(Semi)Classical thermionic emission
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1 unnling - primr Nno oftn pprs in rl tchnology in th form of thin lyrs or brrirs. W r going to look t svrl wys lctrons cn trnsport ovr or through ths brrirs undr vrious conditions. hrmionic mission (clssicl ovr-brrir Pool-Frnkl bhvior (hopping out of tiltd potntil wlls unnling, rctngulr brrir (simpl quntum problm unnling, gnric brrir Fowlr-Nordhim tunnling (tringulr brrir - fild mission Rsonnt tunnling (SmiClssicl thrmionic mission E φ b smiconductor, voltg 0. mtl, voltg V V bi Distribution of incidnt prticl nrgis. hos with high nough nrgy cn mk it ovr clssicl brrir. Nt currnt dnsity J s-m J m-s
2 hrmionic mission: pplictions hrmionic mission oftn govrns: Elctron injction into smiconductors from mtls. Emission of lctrons from hot mtrils into vcuum (th filmnt in vry cthod ry tub, ion gug, tc. Prtly rsponsibl for mission in thrml fild mission sourcs (lctron microscopy. hrmionic mission: clcultion smiconductor to mtl: dn( E ν 3d ( E f ( E, de Assum high (nondgnrt / 4 (m π * E Ec xp( ( E Ec Vn / k de 3 h Rwrit in trms of vlocitis (smiclssicl pictur: 3 m* V m* v n xp xp (4πv dv h k k Add up currnt hding towrd brrir, knowing tht minimum vlocity to gt ovr brrir t bis V is: m* v0, x ( Vbi V
3 hrmionic mission: clcultion Rsult is, skipping som lgbr, J sm πm k 3 h xp k 4 * φ V xp k A *, Richrdson constnt. Hding th othr dirction, on finds J Nt rsult: ms A * φ xp k J tot φ xp V A * xp k k hrmionic mission: dtils J tot φ xp V A * xp k k mprtur dpndnc is strong. Exponntil voltg dpndnc t high bis. Othr prturbing ffcts (som tunnling contribution, for xmpl cn cus msurd A * vlu to dvit from thory prdiction. siclly just clssicl sttisticl mchnics ffct. 3
4 Strongly disordrd systms In strongly disordrd systms, th loch wv pictur brks down. Wvfunctions r loclizd, dcying xponntilly wy from trp sits (potntil wlls. Conduction is du to hopping from loclizd sit to loclizd sit. Hopping cn b thrmlly ctivtd (though it cn lso b du to tunnling. Applying lrg lctric fild cn ffct thrml hopping rts: Pool-Frnkl bhvior Lowring of hopping brrirs in lrg lctric filds is clld Pool-Frnkl bhvior. Assum potntil wll is lctrosttic: V ( r 4πεε r Potntil mximum occurs whn xtrnl fild nd trp fild blnc: 4πεε 0rmx F r mx Amount brrir potntil is lowrd: Fr mx 4 πεε 0F 4 πεε 0F / / 0 4
5 Pool-Frnkl bhvior If hopping rt is ctivtd, σ ~ xp[ U 0 / k ] Lowring of brrir mns 3 σ ~ xp[ ( F / 4πεε / 0 / k ] Scling of conductnc with nd F l this is th signtur of Pool-Frnkl bhvior. Prvlnt in orgnic smiconductors, morphous Si, tc. unnling through rctngulr brrir A F G h m* ( z z V ff - ( z φ ( z Eφ( z φ φ( z, m* ( z z continuous. φ( z A C F z z z D G z z z z < < z < z > Do cs whr E < V 0. 5
6 6 unnling through rctngulr brrir Applying boundry conditions, ] [ ] [ D C A D C A ] [ ] [ G F D C G F D C D C A ( ( ( ( G F D C ( ( ( ( unnling through rctngulr brrir Combining, on gts G F M M M M A whr lmnts com from multiplying mtrics on prv. slid. Incidnt flux: * m k A f inc h trnsmittd flux: * m k F f trn h rnsmission cofficint: ( ( sinh ( M A F f f E k k inc trn Limit of wid brrir: h / ( * 4 ( E V m k k E
