366 PHILlPS TECHNICAL REVIEW VOLUME 28 The, skin .effect

Ill. The skin effect ill superconductors

H. B. G. Casimir and J. Ubbink

In parts I and II of this article we studied the skin tron in an electric field, a new equation for the current. effect in ordinary metals possessing finite conduc- In terms of the skin effect their hypothesis might be for- tivity [1]. In this third part we shall consider the skin mulated by saying that a superconductor, even at zero effect in metals that are in the superconducting state [2]. frequency, behaves like a normal metal in the relaxa- The superconducting state is limited to temperatures tion limit. Thus, even at W = 0, they find a penetration and magnetic fields that do not exceed certain depth equal to the relaxation skin depth Ap if all elec- critical values. Between the critical field He and the trons are superconducting; if, more generally, ns is the temperature Tthere exists the empirical relation: concentration of the superconducting electrons, the "London penetration depth" is AL = V(mjf-lnse2j. Thè , HcjHco = 1- (TjTc)2, (1) fact that this is a new hypothesis, and cannot be where He» is the critical field at T = 0 and Tc the deduced by taking the limit for a -+ 00 of a normal critical temperature at H = O. conductor, is immediately apparent from fig. 1, an The most striking property of a superconductor is its W-7: diagram like fig. 11 in part Il. For 7: -+ 00 at low zero resistance to direct current; not merely extremely frequencies we do not arrive in the relaxation region B low but really zero, as is evident from the existence of or E but in the region of the anomalous skin effect D. persistent currents. From the equation for the classical skin depth (1,10), Ok2 = 2jwf-la, one might at first be inclined to think that the supercurrent would be a purely surface current: with a ~ 00 one finds Ok = O. This was the main line of thought before 1933. In that o year Becker, Heller and Sauter [3] pointed out that a E current layer oî finite thickness is arrived at if the iner- tia of the electrons is taken into account. In fact their reasoning corresponds in broad lines to that given in II for deriving the skin depth in the relaxation limit (in regions Band E in part II), and which therefore leads to the same result: if all electrons are superconducting, the penetration depthis Ap = V(mjf-lnee2), cf. equation (23) in part Il. Both cases involve electrons which are entirely free but possess inertia. Both constitute the _W limiting case for W7: ~ 00; in the extreme relaxation case, however, the emphasis is placed on W -+ 00 for Fig. 1. The 00-. diagram for a normal conductor, with the region moderate 7:, whereas in the reasoning of Becker, of the normal skin effect (A), relaxation (B and E) and the anomalous skin effect (D). See also H, fig. 11. Heller and Sauter it is a, and hence 7:, that tends to infinity while W remains small. All that one can really conclude from the reasoning of Becker, Heller 'and Sauter is that a field variation Supplementing London's theory with the two-fluid cannot penetrate beyond a depth Ap. The fact that also model of Gorter and Casimir [6] (1934), we obtain the a constant magnetic field is expelled from the interior picture of superconductivity already outlined in the of a metal when it changes to the superconducting state introduetion to part I: there are "superconducting (i.e. the Meissner effect [4], 1933), induced the brothers electrons" in addition to "normal electrons" in a ratio London [5] (1935) to postulate ad hoc, in addition to the which is governed by the temperature. The supercon- equation giving the acceleration of an entirely free e1ec- ducting electrons have the effect of screening off elec- tromagnetic fields, even at zero frequency. At zero fre- Prof. Dr. H. B. G. Casimir is a member of the Board of Manage- quency they short-circuit any electric field, but at ment of N. V. Philips' Gloeilampenfabrieken; Dr. J. Ubbink is with Philips Research Laboratories, Eindhoven. non-zero frequencies there exists, owing to the inertia 1967, No. 12 SKIN EFFECT, III 367

of the electrons, an electric field in the penetration closely related (even at zero frequency) to the normal layer which gives rise to energy dissipation via the conductor in the relaxation limit B, E. The screening normal electrons. This model served its purpose for a (-1/ö2) is 1/AL2.If, however, lor the penetration depth long time. Although it has since been superseded by is too small, the latter is no longer equal to AL.We in- more comprehensive theories and now possesses little troduce As to denote, quite generally, the penetration more than historical interest, it is still a useful guide in .depth [81. With what are we now to compare I or As in classifying quite a number of phenomena associated this case, at zero frequency? The quantity v/w is with superconductivity. In discussing the skin effect we shall use this model (the "GeL model") as a guide. In this discussion there are two dominant aspects: the thickness of thepenetra- tion layer and the magnitude of the absorption. These are both comprised in the complex skin depth c5 = ö' + jö", introduced in part 11. Under the conditions c5' « c5" « cko - implying that the absorption is low and the penetration depth small compared with the . wavelength in free space - the penetration depth is equal to c5" and the absorption is given by c5' (see Il, 14). D The results of absorption measurements are usually given in terms of the' "surface-resistance ratio" q = R/Rn = ö'/c5n'; here the subscript n refers to the metal just above the critical temperature. In the following we shall briefly describe a few methods of measuring both B,E the penetration depth and the absorption, and we shall use the results to test the GeL model. In broad lines the GeL model gives a reasonable description of the experimental results. Pippard [71, however, found a marked discrepancy: he discovered A that the skin depth' is dependent on the mean free path Fig. 2. In a normal metal the various limiting cases (see fig. 1) (the degree ofimpurity), whereas according to the GeL can be characterized by indicating which of the three lengths l, 15 and »k» is the shortest. In A it is I, in Band E it is vlo: and in D model the skin depth ought to be determined solely by it is Q. the concentration of the superconducting electrons. This, together with other considerations, led Pippard to propose a modification of London's theory, along obviously ruled out. As a parallel for ulo: we nowhave the lines suggested by the theory of the anomalous skin a new length ~o, introduced by Pippard, called the effect: he argued that it was necessary to take into ac- coherence length of the clean superconductor. If ~o «I count the possibility that the relation between current and ~o «As, the superconducting electrons have their and field is not a purely localone .. full screening effect(fig. 3, B') as predicted by London. To give a rough idea of this line of thought, we shall In a very "dirty" superconductor (I « ~o and I «As) first reconsider for a moment the possible situations in the coherence is "curtailed" to I and the screening effect a normal conductor (fig. 2, see also fig. 2 in part 11). We can distinguish between three limiting cases, depending on whether I, uk», or c5 is much smaller than [1] H. B. G. Casimir and J. Ubbink, The skin effect, I. Introduc- tion; the current distribution for various configurations, the other two quantities. In the normal skin effect (A), I 11. The skin effect at high frequencies, Philips tech. Rev. 28, is the smallest of the three; with extreme relaxation (B, 271-283 and 300-315, 1967 (Nos 9 and 10). These parts are referred to as I and n, and the equations as (1,1), etc. E), the smallest is ulo», and with the anomalous skin [2] A general survey of superconductivity will be given in an effect (D) it is Ö. Provided Ö' «ö", we may take the article by J. Volger, shortly to appear in this journal. . . [3] R. Beeker, G. Heller and F. Sauter, Z. Phys. 85, 772, 1933. 2 quantity -I/ö as a measure of the "screening". In B [4] W. Meissner and R. Ochsenfeld, Naturwiss. 2], 787, 1933. and E -I/ö2 has the value I/Ap2. Disregarding details, [5] F. London and H. London, Proc. Roy. Soc. A 149,71,1935; Physica 2, 341, 1935. the results of II for A and D can now be obtained by [6] C. J. Gorter and H. B. G. Casimir, Physica 1, 306, 1934; saying that, coming from B, E, the screening -1/c52 (or, Phys. Z. 35, 963, 1934; Z. techno Phys. 15, 539, 1934. [7] A. B. Pippard, Proc. Roy. Soc. A 216,547, 1953. rather, its modulus) is reduced in A by a factor I/Cv/w) [8] To link up with Part 11it would be consistent to represent the and in D by a factor c5/(v/w). penetration depth by 15". We prefer here, however, to follow the practice in the literature of superconductivity, where the As already noted, the London superconductor is penetration depth is denoted by Ä. with a subscript. 368 PHILlPS TECHNICAL REVIEW VOLUME 28 is reduced by a factor 111;o (fig. 3, A'). The third case Our subject is the skin effect. We are not, therefore, (fig. 3, D') is the one in which Àsis much smaller than concerned with those situations in which the field pen- 1;0 and I. Compared with B' the screening is now etrates far into the metal. Examples of these in super- reduced by a factor ÀsIl;o. A classification of supercon- conductivity are to be found in a superconductor of ductors made on this basis, i.e. according to thevalues type I in the intermediate state, where the metal divides of 1;0 and I, will be found in figs. 12 and l3. into superconducting and normal regions and the type IJ superconductor in the mixed state, threaded by "fluxoids". The theoretical line ofthought traced above, Gorter & Casimir - London - Pippard, which we shall deal with at greater length in the following sec- tions, is a suitable basis for a discussion of the skin effect, in particular because of the relation to the anomalous skin effect in normal metals. We shall com- pletely disregard another theoretica! development, with . I which in particular the names of Ginzburg, Landau and I Abrikosov are associated, and which predicts, among I • I other things, the mixed state. The theories mentioned so D' I I • far are phenomenological. Since 1957 there has also I been a fundamental microscopie theory of supercon- I I d uctivity, the theory of Bardeen, Cooper and Schrieffer • (BSC) [10]. This, too, we shall disregard, although oc- B' casionally we shall use its results.

The finite penetration depth according to Beeker, Heller and Sauter A' We shall now briefly examine the argument put for- Fig. 3. By analogy with fig. 2 for the normal metal, three limit- ward by Becker, Heller and Sauter [3] showing that the ing cases A', B' and D' can be distinguished in a superconductor, depending on which of the three lengths I, ~o or J,s respectively is absence offriction (a -->- cc) does not lead to zero pene- the shortest. ~o is Pippard's coherence length, which is the formal tration depth, as would be predicted from the equation analogue of vjw in the normal metal. The shaded regions in- for the classical skin depth, but to a finite penetration dica le regions of coherence. depth because of the inertia of the electrons. They con- sider a superconducting sphere which is set into rota-

