Rk44pzpz.com

AppendixAHarmonicOscillatorQuantum-mechanicaldescriptionofsmallvibrationsofaparticleintheone-dimen-sionalpotentialU(x)neartheequilibriumpositionx=x0(determinedbytherequire-mentdU(x)/dx=0)iscarriedoutbyexpandingU(x)inpowersof(xx0)withtheaccuracyuptothesecond-orderterms.
Bychoosingx0=0,wehaveU(x)=U(0)+12d2Udx2x=0x2,(A.
1)andtheeigenstateproblemhoscψ(x)=Eψ(x)isdeterminedbytheHamiltonianoftheharmonicoscillatorhosc=p22m+mω22x2,ω=1md2Udx2x=0,(A.
2)whichisquadraticinbothmomentumandcoordinate.
TheenergyEiscountedfromU(0),andωisthevibrationfrequencyoftheclassicaloscillatorneartheminimumofthepotentialenergyatx=0.
Introducingthedimensionlesscoordinateandenergyaccordingtoq=xmω/andε=E/ω,werewritetheeigenstateproblemintheform12d2dq2+q2εψ(q)=0.
(A.
3)If|q|islargeincomparisontounity,thewavefunctiondecreasesasexp(q2/2),andthemaincontributionfromthekinetic-energytermisequalto(q2/2)exp(q2/2)andcancelsthepotential-energyterm.
Therefore,suchasolutionsatisesEq.
(A.
3)forε1.
Inthegeneralcase,thewavefunctioniswrittenasaproductofexp(q2/2)byanite-orderpolynomial.
Theeigenstateproblem(A.
3)withtheboundarycon-ditionψ||q|→∞=0hasthefollowingsolution:ψn(q)=Nneq2/2Hn(q),εn=n+12,n=0,1,2,A.
4)wherethequantumnumbernnumbersthelevels,thenormalizationconstantNn=[n!
2n√π]1/2isdeterminedbyusingtheidentity∞∞dqexp(q2)[Hn(q)]2=2nn!
√π,733734QUANTUMKINETICTHEORYandtheHermitepolynomialofn-thorderisintroducedaccordingtoHn(q)=(1)neq2dndqneq2.
(A.
5)SincetheHamiltonian(A.
2)iseven(symmetric)withrespecttox,thewavefunctionscanbeeitherevenorodd.
AccordingtoEq.
(A.
4),theirparitycoincideswiththeparityofn,asitisseenfromEq.
(A.
5)orfromtheexpressionsH0(q)=1,H1(q)=2q,H2(q)=4q22,H3(q)=8q312q,H4(q)=16q448q2+12,A.
6)UsingtherecurrencerelationsfortheHermitepolynomials,qHn(q)=nHn1(q)+12Hn+1(q),dHndq=2nHn1(q),(A.
7)weobtainthefollowingconnectionbetweenthewavefunctionsoftheneighboringstates:qψn(q)=n2ψn1(q)+n+12ψn+1(q),dψndq=2n2ψn1(q)qψn(q)=n2ψn1(q)n+12ψn+1(q).
(A.
8)Usingthem,onecanshowthatthematrixelementsofcoordinatearenon-zeroforneighboringstatesonly:n|x|n=2mω√nδn,n1√n+1δn,n+1.
(A.
9)Thesamepropertytakesplaceforthematrixelementsofthemomentumoperator:n|p|n=imω2√nδn,n1√n+1δn,n+1.
(A.
10)Usingtheoperatoridentityq2d2/dq2=(qd/dq)(q+d/dq)+1,onecanreformulatetheeigenstateproblem(A.
3)byintroducingnewoperatorsb=1√2q+ddq,b+=1√2qddq,(A.
11)whichalsoconnecttheneighboringstatesonly.
TheHamiltonianhoscisexpressedintermsoftheseoperatorsashosc=ω2d2dq2+q2=ωb+b+12.
(A.
12)Theoperators(A.
11)areHermitianconjugateandsatisfythecommutationrelation[b,b+]=1.
(A.
13)Theoperatorb+increasesthenumbernoftheoscillatorstate,whiletheoperatorbdecreasesthisnumber:b+ψn(q)=√n+1ψn+1(q),bψn(q)=√nψn1(q).
(A.
14)APPENDIXA:HarmonicOscillator735Thecoordinateandmomentumoperators,whichmixthen-thand(n±1)-thstates,areexpressedthroughb+andbasx=2mω(b++b),p=imωddq=imω2(b+b).
(A.
15)Letusactbytheoperatorb+onthefunctionψ0(q)describingtheground("vacuum")state.
Afternsequentialactions,weobtainthewavefunctionofthen-thstate:ψn(q)=(b+)n√n!
ψ0(q),(A.
16)whileψ0(q)isdeterminedfromtheequationbψ0(q)=0.
UsingtheexplicitformofbfromEq.
(A.
11),wehaveψ0(q)=N0exp(q2/2).
Letusreformulatetheharmonicoscillatorproblembyturningfromthecoordinaterepresentationtotheoccupationnumberrepresentation.
Weintroduceasetofket-vectors|0,|1nn}(A.
17)correspondingtothelevels0,1,.
.
.
n.
.
.
.
TheyareconnectedtoeachotherbytherelationsanalogoustotheonesintroducedinEq.
(A.
14):b+|n=√n+1|n+1,b|n=√n|n1.
(A.
18)Eachelementoftheset{|n}isobtainedfromtheground-stateket-vector|0afteranumberofactionsoftheoperatorb+,asinEq.
(A.
16):|n=(b+)n√n!
|0.
(A.
19)Therefore,theoperatorsb+(b)describecreation(annihilation)ofaquantumwithenergyω.
IftheHamiltonianoftheoscillatorytypedescribesvibrationalmodesofelectromagneticeldorsmallvibrationsofcrystallattice,suchquantacorrespondtoquasiparticles,photonsorphonons.
TheHermitianoperatorN=b+biscalledthequantum(particle)numberoperator,since|nsatisestheeigenstateproblemN|n=n|n.
Tocheckit,onemayusetheoperatorequationNb+=b+(N+1)andtheexplicitformoftheket-vector|n.
ThisleadstoachainofnequationsN|n=b+√n!
(N+1)(b+)n1|0(b+)n√n!
(N+n)|0=n|n.
(A.
20)OnemayrewritetheHamiltonian(A.
12)intermsofNashosc=ω(N+1/2).
Asanexampleoftheoperatoralgebrabasedonthepropertiesofb+andb,wecalculatethematrixelementn|eikx|n.
Suchelementsappear,forexample,incal-culationofthematrixelementsofapotentialV(x)expressedthroughitsFouriertransformaccordingtoV(x)=(2π)1dkeikxV(k).
Expressingthecoordinateop-eratoraccordingtoEq.
(A.
15),werewritethematrixelementasn|eikx|n=n|eiκ(b++b)|n,κ=k2mω,(A.
21)whereκisadimensionlesswavenumber.
Further,weusetheoperatoridentity(knownasWeylidentity)eA+B=eAeBe[A,B]/2(A.
22)736QUANTUMKINETICTHEORYwhichistrueundertherequirementthatthecommutator[A,B]commuteswithbothAandB.
WesubstituteA=iκb+andB=iκbandobtain[A,B]=κ2sothatthisrequirementisfullled.
Nowwehaven|eiκ(b++b)|n=eκ2/2ln|eiκb+|ll|eiκb|n.
(A.
23)Thematrixelementsoftheexponentialoperatorsarecalculatedbyexpandingtheexponentsinseries,exp(A)=∞p=0(A)p/p!
.
ThenweuseEq.
(A.
18)andobtainl|eiκb|n=(iκ)nl(nl)!
n!
l!
,n|eiκb+|l=(iκ)nl(nl)!
n!
l!
,(A.
24)becauseonlythetermswithp=nlandp=nlcontributetothematrixelementsl|eiκb|nandn|eiκb+|l,respectively.
ThesumoverlinEq.
(A.
23)runsuptol=min{n,n}.
Belowweputn>nandchoosethevariableofsummationasm=nl,wheremrunsfrom0ton.
UsingEq.
(A.
24),weobtainn|eiκ(b++b)|n=eκ2/2nm=0√n!
n!
(iκ)nn+2mm!
(nn+m)!
(nm)!
.
(A.
25)ThesumovermcanbewrittenthroughtheLaguerrepolynomialsdenedasLαn(x)=nm=0(1)m(n+α)!
(nm)!
(α+m)!
m!
xm.
(A.
26)Substitutingα=nnandx=κ2,wenallyndn|eikx|n=n!
n!
(iκ)nneκ2/2Lnnn(κ2),κ=k2mω.
(A.
27)Thisresult,ofcourse,canbeobtaineddirectlyaftercalculatingtheintegralswiththewavefunctions(A.
4).
Equation(A.
27)havenumerousapplications,inparticular,inmagnetotransporttheory.
AppendixBMany-BandKP-ApproachLetusconsidertheelectronstatesincrystalsunderexternalelds.
Thedynamicsoftheelectronscanbedescribedbythekp-formalismiftheseeldsaresmoothonthescaleofthelatticeconstant.
Theelectricandmagneticeldstrengths,ErtandHrt,areexpressedthroughthevectorpotentialArtandscalarpotentialΦrtaccordingtoEq.
(4.
3).
Byincludingtheeld-inducedcontributionsintotheone-electronHamiltonian(5.
4),weobtainhcr(t)=hcre2mec(p·Art+Art·p)+(e/c)22meA2rt+Urt+B(σ·[*Art]),(B.
1)wheremeisthefree-electronmass,Urt=eΦrtisthepotentialenergyproportionaltothescalarpotential,andB=|e|/2mecistheBohrmagneton.
