Surface Impedance of Superconductors and Normal Conductors in EM Simulators 1

Transcription

1 hp://wwwmmanraodu/mmos/hml-mmos/mma45/mmo45pdf MMA Mmo No 45 Surfac Impdanc of Suprconducors and Normal Conducors n EM Smulaors 1 A R Krr January 7, 1999 (Rvsd Augus 9, 1999) Th concp of surfac mpdanc Two yps of lcromagnc smulaor Rprsnaon of conducors by surfac mpdancs (a) A hck conducor rprsnd as a sngl conducng sh 3 (b) A conducor of hcknss rprsnd as a par of conducng shs 4 Surfac mpdanc of conducors of fn hcknss 5 (a) Excaon from on sd 5 (b) Symmrc and an-symmrc xcaon from boh sds 5 Modfd surfac mpdanc for hn conducng shs rprsnng a hck conducor 6 Exampls 7 Acknowldgmn 9 Rfrncs 9 APPENDIX: Drvaon of formulas 9 A1 Inducanc of a hn layr conanng a unform magnc fld 9 A Surfac mpdanc and skn dph δ of a normal conducor 9 A3 Surfac mpdanc and pnraon dph of a suprconducor 10 A4 Surfac mpdanc of a normal conducor of fn hcknss 1 A5 Surfac mpdanc of a suprconducor of fn hcknss 14 A6 Effcv surfac mpdanc of a normal conducor of fn hcknss xcd from boh sds 15 A7 Effcv surfac mpdanc of a suprconducor of fn hcknss xcd from boh sds 15 1 Ths s a rvsd vrson of NRAO Elcroncs Dvson Inrnal Rpor No 30, 19 Fbruary 1996

2 Inroducon Elcromagnc smulaors can gv vry accura soluons for mcrowav crcus wh dal conducors Whn h conducors ar non-dal, accura rsuls may sll b oband n many cass by spcfyng maral paramrs or surfac mpdancs Howvr, for srucurs n whch h pnraon dph of h fld no h conducors s of h sam ordr as h conducor hcknss, consdrabl rror can occur Ths s no only a rsul of h conducor hcknss bng nsuffcn o conan h fld complly, bu s du n par o a spara ffc whch arss wh som EM smulaors whn hck conducors ar rprsnd by hn shs wh surfac mpdanc For suprconducng nobum mcrosrp crcus of ypcal dmnsons, such rrors can asly b as gra as 0% n ε ff and 10% n Z 0 In many cass, a smpl corrcon o h surfac mpdanc subsanally mprovs h accuracy Th concp of surfac mpdanc For an dal conducor n an lcromagnc fld, h angnal componn E of h lcrc fld a h surfac s zro A currn flows n a hn sh on h surfac, as rqurd o suppor h magnc fld H angnal o h surfac Ths shor-crcu boundary condon xcluds all flds from h nror of h dal conducor In a ral conducor, flds xnd no h conducor, bu dcras rapdly wh dsanc from h surfac To avod h complcaon of solvng Maxwll's quaons nsd conducors, s usual o mak us of h concp of surfac mpdanc Th surfac mpdanc = E /H provds h boundary condon for flds ousd h conducor, and accouns for h dsspaon and nrgy sord nsd h conducor For a hck plan conducor, h nrnal flds fall xponnally wh dsanc from h surfac, wh 1/ dph For normal conducors, s h classcal skn dph δ = (/ωσµ) ½, and = (1+j)(ωµ/σ) ½ In Au or Cu a 100 GHz and room mpraur, δ K 05 µm, and K 01(1+j) ohms/squar For a suprconducor a a frquncy wll blow s nrgy gap frquncy, s h London pnraon dph,, whch s ndpndn of frquncy For nobum a ~4bK, a frquncs blow ~700 GHz, K 01 µm Th surfac mpdanc = jωµ 0 ohms/squar, corrspondng o a surfac nducanc L S = µ 0 H/squar, whch s ndpndn of frquncy In nobum, L S K 013 ph/squar, gvng K j008 ohms/squar a 100 GHz Two yps of lcromagnc smulaor Two yps of lcromagnc smulaor ar consdrd hr: () fn-lmn solvrs, such as HP/Ansof hfss, whch dvd h spac bwn conducors no a hr dmnsonal msh and solv by marx nvrson for h flds a vry msh pon, usng h boundary condons gvn by h surfac mpdanc; and () mhod-of-momns solvrs, such as Sonn m, whch dvd all conducng surfacs no (wo dmnsonal) clls, and solv by marx nvrson for h currns n h clls, usng h surfac mpdancs as boundary condons Thr s a subl bu fundamnal dffrnc bwn h soluons producd by h wo yps of smulaor for crcus wh hck conducors In boh cass flds nsd h ral conducors ar akn no accoun by h surfac mpdanc whch provds h boundary condons for h soluon Ths mans ha, n h smulaon, h spac corrspondng o h nror of a conducor should b flld wh a prfc magnc conducor o consran nror flds o zro In h cas of fn lmn solvrs hs s accomplshd smply by rmnang h spaal msh a h conducng surfacs;, h msh dos no xnd no h conducors In h cas of mhod-of-momns solvrs hr s no smpl way o achv h sam rsul, and currns n h surfac mpdanc do produc flds n h spac "nsd" h conducors f h surfac mpdanc s no zro I s hrfor ncssary o us a modfd valu of surfac mpdanc whn usng mhod-of momns smulaors for crcus wh hck conducors In many cass h corrcon s nglgbl, bu n som cass (g, suprconducng mcrosrp ransmsson lns), can b subsanal Rprsnaon of conducors by surfac mpdancs To undrsand h way lcromagnc smulaors ra a conducor of fn hcknss, w xamn h dffrnc bwn an acual hck conducor and h modl of h hck conducor whch h smulaor analyzs Th modl of h conducor can b hr a sngl hn sh wh h appropra surfac mpdanc, or a paralll par of hn shs sparad by h hcknss of h acual conducor

