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Nanophysics & Lectures Notes http://www1.na.infn.it/TIMSI/materialicorsi/iavarone/Nanophysics&Nanotechnology/Nano&Nano.htm Nanotechnology Size-Dependent Properties Classes of Materials Metallic Materials Good thermal and electrical conductors, ductile Ceramic Materials Hard, brittle, good insulators Polymeric Materials Good insulators, good resistance to corrosion Composite Materials Matrix-reinforcing phase. Properties vary At the nanometer scale property become size dependent. For example: 1. Mechanical properties-adhesion, capillary forces 2. Chemical Properties-reactivity, catalysis 3. Thermal Properties-melting point 4. Electrical properties-tunneling current 5. Optical properties-absorption, scattering 6. Magnetic properties-superparamagnetic effect New properties enable new applications! Electronic Materials Metal, ceramic, polymers, semiconductors Biomaterials Bio-compatible Nanomaterials Nanoparticles, nanotubes, nanowires, nanocoatings… 1 Thermal Behavior Heat Conduction Heat affects mechanical, electrical, and optical properties . As temperature rises, materials expand, the elastic modulus decreases, the strength falls, and the material start to creep, deforming slowly until the melting point is reached. Electrical resistivity rises with temperature, the refractive index falls, color may change. Heat Conduction Gases: hotter air molecules (with larger kinetic energy) randomly pass their excess energy to cooler molecules. Heat is transported by occasional collisions with each other. The average velocity of a molecule can be as large as 500 m/s. Radiation Solids: Heat is transmitted in three ways, thermal vibrations, movement of free electrons in metals, and if they are transparent by radiation. Thermal waves are really not different from e-m waves that we use in communication (radio/TV signals). There is one fundamental difference which is the wavelength: λ= Average velocity 105106m/s c 3 ×108 m / s 1 = = m f 900 × 10 6 H 3 λ = 0.5µm Much shorter The major difference is not in the propagation but in the generation. generation Thermal radiation typically refers to e-m waves that are generated by the oscillation charges in the atoms, while radio/TV signals are generated by artificial current oscillations in circuits. 2 Phonons Thermal Conductivity: The rate at which heat is conducted in a solid is measured by the thermal conductivity, which is a material property (for macroscopic samples) k[W / m ⋅ K ] Fourier’ Fourier’s Law: Thermal conductivity is calculated from Fourier’s law The phonons travel through the material, and like any elastic wave, they move with the speed of sound (few thousand per meter per second). c0 = E ρ q = −k q = [W / m 2 ] dT (T − T ) = −k 1 2 dx x1 − x2 Heat flux Phonon Conduction Why does heat not diffuse at the same speed of sound? Because of scattering they travel a short distance before bouncing. This distance is called mean free path and it is tipically 0.01 microns. Net flux model: The rate at which heat is conducted in a solid is measured by the thermal conductivity, which is a material property (for macroscopic samples) 1 dT q = ρC p lm c0 3 dx Elastic waves contribute little to conductivity of pure metals such as copper or aluminum because the heat is carried more rapidly by the electrons. The equation shown earlier is still valid, but Cp, c0 and lm become the thermal capacity, the velocity, and the mean free path of the electrons. 1 k = ρC p lm c0 3 3 Thermal Transport of Nanomaterials Classical size effects. In bulk materials the characteristic length of the box is much longer than the mean free path. Therefore, the collisions between the molecules (electrons, phonons) and the wall are neglected and thermal conductivity is regarded as a bulk property. In nanomaterials the mean free path becomes comparable or larger than the size of the system. Quantum size effects. According to quantum mechanics, electrons and phonons are also material waves. In the case of a bulk material the wavelength of phonons are much smaller than the length scale of the sample. However, in nanomaterials the length scale of the material is similar to the wavelength of the phonons. Therefore, quantum confinement occurs. In 0D nanomaterials (nanoparticles) confinement occurs in 3 dimensions In 1D nanomaterials (nanowires, and nanotubes) confinement is restricted to 2 dimensions In 2D nanomaterials (nanofilms, nanocoatings) quantum confinement takes place in one dimension Quantum size effects. The presence of the surface in nanostructures causes a change in the distribution of the phonon frequencies, and creates new modes that are not present in bulk. The same happens for electron. One-dimensional materials behave as phonon waveguide, similar to optical one for light. For example for carbon nanotubes, several authors have predicted very high thermal conductivity along the nanotube, close to 3000 Wm-1K-1. As a comparison we should keep in mind that the thermal conductivity of copper is approximately 400 Wm-1K-1. Despite these results, there are still open questions about phonon transport in 1D nanostructures, particularly regarding phonon/phonon interactions and the role of defects. 