Theelectronasaparticle 1
AndIlaughtoseethemwhirlandflee, Likeaswarmofgoldenbees. Shelley
1.1Introduction
TheCloud
Inthepopularmindtheelectronlivesassomethingverysmallthathassomethingtodowithelectricity.Studyingelectromagnetismdoesnotchangethe pictureappreciably.Youlearnthattheelectroncanberegardedasanegative pointchargeanditdulyobeysthelawsofmechanicsandelectromagnetism.It isaparticlethatcanbeacceleratedordeceleratedbutcannotbetakentobits. Isthispicturelikelytobenefitanengineer?Yes,ifithelpshimtoproducea device.Isita correct picture?Well,anengineerisnotconcernedwiththetruth, thatislefttophilosophersandtheologians;theprimeconcernofanengineer istheutilityofthefinalproduct.Ifthisphysicalpicturemakespossiblethe birthofthevacuumtube,wemustdeemituseful;butifitfailstoaccountfor thepropertiesofthetransistorthenwemustregarditsappealaslessalluring. Thereisnodoubt,however,thatwecangoquitefarbyregardingtheelectron asaparticleeveninasolid—thesubjectofourstudy.
Whatdoesasolidlooklike?Itconsistsofatoms.Thisideaoriginatedafew thousandyearsagoinGreece,∗ ∗ From‘
’=indivisible. andhashadsomeupsanddownsinhistory, buttodayitstruthisuniversallyaccepted.Nowifmatterconsistsofatoms,they mustbesomehowpileduponeachother.Thesciencethatisconcernedwiththe spatialarrangementofatomsiscalledcrystallography.Itisasciencegreatly reveredbycrystallographers;engineersarerespectful,butlackenthusiasm. Thisisbecausetheneedtovisualizestructuresinthreedimensionsaddsto thehardenoughtaskofthinkingaboutwhattheelectronwilldonext.Forthis chapter,letusassumethatallmaterialscrystallizeinthesimplecubicstructure ofFig.1.1,withthelatticeionsfixed(itisasolid)andsomeelectronsarefree towanderbetweenthem.ThiswillshortlyenableustoexplainOhm’slaw,the Halleffect,andseveralotherimportantphenomena.Butifyouaresceptical aboutoversimplification,lookforwardtoFig.5.3toseehowtheelemental semiconductorscrystallizeinthediamondstructure,orgetagreatershock withFig.5.4whichshowsaformofcarbonthatwasdiscoveredinmeteorites buthasonlyrecentlybeenfabricatedinlaboratories.
Letusspecifyourmodelalittlemoreclosely.Ifwepostulatetheexistence ofacertainnumberofelectronscapableofconductingelectricity,wemust alsosaythatacorrespondingamountofpositivechargeexistsinthesolid.It mustlookelectricallyneutraltotheoutsideworld.Second,inanalogywith ourpictureofgases,wemayassumethattheelectronsbouncearoundinthe
Fig.1.1
Atomscrystallizinginacubical lattice.
∗ Weshallseelaterthatthisisnotsofor metalsbutitisnearlytrueforconduction electronsinsemiconductors.
interatomicspaces,collidingoccasionallywithlatticeatoms.Wemayevengo furtherwiththisanalogyandclaimthatinequilibriumtheelectronsfollowthe samestatisticaldistributionasgasmolecules(thatis,theMaxwell–Boltzmann distribution)whichdependsstronglyonthetemperatureofthesystem.The averagekineticenergyofeachdegreeoffreedomisthen 1 2 kB T where T is absolutetemperatureand kB isBoltzmann’sconstant.Sowemaysaythatthe meanthermalvelocityofelectronsisgivenbytheformula∗
because vth isthethermalvelocity,and m is themassoftheelectron.
particlesmovinginthreedimensionshavethreedegreesoffreedom. Weshallnowcalculatesomeobservablequantitiesonthebasisofthis simplestmodelandseehowtheresultscomparewithexperiment.Thesuccess ofthissimplemodelissomewhatsurprising,butweshallseeasweproceed thatviewingasolid,oratleastametal,asafixedlatticeofpositiveionsheld togetherbyajelly-likemassofelectronsapproximateswelltothemodernview oftheelectronicstructureofsolids.Somebooksdiscussmechanicalproperties intermsofdislocationsthatcanmoveandspread;thesolidisthenpicturedasa fixeddistributionofnegativechargeinwhichthelatticeionscanmove.These viewsarealmostidentical;onlytheexternalstimuliaredifferent.