3 Mack [1] has shown ha h surfac mpdanc sn by a plan wav normally ncdn on a conducor s h sam as ha sn by a wav ravlng paralll o h conducor, as n a ransmsson ln For smplcy n h prsn dscusson, w consdr xprmns n whch a plan wav s normally ncdn on h surfac of h conducor or modl undr s (a) A hck conducor rprsnd as a sngl conducng sh Consdr a plan wav normally ncdn on a plan (hck) conducor of surfac mpdanc, as n Fg 1(a) Ths s analogous o h crcu shown n Fg 1(b), a long ransmsson ln of characrsc mpdanc Z η = (µ 0 /ε 0 ) ½ = 377 ohms, a whos nd an mpdanc ohms s conncd Fgur 1 Nx, consdr a plan wav normally ncdn on a hn sh of surfac mpdanc, as n Fg (a) Th corrspondng ransmsson ln quvaln crcu s shown n Fg (b) SS a long ransmsson ln of characrsc mpdanc Z η = (µ 0 /ε 0 ) ½ = 377 ohms, a whos mdpon A an mpdanc ohms s conncd n paralll Wh h plan wav ncdn from h lf, h fld on h ln o h rgh of A s zro only f = 0 A A, h ncdn wav ss an mpdanc n paralll wh 377 ohms (h rgh half of h long ransmsson ln), as n Fg (c) Clarly, h hn sh wh surfac mpdanc (Fg 1) s no physcally quvaln o a (hck) conducor of surfac mpdanc (Fg ) unlss = 0 Th apparn surfac mpdanc, sn by h ncdn plan-wav, n Fg s n paralll wh 377 ohms, and som powr s coupld hrough h hn sh no h spac on h ohr sd For cass n whch «377 ohms/squar (, mos praccal cass), h rror s nconsqunal Fgur 3

