Superconductivity of Metals and Alloys (Advanced Book Classics)

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P. G. DE GENNES Facult6 des Sciences Orsay, France

Translated by A. P r ~ c u s University of California Los Angeles, California

"A

/'

"

A Member of the IBerseusBcx~ks.Grc~ug

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Editor's Foreword

Perseus Books's Frontiers in Pfiysics series kas, since 1961, rnade it possible for leading physicists to communicate in coherent fashion their views of recent developmenrs in the most exciting and active fields of physics--without having tci devote the time and energy required to prepare a forlnal review or monograph. Indeed, throughout its nearly brq-year existence, the series has c~nphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these informal accounts would be replaced by more forrnal counterparts-textbooks or monographs-as the cutting-edge topics they treated gmdually becalne integrated into the body of physics knowledge and reader intcrest dwindled. However, this has not proven to be the case for a nulnber of the volumes in the series: Many works have remained in-print on an on-demand basis, while others have such intrinsic value that the physics co~n~nunity has urged us to extend their life span. The AdvuneeJ Book Classics series has been designed to meet this demnd. I t will keep in-print those volulnes in Frontiers in Physics or its sister series, Lecture Notes and Supptemenu in Phy&s, that continue to provicfe a uniyue accstint of a topic of lasting interest. And through a sizable printing, these classics will he ~nadeavailable at 3 comparatively rrlodest cost m the reader, The lectures transcribed in Nobel Laureate 13ierre-Gilles de Gennes's lecture note volume, Superconductivity of Metals and Alloys, contain m unusually lucid, physical and original extension of the inicroscopic theory of Rardren, Cooper, and Schrieffer (BCS theory) to take into account the role played hy inhornogeneity. As a result, they represent a lasting contribution to our understanding of superconductivity, as generation after generation of graduate students and experienced researchers in the field will attest. With the advent of heavy electron and organic superconductors, and especially with the ongoing extraordinarily high

level of activity in the field of high temperature wlperconductivity, it is clear that research on superconductivity will continue to represent a significant portion of research in condensed matter physics for Inany years to come, and that researchers will continue to turn to de Gennes for enlightenment of the behav. ior of superconductors in a variety of interesting, real-world situations. I am accordingly very leased that the publication of Superconductivity of Metals and Alloys as part of the Advanced Book Classics series will make it readily available to present and future generations of interested readers.

David PF"ines Xsuque, NM January f 999

THESE NOTES ABE DEDICATED

TO THE MEMORY OF PROFESSOR EDMOND BAUER

Vita

P.G, de Gemnm Rafmsor of Pbysic~at the Coitege deFrmce, he is also &mtw of the EcoXe de Physique enginwr at the Ammic E n e ~ ySaclay md as a p s t - h m d visitor at the ty of CAifomh, Berkeley. b f ~ m dke r Gennes hats dso k e n a ProE(f.ssorof mysies at the University of &my and is a m m h r of the F~ncIrAcdemy of Sciences,

ay&Soeiety his resach mow coxentrms OR mamdsm, su~rconduc~ors, Eq@2erysMs,plymers, andcoHoi&F;. the D u ~ Acdenxy h of Sciences, the h e r i a d the Na~ond&&ern y of Science. He

Preface

Superconductivityof Metals and Alloys was wrimn during the post-BCS period, a time of great enthusiasm. A long periodfollowed where progress in matr=hlsscience slowed considerably, h a m p e d by the untsmely d e a of ~ B, Nattfiias. There was a hngerous feling of c m f m on the thmretical side; eveqthing seemed t fit, witfi &heexception of such difficuft cases as haw femian systems, whea many insabaities camp&. mee y thif fwlhg of cornfoa was sMen up, with cupmk systems nt-husiasmfrom nrakriafs sientists. Out of the twenty (or mom) ereating a mw kamtical models o n w p r a p d , most (ineludingmy own) have couapd at a suq~fing pwe,and thasc: which resist &$provingare Rot nmssarily the mmt excitiRg. Xr may be a $ood idea, then, that we ate again to reviw the S-sate Cmpr p&, suibbly eneamged by two-dimensional confinment, &ough the reissue of this bmk, ]if this was natwe*~ choice (much to my reget), repinting &is old b k may well be y tfxhnical details could be irnpoved, but hopfully the spifit is still fiere, I m now W f a =moved from &is field to prduce my&ing which could be emsidmed new, I did accumulate a numkr of ux1Ful wmwtions suggestd by mmy fiends; they have now b n i n c a m m M inm the text. If we som&y get conducting ]polymerswhich we Iiquid, or soluble, md which also suprconduc~then X shall rewfi& &.Wk. In the mewtime, I welcome new redem md affa best wishes tr, those who bldly stack the cumtc=s.

This page intentionally left blank

Preface

The present lecturenotes correspond toan introducrorycoursegiven at Orsay during 1962 and 1%3, The m& of the cow= was to set up a basic howldge of suprcondue~vity for mendists md &wrelicims in our smdI poug (including the lamer), and &am there on,to plan new expaents. It ispossible (and infact &mptbg)io intrexlace suprmnductivityas a Rovd M% of ion8 rmge order, then, from a study of the p h w of the oder pmmeter, to deduce the supduid p r o ~ i e sRux , q w b t i m , and tke Josphson effst. Later &if1would came the;b&u-Ginsburg qwtion and discussion of the magne~cpropaies of supreonmd more spifi: assump~ctns one would rach fhe Badwn ductm, Fmiiffy w

and its applica~ons, r ScMeEer on sup~onductivitymore wgenrf y But in & a y we want& m SWexp~ment;~ ~n we wmted exw~menfsinmcltxing; cowquendy we did not lbry the abve approach. The noes kgin with m elemenw discussion of mapetic propaks of TT f and Type Il suprconduetos. n e n the micro~ogiethtwv is bust up in the Bogolubv Xmguage of SE-mnsistent fields: this i s pweduI enough U)cover che musing sitmcions where the order pmtne&r is madu1aEd in space; it also retains some of the physical ininsight s which we asmiitte: with one-paicle wave functions. At this sage, the p r o w ~ e of days, and in p d c u k of the m-cdX& "&yr' &lays(which, in spite of heir name, we ofkn the cleanat systems on which we cm experiment)are: systematically discussed .in pmltel with &OS of the pure metais. A. numbr of topics have beem p sely negXecW. The discussion of the etweon-elmtron htaactions, for insan=, is r&ucedE to a s ~ cminimum, t since in most metallic superconductors Qwt as in fenamagnegc me@is)we me unable te comput~ aawareiy the EundmenM couplingcansm&,Madels whkh have &R hismfica1Iy very umful but which a e m w of'less cment application are not mention&: examples af this

class am (I) the GamrXmimir &my, md (2) the Win= m&l for the mixed state af Type fI supmnducmrs. Some o&erpblems are mt discus& haufc=they requke a ""song p h , and h e effw&relaw m the desmcgm of the m h m y of h e The notes in thek presnt S@& h a 1 y seminar by C. Blwh on the g m e d k d &f-wnsis@nt field Ii& m finite nuclei was the s@ng pht of G n mspin &it effwu wiis the basis ofour m a p at Orsay, J, P. Bager, C. C mn,togcrkr with ourf ~ e & B.S and &e ~ t ewishes r to expregs his d m p thmb to all of wm-with i4 spid menGon to P, Pincus, who wcepted the mk sfpr&ucing the Engtkh versim, and to M,Thhm, for a cfitieal ra&ng of the resuf~ngmiunuseript. It is h@ thirt =me of the exc;i&ment which we fefc in these p~ds of debae and c m j a c ~ will e firrizlly reach the reader and

find him in rewmance, A very smaH nurnkr of usful referenea m quoM at the end of each ehap~r. They make no at&mptto completeness and @e no weount of histofieal pfio~ties:for ins~nce,I ;Ifways refened &e students, not to the ofiginal BCS d c l e , but r&er to tfxe L e g Houches lectwe notes by Tkkfim, in which hma&daX is p w n d in a mare acleessible fom. The mhnieal cmpmtion of French version), atxi of M,J, B h q u i -thy Pederwn, &chef Shimkin (far the English versian) is gate;EuXlyachawl&gd.

Contents

Chapter I

Fundamental Properties

AE Jm C~ndengedS&te 1 *%l 12 1.3 Absence o f b w Energy Excitations 1-4 Two K i d s ofSyercond1cciors

Chapter 2 Magnetic Properties of First Kind Superconductors Cn"gcalFieCd of a brig GyIinder Pewtradon Deplh 1Magmtr"eBroperrz'esof a S a v k of Arbitr~rySkpe: Enrermdkfe State

Chapter 3

Magnetic Properties of Second Kind Superconductors Magwrz'z~tr"~~ Curves of a brig Cylrit&r VorlexState: Micro~copicDescrbtion h"~~?~?qdIibn"m Pr~perties

Chapter 4 4.1

Description of the Condensed State Imubility of the &rml State in the Prege~cecpf an Attractive Interacfic,n Origin sfrk Atpmtlw Intmction

Ground State and Elemntav Exirm.ons Calculatiofisat Finite reyeratare

Chapter 5 5.1 5.2

Chapter 6 6.1 62 6.3 6.4 6.5 6.6

6.7

The SeIfiCansistent Field Xllethd The Bogolubov E q w t h s

Theorems on the P&r P~renn'aland t k Exitatron S p e c ~ m T k Meissner Efect ii Metals and A l l ~ y s

~henomenolo~icai Landau-Ginsburg Equations In&Qaltcclion Constr~crionG$' the Free Energy Equilibrium Eqlcati~ns T k TWQCharacteristicLengths Siruatiom w k r e Jl;l is Consta~r Situaff"onrwhere I$/ VariesSpatlate S t ~ ~ c f~wf et h eV o r mP h s e in S ~ o n gFields (H-Hc2)

Chapter 7 Microscopic Analysis of the Landau-Ginsburg Equations Liw~rizedSe@uConskteqEqurtt"on bndarr-Ginsburg Eqwtions Sugaee Problem in the hn&u-Gimburg Region

Chapter 8

Effects of Strong Magnetic Fields and of Magnetic Impurities ReIt2fiol"tk ~ e Transition n Tewerafure and Tin-Reversal Properries Ergodie versus Nonergodic BehaviarCapless Superco~dactivity Dir@Supereod~torsin High M~gneticFieldf

l

FUNDAMENTAL PROPERTIES

1-1

A NEW CONDENSED STATE

We take a piece of tin and cool it down; at a temperature T, = 3.7 "K we find a specjrfic heat anomaly (Fig. l -l&). Below T, the tin is in a new thermodynamical state. What has happened? Xt is not a change in the crystallogr;%phic structure, as far a s x rays can tell. It is lzot a ferrornagnetic, o r antiferramagnetic, transition. (It can be seen by m a a e t i c scattering of neutrons, that tin carries no magnetic moment on a n atomic scale .) The striking new property is that the tin has zero electrical resistance. (For instance, a current induced in a tin ring has been observed-to persist over times > f year.) We say that tin, in this particular phase, is a superconductor, and we call the permanent current a supercurrent. A large number of metals and alloys a r e superconductors, with critical temperatures T, ranging from less than 1°K to 18°K. Even some heavily doped semiconductors have been found to be supttrconctuctors. Historically, the first superconductor (mercury) was discovered by Kammerling Onnes in 2 911 , The f r e e energy Fs in the superconducting phase can be derived from the speclfic heat data and is represented on Fig. l - l b (solid line). The dotted line gives the corresponding curve F, for the normal metal, The difference (Fs - F, )T, is called the condensation energy. It is not of o r d e r kg T, per electron; it is, in fact, much s m a l l e r , of order (kg T, l2/EF (where EF is the F e r m i energy of the conduction electrons in the normal metal). Typically E F -- 1 eV and kg T, 10a%eY. Only a fraction kg T,/EF ( M IP3) of the metallic electrons have their energy significantly modified by the condensation process,

(per

(kg T,)~ m-

EF

Figure 1-31 (a) Tbe electrorzic specific heat C of a supercanduetor (in zero mapedic field) as a function of temperature Iqualitative plot). Above To (in the normal phase) C(p,, m k2 TIEF where EF is the Fermi energy. At the transition B p i n t To,C has a discontinuity, A t T X, the field does not have a constant sign, but the ratio p can become negative; this has been observed experimentally (Drangeid and Sommerhalder , 1962).

