the substrate i n place by a screw. On account of the small size of the rods (0,2 mm in diameter) a small size (no 00 - 90) of the screw was used. A tiny pellet of indium was placed between the screw and the glass rod before tightening the screw. This arrangement was found to be satisfactory for holding the substrate but the continuity in the films was s t i l l lacking. Greater precautions were taken for cleaning the substrate. These included using double d i s t i l l e d water and degreasing the rods in chlorethane vapour bath. The substrate and the holder were handled with clean tweezers. The rods were dipped i n a bath of pure propyl-alcohol and mounted in the vacuum system to leave a protective film of propyl-alcohol which was removed by ion bombardment in vacuum by Tesla c o i l . But s t i l l the films were discontinuous. ( i i ) It has been shown by Holland (1953) that the atoms coming at oblique angles of incidence lead to the development of a coarse texture, because the condensation is confined to the peaks of the bigger grains. It was decided to design a s l i t adjustable i n width which would stop oblique incidence. The size of the substrates was also increased to 3 mm. In order that the film be uniform i n thickness along the length i t was necessary that the rotation of the substrate be free from any movement perpendicular to the axis of rotation. This was achieved by constraining the free end of the rod to move in a circular slot. These changes s t i l l l e f t the films no better. ( i i i ) The third possibility, that formation of agglomerates might be the result of a temperature rise of the substrate, was investigated next. The evaporation was done in bursts of one second each. Different kinds of glass rods and quartz rods were tried but without any effect. The alumina painted tungsten boat was abandoned and replaced by tungsten and tanalum boats. This needed more current and the leads which hitherto had been of brass were made of stainless steel rods so that they could withstand higher temperature, at the same time providing a low heat leakage to the kovar seals. Thicker kovar seals for carrying currents of 50 amperes were used. A heavy duty transformer which could deliver currents up to 75 amperes was used to supply power to the boats. This enabled higher rates of deposition. Finally when a l l these changes proved to be f r u i t l e s s , i t was decided to try sapphire as substrate which, as is well known, has a high thermal conductivity. The films deposited on sapphire rods were found to be conducting. The flame polished sapphire rods of different sizes have been supplied by Adolf Meller & Co., Providence, R.I, In most cases the vacuum before the start of deposition was of the order of 10"^ mm of mercury. During vaporisation i t rose to about 6 x 10™' mm but was never worse than 10"*^ mm. Before the sapphire rods were cooled 51 to liquid nitrogen temperatures, the charge, indium or t i n , was heated to get r i d of absorbed gases. The spherical drop l e f t after heating looked clean and free from any oxide. During this preliminary heating, the shield was interposed between the boat and the substrate to protect the latter from any volatile impurities or gases. The indium or t i n used was 99.999$. pure and was obtained from the Consolidated Mining and Smelting Company, T r a i l , B.C. The film deposition took place i n 30 to 150 seconds, the time depending upon the film thickness. The speed of rotation was kept at 3 revolutions per second. The charge was vaporised to completion. For a number of specimens the sapphire rods were weighed before and after the deposition and the film thickness was calculated from the dimensions of the substrate and the mass of the deposit. The ratio of the mass of metal v a l o r i s e d to that collected on the substrate was found to be constant. The weighings were done with the help of a micro balance kept in an air conditioned room and mounted on shock proof supports. The weighing accuracy with the micro balance was t 2^*v»grams and the minimum mass of the deposit for the thinnest film was 67/^gms. Thus the determination of mass deposited by weighing the mass evaporated from the molybdenum boats, provided dependable results. Before breaking the vacuum, liquid nitrogen i n the trap was driven off by using compressed a i r and the system was allowed to return to room temperature. In order to make thicker deposits at the end, the specimen, after weighing, was mounted back i n the vacuum system and a shield was Inserted to allow deposition only at the ends,; After the fabrication of the film was complete, the specimen was Screw A-B Old Spec/men Holder V77\ R71\ U77[ V/// L A V//A 4 4 5 00x90 Screws C-D 2^ A^ tv Spec/men hfolder A-B i — 1 n V/A m V 71 #r»/thenol Connects i n l F I G , . 