Surface Impedance Theory of Superconductors in Large Static Magnetic Fields Citation Sherman, Richard David (1971) Surface Impedance Theory of Superconductors in Large Static Magnetic Fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R9YN-V750. https://resolver.caltech.edu/CaltechTHESIS:03022016-130412002 Abstract A theory of electromagnetic absorption is presented to explain the changes in surface impedance for Pippard superconductors (ξ o ≫λ) due to large static magnetic fields. The static magnetic field penetrating the metal near the surface induces a momentum dependent potential in Bogolubov's equations. Such a potential modifies a quasiparticle's wavefunction and excitation spectrum. These changes affect the behavior of the surface impedance in a way that in large measure agrees with available observations. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Feynman, Richard Phillips Thesis Committee: Unknown, Unknown Defense Date: 28 January 1971 Funders: Funding Agency Grant Number NSF UNSPECIFIED NASA UNSPECIFIED Record Number: CaltechTHESIS:03022016-130412002 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03022016-130412002 DOI: 10.7907/R9YN-V750 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9599 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 03 Mar 2016 21:45 Last Modified: 26 Jun 2024 23:28 Thesis Files Preview PDF - Final Version See Usage Policy. 26MB Repository Staff Only: item control page

SURF ACE IMPEDANCE THEORY FOR SUPERCONDUCTORS IN LARGE STATIC MAGNETIC FIELDS Thesis by Richard David Sherman In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1971 ~ubmitted January 28 , 1971) ii ACKNOWLEDGMENTS I am very grateful for the assistance, insight, and patience of my thesis advisor, Professor Richard P. Feynman, throughout the course of this research. I wish to thank the National Science Foundation and the National Space and Aeronautics Administration for grants that have supported me during the first three years at the California Institute of Technology. iii ABSTRACT A theory of electromagnetic absorption is presented to explain the changes in surface impedance for Pippard superconductors ( s 0>>A.) . due to large static magnetic fields. The static magnetic field pene- trating the metal near the surface induces a momentum dependent potential in Bogolubov' s equations. Such a potential modifies a quasi- particle's wavefunction and excitation spectrum. These changes affect the behavior of the surface impedance in a way that in large m~asure agrees with available observations. lV TABLE CHAPTER OF CONTENTS TITLE INTRODUCTION 1 PAGE 1 BOGOLUBOV 1 S EQUATIONS _, 5 -> 2 j (A) RELATIONS . 9 3 SOLUTIONS OF BOGOLUBOV 1S EQUATIONS IN A LARGE STATIC MAGNETIC FIELD 21 4 FORMULAS FOR SURF ACE IMPEDANCE 39 5 THEORY OF ANOMALOUS SKI N EFFECT 56 6 SURFACE IMPEDANCE THEORY FOR SUPERCONDUCTORS WITH Hdc l 0 AND COMPARISON TO EXPERIMENTAL DATA 63 7 SURFACE EFFECTS 84 8 GAP SELF-CONSISTENCY 92 REFERENCES 98 -1- INTRODUCTION Over the past tw e nty years n1any experimentalists have been measuring the chang es in surface impedance of Pippard (or type I) superconductors due to a large static magne tic field near the surface. (See papers by Pippard(l), Spiewak( 2 ), Glosser( 3 ), Lewis( 4 ), Richards(S), Sharvin and Gantmakher( 6 ).) Although the trends in b e havior ar e now obvious, no theory h a s ye t appeared which can expla in the pat te r n s of the s e r e sult s. This thesis pres e nts j ust such a the o r y. The pri n cipl e f e a t ures obs e rv e d in th e experim ents are that the magnetic fi e ld increases the surfac e r e sistance a t "low t emperatures 11 (T ~. 5 T ) and 11 hit:;h temperatures'' (T ~. 9 T ) but can strongly c c decrease absorp t ion for "intermediate temperatures". At "low tern- peratures 11 the surface reactance is always increased while at "intermediate and high temperatures" the changes can be positive or negative depending on the fr e qu e ncy of the radiation. As usual for solid state problems, everything can change the sizes and proportions of these curves - the actual metal used, surface preparation, impurities in the metal, polarization and frequency of the incid e nt radiation, and orientation of the magnetic field with respect to the crystal axes . We propose that the above phenomena can be explained as follows: When a superconductor is placed in a magnetic field the ele- rnentary excitation properties are strongly modified for quasi-particles near the Fermi surface. In particular, this field creates a momentum -2- dependent potential for the quasi-particles which can look either like i a well or a barrier. The existen ce of the well gives rise to new dis- crete states whose excitation energies can be less than the superconducting gap energy. Such states are responsible for the increased surface resistance at 11 low T 11 • The potential barrier, on the other hand, has the effect of confining quasi-particles near the superconducting gap edge to regions of space where the magnetic field is weak or non-existent. This latter phenomenon accounts for the increased surface reactance at 11 low T 11 , the decreased surface resistance at 11 intermediate T 11 , and the complicated frequency dependence of surface reactance at 11 intermediate and high T 11 • To calculate surface impedance~ for this situation is very complicated for the following reasons: (1) The equations of super- conductivity (Bogolubov' s equations) are two coupled second order differential equations whose solutions must satisfy a self-consistency _, relation for the energy gap or order parameter function, 6 ( r ). Only one case has been solved - the infinite homogeneous metal which is not the situation here due to the static magnetic field. ~ tion between the current density, j (2) The rela~ and the electric field, E is non- local; current at one point inside the superconductor depends on the electric field strength in the surrounding region, which means that to find the field distribution requires solving a c 'omplicated integrodifferential equation. And the kernel of this integro-differential equa- tion must be found from (1) above. Once again this can be solved only approximately analytically for an infinite metal (Mattis and BardeenlJ).) -3- For general spatial variations as created by the static magnetic fields here, some other approximate methods must be used. Once the electric field distribution is known, it is trivial to calculate the surface irnpedance. The vehicle for discus sing this problem is the Bogolubov' canonical transformation fr om interacting electron states to noninteracting quasi-particle states. 1. This is discussed briefly in Chapter Using the Bogolubov transformation, it is possible to derive a general form of the current versus field relation for spatially-varying situations. This is done in Chapter 2. In Chapter 3, we solve the Bogolubov equations for the coefficients in the Bogolubov transformation for the situation of a superconductor in a large static magnetic field. In Chapter 4, we explain a method to calculate the surface impedance while maintaining self-consistency in the field distribution. In the .limit of no static magnetic fields, and vanishing superconducting energy gap, the normal metal case is obtained. We dis- cuss the theory of the anomalous skin effect in normal metals (Chapter 5) to under stand the physics of more complicated cases. In Chapter 6, we use all the previously developed machinery to find the predictions of the theory for HdcrfO and compare with available data. In Chapters 7 and 8, we discuss two left-over topics on the way to finding surface impedances. One is the effect of surfaces them- selves on the absorption processes in metals at low temperatures and -4- the other is the question of whether a large static magnetic field can decrease the superconducting order parameter or energy gap. For simplicity we have restricted ourselves to radiation fields normally incident upon a plane superconducting surface. -5- CHAPTER l BOGOLUBOV'S EQUATIONS Bardeen, Cooper, and Schrieffer (BCS)(S) have proposed a theory which explains the behavior of infinite homogeneous superconductorso Bogolubov generalized and extended their theory to c.over situations with spatial variations o by De Gennes( 9 ). The Bogolubov method is de scribed We briefly· review the results and some salient features below. The electrons in the metal are d e scribed by the Hamiltonian r _, t _, r i r-. _, _, -12 . _, } _, H= ~ \ d r w (r) ~ -? - :-1 \J - e A (r) ~ + U. a...; ~a t. ~m;_ __.:. impur1.t y (r)-E.cJ. '¥ a (r) 1 -y v BCS :0 o;a 1 , _, +_, t-> \ dr '¥' (r) '¥ (r) '¥ j Q Q I ( L 1) -> I Q ..... (r) '¥ (r) 0. where the '¥ ' s are anti-commuting electron field operators, U .t (;) impun y is the impurity scatter i ng potential, Ef is the Fermi en e rgy of the electron ensemble, V BCS is the electron-electron interaction potential, a, a 1 are spin indices, A(;_:') is the vector potential of any external fields, and hereafter take units where :h= c = 1. If only the first term wer.e pres e nt in this Hamiltonian, we would recover the theory of electrons in a normal metal interacting only via the Exclusion Principle and hence obeying Fermi-Dirac statistic So It is the addition of the second term in the Hamiltonian ( 1. 1) which accounts for superconductivity. This term takes into consider a- -6- tion an extra interaction - that between two electrons due to the exchange of virtual phonons. For two electrons at the Fermi surface whose energies differ by an amount less than the Debye energy of the phonon spectrum, there is a net attraction. And BCS have shown that this attraction leads to the formation of bound pairs of electrons in a metal. Since this interaction occurs over a small range of momenta, it must have a long spatial range. Hence BCS and Bogolubov approximate the interaction potential by a constant, V BCS' independent of range. Improvements can be made on these assumptions, but for most purposes this is adequate. The Hamiltonian (1. 1) is t oo difficult to handle and so is replaced by an approximate Hamiltonian, Heff' H f~= e I ~ _,{ dr t- t->(1 _, 2 _, ~ 1 ':i' (r) ~ [-i 7 - e A J + U: (r ) - Ef a. L a. L.m 1m µ ) _,-! ':i' ( r) j ' a. ( 1. 2) where 6(;), the superconducting energy gap or order parameter, is determined later by a self-consistency relation. Bogolubov diagonalized the Hamiltonian Heff by the transformation (1. 3a) ( 1. 3 b) -7- where the y's are anti-commuting annihilation and creation operators. Then H eff =E+6eyty g n,a. n no. no. ( 1. 4) which is just the Hamiltonian of a gas of non-interacting Fermions. The excitations created by y t are the quasi-particles. a. E g is the ground state energy of the system of quasi-particles. The Hamiltonians (1. 2) and (1. 4) are equivalent if the functions u, v satisfy the set of equations -> -l 1 . 2 1 > u (r)= ...,,- (-1 \/ - e A) + U. -Ef j ' u (r ) +ll(~)v (;:') n n LL.m imp. n n -4 t: -4 ( 1. Sa) (1. Sb) which are called Bogolubov's equations. The order parameter, 6(;!), is determined by requiring the free energy to be minimized when calculated with Heff· This leads to The u, v's must also satisfy a normalization condition ( 1. 7) To develop some feel for this method of description, consider the application of Bogolubov's· method to a free electron gas. u, v' s are just plane waves and the y +, y n n The are identical to the a +,a p p -8- operators which create and annihilate electrons in plane wave states. I .lr Yptt , p > pf' acts on t h e groun d state o f t 11e system , an e l ectron is - • -+ --. ip r put above the Fermi surface in the state up(r) = e • If Ypt ' p<pf, acts on the ground state of the system , an electron is removed from the Fermi sea in t he state v (;_:')=eip• r which is equivalent to 11 adding p a hole 11 in the Fermi sea. Hence, we can call the u 1 s wave functions of electron type quasi-particles and the v 1 s wave functions of hole type quasi-particles. Note that, if!:.= 0, then eqns. ( 1. Sa, b) are just the Schroedinger equations for free electrons and free holes, respectively. u(x) is the electron wavefunction and v(x) is the hole wavefunction. When u d:. 0, v = 0 and vice versa so that eqn. (l. 6) is consistent with!:.= 0. The new feature of Bogolubov ' s equations which explains superconductivity is the term with!;.. Electron and hole excitations are no longer independent of each other and the degree of influence depends on 6, constrained by eqn. ( 1. 6). - 9- CHAPTER 2 j(A) RELATION Any discussion of electrodynamics inside a metal requires knowledge of a current versus field relation. In this chapter we derive a particular form for the current as a function of the vector potential useful for surface impedance calculations. Quite generally, the current density operator in secondquantized form evolving in time according to the Heisenber g picture is T a. op (x)= zie m where ('V - • -'V _,, )..... x x x 1 _, x -x -yta. (xi ) ~ a. (x) - me2 A. (x) ~a.t (x) ':; a. (x) ( 2. 1) = (;, t), vector position and time a. = t, l the spin index denotes an operator in the Heisenberg representation. 1/ t (x) is an electron field c r ea ti on operator satisfying Fermion anti-commutation rules. 1 See Abrikosov, Gorkov, and Dzyaloshinski (AGD) ( O) for a derivation. Eqn. (2. 