The coherence length 1;0, which also appears In the tion about an axis through its centre. At first sight we theory of superconductivity in other ways with various might think that, owing to the complete absence of interpretations, plays in Pippard's theory the role of friction and to the inertia of the electrons, it is not pos- the distance over which the field at a given point in- sible to set the electrons in motion in this way. How- fluences the current in the environment. The situation ever, since initially the electrons do not move, whereas D' has a typical non-local character: the current den- the lattice ions do, a time-varying current exists and as sity in the middle of the skin layer is partly determined a result of the magnetic and electric fields associated by the (stronger) field at the surface and the (weaker) with it the electrons in the sphere are finally swept into field in the interior of the metal. This situation is char- motion too, all except those in a thin shell under the acteristic of nearly all elementary metals that become surface. Becker, Heller and Sauter also showed that a superconductive. close analogy exists between the setting in motion of a Finally, it may be asked how Pippard's modification superconducting body and the switching phenomena affects the theory in its prediction of absorption. The in superconductors. We shall briefly recapitulate their surface-resistance ratio depends on the temperature reasoning here, adapted to the case of a supercond ucting and on the frequency. Confining ourselves to the situa- tion D', then by means of a simple estimate based on [9J See T. E. Faber and A. B. Pippard, Proc. Ray. Soc. A 231, 336, 1955. the two-fluid model and Pippard's theory, we arrive at [101 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev.IOS, the interesting conclusion that the freq uency-dependent 1175,1957. [11] This "perfect diarnagnetism" is sometimes expressed as part of q is a "universa!" function [9] of the "red uced zero permeability. In our description, however, zero induc- frequency" I1wlkTc, that is to say, independent of I, 1;0, tion inside the body is due to zero net magnetic field (due to external coils and induced currents) and [i is just the Àp and v. This conclusion seems to be supported by perrneability of the "medium" (see I, Introduet ion), being the experimental results (see fig. 8). usually eq ua! to [io = 1.26 x 10-6 Him. 1967, No. 12 SKIN EFFECT, III 369

cylinder situated in a varying magnetic field parallel trons. From the equations (Il, 18, 19) a complex fre- to the axis of the cylinder. The crux of thev- argument quency-dependent conductivity can be introduced: is that, although in a stationary state there can be no a* = a/(l + jW7:) = nee27:/m{l +jW7:). electric fields present in a perfect conductor, this does not apply to changing fields, owing to the inertia of the The limit of this expression for a -)- 00 (i.e. for 7: -+ 00) electrons. For electrons without friction but possessing is: inertia the equation of motion in an electric field E is a* = nee2/jwm. mVe = eE, (2) In the classical region, the complex skin depth (see Il, which, with J = neeVe, yields 27) is given by 02 = j/w,ua. Substituting for a in this E = ,uÀp2j, ...... (3) equation the above value for a* one finds:

where 02 = -m/,unee2 = -Àp2, (cf. eq. 4), Àp2 = m/ ,unee2 ..... (4) or

0" = Àp, 0' = O. . . (9) (see Il, 23). With this equation, together with the Max- well equations, the finite penetration depth ofthe vary- . The skin effect is "purely reactive": the alternating ing part ofthe magnetic field is easilyfound. Substitution fields are indeed shielded but there is no energy of (3) in Maxwell's second relation: curl E = -:8, with dissipation. B = ,uH, yields: As already remarked, this reasoning is basically iden- tical with the theory of extreme relaxation presented in curl j (5) part Il. In view of the complications of the anomalous Integration with respect to time - assuming that at a skin effect, however, the step proposed by Becker, given instant the magnetic field is Ha and the current Heller and Sauter is not permissible at low frequencies. density Ja (both being then functions of place but not For this reason alone one may not consider the super- of time) - we find: conductor as the limiting case of a normal conductor for 7: -+ 00. curl (J - Ja) = -(H - Ha)/Àp2. (6) The Meissner effect, the London equations and the two- With curl (H - Ha) = J - Ja (since both curl H = J fluid model and curl Ha = Ja) this results in: Although before 1933 it was taken as established that it is not possible to change the magnetic induction in curl curl (H - Ha) = -(H - Ha)/Àp2, the interior of a superconductor, there was at the time or (as curl curl = grad div -IJ. and div (H - Ha) = 0): no reason to suppose that it would not be possible to IJ.(H - Ha) = (H - Ha)/Àp2. .. (7) "freeze in" the :flux by first placing the body in a magnetic field and then making it superconducting by If the radius of the cylinder is large compared with Àp, cooling it. In fact, however, the :fluxis excluded in such then to a good approximation we can treat the surface an experiment; this is the Meissner effect already of the cylinder as a plane surface. For the simple con- referred to, which Meissner and Ochsenfeld [4] dis- figuration introduced earlier (H and J parallel to the covered by accurately measuring the magnetic field plane surface of the conductor and mutually perpen- around a superconducting body. Since 1933, therefore, dicular) eq. (7) leads to: it has been considered that, apart from perfect con- ductivity, zero magnetic induction inside the body H - Ha = (H - Ha)o exp (-z/Àp), (8a) ("perfect diamagnetism") is an essential property of a J - Ja = (J - Ja)o exp (-z/Àp). (8b) superconductor [11]. The London theory [5] gives a description of these The suffix 0 relates to the values at the surface. facts._F. and H. London postulated that, in addition The current and field changes therefore effectively to the Maxwell equations, the following two equations penetrate to a depth Àp• are applicable in a superconductor: Becker, Heller and Sauter show that this result can also be obtained if, starting from a normal conductor, j = E/,uÀL2 (10) and one goes to the limit a -+ 00, taking the inertia of the electrons into account right from the beginning. In curl J = -H/ÀL2 (11a) shortened form the reasoning is as follows. In Drude's or theory allowance is made for the inertia of the elec- J = -A/,uÀL2, ... . (11b) 370 PHILlPS TECHNICAL REVIEW VOLUME 28

where A is the vector potential defined by curl A = B, Superconductivity is a new phase of the metal, and a div A = O,and well-defined state of the superconductor is associated (12) with each point of the H-T diagram. By means of the two-fluid model of Gorter and Casi- ns being the concentration of the superconducting mir [6l, it is possible to describe the properties of the electrons. Eq. (10) is exactly analogous to the accelera- superconducting phase in simple terms. In this' model tion equation (3) for frictionless electrons. Eq. (lla) the electrons can be in either of two kinds of state; a would follow from (6) if the time-independent part of fraction x are in the "excited" or "normal" states and field and current were zero. The London hypothesis - a fraction 1- x in the "condensed" or "superconduct- which, incidentally, was also supported by quantum- ing" states. The thermodynamic functions - as re- mechanical speculations - therefore boils down to gards their dependence on x - can be chosen in a taking the integration constant equal to zero when simple manner such that a number of thermodynamic integrating eq. (5). With these London equations, in properties of the superconductivity - in particular the same way as with (7), it now follows that the form' of the critical field curve (1) - can be satisfactorily described. It is then found that (13) x = {}4, • (14) from which it again follows, just as (7) leads to (8), where that the field in a superconductor, apart from a skin {}= T/Tc• (15) of thickness h, is virtually zero. The London brothers thus made no attempt to explain superconductivity as a In this two-fluid model the concentration of the limiting case of normal conductivity, but postulated superconducting electrons is n« = ne (1 - x). In eqs. (10) and (11) as electromagnetic relations for a combination with the London theory this now gives superconductor, in the place of Ohm's law for a nor- for the penetration depth AL: mal conductor. AL 2 = m/ ftnse2 = m/ ft(l - x)nee2 = Ap2/(1 - x). (16) The Meissner effect implies that the superconduct- ing-to-normal transition is reversible. Fig. 4 shows an From this model we expect the penetration depth AL H- T diagram. If we change a sphere from A(T > Tc, to be dependent upon temperature in accordance with: 'H = 0) via B(T= 0, H = 0) to C(T= 0, °< H < Hco), then the flux is excluded. If we now raise the tem- (17) perature again (-+ D) the flux penetrates the sphere. Be- fore the Meissner effect was discovered it was thought The complex skin depth in the GCL model that cooling from state D would bring about a super- With the model described above it is a simple matter conductive state C' with frozen-in flux. The transi- to calculate the complex skin depth, giving both the tion from C to D would therefore not be reversible. penetration and the absorption. According to the Meissner effect it is not C' that is ob- We again take the simpleconfiguration given in fig. 1 tained upon cooling from D, but C; in other words, the of part IJ (E and J//x-axis, H//y-axis, z-axis sur- transition C-D is reversible. This is of great importance face) and we again assume the fields and currents to because it gives confidence that reversible thermo- be proportional to expj(wt - z/I). We disregard the dynamics mayalso be applied to the superconductor. displacement current, but we do take into account the relaxation of the normal electrons. Under these conditions the Maxwell equations (I, 1 and 2) give jH/1) = J and -jE/I) = -jwftH. The current J is now composed of Js, the current of the superconducting electrons, and Jn, the current of the

,C'C' ------.D-=9=. I normal electrons: J = J« + Js. The London equations -1r for the superconducting electrons give jwJs.= E/ftAL2 i and -jJs/1) = -H/AL2. The concentration of the nor- ~r mal electrons is nn = xn«. The normal electrons re- o r: B'.__---_---T-~7ê.----+1 A- 0 spond to a given electric field in the same way as in the normal conductor; only their number is reduced. Fig. 4. The phase diagram of a superconductor in the H-T plane. Introducing An as a measure of the concentratien of Inside the parabolic curve (eq. 1) the metal is superconducting. If the metal is cooled from D (with flux through the sphere) to normal electrons, o "K, the result is not C' (with "frozen-in flux") but C (with excluded flux). This is the Meissner effect. (18) 1967, No. 12 SKI1:-l"EFFECT, III 371

we then have (cf. II,I9): from zero to infinity at a given x (solid lines) and when x goes from 1 to 0 at a given an; (broken lines). The J« = (1nE/(1 + jcor), (19) line "x = 1" gives 0 for a conductor in the normal where (1n = 't"/p,Àn2 = x't"/p,Àp2 (cf. II,25). Elimination state. For w increasing, 0 goes in the direction of the of the fields and currents, and making use of (16), arrow. If we now keep ort constant when P is reached, yields the result: and we cool the superconductor to below the critical 1 jcorx 1 1- x temperature, then 0 follows the curve "ort = t". If, having arrived at Q we now keep the temperature con- - Tz = 1 + jw't" Àp2 + Àp2 • (20) stant but let the frequency drop to t» = 0, then 0 The "screening" -I/ö2 is thus composed of two addi- .follows the curve "x = t" (opposite to the direction of tive terms: the first is the -1/02 that would be obtained the arrow) and arrives at R: this is "pure London only for normal electrons in a concentratien xn« (see penetration" without absorption. 11,30; the first term there originates from the displace- The figure illustrates the relation between supercon- ment current, here neglected), and the second term is ductivity and extreme relaxation: the point jÀp can be 2 the -1/0 that would be obtained only for supercon- reached both by x ---+ 0 (100% superconducting elec- ducting electrons in a concentration (1 - x)ne (see trons) and by err ---+ 00 (extreme relaxation). In both eq.9). cases the electrons behave as completely free particles. Eq. (20) can be written in a more abbreviated form: For ort « 1 - x (low frequencies and temperatures '

