z = - 2rriw ctg k!! = - 4rrioo [1 - _!_ (koh)2... ] '

Transcription

1 SOVIET PHYSICS JETP VOLUME 5, NUMBER 3 OCTOBER, 957 Skin Effect in Tin Films and Wires B. M.BOLOTOVSKII P. N. Lebedev Pysical Institute, Academy of Sciences, USSR (Submitted to JETP editor February 9, 956) J, Exptl. Teoret. Pys. (U.S.S.R.) 32, (Marc, 95 7) Equations are obtained for te skin effect in tin films and wires by using kinetic teory. A metod is developed for te approximate solution of tese equations wic gives expressions for te impedance of tin films and wires. A KINETIC teory of skin effect for bulk conduce tors was developed by Reuter and Sondeimer. If l «o 0, were lis te electron free pat in a metal and 8 0 is te classic skin layer dept, ten te usual or normal skin effect occurs. If l >v 8 0 or l» o 0, te skin effect is anomalous. In te case of samples of small size (wires of radius R~ l or films of tickness,:(,.!), just as in te 8 0 ~ l case, it is impossible to te use te normal skin effect teory. Hence, te kinetic approac for tin films and wires is necessary for direct current2-6. Also te question arises of te beavior of small-size samples in an alternating field. Dingle 7 considered te question for te optical frequency case, taking only diffusion reflection of te electrons from te metal surface into account. Te work of Azbel' and Kaganov 8 is also devoted to tis question, were, as in Ref. 7, te passage of a plane electromagnetic wave troug a film is analyzed. Hence, te impedance for tis plane wave passage is calculated in tese works. It is also interesting to determine te impedance of tin films and wires wit respect to te ig-frequency current flowing terein. l3elow, we consider te conclusions to wic kinetic teory leads for te case of skin-effect in tin conductors, i.e., in suc conductors as are muc less tick tan te free pat lengt of te electrons in an unlimited metal. To be specific, let us visualize cylindrical wires of radius R and plane films of tickness. 2. Let us direct te z axis parallel to te wire axis (or parallel to te film surface). Let us impose an electric field E = E z ei wt, directed along te z axis, on te sample. Te quantity E z is a function of te points in te wire cross section. If tis function is known, ten te energy losses in te conductor are caracterized by te impedance* Z = R + ix = (4dw I coc) E (P) IE' (P), were P is a point on te conductor surface, E' (P) is te normal derivative relative to te conductor surface, and IX= 2 2rcR (l) for a conducting alf-space for films (2) for wires. Te real part of te impedance R gives te omic resistance of te conductor and te imaginary part X gives te inductance. Te classical skin effect teory, valid for l «, l«o 0, leads to te following expression for films of tickness : (3) z = - 2rriw ctg k!! = - 4rrioo [ - _!_ (ko)2... ] ' c2ko o 2 c2k2 2 0 k~ =-2i I o~ = - 47:iwa 0 / c 2, (4) were a is te conductivity of te unlimited sample. 3. As 0 sown in Refs. 2-6, if te lateral dimensions of te conductor become comparable to te free pat lengt l, te direct current sample conductivity (w=o) starts to depend on te lateral dimensions. For tin films wit «l, we ave 2 : (if s:(;); (5) (if s = )' were 8 is te portion of te electrons reflected specularly from te sample surface. For tin wires wit R <<l, we ave4 : + e:2r cr = cro - e: T (if s :(; ); a = O'o (if s = ). (6) *We use te impedance definition Z =E (P) I I, wic leads to (I) and (2) under te assumption of field symmetry over te wire cross-section. Consequently, we will always select te solutions corresponding to field symmetry subsequently. 4. In deriving te skin effect equations, we used te Cambers metod5 wic, being equivalent to te kinetic equations metod, permits a wole set of intermediate calculations to be avoided. Te gen- 465