1) is to be evaluated up to terms first order in A. The final result will then yield the current density at some point in space and time due to an arbitrary A field distribution surrounding this point. Assume A (x) = A(;, t) to have been turned on adiabatically from some time t in the distant past, t -• -oo. Now switch over to the interaction representation which is the same as the Heisenberg picture fort .... -co . .Any operator 0 is related between these two -iO- pictures by 0 Heisenberg = S -1 0. · . S(t) interaction (t) (2. 2) where S(t) is the S matrix ·. t S(t) = Tt ei ~ - co ] op (y) "A (y) d y 4 ( 2. 3) where Tt is the time ordering symbol. Expanding S(t) to terms first order in A yields t s(t) ';! l+i 'j -co r 4 (2. 4) op (y) • A(y) d y And hence the following is true to first order in A(x). - '1' a. (x)~ '1' a. (x)-i ~t r· Ljop(y) 'l' a.t (x )-'¥·a.(x)Jop (y) 1 .J ·A(y) d 4 y (2. 5) -oo 1 a.t (x1 )=,...., 'l' a.t(x1 )-i ~t [....j op (y)'i'ta.(x' )-'1' ta. (x1 ) .....j op (y) -,...l •A-· (y) d 4 y ( 2. 6) -co where }' (y) is just the zero order approximation op t t :6 (2. 7) 'f R ( yl ) 'J! R ( y ) ~= i !-' !-' Putting eqns. (2. 5), (2. 6), and (2. 7) into eqn . (2. 1) keeping terms _, first order in A (x) and summing over spins yields 2 '::; ie Jop(x) = 4m2 ( 'Y_, - 'V_, x x1 ) ..... _, x 1 - •x (' t j -oo _, A(y) • ( 'V_,- 'V_,, y . yi ) ...., _, yl ->y -11- t e:? _, -rn :0 '±' (x) '±' (x) A (x) a a a (2. 8) To obtain from eqn. (2. 8) a physical ~urrent, it is necessary to average -:-+ Jop over a grand canonical ensemble. r(x) = <1op (x) > T = :0m e 8 (0- m) r( IT <m I· op x) l m > (2. 9) where 0 is defined by (2. 10) E: m is simply the excitation energy of quasi-particles in the Bogolubov equations. That is, the diagonalized Hamiltonian of the system is t HM= :0 € y y + ground state energy to be ignored n,a n na. na (2. l la) and (2.llb) HM is the complete Hamiltonian of the sy,stem at t _, - oo when the radiation field is off. 7 ie2 ( ) J (x) = 4m 2 'V_, -\J _, _, _, 1 x x x 1 ->x \ t .) - oo _, ( A(y) • . 'V _, - 'V _,1 y y ) _,i -• y ->y 2 J?f3 <'i';(y' )'i'f3(y)/a(x1 )'i'a.(x)-'i':(x 7 )'l'a.(x)'i';(yl )'l'f3(y)>Td 4 y-~ A(x) (2. 12) -12- where N = ~ < 'i' t (x) 'f (x)>T is the density of electrons in the system. a a a N is written without its spatial dependence since for perturbations of interest here, N is a constant. Impurities, magnetic and electric fields, etc. applied to a metal only change the electron distribution at the Fermi surface, but this is only a very tiny fraction of the total nwn ber of electrons present. Considerable simplification in taking the remaining thermal average occurs if we transform from electron field operators to quasi-particle operators via the Bogolubov canonical variable transformation, which in the Sehr oedinger representation is · !' (x) _, = :BIr y. u.(x) .. + S a J ._ Ja J a 1 y. t 1 Ja vJ. >:<-1 (x) _\ (2. 13) a f. a where S aI 1 +l ifa1 = t ={ 'l.' t 6~) is obtained from eqn. (2. 13) by taking the Hermitian adjoint. a In the interaction representation the transformation is then '±' (x) = ~ a J [y. (t)u.(x) + S y. ~ (t) Ja J a i Ja (2. 14) -> where x = (x, t) -iI\.,i t iHMt y. JCl Writing (t) =e y. e JCl J (x) in terms of y s yields sums over many terms each of which is a product of four y' s. 1 But only some of these terms are non-vanishing upon takin g thermal averages. Since the thermal average is a weighted sum of diagonal matrix-elements, only terms of the form t t yi yi yj yj (or permutations in ordering) can contribute. < A typical term from >T in eqn. (2. 12) is u .• !- t s 1./T I where fT(t:i) = <yi yi >T = 1 /Le +1 J is just the Fermi distribu- tion function for non-interacting Fermions, and the exponential in t im e r e s ult s fr om <. . 1'j(t) >T = iHMt <. . e e 0 -iHMt >T Yj e = -it: ·t J <. . . Yj > T • For our adiabatically turned on .A (y) = .A (Y) e i(w-io)t A field of frequency w o . . . o+ Y (2. 15) which allows us in each term to trivially perform the time integration \ t J x dt e -oo Y iE;, (t m -t x 1 iwtx ) i(w-io)t y e .,...e y = .....,1=------:-0-=-E .em +,.u-io {2. 16) -14- _ .... __. iwt x j (x) ......j (x)e So all terms rn Proceeding in the above manner for all terms in eqn. (2. 12), doing the trivial sums over 6 functions, then puts the r(~) into the following form: cur rent density -. Ne 2 -~ _.. j(x) = j 1 (x) - - - A(x) --+-> -Jo (2. 17) m where u /i;)u,,;"(;;' )um (j)u [IJ 1 ) - [ v;Fiv t (;; 1 )um lYlu ;'(Y 1 ) e: .e,..:.e:m +rJ.)-i6 _., .,IC: - t ;>'.:; -> -+ -+ ..,r,. _,. -t -J + um(x)u~(x 1 )v~(y)v_e(y 1 ) -v;:'(x)vm(xl )v~:'(y)v_e(x') J -(e: ,e-e:m)+w-io + [1-fT(e: _e)-fT(e:m) r I· u m J (~)v (;;_ 1 )v >:<(y)u ':'(y' ) + u (~)v £ t m ,R, m (~ 1 )v ':'(y"")u >:< (--y' ) m ,R, e +e £ m +w-io 1.-.- ' -+ ,... ~ -+ ~ ..,,, --+ -> -> -+ + _v_m_':'_(x_.)_u_.t_,,(_x_'_)_um_(_y_)v_.t_(y_'_)_+_v_£_-._(x_)_u_m_':'_(x_-i_)_u_m_(y_)_v_;,_(_Y_1_)_ J} - (e ;,+em )+w-io (2. 18) -15- Eqn. (2. 18) is quite general and can be applied to numerous physical situations. If the u, v functions appropriate for an infinite pure superconductor (no static magnetic or electric fields present) _,_, are substituted in eqn. (2. 18) and the Fourier transform of j(x) is taken, we obtain the same expression as Abrikosov et.al. ( calculated the surface impedance for superconductors. be the case, of course. 13 )who This must We will refer quite frequently to this paper since as all perturbations are shut off, all results must tend to their case. We proceed now by considering the geometry in Figure 2. 1. A metal surface lies in the y-z plane. semi-infinite in the +x-direction. It is infinite in y and z, and Note that there are two sets of fields to consider - those from the static magnetic fields to be applied to the surface and those from the high frequency weak radia~ tion impingent upon the surface. FIGURE 2. 1 GEOMETRY OF FIELDS -16- .... ...... The vector potential, Ade' due to the Hdc defines the y-axis . The _, vector potential, A , due to the radiation is linearly polarized in ac the y-z plane and makes an angle 8 with respect to the y-axis. - A 11 Each field has associated with it a current ...... j in the metal and these screening 11 currents are parallel to their re spec ti ve -> -> to be noted here that we aretreating A , and Ad ac c A It is field. separately. The -. present chapter considers A ac as a small perturbation on the system and the current versus field relations are j ac as a function of A ac Ade is not small and its effects are included in the u, v functions directly through Bogolubov's equations which are solved in Chapter 3 . Getting back to further processing of the j ...... versus A relation, we write _, A _,. ac ...... where e y,z -> (x) = (e -? y cos8 + e z sin8) A ac (2. 19) (x) are unit vectors in the y or z direction and x refers to the coordinate on the x-axis. (There will no longer be any reference to x = (~, t) as a four-vector.) , Note that A ac (x) falls off only with x; we have an infinite plane wave impinging upon the surface. It will be shown later that u, v must be of the form u(~) = u(x)e i(K y+K z) y z (2. 20a} v(~) = v(x)e i(K y+K z) Y z (2. 2 Ob} If eqns. (2. 20a, b} and (2. 19) are inserted in eqn. (2. 18), the differentiations A•\J performed, limits taken, integrations performed over the pure exponentials (giving 21T o -functions), and some simple -17- rearrangements made depending on£, m being durnmy indices, we can obtain the following result: j 1 (x) = (• 00 J Q (x, x' )Aac(x' )dx' (2. 21a) 1 0 2e 2 m2 .R, .R, (21T) 2 ~ (K cos8+K sin8) 2 x,, m y z n)-fT(E: ) ] 6(K.R, - Km) 6 (Ke- Km) { [fT(e x, m y y z z [ u (x)u ';'(x}+v (x)v ';'(x) 1 [ u m X, 2 m - e [ m m (x1 )u ':'(xl )+v 2 m (xi )v >!'(xr ) 2 J -e -u;ti6 2 [u,e(x)vm(x)-um(x)v.!(x)] [u;'(x1 )v,:(x' )-um*(x' )v;(x' ): em +E: 1,+w-i6 + [u,;'(x)v;(x)-u; (x)v,.,;'(x)] [um(x' )v ,e(x')-u .!(xl )vm(x' )] J} E:m +e .R,-(J,)ti6 (2.2lb) J1 (x) = ({; cose+; sin8) jl (x) y z (2. 2lc) The last equation says that the current flows in the same direction as the A field 1 s linear polarization. Eqn. (2. 21) for the j versus A amenable to physical interpretation. are represented: The A relation is now in a form ' Two distinct physical processes field causes thermally excited quasi- -18- particles to make transitions frorn one state to another state. A The field causes the creation or destruction of a pair of excitations; this pair is made up of a quasi-particle and an anti-quasi-particle. (These processes are directly analogous to electron-position theory in quantum electrodynamics; the rest mass energy mc 2 plays a role very similar to the _superconducting energy gap, 6.) Consider first the scattering process. Since the incident photon has momentum only normal to the surface and none in the plane of the surface, only the x-component of a quasi-particle 1 s momentum can change in a transition. remain the same. 6(K J., -Km) y y So they and z- component This is expressed through the 6-functions , 6(K zJ., -Km). z If th e transition is from a state of lower energy to one of higher energy, there must be some non-zero probability for the lower state to contain a quasi-particle and some nonzero probability for the upper state to have an empty slot available. The net current, though, is proportional to the number of upward transitions minus the number of downward transitions. This fact is expressed through the Fermi factor difference fT(e 1,)-fT(em). (Recall that if fT(e) is the probability for e to be occupied, then 1-fT(e) is the probability for e to be empty.) Now suppose the incident photon annihilates at xi. There is an amplitude for it to scatter a quasi- particle from state 1, to state m. u to be particle-like and amplitude But since a state has amplitude v to be anti-particle-like, we must consider the quantum mechanical interference and add the amplitudes of the processes: Particle to particle um(x 1 )uz(x:1 ) plus -19- anti-particle to anti- par tic le v m (x~ )v z(xi ) transitions. Similarly, a current found at x g e nerated by th e transition from state .e, to state m could be due either to a particle-particle transition or an anti-particle v .e, (x)vr::(x). anti-particle transition . The factor ~'< Hence the factor u i, (x)u~ (x) + 1/ [E:m-8 .e,-wti6] splits into a real part P(l/[em-8 .e,-W]) and an imaginary part -irr6 (em-8 .e,-W) via a familiar id e n t ity. It will b e s een later that the im ag inary part is rel at ed to po we r absorb e d by t he m e tal i n whic h c ase ene rgy mu st b e cons e rved in th e s cattering proce s s . T h is i s expres se d via the 6-function part. T he real part w h ich do e s not cons e rv e ener g y thus considers only virtual processes and these are related to the diamagn.etic or screening properties of a material. We discuss this later after developing more formalism. Next consider the pair processes. momentum must be conserved. Once again transverse If initially there is no pair, transverse momentum is zero; after a pair is created, there is total y-momentum K .e,+K m for the excitations put in states £, m. y y factors 6(K.e,+Km ), y y 6(K.e,+Km). z z Thus there are the In order to send a pair of excitations into states .e,, m, these states must be empty or if a pair is to be annihilated from .e,, m, there must be a pair sitting in .e,, m. the factor [1-fT(E: ;,)-fT(E:m) ]. tions, as before. Hence This includes up minus down transi- The energy denominators express conservation of energy for absorption processes and yield virtual processes for nonenergy conserving processes. Note that since E:m, 8 .e,• and ware positive, always, the term involving 6(em +e .e,+t..I)) never contributes -20- to the absorption. In fact, this term more nearly corresponds to the destruction of a pair creating a photon and hence depleting current in the material. A photon annihilating ,a t xl can put a particle in m and an anti-particle in J, or a particle in J, and an anti-particle in m. The amplitude for this is um(xi) v£(x 1 ) - uf,(x' )vm(x 1 ). And the explanation of the other matrix elements is obvious. The factor (K: c OS e + KZJ, sine) 2 is the coupling between the photon 1 s polarization and an absorbing quasi-particle's transverse motion. A quasi-particle has the strongest interaction when it is moving exactly parallel or anti-parallel to the direction of linear polarization. -21- CHAPTER 3 SOLUTIONS OF BOGOLUBOV 1 S EQUATIONS IN A LARGE STATIC MAGNETIC FIELD Further evaluation of the j ac versus A ac relation in Chapter 2 requires explicit knowledge of the u, v functions satisfying Bogolubov's equations with large static magnetic fields, Hd ductor 1 s critical field. since ,..., H , the superconc Once again we consider the geometry of ~ Figure 2. 1. c ~ If Hdc is along the z-axis, then Ade is along the y-axis H= \l x A or for our geometry Hd c = ~ z oAY(x) /ax. We are to solve Bogolubov's equations (3. la) (3. lb) (3. le) for the geometry of Figure 2. 1. Outside the metal (x < 0) all the fields are uniform; inside the fields are approximately e -x/ 5 or e -x//... where 6 is called the ac penetration depth and /... is the static field penetration depth. Although these penetration depths have been calculated only for weak fields it is an experimental fact that the penetration depth is independent of field intensity for field strengths right up to the critical field, H • c -22- x ::::>: 0 Thus for (3. 2) ( 3. 3) .... 'V •A= 0 or "transverse gauge" is derivable from the which in the vector potential (3. 4a) (3. 4b) using H = 'V x A . Since the metal is infinite in y constant for translational symmetry. and z, l u \, l v \, \ 6. \ must be Then we try solutions of the form -= u(r) -= v(r) u(x)e i(K y+K z) Y z (3. 