Elcron-lcron inracion and dcohrnc in mallic wirs Chrisoph Txir, Gills Monambaux, Univrsié Paris-Sud, Orsay www.lps.u-psud.fr/usrs/gills
Elcron-lcron inracion and dcohrnc in mallic wirs ( T ) Msoscopic Quanum ( T) = D Macroscopic Classical Cohrn ranspor on mallic wirs and nworks Tmpraur dpndnc of h phas cohrnc im ( T ) a low T Diffusiv rgim : r () = D
Physics wll undrsood for wirs - inracions in wirs ( T) /3 1/3 T ( T) = D T Alshulr, Aronov, Khmlniskii (8) Ag Wak localizaion masurmns on mallic wirs Au Ag T /3 Sauraion du o addiional dgrs of frdom, Cu magnic impuriis 3 T (phonons Saclay, 003
Som complicaions on nworks In nworks, quanum oscillaions ( T ) Analysis of h ampliud of h quanum oscillaions may giv diffrn!!! Which is masurd in xprimns?
Oulin Wak localizaion and phas cohrnc Wha is and how o xrac i from wak localizaion xprimns? Diffusion on nworks Phas cohrnc, - inracions and gomry ffcs w xprimns
Wak localizaion = corrcion o classical ranspor Classical conducanc G cl Quanum corrcion Tim rvrsd rajcoris Phas cohrnc G G cl d 1 λf v Vol F P () / d P () = d (4 π D) d / Cohrn ffs ar mor imporan in low dimnsion Incras of h rsisanc, supprssd by a magnic fild ngaiv magnorsisanc
B Exampl 1: wak localizaion in D G P() G cl / d P () = S 4π D G ln l In a magnic fild : P () = BS / 0 sinh 4 π BD / 0 G ln ( B ) min, l B B 0 Brgmann, 84 Wak localizaion corrcion is supprssd whn R * B * B 0 ( T) 1/ T
G P() G Exampl : wak localizaion in a quasi-1d wir / cl d P() = ( 4π D) 0 1/ G In a magnic fild : min (, B ) = P P / B () () G B 0 BW Wak localizaion corrcion is supprssd whn * B * BW 0 T 1/3 agrs wih AAK
Exampl 3 : wak localizaion in a ring G P() G cl / d wir : P() = ( 4π D) 0 1/ / B G min (, B ) Ring in a Aharonov-Bohm flux : /4 m D P () = cos4π m 4π D m 0 / B G m min (, B ) m Alshulr, Aronov, Spivak, 81 may b xracd ihr from h nvlop * BW 0 Or from h ampliud of h oscillaions m / T 1/3 Dolan,icini,Bishop, 86
works : analysis of quanum oscillaions 3.5 300mK 500mK 1.4K.K 3.3K ĘR/R*10^4 (Ohm) 1.5 1 0.5 wir 0 cr304-0.5-400 -300-00 -100 0 100 00 300 400 nvlop B (G) (T) can b xracd from h fild dpndnc of h W corrcion (nvlop) from h harmonics of h quanum oscillaions harmonics Diffrn (T)!? Orsay, GaAs Wha is = D?
Wha is = D? Dphasing bwn im rvrsd rajcoris ( ) i Φ () / = ( ) yquis im Quaniy xracd from xprimns =??
