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**Nanomaterial Properties Plasmonics, Lorentz Oscillator Model,**

Lecture 12 Continue Nanomaterial Properties Plasmonics, Lorentz Oscillator Model, Mie Theory

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**Discussions/Questions**

Last Homework Discussions/Questions

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**Write down 5 different physical properties that change and a reason why other than size.**

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**Write down the primary reason why the crystal lattice constant becomes smaller.**

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**If you keep cutting you increase/reduce the **

Correct the sentence If you keep cutting you increase/reduce the overall surface free energy and Gibbs free energy by reducing/increasing the number of free bonds. Did you create free radical surface groups? No/Yes Does the stored energy go up? Yes/No Does this relate to powder explosives? Yes/No Do you expect surface reconstructions over time? Yes/No Will this increase the free energy? Yes/No Does this have an effect on the melting point as you go to nanoparticles? NO/Yes Does this have an effect on the lattice constant as you go to nanoparticles? NO/Yes

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**Self-Assembled Monolayers can Change the Surface Free Energy**

Self-Assembled Monolayers can Change the Surface Free Energy. A nanocrystal has been passivated and the surface free energy has been reduced. Do you expect it to be under reduced "compression"? What do you expect in terms of lattice constant?

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**Name the primary two reasons for an increased hardness in the case of nanowiskers?**

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**Nanomaterial Properties Plasmonics, Lorentz Oscillator Model,**

Lecture 12 Continue Nanomaterial Properties Plasmonics, Lorentz Oscillator Model, Mie Theory

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**Most are tunable with size!**

Physical Properties of Nanomaterials Reduced Melting Point -- Nanomaterials may have a significantly lower melting point or phase transition temperature and appreciably reduced lattice constants (spacing between atoms is reduced), due to a huge fraction of surface atoms in the total amount of atoms. Ultra Hard -- Mechanical properties of nanomaterials may reach the theoretical strength, which are one or two orders of magnitude higher than that of single crystals in the bulk form. The enhancement in mechanical strength is simply due to the reduced probability of defects. Optical properties of nanomaterials can be significantly different from bulk crystals Semiconductor Blue Shift in adsorption and emission due to an increased band gap Quantum Size Effects, Particle in a box Metallic Nanoparticles Color Changes in spectra due to Surface Plasmons Resonances Lorentz Oscillator Model. Electrical conductivity decreases with a reduced dimension due to increased surface scattering. Electrical conductivity increases due to the better ordering and ballistic transport. Magnetic properties of nanostructured materials are distinctly different from that of bulk materials. Ferromagnetism disappears and transfers to superparamagnetism in the nanometer scale due to the huge surface energy. Self-purification is an intrinsic thermodynamic property of nanostructures and nanomaterials due to enhanced diffusion of impurities/defects/dislocations to the nearby surface. Increased perfection enhances chemical stability. Most are tunable with size!

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Optical Properties this element is based in part on slides from Rajeev J. Ram, and Vladimir Shalaev, ECE Purdue

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Optical Properties The reduction of materials' dimension has pronounced effects on the optical properties. The size dependence can be generally classified into two groups. One is due to the increased energy level spacing as the system becomes more confined, and the other is related to surface plasmon resonance.

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**What is the Effect - Energy Confinement of States or Surface Plasmons?**

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**Mesoscopic oscillation of charge around a positive lattice**

Bandgap (s & p band) + Envelope (curvature of wavefunction) Mesoscopic oscillation of charge around a positive lattice

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**Theoretical Tools Metals: Semiconductors:**

Lorentz Oscillator Theory of Materials Plasmons and Plasmonics Semiconductors: Band Theory for Crystals Band Transport at Nanoscales: Molecular Metals and Semiconductors Microscopic Description of Optical Properties

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**Optical Properties: Surface Plasmon Resonance**

Surface plasmon resonance is the coherent excitation of all the "free" electrons within the conduction band, leading to an in-phase oscillation. When the size of a metal nanocrystal is smaller than the wavelength of incident radiation, a surface plasmon resonance is generated. The figure shows schematically how a surface plasmon oscillation is created. The electric field of an incoming light induces a polarization of the free electrons relative to the cationic lattice. The net charge difference occurs at the nanoparticle boundaries (the surface), which in turn acts as a restoring force. In this manner a dipolar oscillation of electrons is created with a certain frequency.

