FIELD THEORETICAL DESCRIPTION OF THE SUPERCONDUCTING STATE ROBERT EDWARD PUGH A Thesis Submitted i n Partial Fulfilment of the Requirements for the Degree of MASTER OF ARTS In the Department of Physics We accept this thesis as conforming to the standard required for the degree of MASTER OF ARTS Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA April, 1955 ABSTRACT The superconducting s t a t e i s described i n terms of the e l e c t r o n i c plasma i n i n t e r a c t i o n w i t h the electromagnetic p o t e n t i a l s . The i n t e r a c t i o n of the plasma w i t h i t s e l f and w i t h the l a t i c e , Is represented by a p o t e n t i a l energy deter-mined such that the equations of motion are s e l f - c o n s i s t e n t . The r e s u l t i n g n o n - l i n e a r f i e l d equations are solved i n the l i n e a r approximation. There e x i s t f o u r independent modes of v i b r a t i o n of the f i e l d v a r i a b l e s corresponding t o the plasma o s c i l l a t i o n s , and one l o n g i t u d i n a l and two transv e r s e o s c i l l a t i o n s of the electromagnetic p o t e n t i a l s . The c o n t r i -b u t i o n of these modes to the s p e c i f i c heat i s discussed. ACKNOWLEDGEMENT I wish t o thank P r o f e s s o r F. A. Kaempffer f o r suggesting the research problem and f o r h i s advice and encouragement throughout the performance of t h i s work. I am g r a t e f u l t o the N a t i o n a l Research C o u n c i l of Canada f o r the donation of a Bursary (1954-55) i n support of the research. TABLE OF CONTENTS I INTRODUCTION 1 A. The Experimental Facts 1 B. The Present State of the Theory of Sup e r c o n d u c t i v i t y 4 I I THE LAGRANGIAN FORMULATION OF THE SUPERCONDUCTING STATE AND THE RESULTING EQUATIONS OF MOTION 7 I I I SPECIFIC HEAT 14 IV CONCLUSIONS 21 APPENDIX A 22 REFERENCES 26 LIST OF FIGURES FIGURE 1. Molar Heat C a p a c i t i e s of Niobium 3 FIGURE 2. V a r i a t i o n of the S p e c i f i c Heat from a T^ Law 18 I INTRODUCTION: I n t h i s work an attempt i s made to describe the phenomenon of s u p e r c o n d u c t i v i t y by repr e s e n t i n g the s o - c a l l e d "superconducting e l e c t r o n s " as a c l a s s i c a l charged f l u i d . I t w i l l be shown that the hydrodynamle equations d e s c r i b i n g the f l u i d are c o n s i s t e n t w i t h London's phenomenological equations. Since as yet the behaviour of the s p e c i f i c heat of a superconducting metal below i t s t r a n s i t i o n temperature has not been s a t i s f a c t o r i l y e x p l a i n e d , we s h a l l t r y to des-c r i b e t h i s phenomenon i n terms of the energies of the wave motions of the e l e c t r i c and magnetic f i e l d p o t e n t i a l s . We s h a l l begin our d i s c u s s i o n by reviewing b r i e f l y some of the experimental f a c t s associated w i t h the phenomenon of super-c o n d u c t i v i t y . A The Experimental F a c t s . I n 1911» Kamerlingh Onnes discovered that at a c e r t a i n very low temperature, T , the r e s i s t a n c e of c e r t a i n conductors dropped s h a r p l y to zero, i n d i c a t i n g that no e l e c -t r i c f i e l d could be maintained i n s i d e t h i s 'superconductor'. Kamerlingh Onnes f u r t h e r discovered t h a t t h i s phenomenon of su p e r c o n d u c t i v i t y was destroyed by the a p p l i c a t i o n of a s u f f i c i e n t l y strong magnetic f i e l d , H C ( T ) . Considering the superconductor to be a p e r f e c t conductor ( i n f i n i t e c o n d u c t i v i t y ) , then i t f o l l o w s from Maxwell's equations that the magnetic f i e l d i n s i d e a super-conductor i s constant (B = 0 ) . One would then expect that i f a superconducting metal were placed i n a magnetic f i e l d and then cooled below i t s t r a n s i t i o n temperature, t h a t the magnetic f i e l d would be 'frozen i n 1 . I t was found, however, by Meissner i n 1933 that when the metal became superconducting the f i e l d i n the neighborhood of the specimen i n c r e a s e d , thus showing that the f i e l d i n s i d e the superconductor was pushed out, l e a v i n g no magnetic f i e l d i n s i d e . I t was also shown that when a superconductor i n i t i a l l y i n zero f i e l d i s placed i n a magnetic f i e l d , the f i e l d i n s i d e the superconductor remains zero. Hence one concludes that t h i s process i s a r e v e r s i b l e one and that the magnetic f i e l d may t h e r e f o r e be t r e a t e d as a thermodynamic v a r i a b l e . T r e a t i n g H as a thermodynamic v a r i a b l e of the s t a t e of the superconductor, one can e a s i l y d e rive from the f i r s t and second