4 Thermal conductivity of suspended Platinum nanofilms Thermal conductivity of materials with nanoscale grains Polycrystalline silicon films show a strong reduction of thermal conductivity with respect to the bulk. Pt 28 nm X. Zhang et al., Applied Physics Letters, 86, 171912 (2005) Thermal conductivity is less than those of the corresponding bulk materials. In fact, the thinner the film the lower the thermal conductivity For nanoporous materials, the nanosize effect is determined by the number and the size of the pores. S. Uma et al., Int. J. Thermophys., 22 , 605 (2001) Low thermal conductivity, in the case of microelectronic components, leads to an increase in the operation temperature and earlier circuit failure. A brief review of quantum mechanics Quantum mechanics tells us that the fundamental building blocks of nature can be thought as both particles and waves. What determines when we must take into account the wave nature of particles? λ= h p de Broglie wavelength If the de Broglie wavelength of the particle is less than its physical dimensions, then the classical mechanics provides a good description. 5 Electrical Behavior Resistivity and Conductivity I Electrical Materials Normal metal S l Conductors Insulators Non-piezoelectric dielectrics Quartz (Ultrasonic oscillator ink-jet print-heads) Piezoelectrics Non-pyroelectric Pyroelectric (Barium Titanate) Non-ferroelectric Ferroelectric R= V i ρ =R ΘD S l ρ [Ω ⋅ m ] For good conductors the resistivity is of the order of 1 µΩcm while for best insulators is more than 1024 µΩcm. The electrical conductivity is simply the reciprocal of the resistivity. Its units are Siemens per meter (S/m or (Ωm)-1). PZT ( Diesel injector) Electrical Behavior How come some materials have mobile electrons and some do not? To answer this question we need quantum mechanics Electrical Behavior When n atoms are brought together to form a solid the inner electrons remain in within the atom, the outer ones start interacting. As a consequence the discrete level of an isolated atom broadens to form bands (of closely spaced energy levels). The electrons fill the bands from the bottom. The topmost filled energy level is called the Fermi level. The electrons of an atom occupy discrete energy levels. The electrons fills the shells with the lowest energy according to the Pauli principle (two electrons of opposite spin in each orbit). 6 Electrical Resistance If a field E exerts a force Ee on an electron, why does it not accelerate forever, giving a current that continuously increases with time? Electrical Behavior: Quantum Size Effects This overall behavior of bulk materials changes when the dimensions are reduced to nanoscale. For 0-D nanomaterials an electron is confined in 3D space For 1-D nanomaterials an electron is confined in 2-D, whereas delocalization takes place along the long axis (nanotube…) Normal metal For 2-D nanomaterials the conduction electrons will be confined along the thickness but delocalized in the plane. ΘD Electrical Behavior: Quantum Size Effects For example let’s consider electrons in a thin film. They can be approximated as standing waves inside a potential well. The condition for the formation of such a standing wave is that the wavelength, λ, satisfies the following relation: Electrical Behavior: Quantum Size Effects λ= nh 2D The energy of the electron is: E= D = λn 2 ( n = 1,2...) p= h 2D = p n p2 h2n2 h2 = = n2 2m (2 D )2 2m 8mD 2 Keeping in mind that me=9.1x10-31Kg, h=6.6 x 10-34 : 1) D=1mm 2 E= where D is the width of the potential well (thickness of the film). According to the de Broglie relation, the wavelength is: λ= h p 6.6 × 10 −34 n2 ≈ 10 −34 n 2 J << k BT = 4.14 × 10 −21 J 8 × 9.1× 10 −31 10 −3 room temperature 2) D=10-8 m 2 E= 6.6 ×10 −34 n2 ≈ 10 −21 n 2 J ≥ k BT = 4.14 × 10 −21 J 8 × 9.1× 10−31 10−8 7 Electrical Behavior: Quantum Size Effects There are critical questions to answer. When to consider classical size effect or quantum size effect? 