eue 9. anspo Popeies in Mesosopi Sysems Ove he las - deades, vaious ehniques have been developed o synhesize nanosuued maeials and o fabiae nanosale devies ha exhibi popeies midway beween he puely quanum behavio of aoms and moleules and he lassial behavio of bulk o maosopi maeials. Suh sysems ae ofen efeed o as mesosopi sysems o emphasize he fa ha hey ae inemediae beween miosopi and maosopi sysems. A ypial size of a mesosopi sysem is aound - nm, whih is lage han he miosopi o aomi size. nm, and smalle han he maosopi size whih is geae han µm. Some of basi oneps developed fo mesosopi sysems ae useful fo us o undesand anspo popeies in moleula sysems. One example is andaue fomula ha we disussed in leue 8. Hee we disuss a few moe basi oneps. Chaaeisi lenghs In a lassial (maosopi) meal wie, he onduion eleons expeiene muliple diffusive saeings when hey avese hough he wie. he esisane,, of he wie is given by, ρ () A whee ρ is esisiviy, and A ae he lengh and he oss seional aea of he wie, espeively. he invese of in Eq. is defined as he onduane whih is elaed o onduiviy σ/ρ by A G σ. () In he lassial wie, he onduiviy, σ, is a physial popey ha depends on he maeial of he wie, bu i does no geneally depends on he lengh of he wie. When deeasing he lengh below
he eain haaeisi lenghs, σ depends on he lengh due o quanum effes. In he quanum egime, he eleons a like waves ha exhibi inefeene effes, depending on he bounday ondiions, impuiies and defes in he sysems. In ode o undesand he quanum phenomena, le us sa wih seveal haaeisi lenghs. In mesosopi sysems, he following hee haaeisi lenghs have been used. he fis one is he Femi wavelengh, λ F, he de Boglie wavelengh of eleons a he Femi enegy. he seond is he momenum elaxaion lengh, m, whih desibes he aveage disane an eleon an avel befoe olliding wih impuiies o defes. his is ofen efeed o as eleon mean fee pah. he seond haaeisi lengh is he phase-elaxaion lengh, φ, whih is he lengh ove whih an eleon eains is oheene as a wave. his lengh ells us how well defined is he phase of he eleon waves along he popagaion pahs. Wihin he phase-elaxaion lengh (o oheene lengh), he eleon waves an inefee wih hemselves and wih ohe eleon waves, like ligh waves. Fo an elasi saeing in whih eleons do no lose o gain enegy, a phase shif of he wavefunion an be expeed afe he elasi saeing bu he saeed wave emains oheen. So elasi saeing does no onibue o φ bu only o m. If he saeing esuls in a hange in he eleon enegy by eihe exiing oe eleons, phonons o spins assoiaed wih magnei impuiies, he phase of he eleon wave is los in addiion o a hange in he eleon momenum. hus inelasi saeing onibues boh o φ and m. e s now onside saeing evens beween eleons. In geneal, he saeing beween wo valene eleons does no onibue o m bu only o φ beause of he following onsideaions. Afe wo eleons ae saeed by hei muual Coulomb poenial ha is dynami, he phase infomaion of he wavefunion as a funion of he posiion is los. ha is why he eleon-eleon saeing deeases φ. On he ohe hand, when we onside N eleons in he
Femi sphee, whee eleons oupy saes fom he boom of he valene band all he way o he Femi enegy, he saeing beween wo of he N eleons will NO hange he eleoni sae of he N-eleon sysem simply beause eleons no disinguishable aoding o quanum mehanis. In ohe wods, hee is no hange in he Femi sphee due o he saeing. In expeimens whee only eleons nea he Femi enegy onibue o he anspo poess, m and φ an be elaed o m and φ, alled mean momenum and phase elaxaion imes, espeively. In ems of m and φ, a signifianly hange means, fo example, ha he sum of he hanges in momenum and phase ove he imes m and φ eah h k F and π, espeively. he above haaeisi lenghs an be hanged by vaying expeimenal paamees o by hanging he maeial popeies. Fo example, by loweing he empeaue he exiaion of phonons an be suppessed, whih an inease boh m and φ. In elasi and magnei saeing, he onenaions of ioni and magnei impuiies vay fom sample o sample, whih may also