ThelastterminEq.
(B.
1)describesthePauliinteractionofelectronswiththemagneticeld,andwehaveneglectedthecontribution(B/2mec2)σ·[Ucr(r)*Art]comingfromthespin-orbitinteractiontermforthereasonofitssmallness.
Neartheextremump=0(ageneralizationtothecaseofanarbitraryextremump=p0isstraightforward),wewriteacompletesetofeigenfunctionsasψlp(r)=V1/2exp(ip·r/)ul(r);seeEq.
(5.
5).
Weremindthattheindexlcontainsboththebandnumbernandthespinnumberσ.
ThematrixelementsofArtandUrtcanbewrittenthroughtheirspatialFouriertransforms,whereq=(pp)/:lp|Art|lpδllAqt,lp|Urt|lpδllUqt,lp|A2rt|lpδllA2qt.
(B.
2)Therefore,withinthekp-approachthematrixelementsoftheHamiltonian(B.
1)taketheformlp|hcr(t)|lp=δppHll(p)+δlle2mec(p+p)·Aqt+e22mec2A2qt+Uqt+B(σll·Hqt)ecAqt·vll,(B.
3)737738QUANTUMKINETICTHEORYwhereHll(p)isintroducedbyEq.
(5.
6),σllistheinterbandmatrixelementofthespinoperator,andthematrixelementsofthevelocityoperator,vll,aregivenbyEq.
(5.
7).
Thetime-dependentenvelopefunctionslptintheexpansion(5.
8)isdeterminedfromthesystemofequationsitlpt=lplp|hcr(t)|lplpt(B.
4)describingthedynamicsofmany-bandelectronstates.
Sincetheproblemisspatiallyinhomogeneous,itisconvenienttousethecoordi-naterepresentationoftheenvelopefunctioninsteadofthemomentumrepresentation.
ExpandingtheexactwavefunctionasinEq.
(5.
8),Ψ(rt)=lplptψlp(r),wemaywriteitasΨ(rt)=llrtul(r),wherelrtisconnectedtolptbytheFouriertrans-formationslrt=1√Vpeip·r/lpt,lpt=1√Vdreip·r/lrt.
(B.
5)ThesumoverpmustbetakeninsidetherstBrillouinzone.
Thecoordinate-dependentenvelopefunctionlrtisgovernedbythefollowingequation:itlrt=lHlllrt,Hll=δll(εl+Urt)+vll·πrt(B.
6)+14αβDαβll(παrtπβrt+πβrtπαrt)+BGll·Hrt,whereπrt=peArt/cisthekinematicmomentumoperator.
Equation(B.
6)de-scribestheelectrondynamicsinexternaleldswhenthereareseveralbands,num-beredbytheindiceslandl,closetoeachotherinenergy.
Thecontributionoftheother,remotebandsistakenintoaccountthroughthesymmetricinverseeectivemasstensorDαβll=δllδαβme+12s=l,l(vαlsvβsl+vβlsvαsl)[(εlεs)1+(εlεs)1],(B.
7)whichgeneralizesthetensor(5.
11)tothemany-bandcase.
Theeectivespinvector,Gαll=σαllime2s=l,l[vls*vsl]α[(εlεs)1+(εlεs)1],(B.
8)isalsodeterminedbytheremotebandcontributionsanddescribesmodicationoftheg-factorofelectronsinthecrystal.
InordertodescribetheremotebandsinthewaygivenbyEqs.
(B.
6)(B.
8),oneneeds,apartfromthecondition|εlεs||εlεl|,therequirementsofsmoothnessofexternaleldsonthescaleofthelatticeconstantandoflowfrequencyoftheseelds,toensureω|εlεs|/.
Themany-bandcurrentdensityoperatorIll(r,t)isintroducedasaproportionalitycoecientdeterminingthecorrectiontotheHamiltonianduetoasmallvariationofthevectorpotential,δArt,accordingtotheexpressionδHll=1cdrIll(r,t)·δArt.
(B.
9)APPENDIXB:Many-BandKP-Approach739Inotherwords,thecurrentdensityoperatorisafunctionalderivativeofthematrixHamiltonian(B.
3)overthevectorpotential.
ComparingEqs.
(B.
9)and(B.
6),weobtaintheexplicitexpressionforIll(r,t):Iαll(r,t)=evαllδ(xr)+e2βDαβll[pβδ(xr)+δ(xr)pβ](B.
10)e2cβDαβllAβxtδ(xr)ie2me{[p*Gll]αδ(xr)δ(xr)[p*Gll]α}.
ThersttermisthecontributionofthebandslandlwhichareincludedinthematrixHamiltonian(B.
6).
Thenexttwotermsdescribetheremote-bandcontributions.
Thelasttermisaspin-dependentcontributiontothecurrent.
Inthesimplestcaseofanon-degenerateband,whenthereisonlyonestateinthesetl,thematrixDαβllisreducedtotheinverseeectivemasstensor(5.
11),while2Gαllbecomesascalareectiveg-factormultipliedbythevectorofPaulimatricessothatthePaulicontributiontotheelectronHamiltonianiswrittenasgBσ·Hrt/2.
Thersttermintheexpression(B.
10)isomittedinthiscase.
Inthedipoleapproximation,onehastoconsideronlytheFouriercomponentIll(q,t)=drIll(r,t)exp(iq·r)atq=0.
SincethespincontributioninEq.
(B.
10)isequaltozerointhiscase,thecurrentdensityoperatorIll(q=0,t)≡Ill(t)iswrittenasIαll(t)=evαll+eβDαβllpβecAβt,(B.
11)whereonlythevectorpotentialofthehomogeneouseldremains.
Iftheenergiesofelectronsinthebandslandlaresmallincomparisontotheinterbandenergy|εlεl|,onlytherstterminEq.
(B.
11)isessential.
Considertheelectronstatesindeformedcrystals.
Smalldeformationsofelasticmaterialsaredescribedbyasymmetrictensorofdeformation,εαβ=εαβ(r):εαβ=12uαrβ+uβrα,(B.
12)whereu=u(r)isthedisplacementvectoratthepointr.
ThedeformationchangesthesymmetryofthecrystallatticeandthepotentialenergyW(r)intheelementarycell.
Sincethepointrunderthedeformationisshiftedtothepoint(1+ε)r,themomentumistransformedto(1ε)p,withinthelinearaccuracy.
Thelinearinu(r)contributiontothecrystalHamiltonian(5.
4)iswrittenasδH(ε)=αβpαεαβpβme+αβVαβ(r)εαβ.
(B.
13)Thespin-orbitcontributionisneglectedinthisexpressionbecausetherelativisticcorrectionstoδH(ε)aresmall.
ThematrixVαβdescribesthedeformation-inducedmodicationofthepotentialenergyW(r)accordingtoWε[(1+ε)r]W(r)=αβVαβ(r)εαβ.
(B.
14)Usingthesetofeigenfunctions(5.
5),wendthedeformation-inducedcontributionstothekp-Hamiltonian(5.
6):δHll(ε)=αβΞαβllεαβ,Ξαβll=(pαpβ)llme+Vαβll.
(B.
15)740QUANTUMKINETICTHEORYToestimatethedeformation-potentialtensorΞαβll,onemayusethemodelsofcrystallatticewitheitherrigidordeformedions.
Inthedeformedionapproximation,thechangeofthepotentialenergyissmallsothatΞαβllisdeterminedonlybythersttermofEq.
(B.
15).
Intherigidionmodel,thecrystalpotentialW(r)isapproximatedbyasumofatomicpotentialsVa(rRi)placedatthelatticesitesRi,andthedeformationmerelyshiftsthesitepositionsto(1+ε)RkwithoutanychangeofVa(r).
Forbothapproximations,thetensorsΞαβllappeartobeoftheorderofatomicenergies,thoughthedeviationsoftheirvaluesfromexperimentaldataareconsiderable.
OneshoulduseamoredetaileddescriptionofthebandstructureinordertocalculateΞαβll.
Inthevicinityofanon-degenerateextremumoftheconductionband,itisconvenienttoconsiderthedeformation-potentialtensorinthemainaxeswhoseorientationisdeterminedbythecrystalsymmetry.
AccordingtoEq.
(B.
15),thesymmetryofthistensoristhesameasfortheeectivemasstensor(5.
11).
Ifthesurfacesofequalenergyareuniaxialellipsoids,thedeformation-inducedcontribution(B.
15)totheHamiltonianisexpressedthroughtwoconstants,thelongitudinal,d,andtransverse,d⊥,deformationpotentials,asδHcc(ε)=d⊥(εxx+εyy)+dεzz.
Inthespherically-symmetriccase,onehasd=d⊥=D,andtheinducedenergyδHcc(ε)=Dαεαα=Ddivu(r)isproportionaltothechangeofthecrystalvolumeduetothedeformations.
Inotherwords,theisotropicconductionbandissimplyshiftedinenergyduetothehydrostaticcomponentofthedeformationanddoesnotfeelthedisplacementsinducedbyuniaxialstresses.
Considerthesimplestcasedescribedbythemany-bandkp-approach,whentherearetwospin-degeneratebands(conductionandvalencebands)closeinenergy.
TheircontributionstotheHamiltonian(5.
6)shouldbeconsideredintheframesofatwo-bandkp-model.
Theenvelopewavefunctionhasfourcomponentsnumberedbythebandindexn=c,vandspinindexσ=±1.
Thereareonlytwonon-zerocomponentsofthevelocitymatrix(5.
7),theinterbandvelocitiesvcvandvvc.
SincethevelocityoperatorisHermitian,vcv=vvc.
