Superconducting nanostructures

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2 d flco a oal al spcodco fac d flco s basc pocss fo cag coso. cd p-sp lco w gy dags w a dow-sp lco of gy o fo a Coop pa spcodco lag bd a ol sp-dow bad. a bl spcodco pad d flcos lad o odfcao of gy gsas ad foao of spcodcg codsa w Coop pas. a oal al copld o a spcodco d flcos odc spcodcg colaos s a calld poy ffcs. T pocss of d flco ca b dscbd by Bogolbo d s Bd aos. T Bd aos cold b foally dd fo BC oy of spcodcy. "lco - ol dscpo" of spcodcy pcodcy s dscbd by BC a fld Haloa: c c c c c c c c V c c H ˆ pag V dscbs aac aco s asd fo gy. Ts aloa s sag ss a pag s do o cos pacl b. Usally Haloas coa c c -yp s. Ts sc ca b cod w asfoao: c c T " " opao alas a lco.. cas a ol. T sp labl s o cssay ay o as opaos "" sad fo sp p lcos ad opaos "" sad fo sp dow "ols". s fs loo a oal sa Haloa w w s of opaos: Hˆ - dow sp lco sa s cod o dow sp ol sa. T aspacl gy of ol sa s oppos o a of lco sa. lco sa: - cag > >; < < sg of ad ~dd a sa - Hol sa: cag > <; < > sg of ad ~dd a oppos! pos ol sa popagas ga dco!

3 Vac sa god sa oal al lco dscpo: > ' < Vac sa: o lco sas a flld c ' ' T god sa of a oal al sp s obad by fllg all lco sas blow gy Π c c < ' ' lco-ol dscpo: w ac sa lco-ol pc: Tasfoao fo o - ac sa: Π c ' ' ' ' py sas occpd sas By fllg all sas blow gy bo lco ad ol bad w g sa god sa: Π > Π c - < Π c < ' ' Qaspacls spcodcg sa T Haloa lco-ol psao cosa Σ s od : Hˆ d flco s w psao pacl b s cosd b cag s o!. T aco cospods o d flco w a lco s cod o a ol allg o sd pa o ca sa. od o dagoalz Haloa w opaos a odcd: γ γ { γ γ } { γ γ } c { γ γ } oogoal sas Hˆ γ γ γ γ η γ γ η η γ γ T dagoalzao codo s η. Tag as a al b ad posg >: T sa pas fo all! T aspacls of spcodco γ ad γ a co spposos of lco ad ol oal aspacls. o >> oal sa aspacls a cod. o aspacls a a ally wgd spposo of lco ad ol oal aspacls. gap of ops aspacl spc! 3

4 pcodcg god sa γ γ Q ~ Occpao of lco-ol sas: aod gy lco sas a copld o sd ol sas d flco T spcodcg god sa: all ga gy sas a occpd Πγ Π C... Π c c BC god sa T olo of aspacl cag as a fco of Qaspacl pc: T d pas a ol-l sas ss a sg of s oppos o sg of ~d d. How all sas lco ad ol aspacls a spposd ad gy s gad by pag. laly bl pas a lco-l sas. - - T ga gy sas a cad by γ opao Ε T pos gy sas a cad by γ opao ap spc ao of Bd ao fo C Haloa fo a -sa ψ Hˆ γ o ow o s placd by * *! Hˆ ψ ψ ψ lo ol H ψ ψ H ˆ p ˆ H μ Hlp: f so algba: { } laly: ψ γ ψ ψ ψ Hˆ Hˆ 4

5 5 Ε p p W cosd pos gy solos ga gy sas a occpd sp. lco gfco: a > > olo a oal al - p p Hol gfco: a > < olo a spcodco p p > l lco " " o Ε < : ± > l lco < l ol " " Ε ± fo < s c sa << gcos:? acally s og o calcla sg o Ε > : o Ε < : sg opl pa of -> asc od l ol " " l lco " "