2 466 B. M. BOLOTOVSKII eralization of te Cambers metod to te case of a field- dependent on te coordinates is made witout difficulty. Tis metod yields for te average velocity increment (in te z direction) taken by electrons arriving at a point M from P in te volume of te conductor (Fig. l): n LlVz =,:v E (s) e-s/l ds+ + :e-~l ~ E (s) e-s/l ds], -&e bfl b were a=pm and te same notation is used as in Ref. 5. If E is independent of s, te result of Ref. 5 is obtained from (7). Te expression (7) permits te current at te point M to be expressed troug E z: (7) (8) (x) ~ e-t/l xt ( ).p2 =, 2 t/l c =- Ta dt. -&e For wires of radius R 2" R ~ d:p ~ E (r') (IO) x [:p (I r- r' [) + s~2 (I r- r' i)] r' dr', (ll) :p (! r I) = ~ \ e-rtfl V f2- t!! = 2 S (! ). ', r J ta r 3 \ l '., -- _2 ~ e-a!fl + e-bt/l dt I '?2 (! r I ) - e-rt/lvf2- ~~bt/l t3 r - ee (2) In te expressions (2), S 3 is a function introduced in Ref. 4, r and r' are points in te wire cross section, b is te lengt of te cord drawn troug r and r', b-a and a are segments into wic te point r M FIG. l were { 0 is te equilibrium distribution of te electrons in te metal. Substituting (8) in te equation for Ez we obtain an integra-differential equation for E z. Let us give te final form of te equation. For films of tickness : + s:p 2 (x- x')] dx', were!fll (x) = ~ e-t/xl/ ( +- -j 3 ) dt, FIG. 2 divides te cord (Fig. 2). We neglected te relaxation time in deriving tese equations. Te relaxation time can be taken into account if l/0- icv 'C) is written instead of l in (0) and (2) for cp and cp 2 We will neglect te quantity cv'l" in comparison wit unity below. 5. Let us turn to te solution of te equations obtained. Let us first analyze Eq. (9) for te tin films. Tis equation is easily reduced to te integral equation (9) E (x) =cos c:!. (\x- ~ \) + ~ \ K (x,x') E (x') dx'; c ~ 2 82[ ~ X K (x, x') = -~ ~ sin; (x- t) f:p (t- x') /2 + s~ 2 (t- x')l dt. (3) (4)

3 SKIN EFFECT IN THIN FILMS AND WIRES 467 It is easy to be convinced of te equivalence of (9) and (3) by calculating te quantity E" +(w/c) 2 E using {3). Tis quantity gives te rigt side of {9). Wile te kernel of te original equation (9) would ave a logaritmic singularity at x =X, te kernel of Eq. (3) is bounded. If I 3 i~ max K (x, x') j <, I./)0[ (5) Eq. (3) can be solved by an iteration metod. Putting cos 7'(x-.!p= in te «c/w case, and taking E 0 = for te zero approximation, we obtain E(x) =I+ 2 :~: ~ K (x,x')dx' 0 {6) + (_ 2 3 ~2 y \ \ K (x,x') K (x',x") dx' dx" Substituting (6) in () for te impedance, we find te latter as te ratio of two power series in te parameter (5). In its turn, tis ratio can be expanded in a power series in te same parameter. Te first term of te expansion is independent of te field frequency w and gives te omic resistance of te film to direct current: (7) :~ co~ (-~-- L) i- e-til dt] ;; (- s) "- \3 t 5 -I-c:e -f/ - cr Tis expression is equivalent to tat obtained earlier by Sondeimer6 It can be used as a definition of te film conductivity a in a direct current. Te expression (5) for a is obtained from (7) for «l. Te evaluation of te next terms of te expansion of Z in powers of te parameter {5) appears to be somewat complex. Let us give te results for te case of purely diffuse (8=0) and purely specular _ (8=) reflection for «l. In bot cases, Z=Z0 [--(k) 2 j2+ ], (8) were Z is defined by {7) and k by (4) if a 0 is replaced by a ( 8) according to (5). It is easy to see tat te parameter (5) is te square of te ratio of te film tickness to te dept of field penetration into an unlimited sample wit conductivity equal to te film conductivity. As is seen from (3) and (8), te first terms of te impedance expansion in te normal skin effect case (, a 0» l) agree wit te terms of te impedance expansion for tin films ( «l, a» l) wit k replaced by k 0 Te region of suc "quasi-normal" skin effect in tin films appears to be broader tan te region of normal skin effect in bulk conductors. Higer terms of te impedance expansion in powers of k were not calculated. It sould not be expected tat te coefficients of te expansions (3) and {8) will agree exactly. As will be clear from te sequel, it can be imagined tat tey will be close in order of magnitude. 