2.9 Landau Levels and Quasiparticles in Magnetic Field
2.9.3 The Integer Quantum Hall Effect
As wesaw in Section2.8.3,a MOSFETcanbe used tocreate atwo-dimensional elec- tron gas. Supposethat weapplya magneticfieldperpendicular to the planeofthe gas.
If we applyavoltage in theplaneof thecarriers, we expect toobserve theHall effect, asillustratedinFigure2.38.Charged particlesmoving perpendiculartoamagneticfield
149 2.9 LandauLevelsandQuasiparticlesinMagneticField
z y x VH
VI I
B
±Fig.2.38 ThegeometryofaHalleffectmeasurement.
are accelerated in the direction perpendicular to their motion, which means that neg- ative charge will tend to pile up along one side of the plane, leading to a voltage in thedirection perpendicular tothe current.Thisis knownasthe Hall voltage.Note that the sign of the Hall voltage depends on the sign of the majority carriers in the sys- tem. Positive carriers (holes) will be accelerated to the right, while negative carriers (free electrons) moving in the oppositedirection (corresponding to current in the same direction) will accelerate to their left, thereby giving the opposite polarity of the Hall voltage.
In the previous sections, we looked at the effect of varying the magnetic field whilekeepingthenumberofcarriersinthe2DEGconstant.Alternatively,wecouldkeep themagneticfieldconstantandvarythenumberofcarriersinthe2DEGbyvaryingthegate voltageoftheFET,whichcontrolstheamountofbandbending.Inthiscase,wealsoexpect toseeoscillationsinthepropertiesofthesystemasthenumberofcarrierspassesthrough thecriticalnumbersdefinedbyfillingallthestatesinanintegernumberofLandaulevels.
Inparticular,weexpecttheconductivitypropertiesofthesystemtohavespecialproperties atthesepoints. Whenalltheelectrons exactlyfill allthe statesinanintegernumber of Landaulevels,thenbyPauliexclusion,noneoftheelectronscanchangestate,justlikea fullvalenceband.
Asurprisingresultoftheexperimentsisthatthepropertiesofthesystemdonotmerely oscillate.AsshowninFigure2.39,thereareplateausinthevoltagesmeasuredinboththe parallelandperpendiculardirectionsatthepointswherethenumberofelectronsequalsthe numberofstatesinanintegernumberofLandaulevels.Evenmoresurprising,whenthe systemisinoneoftheseplateaus,theHallresistanceofthesystemisequaltoaconstant, RH = h/e2ν, where ν is the number ofLandau levels filled and h and e are Planck’s constantandtheelectroncharge, universalconstantsofnature,independentoftheexact sizeandshapeormaterialpropertiesofthesystem.Thisisknownasthe integer quantum Hall effect.
We canunderstandhow thisquantized resistancearises byexaminingthecurrentthat flowsintheplane.TheHalleffectoccurswhenthereisabalanceofthemagneticforceand theelectricforceontheelectrons,
Ey = vxBz, (2.9.29)
where vx is the velocity of the electrons. The velocity is related to the current by the relation,
Jx= nqvx, (2.9.30)
2 0 1
50 100 150 200 250
0 3 4 5 6
Magnetic Field B (T)
7 80
2 4 Vx
Vy 6 10
8
Transverse Hall voltage Vx(mV) Transverse Hall voltage Vy(mV) 6
8
v= 5
v= 3
v= 4
±Fig.2.39 ExperimentalvoltagesmeasuredfortheintegerquantumHalleffect,asafunctionof B,forconstantcurrent.From Cage et al. 1985.
where n = νN/A isthenumberofcarriersperareainthetwo-dimensionalplane,with ν thenumberofoccupiedLandaulevels.Wethereforehave
Jx= νN AqEy
Bz. (2.9.31)
Thetotalmagneticflux à = BzA isequalto Nà0,where à0isthefluxquantumcontained intheorbitofasingleelectron,asdiscussedinSection2.9.1.Therefore
Jx= ν Nq Nà0Ey
= ν
´q|q|
h à
Ey, (2.9.32)
where we have used the definition of the flux quantum (2.9.11). Since Jx has units of amperes/cmin atwo-dimensional system,wecanconvertthis to anequationof current andvoltage.Multiplyingbothsidesbythelengthinthe y-direction,wehave
|Ix|= ν
´q2 h
à
|±Vy|≡ |±Vy|
RH . (2.9.33)
WhentheLandaulevelshavetwospinstates(seethediscussionattheendofSection2.9.1), therightsidewillbemultipliedby2.
Inprinciple,asmagneticfieldis tuned,thereisjust onepointatwhichthenumberof electronsinthe2DEGexactlymatchesthenumberofstatesinaLandaulevel.Naivelyone wouldexpectthatifthenumberofelectronsinthesystemnolongerexactlymatchesthe numberofstatesinanintegernumberofLandaulevels,theconductivitypropertieswillbe different.Experimentally,however,thepropertiesofthesystemarethoseofexactlyfilled Landaulevelsevenasthemagneticfieldorelectrondensityisvariedoverawiderange.
151 2.9 LandauLevelsandQuasiparticlesinMagneticField
h c 2h c 3h c E
D(E)
(a)
(b) E
D(E)
±Fig.2.40 (a)DensityofstatesofLandaulevelsinaperfecttwo-dimensionalsystem.(b)Densityofstatesinatwo-dimensional systemwithdisorder.Theshadedregionsindicatelocalizedstates.
