Physics of Zero- and One-Dimensional Nanoscopic Systems PDF
Preview Physics of Zero- and One-Dimensional Nanoscopic Systems
SpringerSeriesin solid-state sciences SpringerSeriesin solid-state sciences SeriesEditors: M.Cardona P.Fulde K.vonKlitzing R.Merlin H.-J.Queisser H.Sto¨rmer The Springer Series in Solid-State Sciences consists of fundamental scientific books prepared by leading researchers in the field.They strive to communicate,in a systematic andcomprehensiveway,thebasicprinciplesaswellasnewdevelopmentsintheoreticaland experimentalsolid-statephysics. PhaseSeparation ElectronScatteringinSolidMatter inSoftMatterPhysics ATheoretical MicellarSolutions,Microemulsions, andComputationalTreatise CriticalPhenomena ByJ.Zabloudil,R.Hammerling, ByP.K.KhabibullaevandA.A.Saidov L.Szunyogh,andP.Weinberger Optical Response of Nanostructures PhysicalAcousticsintheSolidState Microscopic ByB.Lu¨thi NonlocalTheory SolitaryWaves ByK.Cho inComplexDispersiveMedia FractalConcepts Theory·Simulation·Applications inCondensedMatterPhysics ByV.Yu.BelashovandS.V.Vladimirov ByT.NakayamaandK.Yakubo TopologyinCondensedMatter ExcitonsinLow-Dimensional Editor:M.I.Monastyrsky Semiconductors Particle Penetration and Radiation Theory,NumericalMethods, Effects Applications ByS.Glutsch ByP.Sigmund Two-DimensionalCoulombLiquids Magnetism andSolids FromFundamentals ByY.MonarkhaandK.Kono toNanoscaleDynamics X-RayMultiple-WaveDiffraction ByH.C.SiegmannandJ.Sto¨hr TheoryandApplication QuantumChemistryofSolids ByS.-L.Chang TheLCAOFirstPrinciples PhysicsofTransitionMetalOxides TreatmentofCrystals ByS.Maekawa,T.Tohyama, ByR.A.Evarestov S.E.Barnes,S.Ishihara, Low-DimensionalMolecularMetals W.Koshibae,andG.Khaliullin ByN.Toyota,M.LangandJ.Mu¨ller Point-ContactSpectroscopy DiffusioninSolids ByY.G.NaidyukandI.K.Yanson Fundamentals,Methods,Materials, OpticsofSemiconductors Diffusion-ControlledProcesses andTheirNanostructures ByH.Mehrer Editors:H.KaltandM.Hetterich PhysicsofZero- andOne-Dimensional NanoscopicSystems ByS.N.Karmakar,S.K.Maiti,J.Chowd- hury Volumes–arelistedattheendofthebook. · · Sachindra Nath Karmakar Santanu Kumar Maiti Jayeeta Chowdhury (Eds.) Physics of Zero- and One-Dimensional Nanoscopic Systems WithFigures 123 Prof.Dr.SachindraNathKarmakar SantanuKumarMaiti JayeetaChowdhury SahaInstituteofNuclearPhysics Bidhannagar Kolkata India SeriesEditors: ProfessorDr.,Dres.h.c.ManuelCardona ∗ ProfessorDr.,Dres.h.c.PeterFulde ProfessorDr.,Dres.h.c.KlausvonKlitzing ProfessorDr.,Dres.h.c.Hans-JoachimQueisser Max-Planck-Institutfu¨rFestko¨rperforschung,Heisenbergstrasse,Stuttgart,Germany ∗ Max-Planck-Institutfu¨rPhysikkomplexerSysteme,No¨thnitzerStraße Dresden,Germany ProfessorDr.RobertoMerlin DepartmentofPhysics,EastUniversity,UniversityofMichigan AnnArbor,MI-,USA ProfessorDr.HorstSto¨rmer Dept.Phys.andDept.Appl.Physics,ColumbiaUniversity,NewYork,NYand BellLabs.,LucentTechnologies,MurrayHill,NJ,USA LibraryofCongressControlNumber: ISSN - ISBN ---- SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember, ,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagBerlinHeidelberg Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Digitaldatasuppliedbyauthors Production:LE-TEXJelonek,Schmidt&Vo¨cklerGbR,Leipzig Coverdesign:eStudioCalamarS.L.,F.Steinen-Broo,Girona,Spain SPIN //YL– Printedonacid-freepaper To our parents Preface The idea of this volume emerges from the \International Workshop on the Physics of Zero and One Dimensional Nanoscopic Systems," which was held on 1-9 February 2006 at Saha Institute of Nuclear Physics, India. The theme oftheworkshopwastounderstandphysicallytherecentadvancesinnanoscale systems,like,quantumdots,quantumwires,2Delectrongases,etc. Alimited number of distinguished physicists were invited to give pedagogical lectures anddiscusscoremethodsincludingthelatestdevelopments.Thisvolumecon- sists of self-contained review articles on recent theories of the evolution of Kondo e(cid:11)ect in quantum dots, decoherence and relaxation in chargedqubits, edge-statetransportthroughnanographitesandquantumHallsystems,trans- port through molecular bridges, coherence and interaction in di(cid:11)usive meso- scopicsystems,persistentcurrentinmesoscopicrings,and,thethermoelectric phenomena of nanosystems. As these are rapidly growing subjects, we hope that this book with contributions from the leading experts will serve as a stimulus for new researchers and also become a landmark to the body of the knowledge in the (cid:12)eld. We have presented the articles on quantum dots (cid:12)rst, then on quantum wires and (cid:12)nally on 2D electron gases. A brief account of each chapter is given below: The(cid:12)rstchapterbyAvrahamSchillerstartswithabriefhistoricalnoteon