hall_effect-1

TheHallEect 1Backgr ound Inthisexperimen t,theHallEect willbeusedtostudy someofthephysicsofcharge transp ortin metal andsemiconductor samples. In1879E.H.Hallobserv edthatwhen anelectrical curren tpasses through asample placed ina magnetic eld,apotentialproportional tothecurren tandtothemagnetic eldisdevelopedacross thematerial inadirection perpendicular toboththecurren tandtothemagnetic eld[1].This eect isknownastheHalleect, andisthebasisofmanypractical applications anddevices such asmagnetic eldmeasuremen ts,andposition andmotion detectors. With themeasuremen tshemade, Hallwasabletodetermine forthersttimethesignofcharge carriers inaconductor. Eventoday,Halleect measuremen tscontinuetobeauseful technique forcharacterizing theelectrical transp ortproperties ofmetals andsemiconductors. Indeed, the failure ofthesimple modelofmetallic conductivit y,whichwediscuss below,toaccoun tformany experimen talmeasuremen tsoftheHalleect hasbeenoneoftheprincipal motivatorsleading toa betterunderstanding ofelectronic properties ofmaterials [5,pp.58{62]. 1.1Thesimple theory oftheHalleect Consider aconducting slabasshowninFig.1withlength Linthexdirection, width Winthey direction andthickness Tinthezdirection. Figure 1:Geometry ofelds andsample inHalleect experimen t. Assume theconductor tohavecharge carrier ofcharge q(canbeeither positiveornegativ eorboth, butwetakeittobeofjustonesignhere), charge carrier numberdensit yn(i.e.,numberofcarriers perunitvolume), andcharge carrier driftvelocityvxwhen acurren tIx owsinthepositivex direction. Thedriftvelocityisanaverage velocityofthecharge carriers overthevolume ofthe conductor; eachcharge carrier maymoveinaseemingly random waywithin theconductor, but under thein uence ofapplied elds there willbeanettransp ortofcarriers along thelength ofthe conductor. Thecurren tIxisthecurren tdensit yJxtimes thecross-sectional areaoftheconductor 1 WT.Thecurren tdensit yJxisthecharge densit ynqtimes thedriftvelocityvx.Inother words Ix=JxWT=nqvxWT: (1) Thecurren tIxiscaused bytheapplication ofanelectric eldalong thelength oftheconductor Ex.Inthecasewhere thecurren tisdirectly proportional totheeld, wesaythatthematerial obeysOhm's law,whichmaybewritten Jx=Ex; (2) where istheconductivit yofthematerial intheconductor. Nowassume thattheconductor isplaced inamagnetic eldperpendicular totheplane oftheslab. Thecharge carriers willexperience aLorentzforceq~v~Bthatwillde ect them towardoneside oftheslab. Theresult ofthisde ection istocause anaccum ulation ofcharges along onesideof theslabwhichcreates atransv erseelectric eldEythatcounteracts theforceofthemagnetic eld. (Recall thattheforceofanelectric eldonacharge qisq~E.) When steady state isreached,there willbenonet owofcharge intheydirection, since the electrical andmagnetic forces onthecharge carriers inthatdirection mustbebalanced. Assuming these conditions, itiseasytoshowthat Ey=vxBz; (3) where Eyistheelectric eld,called theHalleld,intheydirection andBzthemagnetic eldin thezdirection. Inanexperimen t,wemeasure thepotentialdierence across thesample|the Hallvoltage VH| whichisrelated totheHalleldby VH=ZW 0Eydy=EyW: (4) Thus,fromequations (1),(3)and(4)weobtain VH=1 nqIxBz T: (5) Theterminparenthesis isknownastheHallcoecien t: RH=1 nq: (6) Itispositiveifthecharge carriers arepositive,andnegativ eifthecharge carriers arenegativ e.In practice, thepolarityofVHdetermines thesignofthecharge carriers. NotethattheSIunits of theHallcoecien tare[m3/C]ormore commonly stated [m3/A-s]. Exercise 1Work throughthemathtoderive Eq.