7 unnling through rctngulr brrir Rflction cofficint -, unsurprisingly. Cn lso do cs whr E > V 0 : ( E whr k' m ( * E V0 k ( k ' sin (k' kk' Combind pictur: img from Frry unnling through gnric brrir: th WK pproximtion clssiclly forbiddn x x dfin locl ffctiv wv vctor - imginry in clssiclly forbiddn rgion. x ( E xp h x { m ( V ( x E } / dx * vlid whn turning points r fr prt (mny wvlngths. vlid whn V(x vris slowly (comprd to wvlngth. 7
8 Rlistic tunnling probbilitis As w sw from rctngulr brrir cs, tunnling only occurs with significnt probbility t vry short lngth scls (vry nrrow brrirs. unnling is xponntilly supprssd with brrir width. Fctoid: mtl-mtl tunnling currnt in vcuum (or ir dcys with distnc roughly l ordr of mgnitud pr Angstrom (!. On of your homwork problms: wht is th probbility of Volkswgon tl tunnling through spd bump? A tringulr brrir: Fowlr-Nordhim tunnling Cn pull brrir down by pplying lctric fild. Fowlr-Nordhim limit: brrir gts thin nough to llow tunnling. E F φ b -Fx Anlyticl xprssion cn b obtind using WK, s long s brrir dosn t gt too thin: ( E ( V ( x E φ φ / F xp h 0 xp 4 3 { m ( φ Fx } * / (m* φ hf Fx 3/ / dx 8
9 Doubl brrirs A F G A F G W cn sily considr th cs of two brrirs in row. his cs gts prticulrly intrsting if th brrirs r sufficintly clos tht thr r bound stts in th spc btwn th brrirs tht hv nrgis clos to thos of incidnt lctrons. Mtrix-wis, w know w cn combin th two singl-brrir problms. Doubl brrirs A F G A F G - z0 b Nd to rlt phss of A, to F, G: A' b F, ' G b Dfin trnsfr mtrix: F G 0 b 0 A' M b ' W A' ' Combind systm: A M LM W F' M R G' whr th M L nd M R r mtrics for ch brrir individully. 9
10 Rsonnt tunnling Writ th combind mtrix s M tot M L M W M R, nd writ th mtrix lmnts s mgnitud nd phs. Finl rsult: tot ( E M 4R cos ( kb θ tot k θ rctn tnh( k k rnsmission through th whol structur hs rsonncs! hs corrspond to css whn th nrgy of th incidnt prticl coincids with bound stt nrgis of th wll. Incidnt rflctd wvs intrfr constructivly in wll! At rsonnc, tot. Minimum trnsmission: tot ~ /4. Rsonnt tunnling Wht bout symmtric cs? Rlvnt t finit bis: Expct to s pk in trnsmission whn bis tilt ligns bound lvl with sourc or drin nrgy lvls. At highr bis, xpct trnsmission to drop gin s rsonnc condition vnishs. Prdicts NDR! 0
11 Rsonnt tunnling Algbr gts bit mssy. img from Frry Rsonnt tunnling diod Finl rsult: tot ( E ( R R 4 R R θ Φ k b Agin, t rsonnc Off rsonnc L θ R θ L cos ( Φ θ R 4 rs for. ( off 4
12 Rsonnt tunnling diod Dos this ctully work? GAs RD, AlGAs brrirs, b 5 nm. img from Frry A vry subtl qustion A philosophy qustion worth pondring, vn though it s not dirctly grmn to th cours: How long dos th tunnling procss tk? ht is, for prticls tht succssfully trvrs brrir, how long r thy in th clssiclly forbiddn rgion? Considr incidnt Gussin wvpckt trnsmittd Gussin wvpckt. Surprising nswr: msuring positions of pks of wvpckt, tunnling vlocity cn grtly xcd c! For mor informtion, rd Lndur nd Mrtin, RMP 66, 7 (994.
