Transport electronique
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- Gavin Briggs
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1 rasport lctroiqu Josph P. Hras Dpartt o Mchaical ad Arospac girig Dpartt o Physics h Ohio Stat Uirsity, Colubus, Ohio 4, USA Hras.@osu.du s ots sot aglais, ais j parlrai racais i accts, i cdills Rrcs: His t Ur, hrolctricity: Scic ad girig, Itrscic, Nw Yor 96 Hras, hrush & Morlli, Phys. R Chasar t Stratto, J. lctroics ad Cotrol J. M. Zia, lctros ad Phoos, Clardo, Oord 96, rprit 97 col CNRS, Vtro p
2 S ut du cours A C probl a rsoudr: optiisr ds quatits cotr-idiqus P S S q S q Cas possibls: A. Mthod d Chasar t Stratto a du coicits S t. Mthod a trois coicits: Hall, S t.6 S, Rlatio d Pisaro S S. Mthod a quatr coicits: Hall, Prir chatillo Nrst, S t? Dopag optial
3 Structur du cours partis. Itroductio. rasport das u ga parait d lctros : quatios ituitis. quatio d olta 4. s quatios d trasport orlls oralis tps d rlaatio 5. Optiiatio d u atriau throlctriqu quatios pratiqus 6. chap agtiqu: ts galaoagtiqus t throagtiqus Si l tps prt: 7. Gralisatio: trasport quad sul la dsit d tats st dii solids dsordos, aorphs: quatio d Mott t ss dris 8. Rlatios d Osagr graliss au atriau agtiqus col CNRS, Vtro p
4 . Itroductio: Osagr hoas J. Sbc 77-8 J.C. Athaas Pltir Sbc ct S - V thropowr S V Sbc coicit = throlctric powr = thropowr
5 Rrsibl thro: hropowr = lctro tropy S dvoltag dvoltag G H S Voltag dvoltag d Kli s Rlatio lctrochical pottial o lctro: d S NROPY S p d U du - pv pdv Sbc S Vdp S ds S d V NROPY
6 _ HA HO COD hrolctric Powr Grator Idal rrsibl grator = Carot Joul Hat, iit hral Coductiity H C H C H C N P C H C C H C igur o Mrit
7 Osagr gralis Oh ad ourir laws j, q j V q is atisytric J Q Q IRRVRSI JOU j V hropowr prits: sapl draws o currt, but w apply ad q lows q V S q V V S V
8 . Ituiti iw o trasport proprtis lctros as a idal gas o chargd particls h Drud odl Hits a roadbloc i that it prdicts a -idpdt Sbc coicit, Whil th rd pricipl o throdyaics stats that li S Still ry usul to plai th cocpts col CNRS, Vtro p 8
9 Phas spac. ach particl has positio r =,y,. ach particl has locity =, y,. Assu particl locitis ar distributd uiorly 4. Nubr o locity ctors = ubr o particls N 5. Ral-spac olu: y V d. dy. d 6. Norali pr uit ral olu V: coctratio =NV 7. Di Phas Spac as, y,, with olu d. d. d y Positio:, y, Vlocity:, y, col CNRS, Vtro p 9
10 Phas spac olus: carthsia ad sphrical coordiats Ca do this or both particl:. positio i ral spac r =,y,. locity i locity spac =, y, Not: drawigs ar or r, but sa holds or col CNRS, Vtro p
11 ri suracs Rprstatios o qui-rgtic suracs i -spac Quatu chaics = p p -spac p-spac -spac, y, y Itgratig or Phas Spac,, d y d y d col CNRS, Vtro p
12 May throlctric sicoductors: llipsoidal ri suracs t l t y May ha llopsoids o rolutio G Pb PbS So dirctios or i, i DOS col CNRS, Vtro p
13 I sphrical coordiats:. Copy locity ctors to a sphr y r y Ara o locity sphr: 4. Coctratio o particls:. Arag coctratio o particls pr uit sphr ara: 4 4. rasor to sphrical coordiats A si.. col CNRS, Vtro p
14 Distributio uctios ach particl is dscribd at ach ti t by its positio r ad its locity Distributio uctio or a particl is i gral out o quilibriu otal ubr o particls N otal olu V Nubr dsity = # particls pr uit olu: Mass dsity is: N t r,, t ddyd ddyd V t r,, t ddyd t r,, t ddyd At quilibriu, th t ad r-dpdc aish, th gas is uior: is iit, positi, cotiuous, ad such that: li bcaus th quilibriu locity distributio is isotropic ad uior r,, t M7, ctur, p 4 col CNRS, Vtro p 4