the Kondo problem. The Anderson Hamiltonian for the ultra-small quantum dotisthenmappedontotheKondoHamiltonianapplyingasuitablecanonical transformationeliminatingcharge(cid:13)uctuations.A detailedstudyof resistivity and conductance for tunneling through ultra-small quantum dots is given. The Toulouse limit, where the model can be solved exactly using standard techniques is studied here using Abelian bosonization. At T = 0 and B = 0, a Lorentzian zero-bias anomaly is observed in the di(cid:11)erential conductance as a function of voltage bias. Nonzero temperature smears out the zero-bias anomaly and nonzero magnetic (cid:12)eld splits the peak into two. In this article, a diagrammaticapproachknownasnoncrossingapproximation(NCA) to the Kondoproblem isalsointroduced within slaveboson representation.There is a sharp Abrikosov-Suhlresonance near the Fermi level in the equilibrium dot VIII Preface densityof states.Thisresonancesplits asthevoltagebiassu(cid:14)cientlyexceeds the Kondo temperature which is also supported by experiments. The second chapter by Yuval Oreg and David Goldhaber-Gordon reviews a theoretical analysis of a system consisting of a large electron droplet cou- pled to a small electron droplet. This system displays two-channel Kondo behavior at experimentally accessible temperatures. Special emphasis is put on the estimate of the two-channel Kondo energy scale using a perturbative renormalizationgroupapproach.Theirpredictionsforthedi(cid:11)erentialconduc- tanceinascalingformisconvenientforexperimentalanalysis.Theyhavealso pointed out some open questions. In the third chapter K. Kikoin and Y. Avishai show that a new ingredi- ent in the study of the Kondo e(cid:11)ect in quantum dots (also called arti(cid:12)cial molecules) is the internal symmetry of the nano-object, which proves to play a crucial role in the construction of the e(cid:11)ective exchange Hamiltonian. This internal symmetry combines continuous spin symmetry (SU(2)) and discrete point symmetry (such as mirror re(cid:13)ections for double dots or discrete C 3v rotation for equilateral triangular dots). When these arti(cid:12)cial molecules are attached to metallic leads, the e(cid:11)ective exchange Hamiltonian contains oper- atorswhich couple states belonging to di(cid:11)erent irreducible representationsof the internalsymmetrygroup.In manycases,the set ofdotoperatorsappear- ing in the e(cid:11)ective exchange Hamiltonian generate a group which is referred to as the dynamical symmetry group of the system dot-leads. These dynam- ical symmetry groups are mostly SO(n) or SU(n). One of the remarkable outcomes of their study is that the pertinent group parameters (such as the value of n) can be controlled by experimentalists. The reason for that is that theKondotemperatureturnsouttobehigheraroundthepointsofaccidental degeneracy where the dynamical symmetry is \more exact" and these points canbetunedbyexperimentalparameterssuchasgatevoltagesandtunneling strength. In this review the authors have clari(cid:12)ed and expanded these con- cepts, and discussed some speci(cid:12)c examples. They go from \light to heavy" starting from a simple quantum dot, moving on to discuss double quantum dot (where only permutation (re(cid:13)ection) symmetry can be considered as in- ternal one) and (cid:12)nally elaborate on a triple quantum dot. In particular they concentrate on the di(cid:11)erence between the chain geometry (where the three dotscomposingthetripledotarearrangedinseries)andthering(triangular) geometry.Whenaperpendicularmagnetic(cid:12)eldisapplied,thetriplequantum dot in the ring geometry displays a remarkable combination of symmetries: U(1) of the electromagnetic (cid:12)eld, SU(2) of the dot spin and C of the dot 3v orbital dynamics. The magnetic (cid:12)eld controls the crossover between SU(2) and SU(4) dynamical symmetries and this feature shows up clearly in the conductance versus magnetic (cid:12)eld curve. The fourth chapter with contribution from Alex Grishin, Igor V. Yurke- vich and Igor V. Lerner describes some essential features of loss of coherence by a qubit (controllable two-level system) coupled to the environment. They (cid:12)rst presented the well-known semiclassical arguments that relate both de- Preface IX coherence and relaxation to the environmental noise. Then they show that models with pure decoherence (but no relaxation in qubit states) are exactly solvable. As an example, they have treated in detail the model of (cid:13)uctuating background charges which is believed to describe one of the most important channels of decoherence for the charge Josephson junction qubit. They show that the decoherencerate is linearin