X well under to the critical temperature) we find from Àp2/02 = -1 + .' (21) (21): 1 + jW't" 0" = Àp(I - x)-1/2 = ÀL, (22) Fig. 5 shows a few curves described by 0 in the com- 3 2 plex plane in accordance with eq. (21) when ort goes 0' = -!wïÀpx(I - x)-3/2 = -!wïÀL /Àn . (23) 0" and 0' depend on the temperature via x only. We see that, under the stated condition, 0" is dependent on- lyon the choice of the metal (Àp) and the temperature (x), while 0' depends in addition upon the purity of the metal (r) and the frequency w. As we shall see, some characteristic properties of superconductivity in an alternating electromagnetic ~r / / field are well described by this theory; on a number of / important points, however, there is a considerable i / difference. / / Nevertheless, it is possible to make two simple / amendments to this theory that bring us closer to a / / correct description of superconductivity in an alter- / nating electromagnetic field. / / With a view to these amendments, let us consider / once again the physical background to equations (22) / / and (23). The penetration depth in the superconductor, / which can have different values in different theories, / / is denoted generally in this article by Às, as above. / / Eq. (22) scarcely requires any further comment: under / the condition stated (w't" « 1 - x), 0' is much less than / / ö", and in that case 0" is effectively the penetration / depth: Àa = 0". Eq. (22) expresses the fact that Às does not depend on frequency and is equal to ÀL, the pene- Fig. 5. The complex skin depth 0 in the-two-fluid model accord- tration depth previously found for w = O. In eq. (23), ing to (21). This equation is derived from the Maxwell equations 2 in conjunction with the London equations (10, 11) for the super- the following will show that the expression -!WûL 3/Àn current and Drude's equation (19) for the normal current. The has a more general significance. 0' is a measure of the solid curves show how 0 changes as W goes from ° to 00 for given fixed values of x (i.e. at specific temperatures). The dashed energy absorbed by the normal electrons in the pene- curves represent 0 as the temperature goes from Tc to zero (i.e. tration layer. The restrietion of the fields and the x from 1 to 0) for given fixed values of WT, i.e. at given fre- currents to a layer (of thickness Às) is entirely due I' quencies. (For WT = 0, the curve is coincident with the oH-axis.) The displacement current is neglected. to the superconducting electrons. Without causing 372 PHILlPS TECHNICAL REVIEW VOLUME 28 dissipation they carry the current which according Measurements of the penetration depth to Maxwell's first relation (1,1), must exist in the In principle, the penetration depth Às can be deter- presence of our (rotational) magnetic field. In the mined by establishing for a given configuration and a layer the varying magnetic field induces an elec- given external field the difference between the magnetic tric field, which according to Maxwell's second rela- _flux actually measured and the flux one would expect tion (1,2) is given by E/ Às = jfJ-wH; E is 90° out of to find if there were no penetration. The latter would phase with H. This electric field causes a current of have to be calculated from the geometry. For small normal electrons: J« = anE (where an = 7:/fJ-Àn2) and penetration depths this method is very difficult, and it thereby performs work. The energy delivered to the has not in fact proved possible to determine Às in this electrons per second and per m3, averagedover thetime, direct way. It is, however, much more feasibl~to meas- is JnE = -!anE2. By integration over the layer Às we find ure small changes of flux, and if Às can be influenced in the power per m2 of surface: ianÀsE02 = ianÀs3fJ-2w2H02 one way or another one can then obtain in this way = 7:Às3fJ-W2H02/4Àn2, where Eo and Ho are the ampli- some information on Às. Here eq. (17) comes to our tudes of the fields at the surface. This power is there- assistance: the penetration depth can be influenced fore proportional to the concentration of the normal by the temperature. The methods of determining Às electrons (I/Àn2) and to 7:, the average time of uninter- therefore all amount to an experimental arrangement rupted interaction between electric field and normal in which some quantity (generally the flux) is a linear electron. We now find a more general form of (23), the function of Às: the procedure is then to establish that expression for 0', by equating absorbed powerr= ab- this quantity is a linear function of (1- ff4)-1!2. sorptivity X incident power: After conversion to Às, the coefficient of (1 - ff4)-1!2 is identified with Ào, the penetration depth at T = O. 2 7:Às3fJ-W2H02/4Àn2 = (4wfJ-0'/Zo)xtZoH0 . I) Since the penetration is a surface effect it is nec- (The incident power is tZOHI2 where HI is the amplitude essary to make the surface relatively large. Shoen- of the magnetic field of the incident wave. Ho = 2Hi berg [121, for example, measured the magnetic suscep- since the magnetic fields of the incident and re- tibility of mercury suspensions in which the mercury flected wave at the surface are in phase. Zo = (fJ-o/eo)1!2 was present in the form of a large number of small is the impedance of free space. For the absorptivity, spheres with diameters ranging from 10 to 100 nm. cf. rr,I4 and I~.) We obtain: Variations on this theme are susceptibility measure- ments on capillary mercury threads [131 and on thin (24) films [141. Although these measurements give a clear From the above we can give eq. (24) a wider inter- qualitative demonstration of the penetration, only the pretation: 7: is now the average time during which the latter has yielded reliable quantitative results. field acts uninterruptedly on a normal electron, Às is the 2) In the Casimir method [151 the magnetic flux out- thickness of the penetration layer and Àn is a measure side the body proper is made as small as possible. The of the number of normal electrons (cf. eq. 18). method consists in measuring, at Iow frequencies, the The first of the amendments referred to above comes mutual induction between two coils; the primary is from Pippard's non-local theory of superconductivity, wound closelyon a cylindrical superconducting body which runs parallel with the anomalous skin effect in and the secondary is wound around the primary. The normal conductors. This theory results in a value for coefficient of mutual induction M is proportional to the thickness Às of the penetration layer which differs the cross-sectional area available to the flux through from the London value ÀL (just as the theory of the the primary coil; in other words, when the cylinder is in anomalous skin effect results in a skin depth differing the superconducting state, it is proportional to D + Às from the c1assicalone). where D is the width of the gap between cylinder and The second amendment takes account of the fact coil. The accuracy with which variations in Às can be that, in general, the mean free path of the normal elec- measured therefore depends on Às/D and on the trons is greater than the penetration depth. We make accuracy with which M is measured. An advantage of allowance for this by filling in for 7:, instead of the the method is that well-defined macroscopie bodies time between two collisions, the time needed to pass can be used. The method, with some refinements, has through the penetration layer. In eq. (24), therefore, we been successfully employed by Laurmann and Sheen- must substitute a new value both for 7: and for Às. berg [161. A variant is the measurement of changes in After briefly discussing the methods of measuring the resonant frequency of an Le circuit in which L the penetration depth and the surface resistance, we is the inductance of a coil wound on a superconducting shall compare the results of the measurements with the cylinder [171. predictions of the theory before and after amendment. 3) The measurement of changes in resonant fre- - - . __ - .--~ .~~~~~~~~~~~~--.

1967, No. 12 SKIN EFFECT, UI 373

quency was applied by Pippard [18] to superconducting exceed this value systematically by a factor of roughly resonators at microwave frequencies. The resonant fre- 2 to 5. Table I [19] gives various values for AO found quency of a microwave cavity is always, owing to from measurements; they may be compared with the field penetration in the wall, somewhat lower than value of Ap for the standard metal introduced in part II would follow from the dimensions if there were no (see II, Table I): z, = 22 nm. penetration. Variations of the complex skin depth <5 cause variations of the resonant frequency propor- Al 50 Table I. Experimental values (in tional to <5". It is possible to vary <5 not only via the Cd 130 nm) for Ao, the penetration depth temperature but also - at a given T < Tc - by means Hg 38-45 at T = 0 in a number of super- In 64 conductors. The variation in the of a magnetic field that disturbs the superconducting Pb 39 values for Hg and Sn is due to 51 anisotropy. The table is taken state; <5", then jumps from the penetration depth in the Sn { 47-60 from Lynton [191. superconductor As to the value <5n" ofthe normal metal. Tl 92 The accompanying jump in the resonant frequency gives a measure of the jump LI <5" at different tempera- 3) Finally, there is Pippard's result, already touched tures. The variations found in LI <5" are entirely attribut- upon, and which is of great importance to our sub- able to Às because <5n" in this temperature range can be sequent considerations: if the mean free path I is very regarded as independent of temperature. One can then small, Às becomes dependent upon it [7]. This conflicts find Ao from the slopeof LI<5" asafunction of(l- 04)-1/2. with London's theory, where As AL is a quantity that An incidental result of this measurement is the deter- = does not contain the "sensitive parameter" I (see eq.12). , mination of <5n", this being the sum of Ao and the jump found at 0 OK by extrapolation (seefig. 6). In this way Measurements of surface resistance Pippard was able to verify experimentally eq. (II,41), There are two principal methods of measuring the . <5n" = <5n'V3, in the extreme anomalous region, <5n' surface resistance: the calorimetrie method and the having been found from absorption measurements method of determining the' bandwidth of resonant (see II). cavities. The surface resistance is proportional to the ..do power absorption (II,14). In the calorimetrie methods the absorbed power is measured directly as the heat t generated per second; this heat appears in the form ...... of a temperature difference across a known "heat leak" ...... between the superconducting material and the helium ...... bath. The second method makes use of the propor- , .... tionality that exists (after correction for 'coupling-hole ...... ------effects) between the line width of the resonance of the i' ...... cavity and the power absorbed in the wall. In discussions of the results at not too high frequen- cies it is usual to try and express the measured surface resistance as a constant plus the product of a function of the temperature only and a function of the fre- quency only: (25) 2 R = RnA(w)Qj(1) + Ro. (The fact that this does not hold at high frequencies is Fig. 6. The solid straight line gives LlÖ"(T) = Ön" - As(T), the evident, for example, from the results offig. 9). In (25), jump in ö" when the superconducting state is perturbed by a magnetic field, as a function of (l_1J4)-1/2. This jump in ö" is R is the measured surface resistance at a chosen determined experimentally from the jump in the resonant fre- quency of a cavity resonator (see text). Ao is deduced from the

H slope of the line. In addition, Ön" is found from ).0 and Llö at [12] D. Shoenberg, Proc. Roy. Soc. A 175, 49, 1940. 1J = 0 using Ön" = Llö"(T = 0) + Ao. [13] M. Désirant and D. Shoenberg, Proc. Phys. Soc. 60, 413, 1948. [14J J. M. Loek, Proc. Roy. Soc. A 208,391, 1951. • The results of the measurements carried out so far [15J H. B. G. Casimir, Physica 7, 887, 1940. may be summarized as follows: [16] E. Laurmann and D. Shoenberg, Proc. Roy. Soc. A 198,560, 1949. I) Eq. (17) gives a good description of the tem- [17] W. L. McLean, Proc. VIIth Int. Conf. on low temperature perature dependence. physics, Toronto 1960, p. 330; Univ. ofToronto Press, 1961. [18] A. B. Pippard, Proc. Roy. Soc. A 191, 399, 1947. 2) The values found for the penetration depth at [19] E. A. Lynton, Superconductivity, Methuen, London 1962, OOK, AO, are indeed close to the theoretical value Ap, but p.37. 374 PHILIPS TECHNICAL REVIEW VOLUME 28

-& < 1, and Rn is the surface resistance of the normal Qualitatively both curves have the .same shape. If the metal just above the critical temperature. (/J( -&) is a curves for low values of -& are made to coincide (both function of -& which is 0 for -& = O. Ro is a "residual then being cc -&4) we see that at .0. = 0.8 there is a value" of R which depends to a great extent on the discrepancy of about ~ factor of 2. quality of the surface and which is generally assumed to be zero for an "ideal surface". Correcting for this (by b) Frequency dependence extrapolating the measured results to -& = 0), we find Fig.8 shows values of A(w) as functions of nw/kTc, for the surface resistance ratio q = R/ R« which, from for various materials, as found by a number of in- (Il,I2), is equal to 0'/ On' : vestigators or derived from their results by the method described above (in some cases by using eq. (26) in a q = A(w) (/J(-&). (26) range where it does not really apply, see caption). In the GCL model in its simplest form, 0' is given Remarkably enough, all points lie more or less on one by (23) and On' by (1I, 15,26) (On' = Àp!V2w-r), so that curve, whereas from (27) one would expect a consider- the surface resistance ratio is found to be given by: able spread owing to the presence of -r in the expression. Differences between the predictions of the frequency q = O'/On' = 2-1/2 (w-r)3/2.o.4(l- -&4)-3/2. (27) dependence of q arise from differences, not in 0', but in On'. With the GCL model, and also after the a) Temperature dependence amendment to it which we shall presently discuss, it is For the low-temperature region Pippard has pro- found that 0' is proportional to w. For On', however, posed for (/J( D) the empirical expression: one can choose between the classical limit (On' = Àp/V2w-r, cf. Il,I5,26) and the extreme anomalous (/Jp( D) = -&4(1 - .0.2) (1 - -&4)-2 .. (28) limit (On' = (Àp2v/4bw)1/3, cf. Il, 40, 41). In the first and in the temperature region -& < 0.8 (i.e. x < 0.4 and case q ex: W3/2 (as in 27), and in the second case q ex: w4/3• (/J( -&) < 0.4) most investigators in fact find that the Provided the superconductors are not unduly "dirty" I experimental values of q plotted against

Fig. 7 gives a plot of

The abrupt rise of A(w) at nw/kTc ~ 3 is attributable

uS ~2 to something that has not yet been touched upon, namely a quantum process. In very simple terms it may 0.7 be said that an energy 2L1is needed in order to raise the cp ~, electrons from the superconducting to the normal state. 0 6 1 . At absolute zero this energy is roughly 3.5 kTc (the BeS US theory predicts 3.52 kTc), and it goes to zero when T goes to Tc. Now if the frequency becomes so high that 0.4 hi» ~ 2L1, the absorption suddenly increases; quanta hos, which raise the superconducting electrons to the U3 normal state, are absorbed.

