πŸ”
brainless
  • What is this?
  • Responsibility
  • Changelog
  • meta
    • Sharing
      • Inspirations
      • Workflows
      • Social Media
    • Geography
      • Life
      • Death
        • Family Death
    • Research
      • Project Index
      • 3D Printing
      • Photogrammetry
      • Drone Building
    • External Websites
    • [unreleased]
      • [Template]
      • [TEMP] CL V0.0.5 or whatever
      • Skincare
      • Travel
      • Working and Staying Busy
      • Stride
      • Funeral Playlist
      • Notes and Ideas
      • Boredom
      • Four Noble Truths of "Thermo"
      • Respect
      • Work
  • STE[A]M
    • [guide]
    • Science
      • Materials Modeling
        • Syllabus Description
        • Lecture Slides
        • Student Notes
        • Assignments
          • index
          • Carbon Nanotubes
    • Technology
      • Computer Science
        • Commands
      • Photogrammetry
      • Quantum Computing
      • Computers
      • Programs
        • Matlab and Octave
        • Audacity
        • Google Chrome
          • Websites
            • Google Suite Sites
            • Github
              • Version Control
            • Product Hunt
            • Twitter
            • Youtube
              • Channels
            • Vimeo
          • Extensions
            • Dark Reader
            • Vimium
        • [miscellaneous]
          • Octave
          • PureRef
          • git
          • gnu stow
          • mermaid.js
        • Excel
        • Blender
        • LaTeX
        • Sublime: Text Editor
        • Spotify
        • VLC Media Player
      • Android and iOS
      • Operating Systems
        • macOS
          • mackup
        • Unix
          • folder structure
        • Windows
          • App Installation
          • Meshroom
          • Corsair Utility Engine
      • 3D Printing
    • Engineering
      • Accreditation
        • Fundamentals of Engineering
        • Professional Engineering
      • Continuum Mechanics
        • Fluid Mechanics
          • Incompressible Flow
            • corona final
            • indexhw4
            • indexhw3
            • index
            • hw2
            • hw1
          • Syllabus Description
          • Lecture Slides
          • Student Notes
            • Dynamic or Kinematic Viscosity
          • Assignments
            • Homeworks
              • Homework 1
              • Homework 2
              • Homework 3
              • Homework 4
              • Homework 5
            • Vortex Project
        • Solid Mechanics
          • Syllabus Description
          • Lecture Slides
          • Student Notes
          • Assignments
        • Incompresible Flow
          • Syllabus Description
          • Lecture Slides
          • Student Notes
          • Assignments
      • Experimental Mechanics
        • Syllabus Description
        • Lecture Slides
        • Student Notes
        • Assignments
      • Finite Element Methods
        • Intro to Finite Elements
          • Syllabus Description
          • Lecture Slides
          • Student Notes
          • Assignments
        • Fundamentals of FEM
          • Syllabus Description
            • index
          • Lecture Slides
          • Student Notes
          • Assignments
            • Project
              • index
              • Untitled
            • Homework 1
            • Homework 4
            • index
      • Heat Transfer
        • Syllabus Description
        • Lecture Slides
        • Student Notes
        • Assignments
          • homework
            • hw10
            • q9
            • q8
            • q7
            • hw7
            • hw6
            • q5
            • q3
            • 1 ec
          • Discussions
            • d11
            • d10
            • d9
            • d8
            • d6
            • d4
            • d3
          • Project Notes
      • Machine Dynamics
        • Syllabus Description
        • Lecture Slides
        • Student Notes
        • Assignments
    • Art
      • Color Theory
      • Origami
        • FolderMath
          • Surveying Origami Math
          • Represent a Folded Object
          • Creating a Crease Pattern
          • Making the Folds
          • Simulating Folding Origami
          • List of Resources
            • Codes
            • Papers, Programs, and Inspirations
    • Mathematics
      • Complex Numbers
        • What is i^i?
      • Analytic Hierarchy Process
      • Probability
      • Conway's Game of Life
      • Metallic Numbers
      • Cauchy's Formula for Repeated Integration
      • Wavelet Transform
      • Laplace Tidal Equation
      • Alternating Summation of Ones
      • Constants
      • Bad Maths
      • Calculus
        • Syllabus Description
        • Miscellaneous
  • Thoughts
    • Marksmanship
      • Archery
    • Schooling
    • ...and Ideas?
      • Perceived Time and Learning
      • Content Comprehension
    • Comics and Games
      • Rubik's Cube
      • Dungeons and Dragons
      • Beyond-All-Reason
      • Sekiro: Shadows Die Twice
      • Super Smash Bros
        • Project M
        • Project +
      • League of Legends
      • Satisfactory
    • Literature and Art
      • Books
      • Reading is Hard
      • Various Words and Phrases
      • Poems
      • Interviews
      • Quotes
        • Phrases
      • Jokes
      • ASCII Art
    • Shows and Films
      • Cowboy Bebop
      • My Hero Academia
      • Sword of the Stranger
    • Working and Life Balance
  • Projects
Powered by GitBook
On this page
  • Problem 1: LDA GGA Pseudopotential Comparison
  • LDA: Raw Data and Reduction
  • GGA: Raw Data and Reduction
  • Tabular Data
  • Problem 2
  • Problem 3: Comparison to Correlation
  • Problem 4: Gallium Arsenide
  • Problem 5: Germanium
  1. STE[A]M
  2. Science
  3. Materials Modeling
  4. Assignments

