Lctur. Coulomb Blockad and Singl Elctron Transistors Anothr intrsting msoscopic systm is a quantum dot (.g., a conductiv or smiconducting grain). Whn th grain is small nough, th addition or rmoval of a singl lctron can caus a chang in th lctrostatic nrgy or Coulomb nrgy that is gratr than th thrmal nrgy and can control th lctron transport into and out of th grain. This snsitivity to individual lctrons has ld to lctronics basd on singl lctrons. For a rally small grain, th discrt nrgy lvls of th lctrons in th grain bcoms pronouncd, lik thos in atoms and molculs, so on can talk about artificial atoms and molculs. Bcaus of th quantum ffct, such grain is oftn rfrrd to as a quantum dot. Whn th wavfunctions btwn two quantum dots ovrlap, th coupld quantum dots xhibit th proprtis of a molcul. To dat, th lctron transport proprtis in a varity of quantum dot systms hav bn studid. Exampls includ smiconductor nanostructurs fabricatd with lctron bam lithography (split gat), a mtal nanoparticls sandwitchd btwn two lctrods or btwn a substrat and a STM tip and vn lctrochmical systms using nanolctrods. Fig.
Coulomb blockad and staircas Lt us considr a mtal nanoparticl sandwichd btwn two mtal lctrods (fig ). Th nanoparticl is sparatd from th lctrods by vacuum or insulation layr (such as oxid or organic molculs) so that only tunnling is allowd btwn thm. So w can modl ach of th nanoparticls-lctrod junctions with a rsistor in paralll with a capacitor. Th rsistanc is dtrmind by th lctron tunnling and th capacitanc dpnds on th siz of th particl. W dnot th rsistors and capacitors by R, R, C and C, and th applid voltag btwn th lctrods by V. W will discuss how th currnt, I, dpnds on V. Whn w start to incras V from zro, no currnt can flow btwn th lctrods bcaus mov an lctron onto (charging) or off (discharging) from an initially nutral nanoparticl cost nrgy by an amount of E = () C whr C is th capacitanc of th nanoparticls. This supprssion of lctron flow is calld Coulomb blockad, first obsrvd in th 6s by Giavr at GE in th tunnling junctions that contain mtal particls. Currnt start to flow through th nanoparticls only whn th applid voltag V is larg nough to stablish a voltag φ at th nanoparticls such that φ E =. () C This voltag is calld thrshold voltag and dnotd by V th. So in th I-V curv, w xpct a flat zro-currnt rgim with a width of V th. Whn th applid voltag rachs V th, an lctron is addd to (rmovd from) th nanoparticls. Furthr incrasing th voltag, th currnt dos not incras proportionally bcaus it rquirs us to add (or rmov) two lctrons onto th nanoparticls, which cost a gratr amount of nrgy. Onc th applid voltag is larg nough
to ovrcom th Coulomb nrgy of two lctrons, th currnt starts to incras again. This lads to a stpwis incras in I-V curv, calld Coulomb staircas. W can now look at th problm in a mor quantitativ fashion. First, lt us considr th lctrostatic nrgy rquird to char a nanoparticl with n lctrons. In th absnc of applid voltag, V, it is simply givn by ( n) E =. (3) C This formula has to b modifid for two rasons. First, a nanoparticl in a ral xprimnt is usually lctrically polarizd by surrounding random chargd impuritis, polar groups and countr ions. Th charg impurity can b modld by an offst charg, Q, in th nanoparticl. Bcaus th offst charg is originatd from polarization, it can tak any valu, and hnc, is usually not an intgr multipl of. Scond, in a ral xprimnt th nanoparticl is coupld with two lctrods and th lctrod-nanoparticl-lctrod systm is dscribd by two capacitors, C and C (s fig. ). Sing from th nanoparticl, C and C ar in paralll, so th quivalnt capacitanc is C +C (In th absnc of V, no currnt flows, so R, R and V can b takn out of th circuit in Fig. ). Whn both th offst charg and coupling with th lctrods ar takn into account, Eq. 3 bcoms ( Q n) E =. (4) ( C + C ) From Eq. 4, w can conclud that in th minimum nrgy stat, n taks an intgr valu, n, closst to Q / so that n. (5) Q
