### Table of Contents

# ARCHIVE of PREVIOUS YEARS

# Spring 2016 (PREVIOUS YEAR)

#### PH 575 Introduction to Solid State Physics, Spring 2016

**Instructor:**(Spring 2016) Prof. Ethan Minot, Office: Weniger 417**TA:**(Spring 2016) Kyle Vogt**Textbook:**See course information**Class Meetings:**MWF 3.00pm - 3.50pm, Weniger 304

## Week 1

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

1 | Mon 3/28 | Overview | intro_presentation2.pdf, Course Information, day1_2016.pdf. Reading: Sutton Ch1, pp. 9-20 | hw1.pdf, due next Wednesday |

2 | Wed 3/30 | day2_2016b.pdf: Summarize key points about building a model of electron density. Discussion the relationship between electron spatial distribution and electron energy: Electrons lower their total energy by getting close to positive charge; Wavefunctions with high curvature (high spatial frequency) have more kinetic energy. | - | |

3 | Fri 4/1 | day3_2016.pdf: Wavefunctions that satisfy the Schrodinger equation with a Coulomb potential. Calculating/visualizing the wavefunctions distribution in space and on an energy scale.Y_lm mathematica notebooks | - |

## Week 2

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

4 | Mon 4/4 | day4_2016.pdf: The superposition of two degenerate wavefunction also satisfies the differential equation that constrains the wavefunction. Hybridized orbitals (chemistry lingo) are the same thing as the superposition of degenerate wavefunctions. Example, the angular dependence of a p_x orbital is made by adding together Y_1,1 and Y_1,-1. Introduction to ket notation. | ||

5 | Wed 4/6 | day5_2016.pdf: Definition of orthonormal basis. Vector notation. Projection in vector notation. Hybridization in vector notation. | hw1soln.pdf, hw2 | |

6 | Fri 4/8 | day6_2016.pdf: Building a molecular orbital using LCAO state. Assume a linear combination of 1s orbital centered on left proton and 1s orbital centered on right proton. Turn differential equation into linear equation. Drop the smallest term. Find energy by matrix algebra. pq6.pdf Interpret result as constructive or destructive interference between atomic orbitals. How beta is calculated. |

## Week 3

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

7 | Mon 4/11 | day7_2016.pdf: Picking the basis to use for matrix representation of Hamiltonian: all the atomic orbitals within a suitable energy range. Covalent and ionic bonding in a diatomic molecule. | ||

8 | Wed 4/13 | day8_2016.pdf: Naming the energies in the Hamiltonian matrix: on-site integrals and hopping integrals. Classifying bonding orbitals as sigma or pi. | ||

9 | Fri 4/15 | day9_2016.pdf: Representing wave functions with play dough. Using color to represent phase. Energy level diagrams for molecular orbitals. | hw2soln.pdf |

## Week 4

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

10 | Mon 4/18 | day10_2016.pdf: LCAO states for a 1d extended solid. Start with 4 protons. Use periodic boundary conditions. | hw3 updated | |

11 | Wed 4/20 | day11_2016.pdf: Continue analyzing the chain of 4 protons. Pop quiz based on a chain of 6 protons. Generalize to chain of N protons. Expression for the energy of the mth molecular orbital. Sketch of the 1s band. | ||

12 | Fri 4/22 | day12_2016.pdf: Introduce k, the dispersion relation, E(k), and the first Brillouin zone. Calculate density of states for a 1d system. |

## Week 5

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

13 | Mon 4/25 | day13_2016.pdf: Special values of k, for example the gamma point and chi point. Bloch theorem applied to LCAO states. 1d example. Extend to 2d and 3d. Dispersion relation for LCAO Bloch wave functions (also called the tight binding model). | ||

14 | Wed 4/27 | Review | 2014 Midterm McIntyre Chpt 15 problems, Pop Quiz Day 1 - 13 | hw3 soln |

15 | Fri 4/29 | MIDTERM |

## Week 6

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

16 | Mon 5/2 | No class | Ethan travelling | |

17 | Wed 5/4 | day14_2016.pdf: Density of states in 2d. Free particle example. LCAO band example, 2d dispersion relation. Effective mass. Distribution of electron energies at T = 0. Calculating the Fermi energy. | hw4 | |

18 | Fri 5/6 | day15_2016.pdf: Pop quiz about calculating Fermi energy in a doped semiconductor. Primitive unit cell. Primitive translation vectors. Bravais Lattice. Wigner-Seitz cell. Block wave functions in graphene. |

## Week 7

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

19 | Mon 5/9 | day16_2016.pdf: How many atomic orbital go into a band structure calculation? Wien2k uses muffin tin radius and energy cut-off to decide. The number of atomic orbital is revealed by counting lines on the dispersion relation, or integrating the density of states per primitive unit cell. Finish the calculation of the graphene band structure. | ||