4 (b) A conducor of hcknss rprsnd as a par of conducng shs Consdr h rflcon of a plan wav from a conducor of hcknss, as shown n Fg 3(a) Th ncdn wav ss an mpdanc a h surfac of h conducor - h valu of s no h sam as n h prvous xampl Th mpdanc sn by h ncdn wav s as dpcd n h crcu of Fg 3(b) Th appropra valu of for fn valus of s gvn n a lar scon Fgur 3 Now consdr a plan wav normally ncdn on a par of hn shs, of surfac mpdanc, sparad by dsanc, as n Fg 4(a) Ths s analogous o h crcu shown n Fg 4(b), a long ransmsson ln of characrsc mpdanc Z η = (µ 0 /ε 0 ) ½ = 377 ohms, a whos mdpon A an mpdanc Zs s conncd, wh a scond mpdanc Zs a dsanc o h rgh If h dsanc s much lss han h wavlngh, h mpdanc sn by a plan wav ncdn from h lf s as dpcd n Fg 4(c) Th nducanc L = µ 0 accouns for h nrgy sord n h magnc fld bwn h conducng shs For a conducor 03 µm hck, a 100 GHz, h racanc ωl = ωµ 0 = 04 ohms/squar Fgur 4 I s clar ha f a conducor s hck nough, ωµ 0», and h wo-sh rprsnaon s suffcnly accura For normal mal conducors, hs rqurs ha» δ/, and for suprconducors» 4

5 Surfac mpdanc of conducors of fn hcknss (a) Excaon from on sd Whn h hcknss of a conducor s no vry much grar han h pnraon dph, a fld on on sd of h conducor pnras parally hrough o h ohr sd For normal conducors h surfac mpdanc sn by h ncdn fld s (s Appndx): k σ k σz η k σz η k k k σz η k σz η k k Hr k = (1 + j)/δ, and Z η = (µ/ε) ½ s h characrsc mpdanc of spac (377 ohms n vacuum) In mos cass Z η» k/σ, and k σ formula = (1+j)(ωµ/σ) ½ k k Whn s larg, hs rducs o h usual surfac mpdanc k k In h cas of a suprconducor, whn h hcknss s no much grar han h London pnraon dph, h surfac mpdanc s (s Appndx): jωµ Z η jωµ Z η jωµ Z η jωµ Z η jωµ λ L, λ L whr agan Z η = (µ/ε) ½ s h characrsc mpdanc of spac (377 ohms n vacuum) In mos cass Z η» ωµ, so = jωµ coh / Whn» hs bcoms h usual formula for suprconducors: = jωµ 0 (b) Symmrc and an-symmrc xcaon from boh sds In h abov, has bn assumd ha h fld s ncdn on h conducor from on sd only Ths s h cas for ground plans, wavgud walls, wd paralll-pla ransmsson lns, and wd mcrosrp lns In cass such as a srpln cnr conducor, qual flds ar prsn on boh sds of h conducor In a fw cass, such as a spum across a wavgud (paralll o h broad walls), qual bu oppos flds ar prsn on h wo sds For no-vryhck conducors n such symmrcal or an-symmrcal flds, h ffcv surfac mpdanc sn from on sd s modfd by h prsnc of h fld on h ohr For a normal mal conducor of hcknss wh symmrcal or an-symmrcal xcaon, h surfac mpdanc s (s Appndx): k σ k k k k ± k k, whr k = (1 + j)/δ Th + sgn s for symmrcal flds on h wo sds, and h - sgn for an-symmrcal flds 5

6 For a suprconducor of hcknss wh symmrcal or an-symmrcal xcaon, h surfac mpdanc s (s Appndx): jωµ coh ± csch Agan, h + sgn s for symmrcal flds on h wo sds, and h - sgn for an-symmrcal flds Th followng abl gvs h valus of h coh and snh rms, and hr sum, for ypcal Nb conducor hcknsss assumng = 1000 Å Å /λ coh(/λ) csch(/λ) coh(/λ) + csch(/λ) Modfd surfac mpdanc for hn conducng shs rprsnng a hck conducor A modfd valu of surfac mpdanc can b usd o corrc h dscrpancy bwn h ral conducor and h wo-sh modl L b h dsrd surfac mpdanc as gvn by h appropra formula abov, and l Z X b h valu of surfac mpdanc of h conducng shs whch wll rsul n an ffcv surfac mpdanc of as sn by an ncdn wav, as dpcd n Fg 5 Fgur 5 In mos praccal cass Z η s larg compard wh h ohr crcu lmns, and can b gnord Thn, analyss of h crcu gvs a quadrac quaon n Z X whos soluon s 6

