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 ovr-brrir Pool-Frnkl bhvior hopping out of tiltd potntil wlls Tunnling, rctngulr brrir simpl quntum problm Tunnling, 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. Thos with high nough nrgy cn mk it ovr clssicl brrir. Nt currnt dnsity J s-m J m-s
Thrmionic mission: pplictions Thrmionic 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.
Thrmionic mission: clcultion smiconductor to mtl: dn E ν d E f E, T de 3 Assum high T nondgnrt 1/ 4 m π * E Ec xp E Ec Vn / kt de 3 h Rwrit in trms of vlocitis smiclssicl pictur: 3 m* V m* v n xp xp 4 v dv h π kt kt Add up currnt hding towrd brrir, knowing tht minimum vlocity to gt ovr brrir t bis V is: 1 m* v0, x Vbi V
Thrmionic mission: clcultion Rsult is, skipping som lgbr, J sm πm k 3 h T xp k 4 * φ T xp V k T A *, Richrdson constnt. Hding th othr dirction, on finds J Nt rsult: ms A T * xp φ kt J tot φ xp V A * T xp 1 kt kt
Thrmionic mission: dtils J tot φ xp V A * T xp 1 kt kt Tmprtur 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.
Strongly disordrd systms In strongly disordrd systms, th loch wv pictur brks down. Wvfunctions r loclid, dcying xponntilly wy from trp sits potntil wlls. Conduction is du to hopping from loclid sit to loclid 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 4πεε0r Potntil mximum occurs whn xtrnl fild nd trp fild blnc: 4πεε r 0 mx F r mx Amount brrir potntil is lowrd: 4 πεε0f 1/ Fr mx 4πεε F 0 1/
Pool-Frnkl bhvior If hopping rt is ctivtd, σ ~ xp[ U 0 / kt ] Lowring of brrir mns σ ~ xp[ 3 F / 4πεε 0 1/ / k T ] Scling of conductnc with T nd F l this is th signtur of Pool-Frnkl bhvior. Prvlnt in orgnic smiconductors, morphous Si, tc.
Tunnling through rctngulr brrir A - * E V m ff φ φ m φ φ 1, * continuous. G F D C A > < < < φ F G Do cs whr E < V 0.
Tunnling 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
Tunnling through rctngulr brrir Combining, on gts G F M M M M A 1 1 11 whr lmnts com from multiplying mtrics on prv. slid. Incidnt flux: * m k A f inc trnsmittd flux: * m k F f trn Trnsmission cofficint: sinh 1 1 1 11 M A F f f E T k k inc trn Limit of wid brrir: / 4 4 0 * 4 E V m k k E T
Tunnling through rctngulr brrir Rflction cofficint 1 - T, unsurprisingly. Cn lso do cs whr E > V 0 : 1 T E ' 1 whr k' m * E V0 k k sin k' kk ' Combind pictur: img from Frry
Tunnling through gnric brrir: th WK pproximtion clssiclly forbiddn x 1 x dfin locl ffctiv wv vctor - imginry in clssiclly forbiddn rgion. T E x xp x1 { m V x E } 1/ dx * vlid whn turning points r fr prt mny wvlngths. vlid whn Vx vris slowly comprd to wvlngth.
Rlistic tunnling probbilitis As w sw from rctngulr brrir cs, tunnling only occurs with significnt probbility t vry short lngth scls vry nrrow brrirs. Tunnling is xponntilly supprssd with brrir width. Fctoid: mtl-mtl tunnling currnt in vcuum or ir dcys with distnc roughly l 1 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: V x E φ Fx T E xp xp 4 3 φ / F 0 1/ m* F { m φ Fx } * φ 3/ 1/ dx
Doubl brrirs A F A F G G W cn sily considr th cs of two brrirs in row. This 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 F G F G A A b 0 - Nd to rlt phss of A, to F, G: b b G F A ', ' Dfin trnsfr mtrix: ' ' ' ' 0 0 A A G F W b b M Combind systm: ' ' G F A R W L M M M whr th M L nd M R r mtrics for ch brrir individully.
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: T tot 1 T1 E M T1 4R1 cos kb θ tot11 k θ rctn tnh k k Trnsmission through th whol structur hs rsonncs! Ths 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, T tot 1. Minimum trnsmission: T tot ~ T 1 /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!
Rsonnt tunnling Algbr gts bit mssy. img from Frry
Rsonnt tunnling diod Finl rsult: T tot E 1 R R 1 T T 1 4 R R 1 cos Φ Φ θ k b 1 L1 θ R1 θ L 11 θ R11 Agin, t rsonnc Off rsonnc 4T1 T T rs 1 for T T T 1 T. 1 T off T T 1 4
Rsonnt tunnling diod Dos this ctully work? GAs RTD, 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? Tht 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, 17 1994.
To summri: Thrmionic 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.