0.2 This behaviour is neatly illustrated by the meas- urements by Biondi and Garfunkel [24] on aluminium 0.1 (fig. 9). Here again q is plotted' against nw/kTc (now [20J C. J. Grebenkemper and J. P. Hagen, Phys. Rev. 86, 673, 00 1952 (including some of Pippard's results quoted by these Q1 Q2 0.3 authors). [2lJ R. Kaplan, A. H. Nethercot, Jr., and H. A. Boorse, Phys, Fig.7. The temperature dependence of the surface resistance Rev. 116, 270, 1959 (including results due to M. D. Sturge ratio q according to the London theory (- 0. 1959. 1967, No. 12 SKIN EFFECT, III 375

5.------~~~-, Fig. 8. The frequency-dependent factor A(w) in the surface resistance i ratio q as a function of tuojk'T« for Sn (crosses [201, dots (211), In [201, - / Pb [201, V [221, Cd [231 and Al [241. The • Sn .M~ #/ /~ plotted values of A are either the x Sn e 0.35 / / values reported by the authors them- a In cb 0.50 / / I / selvesor thevalues derivedfrom their 0Pb ~ 0.60 I I results (the slope ofthecurveqvs rpp, I I .,V cp 0.70 I / eq. (28), in so far as this is straight; v Cd .... 0.80 I / , eq. (26». A slightly different treat- A(w) I / ment has been applied to the results I I I I x of Biondi and Garfunkel [241 (see I I fig. 9). Eq, (26) is evidently no I IX i I I longer applicable to these. Values I I for A(w) have nevertheless been I / .~ derived from these results (for I I I I I!i {},,;:;0.8) by putting A = q/rpp({}). I I. I I • The failure of (26) is reflected in a 0.05 I I spread of the points obtained in this I /' way for AI. Even so, these points I I I I. link up to some extent with the I I. others and also give an idea of the /'./ q '1 r , marked increase of at the energy I '! gap. The straight line 1is the relation 1/' between A(w) and nw/kTc according "1 ,0.01 // to (40), the rough, amended London /1: theory. Curve 2 is the relation given 1/ by the BCS theory in the extreme 0.005 I,' non-local limit for tin (Miller [291). II' ~ A line with slope 3/2 and a line with 9 slope 4/3 have been drawn through ,q, the points at low frequencies (see .l / text, p. 374). I 0.001

0.005 0.05 0.1 0.5 5 -~kTc on a linear scale) for various values of -&. The kink in the curve where the quantum energy is equal to the energy gap lies at lower frequencies the higher the tem- peratures. In fig. JO the energy gap found from these experiments is plotted as a function of -& together with the theoretical curve predicted by the Bes theory. The points for aluminium in fig. 8 were derived from the results of fig. 9 for -& ~ 0.8.

------3.0 2.a kTc t 2.0

1.0

_nw 0.2 .1.0 kTc Fig. 9. Surface resistance ratio q as a function of nw/kTc at dif- Fig. 10. The energy gap 2Ll (expressed in units of kTc) for Al as ferent temperatures for AI, after Biondi and Garfunkel [241. Each a function of the temperature, after Biondi and Garfunkel [241. curve shows a kink, which is sharpest at the lowest temperatures. The experimental values (rectangles) are derived from the results The kink is interpreted as the point where nw is equal to the shown in fig. 9. The dashed curve is the result given by the BCS energy gap. theory. - 376 PHILIPS TECHNICAL REVIEW VOLUME 28

The experimental results and the GCL model influences the state at another point at a distance less In considering the results of the measurements we than the coherence length. Considerations of this na- have already made passing references to the GCL ture led Pippard to propose that the current at a model by way of comparison. We shall now summarize given point is determined not only by the vector poten- this comparison in the following list. tial at that point but also by the vector potential in an 1) The model gives a good prediction of the tempera- environment of the order of the coherence length. In ture dependence ofthe penetration depth Às(eq. 17). other words, there exists a non-local relation between 2) The model predicts the order of magnitude of the current and vector potential analogous to the non-local penetration depth for T = 0 "K, 1.0, but the relation that exists between current and electric field in theoretical value is systematically too small by a the theory of the anomalous skin effect (see II). factor of between 2 to 5. The analogy with the anomalous skin effect is very 3) The experimental result that Àsis dependent on the pronounced in the earlier mentioned experimental re- mean free path I is in contradiction with the theory. sult found by Pippard [73. As the mean free path I of 4) The temperature dependence of the surface resis- the electrons in a superconductor decreases (due to an tance ratio q is qualitatively predicted, but there increasing concentration of impurities), the penetra- are quantitative differences of a factor of 2 (fig. 7). tion depth Àsat first remains the same, but begins to 5) The theory gives a good prediction ofthe frequency increase when I becomes equal to or less than l.s. This dependence of q at low frequencies (fig. 8). increase is in contradiction with the London theory, 6) The results of the measurements indicate that the where the penetration depth, according to (12), is in- frequency-dependent factor in q, A(w), as a function dependent of the "sensitive parameter" I. The behav- of IiwfkTc, is not particularly susceptible to the iour runs parallel, however, with the skin effect in nor- "sensitive parameter" (l or r) or to the choice mal conductors: there too the skin depth is indepen- of metal. This is not in agreement with the theory. dent of I at large values of I (extreme anomalous re- Item 3 was one of the main reasons for the postu- gion), whereas <5 increases with decreasing I when lation of the non-local theory. This theory will be I ~ <5 (classical region). briefly outlined in the following section. It willbe shown By analogy with the skin effect, Pippard proposed that the non-local theory not only clarifies item 3 but that the London equation be replaced by the following also reduces the discrepancy under item 2. non-local relation (cf. II,39): The two amendments referred to (a new value for J = ~ r (A' r) exp (-rfç) dV. (31) 1.5 and a new value for .) will then be applied to eq. (24), f 2 4 and we shall see that this gives some clue to the cause I.tÀL r of the interesting result mentioned under item 6. The quantity ç, whose function here is similar to that of I in the skin effect, is the "coherence length" re- The non-local theory ferred to in the foregoing, being a measure of the dis- In the London equation (l Ib) tance over which the current feels the effect ofthe vector potential. The coherence is disturbed by impurities: at J = (- lff-lÀL2)A (30) smalll the value ç is also small. The coefficient outside the coefficient -lff-lÀL2 is governed by the number of the integral is proportional to the number of super- superconducting electrons ns. London showed that this conducting electrons (lfÀL2) and independent of ç (in equation might be derived by assuming that the super- the same way that the coefficient 3af4nl in the skin effect conducting electrons are in a very specific state ("the is proportional to ne but independent of I); Co is there- quantum state p = 0") in which they remain even when fore independent of ç. a magnetic field is applied. This may be freely inter- In the local limit (ç ~ Às)A may be regarded as preted by saying that the "order parameter" na is a uniform. The integration is easily carried out and the rigid quantity that does not depend on the field and result is: J = (4nf3)ç(coff-lÀL2)A. Comparison with that has the same value throughout a superconductor (30) shows that Co has the dimension of a reciprocal at a particular temperature, even at the edge where the length. Putting Co = -3f4nço, we find the local field penetrates slightly. Later, observations and theor- relation: etical considerations led to the assumption that this ç 1 J=-- - A (32), rigidity is not absolute: although such a thing exists ço f-lÀL2 as a "coherence length" within which the order para- Just as (11b) leads to a penetratión depth ÀL,so (32) meter can undergo scarcely any change, variations are gives for the penetration depth Àain the local limit: nevertheless possible over greater lengths. The re- maining rigidity implies that the state at one given point . . . . (33) 1967, No. 12 SKIN EFFECT, III 377

The length ~o, like co, does not depend on the number of The description then ties up completely with that given superconducting electrons or on the degree ofimpurity. by London: what we have is an intrinsic London super- In other words, ~o is a characteristic length of a clean conductor. The inequality ~«As is then equivalent superconductor at T = O. It now seems reasonable to to ~o «AL. As we shall presently see, such .super- assume - and this is justified by other considerations conductors are scarcely ever found. and results - that even in a clean superconductor the If, on the other hand, we have a "dirty" super- coherence is restricted to finite regions, whose dimen- conductor (1« ~o, so that ~ = I) in the local limit, sions may be related to ~o. In the more general theory it follows from (33) that the penetration depth is: . the ~o introduced above turns out to be, in fact, the (37) coherence length of the clean superconductor. (It would be wrong to conclude from (33) that for a clean From the estimate of ~o and the values of'Znin Table I superconductor, where ~ = ~o, the penetration depth it follows that the locallimit does not apply to many is AL because, as we shall presently see, the relation metals, but that on the contrary ~o ~ As. The penetra- between J and A for a clean superconductor is, in tion depth to be expected in that case may be estimated general, not local.) as follows. The current density at the surface is, in The coherence length may be regarded as a measure Pippard's theory, calculated from (31). If A were uni- of the spatial extension of the wave functions of the form, the integrand would extend roughly over a depth superconducting electrons. With the aid of the uncer- ~,and the result would be (32). However, A, and there- tainty principle ~o can be expressed in terms of the fore the integrand, extend only over the penetration energy gap 2L1(0). Only those electrons with energies depth As. The result is therefore reduced by a factor of within a band ofwidth ±LI(O) about the Fermi energy approximately AsfÇ, so that J = -(As/f-l~oAL2)A. For play a role. The spread in the momentum is therefore the penetration depth As following from this we have limited to a value of roughly Lip = 2L1(0)/v, where v As2 = ~oAL2/As, or As3 = ~oAL2. The rigorous theory is the Fermi velocity. Accordingto theuncertaintyprin- gives for thePippard limit (the non-locallimit, ~o ~ AL) ciple, the spatial extension of the wave function is then a penetration depth As (see also 11,40): at least ~o = /i/Lip, from which it follows that ~o = (38) /iv/2L1(0). The BeS theory leads to almost the same re- sult: If we integrate simply over the volume of the super- conductor, we find b = 4n/V3 (as in the anomalous ~o = /iv/nLl(O) = a/iv/kTc, (34) skin effect with diffuse reflection). Fig. 11 gives a log-log plot of the penetration depth where a = (2/n)kTc/2L1(0) = 2/n X 3.52 = 0.18. As in a clean superconductor as a function of ~o. We Using this expression we can estimate the value of note that the theoretical value of As is always greater ~o: for the "standard metal" (v = 1.4 X 106 mis) with than the values in either of the two limits (locallimit Tc = 1.18 OK (aluminium) we find ~o Rj 1.6 (.Lm. for ~o« AL, non-local limit for ~o~ AL; see also The argument presented is somewhat vague because we do not Table 11). know what exactly we are to understand by "wave functions of the superconducting electrons". In the BCS theory, superconduc- tivity comes about because electrons of opposite momentum and spin join up to form "Cooper pairs", thereby lowering their energy. In this theory ~o is the average distance between the members of a Cooper pair.

