18. Nanostructures Imaging Techniques for nanostructures Electron Microscopy Optical Microscopy Scanning Tunneling Microscopy Atomic Force Microscopy Electronic Structure of 1-D Systems 1-D Subbands Spectroscopy of Van Hove Singularities 1-D Metals – Coulomb Interaction & Lattice Couplings Electrical Transport in 1-D Conductance Quantization & the Landauer Formula Two Barriers in Series-Resonant Tunneling Incoherent Addition & Ohm’s Law Localization Voltage Probes & the Buttiker-Landauer Formulism Electronic Structure of 0-D Systems Quantized Energy Levels Semiconductor Nanocrystals Metallic Dots Discrete Charge States Electrical Transport in 0-D Coulomb Oscillations Spin, Mott Insulators, & the Kondo Effects Cooper Pairing in Superconducting Dots Vibrational & Thermal Properties Quantized Vibrational Modes Transverse Vibrations Heat Capacity & Thermal Transport

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

Model of CdSe nanocrystal TEM image AFM image of crossed C-nanotubes (2nm wide) contacted by Au electrodes (100nm wide) patterned by e beam lithography Model of the crossed C-nanotubes & graphene sheets.

2 categories of nanostructure creation: Lithographic patterns on macroscopic materials (top-down approach). Can’t create structures < 50 μm. Self-assembly from atomic / molecular precusors (bottom-up approach). Can’t create structures > 50 μm. 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 nanostructures Reciprocal 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 microscope: probe beam focused on sample by lenses. Resolution β = numerical aperture Scanning 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. focal microscope DOS

Electron Microscopy Transmission 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 e’s 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 Microscopy For visible light & high numerical aperture ( β  1 ), d ~ 200-400 nm. → Direct optical imaging not useful in nanostructure studies. Useful indirect methods include Rayleigh sacttering, absortion, luminescence, Raman scattering, … Fermi’s golden rule for dipole approximation for light absorption: Emission rate (α = e2 /  c ): Real part of conductivity ( total absorbed power = σ E 2 ): Absorption & emission measurements → electronic spectra.

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

Scanning Tunneling Microscopy Carbon nanotube STM: 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 barrier z = distance between tip & sample Typical setup: Δz = 0.1 nm → Δ I / I = 1. Feedback mode: I maintained constant by changing z. → Δz ~ 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 ε F. ( weighted eDOS at E = εF + eV )

Atomic Force Microscopy Laser photodiode array AFM: Works on both conductor & insulator. Poorer resolution than STM. C ~ 1 N/m F ~ pN – fN Δz ~ pm mm sized cantilever 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 ω0 & Q are sensitive to type & strength of forces between tip & sample. Their values are used to construct an image of the sample.

Magnetic Force Microscopy MFM = AFM with magnetic tip Other 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 Systems Bulk: Independent electron, effective mass model with plane wave wavefunctions. Consider a wire of nanoscale cross section. i, j = quantum numbers in the cross section 1-D subbands Van Hove singularities at ε = εi, j

Spectroscopy of Van Hove Singularities Carbon nanotube Prob. 1 photoluminescence of a collection of nanotubes STM

1-D Metals – Coulomb Interaction & Lattice Couplings Let there be n1D carriers per unit length, then Fermi surface consists of 2 points at k = kF . Coulomb interactions cause e scattering near εF . For 3-D metals, this is strongly suppressed due to E, p conservation & Pauli exclusion principle. τ0 = classical scattering rate Caution: our Δε = Kittel’s ε. →  quasiparticles near εF are well defined → For 1-D metals, Let → for |k|  kF  E & p conservation are satisfied simultaneously.

