Optical Properties of Materials

Materials Spectroscopy and Microscopy - Spectroscopy

• These notes are based on the course, held in A.Y. 2023/2024 at the University of Milano-Bicocca, Master's Degree in Materials Science and Nanotechnology.
• Contact [email protected] to report errors or inaccuracies.

Ranges in the EM Spectrum

• Lower limits are shown for each range
• The EM spectrum includes gamma rays, X-rays, UV, VIS, NIR, MIR, FIR, MW, and Radio waves.
• Gamma rays have a wavelength of 10 pm, a wavenumber of 10^9 cm^-1, a frequency of 30 EHz, and an energy of 124 keV.
• X-rays have a wavelength of 10 nm, a wavenumber of 10^6 cm^-1, a frequency of 30 PHz, and an energy of 124 eV.
• UV has a wavelength of 400 nm, a wavenumber of 2.5x10^5 cm^-1, a frequency of 750 THz, and an energy of 3.1 eV.
• VIS has a wavelength of 700 nm, a wavenumber of 1.43x10^5 cm^-1, a frequency of 480 THz, and an energy of 1.77 eV.
• NIR has a wavelength of 1 μm, a wavenumber of 10^4 cm^-1, a frequency of 300 THz, and an energy of 1.24 eV.
• MIR has a wavelength of 10 μm, a wavenumber of 1000 cm^-1, a frequency of 30 THz, and an energy of 124 meV.
• FIR has a wavelength of 100 μm, a wavenumber of 100 cm^-1, a frequency of 3 THz, and an energy of 12.4 meV.
• MW has a wavelength of 1 m, a wavenumber of 0.01 cm^-1, a frequency of 300 MHz, and an energy of 1.24 μεV.
• Radio has a wavelength of 100 Mm, a wavenumber of 10^-8 cm^-1, a frequency of 3 Hz, and an energy of 12.4 feV.
• Some sources differentiate between types of radiation based on their source rather than the spectral range.
• NIR, MIR, FIR, and MW stand for Near InfraRed, Medium InfraRed, Far InfraRed, and MicroWaves, respectively.

(Gaussian Units)

• Gaussian units use the cgs (centimetre-gram-second) system instead of the MKS (metre-kilogram-second) system.
• Conversions between MKS and cgs include: 1 kg = 10^3 g, 1 m = 10^2 cm, 1 J = 10^7 erg, and 1 N = 10^5 dyn.
• Gaussian units are not rationalized, meaning the factor 4π appears in Maxwell's equations, unlike SI units.
• Unit charge in Gaussian units makes Coulomb's law have a force constant of 1 because ε₀ and µ₀ are artifacts of the SI system.
• Also replace µ₀ in Gaussian units using the relationship: ε₀µ₀ = 1/c².
• A statcoulomb is defined as the charge where two charges at 1 cm distance feel an electrostatic force of 1 dyne.
• The magnetic induction B has an extra factor of c, which leads to derived magnetic quantities having different dimensions in SI vs. Gaussian units.
• In the Gaussian system, all fields (E, P, D, B, M, H) have the same dimensions (cm-1/2 g1/2s−1); scalar and vector A potentials also have the same dimensions (cm1/2g1/2s-1).
• Electric and magnetic susceptibilities are dimensionless in both systems but have different numerical values.

Some Equations

• Gives equations from both Gaussian and SI for:
• Lorentz Force
• Coulomb's law
• Electric field of a stationary point-charge
• Biot-Savart law
• Permittivity
• Magnetic permeability
• Electric & Magnetic field (with potentials)

Some Conversions

• Presents formulas for coverting charge, current, voltage, electric field, electric displacement, and magnetic induction from Gaussian to SI units.

Some Fundamental Constants

• No values are given

Maxwell's Equations

• EM waves are transverse
• Often X-rays and γ-rays classification is by their creation, rather than wavelength; gamma rays are from radioactive decays, X-rays from bremsstrahlung. Both X-rays and gamma rays are ionizing radiation UV light may be ionizing for some materials
• Scintillators absorb high-energy photons and re-emit them in the UV-vis range for detection
• Goal of spectroscopy is to link experimental quantities (transmittance, reflectivity, etc.) with material parameters (dielectric function, refractive index, polarizability)
• Presents Microscopic Maxwell Equations in guassian units

Linear response

• Macroscopic equations use macroscopic quantities defined by microscopic counterparts.
• Electric field strength Ē(r) is the average of the microscopic electric field
• Magnetic induction B(r) is the volume average of the microscopic magnetic induction field b(r)
• Charge density ptot(r) and current density jtot(r) are similarly defined as volume averages.
• The current density in a solid includes contributions from bound electrons and free electrons: ĵtot = ĵbound + ĵfree
• Net charge density can be described with the sum of the medium charge and the external source charge : ptot = ppol + pext
• In an electric field, atoms in a solid polarize, displacing electronic charge relative to the nucleus
• Uniform polarization results in no net charge movement, while non-uniform polarization yields a net charge and ppol = −∇· 𝑃
• Presents how a time-dependant electric field affects a current, and spin arising from electron motion