Wha is masurd in W xprimns? Wak-localizaion corrcion : ( ) ( ) iφ() g P() d Diffusion Dphasing : - inracion i Φ () / = T /3
1) Diffusion : assum i Φ () / = shulr,aronov,spivak, 81 g ( n) n/ n / Harmonics of h quanum oscillaions n 1/ n 1/4 Pn ( ) n Pn ( ) n n 4/3 1/3 1/3 1/6 Subdiffusiv winding C. Txir, G.M., J Phys A 38, 3455 (005)
1) Diffusion : assum i Φ () / = shulr,aronov,spivak, 81 g ( n) n/ n / Assuming n/ = T 1/3 = T 1/6 ( T) T /3 " ( T)" T 1/3 C. Txir, G.M., J Phys A 38, 3455 (005)
) Gomric ffcs on dphasing iφ() g P() d /?? Diffusion Dphasing du o - inracion iφ( ) = { / / c AAK Diffusiv Ergodic c w characrisic im D = / D Thoulss im diffusiv vs rgodic rgim Consqunc on quanum ranspor on nworks = 3/ 1/ c D
( T ) T iφ() /3? : a simpl rmindr Dphasing du o - inracion Alshulr,Aronov,Khmlniskii, 8 quasi-1d wirs ( ) ( ) - inracion = lcric flucuaing ponial Flucuaing phas 1 iφ() Φ () Φ( ) = ( ) ( ) () = V(( r ), ) d 0 Φ ( ) = V( r, ) V( r, ) V( r, ) V( r, ) d 0 d Φ () d V
yquis horm d Φ () V V kbt R kbt d r σ 0 S d Φ () k B d T S σ 0 r Diffusion r D kt D B 3/ Φ () σ0s 3/ σ S 0 kt B D /3 1 T /3 yquis im AAK
Diffusiv quasi-1d sysm Φ () 3/ 1 iφ()?? T, C () T, C = Φ on xponnial im rlaxaion iφ() = f i () Φ 1 π 4 3/ 0.8 0.6 0.4 0. i Φ () /.M., E. Akkrmans, Phys. Rv.. 95, 016403 (005) 1 3 4 5 6 7
Gomric ffcs on dphasing du o - inracion d Φ () d V d Φ () d kt B S σ 0 r
yquis horm d Φ () B V V kbt R r σ 0 S d kt d Φ () d kt B S σ 0 r Dphasing dpnds on h naur of h diffusiv rajcoris Diffusiv Ergodic r D r 3/ 3/ B σ 0 S Φ kt D () Φ () kt B σ S 0 c /3 σ 0 S 1 σ /3 kt T B D c 0 B S kt 1 T yquis im AAK This im probs h gomry of h sysm
Diffusiv (infini sysm) Ergodic (fini sysm) Φ () 3/ Φ () c iφ() / = f(/ ) i Φ () / i Φ () / c =
Gomric ffc on dphasing Quasi-1D wir on-xponnial im dpndnc of h phas rlaxaion ( ) f / / kt = g( ) B ( T ) T /3 Ring Exponnial im dpndnc Diffusiv vs Ergodic / 1 c kt = ( ) c ( T ) T B c g Thoulss im D = / D = 3/ 1/ c D
Tim dpndnc of h phas rlaxaion Th dphasing dpnds on h naur of h diffusiv rajcoris, m = 0 D m= 0 ( ) iφ() / = f / T, C D iφ() / c = T, C m= 0 m 0 i () / c Φ = T, C m Harmonics and nvlop may hav a diffrn T dpndnc
C. Txir, G.M., Phys. Rv. B7, 11537 (005) π Ai(0) m 8 c g = m Ai '(0) = D T c = D T c 1/ 1/3 Insad of g m = AAS, 81 m Compar wih naiv xpcaion from convnional analysis m g = m T 1/3 T 1/
Conclusion : dphasing probs h naur of h diffusiv rajcoris ( T ) T /3 ( ) T c T 1?
.... σ Diffusion iφ() P () d Dphasing ormal diffusion + boundd moion Pn (, ) n iφ() C m c n c nt 1/...... Subdiffusiv winding + unboundd moion Pn (, ) n n 4/3 1/3 1/3 1/6 iφ() C m n 1/ nt 1/6
Subdiffusiv winding + unboundd moion Pn (, ) n n 4/3 1/3 1/3 1/6 iφ() C m n 1/ nt 1/6 Diffusiv winding + unboundd moion Pn (, ) n iφ() C m n nt 1/3
Conclusions *Dcohrnc du o - inracion dpnds dramaically on gomry hrough h naur of h diffusiv rajcoris *Two yps of dphasing (diffusiv vs rgodic rgim) ( ) f / / kt = g ( ) B ( T ) T /3 / c kt B c = g ( ) 1 c ( T ) T = 3/ 1/ c D *Diffrn mpraur dpndncs for apparnly diffrn phas cohrnc lnghs
high T low T < < 1/ nt 1/6 nt 1/3 nt 1/3 nt
high T low T < < 1/ nt 1/ nt diffusiv winding, c diffusiv winding, c 1/ nt diffusiv winding, c 1/6 nt subdiffusiv winding, 1/3 nt diffusiv winding, 1/3 nt diffusiv winding,
Wha abou nworks? σ Diffusion P() iφ() d Dphasing Squar nwork < a ormal diffusion iφ() Cm c 3/ m m c = 3/ 1/ m T > a?
* On quasi-1d wir, phas rlaxaion is almos xponnial * yquis im in on lin : V k T R B R K = h R( D ) D kt kt B B R R K K σ 0 S T /3
......