1 + 2 → 3 + 4 Pauli exclusion favors E, p conservation:  For a given Δε1 , there always exist some Δε2 & Δε4 to satisfy the conservation laws provided Δε1 > Δε3 . → →  quasiparticles near εF not well defined Fermi 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 (we’ll 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-D Conductance Quantization & the Landauer Formula 1-D channel with 1 occupied subband connecting 2 large reservoir. Barrier model for imperfect 1-D channel Let Δn be the excess right-moving carrier density, DR(ε) be the corresponding DOS. q = e → The conductance quantum depends only on fundamental constants. Likewise the resistance quantum

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

Two Barriers in Series-Resonant Tunneling tj, rj = transmission, reflection amplitudes. For wave of unit amplitude incident from the left At left barrier At right barrier → → Resonance condition : n  Integers

At resonance Resonant tunneling For t1 = t2 = t : → For very opaque barriers, r  –1 ( φ  n π ) → resonance condition becomes particle in box condition while the off resonance case gives Using & one gets (see Prob 3) the Breit-Wigner form of resonance where

Incoherent Addition & Ohm’s Law Classical 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/τb . For propagation over distance dL, → (Prob. 4 ) Incoherence addition of each segment gives

Localization →  …  = average over φ* = average over k or ε . larger than incoherent limit Consider a long conductor consisting of a series of elastic scatterers of scattering length le . Let R >>1, i.e., R  1 & T << 1, ( R + T = 1 ) . For an additional length dL, Setting →

→  where C.f. Ohm’s law R  L For a 1-D system with disorder, all states become localized to some length ξ . Absence of extended states → R  exp( a L / ξ ) , a = some constant. For quasi-1-D systems, one finds ξ ~ N le , where N = number of occupied subbands. For T > 0, interactions with phonons or other e’s reduce phase coherence to length lφ = A T −α .  for each coherent segment. Overall R  incoherent addition of L / lφ such segments. For sufficiently high T, lφ  le , coherence is effectively destroyed & ohmic law is recovered. All states in disordered 2-D systems are also localized. Only some states (near band edges) in disordered 3-D systems are localized.

Voltage Probes & the Buttiker-Landauer Formulism T(n,m) = total transmission probability for an e to go from m to n contact. 1,2 are current probes; 3 is voltage probe. For a current probe n with N channels, µ of contact is fixed by V. Net current thru contact is Setting → For the voltage probe n, Vn adjusts itself so that In = 0. →

In , Vn depend on T(n,m) → their values are path dependent. Voltage probe can disturb existent paths. Let every e leaving 1 always arrive either at 2 or 3 with no back scattering. if Current out of 1: no probe

Mesoscopic regime: le < L < lφ . Semi-classical picture: App. G Aharonov-Bohm effect

Electronic Structure of 0-D Systems Quantum dots: Quantized energy levels. e in spherical potential well: For an infinite well with V = 0 for r < R : for r < R βn, l = nth root of jl (x). β0,0 = π (1S), β0,1 = 4.5 (1P), β0,2 = 5.8 (1D) β1,0 = 2π (2S), β1,1 = 7.7 (2P)

Semiconductor Nanocrystals CdSe nanocrystals For CdSe: For R = 2 nm, For e, ε 0,0 increases as R decreases. For h, ε 0,0 decreases as R decreases. → Eg increases as R decreases. Optical spectra of nanocrystals can be tuned continuously in visible region. Applications: fluorescent labeling, LED. Kramers-Kronig relation: same as bulk For ω →  → Strong transition at some ω in quantum dots → laser ?

Metallic Dots Mass spectroscopy (abundance spectra): Large abundance at cluster size of magic numbers ( 8, 20, 40, 58, … ) → enhanced stability for filled e-shells. Average level spacing at εF : For Au nanoparticles with R = 2 nm, Δε  2 meV. whereas semiC CdSe gives Δε  0.76 eV. → ε quantization more influential in semiC. Small spherical alkali metallic cluster Na mass spectroscopy

Optical properties of metallic dots dominated by surface plasmon resonance. If retardation effects are negligible, → Surface plasma mode at singularity: indep of R. For Au or Ag, ωp ~ UV, ωsp ~ Visible. → liquid / glass containing metallic nanoparticles are brilliantly colored. Large E just outside nanoparticles near resonance enhances weak optical processes. This is made use of in Surface Enhanced Raman Scattering (SERS), & Second Harmonic Generation (SHG).