-

"After Barclesn and Scfirieffer, Lotv Tetrzperatu?*ePh~fsics, edited by C , J . Garter (Axnsterdm: North Holland, 1961), Vol. 111, p, 170,

-

EXTENSION TO ALLOYS

When the electronic mean f r e e path B is limited by the presence of impurities, i t is natural to expect that the relation between current (at the point r) and vector potential (at the point r r ) contains an at-~r-rf\/t. One is therefore led to assume, with tenuation factor e Pippard,

The normalization coefficient C is assumed to be independent of B (therefore equal. to its pure metal. value - 3ne2/4%mc4,); in other words, one assumes that the contributions ta 1(r) coming from points rf in the neighbarhood of r, ( f rL r 17 g) aar not modified by the impurities. This hypothesis has permitted Pippard to explain a s e r i e s of experiments on the dilute S ~ alloy R system. Qne finds that X increases with the concentration of impurities (Fig. 2-6) and the results a r e well interpreted by the formula, (2-19). One limiting case is partieulary r e markable. When X >> P we can neglect the variations of A(r9 in Eq. (2-19) and perform the integration

Thfa is a London. type equation, but with a coefficient mo&fied with . respect to the pure metal, In garticular, if B 5,)

This, as we h o w , e x p r e s s e s that the field in the normal regions is e q w l to H e . Then we minimize 5 w i t h respect to d. The d-dependent t e r m s a r e the three correction t e r m s

=

macroscopic

H:: +

eh Ly

The optimum d Is given b y

where

U@ +

- p S ) Z V o l Typically, for H,/H,

-

0.7, and thus d 10J6E;. Taking e = l cm, 6 = 3000K, we a r r i v e at d -- 0.6 mm. Near the ends (p, -- O o r p, -- l), @ tends toward O and d becomes still l a r g e r . Now can we o b s e r v e this damain s t r u c t u r e ? Various methods have been used: (I) A fine bismuth wire (whose resistance is strongly dependent on

-

$i=

p: (l

-

FIRST KIND SUPERCONDUCTORS

43

field) is moved close to the surface of the sample. The resistance i s large in the N regions and small in the S regians (Meshkovsky and Shlnikov, 1947). (2) Niobium powder is placed s n the sample. Niobium has a high critical field (-2000 E). Thus the grains a r e always superconducting and diamagnetic, they tend to avoid the lines of force, and they gather on the S regians, (3) The specimen is covered with a thin layer of cerium glass (typical I h i c h e s ~-0.1 mm). This glass has a large Faraday rotation. Polarized light traveling along the magnetic field W , (normal to the sample surface) has its plane of polarization rotated by an angle 6 when it traverses the glass; the total ratation a f b r being reflected from the sample is then 28, Since, in general, 8 Is proportional to the field (B Q.OBO/nrrm/G), if the specimen i s observed between crossed analyzer and polarizer, the N regions, where h 0, appear bright. Ultimately, these measurements furnish a determimtion of the wall energy (characterized by the length 6).

-

Problem, What are the corrections to the critical field of the plate due tothe fact that the thickness e i s finite? Sofution. U: we minimize the free energy with respect to d, keepin& as other independent variables the reduced field hr = Ho/Hc and ps = ds /d we find

i n the region of interest, hr i s close to 1, p, i s close to 0, and we may write for $I

-

where 8' I Ud(0)t Vo(0) / = (l/n) In 2 = 0.22 (this numerical value i s obtained i n the previously mentioned calculation of Landau and Lifshitz). Minimizing 5 with respect to p s , we obtain the condition

The critical field of the plate i s reached when is equal to the normal s t a t e f r e e energy (H: ISn) eLy Lx . This leads to the condition

Both concllitism are satisfied when

By takiw e = l mm and 8 = 10' A, we get a 3% deerease of the critieal field. Similar deviaMons from the maeroseopic theory occBr in all geometriee; very often they are imporbnt in the discussion of e x p e r i r n e ~ kresult& l in type I ~ u per conduetors ,

A PARADOX If we look carefully at the domain structure of Fig, 2-14a, we notice that at a paint such a s A, in the N regions, the lines of farce have ""opened up," Thus the field is significantly lower than at point P on the boundary, that is, lower than H,. At f i r s t sight this is troublesome. We would expect a region near A to become supercandueting again. This complication has been considered by hndau, He concluded t h t the nor m 1 regions should L'brainch" a s shown on Fig, 2-1 43, passibly up to such a fine scale that the &main s t r u c k r e would become unobservable. h fact, the simple domain structure is often abserved, Branching does not take place in clean samples with dimensions of arder 1 cm, The critical field at point A is reduced below He by an effect similar to &at discussed in the problem above (but here the dimension of interest is d, not e, thus the effect is large). Thus A can remain n o r m 1 in rather low fields. IEf we had a superconductor of t h i e h e s s e = 1 mile, d would be of order 1 em, we could still, apply mcroscopie considerations in the neighborhood oi P, and some branching would take pkce. The brmnching ma&l i s not wrong; it is simply not adequate for the usual scale of sample dimensions, ( h o l f i e r way to favor brmching is not to increase e, but rather to reduce 6--this could be done In suitable allay systems ,)

ORIGIN O F THE S U R F A C E E N E R G Y

We qmlitatively discuss two extreme eases: >> h : In aur previous macroscopic discussion, there was a (1) sharp boundary between the normal region N and the superconducting region S (to fix ideas, we shall t a b this baundary as being the yzplane, the fields being along z and the N region csrresponding to x < 0). On the N side, the thermodynamic: potential was lowered by the magnetic

FIRST KIND SUPERCONDUCTORS

vacuum

supere onduc tor

surface

I bl Figwe 2-14 (a) The domain structure Rear the surface, Note that, a t a point such as A, the field i s smaller than a t paint P (as is i d i c a t e d by the curvature af the lines of flux): hla) c h(P) = H e . There a r e some small normal regions near the ends where h c H,! (b) Landata's branching model. Sueh a branching i s required if one wants to keep h > Hc everywhere in t h e normal regions. In fact branching does ,m/ take place: The condition h > Hc applies only tb macroscopic normal regions, Here the small regions of interest near the surface have critical fields signgicantly lower than H, ; they can remain normal for h Hc and the simpler model of Fig. 2- f 4a is the correct one.

field t e r m s H: /8n - H : /4n (see Eq. 2-49). On the S side, S was lowered by the condensation energy -H:/8n. Now, what does happen 0x1 a, microscopic scale? E X is small, it is still carreet ta assume that the field drops abruptly at the limiting plane x = 0, The new Eeat a r e is that on, the S side ( X > O), superconductivity is 66&maged'' in a, region of t h i c h e s s near the boundary, Thus we lose the eondensation energy Hi/@@ on an interval -C, and this gives a wall en-

-

-

-

(HE/8n) 6, (i.e., 6 4 0 L ergy y (2) 5, X, we may keep only the nearest neighbar contributions to the interaction. term in (3-53) and write

where z is the number of nearest neighbors of one line (z = 6 for the t r i a q u l a r lattice), d is related to the induction B through the relation B

-

@@"L

=

@' (triangular lattice) -4-3- dZ

(3-59)

Equation (3-59) can be easily verified on Fig. 3-5. The function G(B) is represented in Fig. 3-43, Since H > Het , the initial slope (aC/8 B ) ~ = ~

vortex cores

Figure 3-5 A triangular lattice af vortex line (after Kleinar, Roth, and Autler, P b f s . Rcrr., 133A, 1226 (1964). The plane of the figure is normal to the field direetion* The contaurs give the lines of constant ns. This figure describes the situation at high fields (nearly overlapping cores),

initial slope

M,,

-

Pipre 3-6 The thermodynamic potential S as a function of the induction B ( B = nL Qlo measures the number of vortices per cm2nL 1. The equilibrium value of B [B, (H)] corresponds to the minimum of I;,

is negative. *As B increases, the interaction term begins to contribute but rather slowly, since it is proportional to K,(d/X). When d > X we may write, according to (3-36)

Thus the interaction term i s exponentially small at small B's. At larger B% hbowever, it dominates the over-all behaviar and G(B) increases. There i s a minimum of: C for some value B = B(W), B(H) i s the induction found at q u i l i b r i u m in the field W. The theoretical B(W) o r M(H) has been computed along these lines by Goodman and i s shown on Fig. 3-1, together with experimental results on a particularly good MoRe alloy, The following points must be noticed: The theoretical curve has an infinite slope (8M/8 HIH= Hcl = 00 at the f i r s t penetratian field, Physically, this reflects the fact that the

that i s , we may think of their inlines repel each other like teraction as having a finite range X . At field slightly larger than Hcl it is thus possible to farm many lines in the sample without competing against the interaction energy, The experimental curve daes not show

H( Oe)

Fiefure 3-7 Experimental magnetization of a molybdenum-rhenium r?tlfoy a t T = 0.52To (after Joiner and Blaugker, Rc.rt, MafI, Pk~cs., 36, 61 (1964). Also shown. a r e two theoretical magnetization curves {afterB. B. Goodman). The broken curve i s for a laminar model, the continuous one for vortex lines,

a very large slope ( a ~ / a ~ ) ~ , ;~this , ~ is not very surprising since

.

in the region of i d e r e s t the interactions between lines a r e very weak and the lines can easilry be pinned by structural defects. However, a s we depart from Hcl by more than 10%, we get good agreement between theory and experiment. A similar theoretical curve can be drawn for another model where the flux-carrying units a r e not vortex lines, but laminas (see problem, p. '71). E the distance between laminas is d, we again find a repulsion between units proportional to e-d'X. However, i n this case, the induction B is proportional to d-l, while in the fine case it is proportional to d * % ~ shown by Eq. (3-59). Thus the fall of M(E) for NI > H,, is more rapid in the laminar model than in the vortex line model. The two theoretical curves a r e compared on Fig. 3-7; it i s apparent that the vortex tine gives a better fit, as emphasized by Goodman, 1. D O M A I N --> 1 in the domainof interest. Theref o r e 1/(1 + X 2 J" can be replaced by 1/h2JZ, Finally we must perform the sum CJ$ l/JZ,which depends on the particular lattice con@

sidered, Here, we will simplify the cailculation by replacing the sum by an ixllegral

with J min -. 1/d and J max .- 115 (the Fourier components relative to the interior of the hard core must be excluded), W e finally find

SECOND K I N D SUPERGOMBUCTQRS

In 13-64) P is a numerical constant of the order unity (for the triangular lattice, Matricon has caleulialed 6 = 0.381). The B(H) relation is sbtained as usual by imposing BG/alE3 = 0. This gives

= j3 ee-I"2nd where d i s always related to B by Eq, (3-59). where The logarithmic dependences predicted by (3-65) a r e in rather e;ood agreement with the experimental data on rev6rsfble magnetization curves in materials with h >> 5.

