Disorder and mesoscopic physics. Lecture 2. Quantum transport, Landauer-Büttiker weak-localization G 2. Landauer-Büttiker formulae.

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1 isordr and msoscopic pysics Landaur-Büttikr formula Lctur wo-trminal conductanc G Quantum transport, Landaur-Büttikr wak-localization our-trminal conductanc non-invasiv voltag probs R Gills Montambaux, Univrsité Paris-Sud, Orsay, ranc usrs.lps.u-psud.fr/montambaux our trminal rsistanc of a ballistic quantum wir (00) Sourc rain «avd-dg ovrgrowt» G G Sourc rain R. Picciotto t al., our trminal rsistanc of a ballistic quantum wir, Natur, 5 (00) A B

2 Multicannl Landaur formula R pt /R pt b t a k y k y -trminal rsistanc is 0 -trminal rsistanc is quantizd or non invasiv contacts R. Picciotto t al., our trminal rsistanc of a ballistic quantum wir, Natur, 5 (00) 5 currnt is t sum of t contribution of t diffrnt cannls mods 6 Multicannl Landaur formula Multicannl Landaur formula t t b a b a t Currnt rsulting from t transmission of a cannl b to a cannl a otal currnt I ( ) t Multicannl Landaur formula G, G tr tt I, ( ) 7 8

3 Multicannl Landaur formula : an wav guid Numbr of transvrs cannls t k k k x y k y n / y k n ( nk, x ) m m x b b k y ny, ny 0 k x k x G, G M k y k y M k k int int / M is t numbr of transvrs cannls 9 0 Numbr of transvrs cannls Quantization of t conductanc (988) Quantum point contact QPC d= M k k int int / k G M int[ ] d= M k / ks int int ( / ) (d) M d d Ad k / Ad d int int ( k ) d d ( / ) ( ) B.J van s t al., Quantizd conductanc of point contacts in a lctron gas, Pys. Rv. Ltt. 60, 88 (988)

4 Quantization of t conductanc : tmpratur ffct Quantization of t conductanc : tmpratur ffct ( ) f G d, r : f G M( ) d ( ) f G d, r : f G M( ) d M ( ) ( n ) M ( ) ( n ) G n f( ) n G n f( ) n Caractristic nrgy : m * * K 50nm Caractristic nrgy : m * * K 50nm Conductanc = transmission Landaur-Büttikr formula Landaur formula G 0 R. Landaur (97-999) wo-trminal conductanc our-trminal conductanc G R Currnt probs 0 oltag probs G? Landaur-Büttikr formalism M. Büttikr (950-0) 5 6

5 Ii ( Mi Rii) i i i M. Büttikr Landaur-Büttikr formula wo-trminal conductanc G 0 0 I G Conductanc matrix our-trminal conductanc R G our trminals MR M R M R M R 7 a priori dpnds on 9 transmission cofficints (6 in zro fild). 8 Landaur-Büttikr formula Landaur-Büttikr formula wo-trminal conductanc G our-trminal conductanc R In d Picciotto xprimnt, 6 cofficints rduc to on ( ) ( ) G ( ) a priori dpnds on 9 transmission cofficints (6 in zro fild). 9 0

6 Landaur-Büttikr formula Potntial barrir + non invasiv prob Symmtry of t two-trminal conductanc ( B) ( B) G B (G) GB ( ) G( B) L. Angrs t al., Pys. Rv. B 75, 509 (007) Symmtry of t four-trminal conductanc? Symmtry of t four-trminal conductanc? B( ) G, I B -B B( ) G ( B) G ( B),, A. Bnoit t al., Asymmtry in t magntoconductanc of mtal wirs and loops, Pys. Rv. Ltt. 57, 765 (986) A. Bnoit t al., Asymmtry in t magntoconductanc of mtal wirs and loops, Pys. Rv. Ltt. 57, 765 (986)