6 6 olo fo a -fac Ε Ε fac a wc as pfc aspacy T f spcodcy s sppssd a > spcodco Ν a < oal al - - Ψ Ψ cog lco l fl d l d flcd ol Tasd lco ad ol-l aspacls oal flcd l. Ψ Ψ ' ' Ψ Ψ Macg wa fcos: d appoao: << d flco apld fo pfcly asg ba < > sg π o Ε > : < o < : ω ω ω accos ag s cos sg ω ω sω cos ω

7 d flco apld fo pfcly asg ba -say o a pfcly asg cal s o oal flco o < d flco pobably s. a pas sf of -π occs o a cog lco d flco ap. s was fo a cog ol s - a jco w pfc assso oal flco s pobd ad all cog lcos a d flcd w a pobably of o. ag - o ac cog lco a cag of s asd -π -π a π pas sf occs! Oppos l: a waly asg l ba bw ad lcod o a l ba w low assso T<< d flco s sppssd pobably fo assso of cags ~T s aspo s doad by dc aspacl lg a V >. T ddv c of l jco dcly ass spcodcg dsy of sas! V V ddv µ 4 g-l ~ T ~ T d ~ T g dv d g V f d g f g ~ T g d g ' d g f V g V f f V f V f T V T g g V V V 7

8 BTK oy codcac of a ballsc jco.. Blod M.Ta T.M.Klapwj PB Ν a < oal al Ε a alsc jco bw a oal al ad a spcodco a f fac scag as o b cosdd. T BTK oy calclas -V c of a jco by odlg fac scag w a ac-dla poal dscbd by a dsolss scag sg Z. V Z δ dsolss ba sg a > spcodco Ε - - Ψ Ψ d flco oal flco aspacl assso Macg wa fcos: Ψ Ψ Ψ Ψ' Ψ' Z Ψ flco pobabls: T pobably fo d flco: T pobably fo oal flco: B B < > Z B [ Z ] 4Z Z [ Z ] oc: C.H. Ka P.. ss -o Z ad < all cog lcos a d flcd - < pobably fo aspacl assso s zo.. B. - >> ad pobably fo oal flco s: Z B >> T Z T T ; Z Z T -T d flco pobably a zo gy: T Z T 8

9 Calclao of c s calcla c a oal sd: [ B ][ f V f ] d ρ T aa of coac T codcac ddv: ρ T oal sa codcac : Z [ B ] f ' V d ρ Z [ B ] f ' V d Z l: V<< fo ac cog lco a ol s flcd ad a cag of s asd Z>> l: cooal lg c V< sap pa a oc: C.H. Ka P.. ss pplcao: as of sp-polazao w coac scw-ad b p C p sapl BUT 5 pzo acao o a oal al w P a cog lco s d flcd s fo ac cog - a cag of s asd foagc sapl a alf-al P d flco s pobd T f of -V cs lls sp-polazao! s a fs appo.: V-P C 9

p polazao of c ada pc Typcal sp-polazd p bad sc: o: agzao accos fo all sps blow gy was sp polazao s a sfac popy. Ty ca a oppos sg.

M T sp polazao P C of c: o lag M ad o sp scag: oally: M M π g V s aag T M T M T M T T T g M PC M M M π ag locy wgd by O of cals: g g oal O g g

P Z P V T Z P V T Z C C pol C pol o polazd c ogal BTK sl s sd.

10 p polazao of c ada pc Typcal sp-polazd p bad sc: o: agzao accos fo all sps blow gy was sp polazao s a sfac popy. Ty ca a oppos sg. a foag a cag splg ass bw wo sp sbbads: T c fo wo sp sbbads: ± M T T w M s b of op cals ad assso. M T sp polazao P C of c: o lag M ad o sp scag: oally: M M π g V s aag T M T M T M T T T g M PC M M M π ag locy wgd by O of cals: g g oal O g g W s c sp polazao: g PC g g g clso of sp polazao BTK oy p polazao o sd ca b cosdd as a s of flly polazd ad polazd cs: pol { P 443 C pol pol V T P Z P V T Z P V T Z C C pol C pol o polazd c ogal BTK sl s sd. polazd c d flco s sppssd ~ T pobably fo oal flco s scald o ps c cosao: B ~ ~ B B W s flco pobably fo os lcos a a o d flcd: B B ~ o o dals s:. J. js al. Pys.. B Maz al. J. ppl. Pys Y. J al. Pys.. B P Z C P Z C ~ [ B ] f ' V d [ B ] f ' V d oc: C.H. Ka P.. ss

s ass: p-sapl appoac. J. ol J. J. M. Bys* M.. Osofsy B. adgoy T. bos.. Cg P.. Bossad C. T. Taaa J. owa J.. Mooda. Bay J. M.. Coy cc 8 85 998 O of fs sds dosag d spcoscopy c fo aos foagc als.

