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Nanostructured materials • 0D: quantum dots • 1D: Nanowires • 2D: superlattices and heterostructures • Nano-Photonics • Magnetic nanostructures • Nanofluidic devices and surfaces Copyright Stuart Lindsay 2009 Nanostructured materials derive their special properties from having one or more dimensions made small compared to a length scale critical to the physics of the process. Copyright Stuart Lindsay 2009 Development of electronic properties as a function of cluster size Each band has a width that reflects the interaction between atoms, with a bandgap between the conduction and the valence bands that reflects the original separation of the bonding ad antibonding states. Electronic DOS and dimensionality Size effects are most evident at band edges (semiconductor NPs). DOS (dn/dE) as a function of dimensionality. 3D case is for free particles. Copyright Stuart Lindsay 2009 k-space is filled with an uniform grid of points each separated in units of 2π/L along any axis. The volume of k-space occupied by each point is: 2 L r-space: 4 r 2 dr V 3 k-space: 4 k 2 dk 4 L3 k 2 dk Vk 8 3 3D DOS Density of states in a volume V per unit wave vector: For a free electron gas: dn Vk 2 dk 2 2 2k 2 E 2m dn dn dk Vk 2 m Vm 2 2 2 2 dE dk dE 2 k 2 dE 2 k dk m 1 2mE 2 E 2 Copyright Stuart Lindsay 2009 2D DOS dn A2k dk 2 2 dE 2 k dk m dn dn dk Am dE dk dE 2 2 Constant for each electronic band Copyright Stuart Lindsay 2009 1D DOS dn L dk 2 dE 2 k dk m dn Lm E 2 dE 2 k 1 2 At each atomic level, the DOS in the 1D solid decreases as the reciprocal of the square root of energy. Copyright Stuart Lindsay 2009 8 0 D DOS In zero dimensions the energy states are sharp levels corresponding to the eigenstates of the system. Copyright Stuart Lindsay 2009 0D Electronic Structures: Quantum Dots Light incident on a semiconductor at an energy greater than the bandgap forms an exciton, i.e. an electron-hole quasiparticle, representing a bound state. Excitons can be treated as “Bohr atoms” 2 2 mV 1 e r 40 r 2 40 r 2 * e m 2 When the size of the nanoparticle approaches that of an exciton, size quantization occurs. Electronic energy gap 2 2 2 1 1 e * * 1.8 E E 0 2 2 R me mh R Intrinsic band gap NP radius electrostatic correction 1-D Electronic Structures: Carbon Nanotubes Wrapping vector: Diameter: n n1a1 n2a2 d n12 n22 n1n2 0.0783nm The folding of the sheet controls the electronic properties of the nanotubes. Conduction in CNTs pz electrons hybridize to form π e π* valence and conduction bands that are separated by an energy gap of about 1V (semiconductor). For certain high simmetry directions (the K points in the reciprocal lattice) the material behaves like a metal. k wave vector perpendicular to the CNT long axis Apex: at this point CB meets VB for graphene sheets (metal-like behavior) Allowed K states The component of the wave vector perpendicular to the CNT long axis is quantized 2n k D D = diameter of the nanotube Metallic behavior: the allowed values of k intersect the k points at which the conduction and valence bands meet. CNTs can be either metals or semiconductors depending on their chirality. Field effect transistor made from a single semiconducting CNT connecting source and drain connectors. Semiconductor Nanowires • Ga-P/Ga-As p/n nanojunctions TEM images Line profiles of (IOP) the composition through the junction region Copyright Stuart Lindsay 2009 2D Electronic Structures: superlattices and heterostructures Superlattice: alternating layers of small bandgap semiconductors (GaAs) interdispersed with layers of wide bandgap semiconductors (GaAlAs). Variation of electron energy in an MBE grown superlattice The thickness of each layer is considerably smaller than the electron mean free path. 17 Modulation of the structure on the length scale d (thickness of the layer in the superlattice) gives rise to the formation of new bands inside the original Brillouin zone . Electrons can pass freely from one small bandgap region to another without scattering. Band splitting into sub-bands Low scattering in 2-D means reaching zone boundaries at reasonable fields, accelerating electrons at the band edges. Resonant tunneling through different sub-bands Negative differential resistance: electrons slow down with increasing bias when approaching the first sub-band boundary (≈20mV). Quantum Hall resistance of 2D electron gas Electrons in a layer are accelerated by an applied magnetic field at a frequency: eB C m Magnetic quantization in 2D electron gas A series of steps in the Hall resistance corresponding exactly at twice the Landauer frequency were observed. Copyright Stuart Lindsay 2009 h RH 2 ne von Klitzing resistance Nobel in Physics 