4--3 b ' Oox90 Screws Scale M 52 mounted on a specimen holder. The details of the specimen holders are shown in figure 4.3 a, b. Figure 4.3 a i s the earlier design used for work with pulses of long rise time of 1;*A$. Essentially the specimen holder consisted of a rod of bakelite milled in the middle as shown in the figure to accommodate the specimen on four no. 30 copper wire hooks which served as the current and potential leads. To one end of the bakelite rod was screwed a circular disc of brass which served as one current lead. After the specimen was mounted as described below, the holder was s l i d into a coaxial copper tube having a circular brass disc soldered to one end. The coaxial mounting was used to protect the specimen from the magnetic fields of the currents in the return lead when the c r i t i c a l currents were being measured. .A couple of holes were d r i l l e d into the brass plates to maintain a circulation of liquid helium. The thin copper hooks, supported by brass screws through bakelite, were used to provide strain free mounting. In earlier mountings when the specimen was la i d along the grooves i n copper cups which were used as current and potential leads, the specimen was bent considerably when dipped i n liquid nitrogen. The copper hook mounting proved quite satisfactory and was also adopted in the modified design to be described below. The modification i n the specimen holder was necessary to enable the use of fast rising pulses, having a rise time of 7 nano seconds. In the previous holder the current and potential leads from the specimen were some distance apart, A fast rising current pulse passing through the specimen produces magnetic flux as shown in the f i g , 4.4a, This magnetic flux gives rise to a back e.m,f. Due to the inductance common to the current and voltage circuits, the back e.m.f. has the same direction i n the voltage circuit as well. Let the specimen be approximated by a combination of R and L i n series where R is the resistance of the specimen and L, the self-inductance of the section of the specimen between the f i r s t potential contact and the short c i r c u i t , as shown in the figure. This approximation i s j u s t i f i e d , as the end of the specimen holder is a short c i r c u i t and we are considering only a small section of the line. For a 3 cm. long section the inductance L i s ~0.02^-H and for a pulse of a rise time of 7 nano seconds., V-^ 3 volts per ampere of current pulse. At helium temperatures, in a state when the specimen is being switched from the superconducting to the normal state R may be »^0,01 ohm. Thus the inductive voltage may be larger than the resistance one by two orders of magnitude. Two modifications in the specimen holder were introduced, one was to use Amphenol type 27-10, 50-^subminax connectors, these being screwed to the brass plate, and the second was to compensate the inductive pulse by introducing a loop i n the voltage leads, wound so as to produce a pulse equal and opposite to the one being compensatod. In fact, by winding the loop i n the opposite sense, i t was possible to enhance the inductive pick up. The exact compensation could be achieved by suitably choosing the number of turns in the loop and adjusting their position relative to the specimen, a change i n either could lead to under or over compensation. The holder thus modified is shown in the f i g . 