∇⋅E=4πρtot

• In presence of an electric field there is also a macroscopic magnetization M, defined as the magnetic dipole moment per unit volume.
• The free current density comes from the motion of conduction electrons, influenced by the field and external sources. Formula to calculate the 1stand 4th Maxwell equations
• Displacement and Magnetic field formula:
• D=Ē+4πP.
• H=B−4πM
• Electrical and magnetic susceptibilities:
• P=xeE
• M=xmH.
• The transverse conductivity describes the system's response to a transverse electric field, it is very different from DC.
• The material's dielectric and magnetic properties are related to the stimulus, linking materials to linear behaviours, in cases of high electricit these should be proportional to higher powers of the fields.
• Dielectric function for electric induction.
• Magnetic permeability for magnetic induction.
• Dielectric function expresses the capability of materials to polarize, crucial at expressing frequency
• Discusses how to split the totoal field into a total electric field within an external stimulus
• Discusses induced currents from the charge density and currents in the charge, with same true for external density

(Formal Solution Of Maxwell's Equations)

• Recall Maxwell's equations, used for vector and scalar potential
• Possible to define a new vector potential (gauge transformation) for the magnetic induction remains the same
• Common restrictions are Coloumb gauges for transformation
• The solution implies the use of resolution of fields and currents in brief and into longitudinal and transverse field for longitudinal media.
• Solution of equation is Green's function
• Conclucions of relation of vectors

Wave Equation

• Gives vector identity formula
• How to write the 2nd and 4th of Maxwell's equations without external charges
• Recalling that the divergence formula, solutions that restrict transverse plane waves will lack a net charge density The conductivity describes the response of the system. at wavelengths the optical conductivity approaches the ordinary electrical conductivity for isotropic materials. Anisotropic materials will have their dielectric functions treated as tensors
• A plane wave is a complex equation
• By substituting this solution one can find the dispersion relation that includes frequency.
• By defining complex index way one can relate frequency

η is the usual refractive index. k i instead is the reflection coefficient.

• Thus real and imaginary forms can be equal
• New definitions for what values is what is often used in the equations.
• The first term describes the material waves while the second term describes that wave velocity to phase.
• Absorption coefficient is the relative intensity function
• Isotropic quantities can't depend on the directions, anisotropic have factors dependent on vectors and tensors

Lorentz Model

• Applicable for insulators and semiconductors (materials with a band gap and no free charges)
• Depicts material as ensembl of harmonic oscillators with optical electrons.
• Classical wave can oscillate.
• There are 2 approximation (quantum mechanical, anharmonicty) that states linearity is valid. The electron motion will give it's displace in equilibrium. Damping will occur as a result of electrons that hampers the motin.
• The local E will be the stimulu to the electron and equation is harmonic.
• Two masses can move into bodies and can have the two one the electron and another for nucleus, this becomes approach to the speed's light.
• Can solve the solution of differentiaton
• The dipole moment is the expression of polarization
• Stimuli are a response by which we use the electric field to relate diopole. And then atomic polarizability (α), we need additional terms.

Dielectric Function

• Relates to frequency when the polarization is dependent on frequency
• How atoms for unit values are related in macroscopic polarization P
• Replacing local fields now can recover the dielectricity in other cases
• Triviality of replacements is reasonable for when electrons are free in a system.
• All are related with their imaginary and real dielectric of each one, these functions mix all of the functions and therefore are independent
• This leads only to considerations of electron types by which in such cases any natural frequency is okay in description of systems and analogies
• Can correlate with quantum mechanic analogues of two separations of an electronic transition between.
• When the relation to reflection strength has amplitudes it does have a refraction
• All are related with their expression to the components of the dielectric at points in space, while frequencies will vary

Drude Model

• Metals, materials in which some e- are able to respond to EM radiation by moving through the material. Inner bound e-'s are not considered in the model.
• The starting point is the Lorentz model: to neglect the existence of e- and the mean value of the "local" field is assumed to be the real/total field value. Also the mean time between e- collision tdefined as Γ=τ-1.
• Easily obtained from the Lorentz model when wo=0
• Recovers the definition of plasma frequency by which we can simplify all of the properties
• In principle an ideal metal without strong damping terms should be reflective but then transparent for high temperatures.
• In metals, with plasma frequency is on the order 𝜔𝑝1015𝑠^−1, while the meantime between collisions is 𝜏10−14 𝑠, so 𝜔𝑝 τ≫1
• ω>ωp we write this in expression In any region metal is transparent or well we have ≫k
• Means that for these properties there is always ~0 , since velocity phase between media

* Let loose metal particles: due to balance charge, there is overshoot

Real Materials

• Silicon resembles ideal reflectivity, its peaks are due to the response of deeper laying electrons
• KCl is much more complex Transparance region 1
• Only taking real waves will get something

Drude

• One and other equations are used to compute a spectrum
• Transitions of bands, energies and light is used to take transitions that occur between (𝑛−1𝑑) and 𝑛𝑠 orbitals, these are for copper

Kramers-Kronig Relations

• Here we are going to show that the quantities 𝜀1,𝜀2,n,k are dependent of other values from functions (𝑟′,𝑡′) to response 𝑋(𝑟,𝑡) with function 𝐺(𝑟,𝑟′,𝑡,𝑡′):

𝑋(𝑟,𝑡)

• From above the response functions is Green's function which shows the system how and why is is that system has an early time.

