Handout 28. Ballistic Quantum Transport in Semiconductor Nanostructures

Transcription

1 Hanout 8 Ballisti Quantum Transport in Smionutor Nanostruturs In this ltur you will larn: ltron transport without sattring (ballisti transport) Th quantum o onutan an th quantum o rsistan Quanti onutan ol anaur (IBM) ( ) str F. astman (Cornll) (198-) C 47 Spring 9 Farhan ana Cornll Unirsity ltron Transport Physis in Nanosal Systms Hyroynami an ballisti transport Quanti onutan Coulomb bloag o tunnling an singl ltron transistors Cohrnt arrir transport Unirsal onutan lutuations Intgr an rational quantum Hall ts Charg nsity wa an spin nsity wa transport nrson loaliation an wa loaliation Mtal-insulator transitions an Mott insulators Molular ltronis an polarons Strongly orrlat ltrons: Frromagnts, anitrromagnts, an high-t supronutors, spin liquis, topologial insulators C 47 Spring 9 Farhan ana Cornll Unirsity 1

2 C 47 Spring 9 Farhan ana Cornll Unirsity Conutors an Dissipation Traitional iw o Conutors: I G I Powr Dissipation in Conutors: &M (nrgy ontinuity quation) tlls us that th powr issipation pr unit olum o a rsistor is: Powr issipation in th ntir rsistor is: G I I P G 1 C 47 Spring 9 Farhan ana Cornll Unirsity Charatristi loity or Conution Currnt Dnsity: x y x x ˆ m n... m m 1 Consir a D soli in whih th nrgy isprsion or onution ban nar a ban minimum is gin by: Only ltrons los to th Frmi nrgy ontribut to th onutiity in mtals or haily op smionutors at low tmpraturs

3 Charatristi loity or Conution an Man Fr Path Charatristi loity: Th haratristi loity is th arag loity o thos ltrons that ontribut to th onutiity:. FBZ FBZ For mtals an haily op smionutors at low tmpraturs: F For low op smionutors at high tmpraturs: Man Fr Path: KT m Th man r path l is in as th arag istan an ltron trals bor it sattrs. It is gin by: whr is th sattring tim. C 47 Spring 9 Farhan ana Cornll Unirsity Ballisti ltron Transport Th lngth sals inol in th smallst transistors an nanosal is, suh as arbon nanotubs an molular onutors, an b small nough so that th ltrons o not sattr uring th tim it tas to tral through th i Intl s nm transistor Carbon nanotub transistor Singl atom transistor (Cornll) Ballisti Transport: Whn th lngth o th onutor boms muh smallr than th man r path l th transport is trm ballisti maning that th ltrons o not sattr uring th tim it tas to tral through th onutor Ballisti transport onition: Qustions: What happns whn << l? Th ormulas or onutiity that ha th sattring tim in thm ar larly no longr ali sin thr is no sattring: n What about issipation? m C 47 Spring 9 Farhan ana Cornll Unirsity

4 Ballisti ltron Transport in a 1D Conutor Consir a 1D onutor (xampl, a quantum nanowir) that is ontat at both ns by an xtrnal iruit. Sour Th isprsion rlation or th ltrons insi th quantum wir is: 1 m 1 m ssum only on subban Th ltron nsity (numbr pr unit lngth) is: n Th lngth o th wir is short nough suh that: C 47 Spring 9 Farhan ana Cornll Unirsity 1 ltri Fils, Chmial Potntials, an oltag Sours Now suppos a oltag sour is appli rom outsi: + - Sour In ltronis, on nr applis ltri ils nor n ltrostati potntial irns to iruits but only hmial potntial irns by using oltag sours Th oltag sour will rais th hmial potntial (or th Frmi ll) on on si o th onutor with rspt to th othr by an amount 1 1 C 47 Spring 9 Farhan ana Cornll Unirsity 4