5a) v(x)e i(K y+K z) Y z (3. 5b) ··. which by eqn. (3 . le) is consistent with a gap function 6. varying only . with x. (3.5c) Using this converts eqns. (3 . la, b, c) to 2 fle n - St + 2m l d 2 dx - V(x) 1..J u n (x) - 6.(x)vn (x) = 0. (3. 6a) V(x) l..J v n (x) - 6.(x)un (x) - 0 (3.6b) . , . -·· (3. 6c) -23- K K/ 2 ~t = 2~ where - Zill eHA. -x/A. V(x) = -Ke m K K y z = K cos8 p = K sin9 y p Kf is the Fermi momentum. K p is the transverse momentum. a Ef = Kf /2m so that the model here seems to require a spherical Fermi surface and thus ignores crystal anisotropy. Actually this is not really right; the situation is better as far as surface impedance calculations are concerned. We will elaborate on this point when we discuss the surface impedance calculation for the anomalous skin effect. 2 We have ignored the Ade K y ;;::, 10 -4 Kf and the K y term since eAdc <<Ky for all of interest in all subsequent work is Ky=Kf.· The quantity designated by V(x) is so labelled because it acts like a potential barrier or well depending on the polarity of K • y Since A. is pure real for static fields, V(x) is always pure real. It would now be very helpful if we somehow knew in advance the spatial dependence of the gap function, 6(x). For an infinite superconductor considered by BCS, 6(x) was a constant making solution of Bogolubov 1 s equations trivial. But in the presence of a large static magnetic field we can no longer be so sure that 6(x) -24- is still constant. Bogolubov 1 s equations are sufficiently complicated that we cannot solve for u and v while simultaneously satisfying the selfconsistency relation on 6, in the presence of large static magnetic fields. Therefore, we shall make a guess at what the final self- consistent 6 might be, solve Bogolubov 1 s equations for the u, v functions, and check through the self-consistency relation just how good was our guess. (This latter point will be considered in Chapter 8 . ) We shall say that 6 is independent of Hdc and spatially constant. Even with the assumption of a constant gap, Bogolubov 1 s equations are still unmanageable. The trouble lies with V(x). Any step function approximation to the exponential in V(x) will lead to a · nd set of coupled 2 - order differential equations with constant coefficients and that can be handled reasonably. We shall content ourselves, first, with a single step and thus make the replacement v x;,;; A.e V(x) - V (x) = { e e 0 (3. 7). x > A.e V , the strength of the effective potential, and A. , the extent e e of the effective potential, are the only parameters in this theory. shall pick A. e = 2A., 0 :s: V reasonable to us. e We :s: 6, V ,..., HK ; this is not unique but seems e y If the results are suspect, a many- step approxima- ·. tion to the exponential can be tried; it will not be so sensitive to parameter choices. In order to solve ordinary constant coefficient differential -25- equations, we must form solutions in each region and match values and appropriate derivatives across the boundaries, at the x = 0 surface, and at the x =L > > > A. surface. Take fir st the problem at the x = 0 surface. At the surface of the metal, the current normal to the surface mu s t vanish. But j x ,...., uou/ ox + vov/o x so either u and v both vanish at x = 0, or au/ox and av/ox vanish there, or both do. To pick the proper set, consider the electron density at the surface, N. 2 + Iv I . Since the electrons are principally confined to within an angstrom or so of the surface, we imagine our system confined by a large potential barrier at x = 0, whence at x = O. N ... 0 ::::;:> u, v-+O We thus consider the boundary conditions n, v= 0 while ou/ox, av/ax i o. It is to be noted that imposing these boundary conditions corresponds to a situation known as specular reflection. We discuss the physical significance of this in Chapter 7. Before solving the differential eqns. (3. 6) it is most convenient to renormalize the variables to dimensionless form as follows: 8 ... 8 / 6 -+ 8 St; ... st/fl ... ~ v ... v;6 _, v Note that the dimensionless variables are denoted by the same letter as before. -26- Then eqns. (3. 6) [ E: - E; ·t + Ef 6 read in this normalized form ~ - Ve(x) 1u(x) - v(x) = 0 dx2 ,... (3. 8a) V (x)l v(x) - u(x) = 0 e _! (3.8b) where we have also dropped the subscript labelling the states tern porarily. Denote the solutions of eqns. (3. 8a, b) in the surface region where the effective potential is non-zero generally by the subscript l and the solutions in the metal's interior by the subscript 2.. in each region are of the form eiqx. Solutions Then straightforward substitu- tion in eqns. (3. 8a, b) yields the following general solutions: u1 (x ) = A 1 (e ir+x_ e-irt..x) + C 1 (e ir_x -e -ir_x) v.1 {x) = A 1 (e-s -V) (e 'r e (3. 9a) x ' ir x ir x + -e -ir·+) + C 1 (i::+s r -V) (e - -e- - ) e ir· x (3. 9b) (3. 9c} (3. 9d} where r ± = { E~[-C:'t ±~'::>r (3. 10) f 1 P± = {~ [ - st .± sP J }z- (3. 11) -271 12 2 Sr = [ ( e - Ve ) - 1 j (3. 12) (3. 13) A 1 , C 1 , A 2 , B 2 , C 2 , and D 2 are constants to be fixed later by imposing boundary conditions on u, v and normalization ~ L>>>)...r 0 2 l u(x) l + \ v(x) j 2 l dx = 1 (3. 14) ...l I- We have picked the solutions so that the boundary co_ndition at x = 0 is automatically satisfied. Matching the u, v solutions and their derivatives at x yields four constraints on the coefficients A 1 , C 1 , ip+A. Aze e -ip+)... +B 2 e e +C 2 e ip )... - e +D 2 e = A. e Az, B 2 , C 2 , D 2 • -i p A.e - (3. 15a) (3.15b) (3. 15c) -28- A 2 (e:-s )p+e p . ip+\ e -B 2 (e:-s )p+e p -ip,T Ae +C 2 (e:+s )p e p - -D 2 (e:+~ )e ip - Ae -ip A - e (3. 15d) p Using the states given by {3. 9) in the normalization equation (3. 14) yields the following: ~Ae[!u1 (x)j +jvi(x)j ]dx+ . ~ 2 Thus o 2 00 2 Ae 2 [Ju2(x)j +jv2 (x)j ]dx= 1 a l = IAi l [ 1+{e: - s· r -v e ) (e: - sr>:< - v e ) J 2 +lei I c1+(e:+s r -v) e (e:+s':<_v r e >J ~:<)Ae ~:');\e i(r _ -r -i(r _ -r . {-e_____ -_e____ _ -r >:< r e { i(r +r >:<))._ -i(r +r >:<))._ - _ - _ e-e_ _ - _ - _ e } _e_ i(r +-r _ >:<)\ e -i(r +-r _ ':' )Ae -e r -r ':' + - r +r >:< e i(r + +r ~:<)A.e -i(r + +r ~:<)A.e -e } r +r >:< + - -29- e i ( r _;'.' - r - P1.e - i ( r ,;, - r ) A. -e + - e { ;i:i; r . + -r - 2 +I Az I [ 1 +(e - sp ) (8 _ ~·p':') J e s '•') J e +A2B':'[1-l-( 2 e - ;::':> ) ( 8 _ ,, p p I i(p + -p':'}L + -e "( p -p':'}A. l + + e P+ -p_;'.' i(p++p~')L i(p +p':')A. + + -e e P+ +p:' i(p + -p _':') L ' i(p -p "')).. e + - -e p -p ,., e + ,. . I i(p++p~:')L e i{p +p':')A. + - -e e p + +p ~:< e -i{p + +p ~')L -i{p ++p+':')~ -e -p -p :i~' + + ,., p~"-p+ e -i(p+ +p ~:')L +B 2 C 2>:'[ 1+(e-s ) (e+s '~) J p p -e -i(p '+p >:')). . .., -p -p ... + - + - e -30- * [ l+(e-s p ) (e+ s p*) J .., +B 2 D 2 p '• -p - +c 2. A::i>:, [ l+(e+s ) (e-s >:') J p e p i(p - -p,:')L + p e + -e '( p -p >!< )A. + e l >:< _-P+ i(p_+p~')L -e p + i(p +p>:')' + 11.e >:< - P+ 2 +! C2 l [l+(e+s p ) (e+s p>:') J e i(p - -p ->!<)L -e >!< P -p - e '( p -p >!< )A. e l - i(p _ +p ~')L i(p + >!')' - P 11.e -e p + >:< - P_ e -i(p - +p ;)L -e - P -p - e -i(p - +p ~:')L +D . >!< 2 C2 [ l+(e+s p -i(p +p ':')A. - + e >~<: + -i(p +p ':')A. e >!< - P _-P_ ) (e+s ) J p e -e i(p,:'-p )L l"( p >!< -p )A. · -e - e ~l<: P _ -p _ (3. 16) -31- In general, not all of the constants Az, B 2 , C 2 , D 2 can be simultaneously non-zero. This ar i ses upon considering the u, v solutions (3. 9) for x-+ co. If p + and/ or p _ has a non-zero imaginary part, the coefficient of the appropriate term must be set to zero for the u, v solutions to be bounded as x-+ oo. Regarding e (the excitation energy), transverse momentum), and V e st (a measure of (the effective potential strength) as the independent state naming variables, we consider the consequences of the following possibilities for 1 ..!.. p±={~f [-~t±(e2-l)2J}2 ( l) 0 ~ e < 1, any st and E: :?:: 1, - st + (e l 2 - 1) 2 < 0 For this case both p + and p _ have imaginary parts so that B2 = D2 = 0 necessarily. But then eqns. (3. 15) are inconsistent unless the determinant of the coefficient vanishes. If st and Ve are fixed, there exist solutions only for certain discrete values of e . given by the vanishing of the determinant here. Since p± are complex, these states are localized near the ip±x surface as determined by e in u, v. Generally the distances 0 here are ,..., SOOOA into the sample. BCS used infinite metals, so, of course, they could not obtain any solutions where p + or p _ had any imaginary part. Furthermore, these surface states exist for excitation energies less than the gap width, 6, a fact which will have important consequences for absorption at 1 'low temperatures" (kT < < 6). -32- Eqns. (3. 15) are sufficiently complicated that results can only be obtained numerically. We have examined, therefore, many cases numerically using a computer and find a few general features to be described below. Firstly, surface states exist only for V ; i.e., V e e < 0 and only for must appear to be a potential well and the possible states lie above the bottom of the well. is true irrespective of how V e and /,. e The preceding are chosen. Secondly, surface states exist only for -20 ~st .s;; 1 which depends on "-e· The small values of St indicate a particular direction of quasi-particle travel to be discussed later. The spectrum used in further calculations is shown in Figures 3. 1 below. -33- e/6. v e = -. 26. v e = -. 56 v e = - /: , FIGURE 3. 1 Typical Surface State Spectrum A. 0 e = 2A.= lOOOA On each contour line p± vary continuously with St• -34- It might be thought that these surface states depend critically on the surface at x = 0 being perfectly flat and smooth. However, as long as the surface irregularities are small compared to a penetration depth of the Hdc' this is not so. For then one could make wave packets of sinusoids which would match the boundary conditions, out of sinusoids whose momenta were all peaked sharply about the correct value for a flat surface. In such a case, all the surface state energies would still be nearly the same as before. Thus 0 small scale (~ lOA) irregularities are no problem. -st± (s 2 - i) > o, e : : : i (2) In this realm both P+ and p A.z, B 2 , C2 , D2 are pure real and all the constants can be non-zero, which leaves a dilemma. are more unknown constants than constraining equations. approaches are possible to resolve the problem. system in a large box of length L x x There Two We can put the and requ:l~e u, v to vanish = L x just as we imposed for u, v at x = 0 surface. at This yields a situation like the prior case - solutions exist only if the determinant of the coefficients vanishes. However, . since Lx states are very closely spaced in energy and for L tinuum. is very large these . x .... oo form a con- All this is well defined but extremely difficult to handle numerically. Clearly the results should be essentially independent of the size of the box for L x >>>A. and yet in solving for the u, v functions and the locations of the states in energy slight changes in L x radically affect the numbers involved through fast oscillating terms ip±Lx "' e -35- Fortunately a nother approach is possible that is more. easily implement e d. We irnagine a scattering process. a quasi-particle incident upon the If e ipx represents x = 0 surface from deep inside the metal in the state with x-component momentum then e -ipx p, is the reflected quasi-particle; the transverse momentum is conserved and the momenttllD normal to the surface reverses sign upon collision. Thus it is natural to consider two cases shown below in Figures 3 . Za, b . P+ P+ / ~ p 0 4_ I •t.. ~ ~ ~ x ol e ' P+ !A. I e I I I ~x 'p FIGURE 3. 2 a, b SCHEMATIC OF SCATTERING STATES A quasi-particle can be incident upon the surface in either state P+ or p _. By a change of variables to spherical momentum coordinates it can be shown that P+ states lie above the Fermi momentum while p_ states lie below the Fermi momentum. For each case, there is some amplitude for the quasi-particle to be reflected in its original state with only the direction altered and some amplitude to find the quasiparticle in the other momentum state also with altered direction. This is a new feature of a superconductor when Hdc f. O. In BCS theory -36- where Hdc = 0 a quasi-particle is always reflected in the same state as it wa~ incident; no mixing of p + and p st' 8, and v e states occurs. Regarding as the independent variables of a state, there are two degenerate solutions which we will refer to as "P+ incident" and "p _ incident" continuum states. In the former D 2 = 0 and in the latter The above picture does not depend upon the type of particle being scattered. They can be either electron-like or hole-like excitations; indeed, our argument applies equally well to both u(x) and v(x) solutions. Holes or anti-particle quasi-particles exist both above and below the Fermi momentum and electrons or particle quasi-particle do so too. This latter fact holds in superconductors ±ip x whether Hdc ·is zero or not. For normal metals, though, u- e which says that electron excitations exist only above the Fermi momentum while hole excitations exist only below the Fermi momentum. For these plane wave like scattering states, the normalization condition (3. 16) is particularly simple; in the limit of L diagonal terms in region 2 contribute substantially. x - oo only Thus for P+ incident states (3.17a) and for p incident states . a l~l (3. 