be onolled. Fuhemoe, he magnei saeing an be suppessed by a magnei field. he eleon-eleon ineaion is deemined by he aie onenaion and he band widh of he valene band. Finally, he fequeny of he all saeing poesses an be hanged by hanging he Femi veloiy, whih is implemened by applying a gae volage o he semionduo-meal junion. So we an sudy diffeen anspo popeies and mehanisms by pepaing maeials and by onolling exenal expeimenal paamees. Using he hee haaeisi lenghs, we an divide eleon anspo in mesosopi sysems ino hee egimes: ballisi, diffusive and lassial anspo. Ballisi anspo ous wih no signifian momenum and phase elaxaion. Diffusive anspo involves many elasi saeing evens, bu he eleons emain oheen (no subsanial phase elaxaion). Classial anspo is he ase whee boh eleon momenum and phase elaxaion ou fequenly and hus
an eleon an be onsideed as a paile. he esisane of a lassial onduo obeys he simple Ohm s law. Ballisi, << m and φ his is wha we have disussed in eue 8 and he onduane of suh a sysem is desibed by he andaue fomula, e e G ij ( EF ) MΤ, (3) h h i, j whee M is he numbe of hannels (ansvese quanum modes) beween he Femi levels of he lef and igh eleodes, and is he ansmission pobabiliy fom he lef o igh eleode via all he onduion hannels, given by he esisane of he sysem is ij ( E F ). (4) j h. (5) e MΤ his esisane inludes boh he esisane of he mesosopi onduo (wie), w, and he ona esisane,, beween he wie and he leads needed fo measuemen. he lae one ( ) is h, (6) e M ha is o say if he wie is ideal (), all he esisane muh be due o ona esisane. So he wie esisane is given by w h h h. (7) e MΤ e M e his sepaaion beween ona esisane and wie esisane is quesionable fom boh expeimenal and heoeial onsideaions. Expeimenal, one simply anno pefom he
measuemen wihou leads o onas, so i is no possible o measue he wie and ona esisane independenly. Fom heoeial poin of view, he oal esisane of wo quanum esisos in seies may no be a simple summaion of wo esisos. Classial, >> m >> φ In a maosopi onduo, he eleon wavefunion is no longe oheen houghou he enie sample, so one anno solve he Shodinge equaion ove he enie sample. When we negle he inefeene effe duing a seies of N saeing evens, he oal ansmission pobabiliy is given by he summing ove eah ansmission i and eah efleion i - i, whee i,, N. e us onside wo suh saeing even, he oal ansmission pobabiliy is (...). (8) Eq. 8 an be e-wien as. (9) Aoding o Eq. 7, we have. () If we genealize he above esuls o N saeing evens, he oal esisane is a summaion of N miosopi esisanes, eah is deemined by a saeing even. his is wha Ohm s law saes he esisane of a maosopi onduo is popoional he lengh. oalizaion, > φ >> m We assumed above ha he phase beomes andom afe a saeing even fo a lassial onduo. When φ >> m, he phase of he eleon wave an shif by an amoun of θ bu no
omplee los (andom). In his ase, we need o onside ansmission oeffiiens,, in ode o alulae ansmission pobabiliy, ( ). Again, le us onside wo saeing evens, eah wih a ansmission oeffiien, and, espeively. he oal ansmission oeffiien is given by iθ e, () whee,, ogehe efleion oeffiiens, and fom he so-alled S-maix. he ansmission pobabiliy is os e i θ θ. () he esisane is obained by aveaging ove θ suh ha <osθ> and we an show ha / <, (3) whee h/e.9 kohm, he esisane quanum. I is lea ha is no a simple summaion of and, so he ohm s law is no longe valid. he non-addiive em in Eq. 3 auses an exponenial divegene fo as a funion of, whee () is defined by a sep of ~ m. Eq. 3 leads o. ) ( ) ( d d (4) he soluion of Eq. 4 is a e, ] [ /. (5) So he esisane is vey lage fo lage so ha he maeial beomes an insulao. his is known as he loalizaion phenomenon. I is ineesing o poin ou he exponenial dependene of he
esisane given by Eq. 5 esembles ha esisane of a moleule given by he supeexhange heoy. HW. 9- Show ha he esisane of a lassial onduo due o N saes is given by w N h i e h ~ N e. i i HW. 9- Deive Eqs. 4 and 5 fom Eq. 3. Assuming ha (), () and ( )( ).