Lecture 6: Semiconductor Conduction. Prof. Ali M. Niknejad

Transcription

1 Lecture 6: Semicoductor Coductio Prof. Ai M. ikejad Departmet of EECS Uiversity of Caiforia, Berkeey

2 EE 105 Fa 016 Lecture Outie Physics of coductio Semicoductors Si Diamod Structure Bod Mode Itrisic Carrier Cocetratio Dopig by Io Impatatio Drift Veocity Saturatio Diffusio Uiversity of Caiforia, Berkeey

3 EE 105 Fa 016 Physics of Coductio 3

4 EE 105 Fa 016 Ohm s Law Oe of the first thigs we ear as EECS majors is: V I R Is this trivia? Maybe what s reay goig o is the foowig: V f ( I) f (0) + f '(0) I + f ''(0) I / +...» f '(0) I I the above Tayor exasio, if the votage is zero for zero curret, the this is geeray vaid The rage of vaidity (radius of covergece) is the importat questio. It turs out to be VERY arge! 4 EECS

5 EE 105 Fa 016 Ohm s Law Revisited I Physics we eared: J s E Is this aso trivia? We, it s the same as Ohm s aw, so the questios are reated. For a rectaguar soid: I V L J s V I A L s A R I Is t it strage that curret (veocity) is proportioa to Force? Where does coductivity come from? 5 EECS

6 EE 105 Fa 016 Coductivity of a Gas Eectrica coductio is due to the motio of positive ad egative charges For water with ph7, the cocetratio of hydroge H + ios (ad OH - ) is: 10-7 moe/l moe/cm cm cm -3 Typicay, the cocetratio of charged carriers is much smaer tha the cocetratio of eutra moecues The motio of the charged carriers (eectros, ios, moecues) gives rise to eectrica coductio 6 EECS

7 EE 105 Fa 016 Coisios i Gas At a temperate T, each charged carrier wi move i a radom directio ad veocity uti it ecouters a eutra moecue or aother charged carrier Sice the cocetratio of charged carriers is much ess tha moecues, it wi most ikey ecouter a moecue For a gas, the moecues are widey separated (~ 10 moecuar diameters) After coidig with the moecue, there is some eergy exchage ad the charge carrier wi come out with a ew veocity ad ew directio 7 EECS

8 EE 105 Fa 016 Memory Loss i Coisios Schematicay eutra Moecue Key Poit: The iitia veocity ad directio is ost (radomized) after a few coisios 8 EECS

9 EE 105 Fa 016 Appicatio of Fied Whe we appy a eectric fied, durig each free fight, the carriers wi gai a mometum of Eqt Therefore, after t secods, the mometum is give by: Mu + Eqt 9 If we take the average mometum of a partices at ay give time, we have: Mu umber of Carriers 1 ( Mu Eqt ) å j + j Iitia Mometum Before Coisio j Time from Last Coisio Mometum Gaied from Fied EECS

10 EE 105 Fa 016 Radom Thigs Sum to Zero! Whe we sum over a the radom veocities of the partices, we are averagig over a arge umber of radom variabes with zero mea, the average is zero Mu 1 ( Mu Eqt ) å j + j Mu 1 åeqt j j j Eqt Average Time Betwee Coisios J qu æ qç è Eqt ö M ø q t M E s E 10 EECS

11 EE 105 Fa egative ad Positive Carriers Sice curret is cotributed by positive ad egative charge carriers: E J J J ø ö ç ç è æ M e M e e t t ø ö ç ç è æ M M e t t s EECS

12 EE 105 Fa 016 Coductio i Metas High coductivity of metas is due to arge cocetratio of free eectros These eectros are ot attached to the soid but are free to move about the soid I meta sodium, each atom cotributes a free eectro:.5 10 atoms/cm 3 From the measured vaue of coductivity (easy to do), we ca back cacuate the mea free time: s m t e ( 17 )( ) (.5 10 )( ) sec 1 EECS

13 EE 105 Fa 016 A Deep Puzze This vaue of mea free time is surprisigy og The mea veocity for a eectro at room temperature is about: mv 3 7 kt v 3 10 cm / sec -7 At this speed, the eectro traves vt 3 10 cm The moecuar spacig betwee adjacet ios is oy cm Why is it that the eectro is o average zoomig by 10 positivey charged ios? EECS

14 EE 105 Fa 016 Wave ature of Eectro 14 The free carrier ca peetrate right through positivey charged host atoms! Quatum mechaics expais this! (Take Physics 117A/B) For a periodic arragemet of potetia fuctios, the eectro does ot scatter. The ifuece of the crysta is that it wi trave freey with a effective mass. So why does it scatter at a? EECS

15 EE 105 Fa 016 Scatterig i Metas At temperature T, the atoms are i radom motio ad thus upset periodicity Eve at extremey ow temperatures, the presece of a impurity upsets periodicity 15 EECS