17b) -37- The sum over states is also simple 2: +,:B - ~ro L :B- dp± (3. 18) -oo which will later be transformed to a more useful independent variable. 1 (3) 1 i:::o::1, -st-(i:: -1> 2 <o, -st+(i:: -1) 2 >o 2 2 ±ip+x In this realm P+ is pure real so e are perfectly accept-ip x is complex and hence e - must be dis- able solutions while p carded, so D 2 = O. BCS also could not have considered this case since it depends on a surface being present and a non-zero V . e This case is very similar to the P+ scattering state previously considered except that for p + incident, the p very strongly localized near the surface. reflected solution is So all can be applied except that the normalization condition simplifies even more than before to 2 + I B:a I ) [ 1+(E: - sp) 2 (3. 19) We can summarize graphically the above results as to which states exist where in Figure 3. 4 below. -38- e/6 . A P+r al p _c 1mplex surface states p± both real V e <O FIGURE 3. 4 TYPES OF STATES FOR VARIOUS St AND E -39- CHAPTER 4 FORMULAS FOR SURFACE IMPEDANCE In order to calculate surface impedances, it is necessary to solve Maxwe 11' s equations for the field distributions and then find the complex power dissipated inside the metal. ..... gauge, = 0, Maxwell's equations read 'V • A ... 'V2 A Using the transverse 0 .A - -= at 2 ..... ..... 4'ITj (A) ( 4. 1) 2 ..... B _,. = 'V xA ( 4. 2) ..... E oA = - at ( 4. 3) in the vector potential formalism. the metal and vacuum. These equations apply both to Since all fields vary sinusoidally, A,.., eiwt, eqn. ( 4. 1) can be tr an sf armed to \J 2 A + (.l A = - 4'ITJ (A) (4. 4) andeqn. (4 .3 )to ..... E ..... iwA = ( 4. 5) ..... ..... Solutions of eqn. (4. 4) require the j versus A relation. vacuum, wave, -:> J -e In = 0 everywhere and the A-· field is just a plane traveling iKx In real metals at room temperature, Ohm's law holds and eqn. (4.4) is easily solved. ..... j = .... CJ E; But we are concerned .......... with temperatures near absolute zero in metals and here the j (A) relation is very non-local; the field in a large region determines the -40- current at some point with this region. We found in Chapter Z that which means that eqn. (4. 4) is an integro differential equation which is difficult to solve generally. One case can be solved in _, _, closed form and it is an important result. If the kernel Q (r, r 1 ) -+ _, is a function only of the difference (r - r 1 ) , then eqn. (4. 4) is soluble in closed form. This case corresponds very nearly to the anomalous skin effect in normal metals and superconductors if no magnetic fields are present. We shall consider this in greater detail below. One further simplification should be mentioned here. The displacement current is negligible in a metal for frequencies of w2A < < \/ 2 A so that eqns~ (4. 4) and (4. 6) can be interest since combined to read (4. 7~} or for our planar geometry 4"1!' ~ 00 Q(x, x 1 } A{x 1) dx 1 (4. ?b) 0 Given that .solutions of eqn. (4. 7) are available, the surface impedance, . Z , z = 41T can then be found from ..... E (0) \ _, _, .) j dr The 4ir is conventional. = -i 4irw I A (0) ~~I (Jx - ( 4. 8) 0 means evaluated at x = o+, just inside 0 I I -41- the surface. Although this approach is correct, it is not feasible to calculate the kernel Q (x, x') for our problem. So we develop some other relations useful here. z ~1 ox 41i (.t) :l,c = Iax oA Ia o i = R -iX 0 ( 4. 9) follows trivially from eqn. (4. 8). An integration by parts yields (4. 10) which by eqn . (4. 7b} is also related to Q. ~ oo >'.c A 0 aaA - - dx axa ·1. · . >::; = (4. 11) Q(x, x') A(x') A (x) dxdx' Thus >'C '{ Z'= \ 00 41T J >'< Q A ' A d x d x' - 0 J\IX) 0 I 'OA 0x 12 dx } (4. 12) a form which allows the spatial integrations to be done immediately and Z to be found directly once we know A(x}. We shall now take the following new point of view toward calculating surface impedances. distribution by an exponential e the static field distribution. o, We approximate the A(x) field -x/o one parameter, o, just as for the ac penetration depth is a .... -42- function of Hdc and w. From this assumed form we would change eqn. (4. 12} to read z>'.c = i 4irw l6 la {41T ~ 00 e -x/ 0 >!< e -x' / 0 Qdx dx 1 - ~co e -x/0 >!<e -x/ 0 dx} 0 0 ( 4. 13) which can now reasonably be done numerically using a computer. But this z':< must be consistent with the z>!< calculated from eqn. >'< z · = -i4irw6 (4. 14) Thus it is possible to find the self-consistent 6 whence we trivially know the surface impedance. From eqn. (4. 14} we note that surface reactance measurements are in fact a measure of penetration depths x = 4irw Re(o) (4.15} and from eqn. (4. 9c} we see that surface resistance measurements determine the amount of power dissipated by the metal. R = 4rr(.ulm (6) (4. 16) For superconductors with so that Im(6) < < Re(6). T <::;; • 9T c and w < 2 6 ,R < < X As a consequence it is really necessary to be concerned with self-consistency only for the surface reactance or penetration depth. Once the correct self-consistent value of Re(6) is known we can calculate directly the surface resistance from eqn. (4. 13} using for 6 the Re(6}. Actually the case for our problem is -43- even better as far as surface resistance is concerned. The changes in surface resistance due to static magnetic fields are so large (RH de 10 /RH = de 0 > 1) and the changes in surface reactance sufficiently small (-lO?h ) that using Re(6) for Hdc= 0 is adequate to obtain answers within 2096 and of the right sign. This will become more obvious after further formal developments. The only point here is to say that maintaining self-consistency is less of a problem than might be expected initially. We now continue to evaluate eqn. ( 4. 13 ). split Q -Ne 2 ; Im• In Chapter 2, we Q., and a diamagnetic term, into a paramagnetic term, For the latter, the integrations are immediately done and one finds that -i4irw -if ) ( 4. 1 7) Hence, the diamagnetic term contributes only to the surface reactance. (The two last terms are both important.) The paramagnetic contribution is proportional to ~ 00 dxdx 1 e -x/6 >:< -x!/6 Q · ( ') e • 1 x, x Referring to eqn. (2. 2lb) we see 0 that this integral is composed of the sum of two quantum mechanical matrix elements magnitude squared. -44- a \':io M 1.m = J (4. 18a) 0 Mb= dx 1.m b b Note that Mm.£, = -M ,e,m (4. 18 b) Mn a is the matrix element for processes XJI'D scattering a quasi-particle from one state, ,e,, to another state, m. M )(;m n b is the matrix element for destruction of a quasi-particle. and its anti-quasi-particle from states b>:< Mf;m is the same as .t and m. before except referring to creation of a pair. Below we calculate M 1 : = ~ 00 * [u1 (x)u2 >!<(x)+v1 (x)v;;/!<(x)]e-x/o 'dx 0 , for the most general u, v's. :>'< 0 + C 21 A,..,-, r;7;J . [l+(E: 1 +i:- ) (E: 2 '=' P1 -s P:a *)] I (p 2 -1 ' p +:a ) -45- .,,, . + B2l. B 22 • [ 1+(8 l - S ) (8 2 - S >!<) ] 12 (-P+1 ' -P+) P1 P2 ,,, + D21 n22 ···c1+(c:1 +s · ) (82+s · ':')J 12(-p , -p 2) Pi P2 -1 - ,., + B21 n22 ·· [1+(81- s · ) (E:2+s · ':')J 12 (-p+l, -p_2) P1 P2 ;::(, [l+(e 1 +s + D21 Bza . + A21 B~ ;:!< P1 ) (82-s >!<) P2 J 12 (-p-1' -P+2) [ l + ( t: l - S ) (E: 2 - S >!<)] 12 (P+ 1 ' -P+ 4) Pi P2 ,., + AZJ. D22. [1+(81 -S + B21 A z;z ':' [ 1+(81 - s Pi Pi ) (82+S ) (82 - s P2 P2 ':')] 12 (P+1 ' -p ) -2 >!< ) J 12 (-p+i, P+2) ,,. + B21 C22 [ l + (8 l - SPi ) (8 2 + SPz >!<) J 12 (-p +1, p -2) ,,, ,., ':') ] 12 (p _1 -p +z) + C21 B 22 ' [ l +(E: 1 +S ) (e 2 - s P1 P2 ~- + C21 n22 ···c1+(s1 + s * P1 ) (c:2+s P2 ':')J 12 (p_l , -p-2) . + Dzi Azz . [ 1 + ( 8 l + S ) (e 2 - S ':') ] 12 (-P-1, P+2) Pi P2 (4.19) where (4. 20a) -46- i(n_ -P2 >:'+ ~J\ 12 ( P1, P2) = i _e_ _ _ _ _ __ (4. 20b) for the most general u, v's. s ) - (8 ~ + sr J, )J 13 ( r - n, r -m) + cl ~ c [ (8 + "' 1m . m rm ;v . ;v + A 1 ;vn C 1m [ ( E: m +E;· rm ) - {€ ;vn - ~ r J, ) ] 13 ( r ..1.., n , r _m ) ;v + A:aJ,Czm[(e:m+i;P) - (e: ;,-sp )] 14 (P+;,' P_m) }; + C ..,nA """' [(e: 2rn m -S pm J, ) - (e: n+S "' P;, )] I4 (p n, P+· ) -;v m + D.2;v~D~ [(e +s ) - (e ;vn+S p}; )] I4 (-p - ;vn,-p -m ) ~u, m pm + B~nD [(e +s ) - (e n-S )] I4 (-p+ n• -p ) "'"' zm rn pm "' p}; "' -m ·: -47- + D nB 2,, + A . nD 2,, 2m 2m [(e m. -S pm ) - (e ,,n+S p J, )] l4 (-p_ ,,n, -p+m) [ ( e n - S ) - (e + S )] l4 ( P+ ,,n, - P -m ) ,, pf, m pm + B 2 £A2rn[(e ,e-SP,e) - (em-spm)] 14 (-P+.e' P+m} + B 2 .e,C2m[(e .e,-SPJ,) - (em +s·Pm)] 14 (-P+t• P_m) + C 2,,n B zm [ ( e ,,n+SP .e ) - ( 8 m - Spm )J I4 ( P - ,,n, - P+ m ) + D . nA 2,, 2m [ ( e n+ S ) - (e - S }] I4 ( - P -,,n, P+m ) ,, pf. m pm + D 2 ,,n C :zm [ ( 8 ,,n+ Sp I4 {Pi , P2 } = i J, ) - (e m + Sp )] I 4 { - P - n, P -m ) m )'J {4. 21.) (4.22b} (4. 22a) -48- In terms of these matrix elements >'< Z' is just I I f,,-m (K yJ, cos 8 + K zJ, sin 8 )2 z >:< = i321T 2 w o 2 '8 IM£~ l a 2 [ i [ 1-f (e )-f (e )] I M b i i +. 2 T t T m J,m 1 e +e +w-io + e +e -f.1..i'ho m ,Q, m £ 4 -i 41TW l0 I o + o,:c 2 4e Kf 3 . ( J}.·'· (4.23} 31Tm The u, v amplitudes are invariant under the change K ..... -K z z contained in ~ causes the field-particle coupling cross term K £ K J,cos8sin8 y z contribution to vanish in eqn. (4. 23). ,Q, This just expresses an obvious symmetry - the surface impedance ± e about the direction is the same for field linear polarization of of Ade. . (k ;, y Because of this cos 8 +K .e, z sin8) a _. (K ;,a ;,2 y ) cos 2 8 + (K z 2 ) sin 8 and we can derive the complete angular dependence of the · surface impedance by calculating Z for just two angles, 0 and 1T/2 ; i.e., -49- Z(8) = Z(8 = 0) cos 2 8 + Z (8 = 1T/z) sin 2 8 (4. 24) It should be noted that this relation is independent of the long series of approximations made above. We could have derived this long ago. ~ The sum over states, £,m , in eqn. (4. 23) is actually more than a double sum; this is so since a surface state requires three num.bers to fix it - st' Ve' and the particular branch number, while a continuum state requires four numbers In any case, it's v e' e:, p± incident. always possible to pull out from the sum dK r, y dK ,e, z x, 2rr in terms of St and cp (Ve = V work. st, 0 and express these integrals · cosCO) which are useful in subsequent This is trivially done using the definitions of St and cp in Chapter 3 and the result is ~ J,, m where ~ ... on the right hand side now and hereafter refers to the J,, m remaining variables necessary to specify the state considered. mly surface states are considered, ~1T dcp ... ~/; dcp 0 are no surface states for V e = 0. If since there 2 Putting all ·this together we find Z >l< -50- 1 4e 2 K 3 ( 5.1m f -i47TW - (4~25) For processes involving surface states, it is most useful to switch from the integration variable c.p to the variable V • e The substitution made is v 2 0 1 \ dV {cos 2 9 e V a J_ y e (V a -V a )"2 o o o e + sini38 (V 2 -V 2 ) ±} o e (4. 26) Finally, for continuum states, the sum over states is just ~ +, - \00 dp± j -oo 2 1T . assuming unit volume. ~ means the sum over P+ +, - and p _ incident states. However, the more useful variable is € • Thus ~ +,- (4. 27) -51- can be used in eqn. (4. 25) whenever considering continuum states. Assembling all these bits and pieces, we can now write the surface impedance contributions for the various processes in a form that can be directly evaluated numerically on a computer. Let z 5 be the surface impedance of the superconductor and ZN denote surface impedances of normal metals. Write the total z form and ti's and c 's are reinserted so that 1 7.i.3c3 The familiar identity P as All variables are renormalized to dimensionless ZS1 + ZSa + . . . where 5 5.62 = 1 ~+ s: lv x Z may be found in ohms. 10-S ohms/(.A~ eV) 2 P ( _!.) -iiro(x) is used in the reduction, X denotes principle part of the integration. We merely state the formulas, here, and discuss the consequences later. (1) Transitions between surface states { cos 2 e (4.28a) -52- (~ _ We shall need later 6 W1) /;;, so that ;· . V a {cos ~ e (-T--) a a a - c:o J[ (i-) -( :e) Jz (4.28b) which is further discussed in Chapter 6. We calculate the matrix element M f,; , for integrals (4. 28) as follows: Solve the set of simultaneous equations (3. 15) with A 1 m, C 1 m, Azrn, C 2 m with the same where st and Ve coefficients A 1 , C 1 , etc. (3. 16). B M;,; is f, and m refer to two surface states but different energies, e ;,• em. must satisfy the normalization condition then trivially found from formula (4. 19) n = D 2;v, ~ = O. :a"", m m These with -53- (2) Transitions between surface states and continuum states Zs>:< = i ~ ~a 2 TI uC l i12c3 ~ u E6 f Ef:a branches m { cos~e v 2 2 (~ 2 y2 a -2 (:o) [(:o~ -(:e) ] y2.!. . a r( 0) - (6e) 1j +sin el T 2 } ( 4. 2 9) Since the energy of continuum states is always higher than the energy of surface states, the o - function term contributing to the resistive part contains only the Lj I) branches +, - j, ~- • • terms. The m reactance, though, picks up contributions from both continuum and surface states as final states in the interaction and keeps all sums. We determined M.t:::i for integrals (4. 'Z9) as follows: If .t is a surface state, then solve eqns. (3. 15) with B 2 A 1 J,,, C 1 J,,, C 2 J,, normalized according to eqn. (3. 16). states, m, = D = 0 for 2 The continuum are found by solving eqns. (3. 15) first with D 2 = 0 (for -54- P+ incident scattering states ) and then with B 2 = 0 {for p _ incident scattering states) normalized now according to eqns. (3. 