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Computational Tool The Finite Difference Time Domain method is a time marching algorithm which solves Maxwell’s equations in time and space for arbitrary nanostructures.

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**Bulk Plasmons d(t)=const*cos(wt) Electric field displacement**

A plasmon (independent of surface or bulk) is an incompressible self oscillation of the conduction electrons Metal = ions + electron gas electron gas The rigid displacement of the electrons induces a dipole moment and an electric field opposing the displacement. Electric field displacement Fastest way to derive plasmon frequency: Use Newtons Equation of Motion displacement acceleration Harmonic Oscillator d(t)=const*cos(wt) electron mass

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**Fastest way to derive plasmon frequency: Use Newtons equation**

displacement acceleration electron mass Harmonic Oscillator Question 1: The rigid displacement of the electrons induces a dipole moment and an electric field opposing the displacement. Consider an harmonic oscillator. Derive the oscillation frequency (show two steps not listed above) when the external electric field induced force and the internal electron kinetic force are balanced. with Question 2: The dipolar displacement is at a maximum/minimum EM wave adsorption at this frequency due to inelastic scattering is increased/decreased?

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ħω

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ħω

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**Did you know what Bulk Plasmons are?**

At optical frequencies, the response of metals to electromagnetic fields depends on the illumination wavelength (frequency). This dependence is characterized by the permittivity of the metal, which varies with the frequency (or the wavelength), as illustrated in Fig. 2. The frequency dependence of the permittivity is called the dispersion relation. Figure 1: Plasmons can be brough in resonance and give specific metallic nanoparticles a strong and well defined color due to adsorption. This effect was already used in the Middle Ages to fabricate stained-glass windows (left). More recently, plasmon resonant nanoparticles have been used by S. Scultz and co-workers as biomarkers. Understand Maxwells Equations and the Dispersion Relation Ship

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Basic Theoretical Knowledge EE approach The dispersion relationship changes for nanoscopic structures

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Maxwells Equation

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**The Lorentz Oscillator Model**

Does not give you the size dependence. but a general understanding of the dispersions relation ship for materials and plasmon frequencies: of atoms and rarified gases of insulators and conductors when combined with maxwells eq.

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**The Lorentz Oscillator Model**

The Lorentz Oscillator model offers the simplest picture of atom--field interactions. It is purely classical; however, this model is an elegant tool for visualizing atom--field interactions. The Lorentz Oscillator model also bears a number of basic insights into this problem. Lorentz was a late nineteenth century physicist, and quantum mechanics had not yet been discovered.

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**The Lorentz Oscillator Model**

Lorentz thought of an atom as a mass ( the nucleus ) connected to another smaller mass ( the electron ) by a spring. The spring would be set into motion by an electric field interacting with the charge of the electron. The field would either repel or attract the electron which would result in either compressing or stretching the spring. source Vladimir Shalaev, ECE Purdue

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**The Lorentz Oscillator Model**

The main sources of gama are atomic collisions and spontaneous emission which were not understood at the time. source Vladimir Shalaev, ECE Purdue j

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**The Lorentz Oscillator Model**

j j Volume number concentration proportional to adsorption proportional to adsorption Plasma Frequency Susceptibility Permittivity

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**The Lorentz Oscillator Model**

j j Volume number concentration proportional to adsorption proportional to adsorption Plasma Frequency Susceptibility Permittivity

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**The Lorentz Oscillator Model**

j Susceptibility Plasma Frequency Volume number concentration Permittivity reflective index real part of permittivity relates to adsorption relates to adsorption

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**The Lorentz Oscillator Model**

j Susceptibility Plasma Frequency Volume number concentration Permittivity reflective index real part of permittivity relates to adsorption relates to adsorption