The interpretation of ~o as the coherence length of the dean superconductor means that ~, which is small when I is small, is equal to ~o when I ~ ~o. To obtain this situation we may, for example, write with Pippard:

1/~= 19o + 1/1. (35)

If, now, we have a superconductor which is clean Eig. 11. The penetration depth As in a clean superconductor as a (I ~ ~o, so that ~ = ~o) and to which the local limit function ofthe coherence length ~o(schematic, logarithmicscales). also applies (~ «As) we find for the penetration depth For ~o « AL (left) Äs approaches to AL (local limit); this is the "intrinsic London superconductor". For ~o» AL (right) Äs ap- with the aid of (33): proaches the Pippard limit, given by (38). Whatever the value of ~o, the value of Äs is greater than either of the corresponding (36) limit values. 378 PHILlPS TECHNICAL REVIEW VOLUME 28

Of the experimental re- Table IT. The penetration depth in tin and aluminium as found experimentally and in various theoretical approximations; after Bardeen, Cooper and Schrieffer [lOl. ÄL is the sults giving general corrob- London penetration depth, Äs(P) the penetration depth in the Pippard limit, Äs(th) the oration of the non-local penetration depth predicted by the complete non-local theory and Äs(exp) the experimental value. The penetration depths and the coherence length, ~o, are expressed in nm. v is the theory, we shall mention Fermi velocity, Tc the critical temperature. only the following. First of all, as was to be ex- v ~O I AL Äs(P) Äs(th) Äs(exp) ~~~~~~~~~~~--- pected, the theory provides 6 a good description of the Sn 0.69 X 10 mIs 3.73 oK 250 35 44 57 51 AI 1.26X 106 m/~ 1.18 oK 1500 16 47 52 dependence of Äs on I. For 49 large I we must apply (38); Às is then independent of I. As I becomes smaller we go more in the direction of (37) and Às increases as I easily done. The London screening, 1/ÄL2, is to be decreases. Quantitatively too, Pippard found a good expected in a superconductor that is not dirty and in measure of agreement with a suitable choice of ~o. which no non-local effects occur, in other words a Furthermore, the discrepancy between' measured and superconductor where ~o «I and ~o «Äs (the intrinsic predicted penetration depths in clean superconductors London superconductor). For the screening in a "dirty" . is considerably reduced. F9r our standard metal with· superconductor ü « ~o, Äs) we find by applying the Tc = 1.18 "K we found ~o = 1.6 [Lm. At T = ° in the above procedure, now with rednetion factor lf~o, that Pippard limit (38) this gives Äs= 59 nm, a value that I/Às2 = (l/~o) (l/ÀL2), in agreement with (37). In a comes closer to the measured values than ÄL which, superconductor in the non-local limit (Äs « ~o, I) we at T = 0, is equal to Àp = 22 nm (II, Table I). As an find 1/Äs2 = (Äs/~o) (l/ÄL2), in agreement with (38), illustration we give in Table II a few values for tin except for a factor 2/b. and aluminium. The success of the theory is partic- The formulae for the superconductor in the three ularly striking in the case of aluminium. limiting cases are obtained in detail. from the corre- We shall =.consider here the consequences of sponding equations for the skin effect in normally Pippard's theory on the temperature dependence of conducting metals by the substitutions: Às. We shall merely repeat that the prediction (17), 0-+ jÄs, »[o: -+ j~o, (/-+ I). Às o: (1 - f}4)-1/2 of the GeL model - only to be used as a guide for classifying the phenomena, having The fact that ~o in the superconductor corresponds to -jvlw been superseded by more fundamental considerations in the normal conductor can also be seen from a comparison - is on the whole a very reasonable prediction, not of expression (31) with the corresponding expression (II,39), only qualitatively, but quantitatively as well. provided we make allowance in the latter expression for the "retardation" (see Ir, p. 310). To do so we must substitute Analogy with skin effect in normal metal; classification for E in (II,39) the field at the time t - r!», when calculating of superconductors the current at the moment t. If current and field depend on time as exp(jwt), we then obtain a factor exp (-jrwlv) under The approach described in the introduetion is useful the integral. The exponent of e under the integral is therefore for briefly summarizing the theoretical results given -rtlll + jwlv) in the normal metal and -r/~ = -r(111 + 1/~0) above. in the superconductor, which demonstrates the correspondence mentioned. A superconductor is formally closely related to a normal metal in the relaxation limit. In a normal metal in the relaxation limit (B, E), v/w is much smaller than Infig. 12 we have classified superconductors accord- I and 0 and the screening -1/02 is I/Àp2. The modulus ing to their values of ~o and I, by analogy with the of the screening in the classical region A (I « 0, v/w) (J)-7: diagram in II, fig. 11 (reproduced here as fig. I). can be obtained by reducing the screening in B by a We have plotted ~o from right to left in order to bring factor I/(v/w). In the same way we obtain the modulus out more clearly the analogy with fig. 11 in II. A' is the of the screening in region D of the anomalous skin region of the "dirty" superconductors, B' that of the effect (0 « I, v/w) by reducing the screening in B by a intrinsic London superconductors, and D' that of the factor o/(v/w). superconductors in the Pippard limit. In the D' region Since Pippard's non-local theory for the supercon- ~> Äs; this corresponds to type I superconductors. The ductor runs formally parallel with the theory of the type II superconductors lie in A' en B' (Äs > ~). anomalous skin effect, it should be possible to find the Fig. 13 shows a number of superconductors in the equations for Äs along similar lines. Apart from a few ~o-I diagram. Nearly all superconducting elements fine points such as the factor 2/b in (38), this is quite are to be found in D'. Examples are the elements Al and 1967, No. 12 SKIN EFFECT, III 379

(and which shift the boundary between D' and B' by a factor of V2 to the left) it can be shown to be a \ reasonable assumption that they really lie in B' [25]: they D' I constitute the onlyexamples known thus far ofintrinsic London superconductors. To be such, however, they I have to be of exceptional purity: an "ordinary" piece 8' of material comes into region A'. A striking difference between the go-l and the W-"C diagram (fig. 1)is that, for a given element, there is only one degree of freedom in the first case (I) but two in the "C). diag-am A' second case (wand In the go-/ one can only move vertically via the degree of impurity, whereas in the (V-"C diagram one can move horizontally as well (frequency of the field). Furthermore, the "filling" of 50- the regions in the two cases is quite different. Admitted- Fig. 12. The ~o-l diagram for superconductors. In D', well away ly A and A' are both to be regarded as the regions of from the boundaries, the Pippard limit applies (type 1 supercon- ductors), in A' and B' the local limit (type II superconductors). "practical" materials for electrical engineering pur- A' is the region of "dirty" superconductors, B' the region of poses. However, whereas with the normal metals it "intrinsic London superconductors". In A' the penetration depth Ä. is given by (37), in B' by (36) and in D' by (38). The figure is was possible to reach region D from region A only in formally analogous with 11, fig. 11, apart from region C in the carefully performed experiments, the first super- latter diagram. conductors arrived in D' and it was not until later that the alloys in region A' began to be explored. And, Sn indicated in the diagram. The superconducting while every shining metal surface bears witness to the alloys, which have proved to be much more impor- reflection in the relaxation region B, E, region B' is tant for practical applications, lie in N. Intrinsic Lon- practically empty. don superconductors are almost non-existent: the B' Reversal of the sign of the field in the penetration layer region is virtually empty. Interesting in this connection The non-local theory has a further remarkable consequence as are the elements V and Nb, which are also indicated: regards the magnetic field in the superconductor. In the simple they lie in the border regions between D' and B'. Using configuration shown by fig. 1 in part II (J//x-axis, Hlly-axis, more rigorous criteria than we have employed here z-axis .L surface) one can write for Maxwell's first relation (1,1): 1 o2A --=-J It oz2 '

that is to say, if A is plotted against z then the curvature of the curve is given by J. If, now, a local relation exists between J and II ,11, A, e.g. of the type (11b) or (32), i.e. J is proportional to A but of 1111 I opposite sign, the curvature is proportional to +A, so that the 1111 8' convex side is always directed towards the axis. This implies London D' Pippard limit 'I I' f that, for a given A at the edge, there is only one value of dA/dz at (elements) I I limit the edge that shows the correct behaviour (see fig. 14a): if dA/dz :::::-~'" I1 ...... _ ...... :::::::::, ...... _ I1 100nm is too large, A is deflected so quickly from the axis that it goes ::.-~~- -- ...... :::::.._ type I I1 --..._ -...:::-- -""::::::...... I towards + co for z -+ co; if dA/dz is too small, it overshoots 0 ..._-- -"""':::--..,:::--.._ ---:::::---:::::-l I and bends away from the axis on the negative side. The only -.... ------..._..._..._-- -.::::_------...._~.:: ~ acceptable solution is the exponential one . ..._--::--~ Ifthe relation between J and A is a non-local one, however, the ----~ type J[ curvature of A is proportional to the mean value of A over a -, A' "dirty" limit -, region around the point under consideration. Close to the surface -, 10nm (alloys) this will turn out to be greater than A at the point itself. Initially, type J[ -, -, therefore, there will be a greater curvature than in the first case, -, so that: 1) if A does not reach zero it will certainly bend away 100nm 10nm 1pm from the axis again, and 2) if A does go to zero there will still be 50- some positive curvature left at the point where A = 0 (see fig. Fig. 13. The ~o-l diagram for the standard metal and for AI, Sn, 14b). The acceptable solution, therefore, passes through zero. A V and Nb. The boundaries (dashed lines) between areas A', B' consequence of the non-local relation is therefore that the vector and D' are entirely governed by the value of ÄL(O) = Äp of the material. The bold vertical lines at the top represent the clean potential A (and likewise the field H) undergoes a reversal of sign materials. As the metal becomes more impure it moves vertically in the penetration layer. downwards in the diagram. The values of ÄL(O) and ~o for Al and Sn are from Table I and for Nb and V from Radebaugh and [25] R. Radebaugh and P. H. Keesom, Phys. Rev.149,217, 1966. Keesom [25] and Stromberg and Swenson [26]. Further details are [26] T. F. Stromberg and C. A. Swenson, Phys. Rev. Letters9, explained in the text. 370, 1962. 380 PHILIPS TECHNICAL REVIEW VOLUME 28

The function W2( {}), along with Wp( {}) and WI( {}), is plotted against {}in fig. 7. With

(26): q = O'JOn' = A(w)W({}), (II,40-41): On' = (VAp2J4bw)I/3, _ ..... (34): ço = alivJkTc - we find that . A(w) = 2(aliwJkTc)4/3. . . .. (40)