D O M A I N nL

lj-"

Here, a s already painted out, our simple model breaks down, and w e shall need a more elaborate approaeh based on the Landau-Ginsburg equations (Chapter 6). The upper critical field HC2 is of order POi t 2 . This, physically, c ~ r r e s p a d to s the onset of overlap between the hard cares. Problem, Compare the Gibbs function in the fitarnentary structure described above with that of a possible larninar structure. Solution, As before, we shall limit our considerations to the case h >> 5 . The laminar structure wiff t x formed of planes, for example, perpendicular to the x axis, and quidistant (spacing d) (Fig, 3- 8). En the neighborhood of each of these planes, over a thickness 25, Ihe superconductivity i s strongly perturbed (H regions), in the remainder (S regions), the density of superconducting electrons has the value ns Such a model has k e n discussed in detail by Gaodman (19611, The fields h(x) (parallel to the z axis) a r e determined by the London equation

-

.

except in the thin IN) regions. The solution is of the form h =

m,, cosh ( x / h f / e o s hIJ

where P = d/2h and Hm is the field in the N regions. The f r e e energy of the f 81 regions becomes, from (3-261,

Flwra 3-8 The litminar model far the Schubnikov phase. Thin normal sheets N of t h i c h e s s -25 alternate with superconducting sheets S, The N sheets repel each other, The range of the repulsive forces i s the penetration depth h .

X1 i s necessary to add the formation energy of the

where

K

=

N regions

An. Finally, to obtain the Cibbs function, we must add a term

Qn minimising G with respect to M,

we obtain H = H ,,

SECOND KIND SUPERCONDUCTORS

the minimum G i s oibtafned for infinite P, wbieh comesponds For N H,/ ta a complete M e i ~ s n e reffect, For M r NG / G the minimum =curls for finite P. The initial field for penetration i s therefore He /Gfor the laminar model. This i s ta be compared with the result for the vortex line model,

Eq. (3-56)e H,,

=

n 'c a- I K

for

1, HCI

K

~ (K K 32

f)

H ~ / K . For HCI

H

HC/~1'2, we have

therefore, Gvortex < Glaminar '* that is, the vortex state i s nnwe favorable in the weak induction domain. It i s also possible to make the comparison in the region where H i s larger (H H,, for example), We are then in the region P f the specimen (giving a e o n t r i b t i a n S " ) As dtr (in the limit 5 -- 0) c o m e e f r o m usual the only important term in lcore the singular term In curl h, and the result is ),

The second term

may be written a s

since on the sample surface h = h* = M, Writing curl h = curl ht + curl & we can separate in S" a term involving ht x curl h*, which -isthe energy in the absence of the line, an additive constant we drop from now on. W e a r e left with

g#=

-87r

plane

j.d0*h~xcurl4

W e rewrite thie inbgraf as Jplane = Score + plane

- Jcor e

E3y maMng use of: bndon's qesuatioa, for h, in the region outside the core, we have

Score + plane

dcr * hi x curl h2 = Score + plane doehz x curl hl

-

hi i s not s l n a l a r near the line axis; thus the core contribution to the righthand side vadshes when 5 0. The integral on the plane also vantshes since 0. F f ~ l l y (h2,) = 0 g" = -#Xz /

1 3 ~ core

d@*B1X curl

-

= $oh1 ( ~ L L 87r

LNote incidentally that S = O when xL = 0, that is, when the line is just on the surface, since hz fx = 0) = 0.1 If we analyze h2 ( r ~irrto ) a direct term and an image term, the direct term gives a s a contribution to g the fine self-energy 3 = $@lEfel/4~~ The image term describes an attracMon between line and image, of value -(qo/8s) h (2xL) where h(rf is the function giving the field at distance r aE a single line (Eq, 3-35). Finally

Dlacussian H/4r) exp (-xL / h ) describes the interaction of the line

(1) The term

with the exterm1 field and the associated s c r e e ~ n gcurrents. ft has the same form a s Eq. (3-50). It is a repulsive term, (2) The term qo11(2x~,)J8n represents the attraction between the line and i f s image. The magnitude of this energy differs from Eq. (3-50) by a faetor 4. But the force derived from it has the conventiomf magnitude @@j/e[when differentiating hf 2xL ) with respect to X L , we get a factor 21.

-

(:G The aspect af fl"(xl, f for various values of the npplied field Eli i s shown

-

on Fig. 3-10. When H MC, there i s a strong barrier apposing the entry of a iine. We can understand this barrier as follows: When H = HcI, S(xL = 0) =

S(xL =

W)

Q. if we start trom

;

XL

large and bring the line closer to the surface,

S E C O N D KIND S U P E R C O N D U C T O R S

vacuum

superconductor

Flpra 8-80 Surface barrier impeding the entry of the first flux line i s a, Type ff superconductor, ( a ) When H < HGI, the force on the line alwalys points towards the surface; no lines can exist (in an Ideal specimen), (b) When H, 3 H Het the tine gains an energy ( $fie/ln)011 Me$1 as it reaches the deep inside of the sample, But there i s a barrier near the surface, and the tine will not e n k r if the surface i s clean. (c) When H fZs, the barrier disappears.

-

the repulsive term (- expt-xL / h 1 dominates the image term (-expf - 2xL / h ) .

Thus S becomes positive and we have a barrier, The barrier disappears, however, in high fielefis: 8s is clear in Fig. 3-liO4cr- m e a H > W s = @@ /4nG, It can be seen from the equation for 9 that the slope (B6/ 8 x L 1X L= 6 becomes nega-

b e shall see later from the microscopic analyeis that the field

defined is of the order of the thermodynamic critical field H,

.

Hs thus

The conelusion i s that, at field If < Ha, the lines earnot enter in an ideal specimen (although their entry i s therm&ymmfcally allowed asr soon as H 2 Hcl). These surface barrier effects have been predicted independently by Bean and Livingaton and by the Oraay group, They have been sbsesved wperlmenblly an lead thatliurn allays (Tomahiih and Joseph) a& on niobium metal (&.E? Blais and de Sorba). (The sample surface must have very few irregularities an the aeaile of h.)

Vortex tine Motions Consider the twa antiparallel vortices of Fig. 3-11, According to

Eq, (3-50) they attract each ather, Will they move wder the action of this foree, or will they stand still?" This question is very much debated at the present time. My personal belief is that in a pure metar! each line will drgt In the other" velocity field, They will thus bath move at right angle from their common. plane, with a velocity

where v,, is the superfluid velocity at point 2 due to the presence of fine 1.

Fimre 9- l% T w o antiparallel vortex line8 In a pure superconductor of Type PX, vs, (vsZ) is the superfluid velocity induced by line X (2)" Each line drifts with the local superfluid velocity v. F o r that particular geometry b t h line8 go with Ure same vafaclEy. Hate thsct V Ss asrmal La the plane ABCB of the lines,

Suck drift motions should lead to amusing collective modes for an assembly of vortex lines in a very pure metal of Q p e II. (P. G . d e Cennes, 5. Matrieon, 1962,) In d i r t y superconductors, on the other hand, friction between the lines and the lattice will dominate the motion. The two antiparallel vortex lines AID and CB will then move toward one another, as shown on Fig, 3-12, with a velocity

where f i s the attractive force between the lines, a s given b_y Eq. (3-52) and q is a viscosity coefficient, W e can estimate 72 with the following assumptions: suppse that the currents due to fine I a r e not, distorted near the c o r e of line 2, This c o r e then e a s r t e s a current density j = nev,,. But this core is essentially normal. Thus we expect a loss (per unit length of line 2)

where a = ne2r/m is the 11orrnalstale conductivity and 6 the core radius, This power dissipation must also be equal to fvdrift (2) = l/qfz. Recalling from Eq, 63-52] that f = -$ nhv,,, we obtain

Viscous motions which are reasonably well described by this type of damping have been observed in dirty m a t e r i a l s by Kim and eoworkers.

Ffgurb, 3- 12 Two antiparallel vortex lines in a dirty superon duet or of Type E: the lines move toward each other with, a drift veloci@ eontrofled by fri~tian with the lattice.

Up to now, we have restricted our attention to the reversible behavior of second type superconductors. We have seen that when the coherence 1enGh 6 is small, they can remain superconducting up to very high fields, of order .H IZ From a technical point of view, however, what is most interesting is to obtain superconducting wires that can carry high currents. But this condition carnot be realized a t thermal. .~tquilibriurn,a s shown by the fallowing argument: Consider a cylindrical wire of radius a; earrying a total current X, (When I I s we&, this current is, in fact, entirely carried by a surface sheet, of t h i c h e s s X , around the cylirmder,) The field at the surface of the wire is

.

.