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**The Lorentz Oscillator Model**

j Susceptibility Plasma Frequency Volume number concentration Permittivity

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**The Lorentz Oscillator Model**

j Susceptibility Plasma Frequency Volume number concentration Permittivity

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**The Lorentz Oscillator Model**

j Susceptibility Plasma Frequency Volume number concentration Permittivity

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**Difference between Bound (insulators and rarified gases) and Free Charges (metals)**

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**What happens in metals? Relaxation time TASK (determine ε) RESULT**

COMPLEX CONDUCTIVITY RESULT Relaxation time

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**What happens in metals? Relaxation time TASK (determine ε) RESULT**

COMPLEX CONDUCTIVITY RESULT Relaxation time

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**Note: There is an important difference in the Lorenz oscillator model between metals and insulators.**

Which statements are correct? The adsorption for metal and insulators is a function of frequency? (yes/no) The transmissivity for metal and insulators is a function of frequency? (yes/no) The refectivity for metal and insulators is a function of frequency? (yes/no) The "spring constant C" in the lorenz oscillator model for bulk metals is assumed to be zero because there is not restoring force? (yes/no) The damping in bulk metal is due to elastic scattering? (yes/no) The damping in bulk metal is due to inelastic scattering? (yes/no) For insulators we can define a resonance frequency w0? (yes/no) The resonance frequency w0 in insulators is close to a maximum adsorption? (yes/no) w0 is zero in bulk metals. There are four distinctive regions in metals referred to as TART and two in insulators referred to as RT. The transition between RT occurs at the plasmon resonance in metals Relaxation time

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**Note: Read the text marked as "additional reading" **

to answer the following questions. Describe why the widths of the resonance is increased with decreasing particle size. Will the scattering be increased and why as you reduce the particle size. The MIE theory describes the oscillation modes of a spherical metal particle in an insulating matrix and is based on Maxwells equation. (yes/no) It does not account for the size dependence in the adsorption that is observed for metal nanoparticles unless g is modified to account for increased scattering effects. (yes/no) Relaxation time

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**Homework: Additional Reading**

Surface plasmon resonance is the coherent excitation of all the "free" electrons within the conduction band, leading to an in-phase oscillation. When the size of a metal nanocrystal is smaller than the wavelength of incident radiation, a surface plasmon resonance is generated and Fig shows schematically how a surface plasmon oscillation of a metallic particle is created in a simple manner. The electric field of an incoming light induces a polarization of the free electrons relative to the cationic lattice. The net charge difference occurs at the nanoparticle boundaries (the surface), which in turn acts as a restoring force.

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In this manner a dipolar oscillation of electrons is created with a certain frequency. The energy of the surface plasmon resonance depends on both the free electron density and the dielectric medium surrounding the nanoparticle. The resonance frequency increases with decreasing particle size if the size of the particles is smaller than the wavelength of the particles. The width of the resonance varies with the characteristic time before electron scattering. For larger nanoparticle, the resonance sharpens as the scattering length increases. Noble metals have the resonance frequency in the visible light range. Mie was the first to explain the red color of gold nanoparticle colloidal in 1908 by solving Maxwell's equation for an electromagnetic light wave interacting with small metallic spheres.

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**Look at the book if you need to know more about this!**

The solution of this electrodynamic calculation leads to a series of multi-pole oscillations: Look at the book if you need to know more about this! Extinction coefficient Ability to adsorb Scattering coefficient Ability to scatter

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**Extinction coefficient**

The equations clearly indicate that the plasmon resonance depends explicitly on the particle size, r. The larger the particles, the more important the higher-order modes as the light can no longer polarize the nanoparticles homogeneously. These higher-order modes peak at lower energies. Therefore, the plasmon band red shifts with increasing particle size. At the same time, the plasmon bandwidth increases with increasing particle size. The increase of both absorption wavelength and peak width with increasing particle size has been clearly demonstrated experimentally. The situation concerning the size dependence of the optical absorption spectrum is more complicated for smaller nanoparticles for which only the dipole term is important. For nanoparticles much smaller than the wavelength of incident light, only the dipole oscillation contributes to the extinction cross-section. Mie theory can be simplified to the following relationship (dipole approximation): Extinction coefficient dielectric constant of the surround material and the particle, respectively.

