Control of Magnetotransport in Quantum Billiards: Theory, Computation and Applications PDF
Preview Control of Magnetotransport in Quantum Billiards: Theory, Computation and Applications
Lecture Notes in Physics 927 Christian V. Morfonios Peter Schmelcher Control of Magnetotransport in Quantum Billiards Theory, Computation and Applications Lecture Notes in Physics Volume 927 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard M.Bartelmann,Heidelberg,Germany B.-G.Englert,Singapore,Singapore P.HaRnggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLoRhneysen,Karlsruhe,Germany J.-M.Raimond,Paris,France A.Rubio,Hamburg,Germany M.Salmhofer,Heidelberg,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. 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Morfonios (cid:129) Peter Schmelcher Control of Magnetotransport in Quantum Billiards Theory, Computation and Applications 123 ChristianV.Morfonios PeterSchmelcher CenterforOpticalQuantumTechnologies CenterforOpticalQuantumTechnologies UniversityofHamburg UniversityofHamburg Hamburg,Germany Hamburg,Germany ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-3-319-39831-0 ISBN978-3-319-39833-4 (eBook) DOI10.1007/978-3-319-39833-4 LibraryofCongressControlNumber:2016947165 ©SpringerInternationalPublishingSwitzerland2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface Thepastdecadeshaveseenthecontinuousappearanceofaccountsonmesoscopic transport owing to the ever increasing range of theoretical and experimental advancesinthefield.Itistherebychallengingtopresentnovelaspectsofelectronic nanostructure physics while still leading the reader coherently from fundamental conceptsandmethodsthroughtotheircurrentapplicationinanaccessibleandself- contained manner. A feasible strategy may then be to follow a path through the layers of acquired knowledgedictated by a narrowed perspective under a specific applicational aim. In the present Lecture Notes we have attempted to fulfill such a task, covering the theoretical treatment and computation of electronic quantum transport from the perspective of two-dimensional ‘billiard’ systems, oriented by the aim to explore the controllability of their magnetotransport properties. Emphasisisplacedonilluminatingtheimplicationsofconfinedscatteringbetween terminalsfor the generaltheoreticaltreatmentand for the mechanismsunderlying the response of such transport devices. The main message is then that, based on universalphenomenasuchasmultiplewaveinterference,electrostaticcollimation, andmagneticdeflectionorphasemodulation,efficientmagnetotransportcontrolcan beachievedinsimplesetupsdefinedbythenonuniversalpropertyoftheirgeometry. Starting outwith a top-bottomdescriptionof mesoscopictransportatsemicon- ductorinterfaces,weaddresstheconceptualaswellastechnicalpeculiaritiesarising frombilliard-typeconfinementandfocusontheingredientsneededforconductance control.Athoroughaccountisgivenontheefficientnumericalcomputationofthe electronicpropagatorinthescatteringsystemwithatechniqueparticularlysuitable forthestructurestobestudied.Therestofthebookisdevotedtotheidentification of magnetically induced mechanisms enabling electronic current controllability through their interplay with the confinement geometry.The purpose is to provide a pedagogical presentation of both the theoretical framework and computational approachin a manner adapted to open billiard systems, and to discuss the arising phenomenawithmagnetotransportcontrolasavehicle. If our purpose is even partially fulfilled, it is because of the contribution and supportofseveralcolleaguesandfriends.Inparticular,wehaveenjoyedhelpfuland inspiringdiscussionswithF.K.Diakonos,S.Rotter,F.Dolcini,andP.Giannakeas, v vi Preface amongothers.SpecialthanksgotoD.Buchholz,whopioneeredthecomputational aspects of the presented material and with whom the ideas on magnetotransport controlwerepartiallyinitiated.Finally,weareindebtedtoA.V.ZambetakiandS.I. Mistakidisfortheircriticalreadingofthemanuscript. Hamburg,Germany ChristianV.Morfonios March31,2016 PeterSchmelcher Contents 1 Introduction .................................................................. 1 1.1 ElectronWavesattheNanoscale...................................... 1 1.2 OpenQuantumBilliards ............................................... 3 1.3 TamingWavePropagationintheDeepQuantumRegime ........... 5 1.4 TheNecessityofEfficientComputationalTechniques............... 7 1.5 OutlineoftheBook .................................................... 8 References..................................................................... 9 2 ElectronsinLow-DimensionalMesoscopicSystems..................... 15 2.1 Two-DimensionalElectronSystems................................... 15 2.1.1 BandStructureandEffectiveMass............................ 15 2.1.2 HeterojunctionsandBandEngineering....................... 17 2.1.3 ModulationDopingandBandDiagram....................... 19 2.2 CoherentTransportDevices ........................................... 21 2.2.1 Shapingthe2DElectronSystem.............................. 21 2.2.2 MesoscopicLengthScales..................................... 23 2.2.3 ApproximationstotheHamiltonian........................... 25 2.3 MagnetoelectricSubbandsandTransportChannels.................. 27 2.4 DensityofStates........................................................ 32 References..................................................................... 34 3 CoherentElectronicTransport:Landauer-BüttikerFormalism ....... 37 3.1 LeadsandReservoirs................................................... 37 3.2 ScatteringMatrixandTransmissionFunction ........................ 39 3.2.1 LeadEigenmodes .............................................. 39 3.2.2 TransmissionAmplitudesandCoefficients................... 40 3.2.3 ConnectedScatterers........................................... 43 3.2.4 Two-TerminalSystem.......................................... 46 3.3 Two-TerminalLandauerFormula...................................... 47 3.3.1 GeneralCaseofCoherentTransport.......................... 