1.2Theeffectofanelectricfield—conductivityandOhm’slaw
Supposeapotentialdifference U isappliedbetweenthetwoendsofasolid length L.Thenanelectricfield
ispresentateverypointinthesolid,causinganacceleration
† See,forexample,W.Shockley, Electronsandholesinsemiconductors,D. vanNostrand,NewYork,1950,pp. 191–5.
Thus,theelectrons,inadditiontotheirrandomvelocities,willacquireavelocityinthedirectionoftheelectricfield.Wemayassumethatthisdirected velocityiscompletelylostaftereachcollision,becauseanelectronismuch lighterthanalatticeatom.Thus,onlythepartofthisvelocitythatispickedup inbetweencollisionscounts.Ifwewrite τ fortheaveragetimebetweentwo collisions,thefinalvelocityoftheelectronwillbe aτ andtheaveragevelocity
Thisissimpleenoughbutnotquitecorrect.Weshouldnotusethe average timebetweencollisionstocalculatetheaveragevelocitybuttheactualtimes andthentaketheaverage.Thecorrectderivationisfairlylengthy,butallit givesisafactorof2.†
Numericalfactorslike2or3or π aregenerallynot worthworryingaboutinsimplemodels,butjusttoagreewiththeformulae generallyquotedintheliterature,weshallincorporatethatfactor2,anduse vaverage = aτ .(1.5)
Theeffectofanelectricfield—conductivityandOhm’slaw3
Theaveragetimebetweencollisions, τ ,hasmanyothernames;forexample, meanfreetime,relaxationtime,andcollisiontime.Similarly,theaverage velocityisoftenreferredtoasthemeanvelocityordriftvelocity.Weshall callthem‘collisiontime’and‘driftvelocity’,denotingthelatterby vD Therelationshipbetweendriftvelocityandelectricfieldmaybeobtained fromeqns(1.3)and(1.5),yielding
wheretheproportionalityconstantinparenthesesiscalledthe‘mobility’(μe ). Thisistheonlynameithas,anditisquitealogicalone.
Assumingnowthatallelectronsdriftwiththeirdriftvelocity,thetotalnumberofelectronscrossingaplaneofunitareapersecondmaybeobtained bymultiplyingthedriftvelocitybythedensityofelectrons, Ne .Multiplying furtherbythechargeontheelectronweobtaintheelectriccurrentdensity
Thehigherthemobility,themore mobiletheelectrons.
Wecanderivesimilarlytherelationshipbetweencurrentdensityandelectric fieldfromeqns(1.6)and(1.7)intheform
Noticethatitisonlythedriftvelocity,createdbytheelectricfield,that comesintotheexpression.Therandomvelocitiesdonotcontributetothe electriccurrentbecausetheyaverageouttozero.∗ ∗ Theygiverise,however,to electrical noise inaconductor.Itsvalueisusuallymuchsmallerthanthesignalswe areconcernedwithsoweshallnotworry aboutit,althoughsomeofthemostinterestingengineeringproblemsarisejust whensignalandnoisearecomparable. seeSection1.8onnoise.
ThisisalinearrelationshipwhichyoumayrecognizeasOhm’slaw
where σ istheelectricalconductivity.Whenfirstlearningaboutelectricityyou lookedupon σ asabulkconstant;nowyoucanseewhatitcomprisesof.We canwriteitintheform Inmetals,incidentally,themobilitiesarequitelow,abouttwoordersofmagnitudebelowthoseof semiconductors;sotheirhighconductivityisduetothehighdensity ofelectrons.
Thatis,wemayregardconductivityastheproductoftwofactors,chargedensity(Ne e)andmobility.Thus,wemayhavehighconductivitiesbecausethere arelotsofelectronsaroundorbecausetheycanacquirehighdriftvelocities, byhavinghighmobilities.
Ohm’slawfurtherimpliesthat σ isaconstant,whichmeansthat τ must beindependentofelectricfield.† † Itseemsreasonableatthisstagetoassumethatthechargeandmassofthe electronandthenumberofelectrons presentwillbeindependentoftheelectricfield.