2-D Nanomaterials: Potential Well The electrons become trapped in a so called potential well of width equal to the thickness. Keeping in mind that ћ =h/2pπ 100 En = 10 1 π 2h 2 2 h2 n n 2 = 2 2 8mD 2mD E/kBT 0.1 0.01 MacroMacroscopic Quantum 1E-3 The electrons are free to move in the plane x-y. Therefore the total energy of the lectrons has two components. The unrestricted motion in plane by two wave vectors kx and ky, which are related to the momentum along x and y, px= ћkx and py= ћky. 1E-4 1E-5 Mesoscopic 1E-6 1E-7 1E-8 0 10 1 10 2 3 10 10 4 10 D (nm) π 2h 2 2 h 2 k F2 n + En = 2 2mD 2m The boundaries between these regions will shift to larger sizes as the temperature is reduced and if we use semiconductors (lower effective mass) 1-D and 0-D Nanomaterials: Energy Levels For 1-D nanomaterials an electron is confined in 2-D, whereas delocalization takes place along the long axis (nanotube, rod, wire…) π 2 h 2 n y 2 π 2h 2 n z 2 + Eny ,nz = 2mL 2 2mL 2 y z π 2 h 2 n y 2 π 2 h 2 nz 2 h 2 k x 2 + + E = 2mL 2 2mL 2 2m y z Density of States If we consider a single electron traveling through a conductor. It will have a wave-function of the form ψ(r)=eik r. It is instructive to think of the occupied states as represented by points within a sphere in k-space. The surface of the sphere with k=kF is known as the Fermi surface, and represents states at the Fermi energy. Energy of electrons at the Fermi surface EF = h 2 kF 2m 2 2π V1 = L Volume of one state For 0-D nanomaterials the motion of electrons is totally confined alonfg the three directions Lx, Ly, Lz. Therefore, the energy is given by: Enx , ny ,nz Number of electrons in the conductor 2 π 2 h 2 nx 2 π 2 h 2 n y π 2 h 2 nz 2 + + = 2 2mL 2 2mL 2 2 mL x y z Under these conditions, metallic systems can behave like insulators, due to the formation of an energy band gap! 3 N =2 3π 2 N k F = V 1 4π k F 3 3 = Vk F 3 3 3π 2 (2π L ) 3 EF = h 2 3π 2 N 2m V 2 3 8 Calculated Conduction Parameters Density of States in 3-D The number of states between k and k+dk is: ∆N = 4πk 2 dk Vk 2 ∆k = 2π 2 V1 The density of states is defined as the number of quantum states per unit interval of energy and per unit volume: D( E ) = E= h2k 2 2m 1 ∆N k 2 ∆k k 2 dk = = V ∆E 2π 2 ∆E 2π 2 dE k= 2m E h2 D(E ) = 2 Density of States in Nanomaterials The number of conduction electrons that exist at a given energy level, the number of electrons dN in a narrow energy range dE: D(E)=dN/dE. This quantity strongly depends upon dimensionality of the structure. 3 k 2 dk 1 2m 2 12 = E 2π 2 dE 2π 2 h 2 Size effect on Electrical Resistance How does the resistance change as we make a conductor smaller? What is so interesting about nanoscale that a wire is not simply a passive element! Various physical properties depend upon the density of states: specific heat, thermopower effect, superconductivity energy gap. Overall the ability to control the density of states is crucial for applications such as infrared detectors, lasers, optical memories…… 9 Magnetic Behavior Magnetic Fields in Materials Magnetic fields are created by moving electric chargeelectric current in electromagnets, electron spin in atoms of magnetic materials. If the space inside the coil is filled with a material the induction changes, because its atoms respond to the field by forming magnetic dipoles. The material acquires a macroscopic dipole moment or magnetization. H= n=number of turns in the coil L= length i=current H=magnetic field generated ni L B = µ 0 (H + M ) B = µ0 µ R H H is a vector and is measured in A/m M = (µ R − 1)H = χH Magnetic fields exert forces on a wire carrying an electric current. A current i flowing in a loop of area S generates a dipole moment m where: m = iS M is a vector measured in Am2 B = µ0 H Saturation magnetization decreases with temperature, falling to zero at the Curie temperature B is measured in Tesla Origin of Magnetic Behavior Orbital Motion of Charged Particles Flow of charge gives rise to magnetic field Charges flow in response to magnetic field There is, therefore, a magnetic dipole associated with each orbiting electron. χ is the magnetic susceptibility Different types of Magnetic Behavior Paramagnetic The interaction energy between a magnetic dipole and magnetic field is given by: Emagnetic = µ ⋅ B = µB cos θ Spin angular momentum Spin is a pure quantum mechanical property It can be thought as a particle turning on its own axis To minimize the energy, the dipole moment points along the field 10 Different types of Magnetic Behavior Different types of Magnetic Behavior Ferromagnetism Paramagnetic For a paramagnet the interaction between spin is negligible. There is no magnetization in absence of external magnetic field. Various types of interaction may lead to alignment of adjacent spins. If the interaction prefers parallel alignment, then the interaction is called ferromagnetic. If the interaction prefers antiparallel alignment, then the interaction is called antiferromagnetic. Note: there is no external magnetic field yet ! Energy Scales Energy Scales Paramagnets Ferromagnets Does the fact that the interactions between magnetic moments in a paramagnet is weak mean that all moments line up exactly in even a small external field? Instead of the magnetic energy of a magnetic moment in an external magnetic field, the relevant energy is the interaction energy between the moments, J (exchange energy). The moments will be randomizedJ < k BT no net magnetization