3ENX0 Nanomaterials: Physics and Characterization Examination, April 13, Exercise 1 Semiconductor quantum dots (10 points)

Transcription

1 3ENX0 Nanomateals: Pyscs and Caactezaton Examnaton, Apl 3, 05 Geneal gude lnes: Motvate you answes. Menton te equatons you use, defne paametes, and nclude ntemedate step n you calculatons. An equaton cat as well as ote useful nfomaton can be found on pages 5 7! Questons ae all n Engls. Mae you own coce as to answeng n Engls o n Dutc. Fnal scong: Maxmum numbe of ponts: 50. Fnal ma s scoed ponts / 5. Execse Semconducto quantum dots (0 ponts) Fgue Enegy levels n a quantum dot (not to scale). Consde semconducto quantum dots, n wc electons ae modeled as patcles n a -dmensonal box of wdt L and wt an nfnte bae egt. Te semconducto as a band gap of Eg, and we defne zeo potental enegy (E = 0) at te bottom of te conducton band. Electons n te conducton band ave an effectve mass m0. Electonc states n te conducton band ae labeled by ndex q, wee q = descbes te state wt lowest enegy see Fgue. a) Fnd an expesson fo te enegy of te electonc states n te conducton band as a functon of ndex q. [4 ponts] Holes n te valence band ae also descbed as patcles n a box wt nfnte baes. Te effectve mass s twce as lage as te mass of electons,.e. m0. Hole states ae labeled by ndex. Te quantum dots ae esonantly excted by oange lgt, weeby an electonc tanston fom = 3 to q = taes place. b) Fnd an expesson fo te poton enegy of te ultavolet lgt used. [3 ponts]

2 Te patcles dsplay lumnescence at a wavelengt of 85 nm (nfaed lgt) due to tanstons fom te = to q = state. Te band gap of te semconducto Eg s.40 ev. c) Calculate te damete of te patcle n unts of nanometes. [3 ponts] Execse Vaous nanosystems (0 ponts) Lftng a nanopatcle by adaton pessue An expement s done wt a stongly absobng nanopatcle n a vacuum cambe. An nfaed lase beam (poton enegy ev) snes fom te bottom, exetng an upwad foce on te patcle. Fo smplcty, we assume tat all lase lgt nteactng wt te 7 patcle s absobed. Te absobed lase powe s 0. W ( 0 Watt). Te densty of te patcle s 0 4 g/m 3. Hnt: te lase powe allows you to calculate ow many potons of ev ae absobed pe second. a) Mae a oug estmate of ow small te adus of te patcle sould be n ode to be lfted, by te lase lgt,.e., te wegt of te patcle s compensated by te adaton pessue. [4 ponts] States of a nanosystem A nanosystem s n nteacton wt a eat bat. Dffeent states ae labeled by ndex n =,, 3,. Te lowest state (gound state) as an ntenal enegy of 0.05 ev, and just possble ealzaton. State as an enegy of 0.07 ev and possble ealzatons. An expement s done at oom tempeatue. A eseace measues ove a longe tme n wc state te nanosystem s fequently undegong tanstons between dffeent states. Se fnds a tme-aveaged occupaton of 0% fo state and also 0% fo state (te sum of bot ealzatons). b) Calculate te tempeatue at wc te measuement as been pefomed. [3 ponts] Potoemsson fom a metallc nanopatcle A metallc nanopatcle s adated by ultavolet lgt wt wavelengt. Te wo functon of te patcles s W. An electon s poto-emtted fom te nanopatcle ognally te electon was n a state wt enegy E below te Fem level. c) Fnd an expesson fo te velocty at wc te electon s emtted. [3 ponts] Execse 3 Mcoscopy of semconductng nanowes (0 ponts) Two slcon nanowes ae gown wtn a MOVPE eacto. Te nanowes ave a cylndcal sape wt lengt L 0 m and damete d 50 nm, and ae gown exactly pependcula to te substate. Te sepaaton between te two wes s 00 nm. One of te two wes dsplays a small stuctual featue of nanometes. a) Explan wete () te sepaaton between te two wes and () te small stuctual featue can be esolved usng a. a egula, g-apetue, optcal mcoscope b. a scannng electon mcoscope (SEM) wt 5 V acceleaton voltage.

3 [4 ponts] b) Te wes ae doped by Alumnum (Z = 3) to mae tem conductve. Gve te electon confguaton of Al (n te fomat s n s m ), and ndcate wete conducton n te nanowe s due to electons o oles. [ ponts] 7 Te concentaton of dopant atoms s.00 0 cm -3. All dopant atoms ae fully onzed. Scatteng tmes of te caes s 5.0 ps. Te caes ave an effectve mass equal to tat of a fee electon. c) Calculate te esstvty of te we usng te Dude model equed constants can be found n te equaton cat. [ ponts] A Scannng Electon Mcoscope s used to mage te nanowes. An acceleatng voltage of 5 V and an electon cuent of na s used. Te focused electon beam s amed at te top of one of te two wes. At te bottom te wes n n contact wt a metallc substate, weeby a cuent flows toug te we. d) Calculate te voltage dffeence between te top and te bottom of te we due to te cuent tat flows toug t. [ ponts] Execse 4 STM on Cabon nanotubes (0 ponts) A scentst pefoms expements on a cabon nanotube. Te wo functon of te metallc STM tp and te cabon nanotube ae bot equal to W. Te Fem enegy of te metallc tp s E. A small electcal bas V s appled. Assume tat ev W and F ev E F. We fst consde a smple D model of electons tunnelng fom tp to nanotube. Electons wt enegy n te nteval EF E EF ev ae assumed to move nsde te tp and ave at te tp apex at a ate R 0 V (numbe of electons pe second). Tese electons ete eflect bac nto te tp, o tunnel to te nanotube. a) Vefy tat te wavefuncton nsde te bae can be wtten as ( z) A exp( z) wee z s te spatal coodnate as measued fom te tp apex, by sowng tat (z) s a soluton of te tme-ndependent Scödnge equaton. [3 ponts] b) Expess n tems of paametes le W, m, etc., and deve an expesson fo te tunnel cuent as a functon of tp-sample dstance d. [3 ponts] Te scentst wants to fnd out wete te studed nanotube s metallc o semconductng by pefomng STM n spectoscopy mode. c) Explan n wods ow ts metod wos, usng at least a sngle dagam, and sow ow te sgnal qualtatvely loos le fo a metallc and a semconductng nanotube. [4 ponts] 3