The situation is stable when H < Hcl When H > Hcl, vortex lines begin to appear. They a r e bent in circles (following the lines of force), Once created at the surface, with radius a, they tend to shri& (to decrease their line energy) and fimlly annihilate near the axis af the wire. This process dissipates energy. Thus in an ideal specimen we have O resistance only if H < Hcl o r I < (ca/2) Hc, If we want to carry higher currents with our wire, we need to Pi@ the vortex lines, that is, to quench their motion by suitably chosen kttice defects, and achieve a nonequilibrium situation. While the field Hcz is an intrinsic property of the metal (or alloy), the critical current measured on a wire is extremely sensitive to the metallurgical state af the sample, This distinction between the factors ruling He, and I was stressed f i r s t by Garter, h practice, a favorable defect structure i s obtained by the fallowing procedures: (1) imperfect sintering (e.g., m,Sn) (22 cold work (e,g,, MoRe alloys) (e.g., lead alloys) (3)precipitation processes The resulting materials, with high critical eurrexlts, a r e called b r d supercond~clors, The coupling mechanisms between the lines and the defects are only vawely h a w n at the present time. A rather simple case is met when we have large caviUes, due to imperfect sfntering, in the superconducting material, A vortex lends to remain pinned to the cavity, since this corresponds to a smaller len@h of line in the superconducting materhl, and thus to ~lllitllerline energy. The mechanical stresses realized by cold work impose slight modifications to the condensation

SECOND KIND SUPERCONDUCTORS

83

energy and to the local density of superconducting electrons ns . This results in local modifications of k, 6, and thus of the line energies 3 and interactions U. These interactions a r e rather compf ex, and in the following we only present a phenotnenolagical description of their ef f ects,

-

Critical State at Zero Temperature Consider a hard superconduetor in a n applied m g n e t i c field H (along Oz). In equilibrium the line density would have the value B(H)/@, and be the same a t all points. We now cansider a metastable situation where the induction B is not e m to B(H) but varies from point to goint -say in the x direction. Thus (1)the line density is w t constant, (2) there is a macroscopic current S = (c/.rln)(eB/ax) flowing in the y direction. The forces acting on the line system can be decompased in the Eoffowing way: First, because of the repulsive irrl;eraclions between lines, the re@ons of kfgh Llne density (high B) Lend to expand towards the regions of low density. This may be described in terms of the pressure p in our two-dimensional line system,VThe force (per cm5) is - a p / a x . This has to be balanced by a piming force due to the struclural defects. This pinning f oree, however, eamot become a r bitrarily large, It must stay below a certain threshold value a,

ltl" at some point is larger t h n a , then the lines start moving and dissipation occurs until condition (3-70) is again satisfied, h practice the line density (I/ql,)Bfx) will thus adjust itself s o that the threshold condition is just realized at a l l points [equality in ( 3 - T O ) ] . The state thus realized is called the c~ciliealstale, and was first described by Bean, We can get some physical feeliw for this critical state by t h i m n g of a s a d hill. Xf She slape of the sand hiff exceeds some critical value, the sand s t a r t s flowing downwards (avalanche), The analo~;yis, in fact, rather good, since It has been shown (by careful experirnelll with p i e h p coils) that, when the system becomes overcritical, the lines ds not move by single units, but rather in, the form of avalanches ineludiing tmically 50 lines or more. We now proceed to compute expXicitfy. the p r e s s m e p of the fine system, Ito be inserted in (3-701,We consider a group of N l n e s intersecting a s w f a c e S in the xy plane, Their energy (per cm along Oz) is

'we make an isotropic approximation and neglect the tensor properties of p.

SUPERCONDUCTIVITY

where G is the thermodynamic potential introduced by Eq. (3-64). The pressure is obtained from 9 by differentiating with respect to S (that is, to the volume) at constant N.

For fixed N we have dS/S = -dB/B

Thus when we know the form of the thermodynamic potential G, we know p(B), that is, the pressure-density relation, It is, in fact, possible to write the pressure gradient ap/ax in a very simple way. Set

where F(B) describes the line self-energies and interaction. The equilibrium relation B(H) o r H(B) is obtained by imposing aC/aB = 0. Thus H(B) = 4~(8F/aB). NOW

Thus, if we know H(B), we can immediately compute the pressure gradient as a function of B and a B/&. In fact, in materials where H,& > Hcl, H(B) is nearly equal to B a s is clear f r o m the magnetization curve of Fig. 3-2. In this region we have simply

The critical state i s then defined by

S E C O N D KIND S U P E R C O N D U C T O R

dx

- supercsnductor

vacuum

Figure 3- 13 XrreversiMe penetration of flux in a hard superconductor: (a) %an model; C b) AndersonUrn model (parabolic profile),

To compute &(X) in the critical state we now need to b o w what is the dependence of the maximum pinning force a, on B (that is,on the line density). It was originally assumed by Bean that a, was linear in B; that is, J was a constant in the critical state. (J would typically be Y Y of order t O5 A/cm2.) A series of experiments by Kim and co-workers on NbZr alloys and on Nb,Sn indicates that for these systems (in the particular metallrtrgieal condition realized, and for fields in the 104Qe range), a good approximation amounts to taking a, independent of B. This very simple resuit i s not fully explained theoretically at the present time, fn particular, we would expect the ginning eenters to be most efficient when their size i s comparable to the interline distance, and the latter depends an B fl&e Elwm). The shape of the profile B(x) in the critical state is represented for the Bean model and for the Kim model (Fig, 3-13), For typical hard supereclnductors and applied fields of order 104Oe, the total thf c h e s s Ax of the zone! where fliux lines have penetrated i s of order I mm, To determine B(x) experimentally [or equivalently to determine a, (B)], the s i m p l e ~method t amounts to measuring the flux @ in cylindrical samples of radius R (Fig. 3-14) for increasing values of the applied field H,

P i w e 9-14 Magnetimtion meaeurernents on a cylinder (exterm1 field increasing), The flux lines penetrate only in the hatched area,

(I) Zf R is muck smaller than &X, the indwtlon I s nearly uniform in the sample, B = B(NI), = =:R2 B(W).? (2) E R >> &X we b s e esserrgially a one-dimensional sitwtion. H x denotes the r a a a l distance, we m y write

+

At the edge of the flux ( X = R - Ax), we have B = 0, H(B) = H,& At the surface of the cylinder fx = R), again assuming no surface barriers, we have the equilibrium value clif B correspmding t;o the e ~ e r m field I H, B = BfW). Transfordng dx by (3-72) and ( 3 -TO),

.

we get

? ~assume e %at there i s no surface barrier impeding the entranee of vortex linas in the cylinder. Surface barrier8 do occur sometfmes, but their effects can easily f;M? separated.

S E C O N D K I N D SUrPERGOrJDUCTOftS

Figure 3-25 Principle of the Kirn experiments on hollow cylinders of hard superconductors. An aternat field W i s applied. The field W' iinside the cylinder i s measured,

where a m Hstands for @,[B(H)]. rivative of 4

Of particular interest is the de-

Thus from magnetization measurements in increasing fields, we may derive ar and LY (B), Another method, devised by %m and eomH m workers, makes use of hollow cylinders as shown in Fig. 3-1 5 , A field H is applied on the outside of the cylinder and the field H q i n the eylinder is measured. When H is increased from 0, W-irst stays strictly eqwl to Q, Then, when the flux f rant reaches the inner surface!of the cylnder, H b t a r t s to inerease (ideally H f would first jump abruptly to Hcl, and then grow steadily). The interest of the method i s to give a direet determination of Ax, for that particular value of H where H h t a r t s to inerease, Mare complicated sitmlians a r e met i f the field EX is alternatively inereased and decreased, as shown in Fig, 3-16, Then we meet regions

suprsrconductor

vacuum

mprs 3-16 F l u distribution in a hard superconduetor when the applied field is first raised to H, (broken curve) and deereases to Hb (full curve).

with dB/&

>

O and regions with dB/dx .( 0, but the absolute value a, This permits a detailed calculation of a l l hysteresis cycles when a m (B) is known.

Iep/ax I stays equal to

.

Flux Creep at Finite Temperatures At finite temperatures, if aplax $ 0, the vortex Unes will tend to move (from the regions of high B towards the regions of low 23) by activated jumps across the piwing barriers, We call, the average flaw velocity of the lines (in the x direction) v,. Various methods can be used to detect this flow, or 6gcreep'?: (I) Magnetic measurements, with thick cylinders o r hollow cylinders, In the b t l e r case, for instance, if H has been raised from O t a some value and then kept constant, we observe that H-increases slowly in time. (2) Electrical measurements, E the lines of force are moving, they create electromotive forces thzlC can be measured directly. The mast simple situation is represented in Fig. 3-15, A wire (in the y direc lion) carries a current of density 3, and is submitted to a n external

SECOND K I N D SUPERCOMDUCTQRS

-

\

vortices

vortex drift velocity

Figure 5- 17 Electrical measurements on a hard auperconduetor w i r e , There is a current J =: -(c/4.n)i(aB/@xl in the y direetion, thus BB/ax 0. The lines are more closely packed on the left side af the wire: they drift with a velocity vx towards the positive x axis,

field H in the z direction. Thus we have a nonzero 8 B / @ x =: 4 % ~ / c and the lines (pointing along Oz) tend to drift in the x direction. The resulting electric field E is along the wire axis (Oy)and in the limit Y H >> H c l , it is given by

To prove 13-76) we compute the power dissipation per unit volume; this is the work done by the presence of gradient on the lines,that is.[@p/ax)~~. By setting this equal to Ey J and making use of (3-72'1, we get (3-76). Thus we need an electric field Ey to maintain the current J. As pointed out by Andersan and mm, this dissli;pative effect in the superconducting slate explains many feratures of the resistive behavior of h r d superconductors, The main difference between (1) the magnetic measurements and (2) the electric ones l i e s in the o r d e r of magnitude of the velocities involved. In ease (1) the c r e e p typically is measured over intervals of hours o r days, and the velocities a r e of o r d e r 1. m m / d y or 1Ome cm/sec. In c a s e (2) taking B = IQ4Ey = 1 pV/cm, we get v, 10m2

-

cm/sec. The main difficulty of (2) is related to possible inhomogeneities in the wire; experimentally it is found that different portions of t h s~a m e w i r e have different E y P s .

The results show ummbiguousfy that (a) the creep velocity has an aotivation energy behavior

v, is not very accurately h o w n , but may be in the range of IQ3 em/sec in typical eases. The energy E may be as high as 100°K. (b) the energy E depends on the pressure gradient ap/ax

where p has the dimensions of a length, and i s typically of order 500A. We can relate E, and p to the critical pressure gradient a m , if we notice that f o r T -- O the velocity v, will depart from 0 only when

E = O

These results (mainly obtained by Anderson, Klm, and co-workers) have various important consequences. First, since v, varies rapidly with E according to ( 3 - T T ) , it is possible to extend the crictieal state concept to finite temperatures. Define a limiting velocity vmin below which the line motion cannot be detected, Then, if

E

P

>

v,

log -'min

the line structure i s frozen, Thus the critical state at temperature T corresponds to = kgT log

v, -

'min

fog

v0

'min

In general a, will depend on T (since a, involves h , t , and the condensation energy, which a r e all temperature dependent). But if 1: is mueh smaller than the transjtion point T, this dependence may be neglected and all the temperature variation in (3-80) comes from the

S E C O N D K I N D SUPERGORTDZTCTORS

factor kgT A linear dependence of

1dp/dx

91

on. T' has indeed been

observed experimentally by Kim on various alloys and carnpundsthe critical currents of hard super canduetors a r e strongly temperature dependent even when T > kg T, , it is clear that the integral has the asymptotic form in ( R M ~ T/,~) +~ C since the hyperbolic tangent is equal to unity in the major part of the domain of integration. A dehiled calculation gives C = ln 1.14, Therefore

where BD is the Debye temperature deduced Taking RwD = f ram specific heat measurements, one can determine the coupling canstank N(0)V from T;, The values for the nontransition metals are given in Table 4-1, In most cases, the coupling constant is rather small and To> wD ). As before, we neglect completely the dependence of the matrix Vkp on the angle between k and $, We write

Equation (4-124) f a r the transition temperature i s now replaced by

Unfsrtunatety, even with the simple interaction (4-128), this integral equation is hard to solve, and we shall discuss it in a sloppy wa_y, We separate tAe Coulomb t e r m and call its contribution A

A,

is independent of 5, The q u a t i o n for

A is

When 14 / 5 BwD the integral is not large, because the factors YP and 1/r cannot be simultaneously large, Thus in this region w e may roughly set A(4) = A, On the other hand, the integral is important when 1 6 1 is small. Call B the average value of L\ in the region fiwD > 1 6 1. Then we have

2

.