Diffusion on nworks : gnral formalism E () = n P D ψ = E n ψ n γ 1 g P() d = = ln S( γ ) γ + E γ n n 1 γ = = D S( γ) = ( γ + E ) n n Spcral drminan : conains all informaions abou diffusion M. Pascaud, G.M., PR 8, 451 (1999)
Spcral drminan on a nwork Solv diffusion quaion on ach bond + currn consrvaion a h nods B αβ S( γ ) = sinh d αβ M M = αα coh M αβ β iθ αβ = sinh αβ αβ nods M is a marix bonds cf: suprconducing nworks (Alxandr, D Gnns) scaring on 1D graphs (C. Txir, G.M.) g ln S( γ ) γ M. Pascaud, G.M., PR 8, 451 (1999)
xampls S( γ ) = (cosh / cos 4 πϕ) ϕ = 0 γ = D/ ϕ AAS sinh / g cosh / cos 4πϕ g n n / S ( γ ) = sinh / + (cosh / cos 4 π ϕ ) = (cosh / cos 4 πϕ) ff sam form as for an isolad ring wih ff g n n ff / ff ff g n n / n g + n ( / ) / C. Txir, G.M., J Phys A 38, 3455 (005)
Winding propris n / gn n / g n Pn ( ) n Pn ( ) n n 4/3 1/3 1/3 1/6 n 1/ n 1/4 Anomalous diffusion C. Txir, G.M., J Phys A 38, 3455 (005)
From spcral drminan o winding propris γ P () d= ln S( γ ) γ P P n ( n) ( ), ( ) and n 1/ d d = 1 n 1/4 d = d = 3 n n ln C C. Txir, G.M., J Phys A 38, 3455 (00
Baurl,Saminadayar,Mall,Schopr 100 T 1/3 for h quasi 1D wir 10 T /11 Envlops (µm) =4a 1 =a 0.1 0.001 0.01 0.1 T (K) 1 10 100
QUESTIO : * Phas cohrnc iφ() = f (/ ) on xponnial dphasing on quasi-1d wir Coopron in a flucuaing fild * Wha abou dcay of quasi-paricl? on quasi-1d wir Frmi goldn rul Quasiparicl injcd a nrgy ε abov h Frmi sa, a mpraur T P / () = ( ε, T ) ( T) = ( T)??? P () vs. iφ() Tim dpndnc of quasiparicl dcay?
1 - ifim of quasiparicl ε / P () =? ω Frmi goldn rul ε andau ε T 1 1 ( ε, T = 0) W ωdω ε ( ε = 0, T) T W dω T 0 0 W : marix lmn of h inracion ε ω ε +ω Diffusiv conducors : h ypical marix lmn of h inracion is nrgy dpndn (Alshulr-Aronov) 1 W ( ω) ω d / 1 P () = ( D) d / ε 1 d / ( ε, T = 0) W ( ω) ωdω ε 0 T T 1 1 dω ( T) = ( ε = 0, T) T W ( ω) dω T d / ω 0 0 Infrard divrgnc for d Sinc rprsns h lifim of a quasiparicl sa, nrgy xchang canno b dfind wih a prcision br han 1/ ω Slf-consisn rlaion 1 ( T ) T T 1 dω 3/ ω d=1 ( ) T T /3
Claim : his infrard divrgnc implis ha h q.p. dcay is no xponnial P / () Sinc rprsns h lifim of a quasiparicl sa, nrgy xchang canno b dfind wih a prcision br han 1/ ω Frmi goldn rul : For a givn valu of im, ransiions consrv nrgy wihin 1/ Enrgy ransfr canno b dfind wih a prcision br han 1/ ω. 1 ω. 1 1 ( T) T T 1 ( T ) dω 3/ ω T dω w () T T 3/ ω 1/ 1/ Rlaxaion ra incrass wih im : rlaxaion is no xponnial
1 dp dω w () = T T 3/ d P ω T 1/ 1/ 3/ /3 T ( T ) P () 3/ P () ( / ) 3/ Characrisic im is unchangd ( ) T σ S DT /3 Th xponn «/3» is indd a signaur of h non-xponnial «3/» rlaxaion G.M., E. Akkrmans, Phys. Rv.. 95, 016403 (005)