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Problem. Equation shows that the extinction coefficient does not depend on the particle sizes; however, a size dependence is observed experimentally. This discrepancy arises from the assumption in the Mie theory, that the electronic structure and dielectric constant of nanoparticles are the same as those of its bulk form, which becomes no longer valid when the particle size becomes very small. Therefore, the Mie theory needs to be modified by introducing the quantum size effect in smaller particles. In small particles, electron surface scattering becomes significant, when the mean free path of the conduction electrons is smaller than the physical dimension of the nanoparticles. For example, conduction electrons in silver and gold have a mean free path of 40-50nml15 and will be limited by the particle surfaces in particles of 20 nm. If the electrons scatter with the surface in an elastic but totally random way, the coherence of the overall plasmon oscillation is lost. Inelastic electron-surface collisions would also change the phase. The smaller the particles, the faster the electrons reach the surface of the particles, the electrons can scatter and lose the coherence more quickly. As a result, the plasmon bandwidth increases with decreasing particle size.116

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The reduction of the effective electron mean free path and enhanced electron-surface scattering can also correctly explain the size dependence of the surface plasmon absorption as follows. g is introduced as a phenomenological damping constant and is found to be a function of the particle size118: sic size effect, since the materials' dielectric function itself is size dependent. In this region, the absorption wavelength increases, but the peak width decreases with increasing particle size. The molar extinction coefficient is of the order of 1 X 109 M-1 cm-1for 20 nm gold nanoparticles and increases linearly with increasing volume of the particles.

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These extinction coefficients are three to four orders of magnitude higher than those for the very strong absorbing organic dye molecules. The coloration of nanoparticles renders practical applications and some of the applications have been explored and practically used. For example, the color of gold ruby glass results from an absorption band at about 0.53mm. This band comes from the spherical geometry of the particles and the particular optical properties of gold according to Mie theory as discussed above. The spherical boundary condition of the particles shifts the resonance oscillation to lower frequencies or longer wavelength. The size of the gold particles influences the absorption. For particle larger than about 20 nm in diameter, the band shifts to longer wavelength as the oscillation becomes more complex. For smaller particles, the bandwidth progressively increases because the mean free path of the free electrons in the particles is about 40 nm, and is effectively reduced. Silver particles in glass color it yellow, resulting from a similar absorption band at 0.41mm. Copper has a plasma absorption band at mm for copper particles in glass.

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**Note: There is an important difference in the Lorenz oscillator model between metals and insulators.**

Which statements are correct? The adsorption for metal and insulators is a function of frequency? (yes/no) The transmissivity for metal and insulators is a function of frequency? (yes/no) The refectivity for metal and insulators is a function of frequency? (yes/no) The "spring constant C" in the lorenz oscillator model for bulk metals is assumed to be zero because there is not restoring force? (yes/no) The damping in bulk metal is due to elastic scattering? (yes/no) The damping in bulk metal is due to inelastic scattering? (yes/no) For insulators we can define a resonance frequency w0? (yes/no) The resonance frequency w0 in insulators is close to a maximum adsorption? (yes/no) w0 is zero in bulk metals. There are four distinctive regions in metals referred to as TART and two in insulators referred to as RT. The transition between RT occurs at the plasmon resonance in metals Relaxation time

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**larger scattering length**

Summary The surface plasmon resonance is a dipolar excitation of the entire particle between the negatively charged free electrons and its positively charged lattice. The size of the particle determines the resonance frequency. Smaller particles have a higher resonance frequency. The width of the resonance varies with the characteristic electron scattering time. At scales close to the scattering length larger nanoparticles have a sharper resonance as the scattering length increases. Noble metals have the resonance frequency in the visible light range. Mie was the first to explain the red color of gold nanoparticle colloidal in 1908 by solving Maxwell's equation for an electromagnetic light wave interacting with small metallic spheres. larger scattering length

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46 minutes.....

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