Belowweconsiderthecaseofcubiccrystals,whenthesevelocitiesareisotropic(theisotropy,however,existsonlyifweconsideranextremuminthecenteroftheBrillouinzone).
Deningvcv=vvc≡s,onemaywritethevelocity(5.
7)intheformofa4*4matrixvα=s0σασα0,(B.
16)whereα=x,y,zistheCartesiancoordinateindex,0denesa2*2matrixwhoseelementsarezeros,andσαisa2*2matrixwhichactsonspinvariablesonly.
Oneshouldchoosethismatrixinsuchawaythattheright-handsideofEq.
(5.
11)be-comesspherically-symmetricandindependentofthespinquantumnumberσ.
Theseconditionsarefullledifthesetof2*2matricessatisestheanticommutationrela-tionsσασβ+σβσα=0,α=β,σασα=1,(B.
17)where1istheunit2*2matrix.
Therefore,σαmaybechosenasthePaulimatrices:σx=0110,σy=i0110,andσz=1001.
Letussetthereferencepointofenergyinthemiddleofthegapbetweenthebandsandintroducetheeectivemassmaccordingto2ms2=εg.
Thenwewrite4*4matricesoftheHamiltonianhandvelocityoperatorvash=ms2ρ3+(p·v)+p22M,v=sρ1σ.
(B.
18)APPENDIXB:Many-BandKP-Approach741TheHamiltonianinEq.
(B.
18)diersfromtheDiracHamiltonianforarelativisticelectrononlyduetothepresenceoftheremote-bandcontributionp2/2M,whichappearssincethetensor(B.
7)inthecaseunderconsiderationisreducedtoascalardenotedasM1.
The4*4matricesρi(i=1,2,3)satisfythecommutationrelations(B.
17)andcanbechosenintheformρ1=0110,ρ2=i0110,ρ3=1001.
(B.
19)ThesymbolicproductofρibyaPaulimatrix,usedinEq.
(B.
18)andbelow,simplymeansthateachoftheunitmatricesinρishouldbereplacedbythePaulimatrix.
Wealsonotethatthescalarcontributionstothematrixexpressions,likep2/2MinhofEq.
(B.
18),shouldbeformallyconsideredasthecontributionsstandingattheunitmatrices.
IfthecontributionoftheremotebandsisnotessentialsothatM=∞,theexpressions(B.
18)and(B.
19)describea"relativistic"electronofmassm,whiletheinterbandvelocitysplaystheroleofthevelocityoflight.
Inasimilarwayasintherelativisticquantumtheory,theHamiltonianhisdiag-onalizedbyap-dependentunitarytransformationUp=ηp+12ηp1/2+iρ2σ·ppηp12ηp1/2,ηp=1+(p/ms)2(B.
20)accordingtoUphU+p=ms2ηpρ3+p22M.
(B.
21)Sinceσhasdroppedoutofthisexpression,theelectronstatesappeartobespin-degenerate.
TheenergyspectraaredeterminedfromtheeigenstateproblemfortheHamiltonian(B.
21):εcp=p22M+ms2ηp,εvp=p22Mms2ηp.
(B.
22)Theenergyspectrumisparabolic,±[εg/2+p2/2m]+p2/2M,atpmsandbecomeslinear,±sp,atpms.
Thisbehavior,correspondingtoatransitionfromnon-relativistictorelativisticregimesforaDiracelectron,iscalledthenon-parabolicityofenergyspectrum.
Inthelimitingcaseofzeroenergygap,εg=0,onehasm→0andtheenergyspectraatM→∞arealwayslinear,εc,vp=±sp.
Thissituationtakesplaceinsomesemiconductoralloys,whereachangeinthealloycompositionleadstoinversionofthesignofεg.
SuchmaterialsarecalledthegaplesssemiconductorsoftypeI.
Anotherkindofgaplessmaterials(typeII)formCd1xHgxTealloys,wherethezero-gapsituationisrealizedinacertainrangeofalloycomposition,andtheenergyspectraareparabolic.
Todescribethiscase,oneneedstotakeintoaccountmoresophisticatedbandmodelsinvolvingseveralenergybands,whichisbeyondthescopeofthisbook.
Nevertheless,thetwo-bandHamiltoniandescribedabovereectsessentialfeaturesofnarrow-gapandzero-gapmaterials.
WhenthediagonalizationoftheHamiltonianbytheunitarytransformation(B.
20)iscarriedout,theinterbandvelocitymatrixvandthepotentialenergyUr=V1*peiq·rUqaretransformedaccordingly.
ThevelocitymatrixbecomesUpvU+p=pmηpρ3+sρ1σηp1ηpp(σ·p)p2.
(B.
23)742QUANTUMKINETICTHEORYTheinterband(proportionaltoρ1)partofthisoperatorisoftheorderofs,whilethediagonal,withrespecttothebandindex,partchangesfromp/mforsmallptosp/pforlargep.
Toconsiderthetransformationofthepotentialenergy,weusethemomentumrepresentation.
Namely,thematrixelementp|Ur|pistransformedintoU(pp)/UpU+p,wherethematrixfactorisUpU+p=12(ηp+1)(ηp+1)ηpηp1+(σ·p)(σ·p)ppηp1ηp+1ηp1ηp+1+iρ2(σ·p)pηp1ηp+1(σ·p)pηp1ηp+1.
(B.
24)Thisfactorisclosetounityforsmallmomenta.
Thesecondtermontheright-handsideofEq.
(B.
24)containsthecontributionproportionaltoσ[p*p].
Thiscontri-butionisresponsiblefortheintrabandspin-ipscattering,whichbecomesimportantwithincreasingp/ms.
Theinterband(proportionaltoρ2inEq.
(B.
24))contribu-tionsbecomeessentialwhenthemomentumtransfer|pp|iscomparabletoms,i.
e.
,whenthecharacteristicspatialscaleofthepotentialenergyUriscomparabletotheinterbandlength/ms.
AppendixCWignerTransformationofProductTheproductofoperators,ct=atbt,iswritteninthecoordinaterepresentationasct(x1,x2)=dxat(x1,x)bt(x,x2).
(C.
1)BelowweconsiderthetransformationofEq.
(C.
1)totheWignerrepresentation.
AccordingtothegeneraldenitionoftheWignertransformationinSec.
9,wehavect(r,p)=drexpiPrt·r*dxatr+r2,xbtx,rr2,(C.
2)whereat(.
.
.
)andbt(.
.
.
)canbewrittenbyusingtheinverseWignertransformation(9.
7):atr+r2,x=dp1(2π)3atr+x2+r4,p1*expip1+ecA(r+x)/2+r/4,t·rx+r2,(C.
3)btx,rr2=dp2(2π)3btx+r2r4,p2*expip2+ecA(x+r)/2r/4,t·xr+r2.
(C.
4)InsteadofthevariablesxandrinEq.
(C.
2),weintroducenewcoordinatesr1andr2accordingtor1=r+x2+r4,r2=r+x2r4,(C.
5)sothatr=2(r1r2)andx=r1+r2r.
UsingthesevariablesandEqs.
(C.
3)and(C.
4),werewriteEq.
(C.
2)intheformct(r,p)=dp1(2π)3dp2(2π)3dr1dr2|J3|at(r1,p1)bt(r2,p2)*e(2i/)Prt·(r1r2)e(2i/)Pr1t·(rr2)e(2i/)Pr2t·(r1r),(C.
6)743744QUANTUMKINETICTHEORYwherePrkt≡pk+(e/c)ArktisintroducedinEq.
(9.
6),andtheJacobianofthecoordinatetransformationisintroducedintheusualway,as|J3|=(r,x)(r1,r2).
(C.
7)Theseexpressionsarewrittenforthe3Dcase.
Toconsiderone-ortwo-dimensionalproblems(d=1ord=2)oneshouldwritethephasevolume(2π)dintheintegralsovermomenta.
TocalculatetheJacobian(C.
7),weuserαr1β=2δαβ,rαr2β=2δαβ,xαr1β=xαr2β=δαβ(C.
8)andobtainJd=22d,d=1,2,3.
Theexactformulafortheoperatorproducttrans-formationtakestheformct(r,p)=dp1(2π)ddp2(2π)ddr1dr2|Jd|*at(r1,p1)bt(r2,p2)exp2iS(rp,r1p1,r2p2),(C.
9)wherethefactorS(rp,r1p1,r2p2)intheexponentisdeterminedbyS(rp,r1p1,r2p2)=(PrtPr2t)·(rr1)(PrtPr1t)·(rr2).
(C.
10)InordertosimplifyEq.
(C.
9)forsmoothfunctionsat(r,p)andbt(r,p),weintro-ducenewvariablesr1,2=r1,2randp1,2=p1,2p,andexpandthefunctionintheexpressionundertheintegralsofEq.
(C.
9)byusingArAr+r1,20.
Asaresult,wehavect(r,p)=|Jd|dp1(2π)ddp2(2π)ddr1dr2at(r,p)+atr·r1+atp·p1+.
.
.
bt(r,p)+btr·r2+btp·p2+.
.
.
*exp2i(r1·p2r2·p1).
(C.
11)ThecontributionproportionaltoatbtinEq.
(C.
11)isdeterminedbytheintegral|Jd|dp1(2π)ddp2(2π)ddr1dr2exp2i(r1·p2r2·p1)=22ddp1dp2δ(2p1)δ(2p2)=1.
(C.
12)TheproductsofatandbtbythederivativesofthesefunctionsvanishfromEq.
(C.
11),sincetheyaremultipliedbytheintegralscontainingthecontributionslinearinr1,2orp1,2.
Theseintegralsareequaltozero(thiscanbecheckedbythesubstitutionsr1,2→r1,2orp1,2→p1,2).