PJ(O)V fog R w ~ A

P

kg To

where V i s some average of V ( W ) over the interval -wD C w P F Finally, Eq. (4-130) defining A may be rewritten as

(2) The isotope &feet is modified: If the ionic mass has a relative variation BXII/M, the Debye frequency is changed accordingto 6wD/wD =

-

4 ~ M / Mand the transition temperature shifts according to the law G @c

1 + K& log 92

The amplitude of the isotope effect is reduced, This reduction should be particularly strong in metals with narrow bands (W, small). This may explain why the isotope effect i s strongly reduced in transition metals and in related compounds (Garland, 1963).

Ga, Zn,Cd Al @

\

\

'\

a

Mb-Mo, Nb-Zr

+

Mo-Re Nb-Re, Mo-Re(@-Mn) @-Phases (Elements from 2nd and 3rd transition row)

* Nb-Ru

.

4

* Pb @Q ~2 Ti 0s

Mo Re

\

"9

Mo-Ru, Mo-Pd, Mo-Te Transition Elements

a o

A

c3

Zr

h

+ o e

Ir Ti

B-Metals

@@I '-a\ 0 @

A

~h @ ' @ + ' a .

2%

%@-, **-.

63,-

Nb

-- - ----La

NbSn

-91

V

0

1

2

3

4

5

6

7

8

9

1

0

m KZ

J male

Fiigure 4 4 describing the eleeP tron-electron interaction via phonons, and the electronic specific heat parameter y (courtesy J, Muffer), An empirical relation betvJeen the coupling constmt K

Conversely, from the experimental values of T, and of the isotope effect, one can deduce Kp and KC , assuming that the model is valid. The results for Kp in various metals show a strong correlation be-

tween &$and the electronic specific heat coefficient y in the normal phase8 (Muller, 1963) (Fig, 4-5).

Calculation of the Thermodynamic Functions Combining the expressjons for the kinetic and potential energies (4-114)and (4-115) together with the expressions derived for uk and vk in the previous section, we obtain for the total energy '7 i s propartiaml to the density of states at the Fermi level, There is, however, one difference between y a d N(0): y i s defined per abrn, while N(O) is defined per unit volume.

DESCRIPTION O F T H E C O N D E N S E D STATE

1 29

In particular, at absolute zero (f = O), we recover the energy calculated in Section 4-3, Writing the entropy (4-1f 8) we find

The specific heat, for example, is determined by

Recalling that ~k depends on T in (4-136),we find

In t h e BCS approximation, r k dtk/dT =

L%

d ~ / d Tis independent of

k. Then,

The form of this specific heal is represented in Fig, 1-1. At very low temperatures (@A >> l), the term A d ~ / is d negligible ~ and

w h e r e h, = AT

The integral i s

The dominant factor in C is the term @-pa@, which we have already predicted. A, measurement of the low -temperature speeif jc heat allows us to determine L%,, Also, Eq. (4-1 39) predicts a discontinuity in the specific heat at the trmsition temperature because of the A dAld7i' term

Numerically from (4-125), one obtains (dh2/@) T~= 10.2/@,9 and C,

- Cn

= 10.2kL Tc N(0)

(1-142)

Fimlfy the Gibbs function i s obtained by adding (4-135)and (4-1 36).

This expression can be transform4 by noting 'that

The firt33t term on the right i s determind from the self-consistency condition (4- 124) and i s - h 2 / v , Then

&owing E, we can calculate the critical, thermodyn;tmie field defined in Chapter 2, from the equalion

In particular, for T = 0,we have Hc = He,, %cl

-- ~ R M ( O l) A(@)l2

For finite T, from (4-144), we find a curve Hc (T) rather close to the

empirical approximation

DESCRIPTIQN O F THE CONDENSED STATE

131

E we look at finer details, we find that, for superconductors where N(O)V is not too large (weakly coupled superconductors), the detailed theoretical curve of l& versus T derived from the BCS theory gives even a better fit to the experimental data than does the simplgied law (4-146).

Calculation of Transition Probabilities Suppwe that a time-dependent exterior perturbation is applied to a gas of superconducting elee trons. Examples: (1) U;lf~asonics: A longitudinal, acoustic wwe modifies the potential e n e r u of each electron by a term W(rt), where t? is the local difatation of the lattice and U is a constant (the ""deformation potential") of the order of several electron volts. The matrh elementlt of this perturbation between the electron plane wave states k and k' i s UBk -k, [where Bq is the Fourier transform of ~ ( r ) ]The . perturbation acting on the electron system is then

(2)Microwaves: The flff ee t of an electromagnetic perturbation described by a vector potential A(r , d) i s given by replacing p2/2m by (112m)[p - le/c) N2 in the electronk energy. To first order i n A the perturbatf on is (- e/2mc) (pA t. Ap) and a s a function of a and a* it bcacomes

More generally, the perturbations have the form X, =

2 k"

B(ko! 1 k'a ') aka akt,, ?

W e find two &f ects in the presence of IK1; : (l) X, induces transitions between the dilFfererzt excited states described by the y* To classify these transitions, w e write K, as a funetion of the y , y" operators, Irmverting (4-100)

.

SUPERCONDUCTIVITY

This gives

k'a'

The t e r m s in y' y . and y . y' describe t ~ a n s i t i o n swhere a quasiJ 1 j particle in the state i is scattered into the state j (and vice versa). The y; y; create two quasiparticles and the t e r m s y i y j destroy two of them. (2) X, can a l s o modulate the parameters, such a s A, which describe the structure of the condensed state. and this modulation also leads to an absorption. In many c a s e s this effect i s negligible; consider for instance the ultrasonic attenuation problem. The direct per turbation i s U$(r). If the pair potential is also modulated, it will shift by an amount 6A = CA0(r), where C is a constant of order unity. This gives a modification in the generalized self -consistent field, But U i s of order 1-10 eV and CA i s of order 10m3eV, Thus the modulation of A i s unimportant here. In the present paragraph, we neglect it (but we return to it later when we consider the Meissner effect), We now return to the form (4-151) f o r X, and consider transitions where a quasiparticle passes from the state (k'a' ) into the state (krr ). The matrix element M(ka I k'a ') is the coefficient of y i , y k t a t . The f i r s t two t e r m s of (4-151) contribute since y,yb = -yi y, for 1 Then

The term Zod B(- k'a' I - ka)p,t,

t

p,

2.

is essentially the matrix ele-

ment B where the spins and momenta of the electrons are reversed.

DESCRIPTION O F THE CONDENSED STATE

133

F o r the interactions X,, which we consider, it differs a t mast from B ( h 1 k'a ') by a sigrz

with

=

+l -1

case I

ease11

Then

M(ka 1 k'cu')

= B(ka k'a') [ u p k ,

- qvkvkf]

(4- 154)

The factor [ukukf - qvkvkt] is called the coherence factor of the transition, The number of transitions ( k k t -- ka) per urrit time less the number of inverse transitions is

where we have assumed that X, i s a sinusoidal perturbation of f r e queney w , The purer ab~orbedi s (4-1 ss)

In o r d e r to calculate W,, we f i r s t average over the angles of k and k b a d the spin indices of the matrix element B,

a r e small with respect to EF [this is ensured Since I Ek 1 and I by the factors f in (4-155)], B a n be treated as a constant, and W, = 2 n w ~ " ~

oQ

H N s ( a ) N , ( ~ ' de ) de' (uu'

-q~v')~

In this formula, Ns(e) = N(O) / d k / d ~1 i s the density of states for the Bogolubov excttatians

By using the definitions (4-71) and (4-72) of u and v, we ean write

SUPERCONDUCTIVITY

134

Each value of 6 is obtained for two opposite values of 5 . The term &j4/€€'disappears when summed over these two values. Finally

[ f ( f f )- f ( ~ ) ] 6 ( f-

C'

- hw)

(4-160)

An analogous calculation can be made for the power W, absorbed by creation and annihilation of two quasiparticles. W, is nonzero if hw > 2A. The final formula for W = W, + W, differs from (4-1 60) only by the domain of integration,

Here E and 6' a r e of arbitrary sign, but / E l > A and 16' I > A. In practice one always compares W to the absorption WN, which would be obtained in the normal state[wN is simply obtained from (4-161) by setting A = 01,

X

[f(c ') - f (C)]&(C-

E'

- Ru)

(4- 162)

APPLICATIONS (1) Absorption of sound: In this case q = 1 and the coherence factor i s small when E and E ' a r e simultaneously close to A or -A. If we study the case where bu is small compared to A or kg T, then (4-162) reduces to

The attenuation is very small at low temperatures and increases rapidly a s T -- To. This therefore gives a way to measure A(T),which has been widely applied, notably by Morse and his co-workers (see Fig. 4-6).

D E S C R I P T I O N O F T H E CONDENSED STATE

Fipre 4-43 Oftrascmie measurements in tin compared with the B6S prlEdiieti~n1a;fter R, W, Marse, l B M J , , 6, 58 (1963)j.