4.3, b. The above modification did not prove satisfactory because the introduction of the inductance in the form of the loops in the potential circuit slowed the rise of the potential pulse. Thus the current and the potential pulse shapes differed from each other. The next attempt was to reduce the magnitude of the inductive pulse by bringing the potential lead very close and parallel to the specimen so h & u ^ A L e N T -f V £ I o o o 0 © c4 I o oo or oo CP 70 p b -,b ^ 1 6 i T 5 2 , 6.2 where n i s t h e number o f c o n d u c t i o n e l e c t r o n s p e r c m ' . , v i s t h e F e r m i v e l o c i t y , m* t h e e f f e c t i v e e l e c t r o n m a s s , e t h e e l e c t r o n c h a r g e , ^ and 1 a r e t h e b u l k r e s i s t i v i t y a n d t h e b u l k mean f r e e p a t h r e s p e c t i v e l y . I f we assume t h a t t h e number o f c o n d u c t i o n e l e c t r o n s r e m a i n s c o n s t a n t w i t h t e m p e r -a t u r e , ^ b l b i s c o n s t a n t a n d i n d e p e n d e n t o f t e m p e r a t u r e . E q u a t i o n 6.2 g i v e s t h e d e p e n d e n c e o f r e s i s t i v i t y on mean f r e e p a t h . I n case o f t h i n f i l m s , t h e mean f r e e p a t h i s l i m i t e d b y t h e b o u n d a r y . F u c h ' s e x p r e s s i o n s (1938) f o r two l i m i t i n g c a s e s a r e g i v e n b y e q u a t i o n s 3.1 and 3.2. A t r o o m t e m p e r a t u r e , t h e b u l k mean f r e e p a t h f o r i n d i u m ~ 1 0 0 A 0 and f o r t h e f i l m t h i c k n e s s u n d e r i n v e s t i g a t i o n ~ ^> 1 and t h e r e f o r e e q u a t i o n 3.2 i s a p p l i c a b l e . I n t h e f i g u r e 6.1 ^ h a s b e e n p l o t t e d v e r s u s d " ^ . The s l o p e and t h e i n t e r c e p t o f t h e s t r a i g h t l i n e g i v e 1^273 a n < * ^273* ^ e v a ^ u e s s 0 d e t e r m i n e d a r e , l B 2 7 3 = .119 A 0 , ^ 2 7 3 a 10.07 x 10" 6 ohm cm. <*b273 l b 2 7 3 " 1.2 x 1 0 " ^ ohm c m 2 I n v i e w o f t h e f a c t t h a t t h e e f f e c t i v e l e n g t h o f t h e f i l m was n o t known v e r y a c c u r a t e l y , t h e r e i s some s c a t t e r o n t h e g r a p h . The p o i n t s shown b y c i r c l e s a r e f o r t h e f i l m s f o r w h i c h t h e e f f e c t i v e l e n g t h was d e t e r m i n e d c a r e f u l l y w i t h a m i c r o s c o p e . The s t r a i g h t l i n e f i t i s o n l y f o r s u c h f i l m s . A t h e l i u m t e m p e r a t u r e , 1^4 2 ^ > d and t h e e q u a t i o n 3.1 h o l d s . The f i g u r e 6.2 shows a p l o t o f {% d ) - l v e r s u g l o g d a n d f r o m t h e s l o p e and t h e i n t e r c e p t we g e t , ^ B 4 > 2 . = 0.129 x 10" 6 ohm c m . , l B 4 < 2 = 7940 A 0 and l B 4 > 2 ^ b4,2 = l ' 0 2 8 x I O " 1 1 ohm c m . 2 . Thus ^1 i s . f o u n d t o be p r a c t i c a l l y c o n s t a n t and i n d e p e n d e n t o f t e m p e r a t u r e . I t i s p o s s i b l e t o e s t i m a t e t h e c o h e r e n c e l e n g t h * ^ „ f r o m t h e k n o w l e d g e o f <^ D l b as f o l l o w s . A c c o r d i n g t o t h e B C S t h e o r y , ^ 0 i s g i v e n b y , ^» ~ T\ 6.3 # 4 - & 1 / 3 ? v r r 71 where v is the Fermi' velocity, A*(0) is the. energy gap at 0 °K. Also the effective electron mass is given by, 6.4 According to Keesom and Perlman (1956), we have m* - 7.3 (n v \ / n a ) - 1 / 3 m* 6.5 where V m is the molar volume, m the electronic mass, n a is the number of atoms per unit volume and 7 the electronic specific heat i n units of milli-joules / mole degree . Combining equations 6.2, 6.3, 6.4 and 6.5 we obtain, 6 ' N ^ A i J o ) V?l 6.6 where N is Avogadro's number. If we substitute the various quantities for indium as quoted by Toxen (1961), we have % 0 m 258 x IO" 1 0 / ^ l 6.7 Substituting for ^ 1 i n this equation, we can calculate f ^ 0 . The various estimates are given below i n table I. TABLE I *Dheer (1959) Davies (1960) Roberts Toxen (1961) Toxen (1962) This work . (Room Temperature) '0 o 4300 A 4400 2900 2600 2600 1 400 A 2150 + (Helium Temperature) 2500 + TSo •«• See Toxen (1961) for references. ?1 0.6 x IO" 1 1 ohm cm.2 0.89 x IO" 1 1 " 0,98 x 10" 1 1 " 2.0 x IO" 1 1 " 1.2 x 10" 1 1 " iro . 