laws of thermodynamics that the d i f f e r e n c e between the s p e c i f i c heat i n the superconducting s t a t e and the spec-i f i c heat i n the normal s t a t e i s c - r = i A l ^5 u r v 9T J y i - 3 -The s p e c i f i c heat at constant volume i n the normal state at low temperatures can be represented by C n = « T +• UWfef . The T^ term, which i s h e r e a f t e r denoted by Cj,, i s a t t r i b u t e d to the l a t t i c e v i b r a t i o n s according to the w e l l known Debye theory of s p e c i f i c heats. The l i n e a r term has been shown (3) to r e s u l t from the a p p l i c a t i o n of Fermi-Dirac s t a t i s t i c s t o a f r e e e l e c t r o n gas, and i s hence u s u a l l y r e f e r r e d t o as the e l normal e l e c t r o n i c s p e c i f i c heat and i s designated by C n . I t i s g e n e r a l l y found t h a t at the t r a n s i t i o n temper-ature T c, where the change from normal to superconducting s t a t e s takes p l a c e , the s p e c i f i c heat as a f u n c t i o n of temperature i s discontinuous (see F i g u r e 1 ) . I t i s seen from Figure 1 that the s p e c i f i c heat i n the superconducting s t a t e , C s, i s greater than the l a t t i c e c o n t r i b u t i o n , C L , at a l l p o i n t s , but not always greater than the normal e l e c t r o n i c s p e c i f i c heat. Hence i t i s g e n e r a l l y assumed that the l a t t i c e v i b r a t i o n s s t i l l c o n t r i b u t e to the s p e c i f i c heat i n the superconducting s t a t e , whereas the normally conducting e l e c t r o n s do not c o n t r i b u t e . Since the d i f f e r e n c e C g -e l cannot be accounted f o r by C n , t h i s d i f f e r e n c e i s c a l l e d the superconducting s p e c i f i c heat, and i s designated by C®3-. e l el Experimental measurements of C g g e n e r a l l y i n d i c a t e that C g obeys a T^ law. Recent work on the s p e c i f i c heat of super-e l conducting Niobium (1) shows that i n t h i s case C g r i s e s much f a s t e r than any T^ dependence. We s h a l l l a t e r d i s c u s s the c o n t r i b u t i o n of var i o u s modes of v i b r a t i o n to the supercon-ducting s p e c i f i c heat, and w i l l give a p o s s i b l e reason f o r t h i s d e v i a t i o n from a T^ dependence. B The Present State of the Theory of Su p e r c o n d u c t i v i t y . There e x i s t s at present no a t o m i s t i c theory of su p e r c o n d u c t i v i t y . One can, however, s p e c i f y some of the co n d i t i o n s that a s u c c e s s f u l theory of s u p e r c o n d u c t i v i t y must f u l f i l l . I t must ( i ) agree w i t h the phenomenological theory (London's equ a t i o n s ) , ( i i ) give the t r a n s i t i o n temperature and the c r i t i c a l magnetic f i e l d , and ( i i i ) give the isotope e f f e c t (M^ T. = c o n s t . ) . Taking the usual view that s u p e r c o n d u c t i v i t y i s caused by the conduction e l e c t r o n s , and i s not a property of the l a t t i c e , one can say that a s u c c e s s f u l approach towards the problem has been made i f one can e x p l a i n the o r i g i n of a la r g e gap I n the energy l e v e l s of the conduction e l e c t r o n s near the surface of the Fermi sphere. I n f a c t , i f e l e c t r o n s were Bose p a r t i c l e s , one could construct a f a i r l y reasonable model of the superconductor, i n which the existence of the c r i t i c a l magnetic f i e l d i s explained i n terms of the Zeeman e f f e c t f i e l d which would tend to c l o s e the gap between the ground s t a t e and the f i r s t e x c i t e d s t a t e of the e l e c t r o n s i n a superconductor of reasonable dimensions. This e f f e c t , - 5 -comes out i n the r i g h t order of magnitude i n the approximation i n which the i n t e r a c t i o n between the e l e c t r o n s themselves and between the e l e c t r o n s and the l a t t i c e i s neglected. Such a model i s u n f o r t u n a t e l y of l i t t l e use i n the case of a Fermi gas of e l e c t r o n s , since the l e v e l d e n s i t y near the surface of the Fermi sphere i s so l a r g e that the magnetic f i e l d needed to make neighboring energy l e v e l s c o i n c i d e by Zeeman e f f e c t i s extremely small (10~^ o e r s t e d s ) . I t can th e r e f o r e be only by t a k i n g i n t o account the i n t e r a c t i o n between e l e c t r o n s them-s e l v e s , and between the e l e c t r o n s and l a t t i c e v i b r a t i o n s , that one can hope to o b t a i n a s u c c e s s f u l s t a r t towards an a t o m i s t i c theory of s u p e r c o n d u c t i v i t y . The very existence of the isotope e f f e c t i