15 quilibriu distributio uctio Jas C. Mawll quilibriu statistical distributio uctio ot uctio o r: =,, Idal gas: Ki rs Probability that a particl b at a rgy ri-dirac: os-isti: p p? Mawll-olta: p col CNRS, Vtro p 5
16 lu = # part. ara ti t oo sipl A lu is th ubr o particls arriig o th surac pr uit ara pr uit ti N At Costruct cylidr isid gas, o ara A, ad hight th aout a particl ca tral or ti itral t ut that is too sipl: particls arri ro ry dirctio col CNRS, Vtro p 6
17 Coctratio pr uit olu Ara o locitis i th dirctios btw & ad & A si Iiitsial olu r & rr, & ad & si si si 4 4 y Dsity o particls with spds btw & by diitio Dsity o particls with locitis =, y, =,,,, si 4 col CNRS, Vtro p 7
18 lu: ro all dirctios V N Nubr dsity o particls A si 4 Volu o cylidr cos t Nubr particls i cyl. si cos A 4 t lu = # part. ara ti 4 si cos Slat cylidr with ais alog ad o lgth t col CNRS, Vtro p 8
19 Particl lu: itgrat or all dirctios cos si cos si cos si Itgrat col CNRS, Vtro p 9 Di arag spd 4 4
20 Collisios, a r path Ma r path l = th arag distac particls tral btw collisios quialt cocpts:. Scattrig cross-sctio sall particl blac hittig larg statioary o op circl, or apl: lctro i a gas o utral particls. W do t us this, but othrs do. d. Rlaatio ti = th arag ti btw collisios. Scattrig rqucy l l his is actually th group locity or particls, group locity is istataous locity or was, group is th locity o th wa pact col CNRS, Vtro p
21 lctrical coductiity: diitios rasport o tsi proprty: currt low [A] Udr th iluc o a GRADIN o a itsi proprty pottial gradit, i [V] Syst is NO i quilibriu wh thr ar gradits apl: Oh s law ad lctrical coductiity [] Oh s law y currt lu Currt I V j=> V I j Ara A RV I dv ti A d Voltag Rsistac Rally a ctorial rlatio: j σ -ctor lctrical coductiity -ctor tsor col CNRS, Vtro p
22 Ituiti iw: lctrical coductiity Apply a lctric ild, Particls ha a charg q orc o ach particl Acclratio: du dt Vlocity at t collisio u drit locity Charg currt dsity lctrical coductiity Mobility Gral j qu j u l Oh s law col CNRS, Vtro p
23 Charg lu orc cos ro lctric ild Drit locity Idal gas oly lctrical coductiity, idal gas j u u l V dt du dt l l Group locity col CNRS, Vtro p
24 hral coductiity: diitios rasport o tsi proprty: hat low [W] Udr th iluc o a GRADIN o a itsi proprty tpratur gradit, i [K] Syst is NO i quilibriu wh thr ar gradits apl: ourir law ad hral Coductiity [WK] ourir s law y Ara A Q q=> Hat low Q d q ti A d pratur gradit Hat lu Rally a ctorial rlatio: q κ hral coductiity -ctor -ctor tsor col CNRS, Vtro p 4
25 y l Ituiti iw: hral Coductiity, idal gas UdUdy Diitio o thral coductiity ro hat lu: Particl lu itgratd or : d q dy U rgy gradit: du dy I cos Atr a distac o a r path l, th rgy dirc is l cosdu Part dy lu o itral rgy IS th hat lu q q U du dy Part l cos du dy Part l col CNRS, Vtro p 5
26 y l UdUdy hral Coductiity, idal gas Diitio o thral coductiity ro hat lu: Hat lu itgratd or : d q dy U q du dy Part l l du dy rgy gradit: du dy Itral rgy: U=c * q lc * d dy c * hr pr uit olu Kitic orula c * l col CNRS, Vtro p 6
27 Pltir coicit, idal gas Pltir coicit: Apply lctric ild, calculat hat lu Pltir hat lu orc is th lctric ild Drit locity Pltir hat lu Osagr tsor lt q u q U u V V Ul V q V dt du dt Ul l V Aalogy with charg lu j u l U
28 Pltir ad Sbc coicits, idal gas q Ul V Osagr tsor lt Apply Osagr rlatios: S V Ul U C S l U S hropowr = Spciic hat charg Pltir = Itral rgycharg V S NROPY Cosistt: hropowr = tropycharg