4 -1 1.038 x 10" 1 1 " "t °-72 The data of Dheer and Roberts are based on the measurements of high frequency surface impedance and Davies also used data on surface impedance to e s t i m a t e % 0 > Toxen's estimates are based on the r e s i s t i v i t y measurements and c r i t i c a l magnetic f i e l d measurements. (3) T r a n s i t i o n Temperature and the T r a n s i t i o n Width: As mentioned i n Chapter I I I , the t r a n s i t i o n temperature f o r t h i n f i l m s departs from i t s value i n the bulk material. Toxen (1961) found that f o r the indium f i l m s deposited on vitreous s i l i c a substrate, the r e l a t i o n -ship between T c and d i s given by, C T" 52- 750 * T ° = r " d2-where £T i s the d i f f e r e n c e i n the T„ for the f i l m and bulk material and d c c i s the f i l m thickness i n angstrom units. In f i g . 6.3 i s shown a p l o t of ST„ versus d from the present work and the data i s found to f i t the equation, r T ^ . Z ^ l - i i 3 ^ 6.8 d d 2 The f a c t that T„ f o r the fi l m s i s higher than T for the bulk material c c indicates that the f i l m s are under t e n s i l e stress which of course i s to be expected because the c o e f f i c i e n t of expansion f o r indium i s greater than that f o r sapphire. The t r a n s i t i o n width, defined to be the temperature i n t e r v a l between 10$ and 90$ of the resistance i n the normal state, l i e s between 3 and 8 milli-degrees f o r the indium f i l m s . There i s no systematic v a r i a t i o n of the t r a n s i t i o n width with f i l m thickness. For t i n films the t r a n s i t i o n p i c ture was quite d i f f e r e n t : . For the f i l m s deposited at the l i q u i d nitrogen temperature', the t r a n s i t i o n s were quite broad and were characterised by a marked plateau. However, the films E L E C T I O N M » c « . oG»R* m of F I L M S Figure G.h a l^OO 8 T i n f i l m ( l i q u i d No temp.) X 5000 Figure.6.h b T i n f i l m (room temp.) X 5000 Figure 6.k c Indium f i l m ( l i q u i d No temp.) X 5000 73 deposited at room temperature showed narrow transitions but films thinner than 900 A3, lacked conductivity. The transition temperature for these films o differed from the bulk value considerably. For the two films 946 A and o 3250 A thick T was 3.907° K and 3.862° K respectively as compared to the c value 3.728° K for bulk t i n . Tin films were photographed with an electron o microscope. Fig. 6.4a gives electron micrograph for a 1400 A thick film deposited at the liquid nitrogen temperature, the transition curve for which is given in f i g . 6.5. The structure shows some evidence of twinning. Figure 6.4b gives the electron micrograph for the t i n film deposited at the room temperature. This shows a smooth structure. Figure 6.4c shows an o electron micrograph of an indium film, 5000 A i n thickness, deposited on o glass substrate i n the earlier stages, when the films less than 2000 A thickness were found to be lacking in conductivity. This film shows very coarse structure, perhaps due to the formation of agglomerations. 4. C r i t i c a l Currents: o o The indium films investigated l i e i n the range from 580 A to 3540 A in thickness. The dependence of c r i t i c a l currents on temperature i s governed by the ratio as mentioned in the f i r s t chapter (equations 1.29 and 1.30). The penetration depth depends upon both the film thickness and the tempera-ture, this w i l l be discussed later i n this chapter. The table II below gives the theoretical and the experimental values of Xo, at various temper-atures. The second column gives the theoretical values for the bulk material calculated by taking "Xoo = 640 A "(Lock 1951) and assuming *Xo = \oo/ J l - ( ^ - ) 4 J ^ The third, fourth and f i f t h columns are the experimental values determined from our work and are taken from figure 6.13 for three films whose thicknesses are given at the top of the column. 74 TABLE I I AT ° K Bulk 3540 A 0 2640 A 580 A .050 2690 A 3600 4700 6100 .100 1790 2400 3575 4575 .200 1375 1650 2550 3500 .300 1088 1325 2050 2960 Thus the condition -L°J^ —. ( 1 is satisfied by a l l the films ,except the one o ^» 3540 A thick ;for the region A T ^ 0.150° K . If we assume the c r i t i c a l current I_ being given by I . o ( . ( ^ T ) n , then for the case ^ f a ^ t / l , n =1.5, AO for ^ l , n * 1.12. The measurement of c r i t i c a l currents presents d i f f i c u l t i e s for two main reasons, f i r s t l y due to extremely high speed of the transition and secondly due to very low thermal capacity of the fi l m and the consequent rise in the film temperature. The above reasons become