4 Execse 5 Optcal spectoscopy (0 ponts) Fgue Enegy level sceme of te molecula dye. A scentst pefoms a lase-nduced fluoescence (LIF) expement on a bologcal stuctue labeled by a specfc molecula dye. Fgue sows a level sceme of te electonc gound state S0 and fst excted state S of te dye, ncludng vbatonal sublevels. In you answe you ae allowed to assume tat te splttng of te sub levels s lage tan BT, wee T s te tempeatue at wc te expement s pefomed. a) Setc bot te absopton spectum and te emsson spectum of ts dye. Clealy state wat s plotted along te ozontal and vetcal axes. Use te paametes E, E, and E3 n you dagam. Befly explan you answe. [4 ponts] In 04 te Nobel Pze n Cemsty was awaded to Betzg, Hell and Moene fo te development of so-called supe-esoluton tecnques, bypassng te dffacton lmt. Two tecnques tat can be used fo optcal mages wt a esoluton bette tan te Rayleg cteon ae nea-feld scannng optcal mcoscopy (NSOM a tecnque wee a small apetue s used) and stmulated emsson depleton mcoscopy (STED a tecnque wee an exctaton lase wt egula Gaussan focus and a depleton lase wt donut-saped focus ae used). b) Befly explan te pncple of ete NSOM o STED (so you can coose wc one). [ ponts] Ote tecnques fo supe-esoluton use te fact tat te spatal pecson can be muc moe accuate tan te esoluton. c) Can suc a tecnque be used to esolve two dentcal molecula dyes, sepaated by 50 nm, and bot excted by te same focused lase spot? [ ponts] Te supe-esoluton tecnque s used wt a fluoescence detecton yeld of 0 0 potons pe second. d) How long do you ave to measue n ode to obtan a elatve accuacy of % n te numbe of detected potons? [ ponts] END OF EXAMINATION Equaton cat and fute nfomaton on next 4

5 Addtonal data Fo constants not mentoned (le atomc damete) you sould use a easonable value. Electon cage: Electon mass: Planc constant: e = C ~ 0-9 C m = g ~ 0-30 g = J s ~ 0-33 J s ħ = J s ~ 0-34 J s Boltzmann constant: B = J/K ~ 0-3 J/K Speed of lgt: c = m/s Acceleaton due to gavty at te eat s suface: g 9.8m/s ~ 0 m/s Vscosty of wate: wate = Pa s Equaton Cat Wee, Lectue Electc feld aound pont cage: e 3 E E BT 4 ( N N ) Fluctuatons: ms ~ N / N n Gmm Gavtatonal foce between pont masses: F m Dffacton: sn a Balme: R n DeBogle wavelengt: p Enegy and momentum of lgt and patcles: E f p Wee, Lectue qq Electostatc foce and enegy between pont patcles: F 4 0 Hesenbeg: p x E t 4 Tme-ndependent Scödnge equaton: 0 qq U 4 0 Wee, Lectue 5

6 m( V E) Tunnel pobablty: T exp L ( ) ( ) ( ) 3D Scödnge equaton: V ( ) ( ) E ( ) m x y z Soluton cubc potental well, wt cone at (0,0,0): ( x, y, z) Asn( xx)sn( y y)sn( z z). Atomc obtals: s, p, d, f, states. Wee, Lectue : Temo dynamcs: E Q W Foce on spee n vscous medum: F 6av Boltzmann dstbuton: / T P ) C exp ( g( ) P( E ) exp E Z Entopy as defned by Boltzmann: S( ) B ln( ) E Heat capacty: C V T Enegy of mass-spng system: E mx x Fee enegy: F E TS Fem-Dac and Bose-Ensten: n exp ( ) B E Relatve sze enegy fluctuatons: E Aveage dsplacement unde Bownan moton: E T E C B V x N Z exp E n exp ( ) BTt, wt 6a. Wee 3, Lectue Electcty: R L A I dq J E E V / L F qe R J I. A L A dt mv dvd vd ee Equaton of moton wtn Dude model: ee. dt m ne Cuent denstes and conductvtes: J nqv J J e J. m Fee electons: E F F EF vf. m* m m 6

7 Wee 3, Lectue In 04/05 Wee 4, Lectue Pont contact conductance: Wee 4, Lectue In 04/05 Wee 3, Lectue Dffacton lmt / Rayleg cteon: NA Wee 5, Lectue B Foce on magnetc patcle: F z m (4.35) z Optcal selecton ules: S 0 0 Wee 7, Lectue Molecula bology: DNA bases adenosne (A), tymne (T), guanne (G) and cytosne (C) ae fomng A-T and G-C ydogen bonded pas. RAP RP Magnetoesstance: MR RP Stablty tme supe-paamagnetc patcle: attempt exp( U / B T ) END OF Equaton cat and fute nfomaton 7