29 hrolctric igur o rit, idal gas CV S S U S CV l l S C 8 ; U 9. 9 V l C V l U Prophtic, but sstially wrog rd pricipl o throdyaics Nrst pricipl li S I C V = R, th this iolats throdyaics ad is COMPY WRONG I o uss bttr odl or lctroic C V = => OK or sipl tals
30 . h olta thor I Gra Nrst Strit Arrhius Hic olta Hausaigr Auligr ttigshaus Klčič W us oly th rlaatio-ti approiatio or siplicity col CNRS, Vtro p
31 h olta rasport quatio Particls ha a distributio uctio r,,t = dist. uctio NO at quilibriu Apply a throdyaic acclratio A A has uits o s, or istac graity: A=g lctric ild: A= Atr a ti itral t th distributio uctio is: r r t, At, t t t r,, t r t, At, t t No Collisios: all particls that start at coditio r,,t also a it to coditio r t,a t,t t r,, t r t, At, t t col CNRS, Vtro p
32 olta rs Collisios a it so that th two trs ar slightly dirt, by a aout: t t t t t t t CO, r,, A, r t t t t t t t t t t t CO c A r, r,, A, r W pad this i a aylor s sris: I stady stat, t = CO t A r Diusio tr: Particls diusig Away ro r Drit tr: Particls oig Udr a orc A Scattrig tr: ct o collisios uits: collisio RA Uits: rats sc col CNRS, Vtro p
33 h olta quatio h physics iw: Zia, Pricipls o th hory o Solids, Cabridg, 97 r t DI A t t ORC t CO CO his is a stady stat but NO a quilibriu stat. At quilibriu: = col CNRS, Vtro p
34 h Collisio Itgral Hr: oard ad Marti lctroic Structur Ad rasport Proprtis o Crystals Krigr Publ., Malabar, 987 p.45 with =har I i ad ihard, p 8-44 oralis: ij- jinot practical to try to sol plicitly I practic, thr ar thr actors:. Probability o collisio. Nubr o aailabl iitial stats. Nubr o aailabl ial stats ij ji- col CNRS, Vtro p 4
35 riial cas: olta i quilibriu stat =. h t orcs ar ro A. h gas is uior r. h collisio rats ito ad out o th two stats ust b qual ij ji t CO So th quilibriu coditio triially satisis th olta quatio: r A t CO col CNRS, Vtro p 5
36 iarid olta quatio Assu that th applid acclratios ar sall, ad oly slightly prturb r,,t ro its quilibriu alu r,, t caus w will us ri suracs i solids, prss i trs o r,, t. h orc tr ca b padd i sris, ad th scod tr is i A : A A. h collisio tr : us th rlaatio ti approiatio t CO col CNRS, Vtro p 6
37 Justiicatio or th rlaatio ti approiatio is th arag ti btw collisio h probability that a collisio occurs i a ti itral t is t h collisios will th chag r,,t by a aout: CO Aout Chagd O arag t Aout Pr collisio - sig: rturig probability t CO Collisio rqucy ca b calculatd ro quatios abo, or ca b asurd h thr actors ow bco actors i th collisio rqucy. Probability o collisio. Nubr o aailabl iitial stats. Nubr o aailabl ial stats col CNRS, Vtro p 7
38 h Kitic Pricipl: st r t CO A r A r. How ca dpd o r? diats ro th local quilibriu oly by a sall aout, i = r r. h itic rgy is rlatd to or a idal gas, = d d col CNRS, Vtro p 8
39 h Kitic Pricipl: d r t CO A r A r. Idal gas: h itic rgy is rlatd to A A y y y y,,,, col CNRS, Vtro p 9. Sa thig or a parabolic disprsio rlatio i a bad, but ow th rrc poit is th ri rgy: * * * y y
40 h olta quatio A A A Gralid throdyaic orc Physical aig: pratur gradits act li a gralid orc by odiyig th statistical distributio uctio his justiis th aalogy btw Oh's ad ourir's laws: oltags ad tpratur dircs iduc lus o currt ad hat col CNRS, Vtro p 4 idal gas lctros i solids
41 Classical orc col CNRS, Vtro p 4 hral orc
42 Ituiti iw: Sbc coicit Charg trasport udr thral orc lctro li Cold sid Hot sid J > J h J h g, g li S g
43 Charg trasport udr thral orc hol- li Cold sid Hot sid J < J h J h g, g li S g