specially important when the current transitions are broad. Both direct current and pulses have been used for c r i t i c a l current measurements. The results w i l l be discussed separately. Pulse Measurements In the present work, two types of pulses were used. For most of the films deposited on 1,5 mm. diameter sapphire substrates, either the Rutherford pulser or thyratron pulser was used as described i n Chapter V. The thyratron pulses, which were used for most of the data had a rise time of 1,2 micro second and decay time of 2.5 micro second. These were single shot pulses. The c r i t i c a l currents, obtained by measuring the current needed to restore half the resistance in the normal state, were normalised to the value ^ T = 0,3° K and have been plotted in f i g . 6,6. For the f i r s t region, AT = 0.08° K , 75 n = 1.27 and subsequently f o r 0.OS°K ^ £ T ^ 0.30° K, n = 0.85. These curves were f i t t e d by the method of l e a s t squares with the help of Fortran 1620 IBM computer. I t may be noted that the value of n found by minimising the res i d u a l s i n log I c AT was found to be = 1,41, as compared to 'I'W =. 1*317 V found by minimising those for I c and AT. These values may be compared with those of Alekseevski and Mikheeva who for the pulses of 0.1 second duration f i n d n = 0.6 and for the pulses of r i s e time 250 micro second f i n d n = 1. In view of the f a c t that the t r a n s i t i o n could occur w i t h i n the r i s e time of the pulse, i t was f e l t that the departure from the t h e o r e t i c a l value of n = 1,5 could be due to Joule heating. In f a c t with the pulses of increasing magnitude i t was possible to see the evidence f o r the Joule heating a f t e r a c e r t a i n resistance was restored i n the f i l m . The voltage pulse shape consisted of an i n i t i a l r i s e followed by a f u r t h e r r i s e with change of slope. The l a t t e r was due to Joule heating. In f a c t i n the e a r l i e r r i s e too, i n view of the f a s t speed of t r a n s i t i o n i t i s conceivable that some heating might have occurred. The f a s t pulses used to i n v e s t i g a t e the current t r a n s i t i o n f r e e from Joule heating, had a r i s e time of 7 nano seconds. The specimen was a f i l m o of indium 585 A f t h i c k coated on a sapphire rod of 0.4 mm. diameter. The ends where current leads were attached had a heavy coat of indium, and the current contacts were made with indium metal as described i n Chapter IV. The t h i c k indium coating, though advantageous from the point of view of Joule heating had one disadvantage. The t r a n s i t i o n temperature for the t h i c k e r part i s lower than that for the thinner part. Hence i f the t h i n f i l m i s to be driven normal by current pulses, i n the immediate neighbourhood of the t r a n s i t i o n temperature f o r t h i n f i l m , the t h i c k e r part might be driven normal e a r l i e r , leading to the production of Joule heating i n the t h i c k e r part of the f i l m . O S C I L L O G , ftft^s ° F ^ ^ ^ f L W 5 U T £ J ) " T R A N S I T I O N ILl o r-c cc UJ 1 U J V IU a l <£ Ul v/) d ) O uJ r LL O ci r-z ui r-o: HI X _J O P 0 © 0 o O lO VI <2f 11) O o c r 1 1 1 1 80 ( i ) At the f i r s t i n s t a n t , the appearance of r e s i s t a n c e i s a con-sequence of the t r a n s i t i o n , unaffected by Joule heating. This i s not r i g o r o u s l y true i f we imagine the t r a n s i t i o n t o be very f a s t . ( i i ) During the i n t e r v a l of 20 nano seconds between the f i r s t and the second i n s t a n t s , the e n t i r e Joule heat heats up the f i l m and the substrate or helium does not p a r t i c i p a t e i n the heat d i s s i p a t i o n . This seems to be a f a i r assumption as d e s p i t e the high thermal c o n d u c t i v i t y of sapphire, the heat t r a n s f e r t o the sapphire rod i s governed by the K a p i t z a boundary r e s i s t a n c e between the f i l m and the s u b s t r a t e . I f one makes use of the data by L i t t l e (1959), i t can be shown t h a t the heat communicated t o sapphire rod during the 20 nano second i n t e r v a l i s n e g l i g i b l e . The heat t r a n s f e r t o helium bath i s known t o be a r e l a t i v