(5).Nowconsider thatanelectriccurrentinthe positive xdirectioncanbecreatedbypositive chargesmoving positive alongthexaxisornegative chargesmoving negative alongthexaxis.Drawdiagramsshowing theelectricandmagnetic forces onthechargestoconvinc eyourself that,inspiteofthisfact,positive chargecarriers willproduce apositive Hallcoecient andnegative chargecarriers willproduceanegative Hallcoecient. 2 1.2TheHalleect inmetals andsemiconductors Inorder tounderstand some oftheideas involvedintheory oftheHalleect inrealmaterials, it isinstructiv etoconstruct amore careful modelforelectric curren tsunder electric andmagnetic elds fromaclassical pointofview. Weimagine thatthecharge carriers moveinamedium that oers some resistance. Theresistance isduetoscattering betweenthecarriers andimpurities in thematerial andbetweenthecarriers andvibrations ofthematerial's atoms. Eachcharge carrier isaccelerated bytheapplied elds buteverysooften itscatters andlosesenergy .Ifweassume thattheaverage timebetweenscattering eventsis,thenwehave,onaverage, aretarding force acting onthecarriers of ~Fretard=m~v ; (7) where misthemassofthecarrier. Sounder thein uence ofapplied electric andmagnetic elds, Newton's second lawreads md~v dt=q ~E+~v~B m~v ; (8) where thevelocity~vistakentobeanaverage overallofthecarriers. Atsteady state, thetimederivativeof~vwillvanish. Under theusual conventionthat~Bpoints along thezaxis,weobtain thecomponentequations for~vbysetting thelefthand sideofEq.(8) tozeroandrearranging: vx=q mEx+q mBzvy; vy=q mEyq mBzvx; (9) vz=q mEz: FromEq.(1)wehavethatJx=nqvx(andcorresp ondingly foryandzcomponents).Bysolving theaboveequations forvx,vyandvzinterms ofthecomponentsof~EandBzweget Jx= 1+(!c)2(Ex+!cEy); Jy= 1+(!c)2(Ey!cEx); (10) Jz=Ez; where nq2 m; (11) and !cqBz m: (12) Intheexpression fortheconductivit yitmayseem thatallwehavedone istoreplace one unkno wnquantity~vwithanother unkno wnquantity.Buttheparameter ,called therelaxation time,iswidely usedindiscussions ofelectronic transp ortinmaterials, andcanbejustied ina quantum-mec hanical contextviatheBoltzmann transp ortequation [8,Ch.7] 3 Theangular frequency !cisknownasasthe\cyclotron frequency". Itisthefrequency ofrota- tionofacharge inamagnetic eld, andcanbetakenasameasure ofthestrength oftheeld. Thecombination !cisusedtocharacterize anexperimen talsituation: ifthemagnetic eldis weakand/or therelaxation timeshort, !c1andourexperimen tisinthe\weak-eld limit"; alternately if!c1theexperimen tisinthe\strong-eld limit". Anumberofmaterials show strikingly dieren tbehaviorbetweentheweak-andstrong-eld limits; alumin umisone. Inourclassical modeloftheHalleect withasingle typeofcharge carrier, however,there isno suchcrosso verbetweentheweakandstrong eld. Thiscanbeprovedbyworking outthefollowing exercises. Exercise 2Showthataclassic alchargedparticle ofchargeq,massmandspeedvwould executea circularorbitofangular frequency !cifitmoves under thein uenc eofamagnetic eld~B.Assume thatnootherforcesactontheparticle. Exercise 3Solveequations 10under theconditions thatJy=Jz=0toshowtworesults: Jx= ExandJx=Ey=(!c).Dene thehallcoecient asRHEy=(BJx)andshowthatRH= 1=(nq). Notethatthismodelpredictstwothings: theHallcoecient isindependent ofthemagnetic eld strengthandthatthereisnodependenc eofthesample resistanc eonthemagnetic eldeither. This secondeectiscalled\magnetor esistanc e",anditwasthiseectthatHalloriginal lytried(and failed)tond[1,5]. Theclassical theory oftheHalleect presen tedaboveassumes thattheelectric curren