User:Jogaka/sandbox/exam quiz

Maryam Alamer 14204223

Problem 1 Dynamic light scattering (DLS) or quasi-elastic light scattering" (QELS) or "photon correlation spectroscopy" (PCS) refers to observing scattered light so as to aid in determining defining properties of a particle dispersion or molecular solution like zeta potential, size of the particle, measuring the diffusion rate of the protein particles as well as particle’s molecular weight. This motion data is then processed to obtain size distribution of a given sample. The method of finding the size is given by obtaining "hydrodynamic radius" or "Stokes radius" of the particles. This radius depends on shape and mass. The basic principle behind the study is first by illuminating the sample by passing it in a polarizer using a laser beam. The scattered light is then passed in a second polarizer where it is picked up by the photomultiplier and the final image is casted on a screen creating a speckle pattern. The solution’s molecules are hit with the light and the majority of the particles diffract the light in all angles. This occurs when the particle’s diameter which is far much lesser than the wavelength of the illuminated light. Therefore, every molecule will diffract the incoming light in all courses, a phenomenon known as Rayleigh scattering. The diffracted light from the majority of the particles can either interfere productively (light sections) or destructively (dark sections). This procedure is repeated at brief time intervals and the subsequent speckle patterns are studied by an autocorrelation that compares light intensity at every spot after some time. Figure 2: Dynamic light scattering measures variation in scattered intensity with time at a fixed scattering angle (typically 90o) ,while static light scattering measures scattered intensity as a function of angle

Small particles in solutions are subject to Brownian motion since most of the molecules are not constant but rather they are suspended in the solution. They move haphazardly because of the impact with solvent molecules, this kind of movement is called Brownian Motion. The rate of Brownian movement can be specifically measured from the scattered light pattern produced by the moving particles, a method known as photon correlation spectroscopy (PCS) or quasi-elastic light scattering (QELS) now usually referred to as DLS. The connection between the speed of Brownian movement of a molecule and that molecule's size is characterized by the Stokes-Einstein comparison: Where: D = speed of Diffusion of particles, k = Boltzmann’s constant, T = absolute temperature, η = viscosity, r = hydrodynamic radius. Small particles should diffuse faster than larger ones since (D) is inversely proportional to the radius of particles. This is the principle of the DLS analysis.

In the sample, the separation between particles is continously changing because of a Doppler movement between the frequency of incoming and scattering light. This distance influences the phase overlap of diffracted light because the brightness of the spots on the intensity in speckle pattern will fluctuate with time. The rate of power fluctuating depends on how quick the particles are moving (quick for smaller particles, moderate for bigger particles).

There are several steps undertaken to study the time scale of the fluctuations related to the particles motion that include:

1.	Measure fluctuations and change them into an  Intensity Correlation Function.

2.	Describe the corresponding movement of the particles in relation to molecule size and Electric-Field Correlation Function.

3.	The Seigert Relationship has to be equated to the correlating functions.

4.	Analyze information utilizing cumulants or fitting schedules.

To start with, the Intensity Correlation Function G2(Ԏ) is meant to describe the speed of scattering intensity. This is done by comparing the intensity at various tieme inervals, that is initial time t upto a certain final (t + Ԏ ) this providing a pqoductive quantitative measurement of the flickering light Mathematically,):

The Electric Field Correlation Function, G1(Ԏ ) correlates the particles motion since it is difficult identify how every particles move. The motion of the particles relative to each other can be analyzed using this equation.

E(t) and E(t+ Ԏ ) refers to the scattering electric fields at the initial time t to a later time (t + Ԏ ). G1(t ) will decay exponentially with a decay constant ┌ for a monodisperse undergoing Brownian motion, The diffusivity in the Brownian Motion and the decay constant is related to each other by Where : ɵ= angle at which detector is located Q= magnitude of the scattering wave vector n = refraction index of solution, q2= the distance the particles travels

. The speed of the particles can be determined by measuring how long it takes the function to go to zero.