Fromourmodelsofaritismorereasonable toassumethat l ,thedistancebetweencollisions(usuallycalledthemeanfree path)intheregularlyspacedlattice,ratherthan τ ,isindependentofelectric field.But l mustberelatedto τ bytherelationship,
Inatypicalmetal μe =5 × 10–3 m2 V–1 s–1 ,whichgivesa driftvelocity vD of5 × 10–3 ms–1 foranelectricfieldof1Vm–1 .
∗ Thisislesstrueforsemiconductorsas theyviolateOhm’slawathighelectric fields.
Since vD varieswithelectricfield, τ mustalsovarywiththefieldunless vth vD .(1.12)
AsOhm’slawisaccuratelytrueformostmetals,thisinequalityshouldhold.
Thethermalvelocityatroomtemperatureaccordingtoeqn(1.1)(which actuallygivestoolowavalueformetals)is
th = 3kB T m 1/2 ~ = 105 ms–1 .(1.13)
Thus,therewillbeaconstantrelationshipbetweencurrentandelectricfield accuratetoabout1partin108 . ∗
Thisimportantconsiderationcanbeemphasizedinanotherway.Letusdraw thegraph(Fig.1.2)ofthedistributionofparticlesinvelocityspace,thatiswith rectilinearaxesrepresentingvelocitiesinthreedimensions, vx , vy , vz .Withno electricfieldpresent,thedistributionissphericallysymmetricabouttheorigin. Thesurfaceofasphereofradius vth representsallelectronsmovinginall possibledirectionswiththatr.m.s.speed.Whenafieldisappliedalongthe x-axis(say),thedistributionisminutelyperturbed(theelectronsacquiresome additionalvelocityinthedirectionofthe x-axis)sothatitscentreshiftsfrom (0,0,0)toabout(vth /108 ,0,0).
Takingcopper,afieldof1Vm–1 causesacurrentdensityof108 Am–2 .Itis quiteremarkablethatacurrentdensityofthismagnitudecanbeachievedwith analmostnegligibleperturbationoftheelectronvelocitydistribution.
1.3Thehydrodynamicmodelofelectronflow
Fig.1.2
Distributionsofelectronsinvelocity space.
ζ mayberegardedhereasameasureoftheviscosityofthemedium.
Byconsideringtheflowofachargedfluid,asophisticatedmodelmaybedeveloped.Weshalluseitonlyinitscrudestform,whichdoesnotgivemuchof aphysicalpicturebutleadsquicklytothedesiredresult.
Theequationofmotionforanelectronis
Ifwenowassumethattheelectronmovesinaviscousmedium,thenthe forcestryingtochangethemomentumwillberesisted.Wemayaccountfor thisbyaddinga‘momentum-destroying’term,proportionalto v.Takingthe proportionalityconstantas ζ eqn(1.14)modifiesto
Inthelimit,whenviscositydominates,thetermdv/dt becomesnegligible, resultingintheequation
whichgivesforthevelocityoftheelectron
Itmaybeclearlyseenthatbytaking ζ =1/τ eqn(1.17)agreeswith eqn(1.6);hencewemayregardthetwomodelsasequivalentand,inanygiven case,usewhicheverismoreconvenient.
1.4TheHalleffect
Letusnowinvestigatethecurrentflowinarectangularpieceofmaterial,as showninFig.1.3.Weapplyavoltagesothattheright-handsideispositive. Current,byconvention,flowsfromthepositivesidetothenegativeside,that isinthedirectionofthenegative z-axis.Butelectrons,remember,flowina directionoppositetoconventionalcurrent,thatisfromlefttoright.Having sortedthisoutletusnowapplyamagneticfieldinthepositive y-direction.The forceonanelectronduetothismagneticfieldis e(v × B).(1.18)
Togettheresultantvector,werotatevector v intovector B.Thisisaclockwiserotation,givingavectorinthenegative x-direction.Butthechargeofthe electron, e,isnegative;sotheforcewillpointinthepositive x-direction;the electronsaredeflectedupwards.Theycannotmovefartherthanthetopendof theslab,andtheywillaccumulatethere.
Butifthematerialwaselectrically neutralbefore,andsomeelectronshavemovedupwards,thensomepositive ionsatthebottomwillbedeprivedoftheircompensatingnegativecharge. Henceanelectricfieldwilldevelopbetweenthepositivebottomlayerandthe negativetoplayer.Thus,afterawhile,theupwardmotionoftheelectronswill bepreventedbythisinternalelectricfield.Thishappenswhen
Equilibriumisestablishedwhen theforceduetothetransverse electricfieldjustcancelstheforce duetothemagneticfield.