Fig. 14. The vector potential A as a function of the depth z in the This is a nice result, because we have now found an superconductor, a) in the local theory, b) in the non-local theory. "explanation" for the fact that - according to fig. 8 - A(w) as a function of the "reduced frequency" IiwJkTc An attempt to verify this, and with it the non-local theory in is not very dependent on the impurity of the metal or on a fairly direct way, has been made by Drangeid and Sommer- the choice of metal. Not only the sensitive parameter I, halder [27l, with positive results. In their experiment a hollow but also such characteristic parameters of the metal as cylinder (a film of tin deposited on glass, thickness 1.87 urn) was Ap, v, ço, and even the parameter b, which depends on placed in an alternating field parallel to the axis of the cylinder. the manner in which an electron is reflected at the sur- The intensity of the alternating field was very low (maximum 30 Oe) in order to avoid complications, and the frequency face, have disappeared from the equation. According ("" 105 Hz) was so low that the superconductor was effectively to (40), A(w) is a universal function of IiwJkTc [9]. In subject to a d.c. field (i.e, the first term in (20) is negligible). the BeS theory, a is a constant: a = 0.18 (eq. 34). We The field inside the cylinder was measured with a coil. This field, recall that the calculation given relates only to "extreme which was extremely weak (10-7 times the external field, in superconductors of type I" which, in the normal state, agreement with the theoretical prediction) was indeed found to r<;:verseits phase below a certain temperature, and thus to have just above Tc, are situated in the extreme anomalous the opposite sign from that of the external field. region. Expression (40) is only a rough application of the The amended theory of the surface resistance non-local theory, and the agreement with experiment We shall now apply the two amendments (a new is quantitatively not particularly good. A plot of value for As and a new value for r) to expression (24) A(w) in accordance with (40), for a = 0.18, is given in for ~', confining ourselves to superconductors oftype 1. fig. 8 (line 1). Mattis and Bardeen [28] have presented Let the frequency be so high that the normal metal (just a theoretical treatment of the anomalous skin effect for above the critical temperature) is in the anomalous normal and superconducting metals on the basis of region. the BeS theory, and this theory has been worked out The new value for As is given by (38). For ï we must in more detail by Miller [29]. Curve 2 in fig. 8 is Miller's substitute the time during which the field is in unper- result for the non-local limit for tin. The discrepancy turbed interaction with the electron; in the present between curve 2 and the experimental results indicates case this is the time in which the electron traverses the that the non-local limit for most metals is not a very skin layer, and is roughly bAsJv. After substitution this good approximation. Miller calculated the corrections gives: to this limit for tin and aluminium, and his results 0' = (bJ2) (wJv)AéJAn2. largely explain the discrepancy. At high frequencies the analysis can no longer be The factor IJAn2 is proportional to the concentration used. The hypothesis (26) that q may be written as a of the normal electrons; according to (18) and (14), product of a function of w only and a function of {} IJAn2 = {}4JAp2. Experimental evidence - and also the only is then no longer valid. This appears, for example, more elaborate theory - shows that the dependence in the spread of the points for aluminium in fig.8; this is of the penetration depth on temperature is reasonably not due to spread in the direct results of the measure- represented by prediction (17): As ex: (1 - {}4)-1/2. ments. According to the BeS theory, however, the Keeping to this température dependence, and using sharp rise of q - reflected in the sharp rise of A(w) in (38) and AL({} = 0) = Ap, we find: fig. 8 - at T. 0 does occur in all metals at the same As = (2ÇOAp2Jb)I/3 (1 - {}4)-1/2. value of IiwjkTc, namely at IiwJkTc = 3.52. This part. of the curve too, therefore, has a universal character. For 0' this gives: Superconductive resonant cavities where As regards the applications of superconductivity a . . . . . (39) great deal of interest is centred on type II superco.n- 1967, No. 12 . SKIN EFFECT, III 381 ductors in the mixed state (e.g. for producing strong is for frequency stabilization. The frequency stability magnetic fields or for the large-scale transport of that can be achieved using a resonant cavity as reference electrical energy without losses [2l). These fall outside element is inversely proportional to the Q. For compar- our subject because the field penetrates deep into the ison we note that quartz crystals, widely employed material. Applications in which the field penetration is for frequency stabilization - at lower frequencies, limited to a skin layer are Iow-loss microwave cavity not exceeding about 10 MHz - have Q values of be- resonators and waveguides [30l. tween 104 and 105 [33l. The construction of a high-Q cavity resonator in- Other possible applications have been investigated at volves the following considerations. A high Q is equiv- the Philips laboratory in Hamburg. With only moderate alent to Iow absorption in the wall, and therefore to a power input an intense microwave field can be excited small surface resistance. It can be seen from fig. 7 that in a superconducting resonant cavity: this field can be T should be small with respect to Tc, and from fig. 8 used to set up electron field emission, and as this effect that '(J) should be small with respect to kTc/n. If the T is non-linear, it can be used for frequency multiplica- 32 that can be reached is given, or if the (J) is fixed by the tion [ l. This study, interesting in itself, has not, how- problem, then a material with a large Tcmust be chosen. ever, yielded results that can match those obtained with According to (26) and (28), q should go to zero for conventional methods of frequency multiplication. {}~ 0 but in practice, according to (25) the limit of A highly topical application is the use of supercon- the surface resistance is determined by the "residual ductive resonant cavities or waveguides in linear par- value" Rs: This depends to a very great extent on the ticle accelerators, e.g. electron accelerators. The par- state of the surface. Much ofthe research done on super- ticles in such accelerators are passed through a series of conductive resonant cavities has therefore been directed resonant cavities which are synchronized in such a way towards finding the best method of surface treatment. that a group of particles is always subject to an accel- As an example of what has been achieved in this erating field, or they are passed through a waveguide field we mention the work of Wilson [31l and that of so shaped as to keep a group of particles "surf-riding" Zimmer [32l. Wilson investigated copper cavities with on a travelling wave [34l. For a given accelerating field, a surface layer of tin or lead at a frequency of 2856 the high-frequency power required is greater the higher MHz. The best results were obtained with unpolished the losses in the wall. To keep the average power within electrolytic films. The highest Q he found was 5 X 107 technically and economically acceptable bounds, most for tin and 2 X 108 for lead. These values were reached existing accelerators are pulsed, so that high-energy at a temperature of 1.8"K. They are a factor of between particles are delivered only for, say, 1 % of the time. 103 and 104 greater than the Q of an identical cavity of Such considerations are governed by the capacity ofthe copper at room temperature - a remarkable improve- cooling system as well as the availability of high-fre- ment compared with the factor of merely 4 that would quency energy sources. By drastically cutting down the at the most be achieved by using high-purity copper at wall losses with the aid of superconductors it might well low temperature (see Il, p. 315). Application of eq. (25) be possible, using a much smaller average power, to to the results showed the relative residual resistance for produce particles of the same energy continuously. . tin to be Rel R« = 0.003 and for lead Rol R« = 0.0013. Summary. This last part of a series of articles on the skin effect Using a rectangular copper cavity with an electrolytic- deals with the penetration of an electromagnetic field in the sur- ally deposited layer of lead at 4.2 OK and 9375 MHz, face of a superconductor. A simple model of the superconductor is the two-fluid model in which a fraction ofthe electrons is "super- Zimmer found a Q of more than 8 X 106• A Q of 107 to conducting" and the remainder "normal". Instead of Oh m's law, 108 therefore seems representative of the best that can the London equations apply to the supercurrent; one of these states that the supercurrent is proportional to the vector potential. be achieved at the present time. In this model the superconducting electrons are responsible for One possible application of such a resonant cavity the screening effect, even at zero frequency, while the normal electrons in the skin layer thus formed are responsible for the. [27] K. E. Drangeid and R. Sommerhalder, Phys. Rev. Letters absorption that occurs only at non-zero frequency. The=London 8, 467, 1962. penetration depth" is independent of frequency. Among the [28] D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412, 1958. amendments to this model special attention 'is paid to that of [29] P. B. Miller, Phys, Rev.118, 928, 1960. Pippard, in which theLondon relation between current and vector [30] See E. Maxwell, Superconducting resonant cavities, Pro- potential is replaced by a non-local relation. This theory runs gress in Cryogenics 4, 123-158, 1964. parallel with that of the anomalous skin effect, discussed in part [31] P. B. Wilson, Nucl. Instr, Meth. 20, 336, 1963. II. A characteristic quantity occurring in Pippard's theory (and [32] H. Zimrner, in: Philips; unsere Forschung in Deutschland, elsewhere) is the "coherence length". By comparing this with the Philips Zentrallaboratorium GmbH, Aachen-Hamburg 1964, penetration depth it is possible to classify superconductors into p. 134, 136. different types. Experirnental results relating both to the penetra- [33] See J. C. B. Missel, Philips tech. Rev. 11, 145, 1949/50. tion depth and the surface absorption support this refinement of [34] See the articles on this subject in this journal: D. W. Fry, the theory at several points. An application of this aspect of The linear electron accelerator, Philips tech. Rev.14, 1-12, superconductivity, in which the field penetration is limited to a 1952/53; and C. F. Bareford and M. G. Kelliher, The 15 skin layer, is to be found in the fabrication of high-Q resonant . million electron-volt linear electron accelerator for HarweIl, cavities. Its applications in linear accelerators are at present Philips tecb. Rev. 15, 1-26, 1953/54. . under serious consideration. ~82 PHILlPS TECHNICAL REVIEW VOLUME 28

Recent scieritific publications

These publications are contributed by staff of laboratories and plants which form part of or co-operate with enterprises of the Philips group of companies, particularly by staff of the following res.earch laboratories: Philips Research Laboratories, Eindhoven, Netherlands E Mullard Research Laboratories, Redhill (Surrey), England M Laboratoires d'Electronique et de Physique Appliquée, Limeil-Brévannes (S.O.), France , L PhilipsZentraIIaboratorium GmbH, Aachen laboratory, Weisshausstrasse, 51 Aachen, Germany . A Philips ZentraIlaboratorium GmbH, Hamburg laboratory, Vogt-Kölln- Strasse 30, 2 Hamburg-Stellingen, Germany H MBLE Laboratoire de Recherche, 2 avenue Van Becelaere, Brussels 17 (Boitsfort), Belgium. B Reprints of most of these publications will be available in the near future. Requests for reprints should be addressed to the respective laboratories (see the code letter) or to Philips Research Laboratories, Eindhoven, Netherlands.

E. A. Aagaard: Time division connection network for G. Blasse & A. Bril: An example of disagreement be- combined speech and large-signal transfer. tween site symmetry and the splitting of Eu3+ emission Communications présentées au Colloque International lines. . de Commutation Electronique, Paris 1966, p. 289-300. Solid State Comm. 5, 1-3, 1967 (No. I). E E G. Blasse & A. Bril: Fluorescence of Eu3+-activated G. A. Acket: Determination ofthe negative differential garnets containing pentavalent vanadium. mobility of n-type gallium arsenide using 8 mm micro- J. Electrochem. Soc. 114, 250-252,' 1967 (No. 3). E waves. 3 Physics Letters 24A, 200-202, 1967 (No. 4). E G. Blasse & A. Bril: Fluorescence of Eu +-activated lanthanide oxyhalides LnOX. W. Albers, G. van Aller & C. Haas: Band structure and J. chem. Phys. 46, 2579-2582, 1967 (No. 7). E electrical properties of ternary chromium chalcoge- 3 nides. G. Blasse & A. Bril: Some observations on the Cr + Coll, int. Centre Nat. Recherche Scient. No. 157 fluorescence in the huntite structure. (Dérivés semi-métaIIiques, Orsay 1965), 19-29, 1967. E Phys. Stat. sol. 20, 551-556, 1967 (No. 2). E R. Bleekrode: Negative absorption in chemically react- L. K. H. van Beek, J. Helfferich, H. Jonker & Th. P. G. ing systems: flames and explosions. W. Th'[ssens: Properties of diazosulfonates, Part I. The J. Chimie phys. 64, 141-146, 1967 (No. I). E dissociation of methoxybenzenediazosulfonates. Rec. Trav. chim. Pays-Bas 86, 405-409, 1967 (No. 4). E P. F. Bongers & E. R. van Meurs: Ferromagnetism in compounds with pyrochlore structure. H. J. Bensieck & H. Severin: Ebener Reflektor mit J. appl. Phys. 38, 944-945, 1967 (No. 3). E variablem Reflexionsfaktor für elektromagnetische Zentimeterwellen, G. A. Bootsma & F. Meyer: EIIipsometric investigation Nachfichtentechn. Z. 20, 280-286, 1967 (No .. 5). H of the film growth at the germanium electrolyte inter- face. . M. Berth & R. Petit: Tube analyseur d'images à Surface Sci. 7, 250-254, 1967 (No. 2). E mosaïque de germanium. Acta electronica 10, 123-135, 1966 (No. 2). L G.-A. Boutry & P. Lebon: Divers régimes d'une dé- charge électrique du type Penning aux très basses pres- M. Berth & C. Venger: Technologie des mosaïques de photodiodes. ~m. . C.R. Acad. Sci. Paris 264B, 519-522, 1967 (No. 7). L Acta electronica 10, 157-180, 1966 (No. 2). L G. Bouwhuis: Optische Methoden bei der experimen- G. Blasse: .Concentration quenching of Eu3+ fluores- tellen Untersuchung nichtlinearer Vorgänge im Gas- cence. laser. ' J. chem. Phys. 46,2583-2585, 1967 (No. 7). E Optik 25, 294-298, 1967 (No. 3). E G. Blasse: Crystal structure and fluorescence of com- G. Brédart & Ph. van Bastelaer: Les paramètres matri- positions ATiNbOo, ATiTaOo and ATiSbOo. ciels du transistor et leur emploi dans l'étude du com- Mat. Res. Bull. 2, 497-502, 1967 (No. 5). E portement transitoire des circuits transistorisés (cas des G. Blasse & A. Bril: Crystal structure and fluorescence petits signaux). 2e partie : Etude du comportement of some lanthanide gallium borates. transitoire des circuits transistorisés. J. inorg. nuel. Chem. 29, 266-267, 1967 (No. I). E Rev. MBLE 10, 33-42, 1967 (No. I). B 1967, No. 12 RECENT SCIENTIFIC PUBLICATIONS