The Seigert Relationship can be used to equate the two functions  Where B= baseline Ɓ= instrumental response, the both are constant

In a conductor, the conduction band which is the highest occupied band is not completely full. Hence, electrons can move to and from the neighboring atoms freely in the process conducting electricity. •	EF lies inside an allowed band (1 electron/unit cell). In an insulator, the highest occupied band known as the valence band is usually full. The conduction band is the first unfilled band found above the valence band. The valence band and the conduction are separated by a wide gap which at room temperature, no energy is enough to move the elctrons from the valence band to the conduction band where they would be able to contribute to conduction, hence no electrical conduction in insultors. •	The Fermi level lies at the top of a band (full band). Large gap between bands In a semiconductor, the gap between the valence band and the conduction band is littler, and at room temperature there is adequate energy to move a few electrons from the valence band into the conduction band, permitting some conduction. An increase in temperature builds the conductivity of a semiconductor as more electrons have enough energy to make the bounce to the conduction band. There are two different types of semiconductors: Intrinsic semiconductors: it is semiconductors in pure state. For each electron that hops into the conduction band, the missing electron creates a gap that can move openly in the valence band. The number of gaps will equal to the quantity of electrons that have jumped. Extrinsic semiconductors: The gap in the band is controlled by doping which involves adding little impurities to the material so as to change the electrical conductivity of the lattice structure. For extrinsic semiconductors, the quantity of the holes does not match with the quantity of the electrons jumped. Extrinsic conductors are of two P type (positive charge doped) and n type (negative charge doped)

Explain how the wave function of an electron placed in a one-dimensional periodic potential can be solved using the Bloch wave approach? A Bloch wave is a kind of wave function for a molecule in an intermittently repeating environment, most usually an electron in a crystal. Bloch waves are vital due to the Bloch's theory, which expresses that the energy eigenstates for an electron in a crystal can be expressed as Bloch waves. A wave function ψ is a Bloch wave in the event that it has the structure: Where: r i=position, ψ= Bloch wave, u= a periodic function with the same periodicity as the crystal, k= a vector of real numbers called the crystal wave vector, e= Euler's number, i= the imaginary unit. In other words, thus, after multiplying a plane wave by a periodic function, you get a Bloch wave. •	 Describes lattice periodic parts, (depending on the value of the wave-vector k) •	  describes wave function of a free particle. This factor retains the same form as for free particles.

These Bloch wave energy eigenstates are written with subscripts as ψn k, where: n= discrete index, called the band index, that is usually present since there are many Bloch waves containing similar k (each with a different periodic component u). in a given band (i.e., for constant n), ψn k changes continuously with k, hence its energy too. For any reciprocal lattice vector K, ψn k = ψn (k+K). Hence, clear and distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice. in solid substances, the discussion revolves around crystals - periodic lattices. In this discussion, a 1D lattice of positive ions will be discussed. If the spacing between ions is a, then the potential lattice will supposedly look like illustrated: According to Bloch's theorem, the solution of the wave function using the Schrödinger equation when the potential is periodic, can be written as: Here, u(x) is a periodic function which satisfies u(x + a) = u(x) There are problems with the boundary condition encountered when approaching the edges of the lattice. Following the Born-von Karman boundary conditions, the ion lattice can be represented as a ring. Assuming L is the lattice length and  that L ≫ a, the amount of ions in the lattice will be so big, that if one is comparing one ion, the surrounding of the environment is almost linear, and the electron wave function will remain constant. So now, instead of two boundary conditions we get one circular boundary condition

(b) Describe how the Bloch wave approach uses symmetry aspects of the periodic potential in order to simplify the solution.? The issue of electrons in the solids is a numerous electron issue. The full Hamiltonian of the solid contains not just the one-electron potential depicting the interactions of the electrons with atomic nuclei, however pairing potential additionally portrays the electron-electron interactions. The simplest methodology for use is a free electron model. The next step in building the complexiy is independent electron approximation assuming that each of the interactions are portrayed by a successful potential. A standout amongst the most imperative properites of this potential is it is periodic on a cross section. U(r)= U(r+T) where the T is the lattice vector.

Explain the nature of Excitons in semiconductor? a.)Excitation is a bound condition of an electron and holes, the both are pulled towards each other by electrostatic Coulomb force. The separation between the electron and the hole is called Bohr radices of excitation as it is a couple of nanometers. Because of Coulomb interaction, the electrons and holes existing in a material are known structure excitons. In this manner, the optical nature of semiconductors can be comprehended by researching the properties of the excitons. An exciton is made out of an electron and a hole. In bulk semiconductors, the exciton can move anyhowly in all ways.

At the point when the length of a semiconductor is shortened to the same as the exciton radius, i.e., to a couple of nanometers, quantum confinement impact happens and the exciton properties are adjusted. Depending with the measurement of the confinement, three sorts of confined structures are discussed: 1.	In a QW, exciton can move freely in other two directions when the material size is reduced only in one direction. 2.	In a QWR, the material size is lessened in two direction and the exciton can move unreservedly in one way as only.

3.	In a QD, the material size is lessened in all directions and the exciton cannot move freely in any direction in these restricted structures.