Expressedintermsofcurrentdensity, RH iscalledthe Hallcoefficient.
Inthisexperiment EH , J ,and B aremeasurable;thus RH ,andwithitthedensity ofelectrons,maybedetermined.
Fig.1.3
SchematicrepresentationofthemeasurementoftheHalleffect.
Thecorrespondingelectricfieldin asemiconductorisconsiderably higherbecauseofthehighermobilities.
Whatcanwesayaboutthedirectionof EH ?Well,wehavetakenmeticulouscaretofindthecorrectdirection.Oncethepolarityoftheappliedvoltage andthedirectionofthemagneticfieldarechosen,theelectricfieldiswelland trulydefined.Soifweputintoourmeasuringapparatusoneconductorafter theother,themeasuredtransversevoltageshouldalwayshavethesamepolarity.Yes...thelogicseemsunassailable.Unfortunately,theexperimental factsdonotconform.Forsomeconductorsandsemiconductorsthemeasured transversevoltageisinthe other direction.
Howcouldweaccountforthedifferentsign?Onepossiblewayofexplainingthephenomenonistosaythatincertainconductors(andsemiconductors) electricityiscarriedbypositivelychargedparticles.Wheredotheycomefrom? Weshalldiscussthisprobleminmoredetailsometimelater;forthemoment justacceptthatmobilepositiveparticlesmayexistinasolid.Theybearthe unpretentiousname‘holes’.
Toincorporateholesinourmodelisnotatalldifficult.Therearenowtwo speciesofchargecarriersbouncingaround,whichyoumayimagineasamixtureoftwogases.Takegoodcarethatthenetchargedensityiszero,andthe newmodelisready.Itisactuallyquiteagoodmodel.Wheneveryoucome acrossanewphenomenon,trythismodelfirst.Itmightwork.
ReturningtotheHalleffect,youmaynowappreciatethattheexperimental determinationof RH isofconsiderableimportance.Ifonlyonetypeofcarrier ispresent,themeasurementwillgiveusimmediatelythesignandthedensity ofthecarrier.Ifbothcarriersaresimultaneouslypresentitstillgivesuseful informationbutthephysicsisalittlemorecomplicated(seeExercises1.7 and1.8).
Inourpreviousexamplewetookatypicalmetalwhereconductiontakes placebyelectronsonly,andwegotadriftvelocityof5 × 10–3 ms–1 .Fora magneticfieldof1Tthetransverseelectricfieldis
1.5Electromagneticwavesinsolids
Sofarasthepropagationofelectromagneticwavesisconcerned,ourmodel worksverywellindeed.Allweneedtoassumeisthatourholesandelectrons obeytheequationsofmotion,andwhentheymove,theygiverisetofieldsin accordancewithMaxwell’stheoryofelectrodynamics.
Itisperfectlysimpletotakeholesintoaccount,buttheequations,with holesincluded,wouldbeconsiderablylonger,soweshallconfineourattention toelectrons.
Wecouldstartimmediatelywiththeequationofmotionforelectrons,but letusfirstreviewwhatyoualreadyknowaboutwavepropagationinamediumcharacterizedbytheconstantspermeability, μ,dielectricconstant, ,and conductivity, σ (itwillnotbeawasteoftime).
FirstofallweshallneedMaxwell’sequations:
Second,weshallexpressthecurrentdensityintermsoftheelectricfieldas
Itwouldnowbealittlemoreeleganttoperformallthecalculationsinvector form,butthenyouwouldneedtoknowafewvectoridentities,andtensors (quitesimpleones,actually)wouldalsoappear.Ifweusecoordinatesinstead, itwillmakethetreatmentalittlelengthier,butnottooclumsyifweconsider onlytheone-dimensionalcase,whenthereischangeonlyinthe z-direction
Assumingthattheelectricfieldhasonlyacomponentinthe x-direction(see thecoordinatesysteminFig.1.3),then
where ex , ey , ez aretheunitvectors.Itmaybeseenfromthisequationthat themagneticfieldcanhaveonlya y-component.Thus,eqn(1.23)takesthe simpleform
Weneedfurther
which,combinedwitheqn(1.24),bringseqn(1.22)tothescalarform
Thus,wehavetwofairlysimpledifferentialequationstosolve.Weshall attemptthesolutionintheform
Wehaveherecomefacetofacewitha disputethathasragedbetweenphysicists andengineersforages.Forsomeodd reasonthephysicists(aidedandabettedbymathematicians)usethesymbol ifor √–1andtheexponent–i(ω t – kz) todescribeawavetravellinginthe zdirection.Theengineers’notationisjfor √–1andj(ω t – kz)fortheexponent.In thiscoursewehave,ratherreluctantly, acceptedthephysicists’notationssoas nottoconfuseyoufurtherwhenreading booksonquantummechanics.
