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  • jalves61 7:27 pm on March 23, 2020 Permalink | Reply
    Tags: , Quantum Mechanics   

    Mobility and Saturation Velocity in Semiconductors 

    In solid state physics, mobility describes how quickly a charge carrier can move within a semiconductor device when in the presence of a force (electric field). When an electric field is applied, the particles begin to move at a certain drift velocity, given by the mobility of the carrier (electron or hole) and electric field. The equation can be written as: density

    This is also related to Ohm’s law in point form, which is the conductivity multiplied by the Electric field. This shows that the conductivity of a material is related to the number of charge carriers as well as their mobility within the material. Mobility is heavily dependent on doping, which introduces defects to the material. This means that intrinsic semiconductor material (Si or Ge) has higher mobility, but this is a paradox due to the fact that intrinsic semiconductor has no charge carriers. In addition, mobility is inversely proportional to mass, so a heavier particles will move at a slower rate.

    Phonons also contribute to a loss of mobility due to an effect known as “Lattice Scattering”. When the temperature of semiconductor material is raised above absolute zero, the atoms vibrate and create phonons. The higher the temperature, the more phonon particles which means greater collisions and lower mobility.

    Saturation velocity refers to the maximum velocity a charge carrier can travel within a semiconductor in the presence of a strong electric field. As previously stated, the velocity is proportional to mobility, but with increasing electric field there reaches a point where the velocity saturates. From this point, increasing the field only leads to more collisions with the lattice structure and phonons, which does not help the drift speed. Different semiconductor materials have different saturation velocities and are strong functions of impurities.

     
  • jalves61 10:31 am on March 21, 2020 Permalink | Reply
    Tags: , Quantum Mechanics   

    Quantum Wells in LEDs 

    Previously, the topic of a quantum well’s functionality was discussed. Here, the topic of quantum wells’ function specifically within Light Emitting Diodes is discussed. In fact, quantum wells often implement multiple quantum wells to increase their luminescence, or total light emission.

    Quantum wells are formed when a type of semiconductor (or compound semiconductor) with a more narrow bandgap between its conduction and valence band is placed in between two wider bandgap semiconductors (such as GaN or AlN). The quantum well traps electrons within it at the conduction band, so as to increase recombination. Holes from the valence band will recombine with the conduction band electrons to emit photons which gives the LED its distinct emission of light. The quantum well is the reason why the LED does not function strictly as a diode. If the electrons were not trapped, the current would simply flow normally as in a regular LED. Although a greater number of quantum wells increases the luminescence of the LED, it can also lead to defects in the device.

    LEDs generate different colors of light by using different semiconductor material and different amounts of doping. This changes the energy gaps and leads to a different wavelength of light being produced. Gallium is a common element used in these compound materials.

     
  • mbenkerumass 5:00 am on March 18, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    Hermitian Operators, Time-Shifting Wavefunction 

    It was mentioned in the previous article on Quantum Mechanics [link] that if the integral of a wavefunction over all space at one time is equal to one (thereby meaning that it is normalized and that the probabilility of the particle existing is 100%), then the wavefunction is applicable to a later time, t.

    qm11

    A function in the place of Ψ*Ψ is used as a probability density function, ρ(x,t). The function N(t) is the resultant probability at a given time, given that the probability was found to be equal to 1 at a given time t0. Shown below, it is proposed that for dN/dt to equal zero, the Hamiltonian must be a Hermitian operator.

    qm12

    A Hermitian operator would satisfy the following:

    herm

    Hermiticity in general may referred to as a type of conjugate form of an operator. An operator is hermitian if the hermitian conjugate is equal to itself. One may compare this relationship as to a real number whose complex conjugate is equal to itself.

    herm2

    Returning to the calculation of dN/dt,

    qm13

     
  • mbenkerumass 5:00 am on March 17, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    Ψ Wavefunction Describes Probability 

    Schroedinger’s first interpretation of the wavefunction was that Ψ would describe how a particle dissipates. Where the wavefunction Ψ was the highest, then that was where more of the particle was present. Max Born disagreed saying that a particle would not dissintegrates, choosing another direction to move. Max Born proposed that the wavefunction would actually describe the probability of a particle inhabiting a space. Both Schroedinger and Einstein were initially opposed to the idea of a probabilistic interpretation of the Schroedinger equation. The probabilistic interpretation of Max Born however later became the consensus view of quantum mechanics.