Iftheproductsrα1,2rβ1,2orpα1,2pβ1,2stayunderanintegralofthetype(C.
12),theyalsovanish.
Thismeansthattheproducts(at/rα)(bt/rβ)or(at/pα)(bt/pβ)dropoutofEq.
(C.
11).
Therefore,onlytheintegrals|Jd|dp1(2π)ddp2(2π)ddr1dr2APPENDIXC:WignerTransformationofProduct745*exp2i(r1·p2r2·p1)rα1pβ2pα1rβ2(C.
13)remain.
Theseintegralsstayattheproductsoftherstderivativesovercoordinateandovermomentum.
Ifα=β,weagainhavezeroinEq.
(C.
13).
Onemayprovethisstatementbychangingthesignsofthevariables.
Thecontribution∝δαβiscalculatedanalogoustoEq.
(C.
12),usingtheintegrationbyparts.
Theexpression(C.
13)isequaltoδαβ2ddp2πdre±(2i/)rpd1dp2πdre±(2i/)rprp=δαβ2ddpδ(2p)d1dp2πp2dripe±(2i/)rp=±δαβi2.
(C.
14)Therefore,thequasi-classicalexpressionfortheoperatorproductisgivenasfol-lows:ct(r,p)=at(r,p)bt(r,p)+i2(at,bt)rpC.
15)ThecontributionlinearinthePlanckconstantiswrittenthroughtheclassicalPoissonbrackets(at,bt)rp=atr·btpatp·btr.
(C.
16)TheseexpressionsareconsistentwithEq.
(9.
24),andthequantumcorrectiondeter-minedbyEq.
(C.
16)canbeneglectedunderthecondition/λp,whereλandparethecharacteristicspatialscaleandmomentumforthefunctionsatandbt.
The2-correctionstoEq.
(C.
15)canbewrittenifthenexttermsoftheexpansioninEq.
(C.
11)aretakenintoaccount.
Letususethequasi-classicalexpression(C.
15)inordertocheckthecommutationrelationforcoordinateandmomentum,[rα,pβ]=iδαβ.
AccordingtoEq.
(C.
15),theproductsoftheoperatorsstandinginthecommutatortaketheformrαpβ→rαpβ+i2δαβ,pβrα→pβrαi2δαβ.
(C.
17)Therefore,theclassicalcontributionstothecommutatorannihilateandthecom-mutatorisequaltoiδαβ.
Inasimilarfashion,onemayprovethattherelationvα=(i/)[h,rα]connectingthecoordinateandvelocityoperatorsisconsistentwiththeclassicalexpressionforthegroupvelocity.
AccordingtoEq.
(C.
15),hrα→εrprαi2εrppα,rαh→rαεrp+i2εrppα.
(C.
18)Composingthecommutator,weobtaintheclassicalexpressionvrp=εrp/p.
AppendixDDouble-TimeGreen'sFunctionsTheintroductionoftheGreen'sfunctionsofelectronsinChapter3isbasedupontheaveraging,overtheimpuritydistribution,oftheGreen'sfunctionoftheSchroedingerequation.
BelowwepresentamoregeneraldenitionofGreen'sfunc-tions,whichiswidelyusedinstatisticalphysics.
LetusrstintroducetheHeisenbergrepresentationofanarbitraryoperatorA:A(t)=eiHt/AeiHt/,(D.
1)whereHisthetime-independentHamiltonian.
TheoperatorAcanbeexpressedintermsofcreationandannihilationoperatorsobeyingeitherbosonicorfermioniccommutationrules.
Theretarded,advanced,andcausalGreen'sfunctions,whicharelabeledbytheindicesR,A,andc,respectively,aredenedthroughthecorrelationfunctionsofapairofHeisenbergoperatorsA(t)andB(t)accordingtoGRtt=iθ(tt)A(t)B(t)±B(t)A(t)≡A|BRtt,(D.
2)GAtt=iθ(tt)A(t)B(t)±B(t)A(t)≡A|BAtt,(D.
3)Gctt=iθ(tt)A(t)B(t)±iθ(tt)B(t)A(t)≡A|Bctt.
(D.
4)Hereandbelow,theupperandlowersignsinequationsstandforthefermionandbosonoperators,respectively.
Thisdenitionismadeforthesakeofconvenience,toemploythecommutationrulesforfermionsandbosonsintheequationsofmotionfortheGreen'sfunctions,seebelow.
Thedoubleangularbracketsdenotetheaver-aginginthesenseofEq.
(1.
18)Sp(η.
.
.
),andthestatisticaloperatorηistime-independentintheHeisenbergrepresentation.
Inthecaseofthermodynamicequilibrium,whenthestatisticaloperatorη=ηeqcommuteswiththeHamiltonianH,onecaneasilyshowthatGstt(s=R,A,c)dependonlyoftt,i.
e.
,Gstt=Gstt.
Therefore,itisconvenienttousetheenergyrepresentationoftheGreen'sfunctionsaccordingtoGstt=∞∞dε2πeiε(tt)/Gsε,Gsε=∞∞dteiεt/Gst.
(D.
5)747748QUANTUMKINETICTHEORYTheenergyrepresentationoftheGreen'sfunctionisalsodenotedasA|Bsε.
Inthecaseofthermodynamicequilibrium,thecorrelationfunctionA(t)B(t)canbeexpressedthroughtheretardedandadvancedGreen'sfunctionsintheenergyrepresentation.
LetusintroduceJAB(ω)=d(tt)eiω(tt)A(t)B(t),JBA(ω)=d(tt)eiω(tt)B(t)A(t).
(D.
6)FirstwenotethatJAB(ω)=eω/TJBA(ω).
(D.
7)Thisidentitycanbecheckedeasilyifwecalculatethetracesin.
.
.
intheexacteigenstaterepresentationandusetheequilibriumstatisticaloperatorwhichisex-pressedthroughthetemperatureT.
UsingEq.
(D.
7),werewriteGRεandGAε,whereε=ω,asGRεGAε=i∞∞dtθ(t)θ(t)eiωt∞∞dω2π(eω/T±1)JBA(ω).
(D.
8)Thestepfunctionθ(t)canberepresentedasθ(t)=i2π∞∞dxeixtx+iλ,λ→+0.
(D.
9)WesubstitutethisexpressionintoEq.
(D.
8),integratethisequationovertandx,andobtainGR,Aε=12π∞∞dωωω+iλ(eω/T±1)JBA(ω),(D.
10)whereλ→+0fortheretarded(R)andλ→0fortheadvanced(A)Green'sfunction.
Equation(D.
10)leadstotheexactrelationGAεGRε=i(eε/T±1)JBA(ω).
(D.
11)AssumingthatJBA(ω)isreal,wealsondGAε=GRεsothattheleft-handsideofEq.
(D.
11)canberewrittenas2iImGAεor2iImGRε.
Letusexpress[ωω+iλ]1throughtheprincipalvalueP(ωω)1anddelta-functionδ(ωω)asinproblem1.
4.
WendthatEq.
(D.
10)leadstothefollowingdispersionrelations:ReGRε=1πP∞∞dεεεImGRε,ReGAε=1πP∞∞dεεεImGAε.
(D.
12)Thespectralrepresentation(D.
10)anddispersionrelations(D.
12)aredirectlyrelatedtothespectralrepresentationanddispersionrelationsforthekineticcoecientsde-scribingthelinearresponseofthesystemunderconsideration,sincethesekineticcoecientscanbeexpressedthroughtheretardedGreen'sfunctions.
WenotethatAPPENDIXD:Double-TimeGreen'sFunctions749equationssimilartoEqs.
(D.
10)and(D.
12)canbewrittenforthecausalGreen'sfunctionaswell.
SincetheHeisenbergoperatorsA(t)satisfytheequationofmotionidA(t)/dt=A(t)HHA(t),onecanwritethefollowingequationofmotionfortheGreen'sfunc-tions:idGsttdt=δ(tt)A(t)B(t)±B(t)A(t)+[A,H]|Bstt.
(D.
13)Thedouble-timeGreen'sfunction[A,H]|Bsttstandingontheright-handsideofEq.
(D.
13)isdeterminedfromasimilarequationofmotionandexpressedthroughanotherdouble-timeGreen'sfunctioncontainingthecommutator[[A,H],H].
Inthiswayonegetsachainofcoupledequationswhichcanbecutunderappropriateap-proximations.
Inparticular,forthesystemswithweakinteractiononecanretainonlythetermsofagivenorder(linear,quadratic,etc.
)intheinteractionpartoftheHamiltonian.
TheGreen'sfunctionsofquasiparticlesaredenedbysubstitutingtheeldopera-torsΨxandΨ+x(ortheannihilationandcreationoperatorsofthesequasiparticles)inplaceofAandB.
Theone-particleGreen'sfunctioninthecoordinaterepresentationisintroducedasGstt(x,x)=Ψx|Ψ+xstt.
(D.
14)ExpandingΨxoverasetofquantumstatesasΨx(t)=kψ(k)xak(t),wheretheindexknumbersthesestates,wehaveGstt(x,x)=kkGstt(k,k)ψ(k)xψ(k)x.
Theone-particleGreen'sfunctioninthek-staterepresentationisgivenbyGstt(k,k)=ak|a+kstt.
(D.
15)Ifψ(k)xareexacteigenstatesoftheHamiltonianH,i.
e.
,whenH=kεka+kak,thecorrelationfunctionsarecalculatedwiththeuseofthefollowingidentities:ak(t)=eiεkt/ak,a+k(t)=eiεkt/a+k,a+kak=nkδkk,(D.
16)wherenkaretheoccupationnumbersforthequasiparticles(fermionsorbosons).