No. While the external field would like to align them, finite temperature likes to randomize their directions. Alignment of all magnetic moments depends on the balance between the energy associated with a moment in magnetic field, and the thermal energy. Emagnetic < k BT Emagnetic > k BT The moments will be randomized J > k BT The moments will order spontaneously Phase transition The moments will be aligned 11 Other magnetic systems Paramagnetic Ferromagnetic Antiferromagnetic Even a ferromagnet below the Curie temperature may not have a net magnetization. This is because the ferromagnet breaks up into domains, each with its own magnetization pointing in a different direction. A magnetic field is required to align the moments of all the domains in the same direction. Ferrimagnetic In a system with antiferromagnetic interactions the crystal structure might be such that the system cannot order antiferromagnetically. Ferromagnet: Domains Hysteresis curve of a ferromagnet An external magnetic field progressively aligns the direction of the magnetization of the domains. The magnetization saturates when all the domains are aligned. Moving domains costs energy. Therefore, the magnetization is not a single valued function of the external field. 12 Ferromagnet: Domain Walls Between two magnetic domains, the average magnetization must switch direction: Energy Considerations Etotal = Eexc + Eani + Edem + Eapp The area over which the magnetization reverses is called a domain wall. The domain wall is typically determined by the material properties. Etotal = M ⋅ H Typical domain wall width for conventional ferromagnets are of the order of 20-100 nm. Closure domain structure Open domain structure Nanoferromagnets Nanoferromagnets What happens if the dimensions of a ferromagnet are smaller than a domain wall? Exchanges forces are dominant, therefore there is a critical grain size below which the materials will be single domain. For a spherical grain: Dcri = 9γ B µ 0 M S2 γ B = 4( AK1 )1 2 A exchange stiffness K1 anisotropic constant MS saturation magnet. If the particle size becomes too small (typically few nm), the magnetization can become unstable and loss of magnetization occurs due to thermal fluctuations. This is called superparamagnetic limit. This determines the minimum size of the domains that can be used to store information. This can be controlled by controlling materials parameters SD MD Critical diameter for Co is 70 nm, for Fe is 15 nm. Which way will the magnetization point? Coercitivity field versus particle size HC A number of factors control the direction of the magnetization. Two are particularly important: crystalline anisotropy, shape anisotropy. DSP Dcrit 13 Giant and Colossal Magnetoresistance Optical Behavior Many aspects of radiation are understood by thinking of radiation as a wave. Others need a different picture. Radiation is discrete packets of energy, photons. Change in resistance 5%, while in CMR materials (manganese base perovskite oxides) there are changes of order of magnitudes Why aren’t metals transparent? E ph = hν = hc λ How does light get through dielectrics? Metals absorb photons by promoting an electron in an empty level. When the electron falls back a photon is reemitted. Dielectrics have a full band separated by the conduction band by an energy gap. The material cannot capture photons with an energy less than the gap, for those frequencies the material is transparent 14 Color Excitons Optical Behavior in Nanostructures Optical Behavior in Nanostructures The effect of nanoscale on optical absorption are associated with: 1. The density of states in valence and conduction bands 2. The quantized energy levels of the nanostructures 3. The influence of excitonic effects. When moving from 3D to 0D quantum confinement is more severe, band gap shift to higher energies, i.e. shorter wavelengths. A blue is expected whe the size is decreased and red when the size is increased. Cd-Se quantum dots of various size Also emission of photons can be tuned with the nanoparticle size. Typical trend is a shift of the emission peak toward shorter wavelength (blue shift). 15 Optical Behavior in Metallic Nanostructures: Plasmons Plasmons are quantized waves that propagate in materials through electrons. Plasmons can exist in the bulk and on the surface. Among these surface plasmons are more relevant for nanomaterials. Surface plasmons have lower frequencies than bulk plasmons and thus they can interact with photons. When photons couple with surface plasmons (surface plasmons polaritons), alternating regions of positive and negative charges are produced on at the surface. Surface plasmons-polaritons can be used to cause extraordinary transmission Light absorption and emission can be enhanced by using nanoparticles. As a result, metallic nanoparticles acan be used as structural and chemical label. 16