e l y slow process. The temperature r i s e i n the f i l m can be estimated by c o n s t r u c t i n g curves of r e s i s t a n c e versus temperature with I = constant from the r e s i s t a n c e versus current c h a r a c t e r i s t i c s f o r T = constant. From the knowledge of £ R, the change i n the r e s i s t a n c e between the two i n s t a n t s , the change i n the temperature i s c a l c u l a t e d from the R - T c h a r a c t e r i s t i c s f o r t h a t p a r t i c u l a r c u rrent. This £T may be c a l l e d the observed temperature r i s e . Another way of e s t i m a t i n g the temperature r i s e i s by c a l c u l a t i n g the energy d i s s i p a t e d , and knowing the thermal c a p a c i t y of the f i l m the temperature r i s e i s found out. This may be c a l l e d the c a l c u l a t e d tempera-tu r e r i s e . I n t h i s c a l c u l a t i o n i t i s assumed t h a t the e n t i r e t h i n f i l m (not counting the t h i c k e r p a r t of the f i l m ) i s heated uniformly. Comparison of ^T observed and c a l c u l a t e d r e v e a l s some i n t e r e s t i n g p o i n t s . (See f i g u r e 6.9.) For low values of the c u r r e n t s , $T observed i s found t o be greater than ST c a l c u l a t e d , i n general the r a t i o goes on 81 d i m i n i s h i n g from a value 10 f o r 10 c a l . to a value 1 f o r Q, = 22.5 x 1 0 " ^ c a l . For i n c r e a s i n g currents the decrease continues and f o r Q^ = 450 x 1 0 " c a l , r a t i o = 0.2. Higher value of the r a t i o could be i n t e r p r e t e d by assuming t h a t f o r s m a l l e r power i n p u t , the temperature r i s e occurs only at the r e s i s t i v e p a r t s of the f i l m and i f the v e l o c i t y of propagation of the thermal waves i s s m a l l , the e f f e c t i v e thermal c a p a c i t y of the f i l m i s s m a l l e r than the c a l c u l a t e d value. For l a r g e r energy d i s s i p a t i o n , the r a t i o decreases f o r two reasons, f i r s t l y the r e s i s t a n c e appears at more p o i n t s and secondly due t o the increased v e l o c i t i e s of thermal propagation, not only the e n t i r e t h i n f i l m but also the t h i c k e r f i l m a t the ends acts as a heat s i n k , This e x p l a n a t i o n seems t o amount to a q u a l i t a t i v e evidence f o r the existence of thermal wave f r o n t s , moving w i t h great speeds. From the above d e s c r i p t i o n i t appears t h a t the current pulses r i s i n g w i t h great speed would, at l e a s t i n p r i n c i p l e , enable the study of t r a n s i t i o n without the accompanying thermal e f f e c t s . T h i s , however, i s not c o r r e c t due t o the f a c t t h a t the surface impedance, at the h i g h frequencies i n t r o -duces a d d i t i o n a l power l o s s e s and w i l l tend to set a l i m i t t o the speed f o r the r i s e of the current p u l s e . The surface r e s i s t a n c e becomes p e r c e p t i b l e when the photon energies of the high frequency component of the pulse become comparable or exceed the energy gap at t h a t temperature. I n view of the dependence of the energy gap on temperature, f o r the temperature r e g i o n c l o s e to T , these frequencies are ~ 1 0 ^ c/s. The highest frequencies used i n t h i s work are lower than the above. K o l c h i n e t a l (1961) f i n d that w i t h f a s t e r r i s i n g p u l s e s , the c r i t i -c a l current decreases, t h i s b e i n g i n t e r p r e t e d by them as heating due t o eddy 4 « D I J L t n i )6ts» loot l 4 n 2 L ? « " > ) 3i_en) 3 ^ 82 currents and amounts to an argument f o r not using f a s t r i s i n g pulses f o r observing i s o t h e r m a l t r a n s i t i o n s . I n the present work, no d i r e c t evidence f o r eddy current h e a t i n g could be obtained. D. C, T r a n s i t i o n s Ginzburg and Shalnikov (i960) measured the c r i t i c a l c urrents f o r t h e i r t i n f i l m s using d i r e c t currents and found t h a t f o r the temperature range AT ^ . 