2.9 Landau Levels and Quasiparticles in Magnetic Field
2.9.1 Quantum Mechanical Calculation of Landau Levels
Inthissection,wewilluseafullquantum mechanicalmodelfortheenergystatesofan electroninamagneticfield.Aswewillsee,wegetnearlythesameresultsasthesemiclas- sicalcalculation.Wealsowillbeabletoderivethenumberofallowedquantumstates.
WebeginwiththeHamiltonianforaparticleinanelectromagneticfield, H = 1
2m(±p − q±A)2, (2.9.13)
with q thechargeoftheparticleand m theeffectivemass.Thisgivesthetime-independent Schrửdingerequation,
1
2m(−i±∇− q±A)2ψ = Eψ. (2.9.14)
There are various choices for the gauge of the A-field. For a magnetic field in the z-direction, forconveniencewe picktheCoulombgauge with ±A = Bxˆy.Wethen guess theformofthesolution
ψ = ei(kyy+kzz)φ(x), (2.9.15) which,whensubstitutedinto(2.9.14),gives
− ±2 2m
∂2φ
∂ x2 +1
2mω2c(x − x0)2φ =
ắ
E − ±2kz2 2m
¿
φ, (2.9.16)
wherewehavedefined
x0= ±ky
mωc (2.9.17)
and
ωc = |q|B
m , (2.9.18)
which is the cyclotron frequency, used previously in Section 2.2. Equation (2.9.16) is just the equation for a one-dimensional harmonic oscillator, which has the quantized eigenvalues
E = ±2k2z 2m + ±ωc
´ ν −1
2 à
, (2.9.19)
where ν isanintegerthat runsfrom 1toinfinity(themathematicsofthequantized har- monicoscillatorarereviewedinAppendixD).Figure8.25showstheseenergylevelsasa functionofmagneticfield.Thesearecalled Landau levels.Ifthe B-fieldislargeenough, theenergysplittingbetweentheLandaulevels, ±ωc,willbelargecomparedto kBT,and theelectronswillalllieinthelowestpossibleLandaulevel.Notethat althoughwehave treatedthe y-axisand x-axisdifferently inourchoiceof ±A,theenergyoftheeigenstates
145 2.9 LandauLevelsandQuasiparticlesinMagneticField
E
B
= 1
= 2
= 3
= 4
±Fig.2.36 Landaulevelsofchargedparticlesinamagneticfield.
doesnotdependonthischoice,asispropersincethevalue of ±B doesnotdependonthe choiceofgaugefor ±A.
ThenumberofelectronsthatcanfitinagivenLandauleveldependsonthetotalnumber ofstatesofaLandaulevel.To calculatethis, wecannolongeruse ±k astheappropriate quantumnumber,aswedidinSection1.8,tocalculatethedensityofstates.Theproper quantumnumbersare kzand ν,theLandaulevelnumber.Supposethedimensionsofthe systemare −Lx/2 < x < Lx/2inthe x-directionand −Ly/2 < y < Ly/2inthe y-direction.
Thenfor thequantized motionin theplaneperpendicular to themagneticfield, x0 runs from −Lx/2to Lx/2,andtherefore,fromdefinition(2.9.17), kyrunsfrom −mωcLx/2± to +mωcLx/2±.
Sincetheformofthesolution(2.9.15)isaplanewavealong y,andthisplanewaveis subjecttotheboundarycondition −Ly/2 < y < Ly/2,thesamelogicusedinSection1.8 impliesthat ky canonlyhavevalues
ky= 2πNy
Ly , (2.9.20)
where Ny isaninteger.Thetotalnumberofstatesisthereforegivenbythetotalrangeof kydividedbythedistancein k-spaceper kystate,thatis,
N = mωcLx/±
2π/Ly
= mωcLxLy
2π± . (2.9.21)
This implies that the total number of states in a Landau level in the dimensions perpendiculartothemagneticfieldis
N = |q|BA
h , (2.9.22)
where A = LxLy is the area of the plane. (This does not take into account spin; see the discussion of spin below.) The total density of states is therefore proportional to
the magnetic field, and therefore also proportional to the cyclotron energy of the Lan- dau level. In the z-dimension perpendicular to the plane, the density of states is still givenbytheone-dimensionalformulafromSection2.8.1, D(E)dE = (L/2π)(√
m/2)dE/
(±√
E − E0).
The number of states (2.9.22) has a natural interpretation in terms of the semiclas- sical cyclotron orbits discussed above. Because electrons are fermions, two electrons with the same spin in the same Landau level cannot orbit the same flux quantum. We musthaveone electronofagivenspinperfluxquantum. Thenumber ofelectronstates in a two-dimensional plane is therefore given bythe total flux divided bythe flux per state,
N = BA à0
= |q|BA
h , (2.9.23)
whichisjustthesameastheresultdeducedabove.
ThenumberofstatesperLandaulevelinaplaneperpendiculartothemagneticfieldis aconstantthatdependsonlyonthestrengthofthe B-field.Noticealsothatthenumberof statesdoesnotdependontheeffectivemassoftheparticles,justasthevalueoftheflux quantumdoesnotdependontheeffectivemass.
Exercise2.9.2 CalculatethetotalnumberoffreeelectronsthatcanoccupyasingleLandau levelformagneticfield10Tinasolidcubewithdimension1cm,iftheconducting electronshaveeffectivemass0.1m0,attemperature T = 1K.
Spin splitting in Landau levels.The above calculation for the Landau levels of an electronreliedonlyonthemassandthechargeoftheelectron;wedidnottakeintoaccount thespinoftheelectrons.Whenwetakeintoaccountthespinoftheelectron,theZeeman effect(derivedinAppendixF)leadstoashiftoftheelectronenergygivenby
Es=ág
2μBB =ág
´ ±e 4m0
à
B, (2.9.24)
where B isthemagneticfield, μB = ±e/2m0 is the Bohr magnetonwith e theelectric chargeand m0thevacuumelectronmass,and g isthe Landé g-factor.Theenergyshifts tothepositiveforelectronswithspinintheoppositedirection tothemagneticfieldand tothenegativeforspininthesamedirectionasthemagneticfield.Thisis knownasthe Zeeman splitting.
In vacuum, g ³ 2.002.(The Diracequation, discussedin Appendix F,gives exactly 2; the slight deviation from this value is due to higher-order field theory corrections.) Therefore, in vacuum,the Zeeman shiftdownward of the lowest spin state almost per- fectlycancelstheLandaulevelshiftupward.Insolids,however,boththeeffectivemass oftheelectronusedin(2.9.3)and theg-factorusedin(2.9.24)dependontheproperties oftheelectronbands.Amethodofcalculatingtheg-factorofelectronsinasolidwillbe presentedinSection10.2.Inatypical semiconductor,theZeemansplittingis muchless thantheLandaulevelenergy.Forexample,inGaAs,theeffectivemassofanelectronis
147 2.9 LandauLevelsandQuasiparticlesinMagneticField
N
B
= 1
= 2
= 3
= 4
±Fig.2.37 Thinlines:thenumberofstatesintheLandaulevelsofaconductorasafunctionofmagneticfield.Dashedline:the
criticalnumber,equaltothenumberoffermionsinthesystem.Heavyline:thenumberofavailablestatesinthe lowestoccupiedLandaulevelofthesystemasafunctionofmagneticfield.
about0.07m0andtheg-factorisabout0.4,leadingtospinsplittingabout1/80ofthelow- estLandaulevelenergy.Undermostconditionsforelectronsinsolidsone cantherefore assumethateachLandaulevelhastwospinstateswithnearlyequalenergy.