TheGreen'sfunctionintheexacteigenstaterepresentationisdiagonal,Gstt(k,k)=δkkGstt(k),whereGRtt(k)=iθ(tt)eiεk(tt)/,(D.
17)GAtt(k)=iθ(tt)eiεk(tt)/,(D.
18)andGctt(k)=i[θ(tt)(1nk)±θ(tt)nk]eiεk(tt)/.
(D.
19)WenotethattheretardedandadvancedGreen'sfunctionsintheexacteigenstaterepresentationaretemperature-independent.
Theenergyrepresentationofthesefunc-tionscoincideswiththeonegivenforelectronsinChapter3,seeEq.
(14.
9),wherethemomentumpstandsinplaceofthequantumnumberk.
Incontrast,thecausalGreen'sfunctiondependsontheoccupationnumbernk.
OnehasGRε(k)=GAε(k)=1εεk+iλ,Gcε(k)=±nkεεkiλ+1nkεεk+iλ(D.
20)750QUANTUMKINETICTHEORYwithλ→+0.
Consideringphotonsandphonons,itisconvenienttointroduceGreen'sfunctionsinanotherway.
Wenotethattheobservablephysicalvaluesrelatedtothephotonsandphononsareexpressedthroughthevectorpotentialofelectromagneticeldandthroughtheatomicdisplacementvectors,respectively.
TheHamiltoniansdescribingtheinteractionofphotonsandphononswithelectrons,aswellastheinteractionofphotonswithphononsandphonon-phononinteraction,arealsoexpressedintermsofthesevectors.
SincethespatialFouriertransformsofboththevectorpotentialandthevectorsofatomicdisplacementcontainbosoniccreationandannihilationoperatorsinthecombinationbq+b+q,whereisthemodeindex(polarization),itisnaturaltodenetheGreen'sfunctionofphotons(orphonons)asD,stt(q,q)=bq+b+q|bq+b+qstt.
(D.
21)WeusetheletterDinsteadofGtoemphasizethedierenceofthedenition(D.
21)withrespectto(D.
15).
Ifthephotons(orphonons)aredescribedbythefree-bosonHamiltonianHb=qωq(b+qbq+1/2),wecanuseEq.
(D.
16)rewrittenforthebosonoperatorsbandb+.
Asaresult,D,stt(q,q)=δδqqD,stt(q),whereD,Rtt(q)=iθ(tt)[eiωq(tt)eiωq(tt)],(D.
22)D,Att(q)=iθ(tt)[eiωq(tt)eiωq(tt)],(D.
23)andD,ctt(q)=iθ(tt)[(Nq+1)eiωq(tt)+Nqeiωq(tt)]+θ(tt)[Nqeiωq(tt)+(Nq+1)eiωq(tt)].
(D.
24)HereNqisthedistributionfunctionofphotonsorphonons,whichbecomesthePlanckdistributionfunctioninequilibrium.
Intheenergyrepresentation,D,Rω(q)=D,Aω(q)=1ωωq+iλ1ω+ωq+iλ,(D.
25)whereω=ε/andλ→+0.
InEqs.
(D.
22)(D.
25)wehaveusedthesymmetrypropertyωq=ωqfollowingfromthesymmetrywithrespecttotimereversal.
Forthesamereason,NqinEq.
(D.
24)canbereplacedbyNq.
AppendixEMany-ElectronGreen'sFunctionsTodescribeasystemofmanyelectrons,onecanuseasetofn-particleGreen'sfunctionsdescribingevolutionofthesystemwhennelectronsareaddedtoitattheinstantt1andtakenoutattheinstantt1.
Theone-particleGreen'sfunctionG(r1t1,r1t1)isintroducedasG(r1t1,r1t1)=iTΨr1(t1)Ψ+r1(t1)o=iΨr1(t1)Ψ+r1(t1)o,t1>t1iΨ+r1(t1)Ψr1(t1)o,t10)regions.
LetusdecomposeU(z)intotwoparts,namelyU(z)=Ul(z)+Ur(z),(H.
2)771772QUANTUMKINETICTHEORYUl(z)=U(z),z0,Ur(z)=0,z0,whichmeansthatUlandUrrepresentunpenetrablepotentialwallsfortheelectronsintheleftandrightregions,respectively.
Foreachregion(j=l,r),weintroducethebasisfunctionsFjk(z)≡z|jksatisfyingthefollowingone-dimensionalSchroedingerequations:p2z2m+Uj(z)εjkFjk(z)=0.
(H.
3)Thesefunctionsformcompletesetsforeachregionj.
Theybelongeithertocontinuousspectrum,whenkisacontinuousvariable,ortodiscretespectrum.
Thediscreteelectronstatesexistiftheregionjcontainsapotentialwellwherethesizequantizationoccurs.
Inanycase,thefunctionsFjk(z)areorthogonal,andtheyarenormalizedaccordingto∞∞dzFjk(z)Fjk(z)=δkk.
(H.
4)Thefunctionsfromdierentregions,however,arenotorthogonal.
TheiroverlapintegralsSjk,jk≡jk|jk=∞∞dzFjk(z)Fjk(z),(j=j)(H.
5)arenotequaltozero.
LetuscalculatethematrixelementsoftheHamiltonian(H.
1)inthebasis|jkdescribedabove.
Theelementsdiagonalinjarejk|H|jk=δkkp2x2m+εjk+jk|V(x,z)|jk+jk|Uj(z)|jkj=j.
(H.
6)Thelastterminthisexpressioncontainsanexponentialsmallnessofthesecondorderwithrespecttotheoverlapfactors(H.
5).
Indeed,thistermisformedasanintegralofaproductoftwowavefunctionsoftheregionjovertheregionj=j,wherethesefunctionsdecreaseexponentially.
Weneglectsuchtermsinthefollowing.
TheremainingpartofEq.
(H.
6)representstheeectiveHamiltonianoftheregionj.
Thenon-diagonalelementsofHarerepresentedas(j=j)jk|H|jk=Sjk,jkp2x2m+εjk+εjk2+tjk,jk+jk|V(x,z)|jk+12jk|Uj(z)Uj(z)εjk+εjk|jk,(H.
7)wherewehaveseparatedthecontributiontjk,jk=12jk|Ul(z)+Ur(z)|jk=12jk|U(z)|jk(H.
8)calledthetunnelingmatrixelement.
ThelastterminEq.
(H.
7)isequaltozero.
Onecanprovethisstatementbywritingthisterm,withtheuseofEq.
(H.
3),as12jk|Uj(z)εjk|jk+12jk|Uj(z)εjk|jk=14m∞∞dzFjk(z)p2zFjk(z)Fjk(z)p2zFjk(z).
(H.
9)APPENDIXH:HamiltonianofTunnel-CoupledSystems773Theintegraloverzinthisexpressionshouldbetransformedbyparts.
TakingintoaccountthateitherFjk(z)orFjk(z)isequaltozeroatz=±∞,wecompletetherequiredproof.
LetussearchforthewavefunctionfromEq.
(H.
1)intheformΨ(x,z)=1Lj=l,rkpAjkpeip·x/Fjk(z),(H.
10)whereListhenormalizationlengthintheplaneXOY.
Usingthematrixelements(H.
6)and(H.
7),wewritethefollowingequationforthecoecientsAjkp:(εjkpε)Ajkp+1L2jkpjk|V[(pp)/,z]|jkAjkp+k(j=j)tjk,jk+Sjk,jkεjk+εjk2εAjkp=0,(H.
11)whereεjkp=εjk+p2/2mandV(q,z)isthespatial2DFouriertransformofV(x,z).
Equation(H.
11)isexact,exceptforthefactthatwehavealreadyneglectedthetermscontaininghigherexponentialsmallness(originatingfromthelastterminEq.
(H.
6)).
Itisoftenreasonabletoneglectalsothecontributionnon-diagonalinjinthesecondtermontheleft-handsideofEq.
(H.
11),sincetherandompotentialV(x,z)ismuchsmallerthantheregularpotentialU(z)determiningthetunnelingmatrixelement.
Thelasttermontheleft-handsideofEq.
(H.
11)describesthetunnelingwithconser-vationofthein-planemomentump.
Bynoticingthat|tjk,jk|U0|Sjk,jk|,whereU0isacharacteristicenergyscaleofU(z),weneglectthecontributionproportionaltoSjk,jkinthistermintheregionεεjk+εjk2U0.
(H.
12)Itistheregionofenergiesthatisimportantforconsideringthetunneling.
Intheseapproximations,Eq.
(H.
11)isreducedtotheeectiveeigenstateproblem(HTε)A=0,wheretheHamiltonianHTisdenedbyitsmatrixelementsinthebasis|jkpdescribedbythewavefunctionsL1eip·x/Fjk(z):jkp|HT|jkp=δjjδkkδppεjkp+L2V(j)kk[(pp)/]+δpptjk,jk,(H.
13)whereV(j)kk(q)=jk|V(q,z)|jk.
TheHamiltonianHTistheeectiveHamiltonianoftunnel-coupledsystems.
Itiswrittenwiththeuseoftheoverlledbasisconsistingoftwocompletesetsofl-andr-statesdescribedbyEqs.
(H.
3)(H.
5).
Thenon-diagonalpartofHT,whichdescribesthecouplingoftheleftandrightregions,iscalledthetunnelingHamiltonian.
Inthesecondquantizationrepresentation,werewriteEq.
(H.
13)asfollows:HT=jkpεjkpa+jkpajkp+1L2jkkppV(j)kk[(pp)/]a+jkpajkp+kkp[tlk,rka+lkparkp+trk,lka+rkpalkp],(H.
14)wherea+jkpandajkparethecreationandannihilationoperatorsoftheelectroninthestate|jkp.