J. C. Brice: The effect of arsenic pressure on crystal J. G. C. de Gast: A new type of controlled restrictor .efficiency for injection luminescence in gallium ar- (M.D.R.) for double film hydrostatic bearings and its senide. application to high-precision machine tools. Solid-State Electronics 10, 335-337, 1967 (No. 4). M Advances in machine tool design and research 1966 (Proc. 7th Int.' M.T.D.R. Conf., Univ. Birmingham), J. C. Brice, J. A. Roberts & G. Smith: Mass spectro- p. 273-298; Pergamon Press, Oxford 1967. E metric studies of impurities in gallium arsenide crystals. J. Mat. Sci. 2, 131-138, 1967 (No. 2). M P. Gerthsen, E. Kauer & H. G. Reik: Halbleitung eini- ger Übergangsmetalloxide im Polaronenbild. J. C. Brice & P. A. C. Whiffin: Solute striae in pulled Festkörperprobleme 5, 1-56, 1966. A crystals of zinc tungstate. Brit. J. appl. Phys. 18, 581-585, 1967 (No. 5). M P. J. van Gerwen: Application of pseudo-ternary codes for data transmission. A. Broese van Groenou: Low frequency magnetic relax- Progresso elettronico 1966 (Atti XIII Congr. per ations in manganese ferrous ferrites at low temper- l'Elettronica, Roma), _Vol.I, p. 463-477. E atures. A. A. van der Giessen: Magnetic properties of ultra-fine J. Phys. Chem. Solids 28, 325-331, 1967 (No. 2). E iron (m) oxide-hydrate particles prepared from iron A. Broese van Groenou, J. A. Schulkes & D. A. Annis: (UI) oxide-hydrate gels. Magnetic anisotropy of sonie nickel zinc ferrite crystals. J. Phys. Chem. Solids 28,343-346, 1967 (No. 2). E J. appl. Phys. 38, 1133-1134, 1967 (No. 3). E, M J.-M. Goethals: Factorization of cyclic codes. K. H. J. Buschow & W. A. J. J. Velge: Phase relations IEEE Trans. on information theory IT-13, 242-246, and intermetallic compounds in the lanthanum-cobalt 1967 (No. 2). B system. J. L. Goldstein: Auditory nonlinearity. J. less-common Met. 13, 11-17, 1967 (No. I). E J. Acoust. Soc. Amer. 41, 676-689, 1967 (No. 3). E K. H. J. Buschow & J. H. N. van Vucht: Systematic A. H. Gomes de Mesquita & K. H. J. Buschow: The arrangement of the binary rare-earth-aluminium sys- crystal structure of so-called a-LaA14 (LajAln). tems. Acta cryst. 22, 497-501, 1967 (~o. 4). E Philips Res. Repts. 22, 233-245, 1967 (No. 3). E H. C. de Graaf! & J. A. van Nielen: Temperature in- H.-J. Butterweck: Frequenzabhängige nichtlineare fluence on the channel conductance of m.o.s. transis- Übertragungssysteme. tors. ,E Archiv elektro Übertr. 21, 239-254, 1967 (No. 5). E Electronics Letters 3, 195-196, 1967.(No. 5). T. D. Clark: High energy structure in the d2IjdV2 C. A. A. J. Greebe: Mechanical excitation of helicon versus V characteristic of metal-insulator-thallium waves. junctions. Philips Res. Repts. 22, 133-141, 1967 (No. 2). E Physics Letters 24A, 459-460, 1967(No. 9). M H. G. Grimmeiss: Elektrolumineszenz in III-V-Ver- bindungen. T. D. Clark & J. P. Baldwin: Superconducting memory Festkörperprobleme 5, 221-248, 1966. A device using Josephson junctions. Electronics Letters 3, 178-179, 1967 (No. 5). M G. J. van Gurp: Temperature dependence ofthe critical field in superconducting vanadium. J. P. Cosier & R. F. Pearson: Low-temperature per- Phys. Stat. sol. 19, 173-176, 1967 (No. I). E meability measurements on ferrites. Brit. J. appl. Phys. 18, 615-620, 1967(No. 5). M W. van Haeringen: The effectofsaturationon thepolari- zation parameters of modes in anisotropic gas lasers. A. J. Dekkers, A. van Duuren, F. A. Lootsma & J. Physics Letters 24A, 65-66, 1967 (No. I). E Vlietstra: Berekening van verkeerslichtenprogramma's J. Haisma: Construction and properties of short stable met behulp van lineaire programmering. gas lasers. E Ingenieur 79, V 11-17:,1967 (No. 8). Thesis, Utrecht 1966. E A. van der Drift: Evolutionary selection, a principle J. Haisma: Gebruik van lasers. governing 'growth orientation in vapour-deposited Ingenieur 79, 065-74, 1967 (No. 26). E layers. . N. Hansen: Gerät zur automatischen Bestimmung der Philips Res. Repts. 22, 267-288, 1967 (No. 3). E Oberflächengröûe feinteiliger Substanzen. W. F. Druyvesteyn: Collective modes in type II super- Automatik 11, 253, 1966 (No. 7). A conductors. P. A. H. Hart: Gas discharge as a source of incoherent Physics Letters 24A, 415-416, 1967 (No. 8). E radiation at millimetre and sub-millimetre wavelengths. G. Engelsma: Effect of cycloheximide on the inactiva- Philips Res. Repts. 22, 77-109, 1967 (No. 2). E tion of phenylalanine deaminase in gherkin seedlings. H. F. van Heek: Unusual electrode configuration for Naturwiss. 54, 319-320, 1967 (No. 12). E Hall measurements on thin filmsand field-effectdevices. Solid-State Electronics 10, 268-269, 1967 (No. 3). E A. J. Fox & J. R. ManselI: Tunable microwave-fre- quency light modulator. J. C. M. Henning: Covalency and hyperfine structure of Proc. lnstn. Electr. Engrs. 114,741-744, 1967 (No. 6). (3d)5-ions in crystal fields. M Physics Letters 24A, 40-42, 1967 (No. I). E 384 PHILIPS TECHNICAL REVIEW VOLUME 28

J. C. M. Henning, J. Liebertz & R. P. van Stapele: B. Lersmacher, H. Lydtin, W. F. Knippenberg & A. W. Evidence for Cr3+ in four-coordination: ESR and Moore: Thermodynamische Betrachtungen zur Koh- optical investigations of Cr-doped AIP04 crystals. lenstoffabscheidung bei der Pyrolyse gasförmiger Koh- J. Phys. Chem. Solids 28, 1109-1114, 1967 (No. 7). lenstoffverbindungen. A,E Carbon 5, 205-217, 1967 (No. 3). A, E P. L. Holster: Gaslagers 'met uitwendige' drukbron, P. R. Locher: Cu-NMR in paramagnetic and ferro- .1. Werking en eigenschappen van gaslagers met uit- magnetic CuCr2Se4. wendige drukbron, 2. Berekeningsmethode van aëro- Solid State Comm. 5, 185-187, 1967 (No. 3). E statische lagers. Polytechn. T. Werktuigbouw 22,. 363-370, 415-425, C. Lockyer: A new interferometric method for meas- 1967 (Nos. 9, 10). E uring the thickness of thick films. J. sci. Instr. 44, 393-394, 1967 (No. 5). M G. J. vanHoytema, W. J. Oosterkamp, A. M. C. van den Broek & A. Druppers: La neuroradiologie avec sous- F. A. Lootsma: Logarithmic programming: a method traction utilisant la télévision et une mémoire d'image of solving nonlinear-programming problems. magnétique. Philips Res. Repts. 22, 329-344, 1967 (No. 3). E Neuro-Chirurgie 12, 801-809, 1966 (No. 7). E F. K. Lotgering & R. P. van Stapele: Magnetic and G. H. Jonker: Kristalchemie en kristal fysica ten dienste electrical properties of copper containing sulphides and van de elektrotechniek. selenides with spinel structure. Chem. Weekblad 63, 81-86, 1967 (No. 8). E Solid State Comm. 5, 143-146, 1967 (No. 2). E G. H. Jonker: Optimale keramische Struktur für ver- H. Lydtin: Eine einfache Methode zur Bestimmung des schiedene Anwendungen von Bariumtitanat. Bodenkörpertransportes über die Gasphase. Ber. Dtsch. Keram. Ges. 44, 265-266, 1967(No. 6). E Z. Naturf. 22a, 571, 1967 (No. 4). A W. Kischio: Über Borphosphid. Z. anorg. allgem. Chemie 349, 151-157, 1967 (No. 3/4). M. H. van Maaren & G. M. Schaeffer: Some new superconducting group Va dichalcogenides. A Physics Letters 24A, 645-646, 1967 (No. 12). E R. J. Klein Wassink: Wetting of solid-metal surfaces by molten metals. G. Marie: Un nouveau dispositif de restitution d'ima- J. Inst. Met. 95, 38-43, 1967 (No. 2). E ges utilisant un effet électro-optique: Ie tube TITUS. Philips Res. Repts. 22, 110-132, 1967 (No. 2). L M. Klerk & E. Roeder: Der Elektronenstrahl-Mikro- analysator, ein analytisches Hilfsmittel für den R. Memming & G. Neumann: Formation of surface Chemiker. states and radicals on Ge electrodes. Chemie-Ing.-Technik 39, 567-574, 1967 (No. 9/10). Physics Letters 24A, 19-21, 1967 (No. I). H A,E L. Merten: Zur Ultrarot-Dispersion zweiachsiger und S. Kornblum & W. G. Koster: The effectof signal inten- einachsiger KristalIe, I. sity and training on simple reaction time. Z. Naturf. 22a, 359-364, 1967 (No. 3). A Acta psychol. 27, 71-74, 1967. E H. J. G. Meyer, G. Ahsmann, J. W. van der Laarse & W. G. Koster & J. A. M. Bekker: Some experiments on E. H. A. M. van der Wee: A new class of low-pressure refractoriness. are columns with positive V-I characteristics. Acta psychol. 27, 64-70, 1967. E Philips Res. Repts. 22, 209-218, 1967 (No. 3). E W. Kwestroo, J. de Jonge & P. H. G. M. Vromans: E. J. MilIett: The balanced substitution of mixed Influence of impurities on the formation of red and valency ions in oxide crystals. yellow PbO. Brit. J. appl. Phys. 18, 357-358, 1967 (No. 3). M J. inorg.: nucl, Chem. 29, 39-44, 1967 (No. I). E B. J. Mulder: Anisotropy of light absorption and ex- W. Kwestroo, C. Langereis & H. A. M. van Hal: Basic citon diffusion in anthracene crystals determined from lead nitrates. the externally sensitized fluorescence. J. inorg. nucl. Chem. 29, 33-38, 1967 (No. I). E Philips Res. Repts. 22, 142-149, 1967 (No. 2). E P. van der Laan & J. Oosterhoff: Experimental deter- B. J. Mulder & J. de Jonge: Electronic absorption spec- mination ofthe power functions ofthe two-sample rank trum ofholes in anthracene .. tests of Wilcoxon, Van der Waerden and Terry by Solid State Comm. 5, 203-205, 1967 (No. 4). E Monte Carlo techniques, r. Normal parent distribu- J. Neirynck & Ph. van Bastelaer: La synthèse des filtres tions. par factorisation de la matrice de transfert. Statistica need. 21, 55-68, 1967 (No. 1). E Rev. MBLE 10, 5-32, 1967 (No. I). B H. de Lang: Magnetic phenomena in gas lasers. J. Neirynck & J. P. Thiran: Sensitivity analysis ofloss- Physica 33, 163-173, 1967 (No. I). E less 2-ports by uniform variations of the element values. M. Lemke: Wellentypen im ferritgefüllten Rechteck- Electronics Letters 3, 74-76, 1967 (Nö. 2). B hohlleiter mit transversaIer Magnetisierung parallel zur D. J. van Ooijen: Dynamics of the magnetization of a Hohlleiterbreitseite. thin-walled superconducting cylinder in a parallel field. Archiv elektro Übertr. 21, 440-446, 1967 (No. 8). H Philips Res. Repts, 22, 219-232, 1967 (No. 3). E 1967, No. 12 RECENT SCIENTIFIC PUBLICATIONS 385