5 ltron Currnts t th lt ontat, th urrnt u to ltrons moing in th right irtion is: I t th right ontat, th urrnt u to ltrons moing in th lt irtion is: I I + - I Sour rrows iniat th irtion o ltron low (not th irtion o urrnt, whih is opposit) 1 1 C 47 Spring 9 Farhan ana Cornll Unirsity ltrons o not sattr in th quantum wir. Thror: ll ltrons that ntr th wir rom th lt ontat ma it to th right ontat ll ltrons that ntr th wir rom th right ontat ma it to th lt ontat Total Currnt: Th nt urrnt is th sum o th urrnts u to th right-moing an lt-moing ltrons: Ballisti Transport I I 1 Sour I I I C 47 Spring 9 Farhan ana Cornll Unirsity 5

6 Ballisti Transport Conutan Total Currnt: I 1 1 ssum T K: I Conutan: I + - I Sour ry simpl I G G 1 1 C 47 Spring 9 Farhan ana Cornll Unirsity Th rlation: I G ins th quantum o onutan as: 5 G Q S Quantum o Conutan G Th quantum o rsistan is thror: I Q G Q I Sour Th Quantum o Conutan: Th quantum o onutan is th smallst possibl non-ro onutan o a ompltly ballisti onutor. quialntly, th quantum o rsistan is th highst possibl rsistan o a ompltly ballisti onutor. ll ompltly ballisti onutors (whthr in 1D, D, or D) will ha onutan that is in multipls o th quantum onutan alu (on an thin o ballisti onutan in D an D as a numbr o 1D onutors in paralll) C 47 Spring 9 Farhan ana Cornll Unirsity 6

7 Th rlation: I G Q suggsts that thr shoul b powr issipation in th onutor gin by: P I Q Th Qustion o nrgy Dissipation Q Q I + - I Sour But, as w ha sn, ltrons o not loos any nrgy in th onutor thy o not sattr thy go ballisti. So whr is th nrgy bing issipat? nswr: Th nrgy is issipat in th ontat not in th onutor! 1 1 ltrons loos nrgy an thrmali whn thy rah th ontat C 47 Spring 9 Farhan ana Cornll Unirsity Multipl Subbans: Quanti Conutan Subbans p, p p 1,, m I p1 p GQ Conutan inrass in multipls o G Q (quanti onutan!) I I Sour C 47 Spring 9 Farhan ana Cornll Unirsity 7

8 Quanti Conutan: xprimnts with 1D Quantum Wirs Smionutor Quantum Point Contats: ltrons ar onin in D in th quantum wll Ngati bias on mtal gats rpl ltrons rom unrnath th gats rating a narrow 1D hannl or ltrons in th spaing btwn th gats Th gat oltag an also ontrol how many subbans o th 1D hannl ar blow th Frmi ll Gas Quantum wll lgas gat sour rain gat G G Q Phys.. tt., 6, 848 (1988) Th onutan (an rsistan) is quanti so tily in Quantum Hall t that it an gi a alu o th Plan s onstant to on part in 1 8 g m C 47 Spring 9 Farhan ana Cornll Unirsity Sattring an Conutan in 1D What i thr is on sattrr (li an impurity atom) in th 1D hannl? Quantum Mhanial ltion an Transmission rom a Potntial Barrir: Consir what happns whn thr is a potntial barrir in th path o an ltron in a 1D quantum wir: r t t Si Solution: 1 1 m 1 m ight Si Solution: i i r x, y r x, y i r t x, y 1 1 Transmission probability: ltion probability: T t r 1T ssum thy ar nrgy inpnnt C 47 Spring 9 Farhan ana Cornll Unirsity 8

9 Conutan as Transmission: anaur s Formula I T T G T 1 T T anaur s ormula G T G Q T T 1 Us: +T = 1 Sour C 47 Spring 9 Farhan ana Cornll Unirsity Conutan as Transmission: Highr Dimnsions -imnsional matrial m T xy xy m T G -imnsional matrial x y xy m m x y T 1 x y x y T T T 1 at T~K xy y x C 47 Spring 9 Farhan ana Cornll Unirsity 9

10 Th ast Sli C 47 Spring 9 Farhan ana Cornll Unirsity 1