17). This determines either the non-zero set A m, C 1 m, ~· B 2 m, C:zm or 1 A m, C 1 im' A' C· D. 2m' zrn' :zm respectively and hence M nma is again ~ fixed by expression (4. 19). (3) Transitions between continuum states >;c 8 ~ '11"2 Zc:: .= i 10 l a ~ood(:~ ~ l +, - . f;, m ;, o(p±/Kf) ( 1+ a(·;,; t:. ) i st,t ) (cos 2 8cos 2 cp + sin a f [fT ("6t) - fT(";') JJM t! J"[P(em ~ i W )- 8sin~cp ) irr6 ("~ - :i -~ )J T-6-7:: (4. 30) a M ;,~,in eqn. (4. 30) is calculated from continuum scattering states solutions to eqns. (3. 15), normalized according to eqns. (3. 17) ·as described earlier. However, not all the non-zero terms are kept -55- in the M a ..em expression ( 4. 19); we ignore all terms which vary i(p n+P like e ;;:, ~'< m ')A. e -56- CHAPTER 5 THEORY OF ANOMALOUS SKIN EFFECT In order to understand the modifications that will be produced by applying a static magnetic field to a superconductor, it is necessary to know what happens when Hdc = O. The theory of surface impedances in such superconductors is, in turn, based upon the theory of surface impedances in normal metals at low temperatures which can be found masquerading under the name anomalous skin effect. (The under- standing of these phenomena also re pre sen ts the direction of historical progress in discussing the more complicated situations.) All this can be accommodated in the previously developed formalism. By shutting off Hdc' we obtain the surface impedance of ordinary superconductors and by letting the gap, metal absorptior:i properties. 6, vanish, we can find the normal Since all this work is thoroughly described in the literature, we merely review the basic features necessary for our problem. Begin by considering a normal metal at room temperature. 0 ·The mean free path of conduction electrons (~lOA) is much less 0 than the penetration depth of the electromagnetic field (-5000A ). For such situations, the fields felt by an electron over its free motion between collisions can be approximated as a constant. · Then the net current density, -+ ... j (r ), generated by a collection of electrons is pro- portional to the field strength at that point, which just expresses Ohm's ..... _. -.-. familiar law, j (r) =crE (r) • <J is called the conductivity and depends -57- on the frequency of electron scatterings of the given temperature. This simple point relation means that Maxwell's equations inside the me~al have a trivial solution, A ,..,.. e -x/o :Now let's see what happens as the temperature is lowered to cryogenic levels. The electron mean free path is generally much 0 larger than typical field penetration depths (""' 1 OOOA ) so that an electron between collisions will usually see a field which is anything but constant. cr, -+ _. - -- It makes no sense to talk about the Olunic conductivity, relating j 00 \ -+ and E, A at a point. -Jo -. -+ A non-local relation holds, _. j (r) =.) . Q(r, r 1 ) A(r 1 ) dr', and solution of Maxwell's equations is 0 more difficult than before but still possible. The solution to this proble:n'}· has been carried out by Reuter and Sondheimer( l l), · Charnbers(lZ)• Mattis and Bardeen( 7 ), and Abrikosov, Gorkov, and 3 Khalatnikov AGK(l ) The AGK paper is most closely related to concepts used here. All the solutions, however, for the surface impedance were made for an infinite, homogeneous, isotropic metal. In view ,of the long mean free paths, this makes such results very suspect. We have investigated the question 'and resolved the matter in Chapter 7. Using an infinite metal is acceptable and we continue the discussion here on that basis. -- For an infinite medium Q(r, r ') is a function only of the d1fference coordinates r;_;, I; AGK( 13 ) have calculated the Fourier transform of the kernel and find that for penetration depths much less than the mean free path {extreme anomalous limit) -58- Q (k)= ( 5. 1) l I kl One can then use the Fourier transformed Maxwell's equations to solve for the -+ A field distribution and obtain the surface impedance of a normal metal RN = ZN= RN 3 The AGK(l } result is .a e~f) s (5. 2a} Uk} (5. 2b) 4ir ~ 32 .... , XN = + iXN • 41T --r- ( .. , To find RN in ohms, use (5. 2c} with w m radians/ sec. b . A-1 an d K f in The formulas for surface impedance, here, are provided mainly for reference. Our main considerati'on is the physical mechanism by which absorption takes place. In the anomalous skin effect realm, a new feature enters - - only electrons traveling very nearly parallel to the surface of the metal play an active role in 'interacting with the electromagnetic field. Electrons that collide with the surface at a large angle with respect to the surface run in and out of the field too quickly to absorb any sizeable amount of -59- energy. We are, of course, only considering electrons at the Fermi surface, as usual. Let us refer our discussion to the usual geometry in this work. .... and the A field is polarized along the x-component of momentum K y The surface is in the y - z y-axis . plane The electron has p and transverse momentum K • = K p coscp, K z = K p sincp . p So the effective electrons here have :f". · very small p compared'to Kf and the maximum angle for nonparallelism with th~ y - z plane is of the order P/Kf < < < 1 . This gets smaller as Kf increases which in turn means that the effective number of electrons interacting with the field is decreased. Almost everything said about the normal metal at low temperatures applies directly to the superconductor with no static magnetic fields present. This first became apparent about twenty years ago when Pippard slightly modified Chambers' results on normal metals to explain the magnetic properties of superconductors. By this we do not mean that the physics of both situations is the same; rather that the formalism adapted for anomalous skin effect is applicable also to superconductors . Whereas a normal metal allows a static magnetic field to penetrate completely, a 0 superconductor confines a static magnetic field to within about lOOOA of the metal 1 s surface. However, both normal and superconducting metals use only the electrons in a thin slice of Fermi surface to screen the ac fields from the interior region. do this also for de fields.) (Superconductors -60- Thus in superconductors one can calculate surface impedances using infinite medium kernels, just for the same reasons as Q, in normal metals. It turns out that the kernel, for superconductors has Q, for its Fourier transformed behavior the same dependence of k, So in this way one can see the relation of normal and superconducting metals through the kernel, having separated out the common spatial behavior. the metal. after In the normal metal, Im Q is different from zero and Re Q is zero. Im Q vanishes for Q, Also w = 0 which means that static fields permeate In a normal metal Q is temperature independent for the range of anomalous skin effect. For a superconductor Q has a much more complicated behavior . Most results are only available numerically. w < 26, Re Qi- 0, Im Q = O. When T = 0, It is Re Q f; 0 which gives rise to the Meissner effect - the expulsion of static fields from the bulk interior of a superconductor. Let's point out an important difference between normal and superconducting metals here. Suppose T = 0 and W"'"' !:::., falls somewhere in the microwave region. a 'frequency which generally Then the field in both cases is confined to the surface region since about the same order of magnitude. is . Im Q ~ 0, Re Q IQ I in both cases is But in the normal metal it = 0 while in the superconductor Re Q 'f 0, Im Q = O. So whereas both metal surfaces are highly reflecting the mechanism for each is quite different. It has been pointed out earlier that Im Q 'I 0 corresponds to power absorption while Re Q 'f 0 is related -61- to reactive or non-dissipative processes. Thus the normal metal expels the field by just being a very good absorber while the superconductor expels its ac field without any power absorption -at all. This can all be understood in terms of the excitation spectrum for quasi-particles in normal and superconducting metals:.. In normal metals an arbitrarily small amount of energy can create excitations absorbing power out of an electromagnetic field. For superconductors, though, there is a gap in the excitation spectrum of magnitude !::,. The minimum energy necessary for creating a quasi-particle is 6.. And at absolute zero where no thermal excita- tions are possible, the only way to absorb energy is through the creation of a quasi-particle anti-quasi-particle pair which costs an amount of energy at least 2/::,. Since we therefore, no power can be absorbed. are considering w,...., 6, However, virtual pair creation processes do occur and it is these which give Re Q -:f. 0. As London has suggested, the superconductor wavefunction has a certain rigidity which is not destroyed by these perturbations. . a geriera . l ru l e about the Feynman ( 14) says that there is reflectivity of materials. Whenever a material gets to be a very good absorber at some frequency, it also becomes highly reflecting. Our exa.tnple above for a normal metal in the anomalous skin effect is a perfect case. The larger Im Q, the higher the reflectivity. But, the rule is inappropriate for superconductors. There is no absorption at all and yet the surface is very highly reflecting! -62- The most extreme difference between normal and superconducting metals occurs at T = O. As T is raised the super- conductor tends to behave progressively more like a normal metal until at some critical temperature, turns in!t o a normal metal. T , the superconductor c This happens for two reasons: higher temperatures thermally excite more quasi-particles aero ss the gap into the continuum of ~tates and higher temperatures cause the gap function to decrease. The thermally ex.cited quasi-particles behave much like the quasi-particles in the normal metal. It is for this reason that a superconducti ve state used to be considered a mixture of "normal and superconducting fluids". We should mention that the gap function actually changes very little except in the region T ;;:i .95T. c In terms of the kernel, Q, the trans- formation from superconducting to normal state occurs as fallows: Re Q ... O as T rises while Im Q, which was zero for a superconductor, tends toward Im Q for a normal metal as Since the gap holds constant until T runs from zero to T • c T """'. 95 T , c Re Q also stays strong until this point, too, dropping rapidly to zero only for T fil • 95 T . c Which is to say that virtual pair processes are the dominant mode of flux expulsion throughout the widest range of tern pe rature. All the above and several other interesting cases can be found discussed by AGK(l 3 ). -63- CHAPTER 6 SURFACE IMPEDANCE THEORY FOR SUPERCONDUCTORS WITH Hdc -f 0 AND COMPARISON TO EXPERIMENTAL DATA We now have developed enough formalism to discuss the implications of this surface impedance theory for Hdc -f O. Since everything must be calculated numerically on a computer ranging from single to quadruple integrals, our results are only for a few cases. These are done in realms where the individual processes are separated as clearly as possible from one another. The experimental data are also rather spotty so a complete comparison . of theory and experiment is impossible at this time. Wherever numerical results of calculations are quoted, the following data have been assumed for the model which corresponds roughly to tin and tantalum; 0 -1 1 A ·'· Ef = 5 eV tiEf= lo- 4 ~t::.~150gc 0 o(T = 0) = SOOA 0 A.e = lOOOA Some important experiments have been performed by (1) Pippard(l) in tin at 9. 4gc (2) Spiewak{Z) in tin at 1 gc -64- (3) Richards(S) in tin at 3 gc (4) Lewis( 4 ) in tin at 24 gc (5) 91osser(3) in tantalum at 9 gc (6) 6 Sharvin and Gantmakher( ) in tin at 2 me. The number measured by the experimentalist is usually the ..... z 5 (H) == R 5 (H) + iX 5 (H), ·surface impedance for some size of Hdc' divided by the normal state surface resistance, RN. RN is temperature independent and for the effects considered here also independent of Hdc" (RN varies with Hdc only when in the many kilogauss range.) As we shall soon see, normalizing the superconductor surface resistance changes to RN can be quite misleading in many instances since large changes divided by an even larger number make the effects appear small. (It is necessary to measure this _s urface resistance ratio because the power losses in the cavity walls usually·-far exceed the sample's power losses of interest. ) In Chapter 5 'it was pointed out that the effective quasiparticles had to be traveling very nearly parallel to the surface of the metal. That angle was said to be about P/Kf° In actually per- forming the integrations for surface impedances, p was not found to be a convenient independent variable. 2 A better measure of the ::i angle for our purposes is St== Kp/lm - Kfjzm, the ·transverse momentum. In terms of St' then, the angle between quasi-particle trajectories and the surface is about ·-(-suEf )t. /J ''' We shall refer -65- occasionally to this angle. The reader should have some feeling for the sizes of the effects involved here. Surface reactances are to be interpreted 0 in terms of penetration depths which are about 1 OOOA. Surface resistances are related to power dissipation in am etal and hence to reflecti'vity of the metal surfaces. surface resistances ,..,,, 10 -4 We are concerned with ohms or less. These are extremely "shiny" surfaces reflecting 99. 