Quantum transport in 2D
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1 Quantum transport in D Quantum transport : wat is conductanc? mtallic ring atomic contact nanotub Landaur-üttikr formalism of quantum transport D gas grapn wir ntwork GRAPHENE & CO, Cargès April -3, 08 Gills Montambaux, Univrsité Paris-Sud, Orsay, Franc usrs.lps.u-psud.fr/montambaux Landaur-üttikr : conductanc = transmission Conductanc = transmission D wir G Exampl : carbon nanotub Landaur formula R. Landaur (97-999) scattrr 3 Landaur-üttikr multitrminal formalism rsrvoir contact trminal lad M. üttikr (950-03) M. üttikr, Four trminal Pas-cornt conductor, PRL 986 lctronic transport btwn t two rsrvoirs is a wav transmission troug a potntial barrir
2 D wir scattrr D wir scattrr rsrvoir contact trminal lad Hypotss : Currnt carrid by an lctron in a stat k : v j k L A trminal sorbs lctrons and injct tm at a givn potntial and a givn tmpratur. No pas rlation btwn incoming and outgoing lctrons in ac trminal. t scattrr is lastic. rsistanc of t rsrvoirs is ngliglibl. Summation ovr all filld stats : () : transmission cofficint ( )[ f( 0 ) f( )] d A problm of D quantum mcanics 5 (Rmarkl rsult : t vlocity as disappard! ) 6 D wir D wir scattrr scattrr Linar rgim : f G ( ) d 0 Landaur formula G No scattrr (infinit conductivity?) ( ) F Low tmpratur : Conductanc quantum : G ( ) F (Rmarkl rsult : t vlocity as disappard! ) /(58,807 ) 7 G conductanc is finit and quantizd!!! 8
3 prfct conductor as a finit and quantizd conductanc!!! Potntial profil allistic x G G A G = potntial drop A t contacts r is t potntial drop? r is powr dissipatd? No rsistanc in t sampl «contact» rsistanc R c m How to masur t conductanc of t scattrr itslf? 9 Powr dissipatd in t contacts P ( ) Potntial profil allistic x G G «trminal» rsistanc vs «trminal» rsistanc = potntial drop A t contacts lads sampl rsrvoirs H. Potir t al., Enrgy distribution of lctrons in an out-of-quilibrium mtallic wir, Pys. 03, (997)
4 vs trminals vs trminals A A scattrr Prfct sampl : = it a scattrr = ( ) G ( ) A G 3 G ( ) G ( ) G G??? A Potntial profil allistic On scattrr = x +R No dissipation in t wir < potntial drop A t contacts G G G G 5 vs trminals ( ) G A «trminal» conductanc ( A ) R G R «trminal» conductanc
5 Conductanc = transmission G conductanc r,d= < G F Om-Drud l L R( ) A R A A R Rc R Rc G G R k F l G L l / G Sarvin F two-trminal rsistanc is t addition in sris of t four-trminal rsistanc and t two contact rsistancs. 7 R Rc R Rc allistic L<l L>l Diffusiv in units Landaur-üttikr formula wo-trminal conductanc G Sarvin λ F contact rsistanc Drud-Om λ F L π l R = λ F + λ F L π l Four-trminal conductanc 3 non-invasiv voltag probs G R S. aruca t al., Sarvin rsistanc and its brakdown obsrvd in long ballistic cannls Pys. Rv. 7, 06 (993) 0
6 Four trminal rsistanc of a ballistic quantum wir (00) Sourc Drain «clavd-dg ovrgrowt» G G Sourc 3 Drain R. D Picciotto t al., Four trminal rsistanc of a ballistic quantum wir, Natur, 5 (00) Multicannl Landaur formula R pt /R pt? sin k y y t 3 b a =invasivity 3 k y k y -trminal rsistanc is 0 -trminal rsistanc is quantizd For non invasiv contacts R. D Picciotto t al., Four trminal rsistanc of a ballistic quantum wir, Natur, 5 (00) 3 currnt is t sum of t contribution of t diffrnt cannls mods