Poy ffc: d flco odcs spcodcg colaos a oal sd. T d flcd ol s allg o -sd pa of cog lco s lco ad ol fo pas-cojgad pas.

11 s ass: p-sapl appoac. J. ol J. J. M. Bys* M.. Osofsy B. adgoy T. bos.. Cg P.. Bossad C. T. Taaa J. owa J.. Mooda. Bay J. M.. Coy cc O of fs sds dosag d spcoscopy c fo aos foagc als. Mo labl sl: fg by odfd BTK odl: T sp-polazao s dd by spl fola: V-P C Pobably a lag dffs coacs w sdd as BTK oy wold o g good f o cs. Poy ffc wy sall w s ballsc coacs? Poy ffc: d flco odcs spcodcg colaos a oal sd. T d flcd ol s allg o -sd pa of cog lco s lco ad ol fo pas-cojgad pas. Ballsc coac: ffs coac: V V a ballsc coac flcd ol als bac o so w alzs. T cog sas a fac all a dsbo of lf lcod ad o spcodcg colaos a ps. a dffs coac a lco ad d flcd ol ca boc bac ad fo o sa ajcoy bw dff pos of coac casg a co spposo.

T dsac fo ba a wc pas coc s los oal al: a ballsc coac: τ a dffs coac: τ wos cas coc lg s: Pppad coc lg sadad oy of spcodcy ballsc dffs Poy ffc - gap oás:. Mossy. al. opys.

12 Poy ffc - coc lg T cog lco ad d flcd ol fos a co pa w la aplds. ssg la pas s js dd by pas of a ay poso. pcodcg colaos oal al -> poy ffc. - δ o sall dffc - δ pas coc of lco-ol pa s los af a ca dsac: δ s s δ τ W s s lg alog ajcoy d o loos pas coc ad τ s cospodg. T dsac fo ba a wc pas coc s los oal al: a ballsc coac: τ a dffs coac: τ wos cas coc lg s: Pppad coc lg sadad oy of spcodcy ballsc dffs Poy ffc - gap oás:. Mossy. al. opys oal al a gap s obsd by lg spcoscopy w a TM.. a sall gap a a of bl spcodco. T lag dsac fo ba sall gap. sall < coc s psd a lag dsac - aally a lag dsac sall gy wdow wll sow C colaos a coc lg wll b f: pasg d o pa T: coc d o lasc scag: T pasg d o bas olag V: V T τ V o: a f bas bo ad flcd ga a gy V alog ba.. a V gy dffc ass bw lco-ol pa V V

Poy ffcs - foagc lcod.. Bzd. Mod. Pys. 77 935 5 a foag wo sp sbbads a spld by a cag gy.

Oscllao of od paa Ψ s cosdd o b a al b Poy ffcs.: ac C.W.J. Ba cod-a99993 T.M. Klapwj Joal of pcodcy 7 593 4 C.W.J. Ba. Mod. Pys.

flly ba w assso pas co jco s pcd o b coplly aspa fac Z π π T ps ow sow a codcac cass blow T c b dops a low og pa. H. Coos al.