4 ° K, I c o C ( A T ) . The c r i t i c a l currents using d . c , measured o i n our work f o r the 585 A t h i c k indium f i l m , are shown i n f i g u r e 6.8, With d . c , the t r a n s i t i o n s occur very s n a p p i l y , but show h y s t e r e s i s f o r i n c r e a s i n g and decreasing c u r r e n t s . H y s t e r e s i s f o r d.c. t r a n s i t i o n s was a l s o observed by Bremer and Newhouse (1959). The h y s t e r i s i s was absent i n the pulse t r a n s i t i o n s . A comparison of the c r i t i c a l currents obtained by using pulses and d i r e c t currents ( f i g u r e 6,8) shows t h a t the d.c. values are lower i n d i c a t i n g that these t r a n s i t i o n s belong t o the t h i n n e r p a r t s of the f i l m . For d.c. values n = 1.31 i n the temperature range^T = 0.300° K. Dependence of C r i t i c a l Currents on F i l m Thickness According to equation 1.29, I c o C d . The c r i t i c a l c urrents f o r Indium f i l m s have been p l o t t e d as a f u n c t i o n of f i l m t h i c k n e s s i n f i g u r e 6.10 f o r v a r i ous values of AT. The dependence i s seen to be l i n e a r i n agreement w i t h the theory. C r i t i c a l Currents f o r T i n Films o o For two t i n f i l m s 946 A and 3250 A i n t h i c k n e s s , the c r i t i c a l currents were determined by using the pulses from the thyrabron p u l s e r . The I c -&.T p l o t was s i m i l a r to the one f o r indium f i l m s , f o r the r e g i o n 0 A T £060*. n ^ 1 . 3 and f o r the second r e g i o n n ^ 0 . 9 . C R . \ T \ ^ * L M A G N E T I C % 10 0 I6000 ® 3S"4-o A X a64t> O lo2L* o Soto ZSoo "Boo* ^ "2- » 0 6 ? 8 6 |.o |-z -'/z_ V 32oo 1. ?o« r Zoo* T I N F I L VIS 1^ 2 _ L _ l-o |. x I-6 l-S 2© 83 5, C r i t i c a l Magnetic F i e l d s : According t o equation 1.28, H Q oZ ( A T ) -1"/2. For va r i o u s indium f i l m s , the c r i t i c a l f i e l d s are p l o t t e d as a f u n c t i o n of A T as shown i n f i g u r e 6.11, R"c were normalised to the value corresponding t o A T = .300° K. An equation H = A ( A T ) N was f i t t e d t o the data w i t h the help of IBM F o r t r a n 1620 computer and the best f i t i s obtained f o r n = 0.50. Thus the dependence of the c r i t i c a l f i e l d s on temperature i s i n agreement w i t h the theory. For t i n f i l m s , t h i s dependence i s a l s o v e r i f i e d . I n f i g u r e 6.12, the c r i t i c a l magnetic f i e l d s have been p l o t t e d as a 1 f u n c t i o n of g-, f o r various values of A T . The p o i n t s i n d i c a t e d w i t h l a r g e c i r c l e s do not f i t the l i n e a r dependence. For these f i l m s , T c was higher than f o r the others i n the same th i c k n e s s range. These same f i l m s a l s o show abnormal behaviour i n the v a r i a t i o n of "X w i t h y, where y = f" 1 T """A -> L 1 _ ( v J as w i l l be discussed i n the f o l l o w i n g s e c t i o n . Dependence of \ on F i l m Thickness and Mean Free Path. F o l l o w i n g I t t n e r , the e f f e c t i v e p e n e t r a t i o n depth Xe has been c a l c u l a t e d by s o l v i n g equation 3.6, f o r both indium and t i n f i l m s , )\£ has been p l o t t e d as a f u n c t i o n of y, i n f i g u r e 6,13 f o r the indium f i l m s and i n f i g u r e 6.14 f o r t i n f i l m s . I n the temperature range of A T - 0,4° K, a l l the p o i n t s l i e on s t r a i g h t l i n e s . These s t r a i g h t l i n e s have been e x t r a p o l a t e d t o get \ e ( 0 ) corresponding to y = 1. I n general 'Xf(O) i s found t o increase w i t h d i m i n i s h i n g t h i c k n e s s . I t i s obvious from f i g u r e 6,13 t h a t f o r three 0 0 o f i l m s 810 A, 1020 A and 1250 A i n t h i c k n e s s , the p e n e t r a t i o n depths are unusually h i g h , but these are a l s o the f i l m s which show abnormal behaviour i n H c versus •jj p l o t . C a l c u l a t i o n of the b u l k mean f r e e path from the r e s i s t i v i t y data a l s o shows t h a t f o r these f i l m s the mean f r e e path i s 6 of z> t-UJ h o J X J Iii J a: u r U u. 0 i n o IP C O r o © 4 g -a-g V 1 c4 o o 6 6° ...I o a 4-o o. O o 0 o. o 2.