    The wavefunction Ψ therefore describes the probability of finding a particle at position x at time t, not the amount of the particle that exists there.

    qm6

    Since the Schroedinger equation is both a function of position and time, it can only be solved for one variable at a time. Solving for position is preferable due to the fact that if the wavefunction is known for all x, this can provide information for how the wavefunction is at a later time.

    qm7

    Of the limits regarding the wavefunction, it is also said that the wavefunction must be convergent. The wavefunction therefore does not approach a finite constant as x approaches infinity.

    qm8

    We also recall that a wavefunction may also be multiplied by a number. It would appear that doing so would violate the above expression. The answer regarding this conjecture is that the above formula represents a normalized wavefunction. Yet it turns out that not all wavefunctions are normalizable. The case of multiplying the wavefunction with a magnitude in fact would still be normalizable, however. A wavefunction can be normalized if the integral is a finite number less than infinity using the following method:

    qm10

     

     

     
  • mbenkerumass 5:00 am on March 16, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    Matrices, Multiple Dimensions in Quantum Mechanics 

    There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.

    matwave

    Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:

    matrqm

    This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.

    qmcommutate1

    Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.

    qm4

     

     
  • mbenkerumass 5:00 am on March 15, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    Operators in Quantum Mechanics 

    Before getting into problems relating to the free particle schroedinger equation, let’s review the full Schroedinger equation. The energy operator E_hat appears in the first equation below. Thus far, the euqation relates only kinetic energy. Potential energy however when considered would allow the Schroedinger equation to be applied in a wide range of possible applications, being able to describe the interactions of atoms and molecules and their interactions in free space, wells, and other environments due to the linearity of quantum mechanics. One major point to take from discovering the free particle Schroedinger equation is how important it is in Quantum Mechanics to create energy operators. An operator can be as simple as a constant or as complicated as a partial differential. By allowing an ‘operator’ to take on this wider range of features as opposed to a basic variable makes for the basis of many quantum mechanical calculations. It then follows that the portential, V(x,t) can also be treated as an operator that modifies the system.

    schr3009

    Consider an operator X_hat that when multiplied by a function, results in the function being multiplied by x. Remember that although this may look like a variable, it is useful to consider this as an operator in Quantum mechanics.

    operator

    Does the order in which operators are multiplied matter?

    ops2

    Considering that operators are not always constants or variables, but also sometimes differentials, the order of operations for operators does matter.

    ops3

    A communtator is understood as the difference of linear operators. The commutator of x_hat and p_hat is i*h_bar.

    ops4

     
  • mbenkerumass 5:00 am on March 14, 2020 Permalink | Reply
    Tags: , , Quantum Mechanics,   

    Direct-Bandgap & Indirect-Bandgap Semiconductors 

    Direct Semiconductors

    When light reaches a semiconductor, the light is absorbed if the photon energy is greater than or equal to the band gap, creating electron-hole pairs. In a direct semiconductor, the minimum of the conduction band is aligned with the maximum of the valence band.

    qwell2

    gaas

    One example of a direct semiconductor is GaAs. The band diagram for GaAs is shown to

    the right. As the gap between the valence band and conduction band is 1.42eV, if a

    photon of same or greater energy is applied to the semiconductor, a hole-electron pair is created for each photon. This is termed the photo-excitation of semiconductors. The photon is thereby absorbed into the semiconductor.