16 EE 105 Fa 016 Summary of Coductio + t + s e M + + t M Coductivity determied by: Desity of free charge carriers (both positive ad egative) Charge of carrier (usuay just e) Effective mass of carrier (differet iside soid) Mea reaxatio time (time for memory oss usuay the time betwee coisios) This is determied by severa mechaisms, e.g.: Scatterig by impurities Scatterig due to vibratios i crysta 16 EECS

17 Semicoductors Departmet of EECS Uiversity of Caiforia, Berkeey

18 EE 105 Fa 016 Resistivity for a Few Materias Pure copper, 73K ohm-cm Pure copper, 373 K ohm-cm Pure germaium, 73 K 00 ohm-cm Pure germaium, 500 K.1 ohm-cm Pure water, 91 K ohm-cm Seawater 5 ohm-cm What gives rise to this eormous rage? Why are some materias semi-coductive? Why the strog temp depedece? 18 Uiversity of Caiforia, Berkeey

19 EE 105 Fa 016 Eectroic Properties of Siico Siico is i Group IV Atom eectroic structure: 1s s p 6 3s 3p Crysta eectroic structure: 1s s p 6 3(sp) 4 Diamod attice, with 0.35 m bod egth Very poor coductor at room temperature: why? (1s) (s) (p) 6 (3sp) 4 19 Hybridized State Uiversity of Caiforia, Berkeey

20 EE 105 Fa 016 Periodic Tabe of Eemets 0 Uiversity of Caiforia, Berkeey

21 EE 105 Fa 016 The Diamod Structure 3sp tetrahedra bod!.35 A! 5.43A 1 Uiversity of Caiforia, Berkeey

22 EE 105 Fa 016 States of a Atom Eergy. E 3 E Forbidde Bad Gap Aowed Eergy Leves E 1 Atomic Spacig Lattice Costat Quatum Mechaics: The aowed eergy eves for a atom are discrete ( eectros ca occupy a state sice with opposite spi) Whe atoms are brought ito cose cotact, these eergy eves spit If there are a arge umber of atoms, the discrete eergy eves form a cotiuous bad Uiversity of Caiforia, Berkeey

23 EE 105 Fa 016 Eergy Bad Diagram The gap betwee the coductio ad vaece bad determies the coductive properties of the materia Meta Partiay fied bad Isuator arge bad gap, ~ 8 ev Semicoductor medium sized gap, ~ 1 ev Coductio Bad bad gap Vaece Bad e - 3 e - Eectros ca gai eergy from attice (phoo) or photo to become free Uiversity of Caiforia, Berkeey

24 EE 105 Fa 016 Mode for Good Coductor The atoms are a ioized ad a sea of eectros ca wader about crysta: The eectros are the gue that hods the soid together Sice they are free, they respod to appied fieds ad give rise to coductios O time scae of eectros, attice ooks statioary Uiversity of Caiforia, Berkeey

25 EE 105 Fa 016 Bod Mode for Siico (T0K) Siico Io (+4 q) 5 Four Vaece Eectros Cotributed by each io (-4 q) eectros i each bod Uiversity of Caiforia, Berkeey

26 EE 105 Fa 016 Bod Mode for Siico (T>0K) Some bod are broke: free eectro Leave behid a positive io or trap (a hoe) Uiversity of Caiforia, Berkeey

27 EE 105 Fa 016 Hoes? - + otice that the vacacy (hoe) eft behid ca be fied by a eighborig eectro It ooks ike there is a positive charge traveig aroud! Treat hoes as egitimate partices. 7 Uiversity of Caiforia, Berkeey

28 EE 105 Fa 016 Yes, Hoes! The hoe represets the void after a bod is broke Sice it is eergeticay favorabe for earby eectros to fi this void, the hoe is quicky fied But this eaves a ew void sice it is more ikey that a vaece bad eectro fis the void (much arger desity that coductio bad eectros) The et motio of may eectros i the vaece bad ca be equivaety represeted as the motio of a hoe J vb å( -q) v å - - å i ( q) vi (- vb Fied Bad Empty States q) v i 8 J å vb - (-q) vi Empty States å qv i Empty States Uiversity of Caiforia, Berkeey

29 EE 105 Fa 016 More About Hoes Whe a coductio bad eectro ecouters a hoe, the process is caed recombiatio The eectro ad hoe aihiate oe aother thus depetig the suppy of carriers I therma equiibrium, a geeratio process couterbaaces to produce a steady stream of carriers 9 Uiversity of Caiforia, Berkeey

30 Itrisic Carrier Cocetratio ad Dopig Departmet of EECS Uiversity of Caiforia, Berkeey

31 EE 105 Fa 016 Therma Equiibrium (Pure Si) Baace betwee geeratio ad recombiatio determies o p o 31 Strog fuctio of temperature: T 300 o K i G G ( T ) + th G opt R k( p) G R k( p) G ( T) th p Gth ( T) / k ( T 10 cm 10-3 ( T) i at 300 K Uiversity of Caiforia, Berkeey

32 EE 105 Fa 016 Dopig with Group V Eemets P, As (group 5): extra bodig eectro ost to crysta at room temperature Immobie Charge Left Behid + 3 Uiversity of Caiforia, Berkeey