Tounderstandthesystemproperly,wemustrememberthatthesystemisnotaperfect two-dimensionalplane.AsdiscussedinSections1.8.2and2.8,allrealsystemshavedis- order.OnemightexpectthatthisdisorderwouldmaketheHallresistanceevenlesslikely tohavetheexactvalue h/e2,buttheeffectistheopposite.
Figure2.40(a)showsthedensityofstatesoftwo-dimensionalLandaulevelsinaperfect system,whileFigure2.40(b)showsthedensityofstatesinarealsystem.Asdiscussedin Section1.8.2,areasonable hypothesis allows ustocategorizethe statesina disordered systemintotwoclasses.Inarandompotentiallandscape,low-energystateswillbe local- ized states, confined in energy minima, while abovesome energy cutoff known as the mobility edge,theelectronicstateswillbe extended states.Whenthereisamaximumto thepotentialenergy,statesabovesomethresholdwillnotbeconfined.
Thestatesinamagneticfieldwillhavethesameproperties.ForelectronstohaveLandau orbits,theymustbefreetomoveintheplane.Therefore,localizedstateswillnotcontribute totheLandaulevels.AsillustratedinFigure2.40(b),betweentheLandaulevelsthereis somerangeofenergyinwhichthestatesarealllocalized,knownasa mobility gap.
As discussedin Section1.1.2, adding disorderto asystem does not change thetotal numberofstates.Ifthereare N statesinanidealLandaulevel,therewillbe N statesinthe sameLandaulevelinthepresence ofdisorder.ThismeansthatastheFermilevelvaries throughthisrange,therewillstillbeoneexactlyfilledLandaulevel,untiltheFermilevel hitstherangeofthefreestatesinthenexthigherLandaulevel.
Thedisorderinthesystemisthereforeessentialfortheobservationoftheintegerquan- tumHalleffect.TheexistenceofarangeoflocalizedstatesallowstheFermileveltovary overawiderangewhilekeepinganexactlyfilledLandaulevel.Iftherewerenolocalized states,anintegernumberofLandaulevelswouldbefilledonlyatcertainexactvoltages.
BecauseofthisconspiracyofnaturetoforcetheHallresistancetoexactly h/e2,these measurementscanbeusedasameasureofthisfundamentalratioofconstantsofnature.
ThisratioisnowmeasuredroutinelytoaccuracyofpartsperbillionusingquantumHall measurements.
±Fig.2.41 Theelectronmotionintwo-dimensionalelectrongasinafullLandaulevel,whichoccursintheintegerquantumHall effect.Onlyelectronsbouncingalongtheloweredgeleadtonetcurrent.Edgestatesalongtheupperedge(indicated bydashedlines)aredepletedofelectrons.
Topological considerations. There is an alternative wayof looking at the quantized Hallresistance.Letusreturntothesemiclassicalpictureofelectronsincyclotron orbits, illustratedinFigure2.41. Electronsmoving incircular orbitsin thebulkoftheelectron gaswillnotcontributetoanetcurrent.Theonlyorbitsthatgiveanetflowofcurrentfrom onesidetotheotherarethosethatreflectofftheboundariesofthesystem,knownas edge states.Electronsinthesestateswillkeepbouncingalongoneedge.
Thecurrentisthenconstrainedtomoveinaone-dimensionalchannel.Thisthenlooks just like a one-dimensional quantum wire (QWR), discussed in Section 2.8.4. Recall fromthatdiscussionthatwederivedtheLandauerformulafortheconductionofasingle quantumchannel,accountingfortwodegeneratespinstates,
I = 2e2
h±V. (2.9.34)
ThisisthesameasthequantumHallformula(2.9.33)forasingleLandaulevelwhentwo spinstatesareincluded.
ThefactthatthequantumHallresistanceandtheLandauerresistancearethesameisnot anaccident.Bothcanbeviewedaselementaryexamplesofa topological effect.Theone- dimensionalchannelinboththeQWRandthequantumHalleffectisstableagainstchanges ofitsgeometry,sinceabendinthepathwillstillleaveitone-dimensional.Therefore,its propertiescannotdependongeometricfactors.Ingeneralterms,whenthegeometrydoes notmatter,onecansaythat h/e2isthenaturalunitforone-dimensionalresistance,since thecurrent I ∼ e/±t, where ±t isthetransit timeofanelectronacrossa channel,and
±t ∼ h/±E inquantummechanics.Thisgives I ∼ e(±E)/h ∼ e(e±V)/h ∼ (e2/h)±V.
Topologicaleffectsintheconductionpropertiesofmaterialshavebecomeamajortopic ofstudy;inmorecomplicatedmaterials,conductingsurfacestatescanplaythesamerole relativetonon-conductingbulk, three-dimensionalstatesthattheedge statesplayin the quantumHalleffectinrelationtothenon-conductingbulkofthetwo-dimensionalelectron gas.
153 2.9 LandauLevelsandQuasiparticlesinMagneticField
Exercise2.9.4 Whatkindofcurrentandvoltagesensitivityisrequiredtoobservetheinte- gerquantumHalleffect?Toanswerthis,supposethatatypicalstructureis1micron inwidth,andaHallvoltageof10 àVisobserved.Whatcurrentdoesthiscorrespond to,inamperes,forthefirstLandaulevel?