(2) Nuclear Relaxation: The interaction X, between the nuclear spins and the conduction electrons is complicated, but at any rate Case IX applies: q = - 1, One can measure the time T, far the nuclear spins to equalize their temperature with the electrons in zero field, The ratio T,,tT, of the relaxation rates in the superconducting m& normal states is still, given by 14-1 62). The frequency W, which, comes into play here, is the precession frquency of the nuclear spins in the local field of the other nuclei. This is small (w m f04). Therefore we can let u O and obtain

-

Here the coherence factor doesn? vanish at the =me time a s the denominatsr (for l e t = A): the integral diverges logarithmically, But in a real metal, Ak i s anisotropic, The singularity in density of states FTs (E) is then smeared out somewhat and the integral converges. The results depend in detail on the anistropy of At In practice, A is rather weakly anisotropic; then the variation of T,,/T, with temperature is as shown on Fig. 4-7. The relaxation rate for T slightly less than Tc is greater than i~ tke normal state. This r e m a r b b l e result a r i s e s

0.1 2.0 2,s 3.0 1", 11" Fimrs 4-7 Nuclear relmalion rates in aluminurn [&er A. G, Xtdfieild;, Phys. Reu,, 125, 159 6196211. Notice the dip of Ti at ternperrttures bellow the tr~nsittonpaint, The theoretical curve is calculated on the assumption that the peak in, the BGS density of sLates is smeared out c m an e n s r a interval A/5. 1.0

1.5

from the increase in the density of states N,(E). At low temperatures

T 0. The region z Q i s either empty o r filled with an insulator where the boundary condition (6- 13) applies. In weak fields, and to first order in h. I d: I q n the region z > 0 can be replaced by its equilibrium value in the absence af a field iji, l Z defined by (6-15).Because 30, is independent of position, when we take the c u r l of both sides of (6-12),we obtain a London-type equation: curl j = -

4e2 mc @,2h

On adding Maxwell's equations, we find, a s in Chapter 2, nonvanishing solutions only if h is in the xy plane: for example, taking h along the X axis,

where A, is the equilibrium value of the pair potential. Equation (6-22b)allows C to be related to directly measurable q ~ a n t i t i e s .Note ~ that MT)is proportional to $;l and therefore to (To- T)] For a pure metal in the free electron approximation, the microscopic caiculation in the BCS approximation gives

LT,/

%ate that for a noncubic crystaJ it is necessary to take into account the tensorial nature of C .

where k L ( 0 ) i s the London penetration depth at absolute 0,given in terms of the number of electrons per em3n by the equation h 2 (0)=

4nnez/me2. Exeept far the numerical coclfficient, (5-23) can be predicted f r o m (6-22) if we guess (as stated earlier) that C NN(O)4: and tht

Conclur~ions, The Landau-Ginsburg hyaothesis, in particular the form of the? free energy (6-7) in the presence of a field, has led us to a local relation (6-12) betvveen the current and vector pokntial, We h o w from the microscopic analy s i s of Chapter 5 that in the special case where 1 Li 1 is constant and h small, the exact refation i s nonlocal; that is, the current density jfr) depends on A{r') 6;& in a pure metal, In order for the local approximation to be for I r -. r" valid, i t is necessary that A or the current have a slow variation on the scale of 4 therefore, h (T) ?> 5 ff

F o r a pure metal, this condition will be satisfied if 11" is sufficiently close to T B . However, for certain nontransition metals, notrrbly aluminurn, h~ (0) i s so mueh s m a l e r than t o that the temperature interval, allowed by (6-24) is very

smltl1.f

(c) We have defined two characteristic lengths ((T) and h(T),which determine the behavior af a superconduetor near the transition pinE. They both diverge a s (TO- T)- ln a s T -. T,. It i s themfare partieularly interesting to form their ratio

By using the definitions (6-17) and (6-22a) of t(T) and ~(9'1, one ob-

tains

rr

i s called tlze

dm-G$@sburgparanzekr of the substance. When

? 1 (A < 41, the material is of the f i r s t kind, when n 5 1 ( h > [). material is of the secand kind, W e see later that the emct separation between the two types of behavior occurs for K = 1 / 6 For a pure subsknce we show later that K

It- i~ interesting t o notice a;att K cm be defined directly from expesimekhlfy obkined numbers on the penetration depth a& thefmactynarnie field IXc measured in the region of validity of the LandauCiinsburg quations, By maklng use of (6=22a), (6- Is), (6-91, and (6-81, on@can put (6- 26) in the for rn

Another way of writing (6-28), &ten useful for numerical calculations, is the following

where Q>, = ch/2e is the flux quantum. In this chapter we a l s o shby other methods (Jlat allow tfie determimtion of rc,

SlTUATlOtJS WHERE

6-5

IS CONSTANT

W e now apply the Landtau-Ginsburg equations to some concrete examples, First, consider the particularly simple c a s e s where the amplitude 1 $ 1 of the order parameter is the =me at all p i n t s , Such a situation has already been encountered (the penetration depth of a weak magnetic field in a bulk sample), The emxnples we now consider a r e rather differeat. They coneern thin samples (films, wires, and s o on) where Jt cannot vary rough the depth withaut catastrophically inereasing the term I V@1 in the f r e e energy, But allow the field h (or currents J) to be strong and 1 +J 1, though consbnt, will not necerssarify be equal to i t s anperturb& value +@,

Critical Current in a Thin Film The sitmtion envisaged is that represented in Fig, 6-2a. The film of thickness d c a r r i e s a current density j along the X axis, W e a s aume sat d

d] seem to be in agreement with theory. Wpically jc i n this temperature region, The variation of I @ I o r I h l with current could, in principle, be followed by a tunneling measurement o?f the gap, (Detailed calcuiations show that, for the present g e o m e t y , the gap in the excitation spectrum is nearly q u a l to I A l .) There are two precautions that muslt be taken in the discussion of critical current experiments: (I)When h ( T ) dr i t is necessary to take account of the variation of the current density through the thickness of the film, mis calculation is feasible bczeause in, the region of validity af the Landau-Ginsburg equations when l $ l is constant in space, the current 1 obeys a simple London equation, ( 2 ) When d < ((T = 0) in addition to d < &(T), the bulk LandauGineburg equations no longer apply (see problem, page 225).

-

The Experiment of Little and Parks C~11siderm w a ~ u p e r ~ ~ n d u ~ film t i x l gdeposited on a cylindrical insulating support af radius R a s i s shown in Fig. 6-3, The film has a thickness d 6, the right-hand side inright-hand side as 1 + ( ~ f ~ ) ~ / E c r e a s e s faster than the left. Since the left-hand side diverges for E = l, there must be an intersection between fp = 0 and f e = 1. On the contrary, there will, be no intersection for s < 45 (see:Fig, 6-5b). Therefore, if d > ~ A ( T )we , have a first-order transition with a critical field Hp deduced from (6-61) and (6-60). E d C ~ A ( T )Gs ,

always remains l e s s than G, when the solution exists; 1(, decreases a s H increases, according to (6-54) and finally vanishes far

m

H = H;! = - - ; - H c

The transition that o c c u r s at Hi is of the second order. These two types of behavior, depending an the ratio d\h(T), w e r e predicted by Ginsburg in 1952, They have been qualitatively verified in a s e r i e s of tunneling experiments by Douglass (1961) (met;al used: aluminurn, 5, = 16,000k).He showed that in the excitation spectrum w a s a decreasing (1) The energy gap function af the field. (2) For T S 0.75T,, in films with thicknesses d > 3.500A, the gap dropped abruptly from a finite value to 0 when the critical field was reached. (3) At the same temperature, in films with d C 3.500A. c , dropped smoothly to 0 when the field was increased up to the c r i t i c a l value, These r e s u l t s are represented on Fig. S-5c. The distinction between f i r s t - and second-order transitions i s very apparent, Unfortunately it is not possible to go beyond this qualitative statement far the following reasons: Both &(T)and d are much s m a l l e r than 5, in a l l the films. The c u r r e n t s js have a rapid variation on the scale 5,) (note that j, rev e r s e s i t s sign when one gaes f r o m one side of the film to the other). In such a situation the Landau-Girtsburg equations cannot be applied, ~ the energy even ifone allows f o r a change of K with t h i e k n e s ~ ,Note: space variations of js depend an two parameters d and h(T). Thus, in general, an ""effective K" "far the present geometry would depend on h ( T ) and lose i t s intrinsic significance. Furthermore, even in the limit hfT) > > d, this effective K would be numerically diiferent from the one required by a critical current measurement on the same film.

LATVFDAU-GIPJSBURC; E Q U A T I O N S

Approx. Thickn$ss a3UUOA D rl00sIt

Reduced Temp. 0,7"1

d/A

0.774

:$.G

5.9

Figure 6-6 (a) A plot of the left- ancl right-hand sides of Eq, (6-61)as a function of the normalized o r d e r parameter fl. Left side: broken curve. Right side: continuous curve, The two curves have an intersection if d 2 G h l ~ ) .(b) The dependence of the order parameter a t the first-order transition as a function of E ;= d/A(T). Note that for E 6 there i s a seeand-order transition a d the critical value fa i s zero. (c) Tunneliw measurement of the energy gap versus magnetic field in two aluminurn samples of different thicknesses. For the film with E 2 8, note the first-order transition; for the thinner film we s e e a much less abrupt behavior. [After D. Douglas, J r IBM J . &S, Deuelop., 6, 41 (1962).)

gap 6, is different from 1 A / in thin films under strong magnetic fields. In particufar there is a finite region of the (H,T) plane where e0 = O while A tt O (gapless superconductivity). This complicates somewbat the anailysis of the lunneling results. Pro'ttlem, Consider a superconducting eylilldrical filrn (radius R ) of thickness 2d (d -9:s) h a s been carried out by Kinsel, I,ynton, and Serin on a n alloy lnBi with 2.5% Bi, an example of a Type I l superconductor with a nearly reversible magnetization curve. The phase diagram deduced from their magnetic n~easurementsan this alloy is shown in Fig. 6- 8. From thc experimental values uf Hcz near To, one finds

LANDAU-Gf N S B W EQUATIONS

Fiwrt3 6-8 The phase diagram of a Type IE superconduetor CInBi alloy). [After T,E n s e l , E. A. Lynton, and B. Serin, PCiys. b t t e r s , 8, 30 (1962).j From (6-9% taking BA = 1.16, one finds from the magnetization curves

This very good agreement7 is further c a d i r m e d by a theoretical calculation of K described fater. for

Problem, Show that the wall r?nerm i n a Type X superconductor vanishes K = ~/fi.

Solution. W e follow a method due to G, Sarma, By taMng the fields h to be along the z direction, we first reduce the Landau-Ginsburg equation to a twodimensional form

where

= p

- 2eA/c.