47 3.3.2 LinearResponseRegime ...................................... 50 3.3.3 TransmissionasConductance................................. 52 vii viii Contents 3.4 MultiterminalConductance............................................ 53 3.4.1 CurrentfromScatteringStates ................................ 54 3.4.2 ConductanceMatrix............................................ 55 3.4.3 Currentand(Fictitious)VoltageProbes....................... 56 References..................................................................... 57 4 StationaryScatteringinPlanarConfiningGeometries.................. 59 4.1 In-PlaneHamiltonian................................................... 59 4.2 GreenianFormulationofScattering................................... 61 4.2.1 GreenFunctions................................................ 61 4.2.2 ScatteringMatrixfromGreenian.............................. 66 4.2.3 ElementsofFormalScatteringTheory........................ 72 4.3 Non-HermitianApproachtoScattering ............................... 77 4.3.1 DecompositionofConfigurationSpace....................... 77 4.3.2 EffectiveScatteringHamiltonianforFiniteSystem.......... 79 4.3.3 ConnectiontoElectronicTransport........................... 85 4.4 Multi-stateInterferenceEffects........................................ 90 4.4.1 FanoInterference............................................... 91 4.4.2 Aharonov-BohmOscillations.................................. 95 References..................................................................... 98 5 ComputationalQuantumTransport in Multiterminal andMultiplyConnectedStructures ....................................... 103 5.1 ComputationalSchemesforQuantumTransport..................... 103 5.2 FromOperatorstoMatrices............................................ 105 5.2.1 GridDiscretizationandTight-BindingHamiltonian......... 105 5.2.2 DispersionRelation............................................ 111 5.3 ScatteringviaSpatialDecomposition................................. 112 5.3.1 TruncationoftheHamiltonian................................. 113 5.3.2 OpenSystemPropagator....................................... 117 5.4 ComputationofthePropagator........................................ 123 5.4.1 Block-PartitioningoftheHamiltonian........................ 123 5.4.2 StandardRecursiveGreenFunctionMethod ................. 125 5.4.3 ReorderedBlock-GaussianEliminationScheme............. 126 5.5 Extended Recursive Green Function Method forMultiterminal,MultiplyConnectedStructures.................... 130 5.5.1 ModularPartitioning........................................... 131 5.5.2 Inter-Connection ............................................... 133 5.5.3 Intra-Connection............................................... 135 5.5.4 ComputationalEfficiencyandConsiderations................ 137 5.6 Transport Through Multiterminal and Multiply ConnectedBilliardSystems............................................ 138 5.6.1 SingleThree-TerminalEllipticBilliard....................... 138 5.6.2 Transmission and Localization Patterns inaLoopedMultiterminalStructure.......................... 142 References..................................................................... 146 Contents ix 6 MagnetoconductanceSwitching by Phase Modulation inArraysofOvalQuantumBilliards...................................... 149 6.1 SystemSetup,ApproximationsandComputationalapproach ....... 149 6.2 Single Oval Billiard: Transmission Suppression fromSelectiveEigenstateInterference................................ 152 6.3 Quantum Dot Array: Composite Resonant States andMagneticallyControlledTransmissionBands.................... 158 6.4 ConductanceSwitching................................................ 163 6.5 TheImpactofImpurities............................................... 167 6.6 SummaryandConclusions............................................. 169 References..................................................................... 170 7 Current Control in Soft-Wall Electron Billiards: Energy-PersistentScatteringintheDeepQuantumRegime............ 173 7.1 PersistentSwitchingViaGeometricRescalingatLowEnergies..... 173 7.2 Decoupling of Resonances and Controllable Finite-TemperatureConductance...................................... 176 7.3 ClosedBilliardEigenspectrum ........................................ 179 7.4 SwitchingBetweenCollimatedandBackscatteredWave Propagation............................................................. 182 7.5 ConductanceSwitchinginSoft-WallBilliardArrays ................ 185 7.6 BilliardGeometryandSoft-WallPotentialVariations................ 187 7.7 SummaryandConclusions............................................. 189 References..................................................................... 190 8 DirectionalMagnetotransportControlinMultiterminal FocusingQuantumBilliards................................................ 193 8.1 FromTwo-terminalto MultiterminalConductance Control:DirectionalCouplingbyWaveGuidingandFocusing...... 194 8.2 SetupandComputationalApproach................................... 196 8.3 SymmetriesoftheTransmissionCoefficients......................... 198 8.4 TransmissionSpectraatZeroMagneticField......................... 200 8.5 GeometryDependentMeanTransmission ............................ 202 8.6 TransmissioninaMagneticField ..................................... 207 8.7 BentCoupledWires.................................................... 212 8.8 DirectedConductance.................................................. 214 8.9 SummaryandConclusions............................................. 216 References..................................................................... 217 9 Summary,Conclusions,andPerspectives................................. 219 References..................................................................... 223 A GreenFunctionsofLeads................................................... 225 A.1 GreenFunctionofanInfiniteQuasi-1DWire......................... 225 A.2 InterfaceGreenFunctionofaSemi-InfiniteQuasi-1DWire......... 227
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