index

PreviousAssignmentsNextCarbon Nanotubes

Last updated 4 years ago

Problem 1: LDA GGA Pseudopotential Comparison

From the information taken in the lab, the base energy is a quadratic function of the lattice parameter, which dictates the volume. Therefore, the equation

Ei=AiV2+BiV+Ci,E_{i}=A_{i} V^{2}+B_{i} V+C_{i},Ei​=Ai​V2+Bi​V+Ci​,

where $i$ serves as the index between LDA and GGA methods, $V$ is the volume of the unit cell, and $E$ is the internal base energy

LDA: Raw Data and Reduction

From the shell on tardis.mat.rpi.edu , two folders (one for each potential) were created inside of the template Prob1 folder which contains their associated si.scf.out files.

nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.1
     unit-cell volume          =    1061.2080 (a.u.)^3
!    total energy              =     -15.39800670 Ry
nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.2
     unit-cell volume          =     265.3020 (a.u.)^3
!    total energy              =     -15.85306213 Ry
nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.3
     unit-cell volume          =     530.6040 (a.u.)^3
!    total energy              =     -15.49134351 Ry

Vectorized, the data listed is LDAvol = [1061.2080, 265.3020, 530.6040], LDAeng = [-15.39800670, 15.85306213, 15.49134351].

GGA: Raw Data and Reduction

nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.1
     unit-cell volume          =    1061.2080 (a.u.)^3
!    total energy              =     -15.33775426 Ry
nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.2
     unit-cell volume          =     265.3020 (a.u.)^3
!    total energy              =     -15.74016593 Ry
nkintc@tardis:~/hw1/Prob1$ grep -w "volume" < si.scf.out.3
     unit-cell volume          =     530.6040 (a.u.)^3
!    total energy              =     -15.40367742 Ry

Vectorized, the data listed is saved as GGAvol = [1061.2080, 265.3020, 530.6040], GGAeng = [-15.33775426, -15.74016593, -15.40367742].

Tabular Data

Method

$A_i$

$B_i$

$C_i$

$l_i$ [a.u.]

LDA

-0.000071431

0.055488605

6.159476427

7.2962

GGA

-0.0000014375

0.0024124034

-16.2790057267

9.4320

We define pressure as the energy contained within the cell volume, so that:

However, it is easier to recognize that the energy expression is parabolic, rather than find the roots of the pressure curve. This is because the third term, which varies with the inverse of volume complicates traditinal root finding methods.

The minimum lattice parameter using the expressed coefficients, $ l{i}=\sqrt[3]{\frac{-B{i}}{2 A_{i}}}$. Numerically, LDA produces minimum lattice parameter of 7.2962 a.u. while GGA produces a higher minimum of 9.4320 a.u. The experimental value of Si lattice is 5.4310 Γ… or 10.2631 a.u. which higher than either of the two values calulated, but closer to the GGA value.

Problem 2

To calculate the bulk modulus $B=-V\left(\frac{\partial P}{\partial V}\right)=V\left(\frac{\partial^{2} E}{\partial V^{2}}\right)$, search for the minimum of the base state energy by applying the quadratic equation to the energy equation as before. Or, by recognizing that the second partial derivative of the energy equation would be $2A_i$ by differentiation rules, then:

In terms of the given coefficients, and using magnitude of $A$ so that all coefficients are now in $\mathbb{R}^+$:

Neither which are similar to the experimental 97.6 GPa.