ω representsfrequency,and k is thewavenumber.
Then,
whichreducesourdifferentialequationstothealgebraicequations
and
Thisisahomogeneousequationsystem.Bytherulesofalgebra,thereisa solution,apartfromthetrivial Ex = By =0,onlyifthedeterminantofthe coefficientsvanishes,thatis
Expandingthedeterminantweget
Essentially, Differentpeoplecallthisequationbydifferentnames.Characteristic,determinantal,anddispersionequationareamongthe namesmorefrequentlyused.We shallcallitthe dispersionequation becausethatnamedescribesbest whatishappeningphysically.
cm c isthevelocityoftheelectromagneticwaveinthemedium.
theequationgivesarelationshipbetweenthefrequency, ω ,and thewavenumber, k ,whichisrelatedtophasevelocityby vP = ω/k .Thus, unless ω and k arelinearlyrelated,thevariousfrequenciespropagatewithdifferentvelocitiesandattheboundaryoftwomediaarerefractedatdifferent angles.Hencethenamedispersion.
Amediumforwhich σ =0and μ and areindependentoffrequencyis nondispersive.Therelationshipbetween k and ω issimply
Solvingeqn(1.36)formally,weget
∗ Thenegativesignisalsopermissible thoughitdoesnotgiverisetoanexponentiallyincreasingwaveaswouldfollow fromeqn(1.39).Itwouldbeveryniceto makeanamplifierbyputtingapieceof lossymaterialinthewayoftheelectromagneticwave.Unfortunately,itviolates theprincipleofconservationofenergy. Withoutsomesourceofenergyatitsdisposalnowavecangrow.Sothewave whichseemstobeexponentiallygrowingisineffectadecayingwavewhich travelsinthedirectionofthenegative z-axis.
Thus,whenever σ = 0,thewavenumberiscomplex.Whatismeantbyacomplexwavenumber?Wecanfindthisouteasilybylookingattheexponentof eqn(1.30).Thespatiallyvaryingpartis exp(ikz)=exp i(kreal +ikimag )z =exp(ikreal z)exp(–kimag z).(1.39)
Hence,iftheimaginarypartof k ispositive,theamplitudeoftheelectromagneticwavedeclinesexponentially.∗
Iftheconductivityislargeenough,thesecondtermisthedominantonein eqn(1.38)andwemaywrite
Soifwewishtoknowhowrapidlyanelectromagneticwavedecaysina goodconductor,wemayfindoutfromthisexpression.Since
theamplitudeoftheelectricfieldvariesas
Thedistance δ atwhichtheamplitudedecaysto1/eofitsvalueatthe surfaceiscalledthe skindepth andmaybeobtainedfromtheequation
Youhaveseenthisformulabefore.Youneeditoftentoworkouttheresistanceofwiresathighfrequencies.Ideriveditsolelytoemphasizethemajor stepsthatarecommontoallthesecalculations.
Wecannowgofurther,andinsteadoftakingtheconstant σ ,weshalllook alittlemorecriticallyatthemechanismofconduction.Weexpressthecurrent densityintermsofvelocitybytheequation
Thisisreallythesamethingaseqn(1.7).Thevelocityoftheelectronisrelated totheelectricandmagneticfieldsbytheequationofmotion
Weare
lookingforlinearizedsolutionsleadingtowaves.Inthatapproximationthequadraticterm v × B canclearlybeneglectedandthetotalderivative canbereplacedbythepartialderivativetoyield
Thesymbol v stillmeanstheaveragevelocityofelectrons,butnow itmaybeafunctionofspaceand time,whereasthenotation vD is generallyrestrictedtod.c.phenomena.