6. In order to determine te impedance at iger frequencies, it is convenient to use a less exact approximate metod of solution wic we will illustrate first by an example of a metallic alf-space (a problem analyzed in Ref. ). In tis case, te skin effect equation is (for simplicity, we consider te reflection to be diffuse): E" + :: E = 2 ~~~ } 9 (x- y) E (y) dy. (l9) As is seen from (9), te current intensity at te given point x is determined by te wole region near tis point. Te effective distance caracterizing te size of tis region equals (in order of magnitude): co co I, = ~ ~ (x) xdx / ~ cp (x) dx = ~ l (20) [see Eq. (0)]. In order to determine te impedance, we must know ow te field itself beaves in te dept of te metal, even if values of te field and its derivative on te metal sudace are essential. In te region close to te metal surface (x =0), E satisfies te equation E" + ~= E = 2 3 : 2 ~ 9 (y) E (y) dy. (2) Let us replace te kernel cp (y) by a function wic equals a certain constant f3 in te limits from 0 to.\[ Eq. (20)] and wic becomes zero outside tis range. Ten (2) becomes: A E"+~:E=~\E(y)dy. (22) c" 2 32 J Tis equation admits of a solution E(x)=e kx. Substitution in (22) gives an equation for k k 2 + w 2jc2 = (3 i~;2 o~lk) ( - e-3 kl/8). (23)

4 468 B.M. BOLOTOVSKII If lktl «, i.e., if te field attenuates at distances muc larger tan te free pat lengt l, ten (23) ecomes (24) Inasmuc as te classical skin-effect teory is valid in tis case, te rigt side must e equal to 2i/o 5. Hence, we find: ~ = 32j9. (25) In te oter extreme case, we obtain for lktl», neglecting displacement currents: ~c = (6if30~l)' == cv3 + i) (2/30~)' ', (26) from wic te impedance is According to exact teory, Tese values are very close. If bot,\ and {3 are considered as adjustable parameters, ten exact a greement is obtained wit te Renter-Sondeimer teory for It is understood tat te agreement wit exact teory can old in te two extreme cases: lktl» and I kll «since te exact solution is sligtly different from te exponential e kx only in tese cases (see Ref., Appendix III). 7. Let us use tis roug teory in te tin film case ( «l). In tis case, we can again write (22), were te film tickness sould be substituted as te upper limit of integration instead of,\ [ Eq. (20)]: E" () + (~~ E () ~~ 3 i; \ E (y) dy. (28) c- 2;2 j Let us look for te solution as E =cos k(x- /2). Substituting tis function into (28), again neglecting te term w 2 /c 2 in comparison wit k 2, we obtain As in Sec. 6, te constant {3 can e determined by requiring tat te impedance agree wit (7), obtained in Sec. 5 by te iteration metod, w-+ 0. Tis yields B _ 4/cr 4 I - 3cr (3) I" ~ ( \ - e-tll -2-,-~-~(-c:) -,3--fs,J tlldt. '- <:e- In te lk I» case, we find In tis case, we find for te impedance were a is te static conductivity of te film of tickness. An approximate expression can e obtained from (27a) by replacing l by and a 0 by a terein. 8. Simplifying () for tin wires (R <<l) by te same metod, we obtain: (32) (33) R d 2 E +_!:_de+-~~ E = 3 i~ \ E (r') dr'. (34) dr 2 r dr C 2 27t32 J We look for te solution of tis equation as E(r) = J 0 (kr). Substituting J 0 (kr) in (34), we find: Hence, again neglecting te quantity w 2jc 2, we obtain for lkr I«: (36) Just as in te tin film case, we determine te constant {3 from te condition tat te impedance at zero frequency sould give te known expression 4 for te resistance of a tin wire. Tis yields 0 - i,-;-:!5! = ~ + < ( :;;:::: I) (37) - 3 Rcr 'o: E "" k cos (k/2) =~ (3i~;0~/) sin (k/2). (29) Hence, we obtain for lkr I» z = + i V3 (2t2wzR )', 2tR c4cr Hence, for lk I «, we find (38) (30) 9. Let us discuss te results obtained. Te pararneter caracterizing te skin effect in te condl!cting