99+% of the incident radiation. So while we shall be talking about '''big and small" changes in the surface resistance, the actual changes in reflectivity are very very tiny, indeed! We shall begin by considering the superconducting surface 0 < T ~.ST , and for electroc Rs;. magnetic radiation frequencies, r,JJ.v. It,. If Hdc = 0, Rs (and /RN) resistance at "low temperatures", is negligible. _ This is well understood since for such tempera- tures there are negligibly few quasi-particles in the continuum to absorb power. If Hd c ~ H c ,..,.500 guass (H c is the critical magnetic field where the superconductor reverts to a normal metal}, however, there is a sizeable power absorption, depends on every conceivable factor . involved he_re. Thus Hdc has changed the surface resistance by several orders of magnitude. w ·h at could be the reason? Within our model, there is only one possible candidate for an explanation - -the surface states. As described in Chapter 3, the Hdc field sets up a potential well within which there are discrete energy levels that range from e: """'0 toe: ~/;,. -66- They exist only for I st I < 206 which according to earlier formulas indicate quasi-particle trajectories even more parallel to the surface than the effective electrons in the anomalous skin effect. 0 lOb. corresponds to an angle ,..., 2 . Since there exist states whose excitation energies are much less than the gap, even "low temperatures" yield sufficient the rm al populations so that a photon can kick a quasi-particle from a lower branch into another higher energy branch conserving transverse momentum (initial and final states both have same st and v e ). . Using the surface state spectrum described in Chapter 3, we have calculated the surface resistance at Hlow temperatures" for various T, t.JJ, Hdc' and 8. (Recall that e is the orientation of the radiation field polarization with the y-axis.) to T <.ST c By confining ourselves and w ~. ZD., we can be sure that the surface resistance is totally dominated by transitions between surface states. We must do this calculation numerically, which according to eqn. (4. 28a) amounts to just a single integral over either v e or st as one chooses. However, doing the integral over the o(em -E: J,- w) introduces either i/l a (em -e ;)/ave I or i/l a(em -e .t)/ast I according to which variable is used for integration. And both of these factors are singular for a typical energy spectrum. Although the integrals are well-behaved, it is too difficult to handle such a situation numerically on a computer. A more reasonable approach is to integrate eqn. (4. 28a) over a narrow frequency range from ((J 1 to w2 , where -67- t.Ll 2 -lt.l 1 < < < 6, and obtain the surface resistance per unit frequency range from a double integral over Ve and St° We assume that the surface resistance varies continuously and smoothly with frequency so that for (JJ.i - w1 ""'. 016, we are really calculating the surface resistance at each point in frequency, except for Wva ~minimum energy separation of the nearest pair of surface states ~. 016. (Obviously, there is no absorption when (tJ is less than the minimum energy separation of two surface states, so for some w there is a discontinuous change in the surface resistance to aero.) The formula then used for calculating the surface resistance is given by eqn. (4. 28b). Although there is much extra work in doing _a double integral, one is rewarded by obtaining the surface resistance over a wide frequency range, not just at one frequency, in doing the double integral once! Some results of this calculation are summarized graphically in Figures 6. la, b, c below. -68- 1. T = . 1 Tc 1. w = • 1 !::, 0.5 Figure 6. la 0 •2 Hdc/Hc •4 T = •4 T c w = . 1 ti 0.5 Figure 6. lb _ ___..____.._...,._______, Hd /H 0 .2 .4 .6 .8 c 1.0 8=0° 1.5 c T = . 5 Tc 1. 0 w= 0 3 ti Figure 6. le _....____....__--11._ 0 .2 •4 .6 __.__._~ •8 1. 0 H /H de SURFACE RESISTANCE DUE TO SURFACE STATES AT "LOW T" c -69- :It should be mentioned first that the requirement of self- . •- consistency on 6, the penetration depth, is least stringent for this case. The Im 6 <<<Re 6 here. Re 6 changes by about 10% as Hdc BL varies over its range while Rs/RN runs from zero to about So we can use, to very high accuracy, 6 = Re 6 only . And using 6 ~ for Hdc = 0 merely makes an overall consistent error in Rs/RN of about 20%. The results in Figures 6. 1 use 6 at Hdc = 0 and T = O. (It is well known that penetration depths are essentially independent of T up to T "'•ST • ) Such approximations are adequate for the c qualitative behavior desired. Figure 6. la displays results which can be mo st closely related' to experimental data available. RS I RN at 8 = 0 w :::::. 076. 0 and 90 0 3 Glos ser( ) has measured as a function of Hdc for T ;;;:.:: • 2 Tc' His results are consistent with ours in that Rs/RN - 0 than increases monotonically with Hdc' Rs I RN is larger for a = 0 for 90°, and Rs/RN is ab<;>ut a maximum of 1. S?,h. Glosser 1 s work( 3 ) has the only data taken on the angular (8) dependence and then .only for Hd c ::::! • 9 H c , near the critical field. In Figure 6. 1 b we have obtained some very interesting results regarding the 8 dependence · on Rs/RN for (.I):;::; • 16, T ~ • 4 Tc as a function of Hdc for which there is no experimental check, as yet. Rs/RN is larger for 9 = 0° than for 8 = 90°. For Hdc ~He But for Hdc ·=:::: • SHc• the surface resistance is larger for 8= 90° than for 8 = 0°! And we believe this conclusion to be generally true if surface states exist with the only proviso being that the superconductor show -70- little or no anisotropy in the gap, 6, for Hdc = 0 cases - a basic assumption of this model. This effect mostly depends on the delicate interplay of occupation probability, density of states in K •K y space and the coupling of field to quasi-particle motion z but not critically upon matrix elements, or spectrum shape! It is governed principally by the terms . l + sin 9 {V a - V 2 )-z } 2 o 2 The "8 = 0 2 o piece" is ±and the "8· ,= 90 o piece "is Ve :3 ~ a · /V {V - V ) o e o e 1 . l 2 - (V - V ) 2. which are plotted in Figure 6. 2. o · e FIGURE 6. 2 Although the "9 = 0 piece" has a singularity at V0 , there is no problem since the area for finite width V is finite. e curves. We can interpret the If the field is polarized along the y-axis ( e = o0 ), the -71- quasi-particles traveling most nearly parallel to the y-axis absorb the mo st energy; those traveling along the z-axis absorb none and conversely for polarization along the z-axis (9 = 90°). V0 ""' /:':,.. Suppose The states for e ,.... 0 have a strong occupation probability ' and couple strongly to the e = o0 case but weakly to the e = 90° case, since IV I ,.... /:':,. means K ,..,, Kf and K ,...,; O. e y z The. states higher in energy have such low occupation probability that their effects barely modify the results, 8 = o0 polarization dominates e = 90° Now probability occupation is low but all transitions have about the same value so in each case (0°, 90°) we must consider the whole range of IVe I· But in general there are more states coupling to the 9 = 90° case (small I Ve I 's) than to the e =o 0 case (large I Ve 1 • s ). Hence e = 90° dominates over e = o0 polarization. Next we consider the superconducting surface resistance at "intermediate T", • 5 T ~ T ~ • 8 T , and frequencies w .:;;; • l /:':,.. . c c All the investigators have observed that Hd resistance to decrease anomalously. c ,...., H c causes the surface 1 Pippard ( ) first noticed this in 1949 and it has remained a mystery since that time. R (Hdc 5 = 0) ]/RN ~ - 1%. This It was pointed out earlier that normalizing surface impedance to RN c,ould be misleading and the particular case is the worst. At "intermediate T", R (Hdc = 0) ~ • l RN so that 5 the surface impedance variations are more substantial than one might expect from looking at the data. It would be more useful if -72- experiment~lists plotted [R 5 (HdcfO)...R. 8 (Hdc = O)]/R 8 (Hdc = 0). Typically this latter quantity would be 50% to 1000%. a rather large change in behavior. At "intermediate T 11 and w ~. 16, there seem to be three mechanisms at work to consider, but two are unimportant. First, there are transitions between surface states which increase the surface resistance, but RS1 /RN is only about 1%. Then there are transitions from surface states to continuum states, but once again for sma.ll frequencies, w< < 6 is progressively smaller as 1 RSa /RN.-.-+ 1%. w is lesser. This contribution Finally there are transitions of thermally excited quasi-particles between states in the continuum. (We have had to estimate the sizes of these three different mechanisms since the Fermi factor is su,fficiently .l arge for T f:l • 6 Tc to guarantee occupancy of all types of states.) Hdc This latter case would exist if = 0 whereas the previous two cases depend upon Hdc f. O. 1 It is the onset of continuum occupancy by thermaliy excited quasiparticles at T =. 6 Tc which starts the process whereby a superconductor gradually turns into a normal metal at T = Tc· And our calculations indicate that Hd '¢ 0 has such a marked effect as to . c account for the suppressed surface resistance solely through modifications of the continuum transitions mechanism. RS 3 /RN > > Rs~/RN so that we may ignore the other two processes involving surface states. (Throughout this Chapter we are picking w, T, etc. so as to isolate the various processes from one another.) So let us RS./R 3 .consider the expression for N; we do the integral over e 2 -73- to drop the o -function in eqn. (4. 30) and are left with a non-trivial triple integral that must be evaluated numerically. number of features to be considered here. the question of which There are a Once again there is o to use for the self-consistency relation. R~ o ;;:> i'o Imo and Re o changes by about 10% due to Hdc" RS changes by about 100% using Re o for Hdc However, so that we are once again justified in =0 to deduce the proper qualitative behavior. The reader is reminded that this triple integral is no more than the anomalous skin effect in superconductors, and normal metals if 6 = O. As mentioned in Chapter 5, only quasi-particles traveling nearly parallel to the surface are effective in absorption. Hence only "small St II are important which for OUr Ca Se turns OUt to be -10 6~~';:>t ~26., 3 , 2 1 - (st ~. 26 since for st ::<:: 0 the minimum € = (6+st }z and e: ;;:, 36 can be ignored due to the Fermi factors.) - That only "small st" are involved is guaranteed by the convergence of our integral being very rapid outside this range. For continuum states, there are many terms in the matrix element, Mn a; each term has an exponential factor which either varies as JC/ID , The former class oscillate rapidly as st runs over its range while the latter class have slowly varying phases. Therefore we keep only these terms and drop the other which make little contribution. (It was not necessary to do this for surface state problems since ' the range of st integration is two orders of magnitude less.) -74- ; The triple integral is so expensive to do that only a few important cases were done; viz. • 8 Tc' Hdc = He~. w = O. 16, T = . 6 Tc' . 7 Tc' The result is in excellent agreement with the 3 data of Glosser( ) in that the surface resistance falls monotonically with T over this range and is within 30% of the measured values. Furthermore, for Hdc _, 0, our results are within 10~ of the Glosser data . This serves as a check on the method. The results are displayed below for comparison with Glosser's ( 3 ) data. T/T c [R5(Hdc = He_) -RS(Hdc= O)]/RN % theo. exp. theo. exp • •6 2. 9 3 1. 2-.45 1 7 .8-.75 =. 75 = .05 •7 6. 9 0 •8 9. 8 10 • 3 - 3. 8= - 3. 5 -4 In the fourth column we have written our result as the sum of two numbers to emphasize that the two effects of surface state transitions and continuum state transitions compete. The first number is surface state data. The experimental data originally provided the clue to understanding the negative shifts in surface resistance. R (Hdc = 0) 5 does not start to rise substantially from zero until T 2::. 6 T • c This is, however, also the same point where R (Hdc f. 0) - R 5 (Hdc = 0), 5 -75- which is fairly flat with T negative slope. up to this point, begins to develop a Hence, it is likely that Hdc is modifying that mechanism responsible for absorption when T;::.,. 