13 o summriz: hrmionic mission clssicl thrml ovr-brrir Pool-Frnkl fild-ssistd clssicl thrml hopping Singl-brrir tunnling is strightforwrd. Gnric brrirs: WK pproximtion Fowlr-Nordhim fild-ssistd tunnling Doubl brrirs: must ccount for intrfrnc Rsult: rsonnt tunnling diods w/ NDR Nontrivil intrprttion issus ssocitd with tunnling! Nxt tim: Scttring mtrics Lndur-uttr formlism - conductnc s trnsmission. 3
14 h story so fr: Smiclssicl trnsport prtty good t duplicting xprimntl rsults in rgim of vlidity. Quntum cohrnc lds to intrfrnc ffcts tht cn domint in smpls ~ siz of cohrnc lngth. Cn think of conductnc s trnsmission in this quntum limit. Spcific xmpl: whn considring tunnling conduction through both singl- nd doubl-brrir structurs. In tht cs, cn us mtrix to connct mod mplituds on on sid of smpl to mod mplituds on othr sid of smpl. unnling rvisitd: th scttring mtrix Slightly diffrnt wy of formulting trnsmission problm. Instd of M mtrics rlting mplituds on diffrnt sids of smpl, us S mtrix - rlts incoming mplituds to outgoing ons. F A G - A M M M M F G S F S S S A G LR ( E S R LR ( E S RL ( E S R RL ( E S
15 h scttring mtrix In symmtric cs, cn hv diffrnt vlocitis on L nd R sids. Rsults bcom: v v LR ( E S RL ( E S v v Common wy to dl with this: rscl ch mtrix lmnt: vn Snm Snm vm Rsult: S mtrix is now unitry: S S SS Consrvtion of probbility is th rsult. Lndur-uttr formlism A gnrl pproch to undrstnding conduction proprtis of smll, (nonintrcting quntum cohrnt systms connctd vi contcts to clssicl (dcohring rsrvoirs tht cn srv s currnt sourcs/sinks or voltg (chmicl potntil probs. Commonly usd to nlyz mny molculr lctronics xprimnts. Cn b modifid to includ lctron-lctron intrction ffcts. Mks rmrkbl prdictions confirmd by xprimnts, indpndnt of th microscopic dtils of smpls or msurmnts. prsnttion ftr tht of Dtt nd Frry.
16 llistic conductors Lndur (959 strtd xmining th qustion of bllistic conductor. Is th conductnc of such n objct ctully infinit? No - xprimntlly, limit of conductnc rmins finit vn s w, L << l. contct w contct L -trminl conductnc must b limitd by contct rsistnc. Clculting th currnt Lt s work ssuming nrrow (d conductor, for s. Ech trnsvrs mod hs som minimum nrgy: ε N E N ( k 0 Numbr of mods t nrgy E M(E counts thos with ε N < E. M ( E ϑ( E ε Occuption probbility of stt in lft rsrvoir f (E Currnt crrid lft to right: I L L k k v( E f ( E M ( E E f h k ( E M ( E N L dk k π I f ( E M ( E de h ottom lin: currnt crrid pr mod pr unit nrgy /h N 3
17 h -trminl Lndur formul Nt currnt rsults whn chmicl potntil of L nd R contcts r diffrnt! Mor occupid mods going L-R thn R-L, for xmpl. V I ( µ µ M ( µ µ M h h So, conductnc of M chnnl bllistic conductor G c M h Sinc th conductor is bllistic, th rsistnc (/G must b contct rsistnc, ssocitd somhow with th intrfcs btwn th contcts nd th bllistic wvguid. Conductnc quntiztion h impliction of th Lndur formul is, if w could dform bllistic conductor to vry th numbr of mods with cutoff nrgis blow E F, w should s stps in th conductnc. GAs/AlGAs dg 0.6 K n d 3.6 x 0 / cm Vn Ws t l., PRL 60, 848 (988 4
18 Conductnc quntiztion Cn hppn in th cs of rl mtls, too: From Rd, Yl From vn Ws, Groningn Mtl brk junction Au constriction nrrows nd brks s wfr is bnt. Rsult: conductnc quntiztion s cross sction pprochs tomic scl (d /k F. Quntum ffcts domint vn t room tmprtur! Whr is th voltg droppd? µ contct contct Right-moving crrirs r t chmicl potntil of lft sid until thy rch right contct. Lft-moving crrirs r t chmicl potntil of right sid until thy rch lft contct. old curv rprsnts vrg of chmicl potntil. Looks gin l it s contct rsistnc. Actul lctrosttic potntil smrd out by scrning. 5