1985 Confinement on optical length scales Plasmonics Small (d<<λ) metal particles exhibit a phenomenon called plasma resonance, i.e. plasma-polariton resonance of the free electrons in the metal surface. A resonant metal particle can capture light over a region of many wavelengths in dimension even if the particle itself is only a fraction of a wavelength in diameter (resonant antennas). Free electrons in metals polarize excluding electric fields from the interior of the metal showing a negative dielectric constant. The polarizability of a sphere of volume V and dielectric constant εr is: r 1 0 3V r 2 When εr →-2 →∞ For d<<λ the resonant frequency is independent on the particle size, but depends on particle shape. For a prolate spheroid of eccentricity e: b e 1 a 2 2 0V 1 r L (1 L 1) r where: 1 e2 1 1 e a L 2 1 ln 1 e 2e 1 e b 1.6 For Ag =-2 at 400 nm, but resonance moves to 700 nm for a/b=6. The resonance is tunable throughout the visible by engineering the particle shape. Plasmon enhanced optical absorption Placing a chromophore near a resonant metal nanoparticle: dye layer Electric field surrounding a resonant nanoparticle (E=Ez) Enhanced fluorescence Reduced decay times The plasmon resonance results in local enhancement of the electric field: doubling the electric field quadruples the light absorption. Enhanced fluorescence A single dye molecule is only visible in fluorescence when the gold NP passes over it. The increased absorption cross section is accompanied by a decrease in fluorescence lifetime. Photonic engineering 3D 2D Modern semiconductor lasers are made from semiconductors heterostructures designed to trap excited electrons and holes in the optically active part of the laser. Performance of a solid-state laser material in various geometries. 1D Lasing effect: the gain of the laser medium must exceed the cavity losses. Dashed lines: density of states 0D In quantum dots all of available gain is squeezed into a narrow bandwidth. 25 Optical cavity Quantum dot laser Quantum dots of the right size can place all of the exciton energies at the right value for lasing. The QDs are chosen to have a bandgap that is smaller than that of the medium. Excitons are stabilized in the optical cavity, because the electrons are confined to the lowenergy part of the conduction band and the holes are confined to the top of the valence band. Photonic crystals: concentrating photon energy into bands Opalescent materials from colloidal crystal (polystyrene latex beads) The concentration of modes into bands results in an increase in the density of states in the allowed bands, particularly near bands edges. Optical wavelengths require that materials should be structured on the half-micron scale. Opalescence comes from sharp (Bragg) reflection in only certain directions. Copyright Stuart Lindsay 2009 For a given spacing in some direction in the colloidal crystal lattice , the wavelength of the reflected beam is given by the Bragg law. Bragg’s law 2nd hkl cos int 2d hkl n 2 sin 2 ext Colloidal particles spaced with polymer spacers sin ext n sin int hkl = Miller indices of the colloidal crystal lattice n = refraction index dhkl = spacing between Bragg planes in the hkl direction 3D Photonic crystals Require “non spherical” atoms to give zero “structure factor” in directions where propagation must be suppressed Optical dispersion in a crystal made by non spherical atoms. A 3D bandgap appear when structures are designed to have zero intensity in directions of allowed Bragg reflections. Repeat period is on the order of 30μm (3D optical bandgap in the far IR, limit of today technology). Photonic crystals convert heat into light! See Optics Letters 28 1909, 2003. Magnetic properties • Diamagnetism: Zero-spin systems give rise to circulating currents that oppose the applied field (negative magnetic susceptibility, Larmor diamagnetism). • Paramagnetism: Free-electrons are magnetically polarized by an external magnetic field (positive magnetic susceptibility, Pauli paramagnetism). • Ferromagnetism: Spontaneous magnetic ordering due to electron-electron interactions. Antiferromagnetism: polarization alternates from atom to atom. No net macroscopic magnetic moment arises. Magnetic Interactions • Exchange (electron-electron) interaction (many-particle wavefunction antisymmetry) - atomic scales • Dipole-dipole interactions between locally ordered magnetic regions Dipole interaction energy grows with the volume of the ordered region. The size of the individual domains is set by a competition between volume and surface energy effects. - hundreds of atoms to micron scales • Magnetic Anisotropy energy Magnetization interacts with angular momentum of the atoms in the