tistheresult ofmanycharge carriers movingindependen tlyofeachother andresponding toapplied elds as classical particles. Butweknowthatelectrons arequantumparticles, specically fermions, and theyhavewavelikeproperties. Curiously ,theactofchanging themodelfromclassic alindependen t particles movingfreely toquantum independen tparticles movingfreely changes littleintheresults sofarpresen ted.The\free-electron quantumgas"modelstillpredicts ahallcoecien tof1=nq andzeromagnetoresistance [5]. Thebenet ofusing aquantumapproac hbecomes apparen twhen itiscoupled withamorerealistic modelofsolidmatter, specically ,crystalline. Inacrystal, theatoms arearranged inaperiodic lattice. Electrons inthelattice feeltheeect ofaperiodicpotentialontheirmotion. Thestrongest eect occurs forthose electrons intheouter atomic orbitals|the \valence" electrons, andespe- cially those valence electrons whose deBroglie wavelength isclosetothespacing ofthepotential's periodicity. Within theperiodicpotentialtheallowedenergies ofthevalence electrons arebrokenintointo aseries ofenergy bands withenergy gapsbetweenthem. Ifthenumberofvalence electrons per unitcellofthecrystal isexactly enough tollaband, thesolidwillbeapoorconductor, since bysymmetry ateachenergy there willbelledmomen tumstates pointinginoppositedirections. Conduction canoccuronlyifanelectron canjump agapintoanunoccupied state. Ifthegapsare large, thenecessary energy maybetoohigh,andthesolidisaninsulator .Ifthegapissmall, then thermal energy maybeenough tocause sucien telectrons tojump thegap,andthesolidiscalled asemiconductor. Ontheother hand, ifthenumberofvalence electrons perunitcellisnotenough tollaband, thenmanyunlled momen tumstates liewithin easyenergy reach,andthesolidisa goodconductor|a metal. 4 However,theshapeoftheenergy surfaces|the bandstructur e|has astrong eect onthetypeof conduction thatcanoccur. Electron states withenergies nearthe\bottom" ofaband|the lowest allowedenergy intheband|b ehavelikefreeelectrons, except thattheirresponsetoanapplied eldmaybethatofaheavierorlighterparticle. Itasiftheelectron mass mhasbeenchanged toaneectiv emass m.While thisstatemen tseems strange, stranger stillaretheresponses of electrons withenergies nearthe\top" ofaband; these electrons actasiftheirmassisnegative !In other words,theyaccelerate oppositely afreeelectron when acted uponbyanapplied eld. Indeed, itiseasier toimagine particle states ofthistypeasacting likepositive charges. Suchstates are called \hole" states. Another waytothink ofthese eects isthattheelectron states withwavenumberslyingnearthe band extremes arediractedbytheperiodiclattice inthesame waythatlightisdiracted bya grating: themotion ofanelectron waveisgreatly altered, evenreversed, analogous tothewaythat agrating canre ect lightofaspeciccoloratspecicangles. Inasemiconductor thebandgapisrelativ elysmall, andelectrons maybeexcited bythermal energy tojump thegap.Thisprocessallowsanelectron-lik estatenearthebottom ofanupperbandanda hole-lik estatenearthetopofalowerbandtocome intoexistence. Bothstates carry curren t,with theholestateacting asapositivecharge. Thenumberofthese curren t-carrying states dependson thetemperature inaroughly exponentialway:thenumberisproportional totheBoltzmann factor eEg=kTwhere Egistheenergy ofthegap. Theideaofsimultaneous electron+hole states hasyielded auseful modeloftheHalleect called the\two-band model". Themathematics worksthrough justlikeinExercise 3,except thatthe totalcurren tisthesumofcontributions fromtheholes andtheelectrons. Ifweleteachtypeof carrier haveaHallcoecien t:Re(forelectrons) andRh(forholes), asgivenbytheEq.