7 Z X 1 jωµ 0 ± 4Z S jωµ 0 1 In h cas of a suprconducor xcd from on sd, = jωµ 0 coh(/ ) I follows ha Z x = β, whr β 1 coh 1 coh Fg 6 shows β as a funcon of / 0 18 ba coh(/lambda) coh(/lambda) ba /lambda Fgur 6 Exampls To dmonsra h sgnfcanc of h β and coh corrcons o h surfac mpdanc, consdr a suprconducng Nb mcrosrp ransmsson ln of wdh 6 µm, wh a 0 µm-hck dlcrc layr wh ε r = 38, ovr a Nb ground plan Th London pnraon dph = 01 um In h frs xampl, h Nb conducors ar 01 µm hck, and n h scond xampl hy ar 03 µm hck Th abl blow gvs h ffcv dlcrc consan and characrsc mpdanc of h mcrosrp whn h uppr conducor s rprsnd by a par of conducng shs Sonn m was usd, wh h hck-conducor valu of h surfac mpdanc and h followng corrcons: () boh β and coh(/ ) corrcons, () only h coh(/ ) corrcon, and () no corrcons Corrspondng rsuls ar also shown for h sam mcrosrps (v) wh h uppr conducor characrzd as a sngl conducng sh whos surfac mpdanc ncluds h coh corrcon (bu no h β corrcon, whch appls only whn wo shs ar usd), and, (v) wh prfc conducors ( = 0) Th scond abl gvs h sam rsuls xprssd as prcnag dvaons from h mos accur soluon, () 7

8 Nb hcknss = 01 µm Nb hcknss = 03 µm ε ff Z 0 ε ff Z 0 () Coh & β corrcons () Coh corrcon only () No coh or β corrcons (v) Sngl-sh (v) Prfc conducors Nb hcknss = 01 µm Nb hcknss = 03 µm % rrors wr op ln % rrors wr op ln ε ff Z 0 ε ff Z 0 () Coh & β corrcons 0% 0% 0% 0% () Coh corrcon only -13% -5% -6% -3% () No coh or β corrcons -3% -10% -6% -3% (v) Sngl-sh -1% 3% 3% 4% (v) Prfc conducors -57% -33% -49% -8% I s also of nrs o compar h rsuls oband by Sonn m wh h mos accura analycal rsuls avalabl W us h analycal rsuls from a rcn rpor by Yassn & Whngon [] for Nb mcrosrp lns of wdh, 4, and 6 µm, wh a 03 µm dlcrc layr of ε r = 38, wh a Nb groundplan Th cnr conducor and groundplan ar 03 µm hck, and = 01 µm, so / = 3 For hs valu of /, h β corrcon s sgnfcan, bu h coh corrcon s vry small Th rsuls for h ffcv dlcrc consan and characrsc mpdanc ar compard blow Agrmn s vry clos, xcp for h narrows ln, n whch cas hr s a 4% dsagrmn n Z 0 Mcrosrp wdh µm Mcrosrp wdh 4 µm Mcrosrp wdh 6 µm ε ff Z 0 ε ff Z 0 ε ff Z 0 Rf [] Sonn m % dffrnc 1% 4% 0% % 0% 1% 8