NotethattheHamiltonian(H.
14)isHermitiansincetlk,rk=trk,lk.
774QUANTUMKINETICTHEORYConsideringthetunnelingHamiltonianasaperturbation,onemaywritetheprobabil-ityoftunnelingtransitionbetweenthestates|jkpand|jkpinunittimeaccordingtoFermi'sgoldenrule,seeEq.
(2.
16):Wjkp,jkp=δpp2π|tjk,jk|2δ(εjkpεjkp),j=j.
(H.
15)Thesetransitionsconservethein-planemomentump.
Onemayalsoconsiderhigher-ordercontributionsintothetransitionprobability,whichincludeboththetunnelingandthescattering.
Suchcontributionsdescribethescattering-assistedtunnelinginwhichthein-planemomentumisnotconserved.
Letuswritethematrixelementsoftheoperatorofelectriccurrentthroughthebarrier(thetunnelingcurrent).
Usingthegeneralexpression(4.
15),weobtainthecurrentoperatoratthepointZ:IT(Z)=e2m[pzδ(zZ)+δ(zZ)pz].
(H.
16)Thematrixelementoftheoperatorofthetunnelingcurrentperunitsquareiscalcu-latedasjkp|IT(Z)|jkp=e2mL2Fjk(z)pzFjk(z)(pzFjk(z))Fjk(z)z=Z=ie2mL2Z∞dz[Fjk(z)2zFjk(z)(2zFjk(z))Fjk(z)]=ie2L2Z∞dz[Uj(z)Uj(z)εjk+εjk]Fjk(z)Fjk(z)(H.
17)∞Zdz[Uj(z)Uj(z)εjk+εjk]Fjk(z)Fjk(z).
IfwechooseZ=0andtakeintoaccountEq.
(H.
3)andtheenergyconservationlawfromEq.
(H.
15),weobtainlkp|IT|rkp=ieL2tlk,rk,rkp|IT|lkp=ieL2trk,lk.
(H.
18)Therefore,thematrixelementsofthetunnelingcurrentareexpressedthroughthetunnelingmatrixelements(H.
8).
ThecurrentisindependentofthechoiceofZ,accordingtothecontinuityequation(4.
14).
ThelattercanbewrittenasdIT(Z)/dZ=0inthestationarycaseandintheabsenceofin-planecurrents,whenIr=[0,0,IT(Z)].
ThepresenceofinnitenormalizationlengthsinEq.
(H.
18),aswellasinthetunnelingmatrixelements(H.
19)and(H.
21)below,shouldnotcreateaconfusion,becausetheselengthsvanishintheobservablequantities(suchasthedensityoftunnelingcurrent).
Belowwepresenttheexpressionsfortlk,rkcalculateddirectlyforthreesimplepotentialsshowninFig.
H.
2.
Thiscorrespondstothecasesof3D-3D(a),2D-3D(b),and2D-2D(c)tunneling.
Inthecase(a),tlk,rk=42m√LlLrkkκeκd(κ+ik)(κik),(H.
19)whereLjisthenormalizationlengthalongzintheregionj,whichappearsbecauseofnormalizationofthewavefunctionofcontinuousspectrum.
Theunderbarrierpenetrationlength,κ1,isrelatedtothewavenumberskandkaccordingtoκ=2m|εp2/2m|,k=2m(εU0rp2/2m),APPENDIXH:HamiltonianofTunnel-CoupledSystems775FigureH.
2.
Rectangularpotentialsdeningtunnel-coupledsystemsofdierentdi-mensionalities.
k=2m(εU0lp2/2m),(H.
20)whereεistheenergyofthetunnelingelectronandU0jisthepotentialenergyintheregionj.
Inthecase(b),tlk,rk=22m√Lrkkκ3/2eκd(κik)(κ2+k2)(1+alκ/2).
(H.
21)Therelations(H.
20)remainvalid,thoughkisnowdiscreteanddeterminedbythedispersionrelationcot(kal)=k2κ22kκ.
(H.
22)Thevaluesofkandκarexedbyk.
Finally,inthecase(c)tlk,rk=2mkkκ2eκd(κ2+k2)(κ2+k2)(1+alκ/2)(1+arκ/2).
(H.
23)Thewavenumberkinthiscaseisalsodiscrete.
Therefore,apartfromEqs.
(H.
20)and(H.
22),onehascot(kar)=k2κ22kκ.
(H.
24)Inthiscase,thetunneling(withoutscattering)mayoccuronlyiftheparametersofthesystemareadjustedtosatisfyEqs.
(H.
20),(H.
22),and(H.
24)simultaneously.
Thistunnelingoccursbetweenthestateskandkwithmatchedenergies.
Thevaluesoftlk,rkgivenbyEqs.
(H.
19),(H.
21),and(H.
23)aredenedwiththeaccuracyuptoaphasefactoreiφ,sincethephasesofthewavefunctionsFlkandFrkcanbechoseninanarbitraryway.
Thisphasefactorisnotessential,becausethetransitionprobabilitiesandtunnelingcurrentsareexpressedthroughthesquaredabsolutevaluesofthetunnelingmatrixelements.
Ifboth|lkand|rkarediscretestates,thewavefunctionsFlkandFrkcanbechosenreal.
Thetunnelingmatrixelementsinthiscasearerealandsymmetric,tlk,rk=tlk,rk=trk,lk.
Inthecaseoftunnelingbetweendiscretestates,onemayconsideronlyapairofthestateskandkwithcloselymatchedenergies.
Theindiceskandk,therefore,canbeomitted,andtheHamiltonianofsuchtunnel-coupledelectronsystemsiswrittenintheformofa2*2matrixinthebasis|land|r:HT=Plhl+Prhr+σxtlr,(H.
25)776QUANTUMKINETICTHEORYwherePl=(1+σz)/2andPr=(1σz)/2aretheoperatorsofprojection,σiarethePaulimatrices,andtlristhetunnelingmatrixelementforthepairofstatesunderconsideration.
TheHamiltoniansoftheleftandrightregionsarehj=εj+p2x2m+V(j)(x),(H.
26)whereV(j)(x)=dz|Fj(z)|2V(x,z).
AccordingtoEq.
(H.
18),theoperatorofthetunnelingcurrentinthisbasisisexpressedthroughthePaulimatrixσy:IT=etlrσy.
(H.
27)ItisconvenienttousethePaulimatricesforapplications,sincethematrixalgebra(thecommutationrelations,etc.
)forthesematricesiswellknown.
Forexample,wepresentausefulformulafortheoperatorexponent:exp(iσ·A)=cos|A|+iσ·A|A|sin|A|,(H.
28)whereAisanarbitraryvector.
Althoughwehaveconsideredthecaseofone-dimensionalpotentialbarrierssepa-ratingtworegions,themethodusedabovesuggeststhatintroducingthetunnelingmatrixelementsisfeasibleforthecaseofasystemconsistingofseveralweakly-coupledregionsnumberedbytheindexj.
Assumingthateachregionischaracterizedbyasetofeigenstates|jδ,whicharetheexacteigenstatesintheabsenceoftunneling,onecanwritetheHamiltonianofthesystemintheformHT=jδεjδa+jδajδ+jδjδtjδ,jδa+jδajδ+H.
c.
j=j.
(H.
29)TheprobabilityoftransitionsbetweentheregionsinunittimeisgivenbyWjδ,jδ=2π|tjδ,jδ|2δ(εjδεjδ),j=j.
(H.
30)Calculatingthetunnelingmatrixelementsforeachparticularcaseisacomplicatedprocedure.
Itisconvenienttochoosethemasparameters.
AbouttheAuthorsFedirT.
VaskohasreceivedhisPhDandDoctorofSciencedegreesin1976and1986,respectively.
HeisnowaSeniorResearchScientistattheInstituteofSemiconductorPhysics,NationalAcademyofSciencesofUkraine.
Duringadecade,hetaughttheQuantumKineticTheoryattheKievStateUniversity.
Hehaspublishedover170papersonquantumkineticsandrelatedsubjects.
OlegE.
RaichevhasreceivedhisPhDandDoctorofSciencedegreesin1992and1998,respectively.
HeisnowaSeniorResearchScientistattheInstituteofSemiconductorPhysics,NationalAcademyofSciencesofUkraine.
Hehaspublishedabout60papersonquantumkineticsanditsapplications.