(6)andwe assume theconductivit y=e+h,where eandharegivenbyEq.(11),thenwecanderive anexpression forthetotalHallcoecien tas RH=2 hRh+2 eRe (h+e)2: (13) Intheabove,theeectiv emassmissubstituted formandthecharge qtakenaspositiveforholes andnegativ eforelectrons. Thisequation followsfromEqs.(10)inthelimitthat!c1:the low-eld limit. Inthecaseofsemiconductors, ithasbecome customary toseparate outthecarrier densit ynfrom theoverallconductivit yformulafor,anddene anewquantitycalled themobility : =q m: (14) Thus,inourtwo-band model,theconductivit ywouldlooklike =ejqejne+hjqhjnh; (15) andthelow-eld Hallcoecien twouldlooklike RH=1 jqjnh2 hne2 e (nhh+nee)2; (16) where wehaveassumed thatthecharge isthesame magnitude forbothtypesofcarriers. 5 Exercise 4ShowthatEq.(16)reducestoEq.(6)inthecaseofonetypeofcarrier. Thenshow thatforthiscase,=jvxj=Ex.Thus, themobility canbefound fromaHalleectexperiment: showthat=RHall. Formetals, theexact valueofthecharge carrier densit yandthesignoftheHallcoecien tdepend ontheenergy bandstructure oftheparticular metal. Foralkalimetals (Li,Na,K,etc.)andsome ofthetransition metals (Cu,Ag,Au),thecharge carriers areelectrons (negativ e)andthecharge carrier densit yisapproximately oneelectron peratom. Thus,wecanoften usethesimple result, Eq.(6)forthisclassofmaterials asanapproximate result. Forsemiconductors, theband structure cangiverisetobothnegativ e(electron-lik e)orpositive (\holes", orabsence ofelectrons) eectiv echarge carriers, andtheelectrical conductivit yisdeter- mined bythedensit yandmobilit yofbothkinds. Weneedtooften usethemore complex formof RH,Eq.(16)forthese materials. Inanycase,atroomtemperatures, thecharge carrier densit y ofsemiconductors ismuchsmaller thanthatofmetals, andthusthemagnitude ofVHallismuch larger foragivenIx,Bzandlmthickness, t.Thisdierence willbecome veryclearwhen youtake measuremen tsonthesamples inthelab. Exercise 5Inasemiconductor thecarrier density isstronglydependent onthetemperature,mostly proportional totheBoltzmann factor oftheenergygapbetweenthevalenceandconduction band,as notedabove.Howwould theHallvoltage, foraxedIx,dependontemperatureforsemiconductors? Discuss thisquestion qualitatively andquantitatively. Inrecenttimes, thestudy oftheHalleect inthinlmsatlowtemperatures andhighmagnetic elds hasdemonstrated theexistence ofthequantumHalleect. Thiseect ismostapparen tin verysmall structures andatverycoldtemperatures. Under theseconditions onecandistinguish the quantization oftheenergy levels,asthesmall structures forcetheenergy levelstosplitandthecold temperatures allowthelow-lying levelstobecome lled. Inthiscase,theHallresistance (dened asVHall=Ix)doesnotincrease linearly withBz,butexhibits steps asBzincreases. Themagnitude oftheHallresistance atthesteps ish=(e2i),where iisaninteger. TheHallconductance, the recipro caloftheHallresistance, thusincreases inintegersteps, andthequantitye2=hnowdenes thebasic unitofconductance. In1985Klaus vonKlitzing wasawarded theNobelprize forthe discoveryoftheinteger quantumHalleect. More recently,afractional formofthequantumHalleect hasbeendiscovered,where theconduc- tance stepsareinrational fractions ofthebasice2=hunit.Itisgenerally believedthatthispeculiar eect isduetointeractions among thecharge carriers, requiring aquantumtheory going beyond theeects ofconnemen t. 2Procedure There arethree dieren tsamples forwhichyouwilldetermine thesignanddensit yofthecharge carriers. Some specics forthesamples aregiveninTable1. Aschematic forthecircuit isshowninFig.2. Mounttheprobeintotheclamp attachedtothestand sothatitcanslipbetweenthemagnet polepieces. Makesurethattheprobeiswellaligned betweenthepoles:itshould beperfectly vertical withtheplane ofthepaddle perpendicular tothetable andcentered bothvertically and horizon tallybetweenthepoles. 