For if Q n >, say Q n =. 5, thn an lctron will transfr onto or from th dot, or n n ±, such that n. Q Scond, w considr th chang in lctrostatic nrgy of th systm whn an lctron is addd to th dot. Using Eq. 4, w hav E E = n+ E n = ( Q ( n + ) ) ( Q n) Q = (n + ( C + C ) ( C + C ) ( C + C ) ). (6) Whn Q =n=, that is whn th dot is initially nutral and has no offst charg, th nrgy chang is. Anothr intrsting spcial cas is whn ( C + C ) Q n =, E = - no Coulomb blockad! It is thus clar that th offst charg play an important rol in th lctron transport of th systm. In gnral, w hav to applid a larg nough voltag in ordr to ovrcom th Coulomb blockad and mov lctrons onto/off th particl. How larg voltag is ndd is what w want to dtrmin nxt. For a givn applid bias V, w nd first to dtrmin th potntial φ with rspct to ground at th quantum dot (th voltag across C ). This is a problm of two capacitors in sris, and w can asily show that C φ = V. (7) C + C As w incras V, φ incrass proportionally. Whn th associatd nrgy, φ, quals th charging nrgy givn by Eq. 6, an lctron is thn addd to th nanoparticl, which is dscribd by C C + C V = ( C + C Q (n + ) ) or
V th, = (n + ), (8) C whr V th, corrsponds to th thrshold voltag to mov an lctron onto th nanoparticl. If n= and Q =, th thrshold voltag for transfrring an lctron across C to th nanoparticl is /C. By th sam tokn, th thrshold voltag for transfrring an lctron across C to th nanoparticl is /C. So th obsrvd thrshold voltag is th lowr of /C and /C. Coulomb staircas occurs at th thrshold voltags for changing th avrag lctron numbr from n to n+. According to Eq. 8, th spacing btwn nth and (n+)th stps is /C if charging across C dominats or /C if charging across C dominats. Quantum confinmnt ffcts Th Coulomb blockad ffct discussd abov is purly du to lctrostatic nrgy of th xtra lctron on th nanoparticl. Howvr, according to quantum mchanics, confining an lctron insid of a nanoparticl costs also kintic nrgy. This can b undrstood by looking at an lctron in an infinit dp potntial wll of width L. Th lctron form standing wavs according to th Schordingr quation. Th longst wavlngth (λ) of th allowd lctron wavs is L, which givs ris to th lowst nrgy stat E p (πh / λ) ( πh) = = =, (9) m m ml whr p = π h / λ is th momntum of th lctron. This is minimum xtra nrgy that an lctron must hav to b addd to th nanoparticl. Th lctron can hav longr wavlngth, but it has to satisfy th following condition L λ =, () n whr n=,, 3,
So th allowd kintic nrgy is discrt and givn by E n ( πh) ml = n. () If w modl th nanoparticl with a thr dimnsional cub of dg lngth L, w can gnraliz Eq. to πh) En n n = ( n x + n y + n x y z ml ( z ), () whr nx, ny and nz tak th valus of,, 3, Whn w masur th currnt I as a function of bias voltag V, th currnt is initially small (with th xcption of Q =/) until V is incrasd abov th Coulomb blockad valu. Thn an additional rsonant tunnling channl opns ach tim th availabl nrgy from th voltag sourc passs through of th discrt nrgy lvls, which givs ris to nw stps in th I-V curvs. W can stimat th width of th stps without using Eq.. Th Frmi nrgy E F is E F h m / 3 4 / 3 / 3 = 3 π n, (3) whr n is th numbr of lctrons pr unit volum. From Eq.3 and n=n/v, w can s that def dn EF =. (4) 3 N So th spacing btwn th discrt lvls is roughly th Frmi nrgy dividd by th numbr of lctrons (atoms for monovalnt mtal) in th nanoparticl.
HW. - In ordr to obsrv Coulomb blockad at room tmpratur for a Au nanoparticl, what is th minimum siz of th nanoparticl? HW. - Show Eqs. 3 and 4. HW. -3 In ordr to obsrv quantum confinmnt ffct at room tmpratur for a Au nanoparticl, what is th minimum siz of th nanoparticl?