Accordingly, these structures are a great possibility for creating  optoelectronic gadgets, for example, semiconductor light-discharging diodes and laser diodes. Semiconductor materials have their electronic structures sorted out in groups: a valence band created by the overlap of the occupied energetic levels of the individual structural units and a conduction band produced by the overlapp of the vacant levels. Generally, the conduction and the valence band in many semiconductors are consistent if intraband energetic spacing littler than kBT (T is the temperature).

Once the electron is advanced into the conduction band a hole is left in the valence band; the semiconductor gets to be conductive and the electron and gap move uninhibitedly using their kinetic energy. The move starting from the earliest stage to the energized state happens as an after effect of some external disturbance, e.g. a photon. The electron in the conduction band and the hole in the valence band can be held together by the electrostatic attraction, to shape exciton. The intercation in the middle of electron and opening can be portrayed by a hydrogen like Hamiltonian where the M is the total mass M= me*+mh* and μ is the reduced mass μ= me*mh*/ (me*+mh*); me* and mh* are the effective masses of the electron and hole, respectively

Explain the concept of the effective Bohr radius in semiconductors? a)	The Bohr radius (a0 or r Bohr) is a physical constant, almost equal to the distance between the proton and electron in a hydrogen atom in its groundstate.

value is 5.2917721067(12)×10−11 m. Bohr exciton radius is defined as At the point when quantum dot size is lesser than the exciton Bohr radius, the electron hole pair energy levels in quantum dots can't be dealt with further based on the hydrogen model. The most reduced energy level of the exciton is presently delocalized over the whole quantum dot. The exciton levels are given by taking care of the traditional quantum mechanical issue of a particle in a box. For the situation where the electron and gap are kept in a little space, the coulomb fascination is irrelevantly little contrasted with a potential U(r) that depicts a spherically symmetric potential well of length r. the relating Hamiltonian is

Boher diameter determines the type of confinement for example; •	3-10 times Bohr diameter is weak confinement ∆E≈ 1/M* (M* is effective mass of excitation) •	Lesser than 3 Bohr diameter is a strong confinement ∆E≈ 1/µ* (µ* is effective mass of holes and electron).

What are QDS? Why is their optical response is tunable? b)	b)	Quantum dots (QD) are nanoscale semiconductor gadgets that the material size is decreased in all directions and the exciton can't move in any directions in any course, that implies the electrons and gaps are bound in every one of the three spatial. The electronic properties of the quantum dots fall between those of mass semiconductors and those of discrete particles of practically identical size. Band gap can be tuned as an element of molecule size and shape for a given composition. For instance the photoluminescence of a QD can be controlled to particular wavelengths by controlling molecule width: •	larger QDs (range of 5-6 nm)emit longer wavelengths bringing about discharge colors, for example, orange or red. •	Smaller QDs (range of 2-3 nm) emit shorter wavelengths bringing about colors like blue and green. c)	The capacity to control the emission from Q dots by chancing it is size is known as the size quantization impact. Their optical reaction is tunable on the grounds that evolving size, shape, and structure, quantum dots can change their absorptive and emissive properties significantly d)	Explain how the bandgap is changing with their physical size and explain why smaller Quantum Dots are blue-shifted in their optical response? The band hole can get to be stronger in the solid confinement where the extent of the quantum dot is lesser than the Exciton Bohr range ab* as the energy levels split up. where ab is the Bohr radius=0.053 nm, m is the mass, μ is the decreased mass, and εr is the size-dependent dielectric constant value. Thus, increment in the aggregate emision energy which is the sum of the energy levels in the smaller band gaps in the solid confinement which is bigger than the energy levels in the band holes of the first levels in the weak confinement and the emissions at different wavelengths. For the most part, the littler the extent of the crystal → the bigger the band gap→ the more noteworthy the distinction in energy between the most astounding valence band and the least conduction band gets to be. The bigger size of the crystal→ the littler the band gab→ the littlest the distinction in energy between the most elevated valence band and the least conduction band. In this manner quantum specks with various sizes can radiate light of various colors, the reason is the quantum control impact. So 1-The bigger the spot, the redder (lower energy) its fluorescence range. 2-littler specks radiate bluer (higher energy) light.

Figure: 1 splitting of energy levels in quantum dots due to the quantum confinement effect, semiconductor band gap increases with decrease in size of the nanocrystal.

For (blue-shifted): When the quantum dots are lit up by UV light, a percentage of the electrons get enough energy to break free from the molecules. This capacity permits them to move around the nanoparticle, and make a conduction band in which electrons are allowed to travel through a material and become a conductor.

At the point when these electrons drop once more into the external circle around the particle (the valence band), they radiate light. The shade of that light relies on upon the vitality contrast between the conduction band and the valence band. Figure 2:      Electrons in a quantum dot generating light.