777IndexAbsorptioncoecient,77,84,133,139–144,146,164,166–167,185,241,247,516–519,533,612–614,631–632,636,656Acoustic-phononinstability,169,173Acousticphononemission,177,182–184Activeregion,293–294,296–299,316–317,335,502–503,506–507Adams-Holsteinformula,429,434Aharonov-Bohmoscillations,554AndersonHamiltonian,660–661Anisotropicdistribution,57,60,62–63,92,135,214,290,293,296,487,628,673Atomicdisplacement,35,585Attenuationofsound,214Augerrecombination,273,322,637,639–641,644–645Averageddensitymatrix,52,54,56,73,343Averageddiagram,110–111AveragedGreen'sfunction,110,112,116,125,442,450,572,704,764–765Balanceequation,10,79–81,83,85,162,202,305,307,310–311,314,326,328,335,486,493,495,558,580,637,643–645Balescu-Lenardcollisionintegral,684Ballistictransport,89,91,94,538–539,547,552,560,598Bethe-Salpeterequation,116–117,120,377,449,499,532,711,730,757,759,767–768Blochoscillations,557,566Boltzmann-Langevinequation,681,729Boltzmanndistribution,25,136,292,295,318,320,503Bornapproximation,53,56–57,111–112,115,126,374,440,442,572,630,661Boson-electroncollisionintegral,158,177Bosondistributionfunction,160,177Bosongenerationrate,162–163,165,167–168,170–172,174–177,186,208,212–213Bra-vector,3,357Brillouinzone,27,36,193–194,199,208,211–212,225,464,523,566,738,740Broadeningenergy,56,114,130,147,181,224,330,338–339,452,462,479,487,489,499,501,516,614,616–617,631–632,770Buger'slaw,164Canonicalmomentum,18,34,75–76,309–310Causalityprinciple,103,147,265Chargedensity,19–20,80,86,148,263,372,522,533,682,721Chronologicalordering,6,355–356,358,360,365,709–710,751Circularpolarization,180,611,620Coherentcontrol,490,529Coherentphonons,520–521,523–524Collisionintegral,55,57–58,71,73,177,232,285,287,291,354,387,413,485,488,608–610,618,632,634,651,653,658,679,684,723Combinedresonance,613–614,627,656Conductance,87,90–91,93–94,400,539–540,542,544–548,554,558,560,582–583,585,700,702,706Conductanceuctuations,702–703,708Conductancequantization,539,546–547,560Conductancequantum,122,460,539,549,617Conductivitytensor,63–64,83–84,86–87,101–104,131,139,147,246,259,278,280,381,383,390,400,404,434–436,446–447,463,466,531,542,596,627,634,666,769779780QUANTUMKINETICTHEORYConnementpotential,30,32,235,238,246,249,274,327,329,599,615,656Continuityequation,21,80,83,86,202,292,303,409,573,618,623,727,774Cooperon,378–380,704,706,762Correlationfunction,57,78,95,101,104,112,115,117,126,129,143,149,151,157,220–222,231,251,264,277,338,346,355,360–361,364,377–378,572,630,648,664–666,670,672,677,682–684,695,698,703,705,707–708,710–713,719,721,747,749,764,767–768Correlationoperator,53,73,157,250,283–284,289Coulombblockade,577,583,585Coulombdrag,262Coupledquantumwells,571,587,629,632,659Current-voltagecharacteristics,289,569Currentdensity,11–12,21,62–63,66–67,76,80,82–84,101–102,104,137–138,235,237,241,243,245,259,302,311,335,372,375,377,383,400,463,476,478,488,540–541,593–594,603–604,616,627,633,666,671,739Cyclotronfrequency,30,70,81,129,398,426,434,449,624,634Cyclotronresonance,84,613,636Debyetemperature,194,200,212Deformationpotential,43,182,213,287,411–412,523,674,731–732,739–740Densitymatrix,4–5,17,24,51,58,72,99,137,156–157,160,170,190,231,236,243,245,281–283,287,341,343,348,352,356,369,383,392,395–396,427,463,478,484,490,499,511,520,524,528,603,615,618,655Densityofstates,26,29–30,33,114–115,127,130,151,327,440,519,533,629,631Depolarizationeect,243Diagramtechnique,109–110,112,265–266,654,708,711,754,756,763Dielectricpermittivity,11,19,65,77,103,131,133,135,137,163,215,218–220,230,263,270,385,466,683–685,692–695,724,726Dierentialcross-section,685,687,690,696,726Diusioncoecient,302–304,316,321,378–380,502,619–620,704,762Diusioninenergyspace,316,502Diusivityofscattering,399,405Dinglefactor,445,447Dipoleapproximation,163,178,243,484,486,493,508,513,739Double-branchtimecontour,359Double-timeGreen'sfunction,220,356,372,708,747,749,751Dragresistance,260–261Driftvelocity,136,168–169,172,196,199,202,204,225,255,257,303–304,316,318–319,321,409–410,457,502,619–620,643,645Dynamicalscreening,262,272,364,684Dysonequation,112–113,370,389,441,731,755,765–766Eectivemass,28–29,32,245,389,391,447,475–476,608,689–690,738–740Eigenstateproblem,6,8,31,57,105,160,236,243,253,332,427,435,733–735,741,773Einstein'srelation,302Elasticlightscattering,685,693,724–725Elasticityequation,422Electricneutrality,89,92,304,638Electrochemicalpotential,302,545,547–548,573,610,619,621,657Electron-bosoncollisionintegral,285–287,323,336,388Electron-bosoninteraction,156–157,282–283,286,288,364,388Electron-electroncollisionintegral,252–254,256–257,262,272–273,364,387,636,641,684Electron-electroninteraction,20,230,235,250,257,261–262,273,282,363,636,751Electron-holepairs,164,166,185,243,247–248,275,475Electron-impuritycollisionintegral,54,58–59,71,73,75,256,354,514,653Electron-impurityinteraction,52–53,56–57,108,114,256,342,344–345,351–352,514,516,518,608,646–647,721,729Electron-phononcollisionintegral,291,293,300,310,413,678Electron-phononinteraction,41,43–44,156,184,212,287,290,297,324,364–365,367–368,370–371,373,389,411–412,416,431,449,501,507,512,521–522,585–586,667–668,679,695,717–718Electron-photoninteraction,156,185,323,373Electrondensity,27,55,64,80,83,85,102,165,173,232,236,240–241,260,264,292,302–304,308,326,377,383,418,438–439,445,453–454,458,489,492,507,523,593,614,618,622,624,634,637,640,644,692,694Electrondistributionfunction,58–59,63,68,71,74,76,81,88–92,105–106,132,160,168,170–172,174,178,235,INDEX781238,254,257,264,286,291–292,295,297–300,308–309,314,319–320,324,354,395,397–399,426,456,459,462,476,485–487,491–493,500–501,503–504,506,512–513,516,539,547,564,572,580,590,608–609,611–612,619,624,633,649,651,655,659,666,670,674,699Electronspinresonance,614,656Ellipticpolarization,612Energyconservationlaw,9,58,139,161,175,178,184,193,210,213–214,224,254,309,311,325,370,427,485,487,569,595,660,688,774Energydensity,16,48,82,85,162,195,202,305,308,624,727Energyowdensity,162,183,197–198,200,202,305,405–406,410Energyrelaxation,255,306,333,416,418,506,529,644Envelopefunction,28,139–140,656,690,738,740Equipartitioncondition,291,301,419,503,506–507Evolutionoperator,6–7,54,70,73,124,158,170,252,284,357Exciton,243,246,248–249,276–277,470–471,473,476Faradayeect,84,466Fermi'sgoldenrule,8,288,476,570,774Fermidistribution,25,77,165–166,172,255,257,300,308,332,426,447,458,462,512,518,569,573,576,580,598,638,653Fillingfactor,453,458,460,464Flowdensity,80,86,302,486,618–619,622Fluctuation-dissipationtheorem,667,700,722,729Fluctuationsofcurrentdensity,666,672Fluctuationsofdielectricpermittivity,696,724,726,728Fluctuationsofelectrondensity,682,692,694,721Fluctuationsofelectrondistribution,679,682–683,725Fluctuationsofphonondistribution,679Frantz-Keldysheect,140,147Freesurface,415Frequencydispersion,62,64,78,83,131,134,403,405,605Fresneldrag,137Fundamentalabsorption,137,139,141,144,147,243,248,470Gaussianrandompotential,57,95,112,126,142,277Generalizedkineticequation,351,363,374,411,710,716Generalizedsusceptibility,107,147,265,665–666Green'sfunction,105–106,108–110,112–119,123–125,129–130,142,150,221–223,227,247,249,269,344,348,350–355,359,370,372,374–375,377,413,434–435,442,470,472,499,502,517,519,530,572,630,647–649,654,703,709,715–717,719,747–757,759–760,762–769Greenwood-Peierlsformula,105,435Hallcoecient,404,636Hallconductivity,84,438,453–454,459–460,464,478Halleect,84,381Hallresistance,390,561Harmonicoscillator,12,30,37,337,381,559,733Hartree-Fockapproximation,233,264–265,269,279,721,757Heisenbergrepresentation,149,265,355,432,578,590,664,667,692,697–698,747,751Hoppingtransport,429,476,589–590,593–596,605Impactionization,643,645Inelasticlightscattering,685,692–694,696,698Interactionrepresentation,7,357,361,364–365,711Interbandrecombination,322–324,512,514,620,640Interbandtransitions,131–132,134,138–139,163–165,182,243,322,465,484,490,508,517,567,611,643,645,689–691Interbandvelocity,133,187,323,493,499,508,740–741IntersubbandAugertransitions,641Intersubbandtransitions,181,235,237,240–241,243,322,324,487,490,493–494Irreduciblediagram,112,116,363,707,755–756Irreduciblevertexpart,116,119–122,704,707,757Jointdensityofstates,133,144,182,248,470Kerreect,135Ket-vector,3–4,6–7,14–15,22–23,105,665,735Kinematicmomentum,18,34,73,104,128,170–171,467,470,615,629,681,738Kineticequation,52,56–59,62,65,75,88,91,156,164,166,169,171,177,193,198,208,212,244,256–257,285–286,288,294,297,301,309–310,314,316,782QUANTUMKINETICTHEORY333,413,426,455,457,485–486,491,513,580,592,608,610,636,641