5 alf-space case is te ratio l/ o, were o 0 is te classical skin layer dept. As follows from our computations, te critical parameter in te tin conductors case is te ratio of te size of te conductor cross-section (film tickness or wire radius R) to te dept of field penetration in a bulk conductor wit te same static conductivity as in te tin conductor. Namely, if te dept of field penetration is large in comparison wit te film tickness or wit te wire radius [see inequality (5)], te analog of te classic skin effect olds wit te difference, owever, tat te conductivity is determined by (5) and (6) in te tin conductor case. In tis case, te field and te impedance can be found by a power series expansion in te parameter (5). As is easy to see, te real part of te impedance (te omic resistance) is expanded in even powers of te frequency w in tis case, just as in te classic skin effect case, and te imaginary part of te impedance, in odd powers of w. Te expansions for tick and tin films agree to te accuracy of terms of order (k) 2 (wit te difference tat k 0 replaces k in te tick films case). Te order of magnitude of te next terms of te expansion can be determined by using an approximate metod explained in Sees. 6, 7 and 8. According to tis metod, te impedance of a tin sample can be obtained from te expansion (3) if we speak of a tin film and from a similar expansion for a wire, by replacing te conductivity of te unlimited sample a 0 by te conductivity of a tin conductor a. Hence, te normal skin effect teory can actually be used to estimate te impedance of tin samples in te frequency band under consideration. Let us note tat te frequency band were suc a quasi-classical skin effect occurs appears to be muc broader tan te region of classical skin effect in bulk conductors. Actually, condition (5) for a tin conductor or te corresponding condition lk I «( lkr I «) in te approximate metod, is satisfied, as is easy to see, for a muc larger frequency band tan in te bulk conductor. A case is even possible wen anamalous skin effect conditions are not generally realized in tin conductors. Tis occurs wen condition (5) is violated at suc ig frequencies for wic it is impossible to neglect te relaxation time and to take te relaxation time : into account yields a condition analogous to te inequality (9) of Ref., i.e., again te classical skin effect condition. SKIN EFFECT IN THIN FILMS AND WIRES 469 Wen tere is an anomalous skin effect region {as always occurs if te conductor is not too tin), te impedance of te tin conductors is expressed by te approximate formulas (33) and (38). In tis case, te impedance depends on te tickness of te film wire. Te impedances of tin wires and tin films are, respectively, proportional to Te impedance of a tin film depends on as ';. and te impedance of a tin wire is proportional to R- 2 0, for w=, a=a 0 Te skin effect in a tin film was analyzed earlier by Ginzburg 9, were ae =a 0 y/l wit y= was assumed for te tin film. As is clear from te above, y=ln (lj). It is impossible to compare te conclusions obtained wit te results of Dingle 7 and Azbel' and Kaganov8 since tey determined te impedance for te passage of a plane wave and not for a current as in te present work. Te autor tanks V. L. Ginzburg for formulating te problem and for many discussions, V. P. Silin for discussions and also M. I. Kaganov for a number of interesting remarks. G. E. H. Reuter and E. H. Sondeimer, Proc. Roy Soc. (London) 95A, 366 ( 948) 2 K. Fucs: Pro c. Camb. Pil. Soc. 34, ( 938) 3 D. K. C. MacDonald and K. Sarginson, Proc. Roy. Soc. (London) 203A, 223 (950) 4 R. B. Dingle, Proc. Roy. Soc. (London) A, 20, 545 (950) 5 R. G. Cambers, Proc. Roy. Soc. (London) A, 202, 378 (950) 6 E. H. Sondeimer, Adv. in Pys., (952) 7 R. B. Dingle: Pysic a 9, 87 (953) 8 M. I a. Azbel' and M. I. Kaganov, U cen. zap. Karkov Univ., Works (Trudy), Pysics section of Pysics-Matematica faculty, 6, 59 (955) 9 V. L. Ginzburg, J. Exptl. Teoret. Pys. (U.S.S.R.) 2, 979, 95; Usp. Fiz. N auk. 42, 69, 333 (950); 48, 25 ( 952) 0 V. L. Ginzburg, G. P. Motulevic, Usp. Fiz. Nauk 55, 469 (955) Translated by M. D. Friedman 27