9 Acknowldgmn Th auhor hanks Jm Mrrll of Sonn Sofwar for hs hlpful dscussons on h rol of surfac mpdanc n Sonn m H also hanks S-K Pan of NRAO for ponng ou som rrors n h arlr vrson Rfrncs [1] RE Mack, "Transmsson Lns for Dgal and Communcaon Nworks", Nw York: McGraw-Hll, 1969 [] G Yassn and S Whngon, "Elcromagnc modls for suprconducng mllmr-wav and submllmrwav mcrosrp ransmsson ln," Journal of Physcs D: Appld Physcs, vol 8, no 9, pp , 14 Spmbr 1995 APPENDIX: Drvaon of formulas A1 Inducanc of a hn layr conanng a unform magnc fld If a plan wav, normally ncdn on a prfc plan conducor, producs a currn J A/m n h conducor, hn by Ampr's law, h magnc fld nar h conducor B = Jµ In a layr of hcknss dx paralll o h conducor, h sord magnc nrgy dw = B dx/µ = J µdx/ pr un ara L h nducanc conrbud by h magnc fld n hs layr b dl H/squar Ths nducanc s n srs wh h currn J A/m Thn h nrgy sord n hs nducanc s J dl/ pr un ara I follows ha dl = µdx H/squar A Surfac mpdanc and skn dph δ of a normal conducor Consdr a plan wav ncdn on a hck conducor Th ncdn wav xcs volags and currns n h conducor whch vary wh dph from h surfac An ncrmnal hcknss dx of a un ara of h conducor s characrzd by h quvaln crcu of Fg A1 From A1 abov, h magnc fld n h volum of hcknss dx accouns for a srs nducanc µdx H/squar Th conducvy σ has a paralll conducanc σdx S/squar Hnc dz = jωµdx and dg = σdx For hs crcu, h npu mpdanc s h surfac mpdanc Fg A1 9

10 Snc h conducor s hck, h mpdanc lookng o h rgh a any dph n h conducor s qual o h surfac mpdanc Hnc Z n dz 1 dg 1 jωµdx 1 σdx 1 Solvng for gvs = jωµ/σ, whnc h sandard rsul: (1 j) ωµ σ From h fgur, d 1 v 1 dg 1 dg 1 σ dx Thrfor d 0 σ dx or 0 σ (x x 0 ) Th sgn of h xponn s posv bcaus of h choc of x-drcon n Fg A1 Wh h abov xprsson for, 0 ωσµ (x x 0 ) j ωσµ (x x 0 ), from whch h skn dph s δ ωσµ A3 Surfac mpdanc and pnraon dph of a suprconducor Th analyss for a suprconducor s smlar o ha for a normal conducor, wh h xcpon ha h conducanc lmn dg s rplacd by a suscpanc As h suprconducor s losslss, h currn s lmd only by h nra of h Coopr pars of lcrons, whch manfss slf as a knc nducanc Consdr a layr of suprconducor of hcknss dx In rms of h avrag vlocy of h carrrs S, h currn d = (n * * S)dx A/m, whr n * s h ffcv dnsy of carrrs wh ffcv charg * If an AC volag v = V jω V/m s appld paralll o h surfac, h forc on a carrr s * V jω = m * dv/d, whr m * s h ffcv mass of a carrr Th carrr vlocy S m Vjω d 1 jω m Vjω 10

11 Th currn n h layr of hcknss dx s hrfor m d 1 jω Wrng d = di jω 1 gvs V jω, n dx di n m V jω dx A/m from whch s vdn ha h knc nducanc of h layr s gvn by d 1 L n dx m Now consdr a plan wav ncdn on a hck suprconducor Th ncdn wav xcs volags and currns whch vary wh dph from h surfac An ncrmnal hcknss dx of a un ara of h suprconducor s characrzd by h quvaln crcu of Fg A Fg A From A1 abov, h magnc fld n h volum of hcknss dx accouns for a srs nducanc µdx H/squar Hnc dz = jωµdx Th knc nducanc of h Coopr pars n h sam volum conrbus a paralll admanc dy 1 jω d 1 L n jω m dx Snc h conducor s hck, h mpdanc lookng o h rgh a any dph n h conducor s qual o h surfac mpdanc Hnc, n Fg A, Z n dz 1 dy 1 jωµdx n 1 jω m dx 1 Solvng for gvs jω µm n 11