Problem 3: Comparison to Correlation

Using the Birch Murnaghan equation of state:

Problem 4: Gallium Arsenide

A direct band gap material, this would serve as a better use of solar cell material than Silicon, a indirect band gap semiconductor

Problem 5: Germanium

Germanium is an indirect band gap material just as elemental silicon is. Strain engineering can be used to to allow for that material to behave in a piezoelectic manner, in essence tuning the band structure directly, by deforming the material.

Pi(V)=Ei(V)V=AiV+Bi+CiV.P_i(V)=\frac{E_i(V)}{V} = A_{i} V +B_{i}+\frac{C_{i}}{V}.Pi​(V)=VEi​(V)​=Ai​V+Bi​+VCi​​.
Vi=βˆ’bΒ±Bi2βˆ’4AiCi2Ai=βˆ’Bi2Ai⏞Vi,0Β±Bi2βˆ’4AiCi2Ai⏞DistanceΒ fromΒ MinimaV_{i}=\frac{-b \pm \sqrt{B_{i}^{2}-4 A_{i} C_{i}}}{2 A_{i}} = \overbrace{\frac{-B_{i}}{2 A_{i}}}^{V_{i, 0}} \pm \overbrace{\frac{\sqrt{B_{i}^{2}-4 A_{i} C_{i}}}{2 A_{i}}}^\texttt{Distance from Minima}Vi​=2Aiβ€‹βˆ’bΒ±Bi2β€‹βˆ’4Ai​Ci​​​=2Aiβ€‹βˆ’Bi​​​Vi,0​​±2Ai​Bi2β€‹βˆ’4Ai​Ci​​​​DistanceΒ fromΒ Minima​
B=βˆ’V(βˆ‚Pβˆ‚V)⏞nonlinear=V(βˆ‚2Eβˆ‚V2)=Vβ‹…βˆ‚2βˆ‚V2(AV2+BV+C)=Vβ‹…βˆ‚βˆ‚V(2AV+B)=2AV2.B= \overbrace{-V\left(\frac{\partial P}{\partial V}\right)}^\texttt{nonlinear}= V\left(\frac{\partial^{2} E}{\partial V^{2}}\right) = V \cdot \frac{\partial^{2}}{\partial V^{2}}\left(A V^{2}+B V+C\right) = V \cdot \frac{\partial}{\partial V}\left(2A V+B\right) = 2 AV^2.B=βˆ’V(βˆ‚Vβˆ‚P​)​nonlinear​=V(βˆ‚V2βˆ‚2E​)=Vβ‹…βˆ‚V2βˆ‚2​(AV2+BV+C)=Vβ‹…βˆ‚Vβˆ‚β€‹(2AV+B)=2AV2.
Bi=2AiVi,02=2AiVi,02=2Ai(βˆ’Bi2Ai)2=Bi22Aiβ†’BLDA=21.5521Ryau3,BGGA=2.02424Ryau3B_i = 2A_iV_{i,0}^2 = 2A_iV_{i,0}^2 = 2A_i \left(\frac{-B_{i}}{2 A_{i}}\right)^2 = \frac{B_i^2}{2A_i}\rightarrow B_\texttt{LDA} = 21.5521 \frac{\text{Ry}}{\text{au}^3} , B_\texttt{GGA} = 2.02424 \frac{\text{Ry}}{\text{au}^3}Bi​=2Ai​Vi,02​=2Ai​Vi,02​=2Ai​(2Aiβ€‹βˆ’Bi​​)2=2Ai​Bi2​​→BLDA​=21.5521au3Ry​,BGGA​=2.02424au3Ry​
P(V)=3B02[(V0V)73βˆ’(V0V)53]{1+34(B0β€²βˆ’4)[(V0V)23βˆ’1]}P(V)=\frac{3 B_{0}}{2}\left[\left(\frac{V_{0}}{V}\right)^{\frac{7 }{3}}-\left(\frac{V_{0}}{V}\right)^{\frac{5}{3}}\right]\left\{1+\frac{3}{4}\left(B_{0}^{\prime}-4\right)\left[\left(\frac{V_{0}}{V}\right)^{\frac{2}{3}}-1\right]\right\}P(V)=23B0​​[(VV0​​)37β€‹βˆ’(VV0​​)35​]{1+43​(B0β€²β€‹βˆ’4)[(VV0​​)32β€‹βˆ’1]}