     

    2232

     

    Indirect Semiconductors and Phonons

    indiresemicFor an indirect semiconductor to absorb a photon, the process must be mediated by phonons, which are quanta of sound and in this case refer to the acoustic vibration of crystal lattice. A phonon is also used to provide energy for radiative recombination. When understanding the essence of a phonon, one should recall that sound is not necessarily within hearing range (20 – 20kHz). In fact, the sound vibrations in a semiconductor may well be in the Terrahertz range. The diagram to the right shows how an indirect semiconductor band would appear and also the use of phonon energy to mediate the process of allowing the indirect semiconductor to behave as a semiconductor.

     

    Excitons

    Excitons are bound electron-hole pairs that are created in pure semiconductors when a photon with bandgap energy or larger is absorbed. In bulk semiconductors, these excitons will dissipate rapidly. In quantum wells however, the excitons may remain, even at room temperature. The effect of the quantum well is to force an electron and hole to be very close to each other. This allows for a strong bonding effect to take place and allows the quantum well the ability to generate light as a semiconductor laser.

     

    Quiz

    The band structure of a semiconductor is given by:

    sc1

    Where mc = 0.2 * m0 and mv = 0.8 * m0 and Eg = 1.6 eV. Sketch the E-k Diagram.

    sc2

     
  • mbenkerumass 5:00 am on March 13, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    DeBroglie Relations and the Scale of Quantum Effects (MIT OpenCourseWare) 

    Assignment Sheet MIT OpenCourseWare – Quantum Physics I

    debroglie22333333333

    qmscan1

    PDF of solutions

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     

     
  • mbenkerumass 6:00 am on March 4, 2020 Permalink | Reply
    Tags: , Quantum Mechanics   

    de Broglie’s “Matter Waves” and Full Schroedinger’s Equation 

    Photons were originally assumed to be waves and even after they were found to be particles, photons still exhibited the same qualities that allowed them to be considered as waves before. de Broglie considered that, if a photon could be considered both a particle and a wave, then perhaps other forms of matter could be treated as waves.

    debro

    de Broglie’s finding later became a pillar of quantum mechanics. The Schroedinger equation becomes an equation for such ‘matter waves’ as proposed by de Broglie. The plane wave as mentioned above later becomes the ‘wavefunction’ ψ(x,t)  as is fundamental to Schroedinger’s equaiton. Is the wavefunction ψ(x,t) measurable? What is the meaning of the wavefunction ψ(x,t)? To come to this understanding, we should unpack some of de Broglie’s matter wave formulas.

    Consider that the wavelength is inversely proportional to the momentum in de Broglie’s formula. Further, the momentum is found to be equal to h_bar multiplied by the wavenumber.

    debrog

    de Broglie’s wave formula presented further complications however. It was found that certain aspects of the wave would in fact be correct, but other parts of the wave would not be accurate. For instance, the phase of de Broglie’s matter wave would be accurate. The phase of the wavelength is understood to adhere to Galilean, non-relativistic physics, meaning that the result is not altered according to perspective. An example of a system would be relitivistic is the case for instance where the speed of a vehicle is determined relative to another moving vehicle. The phase of de Broglie’s calculation is therefore is a type of ‘objective’ calculation.

    debro2

    Further calculations below show that the waves proposed by de Broglie are not directly measurable. The wavefunction  ψ(x,t) is also known to be multiplied by an imaginary number and this makes it difficult to measure. Further, the system is not Galilean invariant. This means that matter waves may differ according to reference. Finally it is concluded that the wavefunction ψ is not like sound waves, water waves, mechanical or electromagnetic waves. A difference according to reference however does not mean that all hope is lost in the calculation of the wavefunction, however. By taking account for the difference in reference, two points can be compared to allow for a wavefunction that may vary according to reference.

    debro4

    debro6

     

    What does a matter wave look like?

