### Table of Contents

# PH 575 Introduction to Solid State Physics, Spring 2017

**Instructor:**(Spring 2017) Prof. Ethan Minot, Office: Weniger 417. Office hours Tuesdays 1-3pm and by appointment.**TA:**(Spring 2017) Mitchell Senger**Textbook:**See course information**Class Meetings:**MWF 3.00pm - 3.50pm, Weniger 304

## Week 1

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

1 | Mon 4/3 | Atomic orbitals | Advertisement for solid state physics intro_presentation2017.pdf. Set up key questions: How electrons are distributed in space, energy and momentum. Starting point: Atomic orbitals. “Tutorial” based on hydrogen atom - develop the mathematical language for wave functions, and some visual representation methods day_1_2017.pdf, poster_for_ph575.pdf | Complex number practice |

2 | Wed 4/5 | Atomic orbitals | Quiz question about visualizing an atomic orbital quiz_1.pdf. The analytical expressions for the first 25 H-atom orbitals day2_handout.pdf. Discuss patterns within the set of wavefunctions. Group wavefunctions together that have the same orbital angular momentum (s, p, d, f etc.). Orbital angular momentum is related to degeneracy. Group wavefunctions together that have similar energy. Define the quantum numbers n, l and m. day_2_2017.pdf | hw1 updated |

3 | Fri 4/7 | Atomic orbitals | The patterns within the set of H-atom wavefunctions help us develop chemical intuition. Hybridization of degenerate wavefunctions. Introduce operator notation for Schrodinger equation. day_3_2017.pdf quiz_3.pdf |

## Week 2

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

4 | Mon 4/10 | Atomic orbitals | Orthogonal wavefunctions. Bra-ket notation. Matrix representation of an operator. quiz_2.pdf, day_4_2017.pdf | hw1 question 5 hint |

5 | Wed 4/12 | Molecular orbitals | Vector notation to represent wavefunctions. Two protons and an electron. Finding the ground state wavefuction. day_5_2017.pdf, quiz_4.pdf | hw1 soln |

6 | Fri 4/14 | Molecular orbitals | Nomenclature: overlap integral, on-site integral and hopping integral. Definite integral expressions for on-site integral and hopping integral. Discuss sign and magnitude. Nomenclature bonding orbital, anti-bonding orbitals, symmetric combination, anti-symmetric combination. The idea of constructive/destructive interference when the atomic orbitals in an LCAO state overlap. The idea of bonding orbitals with enhanced electron density between the nuclei. day_6_2017.pdf, quiz_5.pdf | hw2 updated |

## Week 3

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

7 | Mon 4/17 | Molecular orbitals | Practice translating from bra-ket notation to an energy. Dipole nature of water explained by on-site integrals. Sigma bonding/antibonding. Pi bonding/antibonding. Solving the full S. Eqn to demonstrate hopping behavior and the relationship to the hopping integral. day_7_2017.pdf, quiz_6.pdf | |

8 | Wed 4/19 | The chain of atoms | Introduction: the energy level diagram of an extended solid. Chain of 4 protons. Periodic boundary conditions. The elegant solutions when periodic boundary conditions are assumed. Modeling the four lowest-energy wavefunctions using playdough. day_8_2017b.pdf, quiz_7.pdf | hw2soln_2017.pdf |

9 | Fri 4/21 | The chain of atoms | Assigning each LCAO wavefunction a momentum value. Plotting energy as a function of momentum. Generalizing the LCAO solution for a chain of any length. day_9_2017.pdf, quiz_8.pdf | hw3 |

## Week 4

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

10 | Mon 4/24 | The chain of atoms | Quiz question: Phase-colored maps of wavefunctions shown on screen, determine k. Define the first Brillouin zone. Define density of state in 1d. day_10_2017.pdf, moleculear_orbitals_for_chain.pdf | |

11 | Wed 4/26 | 2d crystals | Integrating density of states. quiz_9b.pdf. Small group activity with tessellating paper tiles. Define the primitive unit cell. Define the Bravais lattice. Define the primitive lattice vectors. day_11_2017.pdf | hw3_solutions.pdf |

12 | Fri 4/28 | 2d crystals | Symmetries of the different crystal families. Constructing an LCAO state for a 2d crystal. Visit the computer lab, 412 Weniger, to start using OpenMX. day_12_2017.pdf | hw4_2017b.pdf |

## Week 5

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

13 | Mon 5/1 | 2d crystals | Analytical expression for the dispersion relation of a 2d crystal when there is one atom per PUC and one atomic orbital per atom. Applying periodic boundary conditions to a 2d crystal. The density of state of a 2d crystal when wavefunction energy depends on magnitude of k (and not direction). day_13_2017.pdf, k-states_in_2d.pdf, quiz_10.pdf | |

14 | Wed 5/3 | 2d crystals | The first Brillouin zone for a 2d square lattice. Selecting a path through the first Brillouin zone for the purposes of band dispersion relation calculations. Definition of the reciprocal lattice. day_14_2017.pdf, quiz_11.pdf | hw4 soln |