1/τ isintroducedagainasa‘viscous’or‘damping’term.
Assumingagainthattheelectricfieldisinthe x-direction,eqn(1.47)tells usthattheelectronvelocitymustbeinthesamedirection.Usingtherulesset
outineqn(1.32)wegetthefollowingalgebraicequation
Thecurrentdensityisthenalsointhe x-direction:
where σ isdefinedasbefore.Youmaynoticenowthattheonlydifference fromourprevious(J – E )relationshipisafactor(1–iωτ )inthedenominator. Accordingly,thewholederivationleadingtotheexpressionof k ineqn(1.38) remainsvalidif σ isreplacedby σ/(1–iωτ ).Weget
If ωτ 1,wearebackwherewestartedfrom,butwhathappenswhen ωτ 1?Couldthathappenatall?Yes,itcanhappenifthesignalfrequency ishighenoughorthecollisiontimeislongenough.Then,unityisnegligible incomparisonwithiωτ ineqn(1.50),leadingto
Introducingthenewnotation
weget Equation(1.53)suggestsageneralizationoftheconceptofthe dielectricconstant.Wemayintroduceaneffectiverelativedielectric constantbytherelationship
Itmaynowbeseenthat,dependingonfrequency, εeff maybepositiveornegative.
Hence,aslongas ω>ωp ,thewavenumberisreal.Ifitisreal,ithas(bythe rulesofthegame)noimaginarycomponent;sothewaveisnotattenuated.This isquiteunexpected.Byintroducingaslightmodificationintoourmodel,we maycometoradicallydifferentconclusions.Assumingpreviously J = σ E ,we workedoutthatifanyelectronsarepresentatall,thewaveisboundtodecay. Nowwearesayingthatforsufficientlylarge ωτ anelectromagneticwavemay travelacrossourconductorwithoutattenuation.Isthispossible?Itseemsto contradicttheempiricalfactthatradiowavescannotpenetratemetals.True;
butthatisbecauseradiowaveshavenotgothighenoughfrequencies;letus trylightwaves.Cantheypenetrateametal?No,theycannot.Itisanother empiricalfactthatmetalsarenottransparent.Soweshouldtryevenhigher frequencies.Howhigh?Well,thereisnoneedtogoonguessing,wecanwork outthethresholdfrequencyfromeqn(1.52).Takingtheelectrondensityina typicalmetalas6 × 1028 perm3 ,wethenget
where ε0 isthefree-spacepermittivity.
Atthisfrequencyrangeyouareprobablymorefamiliarwiththe wavelengthsofelectromagneticwaves.Convertingtheabovecalculatedfrequencyintowavelength,weget
where c isthevelocityoflight.
Thus,thethresholdwavelengthiswellbelowtheedgeofthevisibleregion(400nm).Itisgratifyingtonotethatourtheoryisinagreementwithour everydayexperience;metalsarenottransparent.
Thereisonemorethingweneedtocheck.Isthecondition ωτ 1satisfied?Foratypicalmetalatroomtemperature,thevalueof τ isusuallyabove 10–14 s,making ωτ oftheorderofhundredsatthethresholdfrequency. Bymakingtransmissionexperimentsthroughathinsheetofmetal,thecriticalwavelengthcanbedetermined.Themeasuredandcalculatedvaluesare comparedinTable1.1.Theagreementisnottoobad,consideringhowsimple ourmodelis.
BeforegoingfurtherIwouldliketosayalittleabouttherelationshipof transmission,reflection,andabsorptiontoeachother.Theconceptsaresimple andonecanalwaysinvoketheprincipleofconservationofenergyifintrouble.
Table1.1 Thresholdwavelengthsforalkali metals
Incident wave
Reflected
Fig.1.4
Incidentelectromagneticwavepartly reflectedandpartlytransmitted. Medium 1
Incident wave
Reflected wave Medium 2
Fig.1.5
Incidentelectromagneticwave reflectedbytheconductor.
Ifthereisasmalleramplitude transmitted,therewillbealarger amplitudereflected.
Letustakethecasewhen ωτ 1; k isgivenbyeqn(1.53),andourconductorfillshalfthespace,asshowninFig.1.4.Whathappenswhenan electromagneticwaveisincidentfromtheleft?