D. J. van Ooijen & A. S. van der Goot: The internal H. G. Reik & D. Heese: Frequency dependence of the friction of cold-worked niobium and tantalum contain- electrical conductivity of small polarons for high and ing both oxygen and nitrogen. low temperatures. Philips Res. Repts. 22, 150-160, 1967 (No. 2). E J. Phys. Chem. Solids 28, 581-596, 1967 (No. 4). A J. J. C. Oomen: On some electrical properties of H. G. Reik & R. Mühlstroh: Phonon structures in the colloidal systems. polaron enhanced light absorption at low temperatures. Thesis, Utrecht 1966. E Solid State Comm. 5, 105-108, 1967 (No. 2). A G. W. van Oosterhout: The transformation y-FeO(OH) J. Revuz: Variation du facteur de contraste dans les to a-FeO(OH). cibles à mosaïque de jonctions. J. inorg. nucl. Chem. 29, 1235-1238, 1967 (No. 5). E Acta electronica 10, 195-214, 1966 (No. 2). L W. J. Oosterkamp: A survey of scintillation camera C. Rooymans & W. Albers: High pressure polymor- systems. phism of FeCr2S4 and related compounds. Progress in Radiology, XIth int. Congress, Rome 1965, ColI. int. Centre Nat. Recherche Scient. No. 157 Vol. IX, p. 1150-1157; Excerpta Medica Foundation, (Dérivés semi-métalliques, Orsay 1965), 63-66, 1967. E Amsterdam 1967. E D. Saget: Formation de mosaïques de jonctions de W. J. Oosterkamp, A. P. M. van 't Hof & W. J. L. germanium p-n par épitaxie. Scheren: Röntgenfarbbilder: neue Möglichkeiten der Acta electronica 10, 181-194, 1966 (No. 2). L Subtraktion durch die Fernsehtechnik. J. G. van Santen: Monolitische geïntegreerde schake- Deutscher Röntgenkongress 1966, Berlin, Vol. A, p. lingen. 168-170; Thieme, Stuttgart 1967. E Ingenieur 79, E 37-42, 1967 (No. 10). E C. van Op dorp & H. K. J. Kanerva: Current-voltage J. J. Scheer & J. van Laar: Fermi level stabilization at characteristics and capacitance of isotype heterojunc- cesiated semiconductor surfaces. tions. Solid State Comm. 5, 303-306, 1967 (No 4).. E Solid-State Electronics 10, 401-421, 1967 (No. 5). E H. Schemmann: Wirkungsgrad von kleinen zweipoligen A. Op het Veld & P. Bogers: Eine Präparationsmethode Gleichstrommotoren mit dauermagnetischem Läufer. zur Gefügeentwicklung von Hartmetallen (A prepara- Elektrotechn. Z. A 88, 225-229, 1967 (No. 9). .A tion method for revealing the structure of hard metals). W. Schilz: Influence of the sample geometry on fre- Prakt. Metallogr. 4,235-239 (240-242), 1967 (No. 5). E quency and Q-value of the helicon resonance in PbTe. J. Th. G. Overbeek & G. W. Rathenau: E. J. W. Ver- Solid State Comm. 5, 503-507, 1967 (No. 7). H wey; bij zijn aftreden als directeur van het Philips U. J. Schmidt: Anwendungen und Stand der digitalen Natuurkundig Laboratorium. Lichtstrahlablenkung. Chem. Weekblad 63, 3-7, 1967 (No. I). E Int. elektron. Rdsch. 21, 165-168, 1967 (No. 7). H R. Petit: Problèmes d'optique électronique dans les K. J. Schmidt-Tiedemann: On the realizability oftradi- tubes analyseurs d'images infrarouges. tors. Acta electronica 10, 137-155, 1966 (No. 2). L Proc. IEEE 55, 554-555, 1967 (No. 4). H L. J. van de Polder: Target-stabilization effects in tele- K. J. Schmidt-Tiedemann: Improved electronic half- vision pick-up tubes. shadow method for the measurement of birefringence Philips Res. Repts. 22, 178-207, 1967 (No. 2). E and dichroism. Rev. sci. Instr. 38, 625-627, 1967 (No. 5). H J. C. M. A. Ponsioen & J. H. N. van Vucht: The struc- ture of {3-TaC03 and the effect of the substitution of G. R. Schodder: Akustische Verstärker. Ta and Co by related elements. Funkschau 38, 175-178, 1966. A Philips Res. Repts. 22, 161-169, 1967 (No. 2). E J. F. Schouten & J. A. M. Bekker: Reaction time and A. Rabenau, E. Kauer & H. Klotz: Sur l'hexaborure de accuracy. E lanthane LaB6. Acta psychol. 27, 143-153, 1967. CoIl. int. Centre Nat. Recherche Scient. No. 157 (Déri- G. SchuIten & J. P. StolI: A high-precision wide-band vés semi-métalliques, Orsay '1965), 495-498, 1967. A wavemeter for millimetre waves. Philips Res. Repts. 22, 309-314, 1967 (No. 3). H A. Rabenau & H. Rau: Kristallzüchtung mit Hilfe von chemisehen Gleichgewichten zwischen zwei und mehr E. Schwartz: Streu- oder Betriebsmatrizen mit passiven Phasen. und aktiven Bezugsimpedanzen. Z. phys. Chemie Neue Folge 53, 155-162, 1967 (No. Archiv elektr. Übertr. 21, 317-320, 1967 (No. 6). A 1-6). A E. Schwartz: Über nichtreziproke, symmetrische Drei- H. Rau: Defect equilibria in cubic high temperature tore und Zirkulatoren. copper sulfide (Digenite). Nachrichtentechn. Z. 20, 329-332, 1967 (No. 6). A Phys. Chem. Solids 28, 903-916, 1967 (No. 6). J. A A. M. J. H. Senter: The electrical resistivity of MnTe H. Rau & A. Rabenau: Crystal syntheses and growth in at elevated temperatures. strong acid solutions under hydrothermal conditions. ColI. int. Centre Nat. Recherche Scient. No. 157 (Déri- Solid State Comm. 5, 331-332, 1967 (No. 5). A vés semi-métalliques, Orsay 1965), 459-464, 1967. E 386 PHILIPS TECHNICAL REVIEW VOLUME 28

M . .J. Sparnaay: Semiconductor surfaces and the elec- C. H. Weijsenfeld: Yield, energy and angular distribu- trical double layer. tions of sputtered atoms. Adv. Colloid Interface Sci. 1,277-333, 1967 (No. 3). E Thesis, Utrecht 1966. 'E K. van Steensel: Electrical conduction in island-struc- ture films of gold and platinum on insulating substrates. H. o. Westermann: Planning public lighting 'for X- Philips Res. Repts. 22, 246-266, 1967 (No. 3). E town. Int. Lighting Rev. 17, 92-97, 1966 (No. 3). A K. van Steensel, F. van de Burg & C. Kooy: Thin-film switching elements of V02. Philips Res. Repts. 22, 170-177, 1967 (No. 2). E M. V. Whelan: On the nature of interface states in an Si02-Si system, and on the influence of heat treatments .J. P. St01l: Vergleichende Messungen der komplexen on oxide charge. skalaren Permeabilität an Kugeln -und Stäben aus Philips Res. Repts. 22, 289-303, 1967 (No. 3). E Ferriten mit hohen und niedrigen Verlusten bei Mikro- wellen. Z. angew. Physik 22, 214-219, 1967 (No. 3). H M. V. Whelan, A. H. Goemans & L. M. C. Goossens: Residual stresses at an oxide-silicon interface. T. L. Tansley & P. C. Newman: Measurements of Appl, Phys. Letters 10, 262-264, 1967 (No. 10). E heterojunctions alloyed on to GaAs. Solid-State Electronics 10,497-501, 1967 (No. 5). M W. J. Witteman: Dependence of radiation production N. C. de Troye: De invloed van de m.icro-elektronica and population densities on the thermal relaxation op de ontwikkeling van digitale schakelingen. processes in a high power molecular laser. Ingenieur 79, E 42-48, 1967 (No. 10). E J. Chimie phys. 64, 107-110, 1967 (No. I). E .J. Verweel : Magnetic properties offerrite singlecrystals with the Y structure. L. E. Zegers: Error control in telephone.channels by J. appl. Phys. 38,1111-1117, 1967 (No. 3). H means of time diversity. K. Walther: Ultrasonic relaxation at the Néel temper- Philips Res. Repts. 22, 315-328,1967 (No. 3); Progresso ature and nuclear acoustic resonance in MnTe. elettronico 1966 (Atti XIII Congr. per l'Elettronica, Solid State Comm. 5, 399-403, 1967 (No. 5). H Roma), Vol. I, p. 677-694. E W. L. Wanrnaker, .J. W. ter Vrugt & .J. G. C. M. de A. L. Zijlstra & C. M. van der Burgt: Isopaustic glasses Bres: Luminescence of manganese-activated alumin- for ultrasonic delay.lines in colour television receivers ium-substituted magnesium gallate. and in digital applications. Philips Res. Repts. 22, 304-308, 1967 (No. 3). Ultrasonics 5, 29-38, 1967 (Jan.). E .J. D. Wasscher: Weak ferromagnetism in the NiAs structure and galvanomagneticmeasurements onMnTe. H. Zimmer: Parametrie amplification ofmicrowaves in Call. int. Centre Nat. Recherche Scient. No. 157 (Déri- superconducting Josephson tunnel junctions. vés semi-métalliques, Orsay 1965), 465-471, 1967. E Appl. Phys. Letters 10, 193-195, 1967 (No. 7). H

Volume 28, 1967, No. 12 pages 355-386 Published 6th December 1967