20 | Wed 5/11 | day17_2016.pdf: The reciprocal lattice: Dispersion relationships repeat in k-space, forming reciprocal lattice. Primitive reciprocal lattice vectors define this repeating pattern. All information about the energy of Bloch wavefunctions is in the primitive unit cell of this reciprocal lattice. By convention, we set the primitive unit cell of the reciprocal lattice to be a Wigner-Sietz cell and call it the first Brillouin zone. | hw4soln.pdf | |

21 | Fri 5/13 | day18_2016.pdf: Outline for last 3 weeks. Defining a conductor. How electricity is conducted: the electric field modifies the distribution of electrons in momentum space. Example: Conductivity of graphene. |

## Week 8

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

22 | Mon 5/16 | day19_2016.pdf: Discuss conventional unit cell. Optical properties of semiconductors. Define absorption coefficient. Conservation of energy and momentum. Vertical transitions. Conversion from eV to wavelength. | ||

23 | Wed 5/18 | day20_2016.pdf: Calculating absorption coefficient: Joint density of states; Fermi golden rule; dipole matrix element. Thickness of a solar cell. Example, calculate the dipole matrix element for the k = 0 transition in the 2pz band of graphene. | ||

24 | Fri 5/20 | day21_2016.pdf: Relationship between absorption coefficient and optical conductivity, complex dielectric constant and complex refractive index. Pop Quiz. Indirect optical transitions enabled by phonons absorption or emission. |

## Week 9

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

25 | Mon 5/23 | day22_2016.pdf: Highest vibrational frequency in crystals ranges (depends on material). Analyze the absorption spectrum of silicon at T = 290 K vs T =4 K. The free electron model for metals. Examples of free electron metals: Na, Mg and Al. | ||

26 | Wed 5/25 | POSTER SESSION | ||

27 | Fri 5/27 | No class | Ethan traveling | hw5 |

## Week 10

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

Mon 5/30 | MEMORIAL DAY | |||

28 | Wed 6/1 | day23_2016.pdf: Beyond band structure. The role of impurities. Impurities control the color of Al2O3 and other precious gems. Donor and acceptor impurities control the conductivity of semiconductors. Emergent behavior. Ferromagnetism emerges when exchange energy is more important than the kinetic energy cost of spin polarization. The Mott Insulator state emerges from a metallic band structure if electron-electrons are sufficiently strong. | term paper is due | |

29 | Fri 6/3 | Review. Pop Quizzes from Day 14 - 22 | hw5soln_2016.pdf |

# Spring 2015 (PREVIOUS YEAR)

#### PH 575 Introduction to Solid State Physics, Spring 2015

**Instructor:**(Spring 2015) Prof. Ethan Minot, Office: Weniger 417**TA:**(Spring 2015) Eric Krebs**Textbook:**See course information**Class Meetings:**MWF 3.00pm - 3.50pm, Weniger 304

## Week 1

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

1 | Mon 3/30 | Overview, H-atom | day1.pdf lecture notes, Advertisement for solid state physics, Course Information. Reading: Sutton Ch1, pp. 9-20 | hw1.pdf |

2 | Wed 4/1 | H-atom | day2.pdf lecture notes: Visualizing H-atom wave functions, demonstrations, understanding the complex phase factor, quantized angular momentum, wavefunction curvature comes from either r-dependence, angular dependence or both. Y_lm mathematica notebooks. S: Ch. 1, pp. 9-20 | - |

3 | Fri 4/3 | H-atom, QM formalisms | day3.pdf lecture notes: Discuss handout_day3.pdf, orthogonal functions and bra-ket notation, linear combinations of degenerate wavefunctions. S: Ch. 2, pp. 21-25 | - |

## Week 2

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

4 | Mon 4/6 | QM formalisms | day4.pdf lecture notes: Freedom to choose a different basis. Projecting a wave fuction onto basis functions. Example with 2py wave function. Matrix representation of operators. S: Ch. 2, pp. 21-25 | |

5 | Wed 4/8 | electron and 2 protons | day5.pdf lecture notes: Forming a bonding orbital and an antibonding orbital using a linear combination of 1s orbitals. Orbital energies. The hopping integral quantifies energy splitting. S: Ch. 2, pp. 25-31 | hw2.pdf |

6 | Fri 4/10 | day6b.pdf lecture notes: Definition of on-site integral and hopping integral. Description of an ionic bond in the QM formalism. Discussion of sigma bonds and pi bonds. |

## Week 3

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

7 | Mon 4/13 | Molecular orbitals, Linear chain | day7.pdf lecture notes: Energy level diagrams for molecular orbitals. s-p hybrid atomic orbitals. Qualitative picture of molecular orbitals for a chain of atoms. Solving the N = 4 chain of atoms. | |

8 | Wed 4/15 | Molecular orbitals in 1d | day8.pdf lecture notes: N = 4 chain of atoms. Periodic boundary conditions. The four molecular orbitals, understanding the molecular orbital coefficients the energies of the orbitals. moleculear_orbitals_for_ring.pdf | hw3.pdf |