1. ω>ωp .Theelectromagneticwavepropagatesintheconductor. Thereisalsosomereflection,dependingontheamountofmismatch.Energy conservationsays
energyintheincidentwave=energyinthetransmittedwave +energyinthereflectedwave.
Isthereanyabsorption?No,because ωτ 1.
2. ω<ωp .Inthiscase k ispurelyimaginary;theelectromagneticwave decaysexponentially.Isthereanyabsorption?No.Cantheelectromagnetic wavedecaythen?Yes,itcan.Isthisnotincontradictionwithsomethingor other?Thecorrectanswermaybeobtainedbywritingouttheenergybalance. Sincethewavedecaysandtheconductorisinfinitelylong,noenergygoesout attheright-handside.Soeverythingmustgoback.Theelectromagneticwave isreflected,asshowninFig.1.5.Theenergybalanceisenergyintheincident wave=energyinthereflectedwave.
3.LetustakenowthecaseshowninFig.1.6whenourconductorisoffinitedimensioninthe z-direction.Whathappensif ω<ωp ?Thewavehasa chancetogetoutattheotherside,sothereisaflowofenergy,forwardsand backwards,intheconductor.Thewidertheslab,thesmalleristheamplitude ofthewavethatappearsattheothersidebecausetheamplitudedecaysexponentiallyintheconductor.Thereisdecay,butnoabsorption.Theamplitudes ofthereflectedandtransmittedwavesrearrangethemselvesinsuchawayas toconserveenergy.
Ifwechooseafrequencysuchthat ωτ 1,then,ofcourse,dissipative processesdooccurandsomeoftheenergyoftheelectromagneticwaveis
Fig.1.6
Incidentelectromagneticwave transmittedtomedium3.The amplitudeofthewavedecaysin medium2butwithoutanyenergy absorptiontakingplace.
convertedintoheat.Theenergybalanceinthemostgeneralcaseis
energyinthewave=energyinthetransmittedwave +energyinthereflectedwave +energyabsorbed.
Agoodexampleofthephenomenaenumeratedaboveisthereflectionof radiowavesfromtheionosphere.Theionosphereisalayerwhich,asthename suggests,containsions.Therearefreeelectronsandpositivelychargedatoms, soourmodelshouldwork.Inametal,atoms,andelectronsarecloselypacked; intheionosphere,thedensityismuchsmaller,sothatthecriticalfrequency ωp isalsosmaller.ItsvalueisafewhundredMHz.Thus,radiowavesbelow thisfrequencyarereflectedbytheionosphere(thisiswhyshortradiowaves canbeusedforlong-distancecommunication)andthoseabovethisfrequency aretransmittedintospace(andsocanbeusedforspaceorsatellitecommunication).Thewidthoftheionospherealsocomesintoconsideration,butat thewavelengthsused(itisthewidthinwavelengthsthatcounts)itcanwellbe regardedasinfinitelywide.
1.6Wavesinthepresenceofanappliedmagneticfield: cyclotronresonance
Inthepresenceofaconstantmagneticfield,thecharacteristicsofelectromagneticwaveswillbemodified,butthesolutioncanbeobtainedbyexactlythe sametechniqueasbefore.Theelectromagneticeqns(1.22)and(1.23)arestill validforthea.c.quantities;theequationofmotionshould,however,contain theconstantmagneticfield,whichweshalltakeinthepositive z-direction.The appliedmagneticfield, B0 ,maybelarge,hence v × B0 isnotnegligible;itisa first-orderquantity.Thus,thelinearizedequationofmotionforthiscaseis
Inordertosatisfythisvectorequation,weneedboththe vx and vy components.Thatmeansthatthe currentdensity,andthroughthat theelectricandmagneticfields, willalsohaveboth x and y components. m ∂ v ∂ t + v τ = e(E E E + v × B0 ).(1.56)
Writingdownalltheequationsisalittlelengthy,butthesolutionisnot moredifficultinprinciple.Itmayagainbeattemptedintheexponentialform, and ∂/∂ z and ∂/∂ t mayagainbereplacedbyik and–iω ,respectively.Allthe
differentialequationsarethenconvertedintoalgebraicequations,andbymakingthedeterminantofthecoefficientszerowegetthedispersionequation.I shallnotgothroughthedetailedderivationherebecauseitwouldtakeupa greatdealoftime,andtheresultingdispersionequationishardlymorecomplicatedthaneqn(1.50).Allthathappensisthat ω inthe ωτ termisreplaced by ω ± ωc .Thus,thedispersionequationfortransverseelectromagneticwaves inthepresenceofalongitudinald.c.magneticfieldis
Theplusandminussignsgivecircularlypolarizedelectromagneticwaves rotatinginoppositedirections.Toseemoreclearlywhathappens,letussplit theexpressionunderthesquarerootintoitsrealandimaginaryparts.Weget
Thislooksabitcomplicated.Inordertogetasimpleanalyticalexpression,let usconfineourattentiontosemiconductorswhere ωp isnottoolargeandthe appliedmagneticfieldmaybelargeenoughtosatisfytheconditions,
Weintendtoinvestigatenowwhathappenswhen ωc iscloseto ω .The secondandthirdtermsineqn(1.59)arethensmallincomparisonwithunity; sothesquarerootmaybeexpandedtogive
Theattenuationoftheelectromagneticwaveisgivenbytheimaginarypart of k .Itmaybeseenthatithasamaximumwhen ωc = ω .Since ωc iscalled thecyclotron∗ ∗ Afteranacceleratingdevice,the cyclotron,whichworksbyaccelerating particlesinincreasingradiiinafixed magneticfield.