crystal. – many microns Super-paramagnetic particles • Ferromagnetic domains, created by d-electrons exchange interactions, develop only when a cluster of iron atoms reaches a critical size (ca. microns). The magnetic moment per atom decreases toward the bulk value as cluster size is increased. Stable domains cannot be established in crystals that are smaller than the intrinsic domain size. • Small particles can have very high magnetic susceptibility with permanent magnetic dipole. Small clusters consisting of a single ferromagnetic domain follow the applied field freely (super-paramagnetism). The magnetic susceptibility of superparamagnetic particles is orders of magnitude larger than bulk paramagnetic materials. Ferromagnetic limit Magnetic response for particles of increasing size (Gd clusters) Superparamagnetic separations Induced magnetic moment: M H Magnetic force: B Fz M z z B 0 ( H M) Magnetic sorting of cells labeled with superparamagnetic beads MFS: microfabricated ferromagnetic strips Particle were pulled to point of highest field gradient Copyright Stuart Lindsay 2009 Giant Magnetoresistance Magnetic hard drives are based on a nanostructured device, called giant magnetoresistance sensor. Albert Fert, Peter Grünbers Nobel Prize in Physics 2007 Hitachi hard drive reading head The magnetization on the surface of the disk can be read out as fluctuations in the resistance of the conducting layer. Co, magnetic layer Layers have a width that is smaller than electron scattering length. Cu, electrically conducting layer NiFe alloy, magnetic layer An easily re-alignable magnetization Giant magnetoresistance occurs when the magnetic layers above and below the conductor are magnetized in opposite direction. Electron scattering in magnetic media is strongly dependent on spin polarization. I II III When magnetic layers are parallely magnetized, only one spin polarization is scattered (I,III). For antiparallel magnetic layers both spin polarizations are scattered, giving rise to super-resistance (II). Nanofluidics Fluid flow in small structures is entirely laminar and dominated by the chemical boundaries of the channel. Reynolds number (Re), a dimensionless number quantifying the ratio of inertial to viscous forces that act on the volume of a liquid: density uL Re Re >> 1: turbulence regime Re << 1: viscous regime channel narrowest dimension viscosity Kinematic viscosity u For water: ν =1·10-6 m2s-1 (25°C) Re in a L=100nm channel with <u> = 1mm/s not exceeds 10-4 Flow in nanoscale channels is dominated by viscosity! Fluids do not mix in a nanofluidic device. The chemistry of the interface becomes critical and aqueous fluids will not generally enter a channel with hydrophobic surfaces. 1-D Nanochannel devices A significant stretching of large molecules can occur in a large ion gradient or electric field in a channel that is comparable to the radius of giration of the molecule. Ex. A 17μm DNA (fully stretched length) is equivalent to 340 freely jointed segments each of 5 nm length. The relative giration radius is: r 340 5 92nm It can be significantly extended in a channel of 100 nm diameter, owing to the strong interaction between the fluid and the wall. DNA is introduced through the microchannel and then transported through the nanochannels by an applied voltage. DNA cutting starts Mg ions introduced Time Fluorescently labeled DNA at various times Microchannels DNA + cutting enzymes introduced distance Continous time course of the cutting process Nanopores: 0D fluidic nanostructure 2-nm diameter holes: only a single DNA molecule can pass through it at a given time. electrophoresis Polystyrene bead Optical tweezer The passage of DNA molecules can be measured by the drops in currents that occur when a single DNA molecule occludes the hole. Flow in Carbon Nanotubes A. Fabrication of a layer of CNTs penetrating a silicon nitride membrane; B. and C. SEM images before and after filling with silicon nitride; D. Individual finished device: E. An array of devices The measured rate of water flow through 2nm-diameter CNTs was found to be 1000 times higher than predicted by classical hydrodynamics. 42 2D Nanostructures: Superhydrophobic surfaces The angle formed by a tangent to a flat surface of a drop of water at the point of contact (contact angle) is given in terms of the interfacial energies of the system by the Young equation: AB AC cos c BC γAB= air/surface interfacial tension γAC= water/surface interfacial tension γBC= air/water interfacial tension cos c 1 Water/surface repulsion (large interfacial tension) Water drop Si Nanowires Coated Si surface (planar) Coated nanostructured surface (rough) Roughening on the nanoscale can greatly increase hydrophobicity.