18. Nanostructures Imaging Techniques for nanostructures Electron Microscopy Optical Microscopy...

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Transcript of 18. Nanostructures Imaging Techniques for nanostructures Electron Microscopy Optical Microscopy...

  • 18. NanostructuresImaging Techniques for nanostructuresElectron MicroscopyOptical MicroscopyScanning Tunneling MicroscopyAtomic Force MicroscopyElectronic Structure of 1-D Systems1-D SubbandsSpectroscopy of Van Hove Singularities1-D Metals Coulomb Interaction & Lattice CouplingsElectrical Transport in 1-DConductance Quantization & the Landauer FormulaTwo Barriers in Series-Resonant TunnelingIncoherent Addition & Ohms LawLocalizationVoltage Probes & the Buttiker-Landauer FormulismElectronic Structure of 0-D SystemsQuantized Energy LevelsSemiconductor NanocrystalsMetallic DotsDiscrete Charge StatesElectrical Transport in 0-DCoulomb OscillationsSpin, Mott Insulators, & the Kondo EffectsCooper Pairing in Superconducting DotsVibrational & Thermal PropertiesQuantized Vibrational ModesTransverse VibrationsHeat Capacity & Thermal Transport

  • 1-D nanostructures: carbon nanotubes, quantum wires, conducting polymers, .0-D nanostructures: semiconductor nanocrystals, metal nanoparticles, lithographically patterned quantum dots, .Gate electrode pattern of a quantum dot.SEM imageWell deal only with crystalline nanostructures.

  • Model of CdSe nanocrystalTEM imageAFM image of crossed C-nanotubes (2nm wide) contacted by Au electrodes (100nm wide) patterned by e beam lithographyModel of the crossed C-nanotubes & graphene sheets.

  • 2 categories of nanostructure creation: Lithographic patterns on macroscopic materials (top-down approach).Cant create structures < 50 m. Self-assembly from atomic / molecular precusors (bottom-up approach).Cant create structures > 50 m.Challenge: develop reliable method to make structure of all scales. Rationale for studying nanostructures:Physical, magnetic, electrical, & optical properties can be drastically altered when the extent of the solid is reduced in 1 or more dimensions.1. Large ratios of surface to bulk number of atoms. For a spherical nanoparticle of radius R & lattice constant a:R = 6 a ~ 1 nm Applications: Gas storage, catalysis, reduction of cohesive energy, 2. Quantization of electronic & vibrational properties.

  • Imaging Techniques for nanostructuresReciprocal space (diffraction) measurements are of limited value for nanostructures:small sample size blurred diffraction peaks & small scattered signal. 2 major classes of real space measurements : focal & scanned probes.focal microscopeFocal microscope: probe beam focused on sample by lenses. = numerical aperture ResolutionScanning microscopy: probe scans over sample.Resolution determined by effective range of interaction between probe & sample. Besides imaging, these probes also provide info on electrical, vibrational, optical, & magnetic properties.DOS

  • Electron MicroscopyTransmission Electron Microscope (TEM):100keV e beam travels thru sample & focussed on detector.Resolution d ~ 0.1 nm (kept wel aboved theoretical limit by lens imperfection).Major limitation: only thin samples without substrates can be used. Scanning Electron Microscope (SEM):100~100k eV tight e beam scans sample while backscattered / secondary es are measured.Can be used on any sample.Lower resolution: d > 1 nm.SEM can be used as electron beam lithography.Resolution < 10 nm.Process extremely slow used mainly for prototypying & optical mask fabrication.

  • Optical MicroscopyFor visible light & high numerical aperture ( 1 ), d ~ 200-400 nm. Direct optical imaging not useful in nanostructure studies. Useful indirect methods includeRayleigh sacttering, absortion, luminescence, Raman scattering, Fermis golden rule for dipole approximation for light absorption:Emission rate ( = e2 / c ):Real part of conductivity ( total absorbed power = E 2 ):Absorption & emission measurements electronic spectra.

  • Fluorescence from CdSe nanocrystals at T = 10KSpectra of Fluorescence of individual nanocrystals.Mean peak: CB VBOther peaks involves LO phonon emission.Optical focal system are often used in microfabrication.i.e., projection photolithography.For smaller scales, UV, or X-ray lithographies are used.

  • Scanning Tunneling MicroscopyCarbon nanotubeSTM: Metal tip with single atom end is controlled by piezoelectrics to pm precision.Voltage V is applied to sample & tunneling current I between sample & tip is measured. = tunneling barrierz = distance between tip & sampleTypical setup: z = 0.1 nm I / I = 1.Feedback mode: I maintained constant by changing z. z ~ 1 pm can be detected.