Conclusion : * Quasiparicl dcay in a quasi-1d wir is a non xponnial procss P ( / ) () = 3/ ( ) T σ S = DT /3 * In a ring, nrgy ransfr canno b smallr han dω P() T T if 3/ ω 1/ D 1/ D c = / ( ) P () c c T 1 RT
Abou Frmi goldn rul (1) P () = 1 P () if f ( ω) P U f if () = αγ, βδ ( εα + εγ εβ εδ) f E E = ω i f Enrgy «uncrainy» funcion of widh 1/ : nrgy is consrvd wihin 1/ (Hisnbrg usually f ( ω) π δ ( ω) P d d U f if ( ) ω ω ( ω ) ω, ω, Τ π ω d U ω,τ ω = ε ε Enrgy ransfr β α T dω ω 3/ Howvr, should b : T dω 3/ 3/ max( ω, ω )
Abou Frmi goldn rul () f ( ω) π δ ( ω) P d ω d U f d ω U π if ( ) ω ( ω ) ω, ω, Τ ω,τ Propr ingraion ovr ω h( ω) dω Pif () T h( ) 3/ ω ω 0 Cuoff funcion 4 6 8 ω Propr applicaion of Frmi goldn rul provids naurally a low nrgy cuoff dx P T h x 3/ if () ( ) 3/ x 0 3/
Dphasing on a ring udwig, Mirlin 03 W g m m / 9/4 g m m( / ) 3/ AAS 1/ 3 σ 0DS = D = T Hr : * Analyical soluion for W corrcion * Simpl dscripion and analysis in im rprsnaion ( / ) 3/ i () Φ { / c m( / ) 3/ is h signaur of xponnial rlaxaion c * w gomris
Diffusiv conducors : h ypical marix lmn of h inracion is nrgy dpndn (Alshulr-Aronov) W 1 ( ω) ω d / Th ffciv - inracion is proporional o h im spn in h inracion rgion : 1/ ω W ( ω) P( ) d 1 ) P() rurn probabiliy P () = d ( 4π D) /
i Φ () T, C? 1 1 () iφ() T = Φ T 1 iφ() = Φ T, C () T C i Φ () 1 0.8 d Alshulr,Aronov,Khmlniskii iφ() γ T C 1 Ai, ( γ) d = 4π D DAi'( γ ) 1 () T, Φ C Srn,Imry,Aharonov 0.6 0.4 0. i () Φ T, C 1 3 4 5 6 7
iφ() γ T, C 1 Ai( γ) d = 4π D DAi'( γ ) ( T ) σ S DT /3 1 umrical aplac invrs ransform i Φ () 0.8 0.6 / i () Φ T, C 0.4 0. π 4 3/ 1 3 4 5 6 7 / Phas rlaxaion scals as 3/
i Φ () 1 0.8 0.6 0.4 i () Φ T, C / 0. 1 3 4 5 6 7 iφ() γ 4π D d 0.6 0.5 0.4 γ = DW B 3 0.3 0. 0.1 γ 1 3 4 5
Conclusion Th dcay of quasiparicl and h dphasing of im rvrsd rajcoris ar dscribd by h sam characrisic im = P 3/ ( / ) () = flucuaing fild () Frmi goldn rul + - inracion i () () P () = = = iφ() ( / ) 3/ Coopron in a flucuaing fild () ( ) iφ() i( () ( )) ( () () ( )) = =
Rlaxaion of quasiparicl i () () P () = = T dq DT 3/ () σ q σ 1/ D Phas rlaxaion (coopron) iφ() i( () ( )) ( () () ( )) = = T dq () ( ) σ q 1/ D Dq T dq DT () () ( ) (1 ) Dq 3/ σ q σ 0
From spcral drminan o winding propris γ P () d= ln S( γ ) γ P ( n) () and P () ff ( ) ϕ ϕ = D n 1/ 1/ ff ( ) ff = n 1/ 0 ff = ϕ P (0) ϕ n P ϕ = (0) d = 1 P ϕ (0) ϕ d = P (0) ln ϕ d = 3 P (0) C ϕ ϕ n n n 1/4 ln C
Φ ( ) = V( r, ) V( r, ) V( r, ) V( r, ) d 0 d=1 d= d=3
Ring : Harmonics conn of h phas rlaxaion Harmonics of h W corrcion iφ g P() d C 1 m /4D P() = cos4π m 4π D m Harmonics xpansion of h rurn probabiliy m/4d iφ() g m Cm 4π D d c c m c Alshulr, Aronov, Spivak bu R c = D c = D T insad of = D T Ν 1/3 g m 3/ 3/ m udwig, Mirlin (04)