13 Poy ffcs - foagc lcod.. Bzd. Mod. Pys a foag wo sp sbbads a spld by a cag gy. Ts odcs a wab dffc bw cog lco ad d flcd ol a. Ψ δ Ψ T la apld of ad wafco: T od paa oscllas w dsac fo ba! Oscllao of od paa Ψ s cosdd o b a al b Poy ffcs.: ac C.W.J. Ba cod-a99993 T.M. Klapwj Joal of pcodcy C.W.J. Ba. Mod. Pys a dffs coac cog lco acs fac T dffs go s og a lo of scags ow a sclasscal pc odlld by a sgl d flcd ol cos bac o -sd pa s a flly ba w assso pas co jco s pcd o b coplly aspa fac Z π π T ps ow sow a codcac cass blow T c b dops a low og pa. H. Coos al. placs ad Mcoscs π π pas sf dsc fc! T cog lco acs a pas was d flcd ol o -sd pa acs a pas b d flco cass a pas sf of π s pas bw wo pas s π! low og pa coc lg cass ad dsc fc bcos poa. ca b sow a a T! 3

14 Poy ffcs.: ac C.W.J. Ba cod-a99993 T.M. Klapwj Joal of pcodcy C.W.J. Ba. Mod. Pys T zo-bas codcac of a jco: fac Z π π π a lco cags a asd b of cals T pobably a a cog - s d flcd as a ol π pas sf dsc fc!... T T! < agg w ado a oy: T T T T T T o: wo pas sf of wold co! Poy ff..: flcolss lg T.M. Klapwj Joal of pcodcy C.W.J. Ba. Mod. Pys f fac as sall assso << apld of d flco s sall ~ lco cags coss ba. How lco ca b flcd bac o fac by dsodd go sal s s ca padly ap d flco. T zo-bas codcac of a jco: o a sgl pocss: T TT g p lpl aps: pas s sa fo all!. Kasalsy al. Pys T T T T T codcac s cosdably lag! T T TT... >> TT al sa: Ba. Mod. Pys g T class g T class T >> g B V B > V class class 4

15 Jospso ffc adoal appoac ψ ρ ψ ρ Macoscopc wa fcos. ψ ~ pacl dsy ρ pas dffc δ W apply a olag of V o jco! dψ V V ψ Tψ & ρ ρ & ρ T ρ d ρ dψ V ψ Tψ... d dg by o ad wg aos spaaly fo al ad agay pa: T T & ρ & ρρ sδ ρ ρρ sδ & T ρ V & T ρ V cosδ cosδ ρ ρ T c s popooal o dρ d-dρ d: sδ V bacg aos fo pas: & δ δ δ V d Jospso aos δ pplyg a cosa bas olag: Jospso ffc adoal appoac V s δ C c w ωvħ s flowg. T C c aags o zo. zo bas olag a aal sppc of ca flow bw wo sds! Volag basd jco: C basd jco: wcg a ccal c! - V - V s sppos a C cowa olag o a C olag! V sδ V sδ V V V V V cosω δ δ sω ω V V V V sω s δ s ω cos δ ω ω V V sω δ ω V sω δ f w wold pad g od w wold g a C c ca flow a: w s a g V ±ħω a C c wll flow! ω V ± 5

16 apo sps QU apo soacs a olag ad c bas: ω V - V - V pcodcg a fo dc QU: Two Jospso jcos paalll a "loop" goy. T loop closs a agc fl of Φ T spcodco as a wll-dfd pas a y poso. -> T pas dffc bw ad B s cosa fo all ajcos. B B δ ds B δ ds δ δ ds Φ s a: δ Φ Φ δ δ δ Φ π Φ [ s δ Φ s δ Φ ] sδ cos Φ T aal al of ccal c s d by agc fl: cos Φ a d bod sas a so jco w pas bas gl cal coac w pfc assso: - p ϕ - p accos accos ϕ ϕ T lco-ol pa ca boc bac ad fo oal go. f ad pas s π cosc fc a bod sa s fod. T codo fo bod sa: accos π ± cos δ ± gl cal coac w abay assso: Ba w assso τ o o o o Ts ca b cobd o: oal scag fo lcos: o o a al τ τ δ oal scag fo ols: o o d flco a lf: d flco a g: o o M M T codo o g ozo solo: d M ± ± τ s δ o o 6