By putting Ilt = Ilx

*

illy, we have

he agreement i s even surprising since the experimental magnetization curves a r e not perfectly reversible, which implies a certain uncertainty in

For these particular ""srxna solutiansw the field h a t any point i s a. b o w & function of the order p a r a m e k r JI a t the s a m e point, given by

We muat furthermore e m u r e that the fields and currents associated with the

Sarma ~ o l u t i o na r e self-consistent:

This relation must agree with the h(+) relation obtained earlier, Differentiation of the h(#) refation gives

This may be transformed in the foflowiw way: The condition R + @ = O may be wriLten as

from which we obtain

This does agree with the equation for the current provided that

-

-

-

Consider now mare specifically a wall in the y z plane, F a r x -. - m (nor(superconducting side), h O and mal side), h H, and @ 0, For x

-

l i2 = - - a / @ .These two conditions are in agreement with the h(#) relation for ' the Sarma solution does apply to the present problem. the ~ a r m asolutisn-thus The thermodynamic potential {per unit area in the y z plane) g is given by

The last t e r m is the microscopic analog of the -BH/4% term, the t h e r m d y namie field W being here equal to H e tSince + satisfies the Landau-Cinsburg equation, 6 can be inteirated by parts to give

we subtract f r o m g the t e r m .f (H: /an) dx, which corresponds t o the potential In the normal ( o r in the superconducting) phase: To obtain the wall energy

The h(+) relation can be rewritten as

Thus gwall = 0 for n = 1/a. A similar ea1euIa;tion in t e r m s of Sarnza solutions can be done for t h e energy of an isolated vortex line, when K = 1 1 6 . The conelusion there is that the f i r s t penetration field H i s equal to H, . F o r that particular value of K . Ci

H,$, Hc2, and H, coincide. REFERENCES A mueh more detailed study of the various applications ai the Landau-Ginsburg equation is to be published by Saint-James, Sarma, Thornas (Pergamon).

MICROSCOPIC ANALYSIS O F THE LANDAU-CINSBURG EQUATIONS

7-1

LINEARIZED SELF-CONSISTENCY EQUATION

Xn the preceding chapter we canstrueted the Landau-Cinsburg equations from a postulated form of the free energy F, which introduces an unbown coefficient [for example, the coefficient C of (6-V)] giving the energy associated with spatial variations in the order parameter, Garkov (1959) has shown that it is possible to establish the LandauGinsburg equations from the miicroscopie theory and to cafculate in particular the coefficient C, Here we give a simplified version of this calcufalian:

A Treated as a Perturbation Our starting point will be the self -consistency equation

where un and v, are solutions corresponding to positive eigenvalues of the Bogolubov system

.M.

a

CI

51

-Ef-

C

4a

-52

fir

'i

m

*3

k

Q1

cW .a

9

3,

-4

2Q p

S ;=: B Q) B cu"

CI\

tr

& 3

U

m

.S C

b G

'C"

'"(

.(a,

U)

Q,

2 8

Q1

m

3r

-Q.

E

.

a, 'a 3 C & c d

Q,

g

g

9

*

g

5

g

m Ei

% &-. k

W

at: 3

rr,

k

it=

E

w

-8-

k

yc,

"r.r

11

4 %

8

Q,

E

CI

us"

W

E:

I -

C-)

7

We can now insert u1 and v-defined by (7-6 ) and (7- 7) into the n n self -consistency equations (7- 1). Ws obtain to first order :

where

:U

i s nonzero only for

tn > 0, while

v: is nonzero for

tn < 0. Also

the function 1 - B((,) = tanh (Pt,/2) is odd in [n. These remarks emble us to collect the uu and vv t e r m s into one s h g l e component, Finally, we can symmetrize the expressha with respect ta the indices n and m (since it i s to be summed aver n and m) and abtain

K(s,r) =

C

fZ . m

+ tanh

tanh tn

+

trn

It is sometimes useful to transform. this result in the following manner: Let, us write

t

tanh where E w

=

28kgT ( v + $) and 22 represents a sum over all positive

o r negative integers v , Equation (7-10) can be verified by eomgaring the poles and residues of both sides of (7-10)in the complex ( plane, (Also note that we can use either t i in the denominator of (7-10)since b t h u and -W contribute.) Then

ANALYSIS O F CANBAU-GINSEffRG E Q U A T I O N S

213

and

E q u t i o n (7-8) with the explicit form (1-11) far the kernel constitutes the Iinearized form of the self-consistency condition (7- 7). The enormous advantage over (7-1) is t h t the functions u and v a r e elimimted, and a(r) is then the only u w o v v n in (7-8). However we must remember that the linearized f o r m only applies f o r very small h, that is, in. the immediate neighbrhood of a secand-order transition.

Separation of Magnetic Effects We now assume that the s p a t h l variations in the vector potential, are small. Then the eigenfunctions qn in the normal metal in the presence of A differ f r o m the eigenfunctions wn in the absence of A by only a phase factor.

where the W, a r e taken to be real. It is easy to verify that (7-12) is compatible with (7-4) and the definition of the W, if the spatial variations in A, a r e systematically neglected. What is the range of validity of this appradmation? (1) WE?s e e later thgt the range of the kernel K(a,r) in ;z p u ~ emshE is of the order of 5, = 0.1 8 AvF /kg T, . The spatial variations in the vector p o t e n t b l must therefore be smll on t h i s scale. A condition is thus obtained by requiring the field h to vary slowly, which implies that the penetration depth be large with respect to 6,

SUPERCONDUCTIVITY

214

(2) A. slow variation of h = curl A does not gwrantee that A varies slowly, Over the distance I s - r I -. to, A can vary by -6,h. This results in an uncertainty in the p b s e

This must be small compared to one, and we a r e then led to

where wc = eh/mc is the cyclotron frequency of electrons in the normal metal in the fieid h, For a bulk, first -Mnd super conduelor, h is at most equal to the thermodynamic field H, (T).From Eqs. (6-25) and (6-19) we have, for a pure nnekl,

where K is the hndau-Gixzsburg parameter and G, = hcl2e is the quantum of flux. The condition (7-1 4) then becomes (T, - T)/T, 4 K. This is less restrictive than condition, (7-13) which can be written as x > - T)/T,]U2. For a second-kind superconductor, h is at most of the order of .H (+,/{g)[(T, - T)/T,] and (7-14) becomes (T,- T)/T, C 1. (3) For (7-12)to be correct, it is evident that the radius R = m c v /eh ~

[m,

-

of the electronic orbits in the field h must be large with respect to the range of the kernel K(s,r). This requirement assures that all ef f ects r e h t e d to U n d a u diamagnetism a r e negligible. The condition can also be written R 2

again less restrictive than (7-14). We conclude that for pure metals the substitution (7-12) is valid provided that T is close enough to T , , This, as we shall see, leads directly to the Landau-Ginsburg equations, For "dirty" alloys, we show later that the situation is even more favorable, and that the linear integral equation (7-8) can be replaced at all temperatures by a second-order differential equation of the Undau-Ginsburg form, Finally we a r e led to

ANALYSIS O F LANDAU-GXNSBURG EQUATIONS

215

Infinite Homogeneous Medium We usually consider the kernel K, in an infinite homogeneous rnedium. Then, if this is a pure metal, it is clear that K,(s,r) depends only on S - r. On the contrary, for an alloy, this translational invariance is lost. In this case, the translational invariance i s restored by making an additional approximation. In (7-1 6), the average is taken over all impurity configurations. On the right-hand side of (7-1 6) app e a r s the average K0(8,r)b(r). We approximate this in the following way:

Equation (7-18) is not rigorous because it neglects certain distortions of the pair potential in the immediate neighborhood of each impurity. However, detailed calculations (C. Caroli, 1962) have shown that this approximation is reasonable when the impurity potentials can be treated a s weak perturbations. We therefore limit our discussion to alloys whose constituents are not too different chemically. Then (7-1 8) is applicable and the integral equation f o r ~ ( r )can be written completely in t e r m s of the average kernel K, (s,r), which only depends on S - r ,

Relation between KO and a Correlation Function A p r i o ~ i the , product of the four functions wn that occurs in (7- 17) is rather discouraging. However, it can be shown that this product is related to r a t h e r simple physical concepts. Taking, f o r simplicity, a n infinite homogeneous medium, we study the F o u r i e r t r a n s f o r m

is the volume of the sample. We have taken q along the where direction and set

X

SUPERCONDUCTIVITY

216

It is useful to first discuss the real function g(q,a) =

C m

(nleiqXlm)(mle

-iqx

&ttm- tn - f i Q )

sometimes called the spectral density of the one-electron operator

eiqx. The symbol represents an average over all states of fixed energy 2, (for example, 4n = O corresponds to the Fermi level). In practice, g(q,Sl) depends strongly on $2 but only slightly on 6, and the average will be taken at the Fermi level. If g is known, then K,(q) is immediately given by

NOWg(q,Q) has a simple physical significance, Introduce the Heisenberg opera tor

which describes the evolution of eiqx in time for a n electron in the pure normal metal described by the Hamiltonian Xe = (p2/2m) + U@). In t e r m s of the operator eiqx't', simple form:

the spectral density takes a very

Equation (7-23)can be verified by writing explicitly the matrix elements of eiqx@) between states 1 n) and 1 m). In order to determine g, it is only necessary to determine the correlation function of eiqx,

for an electron at the Fermi energy in a normal metal. (l)First for a puye metal, we can assume that at t = 0 the electron

ANALYSIS Q F LANDAU-GINSBURG EQUATIONS has abscissa

X,

and velocity v F cos 8 along the

X

211

axis ( B is the angle

between q and the electron velocity vector).

e -iqx(O) = e-iqx,

Thus (e

-iqx(o) eiqx(t)

=

)EF

n

J,

sin 8 dB exp(iqvF cos Bt) (7-25)

and R

g(q,Q) =

5 l.

sin 6' dB 6(52

- qvF

cos 8 )

(2) Far an impure metall where the mean f r e e path 1 is smfl com-

pared to the wavelength studied q'l, eiqx(t' is controlled by a diffusion (random walk) process. If D = vF t / 3 i s the diffusion coefficient, we have

Explicit Calculation of the Kernel KO Usiw the forms (7-26) and ( 7 - 2 1 ) for g(q, a), one can explicitly cafcufrzte the kernel K, (q) from (7-21). The variables of integration a r e taken to be 5 and RSZ == C' - [ a d the first integration i s performed by the theory of residues over 6, F o r the pure metal, one finds

--

2nN(0)Vkg T

F"~vF

g 0

tan-"

and for the impure m e h l

- N(0)Vkg T fi

C

1

(is$ c< 1)

(I-28b)

0

mscuseian of theere resuhs ( 1 ) Precautions related to the frequency cut-off in the interaction V. Until now we have neglected the fact that the interaction V in the BGS appraxihw D . This causes the sums mation only couples those states of energy 1 5 j in (7-28) to diverge. However, this i s easily remedied by writing

The only divergent term i s Ko(0), which can be calculated directly from (7-9).

Ka(0) = V

C

t a b ( ~ 1 / 2I )W, ('1

i2

n

where the 8 integration was performed by using t h e orthogonakty of the function w,(8).' (2) Spatial form of the kernel 4.On taking the inverse Fourier transform of ("1-281, we: find

'we had already obtained the result (7-30) in the simpler ease of an infinite homogeneous medium without fields o r currents where w e predicted that A ( r ) is c o n s a n t in space. Then the linearized self-consistency equation is simply written A = Kf,(010. This has a nonzero solution only if &(Q) = l . "rhis condition gives the temperature To a t which at nonzero order parameter can appear, that is, the transition temperature.