Theroleof ωc τ isreallyanalogous tothatof Q inaresonantcircuit. Forgoodresonanceweneedahigh valueof ωc τ .
frequencythisresonantabsorptionofelectromagneticwaves isknownas cyclotronresonance.Thesharpnessoftheresonancedepends stronglyonthevalueof ωc τ ,asshowninFig.1.7,whereIm k ,normalized toitsvalueat ω/ωc =1,isplottedagainst ω/ωc .Itmaybeseenthatthe resonanceishardlynoticeableat ωc τ =1.
Thecurveshavebeenplottedusingtheapproximateeqn(1.61);neverthelesstheconclusionsareroughlyvalidforanyvalueof ωp .Ifyouwantmore accurateresonancecurves,useeqn(1.59).
Whyistheresuchathingascyclotronresonance?Thecalculationfromthe dispersionequationprovidesthefigures,butifwewantthereasons,weshould lookatthefollowingphysicalpicture.
Fig.1.7
Cyclotronresonancecurvescomputed fromeqn(1.61).Thereismaximum absorptionwhenthefrequencyofthe electromagneticwaveagreeswiththe cyclotronfrequency.
Supposethatatacertainpointinspacethea.c.electricfieldisatright anglestotheconstantmagneticfield, B0 .Theelectronthathappenstobeat thatpointwillexperienceaforceatrightanglesto B0 andwillmovealongthe arcofacircle.Wecanwriteaforceequation.Whenthedirectionofmotionis alongthedirectionof E themagneticandcentrifugalforcesarebothatright anglestoit,thus r istheinstantaneousradiusof curvatureoftheelectron’spath.
Consequently,theelectronwillmovewithanangularvelocity
Theorbitswill not becircles,forsuperimposedonthismotionisanaccelerationvaryingwithtimeinthedirectionoftheelectricfield.Nowifthefrequency oftheelectricfield, ω ,andthecyclotronfrequency, ωc ,areequal,theamplitude oftheoscillationbuildsup.Anelectronthatisacceleratednorthinonehalfcyclewillbereadytogosouthwhentheelectricfieldreverses,andthusits speedwillincreaseagain.
Underresonanceconditions,theelectronwilltake upenergyfromtheelectricfield;andthatiswhatcausestheattenuationofthe wave.Whyisthe ωc τ> 1conditionnecessary?Well, τ isthecollisiontime; τ =1/ωc meansthattheelectroncollideswithalatticeatomaftergoinground oneradian.Clearly,iftheelectronisexposedtotheelectricfieldforaconsiderablyshortertimethanacycle,notmuchabsorptioncantakeplace.Thelimit mightbe ωc τ =1.
Nowwemayagainaskthequestion:whatiscyclotronresonancegoodfor? Therehavebeensuggestionsformakingamplifiersandoscillatorswiththe
Noticethatanyincreasein speed mustcomefromtheelectricfield; theaccelerationproducedbya magneticfieldchangesdirection, notspeed,sincetheforceisalways atrightanglestothedirectionof motion.ThisisthebasisofFouriertransformioncyclotronmass spectrometry(FT–ICR–MS).