12 To dduc h pnraon dph n a suprconducor, consdr agan h crcu of Fg A: Wh h abov xprssons for dy and, d 1 v 1 dy 1 dy d µn m dx 0 µn m (x x 0 ) or 0 (x x 0 ) (Th sgn of h xponn s posv bcaus of h choc of x-drcon n Fg A) Th quany m µn s h London pnraon dph, and s ndpndn of frquncy Th xprsson for h surfac mpdanc can b wrn n rms of as jω µ ohms/squar, whch corrsponds o a surfac nducanc L S µ H/squar A4 Surfac mpdanc of a normal conducor of fn hcknss To dduc h surfac mpdanc of a normal conducor of hcknss, consdr frs an ncrmnal hcknss dx of h conducor Ths s rprsnd by h quvaln crcu of Fg A3 Fg A3 In h fgur, d vdg σ vdx and dv dz jω µdx, 1

13 hrfor d dx jωσµ Ths has h soluon kx kx, ωσµ whr k jωσµ (1 j) 1 j, δ and δ s h classcal skn dph as drvd abov Now consdr h quvaln crcu of h conducor, rmnad on h rgh by h mpdanc of spac Z η, as shown n Fg A4 Fg A4 In Fg A4, kx kx and v 1 d σ dx k σ ( kx kx ) A x = 0, v k σ Z η Thrfor σz η k σz η k, 13

14 and hnc, v() () k σ k σz η k σz η k k k σz η k σz η k k, whr k jωσµ (1 j) ωσµ 1 j δ In mos praccal suaons, so Z η» k σ k σ k k k k A5 Surfac mpdanc of a suprconducor of fn hcknss Th analyss n hs cas follows ha for h normal conducor bu wh dg rplacd wh dy n I jω m dx d follows ha, and dx 1 λ L v() () jω µ Z η jω µ Z η jω µ Z η jω µ Z η jω µ λ L, λ L m whr s h London pnraon dph drvd abov Usually, Z η» jωµ, n whch cas w µn oban h usual formula: jω µ coh 14

15 A6 Effcv surfac mpdanc of a normal conducor of fn hcknss xcd from boh sds Whn a conducor of fn hcknss has flds ncdn on boh sds, h apparn surfac mpdanc on hr sd s affcd by h fld on h ohr From abov, and wh rfrnc o Fg A4: Whn h xcaon s on on sd only, kx kx and v k, σ ( kx kx ) ωσµ whr k jωσµ (1 j) 1 j δ For on-sdd xcaon, and assumng Z η >> k/σ : A x = 0, = 0, so - = - + Thrfor, v(0) k σ ( kx kx ) k σ A x = () ( kx kx ) and v() k σ ( kx kx ) Whn h crcu s xcd by qual currn sourcs ( k k ) on boh sds, hn a x =, usng suprposon: v() k σ ( k k ) k σ Thrfor, v() () k k k σ k k k k If h xcaon on h wo sds s ou of phas, h sgn of h scond rm n h squar bracks bcoms ngav A7 Effcv surfac mpdanc of a suprconducor of fn hcknss xcd from boh sds Th approach follows ha usd abov for h normal conducor For sngl-sdd xcaon, rfrrng o Fg A4, x x and v jω µ ( For on-sdd xcaon, and assumng Z η >> ωµ : A x = 0, = 0, so - = - + x x ) 15

16 x λ Thrfor, v(0) jω µ ( L λ L ) jω µ x λ L ) A x = () ( and λ v() jω µ ( L λ L ) Whn h crcu s xcd by qual currn sourcs ( k k ) on boh sds, hn a x =, usng suprposon: v() jω µ ( λ L ) jω µ Thrfor, v() () jω µ coh csch If h xcaon on h wo sds s ou of phas, h sgn of h scond rm n h squar bracks bcoms ngav 16