<£oo Xooo\ t - 4 > I " 5 " •Boof \6I o f t -©—© ©- _© a_ -© GT 2.4 4- o A -O 9 8 9-4oof 3f f«A -Q 9 Q ©- -o cr -O-Zo 40 FiGf. 6-16 60 20 \oo 84 s m a l l e r than expected and consequently the increase i n the p e n e t r a t i o n depth could be due to both the t h i c k n e s s and mean f r e e path e f f e c t s . S t r i c t l y speaking, one i s not j u s t i f i e d t o e x t r a p o l a t e the s t r a i g h t l i n e to get " X^O). As pointed out by M i l l e r (1959) when the t h e o r e t i c a l values of *X (T) are p l o t t e d as a f u n c t i o n of y, a s t r a i g h t l i n e f i t i s obtained which bends below y T 1,5, the d i f f e r e n c e between the slope f o r l a r g e y and the i n t e r c e p t a t y = 1 being of the order 10$. There i s yet another way of c a l c u l a t i n g the p e n e t r a t i o n depth. For t i n f i l m s , the c r i t i c a l magnetic f i e l d i s given by equation 3,11, In f i g u r e 6,15 H c (T) / (At)1/2 has been p l o t t e d as a f u n c t i o n of M / T c from t h i s the value of \(o,d) and € has been c a l c u l a t e d as given i n the t a b l e . I l l These values f o r £ favour the Ginzburg - Landau expression f o r f r e e energies as r e f e r r e d t o i n Chapter I I I , The equation corresponding t o 3.11 f o r indium f i l m s i s not a v a i l a b l e because the expression f o r "X(T, d) i n which the temperature and t h i c k n e s s de-pendence were f a c t o r a b l e could not be found i n the l i t e r a t u r e . However, l e t us assume th a t H Q (T) f o r indium i s given by an expression of the form He ( T ) - t ^ i M } ( A t ) \ l + 6 M ; 6.9 o where b i s an unknown constant. For 3540 A t h i c k f i l m the value of ,^ (o,d) ( i . e . "X^ ) as found by e x t r a p o l a t i n g t o y = 1 i n f i g u r e 6.13 i s 650 A. S u b s t i t u t i n g t h i s value i*n equation 6.9,b can be c a l c u l a t e d and t h i s , value has been used f o r the c a l c u l a t i o n of X(o,d) f o r other f i l m s from the curves of f i g u r e 6.16, These values of "X(o,d) together w i t h *Xf, as c a l c u l a t e d from f i g u r e 6,13 are given i n the t a b l e I I I , 85 TABLE I I I Indium Films F i l m Thickness \ ( 0 , i ) " X £ £ o o o 3540 A 650A 650 A 1.65 ± • 0 • '-4 1 1 5 7 1350 0.41 3 2 5 0 76"0 860 0.44 ± o-.o CHAPTER VII SUMMARY AND CONCLUSIONS Most of the r e s u l t s presented i n the l a s t Chapter are f o r indium o , o f i l m s i n the t h i c k n e s s range of 580 A t o 3540 A. Due t o d i f f i c u l t i e s o encountered i n preparing t h i n f i l m s of t i n , only two t i n f i l m s 910 A and o 3250 A t h i c k have been i n v e s t i g a t e d . The measurement of r e s i s t i v i t y a t room temperature and helium temp-erature enabled the c a l c u l a t i o n of ? 1 and % 0 and these r e s u l t s have been l i s t e d i n Table I along w i t h the r e s u l t s of other workers. Our estimates are i n f a i r agreement w i t h the other values. The t r a n s i t i o n width f o r indium f i l m s , as measured by r e s i s t a n c e measurements, was found t o l i e between 3 and 8 m i l l i - d e g r e e s . The t r a n s i -t i o n temperature was found to increase w i t h decreasing f i l m t h i c k n e s s . < o For a 585 A t h i c k indium f i l m , the c r i t i c a l c u r r e n t s were measured by using pulses of a r i s e time of 7 nano seconds, the t r a n s i t i o n s being observed w i t h the he l p of T e t r o n i x type 661 sampling o s c i l l o s c o p e having a r i s e time of 0.35 nano seconds. These measurements gave n = 1,46 f o r 0 < A T ^ 0.150° K and n = 1.02 f o r 0.150° K CAT^* 0.300° K. The Ginzburg-Landau theory p r e d i c t s n = 1.5 f o r the e n t i r e r e g i o n under observation i . e . 0h i n ngronmrnt with t h r work nf Pronm nnd Rhndm-ink rind Ghnrry nnd Gittlrmnn. The temperature dependence of the c r i t i c a l magnetic f i e l d i s i n 88 complete agreement w i t h the behaviour p r e d i c t e d by the equation 1.28; and so i s the dependence of f i e l d on f i l m t h i c k n e s s . The c r i t i c a l f i e l d data has been used f o r c a l c u l a t i n g the e f f e c t i v e p e n e t r a t i o n depth and i t has been found t h a t the p e n e t r a t i o n depth increases w i t h decrease i n both the f i l m t h i c k n e s s and the mean f r e e path. These r e s u l t s are i n q u a l i t a t i v e agreement w i t h the t h e o r e t i c a l and experimental f i n d i n g s of other workers. 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