33 EE 105 Fa 016 Door Accoutig Each ioized door wi cotribute a extra free eectro The materia is charge eutra, so the tota charge cocetratio must sum to zero: r -q0 + qp0 + q d 0 Free Eectros Free Hoes Ios (Immobie) 33 By Mass-Actio Law: p i i - q0 + q + q d 0 - q ( T) qi + q d 0 0 Uiversity of Caiforia, Berkeey

34 EE 105 Fa Door Accoutig (cot) Sove quadratic: Oy positive root is physicay vaid: For most practica situatios: Uiversity of Caiforia, Berkeey i d d i d + ± i d d + + d i >> d d d i d d +» ø ö ç è æ

35 EE 105 Fa 016 Dopig with Group III Eemets Boro: 3 bodig eectros à oe bod is usaturated Oy free hoe egative io is immobie! - 35 Uiversity of Caiforia, Berkeey

36 EE 105 Fa 016 Mass Actio Law Baace betwee geeratio ad recombiatio: p -type case: o o i 0 + ( T 300 K, i d 10 i d cm 10-3 ) P-type case: p 0 - a p i a 36 Uiversity of Caiforia, Berkeey

37 EE 105 Fa 016 Compesatio Dope with both doors ad acceptors: Create free eectro ad hoe! Uiversity of Caiforia, Berkeey

38 EE 105 Fa Compesatio (cot.) More doors tha acceptors: d > a Uiversity of Caiforia, Berkeey i a d o >> - More acceptors tha doors: a > d a d o p i - i d a o p >> - d a o i -

39 Drift Currets Departmet of EECS Uiversity of Caiforia, Berkeey

40 EE 105 Fa 016 Therma Equiibrium Rapid, radom motio of hoes ad eectros at therma veocity v th 10 7 cm/s with coisios every τ c s. Appy a eectric fied E ad charge carriers acceerate for τ c secods v t zero E fied 1 * m v 1 th kt th cm / s 10 s c cm v th positive E (hoe case) a c 40 x v th Uiversity of Caiforia, Berkeey

41 EE 105 Fa Drift Veocity ad Mobiity Uiversity of Caiforia, Berkeey v pe dr µ E m q m qe m F a v p c c p c p e c dr ø ö ç ç è æ ø ö ç ç è æ ø ö ç ç è æ t t t t For eectros: E m q m qe m F a v p c c p c p e c dr ø ö ç ç è æ - ø ö ç ç è æ - ø ö ç ç è æ t t t t For hoes: v E dr µ -

42 EE 105 Fa 016 Mobiity vs. Dopig i Siico at 300 o K 4 Typica vaues: µ µ p Uiversity of Caiforia, Berkeey

43 Diffusio Currets Departmet of EECS Uiversity of Caiforia, Berkeey

44 EE 105 Fa 016 Diffusio Diffusio occurs whe there exists a cocetratio gradiet I the figure beow, imagie that we fi the eft chamber with a gas at temperate T If we suddey remove the divider, what happes? The gas wi fi the etire voume of the ew chamber. How does this occur? 44 Uiversity of Caiforia, Berkeey

45 EE 105 Fa 016 Diffusio (cot) The et motio of gas moecues to the right chamber was due to the cocetratio gradiet If each partice moves o average eft or right the evetuay haf wi be i the right chamber If the moecues were charged (or eectros), the there woud be a et curret fow The diffusio curret fows from high cocetratio to ow cocetratio: 45 Uiversity of Caiforia, Berkeey

46 EE 105 Fa 016 Diffusio Equatios Assume that the mea free path is λ Fid fux of carriers crossig x0 pae 1 ( -) (-)v th (0) () ( )v th 1 F vth - ( ( -) ( ) ) 1 æ é d ù é ùö ç ê ú ê + ú è ë û - d F vth (0) - (0) dx ë dx ûø d F -vth dx J -qf qv th d dx 46 Uiversity of Caiforia, Berkeey

47 EE 105 Fa 016 Eistei Reatio The therma veocity is give by kt v th 1 * m v 1 th v t th c kt t vtht c kt m c * kt q Mea Free Time qt m c * J d æ kt ö qvth qç µ dx è q ø æ kt ö D ç µ è q ø d dx 47 Uiversity of Caiforia, Berkeey

48 EE 105 Fa 016 Tota Curret ad Boudary Coditios 48 Whe both drift ad diffusio are preset, the tota curret is give by the sum: J J drift + J diff qµ E + qd I resistors, the carrier is approximatey uiform ad the secod term is eary zero For currets fowig uiformy through a iterface (o charge accumuatio), the fied is discotious J 1 ( s 1 ) J ( s ) J 1 J s E s 1 1 E E E 1 s s 1 d dx Uiversity of Caiforia, Berkeey

49 EE 105 Fa 016 Referece Edward Purce, Eectricity ad Magetism, Berkeey Physics Course Voume ( d Editio), pages EECS