19 Rflctionlss contcts Origin of contct rsistnc: contcts contin lrg numbr of mods, whil only fw cn gt through th conductor. Assums contcts r rflctionlss - ll th k crrirs com from th lft contct, nd hv chmicl potntil µ, rgrdlss of wht right contct is doing. Sm for -k crrirs, only for right contct. Worth considring contcts, lds, nd thn th conductor, llowing rflctions from conductor: contct contct ld ld Nonbllistic clssicl conductors: Ohm s Lw? If w join conductors togthr in sris clssiclly, w xpct to find Ohm s lw. Cn w gt this rsult? Considr two conductors in sris: R R Dirct trnsmission: (? his ignors multipl rflctions. Crrir could bounc off nd conductor, thn off st conductor, thn b trnsmittd. R R R R... R R 6
20 Ohm s Lw Rwriting in trms of s, Impliction for N idnticl scttrrs: ( N N( If thr r ζ scttrrs pr unit lngth, thn L0 ( L L L 0 L0 ζ ( For lrg L, this givs Ohmic vrition, s xpctd. Sinc (-/ is dditiv, suggsts tht rsistnc pr scttrr is proportionl to tht quntity for ch scttrr. Rwriting our Lndur formul, G h h h M M M contct pic scttrr pic mn fr pth. Enrgy distribution of lctrons contct contct ld ld µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ - -k stts k stts x 7
21 Whr is th ht dissiptd? µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ W v ssumd nrgy rlxtion tks plc without switching dirctions of crrirs - not lwys tru. Nonthrml distributions thrmliz on som nrgy rlxtion lngth scl. For cs bov, tht hppns in th lds. For cs of point contct, tht hppns in th contcts. Min id: gt nonthrml lctronic distributions; ht dissiption hppns whn lctronic distributions cn rlx. ht s why,.g., bllisticlly conducting nnotubs cn hndl high currnt dnsitis! h corrct conductnc formul. G h h h M M M contct pic dvic pic Sprting rsistncs into contct nd dvic pics. Not tht if scttring is zro, nd th dvic contribution vnishs, s xpctd - tru bllistic conductor (just th conductor! hs infinit two-trminl conductnc. Wht s going on hr? contct contct ld ld 8
22 h corrct conductnc formul contct contct ld ld µ µ µ A µ µ G h dvic M µ A µ I Chmicl potntil in lds diffrs from tht in contcts du to scttrd lctrons. wo-trminl vs. four-trminl If th rsistnc from th two-trminl Lndur formul is rlly contct rsistnc, on might think of prforming 4-trminl msurmnt. ht is, cn on ctully sns th dvic rsistnc contribution, which should truly go to zro for bllistic dvic? Complictions: Voltg probs cn prturb currnts in dvic Voltg probs my not coupl to k nd -k mods qully. contct contct ld ld 9
23 Four-trminl probs P P Voltg msurd (idl: µ P µ P ( ( µ µ Currnt: I M ( µ h 4 rsistnc: R 4 ( µ P µ P I µ / h M contct contct - -k stts ld ld µ k stts Wht if lft prob only coupld to lft-moving stts, nd vic-vrs? x µ µ P P ( ( µ µ, ( µ µ h R 4 M Could vn b ngtiv! wo-trminl vs. four-trminl - xprimnt d Picciotto t l., Ntur 4, 5 (00. Clvd-dg ovrgrowth tchniqu to produc quntum wir with d contcts. Gts usd to crt d wir rgions t dg of wfr. -trminl conductnc shows quntizd conductnc, but with ~
24 wo-trminl vs. four-trminl - xprimnt Cn crfully tun invsivnss of voltg probs. Comprison of - trminl vs. 4-trminl rsistnc shows tht, for idl voltg prob coupling, on rlly dos msur dvic rsistnc of zro in th bllistic cs! o summriz: Scttring mtrix (rlting ingoing nd outgoing fluxs is n ltrntiv wy of dscribing conduction of quntum scttring systms. Lndur formul: -trminl conductnc of som quntum cohrnt systm coupld to clssicl rsrvoirs is givn by ( /hσ M, whr M is th numbr of chnnls nd is th trnsmission cofficint of ch chnnl. Lck of inlstic scttring mns nonthrml lctronic distributions nd subtltis in figuring out voltg drops. Finit conductnc of bllistic systm is rsult of contct rsistnc. Propr 4 conductnc of bllistic systm rlly dos pproch infinity.
25 Nxt tim: Accounting for Coulomb intrctions uttr formul - multitrminl cs Finit tmprtur nd bis
(Semi)Classical thermionic emission
Tunnling - primr Nno oftn pprs in rl tchnology in th form of thin lyrs or brrirs. W r going to look t svrl wys lctrons cn trnsport ovr or through ths brrirs undr vrious conditions. Thrmionic mission clssicl
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