9 | Fri 4/17 | Molecular orbitals in 1d. Crystal structures | day9b.pdf lecture notes: “m” counts the phase oscillations around the ring. A limited range of m describes the unique molecular orbitals. Concept of first Brillioun zone. Wave number k. In limit of large N, E(k) becomes a band structure diagram. Second half of class: describing_crystal_structure.pdf |

## Week 4

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

10 | Mon 4/20 | 2d Band structure | Pop quiz. day10.pdf lecture notes: Bloch's theorem. Band structure of a 2d periodic structure, energy vs. k-vector. | |

11 | Wed 4/22 | 2d Band structure, Density of states | day11.pdf lecture notes: Dispersion relation, E(k), assuming nearest neighbor hopping. Representing the molecular orbitals as dots in k-space. Special points in k-space called M, gamma, X. Density of states in 1d and 2d. | |

12 | Fri 4/24 | Brillioun Zone | moleculear_orbitals_for_grid_multi_choice.pdf. day12.pdf lecture notes: Pop quiz about 1st Brillioun Zone. How to construct the 1st Brillioun Zone. Example: The 2d hexagonal Bravais lattice. | 2014 Midterm McIntyre Chpt 15 problems |

## Week 5

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

13 | Mon 4/27 | Filling states | day13.pdf lecture notes: 1st BZ for a 3d system. State filling. Fermi energy and Fermi surface. Using density of states to figure relationship between E_Fermi and number of electrons in the system. Effective mass. | |

14 | Wed 4/29 | Review | day14.pdf lecture notes: Bookshelf analogy for density of states. density_of_state_questions.pdf | |

15 | Fri 5/1 | MIDTERM | hw4_.pdf |

## Week 6

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

16 | Mon 5/4 | day16.pdf lecture notes: Group velocity describes movement of electron wavepacket. Force changes the crystal momentum. Bloch oscillations. Scattering interrupts Bloch oscillations and explains net flow of current. F = ma can be applied to electrons in crystals, if you calculate m_eff correctly. S: Ch4 p80-86 | ||

17 | Wed 5/6 | pop quiz. day17.pdf lecture notes: Kinematics for electrons/holes in a crystal. DC conductivity. Comments about Si band structure. | homework | |

18 | Fri 5/8 | day18.pdf lecture notes: Pop quiz about graphene band structure. Classification of insulators, semiconductors and metals. Example calculation: estimate the conductivity of intrinsic silicon. Fermi-Dirac statistics: fractional occupancy of states. |

## Week 7

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

19 | Mon 5/11 | day19.pdf lecture notes: Pop quiz: finding the chemical potential in doped silicon. Optical properties: Conservation of energy and momentum in the absorption process. Calculating the absorption coefficient of a direct band gap semiconductor. | ||

20 | Wed 5/13 | Optical properties | day20.pdf lecture notes: Transition rate for direct absorption: Directly-associated density of states, Fermi's Golden Rule. Visualizing a direct transition in real space / k-space handout. How thick should I make a GaAs solar cell? Comments on transition matrix element and selection rules. Translating from complex dielectric constant to absorption coefficient. | |

21 | Fri 5/15 | Optical properties | day21.pdf lecture notes: Indirect transitions. |

## Week 8

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

22 | Mon 5/18 | Link to Figures. Electron density maps. Electrons per cubic Bohr. The special character of d-bands: Relationship between d-bands and color (like the color of gold). Relationship between magnetism and partially-filled d-bands. day22.pdf | ||

23 | Wed 5/20 | POSTER SESSION | ||

24 | Fri 5/22 | Dielectric constant | day24.pdf lecture notes: Dielectric constant: the connection between a sharp absorption peak at ω_0 and epsilon at all other frequencies. Electron-on-a-spring model. Estimating dc dielectric constant of semiconductors and insulators. Limitations of band theory as implemented by Wien2k. Brief discussion of Mott Insulators. |

## Week 9

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

Mon 5/25 | MEMORIAL DAY | |||

25 | Wed 5/27 | Nearly free electron model | Pop quiz about square muffin tins (see handout). day25.pdf lecture notes: Calculation methods to find self-consistent potentials. Begin discussion of free electron model (see day 26) | term paper is due, hw5 |

26 | Fri 5/29 | Nearly free electron model | day26.pdf lecture notes: Positively charged jelly. Free electron wavefunctions with periodic boundary conditions. Fermi energy relative to bottom of conduction band. Estimate E_F. Band structure inside the 1st Brillioun Zone. The “primitive unit” of the Block wavefunctions describing free electrons. Compare to real materials. Compare to LCAO method. |

## Week 10

Day | Topic | What was covered | Assignments | |
---|---|---|---|---|

27 | Mon 6/1 | Graphene | day27.pdf lecture notes: Dispersion relation for graphene, handout | |

28 | Wed 6/3 | ionic_crystals.pdf, reduce-zone_scheme.pdf | hw5 due | |

29 | Fri 6/5 | Review |

**About the image:** The image in the upper-left corner comes from the group of Kyle Shen at Cornell University. Shen's group uses state-of-the-art growth techniques, such as oxide molecular beam epitaxy, to create next generation quantum materials with atom-by-atom precision.