T at low temperatures and saturatesto a T-independentclassicallimit at ‘high’ temperatures,while depending in all theregimesnon-monotonicallyonthecouplingofthequbittothe(cid:13)uctuating backgroundcharges.Theyhavealsoconsidered,albeitonlyperturbatively,the qubit relaxation by the background charges and demonstrated that a quasi- linearbehaviorofthespectraldensityofnoisededucedfromthemeasurements of the relaxation rate can be qualitatively explained. The contribution by Katsunori Wakabayashi in the (cid:12)fth chapter eluci- dates the role of the edge states on the low-energy physical properties of nanographite systems. He (cid:12)rst discussed the basics of the electronic proper- tiesofthenanographteribbonsandpointedouttheexistenceofedge-localized statesnearthezigzagedge.Hethenpresentedtheelectronicpropertiesofthe nanographite systems in the presence of magnetic (cid:12)eld and provides a sim- plepictureforthe originof half-integerquantum Halle(cid:11)ectin graphene.The studyoftheorbitalandPaulimagnetizationshowsthatananographitesystem with zigzag edges exhibits strong paramagnetic response at low-temperature due to the edge states, and there exist a crossover from a weak diamagnetic responseatroomtemperaturetoastrongparamagneticresponseatlowtem- perature. It is also observed that electron-electron interaction can produce a ferrimagnetic spin polarization along the zigzag edge. In this article author also describes the electron transport properties of nanographite ribbon junc- tions. A single edge state cannot contribute to electron conduction due to the non-bonding characterof the edge states. However,in the zigzag ribbons edge states can provide a single-channel for electron conduction in the low- energy region due to the bonding and anti-bonding interaction between the edge states. The remarkable feature is the appearance of zero-conductance dips in the single-channel region where current vortex with Kekul(cid:19)e pattern is observed. Its relation with the asymmetric Aharonov-Bohm ring is also discussed. The sixth chapter by K. A. Chao and Magnus Larsson is a review of thethermoelectricphenomenainnanosystems.Startingfromthediscoveryof thermoelectricphenomenonin 1822bySeebeck,theauthorshavedividedthe developmentofthermoelectricityintothreestages.Theypointedoutthatthe thermodynamic theory was the driving force in the (cid:12)rst stage, during which theSeebecke(cid:11)ect,thePeltiere(cid:11)ect,theThomsoncoe(cid:14)cient,thedualrolesof thermoelectric power generation and refrigeration, and the e(cid:14)ciency of ther- moelectric processes were extensively investigated and understood fairly well qualitatively. For a long time the practical use of thermoelectricity was mea- suring temperature with thermocouples. The beginning of the second stage was markedby the correctcalculation of the e(cid:14)ciency of thermoelectric gen- X Preface erator and refrigerator by Altenkirch in 1909. It was demonstrated that the e(cid:14)ciency depends mainly on a quantity which was later called the (cid:12)gure of merit. A higher value of this (cid:12)gure of merit indicates a better thermoelectric material. Using the free electron gas as a model system, Io(cid:11)e calculated the (cid:12)gure of merit and predicted doped semiconductors as favorable thermoelec- tric materials. Using the (cid:12)gure of merit as an indicator, and guided by the semi-classicaltransporttheory, the searchfor better thermoelectric materials hadlastedforalongtimeuntilaround1980swhenthemodernmaterialtech- nology enabled the fabrication of layer materials with nanometer thickness. This is the end of the second stage. In the second stage the search for new thermoelectricmaterialswasbasedonthesemi-classicalBoltzmanntransport equation, in which the dominating scattering process results in slow di(cid:11)usive transportandsolowvalueofthe(cid:12)gureofmerit.Inlayermaterialsitispossi- ble to reduce the scattering and a new thermoelectric mechanism is found in the so-called thermionic transport. Thermionic emission of electrons from a hotsurfaceisawell-studiedphysicalprocess,andtheemittedcurrentdensity depends on the temperatureandthe workfunction of the emitting materials. In principle, large thermionic current can be achieved if one can reduce the work function to su(cid:14)ciently low. With the advancement of material fabrica- tion technologyto produce high quality layermaterials, there has been much progress in thermionics. The reduction of layer thickness in order to achieve e(cid:14)cient transport processalso inevitably creates new fundamental problems, many of which are of quantum mechanical nature. Therefore, in the present