    Consider it should look like any other wave. It should have some kind of sinusoidal representation. Considering one aspect of matter, it should be that matter is not allowed to exist in any one place. This is the case for instance, when a particular piece of matter is not present in a location. However, to restrict a matter wave by time could be problematic. This is to say that, at a certain moment, no matter is allowed to exist anywhere or for all positions.

    There are four cases listed below that, being sinusoidal, may seen to be possible respresentations for matter waves. In the case however for a sin(kx-wt) or cos(kx-wt), it is implied then that for time wt eual to pi/2, 3*pi/2, etc matter is not allowed to exist anywhere. Therefore these functions are not acceptable representations.

    For the case however of e^(ikx-iwt) + e^(-kx-iwt) and it’s counterpart e(-ikx+iwt)+e^(ikx+iwt), this is not the case. Could both be used together in superposition? The answer is that if they were added together, matter would be restricted to one direction, which is undesireable as well. Therefore, either representations would be acceptable, but not both. The boxed answer below is the normative convention taken by physicists for the matter wave wavefunction for a particle. The other representation would work as well.

    mwave

     

    Principle of Stationary Phase

    Consider the case of a narrow peak, modulated by a sinusoidal function centered about zero. If the sinusoid frequency is too high with relation to the narrowness of the peak or if the phase is rapidly changing, the averaging of the system will cause the narrow peak to disappear. If however the frequency is low – or if the phase is stationary at the narrow peak, any averaging that is done will still allow the narrow peak to exist.

    yu

    As related to quantum mechanics, while determining characteristics of a wave, the principle of stationary phase becomes important. The wave Φ(k) as a function of wavenumber will be detected as a narrow peak. In order to properly detect this narrow peak, the modulating sinusoid must have a phase that varies much less with respect to (in this case, k) the x-axis. Otherwise, the narrow signal will be lost.

    yu7

     

    Towards developing a generalized Wavefunction

    Performing a specific operation to the wavefunction interestingly produces the momentum multiplied by the wavefunction. The operator used is termed the “momentum operator.”

    wf

    As we know from linear algebra, if a matrix multiplied by a vector is equal to a number multiplied by the vector, then the vector is termed an eigenvector of the matrix.

    la1

    Given the above momentum operator relationship, it may be concluded that ψ(x,t) is an eignevector, or more specifically an eigenstate of the momentum operator p_hat. p is then also the eigenvalue.

    la2

    And finally, using the eigenstate condition, Schroedinger’s Equation in general form for a free particle is derived:

    sch3

     

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
  • mbenkerumass 6:00 am on March 3, 2020 Permalink | Reply
    Tags: Quantum Mechanics   

    Compton Wavelength of a Particle, Compton Scattering 

    Planck’s constant h is fundamental to Quantum Mechanics. But what is Planck’s constant?

    Separating the fundamental units from h, it is found to be equal to a distance multiplied by a momentum. If then we consider using the speed of light, c and a given mass, we arive at a new parameter equalling a distance. This parameter is the Compton wavelength of a particle.

    compton

    compton2

    The Compton wavelength of light:

    compton2

    Compton wavelength of an electron:

    compton-e

     

    Compton Scattering

    Physicists had considerable difficulty accepting the existence of the photon. For one, it introduced the non-deterministic nature which collided with classical mechanics and it also went against Maxwell’s theories, which at the time were considered highly successful. The concept of Compton Scattering was one of the final pieces of support that lead to a wider acceptance of the photon as a particle.

    Compton Scattering provided an image of a photon literally colliding with an electron, showing that like the electron, the photon was worthy of being considered a particle as well. The classical version of Compton Scattering was Thompson Scattering, which considered the photon as a wave. The Thompson Scattering experiment applied a photon wave to an electron, which caused the electron to be energized and radiate a photon wave.

    thom

    Compton Scattering, treating the photon as a particle provides a different picture at higher energy levels:

    compt5

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
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