7 Conductanc, transmission and probility Pas cornc Conductanc = transmission Non- locality ransmission troug a disordrd systm= probility to cross t systm G P(0, L), 0 L Appl. Pys. Ltt. 50, 89 (987) 5 6 iffusion probility, microscopic approac P r, r', t probility to find a particul at r, if it as bn inctd at r Quantum amplitud i A (, r r') A (, r r') Grr (, ') A (, rr') ( r, r') r ' rr, ' pdl. r r Cf. Young s slits r probiliy is t modulus squar of t amplitud :, ', ', ' P rr G rr A rr isordr avrag 7 8

8 iffusion probility, microscopic approac Exampl : Young slits I I I int wo contributions isordr avrag I A A * ' ' P(, rr') A(, rr') A(, rr') A(, rr') Classical trm Intrfrnc trm Classical transport : only paird tractoris A A contribut If t tractoris ar diffrnt, t amplituds A t A ar diffrnt Quantum ffcts S uncorrlatd pass In avrag, t intrfrnc trm disappars P(, rr') P (, r r') 9 * * I A A AA A A 0 Exampl : Young slits I I I int iffusion probility, microscopic approac a x I I kax cos I I I int wo contributions * ' ' P(, rr') A(, rr') A(, rr') A(, rr') Classical trm Intrfrnc trm s ' a x I I kas sin ' kax cos kas ' Iint 0 P (, r r') IUSON Quantum ffcts Classical transport : only paird tractoris A A contribut If t tractoris ar diffrnt, t amplituds A t A ar diffrnt P(, rr') uncorrlatd pass In avrag, t intrfrnc trm disappars P (, r r')

9 Classical probility, microscopic approac P(, rr') Probility to go from r to r Classical probility, microscopic approac Itration in ourir spac : Pq (, ) P( q, ) 0 P0 ( q, )/ P(, r r') P(, r r') P(, r r ) P( r, r') P(, r r) P( r, r ) P( r, r') = Probility to go from r to r witout any collision + probility to go from r to r wit on collision + tc. «building block» ( R vt) P0 ( R, t) R t / d= Itrativ structur (Bt-Salptr quation) iffusiv limit ωτ ql P q i q (, ) ( ) 0 v d vl d P(, rr') P(, rr') P(, r r) P( r, r') 0 0 «building block» (, ) i q P q Z δ(r vt) P 0 (~q, ω) = πr t/τ iωt iqr cos θ πr sin θdrdθdt P 0 (~q, ω) = v Z R/l + iωr/v iqr cos θ π sin θ dθ π dr P 0 (~q, ω) = À v /l iω/v + iq cos θ θ Classical probility, diffusion quation P (, r r') is solution of a assical diffusion quation: P (, r r',) t ( rr') () t t Solution in fr spac in d dimnsions: P ( q, t) P ( R, t) d t / q t R t À τ P 0 (~q, ω) = iωτ + iql cos θ θ q l cos θi = q l d = q τ An important rsult, t rturn probility: Prrt (,,) ( t) d / iffusiv limit ωτ ql P 0 (~q, ω) =τ ( + iωτ q τ ) 5 ypical distanc: R () t d t 6

10 Rmarks out assical diffusion : oulss and rcurrnc tims ypical tim to rac t dg of t systm iffusion tim oulss tim τ = L otal tim spnt nar t origin in a finit systm : rcurrnc tim Rmarks out assical diffusion : oulss and rcurrnc tims Probility is normalizd suc as Probility to b in a givn volum aftr tim t Z P (r, r 0,t)dr 0 = Z P (r, r 0,t)dr 0 otal tim spnt in a givn volum aftr tim t = Z t Z dt P (r, r 0,t)dr 0 0 otal tim spnt nar t origin τ R = v Z 0 dtp (r, r,t) 7 In a finit systm : τ R = Z τ τ v dt (πt) d 8 Rmarks out assical diffusion : rcurrnc tim iffusion probility, quantum corrctions ypical tim to rac t dg of t systm iffusion tim oulss tim τ = L P(, rr') otal tim spnt nar t origin in a finit systm : rcurrnc tim P (, r r')? τ R = Z τ τ v (πt) d dt d = d = τ R τ L l τ R τ ln L l * ' ' P( rr, ') A( rr, ') A( rr, ') A( rr, ') d = τ R τ 9 0