Discrete Charge States Thomas-Fermi approximation: U = interaction between 2 e’s on the dot = charging energy. α = rate at which a nearby gate voltage Vg shifts φ of the dot. Neglecting its dependence on state, C = capacitance of dot. Cg = capacitance between gate & dot If dot is in weak contact with reservoir, e’s will tunnel into it until the μ’s are equalized. Change in Vg required to add an e is

U depends on size &shape of dot & its local environment. For a spherical dot of radius R surrounded by a spherical metal shell of radius R + d, Prob. 5 For R = 2 nm, d = 1 nm & ε = 1, we have U = 0.24 eV >> kBT = 0.026eV at T = 300K → Thermal fluctuation strongly supressed. For metallic dots of 2nm radius, Δε  2meV → ΔVg due mostly to U. For semiC dots, e.g., CdSe, Δε  0.76 eV → ΔVg due both to Δε & U. Charging effect is destroyed if tunneling rate is too great. Charge resides in dot for time δt  RC. ( R = resistance ) → Quantum fluctuation smears out charging effect when δε  U, i.e., when R ~ h / e2 . Conditions for well-defined charge states are &

Electrical Transport in 0-D For T < ( U + Δε ) / kB , U & Δε control e flow thru dot. Transport thru dot is suppressed when µL & µR of leads lie between µN & µN+1 (Coulomb blockade) Transport is possible only when µN+1 lies between µL & µR . → Coulomb oscillations of G( Vg ).

Coulomb Oscillations GaAs/AlGaAs T = 0.1K Coulomb oscillation occurs whenever U > kBT, irregardless of Δε . For Δε >> kBT, c.f. resonant tunneling: Thermal broadening Breit-Wigner lineshape

Single Electron Transistor (SET): Based on Coulomb oscillations ( turns on / off depending on N of dot ). → Ultra-sensitive electrometer ( counterpart of SQUID for B ). → Single e turnstiles & pumps: single e thru device per cycle of oscillation. quantized current I = e ω / 2 π. 2-D circular dot dI/dV: Line → tunneling thru given state. White diamonds (dI/dV = 0 ) : Coulomb blockades of fixed charge states ( filled shells for large ones ) N 1 2 3 … 7 (i, j) (0,0) (0,1) or (1,0) (1,1) , (0,2), or (2,0) Δ Vg U /α e (U + ) /α e (U + ) /α e Height of diamonds:

Spin, Mott Insulators, & the Kondo Effects Consider quantum dot with odd number of e’s in blockade region. ~ Mott insulator with a half-filled band. No external leads: degenerated Kondo effect : with external leads & below TK : Ground state = linear combinations of  &  states with virtual transitions between them. (intermediate states involve pairing with an e from leads to form a singlet state → Transmission even in blockade region. For symm barriers & T << TK , T 1. Singlets states in 3-D Kondo effect enhances ρ.

Cooper Pairing in Superconducting Dots Competition between Coulomb charging & Cooper pairing. For dots with odd number of e’s , there must be an unpaired e. Let 2Δ = binding energy of Cooper pairs. For 2Δ > U, e’s will be added to dot in pairs. Coulombe oscillations 2e – periodic.

Vibrational & Thermal Properties Continuum approximation: ω = vs K → ωj upon confinement. Quantized vibrations around circumference of thin cylinder of radius R & thickness t << R. j = 1,2, … Longitudinal compressional mode → Radial breathing mode j = 1,2, … Transverse mode

Measuring ωRBM gives good guess of R. Raman spectrum of individual carbon nanotubes ( 160 cm–1 = 20 meV ) vL = 21 km/s → Measuring ωRBM gives good guess of R.

Transverse Vibrations Transverse mode is not a shearing as in 3-D, but a flexural wave which involves different longitudinal compression between outer & inner arcs of the bend. Transverse standing wave on rectangular beam of thickness h, width w, & length L : → C.f. Torsion / shear mode Si nanoscale beans: f  L–2 Micro / Nano ElectroMechanical systems ( M/N EM)

Heat Capacity & Thermal Transport Quantized vibrational mode energies are much smaller than kBTroom . → Modes in confined directions are excited at Troom.  Lattice thermal properties of nanostructure are similar to those in bulk. For low T <  ω / kB , modes in confined directions are freezed out. → system exhibits lower-dimensional characteristics. E.g. (Prob.6) Gth depends only on fundamental constants if T = 1.