  • STM can be used to manipulate individual surface atoms. Quantum coral of r 7.1 nm formed by moving 48 Fe atoms on Cu (111) surface.Rings = DOS of e in 3 quantum states near F.( weighted eDOS at E = F + eV )

  • Atomic Force MicroscopyLaser mm sized cantileverphotodiode arrayC ~ 1 N/mF ~ pN fNz ~ pmAFM: Works on both conductor & insulator. Poorer resolution than STM.Contact mode: tip in constant contact with sample; may cause damage.Tapping mode:cantilever oscillates near resonant frequency & taps sample at nearest approach.Q = quality factor 0 & Q are sensitive to type & strength of forces between tip & sample.Their values are used to construct an image of the sample.

  • Magnetic Force MicroscopyMFM = AFM with magnetic tipOther scanned probe techniques: Near-field Scanning Optical Microscopy (NSOM) Uses photon tunneling to create optical images with resolution below diffraction limit. Scanning Capacitance Microscopy (SCM)AFM which measures capcitance between tip & sample.

  • Electronic Structure of 1-D SystemsBulk: Independent electron, effective mass model with plane wave wavefunctions.Consider a wire of nanoscale cross section.i, j = quantum numbers in the cross sectionVan Hove singularities at = i, j1-D subbands

  • Spectroscopy of Van Hove SingularitiesSTMphotoluminescence of a collection of nanotubesCarbon nanotubeProb. 1

  • 1-D Metals Coulomb Interaction & Lattice CouplingsLet there be n1D carriers per unit length, thenFermi surface consists of 2 points at k = kF .Coulomb interactions cause e scattering near F .For 3-D metals, this is strongly suppressed due to E, p conservation & Pauli exclusion principle. 0 = classical scattering rate quasiparticles near F are well definedFor 1-D metals, for |k| kF E & p conservation are satisfied simultaneously.LetCaution: our = Kittels .

  • 1 + 2 3 + 4Pauli exclusion favorsE, p conservation: For a given 1 , there always exist some 2 & 4 to satisfy the conservation laws provided 1 > 3 . quasiparticles near F not well definedFermi liquid (quasiparticle) model breaks down.Ground state is a Luttinger liquid with no single-particle-like low energy excitations. Tunneling into a 1-D metal is suppressed at low energies.Independent particle model is still useful for higher excitations (well discuss only such cases).

  • 1-D metals are unstable to perturbations at k = 2kF .E.g., Peierls instability: lattice distortion at k = 2kF turning the metal into an insulator. Polyacetylene: double bonds due to Peierls instability. Eg 1.5eV.Semiconducting polymers can be made into FETs, LEDs, .Proper doping turns them into metals with mechanical flexibility & low T processing. flexible plastic electronics.Nanotubes & wires are less susceptible to Peierls instability.

  • Electrical Transport in 1-DConductance Quantization & the Landauer Formula1-D channel with 1 occupied subband connecting 2 large reservoir.Barrier model for imperfect 1-D channelLet n be the excess right-moving carrier density, DR() be the corresponding DOS.q = e The conductance quantum depends only on fundamental constants.Likewise the resistance quantum

  • Channel fully depleted of carriers at Vg = 2.1 V.If channel is not perfectly conducting,Landauer formulaFor multi-channel quasi-1-D systemsi, j label transverse eigenstates.T = transmission coefficient.For finite T,R = reflection coefficient.

  • Two Barriers in Series-Resonant Tunnelingtj, rj = transmission, reflection amplitudes.For wave of unit amplitude incident from the leftAt left barrierAt right barrierResonance condition :n Integers

  • At resonanceFor t1 = t2 = t :Resonant tunnelingFor very opaque barriers, r 1 ( n ) resonance condition becomes particle in box condition while the off resonance case gives Usingone gets (see Prob 3) the Breit-Wigner form of resonancewhere&

  • Incoherent Addition & Ohms LawClassical treatment: no phase coherence.(Prob. 4 )= Sum of quantized contact resistance & intrinsic resistance at each barrier.Let the resistance be due to back-scattering process of rate 1/b .For propagation over distance dL,(Prob. 4 )Incoherence addition of each segment gives

  • Localizationlarger than incoherent limit = average over * = average over k or . Consider a long conductor consisting of a series of elastic scatterers of scattering length le .Let R >>1, i.e., R 1 & T
  • whereC.f. Ohms law R LFor a 1-D system with disorder, all states become localized to some length .Absence of extended states R exp( a L / ) , a = some constant.For quasi-1-D systems, one finds ~ N le , where N = number of occupied subbands. For T > 0, interactions with phonons or other es reduce phase coherence to length l = A T .for each coherent segment.For sufficiently high T, l le , coherence is effectively destroyed & ohmic law is recovered. Overall R incoherent addition of L / l such segments. All states in disordered 2-D systems are also localized.Only some states (near band edges) in disordered 3