6 Material Thickness t(m) Width w(mm) Length `(mm) Resistance ( ) Au 1:370:1610712:70:05 30:50:05 0:7700:006 Al 2:340:1710712:70:05 30:50:05 0:8720:018 InAs 1:260:021040:6920:004 1:5420:005 1:30:3 Table1:Characteristics oftheHalleect probes.Thethicknesses ofthemetal samples were determined optically bycomparing fringes created bythin-lm interference. (SeeOptics, 2nd edition, byE.Hecht,pp.381,382 foranexplanation ofthismetho d.)Thelength andwidth were measured withcalipers.Thedimensions oftheoftheInAsprobeweredetermined bytheuseofa measuring microscop e.Theresistance oftheprobeswasmeasured along thelength `ofthesample byathefour-p oint-prob emetho dusing alowfrequency ACexcitation andlock-inampliers. Connect theprobewires according tocolorfollowingthediagram. Notethatthevoltage measure- mentwires areattachedtoadualbanana-plug connector thatcantintotheDMM sockets.Note thatthemeter whichmeasures curren tshould usethewhite andblack connection points,while themeter thatmeasures voltage should usetheredandblackconnection points.DoNOTusethe connectionpointsmarkedSENSE 4WIRE. Turnonthemeters andthepowersupply .Setthevoltage measuremen tmeter tomeasure DC voltsbypressing the\DCV" button, andthecurren tmeasuremen tmeter tomeasure DCamps by pressing the\DCI" button. Severalprecautions mustbefollowed: Donotexceed 200mAcurren tthrough either ofthetwometal samples (AuandAl). Donotexceed 100mAcurren tthrough theInAssample. Turndownthecurren tandvoltage settings ontheHP6181B toZERObeforeconnecting ordisconnecting anyoftheHallsamples. Because thecurren tsource attempts tomaintain thesame curren tregardless ofloadresistance, asudden increase intheresistance (say,by disconnecting thewires) cancause aspikeintheoutput voltage. TheHP6181B should beoperated intheconstan tcurren tmode:besurethatthecurren tis beinglimited bythecurren tsetknobandnotthevoltage setknob. Youcantellifthecurren t isbeinglimited bythevoltage knob ifthelightnearthatknob ison.Testthisbyturning thevoltage knoballthewaydownandthenallthewayup.Youshould seethelightturno when thevoltage ishighenough tosupply thedesired curren t. Tothispoint,Vyhasbeenequated withVHall.Inpractice thisdoesnotholdtruebecause of asymmetries intheprobeelemen tandvoltage measuremen tcontacts. Thus,evenforBz=0,when acurren tIx owsavoltage Vywillbemeasured duetothevoltage dropalong thelength ofthe sample ordierences intheresistivit yofvarious parts ofthesensing elemen t. Checkthisoutwiththesample outside themagnet: Turnthecurren tupandnotethatyouseea change inVyeventhough there isanear-zero magnetic eld(relativ etothestrong onebetween themagnet poles). Onewaytoeliminate theeect ofthisoset voltage istomeasure theVyforonedirection ofBz, removethesample fromthemagnet, ipthesample over180,putthesample backinthemagnet andmeasure Vyagain. Thedierence inthetworeadings willbetwiceVHall. 7 KIETHLEY 2000 MULTIMETERVOLTS COM AMPS HP6181BKIETHLEY 2000 MULTIMETER AMPSCOMVOLTS HALL PROBE CURRENT SUPPLYVOLTAGE MEASUREMENT CURRENT MEASUREMENTWHITE GREEN RED REDBLACK+_BETWEEN MAGNET POLES Figure 2:Schematic ofHallEect experimen t.Theblade oftheprobeshould beslippedbetween themagnet polepieces sothatthesensing elemen tiscentered. When using thistechnique itisimportantthatthecurren tthrough theproberemain exactly the same forthetwoorientations oftheprobe.Yourmeasuremen tsshould followthisprocedure: 1.Setthecurren ttothedesired level. 