n d i c a t e s that the width of the energy gap at the surface of the Fermi sphere depends e s s e n t i a l l y on the i n t e r a c t i o n s between the e l e c t r o n s and the l a t t i c e v i b r a t i o n s . I t i s beyond the scope of the present work t o t r y to describe the mechanism which sets up the superconducting s t a t e , which i s presumably due to the i n t e r a c t i o n of e l e c t r o n s among themselves and w i t h the l a t t i c e i o n s . I t w i l l be assumed tha t t h i s i n t e r a c t i o n leads to a r i g i d ground s t a t e f o r the e l e c t r o n s , i . e . , most of the e l e c t r o n s must be i n t h e i r lowest s t a t e s , and the p r o b a b i l i t y that they can r i s e to s t a t e s of higher energy must be s m a l l . I n the f o l l o w i n g , the e l e c t r o n i c d e s c r i p t i o n of the superconductor w i l l be c h a r a c t e r i z e d by a system of continuous - 6 -e l e c t r i c charges which i n t e r a c t w i t h the electromagnetic f i e l d which they themselves produce. In other words, the e l e c t r o n -i c behaviour i s c h a r a c t e r i z e d by a system of simultaneous d i f f e r e n t i a l equations c o n s i s t i n g of Maxwell's equations and the c l a s s i c a l hydrodynamic equations f o r a continuous f l u i d . - 7 -I I THE LAGRANGIAN FORMULATION OF THE SUPERCONDUCTING STATE AND THE RESULTING EQUATIONS OF MOTION: We s h a l l t r y the f o l l o w i n g Lagrangian d e n s i t y : * where H i s the magnetic f i e l d , E the e l e c t r i c f i e l d , (f) and A the e l e c t r i c and magnetic v e c t o r p o t e n t i a l s r e s p e c t i v e l y , i the d i e l e c t r i c constant, and v (defined by equation [2] below) i s the v e l o c i t y of the f l u i d , (p i s the v e l o c i t y p o t e n t i a l , and j o i s the mass d e n s i t y of the f l u i d . The f i r s t two terms are the us u a l c l a s s i c a l hydrodynamic terms f o r a f r e e c o n t i n -uous f l u i d of d e n s i t y ^ ; by themselves, they give upon independent v a r i a t i o n w i t h respect to j o and
^ has been replaced by THJ? w n e r e jf3 Hf a r e the d e v i a t i o n s from the average d e n s i t y j o and the average p o t e n t i a l (jj> r e s p e c t i v e l y . This means tha t the background of p o s i t i v e l a t t i c e charge * This Lagrangian i s chosen f o r s i m p l i c i t y ; one can t r y other Lagrangians which are derived from analogy w i t h the c l a s s i c a l Schroedinger f i e l d , but i t turns out tha t t h i s more complicated Lagrangian leads to the same s o l u t i o n s i f one demands that London's f i r s t equation be s a t i s f i e d . The treatment of t h i s Lagrangian i s out-l i n e d i n Appendix A. - 8 -e f f e c t i v e l y cancels out the Coulomb i n t e r a c t i o n * of the f l u i d l e a v i n g only the i n t e r a c t i o n between the net e l e c t r i c charge densityJ/O, and the net e l e c t r i c f i e l d p o t e n t i a l (|) . j o and (|) are imagined to be w r i t t e n a s j o = i j 3 + j i and ^ = + I n a d d i t i o n we define the f o l l o w i n g : [2] = VOP -' Tne " [3] tf= CJUUXjtrT' [4] ff = -V$ - ±{T . The choice [2] has been made to s a t i s f y i d e n t i c a l l y London's equation [2a] OUUtlj\.J = - ± ht , where ytj= ?pj]L . while [3] and [4] are the usual expressions f o r the e l e c t r i c and magnetic f i e l d s i n terms of the p o t e n t i a l s (|) and A. We w i l l now assume that the d e n s i t y of the f l u i d i s almost constant and that v a r i a t i o n s i n d e n s i t y are s m a l l , i . e . , we assume that i n the expansion jo ~jo +jo , the d e v i a t i o n i s much smaller thanj> , and that JD i s constant. This assumption w i t h the previous assumption t h a t the p o s i t i v e l a t t i c e charge e f f e c t i v e l y cancels out the e l e c t r i c f i e l d due to the e l e c t r o n s , makes i t reasonable to assume that
, ^ , and A , we get the f o l l o w i n g equations of motion; [ 5 ] JO +J?^
" I < O T + R P where [14] K 4= - 11 -For b r e v i t y we put £>' =«~ s,e 7 ' * and < = : ^ ^ _ z c o ^ — ' Equations [5] and [6] y i e l d the f o l l o w i n g s o l u t i o n f o r
has a frequency, - 12 -u) , and a propagation v e c t o r , K, which are r e l a t e d by i A' has a frequency cJ,,and a propagation v e c t o r K-p r e l a t e d by oJ?~ = + c o -I t i s shown i n S e c t i o n I I I , t h a t , f o r thermodynamic consider-a t i o n s , i s n e g l i g i b l e compared w i t h <^J< C7"] We can determine A 2 and C 1 as f o l l o w s : take the divergence of [7" ] and demand that the Lorentz condition;*7./T=-J.