,651–653,658–659,679Kondoeect,654Kondotemperature,654Kramers-Kronigdispersionrelations,103,134,147–148Kuboformula,102,130,149,382,426,434,453–454,541,543,595,605Ladderapproximation,118,375,377,499,532Lagrangeequations,50,216,422Landaudamping,271,280,673,695Landaulevel,30,130,327,337,427,430,432–433,436,438–439,442–443,447,449,451,453–455,458–461,464,467,472,478,559,561,612,614,763–765,767,770Langevinsource,670–671,676,680,682,696,718,721,729Lightscattering,685,687,690,725Lindhardformula,270Linearpolarization,163,180,182Linearresponse,62–63,99–100,104,107,131,147–148,197,220,239,263,361,374,379,398,426,434,490,497,507,540,626,633,653Linearizedkineticequation,63,198,203,236,244,376,398,406,462,615,658,681–682Localizationinone-dimensionalsystems,551Lorentzforce,69,72,618,622,682Magneticedgestate,559,561Magneticux,554–557Magneticuxquantum,553Magnetoexciton,473–474,482Magnetophononoscillations,433–434,449,452Markovianapproximation,58,63,94,171,192,286,485,580,592,609,632,658,678MatrixDysonequation,345,368,630MatrixGreen'sfunction,342–345,348–349,415,518,629–630,709Matrixkineticequation,161,607–608,610,612,618,621,632–634,655,658Maxwelldistribution,136,165,168,312,335,614,642Maxwellrelaxationtime,304Meanfreepath,87,89,91,121,196,205,300,378,398,401,403,405,407,409,538,574,610,646,701,729,761Mesoscopiccylinder,599Mesoscopicring,551–552,554–557Mesoscopicsample,540,545,698,701–702,706Microcontact,87–89,91,93–94,538,545,701,729Mobility,303,489Momentumrelaxation,57,60–61,63,87,291–292,300,399,416,418,507,620,626,633Multi-phononprocesses,322,329–330,596Multi-photonprocesses,72,75,77,312Nanostructure,32,322,324,538Negativeabsorption,160,164,167,182,497,508,518Negativemagnetoresistance,380–381,404,636Negativetransientconductivity,497,507–508Noisepower,666,671,698–699,701,720,729Noisetemperature,673–674Non-equilibriumdiagramtechnique,345–346,355,359,361,364,374,382,708,716Non-equilibriumGreen'sfunction,345,355–356,359,365–366,375,382,411Non-idealsurface,392,398,405–406Non-Markovity,55,58,159,170,485,526,679Normalprocesses,193,196–197,200,202,204,207,405,409–411Normalizationcondition,12,14,43,55,66,76,95,114,196,292,295,298,319,472,477,515,543,596Observable,2–5,24,52,79,147,750,764,774Onsager'ssymmetry,103,107,137,154,541,666Optical-phononinstability,169,176Overlapfactor,139–141,184,245,249,288,290–291,324,476,632,637,660,772Partiallyinverteddistribution,497,507Partitionfunction,17,25,581,583,591Passiveregion,293–298,315–320,501–503,506–507Pathintegral,123,125–127,130,141,144–146,152,248Pauliblocking,133,160,166,184,323–325Pauliprinciple,22,253,279,282,287,646,701,721Permutationoperator,250,252,284–285Persistentcurrent,556–557Phaserelaxationlength,122,378,549Phonon-assistedRamanscattering,696Phonon-phononcollisionintegral,193,212Phonon-phononinteraction,44–45,189,192,195,202,207,212,220–221,224,410,521,524,526Phonon-photoninteraction,38,42,217–218Phonondecay,193,208–209,211,224,526INDEX783Phonondistributionfunction,169,171,174,183,192–193,195–198,200,204,208,210–211,213,225,301,405–406,670,677PhononGreen'sfunction,221,223,366–369,371,412–414,422,424,719,731,750Phononhydrodynamics,205Photogenerationrate,326,484–487,490–491,493–495,498,612Photoluminescence,164–165,168–169,177,180,182Photon-phononresonance,316,320–321Photondistributionfunction,164,166,169,178,180Photondrag,487,489Photonemission,164,173,178–180,187Planck'sformula,48Planckdistribution,17,161,195,285,287,291,332,414,525,591,750Plasmafrequency,65,87,137,270,304,389Plasmon-phononmodes,270,280,534,695,698,726Plasmon,87,98,262,270–271,617,693Poissonbrackets,233,745Poissonequation,19,39,41,86,88,233,235,263,278,304,309,423,682Poissoniannoise,701Polaritons,215,218,227Polarizability,104,148,264,279,475,692,721Polarizationfunction,264–265,267–269,279,363–364,368,370,385–386,756Polaron,586,589Powerlossterm,82,305–306,311,335Poyntingvector,15–16,77,96,242,489,727–728Quantumcorrection,69,122,144,353,374–381,390,549–550,555,599,745Quantumdot,322,327,329–330,338,585QuantumHalleect,452–453,455,460–461,465,560–561,600Quantumkineticequation,10,54,65,73,158,190,233,274,287,395,485Quantumpointcontact,546–547Quantumwell,30–32,177,182,186–188,234–235,241–242,249,274,322,324,326,332,411,418,487,492–493,496,498,570,641,643,726Quantumwire,32,250,322,326,539,598Quasi-classicalkineticequation,71,79,180,234,300,354,384,388,398,457,461,618,682Quasielasticscattering,63,290,294,299,306–307,312,314,316,320,326,333,501,674Quasienergy,34–35,484–485,511Rabioscillations,496,508Randomphaseapproximation,269,271–272,385–386Randompotential,57,72–73,95,108,112,125,128–130,141,248,277,356,361,383–384,429,435,450,461–462,499,611,703,721,764,771,773Rayleighsurfacewaves,420Recombinationrate,323,486,513Reduciblediagram,112,116,267–268,441–442,755–756Reectioncoecient,166–167,548,550–551,565Relativeabsorption,242,326,489,616–617,631Relaxationrate,60–62,118,172–174,176,212–213,238,291,294,297,306–307,324–331,337,399,418–419,426,488,495,506,524,528,610,620,625,659Renormalization,113,160,178,232,234–235,240,351,389,588Resistanceresonance,635–636Resonanttunneling,565,573,585Rigidsurface,415Run-awayeect,293Screenedinteraction,151,262,266,270–272,278,363–364,387,684,755–757Screeninglength,270,272,693Secondquantization,15,23,44,155–156,230,264,431,509,512,586,647,665,692,694,699,773Secondsound,206–207Self-energyfunction,112–113,339,350,363,367–368,370,375,387,411–412,437,439,441–442,448,731,755–756,766Self-energymatrix,345–348,519,630,709Sequentialtunneling,575,577Shotnoise,700–702Shubnikov-deHaasoscillations,447–448,452Single-electrontunneling,577Sizeeect,398,405,410Sommerfeldfactor,248,276–277,473–474Soundvelocity,43,173,194,206,213,291,313,389,414,416,422,503,523Spatialdispersion,86–87,102,137,672Specicheat,203,407Spectraldensityfunction,106,108,572Spin-iptransitions,391,467,607–608,612,614–615,627,691,742Spin-orbitinteraction,26,420,608,614–615,623,627–628,654Spin-orbitterm,615,621–622,656Spin-velocitytensor,623Spindensity,617–620,622–623,626,658–659Spindiusionlength,621Spinowdensity,618,622–623,627784QUANTUMKINETICTHEORYSpinpolarization,611–612,623,625,628,646,659Spinprecession,610–611,618,620–621,623,625–626,655Spinrelaxation,609–611,620,623,626–627Spontaneousemission,158–159,164,167,208,293–295,314–315,326,334,495,504Staticconductivity,64,83,105,117–118,403–404,428–429,432–433,435,439,446,595,634–635,653,703Statisticaloperator,4,156–157,190,356–357,431,435,517,570–571,579,581,590,664–665,747Stimulatedemission,164,486,535Stokesparameters,180,187Streaming,296,299,335Stresstensor,422Superlattice,587,603Thermalconductance,598Thermalconductivity,197,200–201,205,305–306,405,407–411,421Thinlm,380,398,402,404,421Timereversal,103,107–108,655,665–666,750Transitionprobability,8–10,59,254,287–288,290–291,301,328,413–414,429,570,592,685–686,693,774,776Translationalinvariance,29,36,57,62,95,110,159,192,198,247,286,380,470,488,757,763–764Transmissioncoecient,544–546,548,550,554–555,562,564–565,567,699,701Transmissionprobability,544–547,560,564,567,597,700Transporttime,63,82–83,87,119,257–258,300–301,303–304,307,333,443,619,634,653,730Transverserelaxation,215Tunnelcoupling,628–629Tunneling,562,565,567,569–570,575,589,628,773–775Tunnelingconductance,563–565,570,573,575,604Tunnelingcurrent,562–563,565,567–573,774,776TunnelingHamiltonian,563,578,773Tunnelinglength,574Tunnelingmatrixelement,570,629,772–776Two-bandmodel,28,132–133,139,164,186,245,323,466–467,480,490,508,522,531,567,601,608,613–614,636–637,654,656,660,690,740Two-bosontransition,289,329–330,336Two-particleGreen'sfunction,266–267,711,752–754,756Ultraquantumlimit,439,442–443,612Ultrashortpulse,97,490,492–495,520–521Umklappprocesses,193,195–197,200,202,204,207,407,411,526Unitarytransformation,72,186,348,510,517,532,544,615,629,661Urbachtail,147Vectorpotential,11,15,20,29,33,65,100,104,128,156,353,371–372,390,427,509,531,553,556,615,629,680,686,737,739,760,763–764Velocityoperator,49,102,104,106,133,139,164,182,186–187,236,243,274,427–428,436,463,466–467,487,491,493,513,593,605,612,687,738,740,745Vertex,109,345–346,363,367,370,705,767Waveequation,12,47–48,84,206,215,219,241,724,726Weaklocalization,123,380–381,550Weakeningofcorrelations,54,56,158,192,251,284,678Wignerdistribution,66–67,161,183,194,234,286,397,455,610,621,632,657,674,677,680Wignerrepresentation,70,233,377,743Wignertransformation,66–67,69–70,96,179,351–353,377,385,412,486,618,655,743,759Zeemansplitting,438,448,481,607,612,614–615,623,625Zenertunneling,563,566