6 Tc; viz., continuµrn. transitions. But what is the reason for this negative shift in surface resistance? When Hdc = 0, there is uniform pr,obability to find If Hdc -::/- 0, though, the a quasi-particle anywhere in the metal. situation is vastly different. The probability to find quasi-particles, then, varies with space and even the spatial behavior varies with e, st and cp in a complicated way. Ve Hdc sets up the effective potential = V 0 coscp described in Chapter 3. effect of V e (V 0 _. Hdc) The principle is to repel quasi-particles from the surface region ""'A for certain important values of the parameters. and € ~ 6 +Ve• If e l st! ~ 206, Ve> 0, r ± are complex so that the probability to find quasi- particles falls exponentially in the surface region. quasi-particles in the surface region tor But it is the I st I ~ 2ot:. which are the most important absorbers of energy; a significant fraction of quasiparticles no longer plays a role in interacting with the electromagnetic field. For l st I ; ; 206, r ±-+ p ± which are usually pure real. But this does not mean that the u, v amplitudes are spatially invariant. Even here there is a smaller probability to find quasi-particles in the sur:face region, but only fractionally less - not by orders of magnitude as when r± are complex. · This phenomenon is related to the induced static currents created by Hdc and could probably be -76- obtained if the u, v functions were calculated for an infinite medium carrying a uniform current along the y-axis. That is, it is due to transforming the infinite medium excitation spectrum from the coordinate system at rest with respect to the screening current to the lab frame in which the superconductor is mounted. In any case, the net effect of Hdc is to make most of the matrix elements, M f,~' smaller as Hdc increases. The re are no resonance effects where by M f,~ increases substantially. That this occurs is very important for it will allow us to explain several important features of the reactance behavior without resort to further calculations. Finally, there .is the superconducting surface resistance at "high T", T P. 8 T c1 and low frequencies w ~. 16. Here, the experimental data show · that surface resistance starts to increase again with Hd R (Hdc 5 c so that at T ...... 95 T , Hd once more causes . c c f O) > R 5 (Hdc = O). R (Hdc = 0)-> RN. 5 Finally for T ~. 95 Tc, R (Hdc (= O) 5 -+ This last realm is the easiest to understand since !:, -+ 0 rapidly and the superconductor is turning into a normal· metal. But why does the shift in RS change sign and become positive for sufficiently high T? First there is the obvious fact that Hdc or V 0 have decreasing maximum values as . T H (T=O) (1 - T 2 /T 2 )] c c e Thus the potential V increases. [ H c (T) ==: . weakens for higher . T and is less effective in keeping quasi-particles away from the surface region. Also quasi-particles of larger energy are more important -77- at higher T and these feel the effect of V e much less than those of small e. Hence the suppression effect doesn 1t work as well at "high T' 1 • Thus the suppression effect can no longer overwhelm the positive effect of transitions from surface states to continuum states. In fact the latter mechanism dominates once again. We have only estimated the size of this effect but it is sufficient (Rs /RN,...,, 2%) to give a net positive shift in RS for large Hdc ,...,, He. 3 Only one property of this mechanism is important to discuss here related to the experimental data. clearly show The data of the experimentalists that the maximum positive shift in RS at "high T" shifts to lower T as w increases. 4 Lewis( ) summarizes all the For any w, only cases and finds a nice monotonic behavior. surface states within w contribute. As of the lower edge of the continuum can w grows then more surface states absorb giving a larger surface resistance. But further, higher w's use surface states from deeper in the gap and these have higher population/ differences at any given T which also makes RS grow. higher · w 's begin to make a certain size R5 at lower Thus the T than the 2 smaller w's. When added to RS· the net result is as observed. 3 Since the above depends on two competing processes cancelling in a proper manner, a more direct way of observing the surface state to continuum state transition is desirable. We propose the following experiment be performed which, if successful, will be another verification to the existence of surface states. and /::; ~ w ~ 26 in tin, tantalum, indium, or aluminum. Let T > > T · c If Hdc = 0, -78- R 5 = 0 since w < 21::. means there is insufficient energy for pair creation. Now raise Hdc from zero . If RS suddenly becomes non- zero for some Hdc <He, we are observing pair creation with one excitation going to the c6ntinuum edge and the other excitation going to the lowest lying ~mrface state energy. Furthermore, w :;;:, !::. ....,, ,.,., should require Hd < H and w ~ 21::. should require Hd > 0 to c c c observe the discontinuity in Rso This is so since stronger Hdc · pushes the lowest branch of the surface state spectrum to lower energy. Higher T's will smear the RS discontinuity so every attempt should be made to keep T -> O. Our next order of business is the surface reactance behavior of superconductors when Hdc ¥ O. While surface resistance processes are susceptible to a simple physical interpretation, surface reactanc e phenomena are more elusive. Surface resistance is just another name for power absorption; a photon annihilates, kicking a quasi-particle from one state to another, conserving energy in the process. Surface reactance was shown to be a measure of field penetration and depends primarily on processes that always violate energy conservation, although staying within the bounds of the Heisenberg Uncertainty Principle. Nevertheless we have devised a picture that simplifies understanding surface r eactance and is consistent with the predictions of the formalism heretofore developed. Consider only virtual processes . Suppose that the initial configuration of quasi-particles changes to another arrangement in the presence of some A field at a particular frequency w. If the -79- total energy of the final state is greater than the energy of the initial state plus the energy, ·nw • then the effect of such processes is to reduce the penetration depth of the A field. And conversely, if the final state has less energy than the initial plus 'h.r.u, the penetration depth increases. Crudely speaking, we can imagine that if there is insufficient energy to make the transition from one state to another, the extra energy is extracted from the A field so that the penetration depth is decreased; whereas if there is more than enough energy. the surplus energy is returned to the A field increasing the penetration depth. Let us view this in terms of the two particular processes -scattering and pair creation. For scattering the relevant terms in the j(A) relation are [fT(e:a) - fT(e: 1 )]/(e 2 -e: 1 -w ). If e: 2 > e: 1 • the thermal factors favor a net transition of quasi-particles upward from e: 1 to e:a. e: 2 > e: 1 When + w. the generated current causes a field which is in such a direction as to oppose the original A field. For e 2 < e: 1 + w (e: 2 > e 1 ). the opposite holds and the original A field is enhanced. If e: 2 < 81 (e: 2 always less than e: 1 + w. of course). the current opposes the field· Although this transition is energetically favorable, tically undesirable ability it is statis- since the final state has higher population prob- than the initial state. Thus, for e 2 < 8 1 • one might say that it is an anti-particle type quasi-particle which is making the transition and such a particle causes a current that is opposite of particle type quasi-particle currents. -80- Note that in the limit of w _, 0, all the currents always oppose the field so that penetration depth is decreased for any transition. For pair processes, we are concerned with the terms -[ 1-fT(e .e) - fT(E!m)] relation. f 1/(e .e+em +w) + l/(e ,e+E:m -w) } in the j(A) The energies e .e' em here refer to the energies of a particle and anti-particle type quasi-particle either before or after interaction with the A field. When e .e+em > w, the only case of interest presently, virtual creation of particle and anti-particle type quasi-particle into states of energy e .e' em always tend to make currents which further reduce the A field. Virtual annihilation of the two quasi-particles from states e .e' em . would yield current tending to increase the A field if it were possible to have greater population probability in e ,e' e:m than in the "vacuum state 11 - but this cannot be. Thus creation and annihilation always tend to screen the field from the interior when w < e: .e + e:m. In discussing surface resistance we carefully checked the necessity of maintaining self-consistency on o and concluded that this was not too important. That, however, is not true when considering surface reactance changes; we must be very careful here. Suppose using eqn. (4. 30) we found that XS ""o 2 (l+Hdc); it would not imply that XS shifts positive with Hdc because XS,...,. o from eqn (4. 14) and o,...., l/(l+Hdc) which means XS-.. l/(l+Hdc). XS shifts negative with Hdc!! This type of situation generally pre- -81- vails for superconductors. The discussion of surface resistance fdr w < 6 above required con~ideration only of the terms involving scattering of a single quasi-particle from one state to another; the pair terms could make no effects. However, for surface reactance or pene- tration depth problems, it is the pair terms which are the most important. Consider the "low temperatures", T E:; • 5 T • c All the Fermi factors, fT(e), are much less than one for continuum states, e ;;:, !:::.. Hence only the pair terms contribute substantially. w < 26, only the reactive part of this remains. And if But that is sufficient to explain the Meissner Effect -- the expulsion of the A field from the superconductor's interior. The penetration depth of a field due to the pair mechanism has a weak frequency dependence. fact AGK have shown that at T 0::; w< 26. In = 0, o varies by about 7~ for For wt:;;;. I!:::., e: J, +em>> w, so that w= 0 results are perfectly adequate in this "low temperature'' realm . far, applies irrespective of Hdc" only the matrix-elements, All this, so A static 'magnetic field affects M.Q,~ the effect, tho', is identical to what happened to continuum states at "intermediate T" when surface resistance was discussed -- Hdc reduces the size of most matrix elements. Thus the current of virtual pair processes tending to expel the A field is reduced and so the surface reactance or penetration depth increase as Hdc rises. Another way to say this is that Hdc creates an effective potential, Ve, which shoves quasi- -82- particles into the metal 1 s interior. . further to interact with them. has seen. Hence the A field must penetrate And this is what every experimenter The data at 700kc, 2mc, 1 gc, 3gc, 8. 8gc, 24. 5gc in various metals show a positive shift of surface reactance for "low T" with no evident frequency behavior. When T ~ • 5 T c·. the surface reactance shifts with Hd c · are much more complicated. At 700kc and 2mc, the shift is positive. At lgc and 3gc, the shift is negative. shift is again positive. At 8. 8gc and 24. 5gc, the This pattern, too, has an explanation based strictly upon the reduction of matrix elements through Hdc" The pair term works as at "low T" causing frequency independent positive reactance shifts so this can be regarded as a constant· background. But in addition T 1". 5 T c makes the Fermi factors large enough so that reactance effects caused by scattering quasi-particles from one level to another are important. discussed qualitatively above. This was It was noted that for w = 0 the virtual transitions lead only to further flux expulsion; so if the matrix element is squashed by Hdc' then A mechanism is modified for w-:/ O. e1 < 8 2 < 8 1 penetrates further. Transitions from 8 1 to 8 2 where + w lead to "screening" currents that suck the further ,into the metal as shown earlier. more important as w increases. The A field That contribution becomes If Hdc suppresses the matrix element, then the penetration depth or surface reactance shifts negati ve wh~ch is what the experiments at 1 and 3 gc show. When w is -83- further .i ncreased, though, the processes tend to cancel; there are as many transitions above resonance as below. Which in turn means that the pair processes again dominate and the surface reactance shifts positive. This corroborates the data at 8. 8gc and 24. 5gc . Throughout the reactance analysis we have consistently ignored the effects of surface states . Their total contributions are much too small to make any difference . All the reactance changes can be explained by reference only to the modification of continuum states wavefunctions. -84- CHAPTER SURFACE 7 EFFECTS Early in the research questions arose about the effect of a surface on absorption in metals at low temperatures. We discuss the matter in this chapter. Consider a normal metal at room temperature. free path of the electrons is several angstroms long. The mean So the current at some point inside the metal is determined by the electric field distribution only in a region of several angstroms radius about this point. And as long as these points are more than a few angstroms from the surface of the metal, there can be no surface effects, hence it makes no difference whether we use an infinite metal or one which makes explicit mention of a boundary. A typical electric field in this case penetrates the surface to about 5000 angstroms depth. Thus, the vast amount of absorption occurs in the interior away from any surfaces. Negligible error is made in computing surface impedances using infinite extent models. Now, however, suppose that the temperature is lowered to near absolute zero. The mean free path of the electrons can be very long,...., 100, 000 angstroms. Then, the current at a point is determined by the field distribution in a huge region surrounding that point. For typical fields penetrating about l 000 angstroms, most of the electrons arriving at some point will have had an interaction with the surface. The surface effects seem to be playing a large role in determining -85- absorption processes. This phenomenon, the anomalous skin effect, yet, has been treated in the past entirely by considering only infinite medium models! 