2.Waituntilthevoltage reading settles towithin thedesired precision (3{4signican tgures isusually adequate). 3.Record thevoltage, making suretonotetheorientation oftheprobe. 4.Pulltheprobestand back,loosentheclamp, rotate theprobeby180andtightentheclamp. 5.slidetheprobestand backintoposition. 6.Checkthatthecurren thasnotchanged signican tly(i.e.,lessthan1mA). 7.After thevoltage reading settles, record thevoltage. 8.Gobacktostep1withanewcurren tsetting. Abigpiece ofuseful advice: When taking datadonotattempt togure outalloftheangles andorientations youwillneedtodetermine thesignofthecharge carriers asyougo.Itismuch morereliable torecord thedatabymeans ofunambiguous visual markers,suchas\green wireup", \green wiredown",andthentodrawacleardiagram oftheprobelayoutwithallwires labeled according tocolor, along withtheprobe'sorientation tothemagnet. Don't forget topayattention tothesignofthevoltage thatyoumeasure aswellastheconnections toalloftheelectronics. With thisinformation youcanreconstruct therelationships among thecurren tdirection, voltage sign, andmagnetic elddirection when youcanconcen trateonthedataanalysis. Because youneedto iptheprobeorientation ateachsetting ofthecurren t,itishelpful touse thewoodpieces onthetable andthemetal lockingringsonthestand andprobehandle tosetup theprobeposition. Then itbecomes asimple matter tomovethestand and iptheprobe. 8 Indoing thisexperimen t,youwillmeasure Ix,Vy(fortwoprobeorientations) andBzforthethree dieren tprobes.Youshould takereadings ofVyfor6to8dieren tvalues ofIxforeachsample. Forthetwometal probestheHallvoltage isverysmall|in themicrovoltrange|so youwillneed tosetthevoltmeter onitsmost sensitiv escale. Usethetriangular RANGE buttons forthis. It isalsohelpful tousethe\ltering" capabilities ofthemeters toaverage outnoise uctuations inthemeasuremen ts.Turnthisonbypressing theFILTERbutton (note the\FLT"indicator on thepanel belowthereading), andthenchoosing toaverage manyreadings together (a\running average" with50readings workswell). FortheInAsprobe,theHallvoltage ismuchgreater. Thisistypical ofsemiconductor probes,and isthereason fortheuseofsemiconductors incommercial Hall-eect devices. Butthetemperature sensitivit yismuchgreater too.Itwilltakeabitlonger fortheInAsvoltage tosettle, since the probeelemen tisheated bytheapplied curren t. Themagnetic eldBzremains xedduring theexperimen t.Itcanbevariedbyadjusting theshunt rings around thepolepieces: turning therings towardthepolefacesreduces theeldstrength. They arenormally setatthefarthest pointbackinorder tomaximize theeld. Curvesofeld strength versus pole-face type,gapandringsetting areavailable inthelab.Theeldismeasured withtheSypris (Bell) gauss meter. Thismeter isitself aHalleect device, andthetransducer islocated atthetipoftheblade. Before measuring themagnetic eld, thegauss meter should bezeroedbyplacing theendoftheprobeinside thezero-eld chamberandadjusting thereading tozerobypushing thezeroing button ontheSypris (andwaiting untilitisdone). Explore the magnetic eldvariation overtheregion betweenthetwopolefaces. Usebothorientations ofthe probetomeasure theeldstrength. Calculate amean valueandstandard deviation fromyour measuremen ts.Explain inyourreportyourmetho dfornding themean eldanditsuncertain ty. 3Analysis Fromyourmeasuremen