• Since Green's function is not practical with frequency by performing a full numerical formula it will also perform a numerical method too Relate to monochromatic stimuli. with each stimuli is some number at points. This depends the frequency of which is at some point.

• This with upper half means that we have an according integral.

• Where 𝑃 denotes principal value of the results for containing a semicircular loop in this contour

• If then we can see semicircle has infinite the approach to integrals,

• These can connect how real and image in functions and expression of polarization based

• This could be used in all cases due to system and polarization on before . Model for polarization needs to be causal with a value

• If there is an understanding if system will need to be what is the system.

• Can be in output/input and has important relation on expression and vice versa.

The causality principle

• It means absorption at all time on the shift frequency and has linearity From superposition each function real has related function
• Must understand why responses functions on the way it has all of the other frequencies on which real or virtual that

Green's Function

Differential Equation

An operator that defines the function and operations of the differentiation a linear combination also

An Ordinary Equation

Functions that are both variables in functions for different and non homogones values for r.

There a green impulse function for when a limit depends and function L for Delta

• Also green is for the solution in a euclidean space

Exploiting Green to get a better equation it becomes integration

General Formula

• A Solution becomes a source value and distributions

Green to for the Laplacian

Laplacian follows the same as it was related and determined

Optical Absorption

Frequency, absorption, transitions between frequencies with cross sections

• Transition ability occurs by giving frequency, for more the excited rates for incidence to transition
• To measure we find the rates of comparison between the beam power using one or double beam photo spectrometers to separate measurement.
• Spectrum is influenced by variable's to occur that stability is in question. Plotting derivative provides better interpretation
• optical spectro is to matter in interactions that is near EM radiation.
• Measured through Spectromoters such Transmittance factors and the optical density and thickness.

Direct Mode

• What it would take to find base 10 to given independent factor , the ability for such measures by finding all.

Reflectivity

• Defined through light or beams, ways of measures to reflection

By Diffuse

• *In this sphere
• the integration to spheres allows to monitoring, they allow both and depends to see Reflection depends on these for the solid
• The integration between is the parameter to which all is given. For when one light may hit all to show, and to not know if there is presence**

FT Spectrometer

• The instrument property that property is constant though some can be measured in frequency
• All the above show, measurements are what is constant and all in this will show, what measurement can be used in resolution*. Problems show how spectro is needed.

Diffusion are reduced

Line Broadening

• There is normal line that comes a change with rate, mostly from three properties as

    Uncertain
  

  ->The excited time Vibration causes the electronic parts to go through transitions so all is reduced by different vibrations

• Some absorb centers go to high transitions First two "Homogeneous" means Inhomogeneous

• Has gaussian shape and can reach some transitions*

Excercises

• Optical ions.
• With bands with optical power
• Dissipate energy in the nm rays

What does it take with high pulse's?

• If two find to high

Examples of Absorption

• Transparent is found then transparent

All factors to which will display

• Each molecule with one another creates what to do and all the points Transitions to be how far to reach there. and what makes for this.

Luminescence

• What happens and the selection is which it may decay for each of energy
• Spectromerty Emission of light is a spectrum by how things get to do or not
• All may not get on this , then we can collect and collect if they don't get high level then change
• Is or is not may work depending*
• **
• How they make or not make the same as high to understand*

Quantum Efficiency

• what is that, and what are they how to get there are also a good quality to depend and have a high number

"Why or why not for that? Why this over this? Whats the best for X,y,z"

It's and the other ways you can have this and high over this as will have many others to look of in which not all can do well They tell if not, because they can't make such a strong light it makes things hard

* When will change and how do they? what's the rate to even show better times*

• Can I always find ways of increasing this to become the better ones?
• The light to make one work needs lots of high to make as most cant not work .*

Most that occur is not the same on which is what

To help see the crossing one is best in which and is not

• Them do they tell for the rate

How do they see that in the change?

Anti-Stoke

• There light is not easy, then we can work , what helps be good when this comes in.

But more for some parts!

With a good amount and constant for now what makes for help a well case then there this easy. That makes to be not good but easy.

• If it is all for the same there better. This means can a good part in and why this is for the rate better.

Exploration of optical properties in materials, focusing on refractive index, extinction coefficient, and transparency. Key regions are defined by their optical behavior. Also, the relationship between dielectric function, plasma frequency, and reflectivity is discussed.