44 Calculatios o th lu o charg: Drit j u d d,r 4 d dj,r 4 j,rd 4 h tric: rasport o charg ad hat Istataous d y d Calculatios o th lu o hat: q Uu u d d,r 4 d dq,r 4 q 4,rd Wh cosidrig collcti otio, o ust us th drit locity u Wh cosidrig th otio o o sigl particl, o ust oly cosidr its istataous locity d y d Now w ha all th tools dd to calculat Osagr trs,,, isotropic rlaatio ti approiatio col CNRS, Vtro p 44
45 apl: lctrical coductiity j j Diitios: r, d r, d d j A Calculatios o th lu: r, d d olta: A j d d col CNRS, Vtro p 45
46 Cosidr ow both lcrical ad thral orc q j j d d d d col CNRS, Vtro p 46 lctrical orc hral orc rasport o charg rasport o hat
47 4. oral trasport quatios rlaatio ti approiatio 4. Dri th oral trasport quatios ro olta 4. hy iol itgrals or -spac: trasor that ito scalar itgrals or rgy 4. h cocpt o dsity o stats col CNRS, Vtro p 47
48 Gral cas: col CNRS, Vtro p 48 y d d d K d K Di th trasport itgrals K K K S V q j Zia, p 84, q apl: Wida-ra law: K K K Cra patitly through th algbra
49 Dsity o stats g sipl parabolic bad odl Dsity o stats is, by diitio, th ubr o aailabl stats i uit olu o sapl at rgy btw ad d y rasport itgrals ar i -spac aps o to locity spac ach poit rprsts o stat Volu o th sapl V = uity c Volu o -spac is Nubr o allowd -alus pr uit olu o -spac is, y, V 8 8 col CNRS, Vtro p 49
50 rgy Dpdc o th Dsity o stats g D d d g R R R R y y rgy Di Radius How ay lattic poits li at a id distac ro th origi i this locity spac? Nubr o allowd -alus pr uit olu o - spac is Volu o sphr o phas spac o radius R: Dsity o stats = th dsity o stats btw rgis & d col CNRS, Vtro p 5 y 8 ach -stat ca accoodat lctros o opposit spi
51 h gral cas: g as trasoratio o ariabls a ay proprty o lctros, which is th su o that proprty or ach stat I thr ar ay poits i -spac I w wat to prss i trs o a scalar rgy oly, w ha to us a trasoratio o ariabls to gt: h uctio that dos th trasoratio is th Wor or th ri surac istad o or its olu g d g d 4 d Ashcrot ad Mri, Solid Stat Physics, Holt, Rhihart ad Wisto 976 g risurac ds 4 col CNRS, Vtro p 5
52 Carrir coctratio -Di., ri statistics col CNRS, Vtro p 5 Dgrat statistics: bad illd up to Sphrical bad Coctratio o carrirs i -spac DOS: Carrir coctratio ro scalar -itgral With ull ri statistics: Dsity: Itral rgy: d g g 4 d g U d g d
53 Parabolic disprsio rlatio, -D, llipsoidal bads col CNRS, Vtro p d d U d d d d g DOS DOS DOS DOS DOS ; y DOS y g DOS Itgratio by part
54 Apply to trasport: tur th cra col CNRS, Vtro p 54 d g K d d d K y K K K K K K K K S Zia, p 44, q... Assuptio: Rlaatio ti is scalar isotropic
55 rasport itgrals col CNRS, Vtro p 55 g y DOS y DOS currt dirctio d d d d d d S d d d DOS K K K K K S K g K K K K
56 Rlaatio tis i sicoductors I parabolic bads, w usually assu is calld th scattrig pot Mg M with M Itractio Probability atri lt Dsity o aailabl stats Scattrig Mchais Raso dscriptio Acoustic phoos - Motu chag oly Polar optical phoos Oly at > Dby Grai boudaris - Ma r path = grai si = costat Itr-ally ro o poct o ri surac to aothr Ioid ipuritis og-rag Coulob itractios his is ry ry crud, but th bst w ca do. itratur aailabl upo rqust. col CNRS, Vtro p 56
57 h cts o th DOS-ass Pb-li atrial, K, acouststic phoo scattrig; idpdt ariabl = col CNRS, Vtro p 57
58 h cts o th scattrig paratr Pb-li DOS, K, obility st qual at 9 c - but this r wors col CNRS, Vtro p 58
59 With powr law or rlaatio ti, th trasport itgrals ar ot prssd as uctio o rducd itgral Optiiatio o dopig ll j j d or tals : j j j 6 j j or udopd sicoductors : j j! H. J. Goldsid With urical thods, Do o logr bothr to do that, Us ull quatios col CNRS, Vtro p 59
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