Some analytic results in coherent quantum transpor

SomeanalyticresultsincoherentquantumtransportMohammadMardaani,KeivanEsfarjaniDepartmentofPhysics,SharifUniversityofTechnology,11365-9161Tehran,IranEmail:[e-mail],[e-mail]funiformcrosssection(butwitheventualdisorder)withthreeregions:dot,leftlead,andrightlead,isconsidered.Assumingthatthesameunitarytransformationdiagonalizesallunitcellsofthiswire,weproposeanewformulaforthecalculationoftheGreen’sfunction(GF)andthecoherenttransmissioncoefficient.Thisformulaallowstocalculatethesequantititesmuchfasterthanthestandardmethods.Inparticular,theproblemofauniformdot(simplecubicuniformdotattachedtothesimplecubicwire),withallonsitesequalandallhoppingsequalissolvedfullyanalytically.Theenergyanddot-lengthdependenceoftheGF,localdensityofstates(LDOS),thetransmissioncoefficientandboundstateenergiesarealsoderived.IntroductionThestudyofelectrontransportinmesoscopicsystemsisoneofthemostfundamentalproblemsinnanostructurephysics;ithaschangedourunderstandingoftransportphenomenaincondensedmattersystems.Inrecentyearstherehasbeenagrowinginterestinelectricaltransportinquantumdots,nanowires,nanocrystals,andsinglemolecules.Quantumwiresandquantumdotsarethefundamentalbuildingblocksfornanoelectronicdevices.Someauthorsconsiderdevicesthatconsistofadotconnectedtotwocontinuumreservoirsofelectrons(usuallymetallicleads)[1-15].Therehasalsobeenanincreasinginterestinthetheoreticalmodelingofmolecularwiresystems.Transportmeasurementisapowerfultoolfortheinvestigationofelectronicpropertiesofmaterials.Experimentalresearchonmolecularwireshasincreasedoverthepastfewyearslookingintothepossibilityofrectificationandotherphenomena[16-23].Twodifferentmethodshavebeendevelopedforthestudyofelectron-impuritypotentialscattering,wherebyimpuritypotential,wemeananyexternalinhomogeneitycausedbyimpuritiesoranychangeintheshapeofthewallsetc,whichmaketheHamiltonianofthedotchangeunderlatticetranslations.OnemethodisbasedonasolutionoftheLippmann-Schwingerequationandtheothersolvesthet(transmission)matrix[5,6,7].AfundamentalideabehindtheseadvanceshasbeenduetoLandauer[13]whostatedthatconductanceisproportionaltotransmittance.WerelyonLandauertheory[15]asthebasisforstudyingtheconductancepropertiesofquantumdotsystems.Thisrelatestheleadtoleadcurrenttothetransmissionprobabilityforanelectrontoscatterthroughthequantumdot.Inthispaper,westudythetransportpropertiesintheballisticregimeofacubicuniformquantumdotinthepresenceofmetallicleads.(Thewordsleadandwirehavebeenusedinterchangeablyinthispaper).1ModelInthissection,wepresentourmodelandassumptions,andderiveananalyticformulaallowingtocalculatetheconductanceofauniformdotofarbitrarysizeandshapeanalytically,andinthecaseofadisordereddot,wepresentanalgorithmtocalculateitinamuchfasterway.ActuallyTheusualtaskofmatrixinversionofthedotHamiltonianisoftheorderofthecubeofthenumberoforbitals(basisfunctions)inthedot.Inthiswork,however,weshowthatundersomeassumptions,thistaskisreducedtothecubeofthenumberofunitcells,theGFandthetransmissioncoefficientofaone-bandchaincanbecalculatedanalytically.Assumptionsofthemodel:Boththedotandtheleadsareassumedtohavethesamegeometrynamelythatofacrosssectionofarbitraryshaperepeatedperiodically,thisunitcellbeingthesameforbothleadsandthedot.Wefurtherassumethatallhoppingintegralsfromoneunitcelltothenextarethesameregardlessofatompositionswithintheunitcell,buttheonsiteenergiesneednotbethesamewithinacell,orfromoneunitcelltothenextinthedotregion.Theyneed,however,tobeonlyscaledfromonecelltothenext.Putinmoremathematicalterms,werequirethatthesameunitarytransformationdiagonalizesanyunitcellofthesystem.Basedontheseassumptions,wecanwritethethreeHamiltoniansintheleftandtherightleadsandthedotinthefollowingform:()(),,,,,'',,'',1,,|,,,,||,,',',|.|,,,,1|.xyzxyzxyzxyzxyxyzxyzxyzxyzHExyzxyzVxyzxyzcWxyzxyz+=++∑∑∑ccc+++1(2)(1)wherereferstounitcellnumberinthedirectionalongthewireandccreferstocomplexconjugateof|,anddenotetransversedegreesoffreedomwithinthecell.EistheonsiteenergyandVreferstohoppingintegrals.Thefirsthoppingintegralintheaboveformulacorrespondstointracell,andthesecondonetointercellhoppings.ItisassumedthatintercellhoppingsareallthesameVv.OnecanconsiderasinglecellHamiltonianandrewriteitinabasiswhereitisdiagonal.z.xyzxyz,,,,|xyzxyz′′,xy,1,zz++=where,,1()(,1),,1celltztzzttHztztzandVzzvtztzε+=+=∑∑+pc+[]()(,1).cellzHHzVzzc=++∑Here,labelsthediagonalizedtransversedegreesoffreedomobtainedfromtheoriginalbasisbyaunitarytransformationsothatthefirsttwotermsinequation(1)aretransformedintoH1,...,t=,xycell.Furthermore,thecellhasbeenchosensothatelectronhoppingonlyoccursbetweennearestneighborcells.2Thus,underourassumptions,thesystemof“thick”wirescanbeseenasindependentone-bandchain,wheremostcalculationscanbedoneanalytically.Thisassumptionisvalidforthecaseofmostquantumwiressuchascarbonnanotubes.Wecanalsoseehere,thatthegeneraltaskofinvertingaNNmatrix(whichisoforder()can,eveninthecaseofadisorderedsystem,bereducedtothatofinvertingNNpNNxyzN3xyzNxy3()ON))timesamatrixofsizeonly.Thislattertaskisoforder(ifstandardmatri