15 | Fri 5/5 | The 1st Brillouin Zone | Determine primitive reciprocal lattice vectors for triangular 2d lattice. Construct 1st BZ for a rectangular 2d lattice. Construct 1st BZ for an fcc 3d lattice. day_15_2017.pdf, quiz_12.pdf | Practice: midterm_2016.pdf |

## Week 6

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

16 | Mon 5/8 | Graphene and diamond | Predicting the number of bands within an energy range of a dispersion relation. Calculating the dispersion relation for the two bands in graphene that are formed from 2pz orbitals (an example of band structure calculations when there is more than one atomic orbital in a primitive unit cell). day_16_2017.pdf, graphene dispersion relation | |

17 | Wed 5/10 | midterm | ||

18 | Fri 5/12 | Bloch Theorem | Results from the midterm. Dispersion relation that includes next-nearest neighbor interactions. Features of Bloch's theorem: Electron probability density has same periodicity as the lattice. Energy eigenstates can be written using phase factor exp(ik.r) day_18_2017.pdf quiz_14.pdf | hw5_2017.pdf |

## Week 7

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

19 | Mon 5/15 | Nearly-free electron model | LCAO/Bloch Video of proof. Free electron wavefunctions also satisfy the Bloch theorem. Free electron wavefunctions can be forced into the first Brillouin zone (the reduced-zone scheme). A simple dispersion relation looks much more complicated when graphed in the reduced-zone scheme. There is a beautiful evolution of band structure if we change from free electrons to tightly bound electrons (LCAO wavefunctions). New topic: Quiz question about calculating the Fermi energy. day_19_2017.pdf, quiz_15.pdf | |

20 | Wed 5/17 | Simple metals | Wine glass analogy for calculating Fermi energy. Another approach to finding Fermi energy (using k_F). Apply the k_F approach to calculate the Fermi energy of sodium, magnesium and aluminum, in units of electron volts. day_20_2017.pdf, quiz_16.pdf | hw5_solutions.pdf |

21 | Fri 5/19 | Interpreting band structures | The atomic orbitals in a primitive unit cell should be consistent with number of bands. Partial density of states will show you the character of the Bloch wavefunctions at a given energy. Deciding where the Fermi energy will lie. Metal vs. insulator/semiconductor. How doping affects the Fermi energy. The effective mass model for predicting how free carriers will behave in a doped semiconductor. day_21_2017.pdf. Crystallography Open Database | Work on class project |

## Week 8

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

22 | Mon 5/22 | Interpreting band structures | Practice calculating effective mass for a dispersion relationship that has an analytical formula. The importance of effective mass in material properties (comparison of Si and GaAs). Resistivity of materials. The role of effective mass, scattering time and carrier concentration. day_22_2017.pdf, quiz_17.pdf | |

23 | Wed 5/24 | Interpreting band structures | Practice calculating mobility and resistivity with realistic numbers. Discuss the correspondence between quantum description of electron transport and classical description. Optical properties of metals. day_23_2017.pdf, quiz_18.pdf | |

24 | Fri 5/26 | Interpreting band structures | Examples of crystals with fcc Bravais lattice and 2 atoms per PUC. Optical properties of semiconductors. Energy and momentum conservation. Selection rules. Example calculation using joint density of state and Fermi's golden rule. Magnetic properties. The role of exchange interaction and competition with kinetic energy. Ferromagnetism occurs in partially-filled bands with large effective mass. day_24_2017b.pdf |

## Week 9

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

Mon 5/29 | MEMORIAL DAY | no class | ||

25 | Wed 5/31 | Poster session | ||

26 | Fri 6/2 | Interpreting band structures | Light absorption by a direct gap semiconductor. Typical values for the transition matrix element. Relationship between optical transition rate and the absorption coefficient. day_25_2017.pdf quiz_19.pdf |

## Week 10

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

27 | Mon 6/5 | DFT calculations | Color centers in insulators (why rubies are red). Introduction to density functional theory. The calculation loop that seeks a self-consistent electron density. Methods to determine the effective potential from the electron density. The Hartree term. The exchange-correlation term. The LDA method and the GGA method. day_26_2017.pdf, quiz_20.pdf | |

28 | Wed 6/7 | Indirect optical transitions | Definition of indirect transition. Silicon is an example of a material with an indirect gap. Phonon-absorption process and phonon-emission process. Compare the absorption coefficient spectrum of silicon and GaAs. The thickness of solar absorbing layers. Introduction to BCS theory of superconductivity. Cooper pair wavefunction, size and binding energy. day_27_2017.pdf, day_27_superconductivity.pdf | term paper due |

29 | Fri 6/9 | Review | quiz_21.pdf |

## Final Exam

Day | Time |
---|---|

Tues 6/13 | 2pm - 4pm |

links to previous years: 2015 and 2016

**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.