11 (See work of Reuters and Sondheimer ( ), Chambers(IZ>, Mattis and Bardeen{ 7 !} An exactly analogous situation prevails for superconductors and there, toq infinite medium. {See Mattis and Bardeen{ 7 ), Pippard{l), results have been applied. AGK{l 3 ).) These calculations agree very well with the data so what is the paradox.? We consider here only the case of the normal metal; the superconductor is handled similarly. Bogolubov's equations adequately describe a normal metal under present circumstances if the gap, 6, is set to zero. This is then just a free electron gas; the elementary excitations are just electrons and holes. We consider the same geometry as always with the surface of the metal in the y-z plane and extending infinitely in the +x direction. only. For such a case the u, v functions are simply s> 0 s< 0 s <0 s> 0 1 u= z sin px, = 0 1 v= z sin px, , =0 K 2 pa __Y_ where S = Zm + 2m K (7. la) {7. lb) (7. le) {7. ld) 2 z Zm - Ef The u, v here satisfy all the boundary conditions imposed earlier in Chapter 2 and 3. -86- Using these u, v functions, the electromagnetic kernel Q(x, x 1 ), defined by j(x) = Soo Q(x, x 1 ) A(x') dx 1 , as in Chapter 2 can 0 be grouped into three terms Q(x, x 1 ) = Q 00 (x-x') + Q 00 (x+x') + Q 5 (x, x') (7. 2) Those terms that depend only on the difference (x-x') have a functional behavior called Q 00 ; those depending on the sum (x+x') also behave according to Q ; while the remaining terms are 00 It is not important presently to display any of lumped into Q • 5 those Q's more explicitly. The subscripts on the Q's refer to their physical significance which will become obvious later. is the infinite medium kernel while Q Q 00 is the correction kernel for 5 surface effects. Suppose for the moment that Q 5 can be ignored. Then the current at x, j(x), produced by the field distribution, A(x}, is given by 00 [Q (x-x 1 ) j(x) = \ ' .) 0 00 +Q (x+x')] A(x dx' 1 00 . ) ' (7. 3) Now suppose we define A(x) for x < 0 to be A{-x) = A(x); i.e., A(x) is symme.trical about the plane at x = O. Then a simple change of variables in the second tern of eqn. (7. 3) yields after some rearrangement ' j(x) 00 =S -oo Q 00 (x,x 1 ) A(x') dx' (7. 4) This transformation and final result is of large significance and in -87- part res,olves the paradox of surface effects mentioned above because :Q oo (x-x') turns out to be identical to the total electro- magnetic kernel calculated in an infinite medium! If we had calculated the kernel Q(x, x') using infinite medium u, v functions u= e ipx u= 0 v= e ipx v= 0 , s> 0 (7. 5a) , s<0 (7.5b) , s< 0 (7. 5c). , s>0 (7. 5d) we would have found a Q (f{:, x 1 ) that depends only on the difference (x-x 1 ) and which is the same function as Q in a semi-infinite metal. subscript 00 determined with u, v 1 s And that is why we labelled Q with the 00 " OO• Eqn. (7. 3) and its mathematical equivalent eqn. (7. 4) can be neatly interpreted with the aid of some simple physical pictures. Refer to Figures 7. la, b belowo ·Y FIGURE 7. la FIGURE 7. lb -88- Figure 7. la is a diagram of eqn. (7. 3) and Figure 7. lb is a diagram of eqn. (7. 4). Consider the former. A photon annihilates at x. 1 creating an excitation which produces a current at x. This excita- tion could arrive at x via two different journeys - - it could go directly from x' to x, a distance (x-x'), or it could get to x afte·r a collision with the surface of the metal, a distance (x+x'). the same effect is achieved if the photon annihilates at But x' and goes directly to x in an infinite medium through a symmetrized A(x) field. In other words, an obse.rver at x cannot distinguish whether the received excitation at x was generated at x 1 in the semiinfinite metal or at -x' in the infinite metal. Yet still another way -- an excitation going from x' to x via the indirect path feels the same forces as one traveling from -x' to x directly. The processes just described are referred to as specular reflection and depends on the surface being perfectly flat. An opposite extreme is the perfectly rough surface so that an incident excitation comes off in any random direction thus making no net contribution to the current. For such a case, Q 00 (x+x') is ignored in eqn. (7. 3) and the current is j(x) =~ 00 QOC>(x-x 1 ) A(x 1 )dx 1 (7. 6) 0 a situation referred to as diffuse reflection. Reuter and Sondheimer have calculated the . surface impedance for specular and diffuse reflection in normal metals. The amazing result is that they differ only by about 101h for those two extreme cases. -89- Thus, it is plausible that infinite medium kernels are capable of yielding sensible surface impedance results g_ the terms lumped into Q (x, x') are somehow small compared to those 5 in Q 00 • That question, though, has not been dealt with heretofore. QS is far too difficult to calculate and even if available, the integro-+ . differential equation for A would be hard to solve. use an iterative approach. Using the surface impedance determined from Q , co 4. we find a We therefore 6 to use in A,..., e -x/o as explained in Chapter Then with this A distribution, we calculate a new surface impedance but include in the matrix element Mn a the surface terms . ..11m which appear in Q • 5 (M ;,!= 0 in a normal metal) The complete matrix a element ¥;,m looks like a (' OCl M ~m "'Jo 1 ,..., The first • • sin p J,x sin pmx e P.e-Pm+i/6~< 1 -x/f/:<d x 1 1 P;,-Pm-i/o>!<+ P.e,+Pm-i/o~< - p J, +pm +i/[)>l< two terms are just those that would appear for an infinite medium while the last two depend strictly u,pon the existence of a surface at x = O. Further evaluation of the above requires the sizes of typical variables. -90- and the important range of integrations is O~e~ .1:1,m ~2 so over most of the range of integration Thus usually So the first two terms in M f,:i are each about 6*,,..., 10 3 ; while the 2 l. last two terms are about l/p.e,+Pm"'"' 10 /(-st)z and subtract from each other. The conclusion is that the surface terms make a negligible contribution to the surface impedance! The only possible way to see a surface effect is for penetration 0 depths 6 E:; SOA. Then the range of St integration is much smaller since the quasi-particles must make extremely small angles of nonparallelism with the surface to be effective. But no such metal is known where 6 is so small. Although the above treatment concerned normal metals, the -91- reasoning and conclusions are identical for pure superconductors with no static magnetic fields present. When a superconductor with a static magnetic field is involved, . none of the above conclusions are relevant. In fact, as has been seen, it .is the spatial inhomogeneity which gives use to the many interesting features. -92- CHAPTER 8 GAP SELF-CONSISTENCY A critical requirement for this theory is that the gap function, ti(x), be independent of Hdc and spatially constant. by seeing if the u, v wavefunctions for a This can be checked step potential and constant gap are self-consistent. Of course, within a few angstroms of the surface the gap rapidly falls to zero, since u, v do, but this isn't important. with the changes over 1000 We are concerned A distances. The self-consistency relation was last encountered in Chapter 3 andi we found that (8. 1) V BCS• the electron-electron interaction, is independent of Hdc and T and so can be ignored here. Whereas the expressions for surface impedance involved only excitations traveling nearly parallel to the surface, the gap relation eqn. (8. 1) is strongly sensitive to excitations traveling in every direction. Hence surface states make very little effect, here; only continuum states are important and we make little error in considering continuum states in the approximation -10 where r + ::::: r - ~ p + :::= p , all pure real. - 3 6~St 6;106 The St range is wide since we are no longer trying to couple to a radiation field; the electronelectron interaction is mediated by an isotropic phonon exchange -93- mechanism • . With these simplifications in mind it is useful to consider the product v=i<(x) u(x) for various regions and types of scattering states. ( 1) x ::::: A., p+ incident scattering states v*(x)u(x) = A 2 Aa =i< ( e - S ) + B 2 B 2 =i< (e - S ) + Ca Ca >:< ( e + S ) p p p -iZp+x + B 2 A 2 *(e-sP)e . , i(p _ -p+)x + C 2 A:z"<(e-sp}e (8. 2) Integration over the fast oscillating terms will give a negligible contribution so eqn. (8. 2) is simplified to v*(x)u(x) = A;aA2 *(e-s p ) + B 2 B2>:<(e-s . p ) +C2C 2 *(e+s p ) i(p_-P+)x + C 2 A 2 ':<(e-sP}e , i(p+-P_)x + A;aC 2 "<(11:+sP)e Around x,..., /.., the last two terms are very slowly varying. (8. 3) But deep inside the metal, x > > > A., even the last two terms are fast varying; thus they can be neglected implying that the equilibrium gap is spatially constant c;s expected and determined in part by the terms (8. 4) -94- (2) x ~ A., p _ incident scattering states Using the same approximations as before for x >>>A. (8. 5) (3) x ~ >.., p± incident Using continumn states for the surface region and disregarding fast oscillating terms, which requires staying several angstroms away from the surface, gives +C 1 A 1 •!< ( € - S - V ) [ e r e i(r -r +)x i(r + -r )x +e - J (8. 6) In the surface region for the range of integration concerned, the exponentials can sometimes be expanded and only the lowest order terms retained; hence ' ' v*(x}u(x) ~ 2 ([A1 A 1 >:< + C 1 A 1 •!<] (e-s -V) r e (8. 7) which is spatially constant. Let us consider this last transformation carefully. Whereas 0 Q most superconductors have penetration depths, \,,...... SOOA-1 OOOA, the -95- parameter 6/Ef has a range of a couple orders of magnitude. Typically o/Ef,...,, 10- 4 for tin, 6 5 /Ef ....... 10- for aluminum. Consider the exponentials in eqn. (8. 6). The phase varies most rapidly at x,.,., 'A and least rapidly as x-+ 0. At x""' A., the phase varies in tin from about 1 radian to O. 1 radians as i;t runs its full range; and Ai Ci*, Ci Ai* are large st when the phase is large st (near small st}. So the neglect of spatial variation in eqn. (8. 7} is somewhat questionable. That was for tin; but in aluminum the phase varies from about 1/3 radians to O. 03 radians and it is a good approximation to neglect spatial behavior. The smaller 6 /Ef and the smaller A., the less is the maximum range of the phase; but furthermore, 6 for smaller /Ef and A., the smaller is the range of st over which Ai Ci>:<, Ci Ai>:< are non-zero so the spatial variation terms can be forgotten irrespective of the behavior. Conversely, superconductors with large 6/Ef and 'A have large phase variations and large ranges of St where A 1 Ci >:c, Ci Ai>:< are non-zero. Thus it is likely that the gap is really constant, independent of Hdc for superconductors with 6/Ef ~ 10- 5 , A._. 10 due to Hdc if 3 .A. And that the gap has strong spatial variations t 3 3 6 /Ef ;o 10- ?. 'A 1310 . From the above discussion, one might suspect that 6 /E[ is of some fundamental significance in the theory of superconductors. The original BCS theory of infinite homogeneous superconductors contained a quantity, s , called the 'coherence length" which was 0 identified as the average distance between two paired electrons in ... ... the momentum-spin states kt, - k i. Just as the electron is the -96- basic entity in the normal metal, the 11 pair 11 of size the analogous particle in a superconductor. - typically s0 . . ., 3•10 3Ain tin and S0 rv 2• 10 the trend in gap variation for various 4 s . . , Ki"/m 6 is 0 l:"'fEf .....,l/K/; Thus our 0 ; A in aluminum. When 6 /Ef values quoted above is translated into the coherence length language, the behavior is immediately comprehensible. varies strongly if S < < A· 0 6 varies little if i; 0 > > A and 6 . In the former case the A(x) field is strong only in a region much smaller than the size of a pair so the pairing is almost totally unaffected and unperturbed value. 6 remains unchanged from its In the latter case' A(x) is uniform across a pair so pairing is modified in proportion to the strength of A(x) and hence 6 has strong spatial variations where A(x) has strong spatial variations. This result is not new. the Landau-Gingburg equations that behavior with Hdc' i; 0 , and A· Caroli(lS) has shown using 6 does indeed have this However, the L-G equations only apply to situations where ~ < < A which is the same as requiring local electrodynamics to be true. Our surface impedance theory s >>A, non-local electrodynamics. Hence Caroli s results are only useful for s <<A· is primarily concerned with the opposite limit, 0 1 0 The self-consistent method used in this thesis does not suffer from any restrictions on the relative sizes of s and A, and so provides 0 an independent test of how various perturbations affect the gap, 6. That the L-G equations yield the correct qualitative behavior here is still quite a mystery. -97- In conclusion, we simply say that for the metals considered in this the sis, tin, indium, aluminum, tantalum, s />. . is 0 sufficiently large that we expect the qualitative results of our surface impedance theory to always be valid. -98REFERENCES (1) A. B. Pippard, Proc. Roy. Soc. (London) A203, (1950) (2) M. Spiewak, Phys. Rev. 113, 1479 (1959) (3) R. Glosser, Phys. Rev. 156, 500 (1967) (4) R. T. Lewis, Phys. Rev. 134, Al (1964) (5) P. L . Richards, Phys. Rev. 126, 912 (6) Y. V. Sharvin and V. F. Gantmakher, JETP 12, 866, (7) D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (8) J. 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