17 C - pas lao V pas basd jco d: calclao of c a codco: a d bod sa a sgly occpd gy sa s cosdd wc cas cag: p f f p p T locy ca b obad as: s How ad spcodcg pas a o aa fo a gag asfoao. T gy s b a fco of a gag aa fco of ad. ' χ ' χ g s pop gag aa ay { δ δ ± ± τ s δ C pas lao fo occpd ga gy d bod sa: δ τ sδ τ s δ o a l jco τ<<: δ τ sδ W g bac Jospso lao! o copl dsco a f bas : Mlpl d flcos V - V V V V T C c: C f bas lcos ad ols ga o loos a gy of V w y coss ba. W sa a aspacl w < fo lf sd. gas a gy V b f V< ca oly b d flcd as a ol. og bacwad ol gas a gy of V ad so o. ally gy wll cas abo ad a aspacl ca la o lcods. T apld of cog T d flco apld: s aspacl: cldd dpdc! J V Wafcos a lf ad g sd of ba: V J δ ψ V V V ψ V cag o ba: ; V V d J 7

Mlpl d flcos - ala pc f V> aspacl sad fo lf sd gas og gy o ac a py aspacl sa o g sd. s pocss a sgl lco cag s asd w pobably τ.

s pocss cags a asd w pobably of τ cas coss wc ba V> a pocss w a cag asf of ad a pobably τ bcos aalabl ccodgly a -V c sows sglas a V.

Mas of c-pas lao Jospso jco gl ao l coac T dc s a "aoc QU" QU.. a C aoc coac C ad a Jospso jco JJ paalll. T ccal c s c lag fo JJ a fo C.

18 Mlpl d flcos - ala pc f V> aspacl sad fo lf sd gas og gy o ac a py aspacl sa o g sd. s pocss a sgl lco cag s asd w pobably τ. f V> sgl aspacl assso s pobd b w a sgl d flco cag asf s alady possbl. s pocss cags a asd w pobably of τ cas coss wc ba V> a pocss w a cag asf of ad a pobably τ bcos aalabl ccodgly a -V c sows sglas a V. f w a a jco w a fw codcac cals all assso pobabls ca b dd by placg jco bw spcodcg lcods ad fg "sbgap" scs -V c. Mas of c-pas lao Jospso jco gl ao l coac T dc s a "aoc QU" QU.. a C aoc coac C ad a Jospso jco JJ paalll. T ccal c s c lag fo JJ a fo C. To ow assso pobabls of C sbgap scs s b asd -V c. Ts s do w a olag basd as. pcpl C c of JJ sold b zo a f C bas s -V c wold co ply fo C. aly JJ also as scs - V c d o fcs w o. T -V c of JJ ca b spaaly asd by coplly bag jco. C V QU V JJ V f sbacg -V c of JJ fo a of QU assso pobabls of C ca b dd. 8

. ccal c of JJ s odlad by c pas lao of C... ccal c of JJ s odlad by c pas lao of C. p w as a swcg c sad of ccal c wc s a ally acad socasc aabl.

19 γ δ Mas of c-pas lao M. Ca T Jospso ffc oc Coacs P.. ss 5 T pas dffc o aoc coac s: zo pa JJ swcs o of s zo bas sa a γπ a a ccal c δ γ 4π Φ Φ 443 ϕ fo ccal c of QU s: QU ϕ C ϕ π.. ccal c of JJ s odlad by c pas lao of C... ccal c of JJ s odlad by c pas lao of C. p w as a swcg c sad of ccal c wc s a ally acad socasc aabl. o c plss a appld o sapl ad dsbo of swcg c s asd. sw sw QU ϕ JJ C ϕ π T aag al of JJ swcg c ca b dpdly asd w a op C. T sc dd c-pas lao sows aabl ag w oy: sw QU ϕ T c pas l. a.. swcg oy w T ff 6K ϕπ ϕπ 9

Phys Nov. 3, 2017 Today s Topics. Continue Chapter 2: Electromagnetic Theory, Photons, and Light Reading for Next Time

Phys Nov. 3, 2017 Today s Topics. Continue Chapter 2: Electromagnetic Theory, Photons, and Light Reading for Next Time Phys 31. No. 3, 17 Today s Topcs Cou Chap : lcomagc Thoy, Phoos, ad Lgh Radg fo Nx Tm 1 By Wdsday: Radg hs Wk Fsh Fowls Ch. (.3.11 Polazao Thoy, Jos Macs, Fsl uaos ad Bws s Agl Homwok hs Wk Chap Homwok