7 Multicannl Landaur formula Multicannl Landaur formula b t a b t a t Currnt rsulting from t transmission of a cannl b to a cannl a otal currnt ( ) t Multicannl Landaur formula G,, ( ) 5 6 Multicannl Landaur formula : clan wav guid b t b Conductanc of a cornt ballistic systm G q = M = nt λ F Quantum point contact QPC an s t al. PRL 988; aram t al. J. Pys. C 988 G, G M wav guid pr mod M is t numbr of transvrs cannls 7 M transvrs cannls mods s tomorrow s lctur on Landaur formula
8 Conductanc of a cornt ballistic systm (finit ) Quantization of t conductanc : tmpratur ffct G q = M = nt λ F ( ) f G d, r : f G M( ) d Quantum point contact QPC an s t al. PRL 988; aram t al. J. Pys. C 988 M ( ) ( n ) n n m wav guid G n f( ) n M transvrs cannls mods Caractristic nrgy : m * * K 50nm Landaur-üttikr multitrminal formalism Landaur-üttikr formula Landaur formula G 0 R. Landaur (97-999) wo-trminal conductanc Four-trminal conductanc 3 G G R Currnt probs G, oltag probs M. üttikr (950-03) M. üttikr, Four trminal Pas-cornt conductor, PRL How many cofficints to caractriz t «black box»? G? 3
9 i ( Mi Rii) i ij j ji M. üttikr MR 3 M R M3 R M R G Conductanc matrix M R 3 0 M R G Four trminals MR 3 M R M3 R33 3 M R 3 3 D transmission cofficints 33, G Landaur-üttikr formula im Rvrsal Symmtry M. üttikr wo-trminal conductanc Four-trminal conductanc G G R G MR 3 M R M3 R33 3 M R n gnral dpnds on 9 transmission cofficints G D 3 3 can b ngativ! 0 ( ) ( ) ij ij ji ji 35 36
10 G MR 3 M R M3 R33 3 M R 3 Landaur-üttikr formula wo-trminal conductanc G trminals 9 transmission cofficints N trminals ( N ) Four-trminal conductanc 3 G R n zro fild, t 3 x 3 submatrix is symmtric 6 transmission cofficints N( N ) ( ) ( ) ij ji 3 G D n gnral dpnds on 9 transmission cofficints ( 6 in zro fild) ij ji trminal rsistanc in a carbon nanotub trminal rsistanc in a carbon nanotub 3 3 R,3 D 3 3 P = X ( ij + ji )( i j ) i,j Low : t trminal rsistanc can b ngativ. Gao t al., Four-point rsistanc of individual singl-wall carbon nanotubs, Pys. Rv. Ltt. 95, 9680 (005) Low : t trminal rsistanc can b ngativ, but powr dissipat is positiv
11 Symmtry of t two-trminal conductanc Symmtry of t four-trminal conductanc? ( ) ( ) (G) G ( ) G( ) ( ) G, 3 3 D 3 3 L. Angrs t al., Pys. Rv. 75, 5309 (007) A. noit t al., Asymmtry in t magntoconductanc of mtal wirs and loops, Pys. Rv. Ltt. 57, 765 (986) Symmtry of t four-trminal conductanc? Landaur-üttikr formula G D ( ) G ( ) G ( ),3 3, n d Picciotto xprimnt, 6 cofficints rduc to on 3 =invasivity ( ) ( ) G ( ) A. noit t al., Asymmtry in t magntoconductanc of mtal wirs and loops, Pys. Rv. Ltt. 57, 765 (986)
12 Quantum Hall Effct Pas cornc i= R K =5 8, 807 Non- locality i=3 i= R R H L 0 M 3 5 R H M Appl. Pys. Ltt. 50, 89 (987) R L 0 6 Quantum Hall ffct ulk trajctoris ar pinnd by disordr Ciral dg trajctoris propagat frly ulk insulator Prfct «ciral» conductor at t dgs Quantum Hall ffct Lft-going and rigt-going lctrons ar spatially sparatd Dissipation in t arrival trminal «opological insulator» 7 Dissipation in t arrival trminal
13 Quantum Hall ffct Lft-going and rigt-going lctrons ar spatially sparatd Quantum Hall ffct Lft-going and rigt-going lctrons ar spatially sparatd Dissipation in t arrival trminal is xprimnt sows tat lctrons stay at t cmical potntial of t injction rsrvoir and xcang tir nrgy at t arrival rsrvoir =0 =0 dissipation Dissipation in t arrival trminal maging of t dissipation in quantum Hall ffct xprimnts U. Klass t al., Z. Pys. 8, 35 (99) 50
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