11 Conductanc, quantum corrctions conductanc is proportional to t probility to transfr lctrons from on sid of t sampl to t otr sid (Landaur-Buttikr) G P(0, L) Quantum corrctions P(, rr') * ' ' P(, rr') A(, rr') A(, rr') A(, rr') Classical condutanc : G P (0, L) Exampls : Byond t assical contribution? << << << P(, rr') +.. l IUSON P (, r r') + quantum corrctions Quantum corrctions Q: wn do quantum ffcts appar? A : n tractoris cross Exampl : particuls from r,r to r,r P (, rr') Quantum crossing ( r, r, r, r ) Gr (, r) G( r, r) Gr (, r) G ( r, r ),, *,, *,, i r r i r r Quantum corrction : *,, *,, Ai( r, r ) Ai' ( r, r ) A( r, r ) A' ( r, r ) ii' ' Classically : product of probilitis ( r,,, ) (, ),,,, r r r P r r P ( r, r) l olum l d r r i i = r i Quantum crossing = xcang of amplituds Simpl pictur : t diffuson P (, r r',) t is an obct of lngt v t and cross-sction d r i r = r if r =r «Hikami box» * Quantum ffcts ar du to crossings * Importanc of quantum ffcts => valuat t probility of crossing

12 Evaluation of t probility of quantum crossing Conductanc : ratio of two volums Simpl pictur : t diffuson P (, r r',) t is an obct of lngt v t and cross-sction d Probility of crossing during a tim t, in a volum =L d p () t d v d L t ransport pnomna : in a sampl of siz L, t wav spnds a tim =L / probility of quantum crossings wic affcts tranport proprtis is tus g v d olum of t systm olum of a tub of sction wic diffuss across t systm d v p ( ) d L g!!! g imnsionlss conductanc! g G/( / ) 5 6 Cornt ffcts and quantum crossings vrsus quasi- probility of quantum crossings is /g Quantum crossings Quantum corrctions luctuations, corrlations on cannl: strictly motion wir Classical transport Quantum ffcts ar of ordr (fluctuations, oscillations, corrctions) G g G g many cannls, motion, but diffusion quasi Quasi- wir xampl: Aaronov-Bom (or Sarvin-Sarvin) oscillations in a mtal ar of ordr / In a good mtal (g >>), quantum ffcts ar small 7 Numbr of cannls : ( k ) M 8

13 Summary of prvious lctur ak localization G = G + δg(b) conductanc ~ transmission ~ probility P (, rr') ariation of t rsistanc R(B) of a mtallic film as a function of t applid magntic fild magntic fild scal dpnds on tmpratur A cornt contribution, dviation to Om s law, incrass t rsistanc and is dstroyd by a magntic fild: assical diffuson quantum corrctions Ngativ magntorsistanc : signatur of wak localization magntorsistanc of Mg film (Brgmann 98) d v p ( ) g quantum crossing /g corrction assical transport quantum ffcts g 9 In a cylindr, t rsistanc is priodically modulatd wit t flux troug t cylindr, wit a priod A pas cornc ffct wic rsists disordr 50 «Sarvin-Sarvin» oscillations (98) ak localization ak localization : (-) sign Classical conductanc G P (0, L) (-) sign Quantum corrction G Pint () t G g G Pint () t On crossing On loop im rvrsd tractoris Crossing P(0, L) k k ' k' k k k ' l P () t int = distribution of numbr of loops wit tim t = rturn probililty 5 backscattring 5

14 ak-localization Loops and quantum crossings Nb loops, probility P int (r,r,t), rturn probility Magntic fild, pas cornc ak-localisation in dimnsion d A fw solutions of t diffusion quation and L Magntic fild and ngativ magntorsistanc Magntic fild in quasi- wirs AAS oscillations 5