thirdstageofthermoelectricity,wefacethechallengeofanentirelynew(cid:12)eldto which the macro-scalethermoelectrictheorydoes notapply. Thisnew (cid:12)eld is thenano-scalethermoelectricity.Themainthemeofthischapteristoprovide a smooth transition of thermoelectric phenomena from macro-scale systems to nano-scalesystems. The review article by Gilles Montambaux in the seventh chapter gives a nice introduction to coherent e(cid:11)ects in disordered electronic systems. Avoid- ing technicalities as most as possible, he presented some personal points of view to describe well-knownsignaturesof phase coherencelike weaklocaliza- tion correction or universal conductance (cid:13)uctuations. He showed that these physicalpropertiesofphasecoherentconductorscanbe simplyrelatedtothe classical return probability for a di(cid:11)usive particle. The di(cid:11)usion equation is then solved in various appropriate geometries and in the presence of a mag- netic(cid:12)eld.Theimportantnotionofquantumcrossingisdeveloped,whichisat theoriginofthequantume(cid:11)ects.Theanalogywithopticsisexploitedandthe relation between universal conductance (cid:13)uctuations and speckle (cid:13)uctuations inopticsisexplained.Thelastpartconcernsthee(cid:11)ectofelectron-electronin- teractions.Usingthesamesimpledescription,theauthorderivedqualitatively the expressionsofthe Altshuler-Aronovanomalyofthe densityof states,and of the correction to the conductivity. The last part, slightly more technical, addressesthequestionofthelifetimeofaquasi-particleinadisorderedmetal. Preface XI The eighth chapter by Georges Bouzerar is on the phenomenon of per- sistent current in mesoscopic normal metal rings. With a brief introductory note he (cid:12)rst showed that the single particle picture can neither explain the magnitudenorthesignofthepersistentcurrentmeasuredindi(cid:11)usivemetallic mesoscopicrings.Thisnaturallyleadhimtothemainpartofthearticle{the interplay between electron-electron interaction and disorder. One important resultisthatelectron-electroninteractioncaneitherenhanceorsuppressper- sistent current depending on the strength of the interaction. The underlying physics has been discussed in details. The ninth chapter by Santanu K. Maiti and S. N. Karmakar focuses on electrontransportthroughnanostructures.Theauthors(cid:12)rstbrie(cid:13)yintroduce the Green’s function technique in this study. Electron transmission through various molecular bridges are investigated in detail within the tight-binding framework. They show that the transport properties through such bridges are highly sensitive to relative position of the atoms in the molecule, cou- pling between molecule and electrodes, and also to the external magnetic or electric (cid:12)elds. The theoretical results are in qualitative agreement with the experimental observations. These model calculations provide better physical understanding of the transport problems through nanostructures. The au- thors have suggested some molecular devices in which electron transport can be tuned e(cid:14)ciently. Finally, the tenth chapter by S. Sil, S. N. Karmakarand Efrat Shimshoni is on quantum Hall e(cid:11)ect. This article provides an account of the exotic sta- tisticalnatureofthequasi-particlesinquantumHallsystem.Forinstance,an electron in the presence of electron-electron interaction and strong magnetic (cid:12)eldmayundergoBosecondensationbycharge-(cid:13)uxcomposite,andfractional chargeexcitationsemergeasquasi-particles.Thesequasi-particlesmanifestlot ofsurprisesinthestudiesofquantumHallsystems.Thisreviewisonboththe integer and fractional quantum Hall e(cid:11)ects within the (cid:12)eld theoretic frame- work.Inthisreview,theauthorshavealsodiscussedtheroleoftheedgestates onintegerandfractionalquantumHalle(cid:11)ectstounderstandtheexperimental results. It is our great pleasure to thank Prof. Yuval Gefen and Prof. Bikas K. Chakrabarti for their invaluable cooperation and support in organizing the internationalworkshopwithoutwhichthisbookmighthavenotseenthelight of the day. In this context, we also thank Prof. Hans Weidenmueller, Prof. Yoseph Imry, Prof. Markus Buttiker, Prof. Amnon Aharony and Prof. Yigal Meir for their advices and encouragements. We wish to thank all the invited speakers who made the workshop successful and all the authors who have contributedtothisvolume.WeareverymuchgratefultoProf.PeterFuldeand Dr. Claus Ascheron for recommending the publication of this book. Thanks are also due to Dr. Angela Lahee and Dr. Elke Sauer from Springer-Verlag for friendly collaboration. Fine help and constant encouragement from our colleagues in this endeavor is also highly appreciated. Finally, we thank the
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