A N A L Y S I S O F LANDA'EJ-GXNSBURG E Q U A T I O N S

219

For R f 0, the sums converge, 11 i s particularly interesting to study the asymptotic form of KO(R) at large distances. The only important terms a r e then Rw--rtak T f w f z k Tg). Thenwe f i n d f o r T = T o B B 2akg TOR

=-

%vF

EV F

=-

kg To

EDR

1 2

e x p (-1.13R/5,)

exp I- 1.8R

-

Conclusion. For a pure metal and T T, , the range of (R) i s of the order of 6, = 0.18RvF/kT, a s we previously stated. On the conThe range of vatrary when P is small, (7-32b)gives a range gP. > P or lidity of (7-3230)is limited by R >> P, which innplies (6, lja >=>

P

(dirty metal)

(7-33)

[when 67-33) is satisfied, the alloy is said to be dz'p-ty following the terminolom introduced by ~ n d e r s o n , )E the matrix is a nontransition metal, 6, is generally high (for example, 4, = 16 x lo3A for At), and a small fraction of impurities is sufficient to m&e the metal dirty. It is interesting to compare K, with the kernel SF, , studied in Chapter 5, which gives the c u r r e n t respands of a superconductor in the presence of ai vector patenthl A. We have seen that in a dirty allay the range of SPY is P, but the range of K, i s $(, What !l.is the reason for this difference ? Answer, The kernels and 4 can both be expressed as average oneelectron correlation functions in a normal rnetal in the absence of fields:

where ( j j,) is a transverse current correlation function and (6 S) is a denbb sity correlation function. Eauaticrns (7-34) and (7-35) can easily be verified by explicitly calculating the matrix elements that appear irr products such as j f P

-

V

.

For T .- To,the integrations over 5 and 5 ' i n 17-34) and 47-35) firnit the useful interval to t h/kgTo. If there are no collisions, the distance traveled by an electron in the time t is v~ t 5 Then both SF and % have the range

-

,

S,. ff the afloy is dirty, the velocity correlation function Ealls to zero after the electron travels a distance 1: the range of SF, is l . However the density correlation funetion is not destroyed by a colllsisn. It obeys a diffusion equation, The average disbnce travelcd in time t isvE,where R I s the diffusion coef-becomes ficient D = vFP Thus the range of the kernel


@,/(hs + X,,) 6 , the field becomes nearly uniform along the junction and h,f y) -- H, tt, can be determined from (7-851,

+

+ const

L =

$0

( h s + h,f1H

The structure of the currents i s then periodic with period L I, = I,

sin q5

2

Im sin

By placing (7-91) into the right-hand side of (7-871, it i s easy to verify that az+/8y2 and ah,/ay a r e essentially negligible for L > 51, the situation is similar to a Type 11 superconductor where

SUPERCONDUCTIVITY

2 44

X > 5. Xn the limit L 0,

We insert (8-36) in (8-1) and perform the integration over t, obtaining

Fimlfy, if we make use of (8-371, our starting equation (8-8) becomes

The eigenfunetions of this l inear integral equation for A a r e simply the g's. F o r , if w e choose A ( r ) = gq(r), the orthogonality of the g's leaves only one term p = q in the sum Z; and (8-38) is satisfied proP vided that

SUPERCONDUCTIVITY

270

Equation (8-39) has a familiar farm, and leads to

where the universal function Un(x) has been discussed earlier, For a fixed field H and a given sample shape, the E ' s art: fixed [and can g

be computed, more or less laboriously, by solving Eq. (8-3511. What we want to b o w is the highest temperature T at which (8-38) has a nonzero solution A fr). Ua(x) Is a decreasing function of X. Thus we must choose the lowest ez'genzrala@ Thus the recipe to compute a nucleation field for a ""dirty" sample of arbitrary shape is the following: W e find the lowest eigenzt.cllw E , of Eq. (8-35) plus the boundary condition. carresponding to (8-33). This i s then a h o w n function of the field E, (H). To obtain the relation between nucleation field and temperature, we write

X)rob;t@m,Derive a formula for the critical field Hc2 in a dirty superconductor, valid at all temperatures (K, Maki, P. G . de Cennes, and N, Werthamer, 1964). Solution. Equation (8-35) has the farm of a Sehrijdiager equation for a particle of mass R12D and charge 2e. The cyclotron frequency for such a particle is

where rpo i s the flux quantum, The lowest eigenvalue i s

Writing Ghat H = Efc2 and

60

= UntT/"T,),

we arrive at the implicit equation

Discussion of this equation: First, for T ctose IQ 1"@, Wez close to O, we can make USE? of the aforementioned expansion of tfn(T) and $et

ft is easily verified that this agrees with the Landau-masburg formula HGz =

MAGNETIC F I E LDS A N D IMPURITXES

271

&f Hc when we take for n the Gorhov expression in the dirty superconductor limit (Eq, 7-41), that is,

x

For T -- O on the other hand, we have

while the bulk critical field is given by H: /8n =

4 N ( o ) A ~ ~o,r

This leads to

Thus the ratio HC2 /*Hc increases only by 20 p e r cent when one goes f r o m T = TII to T = 0. A similar calculation for a plane boundary shows that in a supereollductor the ratio Hc3 / Hcz shauld remain equal to 1.69 mt all temperatures, Also, when the Eield i s applied a t a finite angle 6 from the surface, the angular dependence of HC3(B)/Nc2(0) has a form independent of temperature. Both these properties appear t a be well, confirmed by experiment. (In fact, it had k e n a surprise to notice that in dirty supereond-uctors the Landau-Cinsburg linearlzed equation applied so well at low temperature,)

A T H E O R E M O N T H E L O C A L DEXiTSfTY O F S T A T E S Just as in Section 8-2, we can, if we wish, study our system at ternperatures slightly below the nucleation paint 71". In particular we can compute the density of states Ns ( 2 6 ) for excitation energy E . It is now a function af the observation point r since the order parameter b ( r ) is not constant in space. This function can be measured by tunneling experiments. When Ns is computed to order \ A I? , the results a r e remairlrabliy simple (P, G , de Gennes, 1964). (1) N, (rE ) depends only on the value of ~ ( rat) the observation point P. This local relationship is at first sight very surprising, In the initial equation it is found in fact that N, (rE) depends on the values of b ( r f ) within a radius -. ((T)f r m F. But just below the nueIeation temperature, we know that the function A ( T ) has 8 simple shape, It is proportional to one of the eigenfunetions g,(r) of EQ,(8-35). This r e m r k permits one to c a r r y out explicitly the required integraions in N, (rE f and the f i d result invotves only ~ ( r ) .

(2) N, (rE ) is given by Eq. (8-30) where A is now the local value ~ ( r )and , where E / T ~= E~ is related to the temperature by Eq.(8-40). Thus, in the region of validity of the calculation [XT~ /B 2 1, '0 I] , mr dz'rty systn?ms are alE gQpless super(M nucleation - M)/N

cand@clom, These predictions have recently been confirmed by tunneXing experiments on various allays i n both the Hc, and Wcz regions (E, Guyon and A. Martinet, 1964). In particular, these experiments show that the energy scale (given by A/rK = E @ ) on which the density of states deviates from N(O), is finite even when the order parameter d(r) becomes very small (that is, whcm H becomes close to the nuef eation field). These measurements give in fact an accurate measurement of T~ . REFERENCES

Magnetic impurity effects and gapless superconductivity: Theory : A. Abrikosov and L, P. Gor%ov, Zh, Eksgerim, z' reor, Fiz,, 38, 1181 (l9601, Translation Soviet Pkys,- JBTP, 18, 593 (l960). Experiment : M, A. Woolf and E" Reif, Phys, Rev,, 13"t, 557 (1965). Wgnetic field effects: Theory: K, Maki, Phys,, l, 21, 127 (1964). P. G , de Gennes and M, T i n u a m , Phys,, 1, 107 (1964). P. G . be Eennes, Phys, Cmdmsed Mcktter, 3, 79 (1964). Experiment : E. Guyon, A, Martinet, J, Matricon, and P. Bincus, Phys. 138h, 146 (1965).

&v,,

INDEX BCS wavefunction, 105

in zero field, 2-3

Boundary effect, 227- 232 Gapless superconductivity, 1Q, Coherence factors, 133 Coherence length, and other efiaracteristic lenfihs, 225 Plppard- BGS, 12 temperature dependent, L 71- 178 Cooper pairs, 93 Critical fields, He, 14

265- 261, 271- 272

Gauge transformations, 145- 149 Intermediate state, domain structure of, 40-46 in a sphere, 35 origin of, 26

48, 66

HGz, 49, 195-196 H,,

, 196-201

for thin films, 189- 192 Electron-plronon interaction, 96- 104 relation with Te and isotope effect, 124- 128 Energy gap, relation to slxperfluidity, 10 a t T = 0, 123 table of values, 10 temperature dependence of, 523

Josephson effect, a t T = 0, 118-121. from Landau- Ginsburg wuations, 234- 238 under fields, 240- 244 Landau- Ginsburg parameter, definition, 18f G o r b v formula f a r dirty limit, 224 values in Type I materials, 201 London equation, 3, 58, 169, 171, 180 Mefssner effect, and Loxrdton equation,

Flux quantization, 149- 152 and Litgle- Parks experiment, 185 F r e e enc;rg;y, Landau-Cinsburg from, 172-175

4- 7

microscopic calculatlsn, 160- 170 PIJucIear relaxation, 136

Pair potential, def faition, X39 guage invariance of, 145 linearized quation, 160- 162, 210 seff-consistency eqclalion, 143 s i w l e valuedness, 149 Paramagnetic impurities, 241, 251-253, 263 Penetration deptfis, for dirty miiterlals, 26 London value, 6, 12 measurements, 22-24 and ather characteristic lenghs, 225

Pippard limit, 21-23 Pigpard nonlocal theory, 19 comparison with rnicrorscopic calculations, 569 wanturn interf erometers, 244- 245 Quasigarticlea, &goXubv muation far amplitudes, l40 In gapfess situation&, 265 e c a t t e r i q and creation by exterm1 perturbations, 131- 136 for an SNS clean system, 155-157 in ungarm superconductor, 114-1143

near a. vortex, 153-155 Specific heat, discontfnuity a t Hcz, 54-55

discontinuity at Tc , 18 In zero field, 2-3 Superfluidity, and Josephson current, 120

relation L s the energy gap, 10 Surface supercondueMvity, 11, 50, 196-199, 211

Tumel effect, in dir'ty superconductors under high fields, 272 dissipatIve, 10, 117 in films u d e r currents, 144- 145 nondissipative (Josephson), 117-121, 234-244 Vortex lines, from the Landau-Ginsburg quations, 201-207 in the b n d o n approximation, 55- 80 low energy exeitations, 153 motions, 80- 81 neutron diffraction by, 74 Vortices, flat, 60-63

Wall a n e r n , Landau-Ginsburg caiculation of, 178 limiting bellavior for large and K, 44-46 207 vanishi= for K = 1/'