tsonthethree samples calculate 2VHbysubtracting thevoltage measure- mentsforthetwodieren tprobeorientations. Becarefulaboutthesignofthevoltage! Then plot 2VHversus Ix,andtthelinewithacomputer toobtain theslopeanditsuncertain ty. Usetheresult, along withtheprobedatagiveninTable1todetermine themagnitude andsignof theHallcoecien tforeachofthesamples, along withtheexperimen talerrorforthese quantities. Fromthestated values ofthedimensions andresistances oftheInAs sample, itispossible to determine thesample conductivit y: =` Rwt; (17) where Ristheresistance ofthesample. Thisformulaassumes thattheresistivit yofthematerial isuniform overthesample andthatthedimensions ofthesample accurately re ect itsgeometry . Fromtheconductivit yandtheHallcoecien tRHallitispossible tocompute the\Hall mobilit y" Hofthecharge carriers, dened as Hjvxj jExj: (18) Useyourvalues ofRHallandtocompute H(andtheerror) fortheInAssample. Then lookup mobilities forInAsintextsorhandb ooksandcompare yourvaluewiththeirs. Some questions toresearc h,ponder anddiscuss inyourreport: 9 1.Find some values formobilit yintables thatarelisted intexts, suchasKittel's orother references, suchastheLandolt-B ornstein series ofbooksavailable inthelibrary .Compare yourresults withthetabulated values. Howcloseareyourresults totheonescitedforInAs? Canyouexplain anydierences? 2.Inthecaseofthealumin umsample, youwillndinsuchtexts asKittel's \Introduction to SolidState Physics" thattheassumed charge carriers are\1-hole peratom". Whyshould we assume thatthecharge carrier inalumin umisahole? Thisisnotpredicted bythesimple \Drude" theory ofelectrons inmetals. 3.Yourresults foralumin umarelikelytobedieren tfromthepredicted ortabulated values, whichareusually quoted asbeingmeasured atlowtemperature andhighmagnetic eld. Seewhat youcanndoutabouttheHallcoecien tforarange oftemperatures andeld strengths, anddiscuss howyourmeasuremen ttsinwiththese. References [1]Hall,E.H.,\Onanewaction ofthemagnet onelectric curren ts",American Journal ofMath- ematics, 2,No.3,pages 287{292 (1879). Theoriginal paperbyHallwhichdescrib estheeect. Aninteresting historical read. [2]Preston, D.W.andE.R.Dietz, TheArtofExperimental Physics, JohnWiley &Sons(1991), pages 303{315. Excellen tdiscussion oftheHalleect insemiconductors. [3]Melissinos, A.C.,Experiments inModernPhysics, Academic Press (1966), pages 85{87. [4]Kittel, C.,Introduction toSolidStatePhysics, 6thedition, John Wiley &Sons(1986), pages 147{151. Standard undergraduate text.Contains useful tables, butdoesnotgiveadetailed accoun toftheHalleect formetals. [5]Ashcroft, NeilW.,andN.DavidMermin, SolidStatePhysics, Saunders College, Philadelphia (1976). Chapter 1givestheelemen tarytheory oftheHallEect. Chapter 3discusses thefail- uresofthefree-electron model.Chapters 12and15treattheHalleect inthehigheldlow temperature limitformetals. [6]Hurd, Colin M.,TheHallEectinMetals andAlloys,PlenumPress, NewYork(1972). A thorough overview oftheHalleect inmetals. Predates thequantumHalleect. [7]Silsbee,RobertH.,andJorgDrager,Simulations forSolidStatePhysics, Cambridge University Press (1997). Thebookandcomputer simulations illustrate manyimportantconcepts insolid statephysics. Thesimulations areavailable onacomputer inthelab. [8]Ziman, J.M.,Principles oftheTheoryofSolids, Cambridge UniversityPress (1969). Anexcel- lent,readable andcleartextcovering muchofthesame material asAshcroft andMermin. PreparedbyD.B.Pengra,J.Stoltenb erg,R.VanDyckandO.Vilches hall_effect_10-0 7.tex--Updated 4October 2007 10