Band Theory of Electronic Structure in Solids

Slides:



Advertisements
Similar presentations
Free Electron Fermi Gas
Advertisements

ELECTRICAL CONDUCTIVITY
LECTURE 2 CONTENTS MAXWELL BOLTZMANN STATISTICS
Lectures Solid state materials
Energy Band View of Semiconductors Conductors, semiconductors, insulators: Why is it that when individual atoms get close together to form a solid – such.
CHAPTER 3 Introduction to the Quantum Theory of Solids
P461 - Solids1 Solids - types MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van der.
Exam Study Practice Do all the reading assignments. Be able to solve all the homework problems without your notes. Re-do the derivations we did in class.
1 Motivation (Why is this course required?) Computers –Human based –Tube based –Solid state based Why do we need computers? –Modeling Analytical- great.
Band Theory & Optical Properties in solids
Laser Physics I Dr. Salah Hassab Elnaby Lecture(2)
AME Int. Heat Trans. D. B. GoSlide 1 Non-Continuum Energy Transfer: Electrons.
Project topics due today. Next HW due in one week
SEMICONDUCTOR PHYSICS. BAND THEORY OF SOLIDS  Ge and Si are pure semiconductors  Electronic configuration of Si is  1S 2, 2S 2, 2P 6, 3S 2, 3P 2.
Physics 355. Consider the available energies for electrons in the materials. As two atoms are brought close together, electrons must occupy different.
Chapter 6: Free Electron Fermi Gas
Microscopic Ohm’s Law Outline Semiconductor Review Electron Scattering and Effective Mass Microscopic Derivation of Ohm’s Law.
ECE 4339 L. Trombetta ECE 4339: Physical Principles of Solid State Devices Len Trombetta Summer 2007 Chapter 2: Carrier Modeling Goal: To understand what.
1 Free Electron Model for Metals Metals are very good at conducting both heat and electricity. A lattice of in a “sea of electrons” shared between all.
UNIT 1 FREE ELECTRON THEORY.
ECE 340 Lecture 6 Intrinsic Material, Doping, Carrier Concentrations
EEE 3394 Electronic Materials
Electronic Bandstructures Information from Kittel’s book (Ch. 7) + many outside sources. Some lectures on energy bands will be based on those prepared.
SOLIDS AND SEMICONDUCTOR DEVICES - I
Free Electron Model for Metals
1 Lecture VIII Band theory dr hab. Ewa Popko. 2 Band Theory The calculation of the allowed electron states in a solid is referred to as band theory or.
Last Time The# of allowed k states (dots) is equal to the number of primitive cells in the crystal.
Topic #1: Bonding – What Holds Atoms Together?
Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch
BASICS OF SEMICONDUCTOR
المملكة العربية السعودية وزارة التعليم العالي - جامعة أم القرى كلية الهندسة و العمارة الإسلامية قسم الهندسة الكهربائية ELECTRONIC DEVICES K INGDOM.
1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2012 DEE4521 Semiconductor Device Physics Lecture.
Physics of semiconductor devices Nihar R. Mahapatra.
Origin of energy band formation:
Condensed Matter Physics: Quantum Statistics & Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics)
Band Theory of Electronic Structure in Solids
Question on Van der Waals Interactions
Electrical Engineering Materials
Metallic Solids Metallic bond: The valence electrons are loosely bound. Free valence electrons may be shared by the lattice. The common structures for.
Announcements Added a final homework assignment on Chapter 44, particle physics and cosmology. Chap 42 Homework note: Binding energy is by convention positive.
Manipulation of Carrier Numbers – Doping
Do all the reading assignments.
Prof. Jang-Ung Park (박장웅)
Band Theory of Electronic Structure in Solids
Schrödinger's Cat A cat is placed in an airtight box with an oxygen supply and with a glass vial containing cyanide gas to be released if a radiation detector.
SEMICONDUCTORS Semiconductors Semiconductor devices
Insulators, Semiconductors, Metals
Tightbinding (LCAO) Approach to Bandstructure Theory
Band Theory of Solids So far we have neglected the lattice of positively charged ions Moreover, we have ignored the Coulomb repulsion between the electrons.
Elementary Particles (last bit); Start Review for Final
Polytetrafluoroethylene
Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch
Condensed Matter Physics: review
Band Theory The other approach to band theory solves the Schrodinger equation using a periodic potential to represent the Coulomb attraction of the positive.
Free electron theory of metals (ctd)
Physics 342 Lecture 28 Band Theory of Electronic Structure in Solids
Condensed Matter Physics: Quantum Statistics & Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics)
Lecture 2 OUTLINE Semiconductor Fundamentals (cont’d)
Band Theory of Solids 1.
Elementary Particles (the last bit)
More Wave Equation Solutions Leading To Energy Bands 23 and 25 January 2017.
More Wave Equation Solutions Leading To Energy Bands 3 and 5 February 2014.
ECE 340 Lecture 6 Intrinsic Material, Doping, Carrier Concentrations
Spin quantum number – ms
PHY 752 Solid State Physics Plan for Lecture 30: Chap. 13 of GGGPP
Energy Band 7 In free electron model, electrons occupy positive energy levels from E=0 to higher values of energy. They are valence electron so called.
More Wave Equation Solutions Leading To Energy Bands 2 and 4 February 2015.
Energy Band View of Semiconductors
More Wave Equation Solutions Leading To Energy Bands 30 January And 1 February 2019.
Conduction of Electricity in Solids
Presentation transcript:

Band Theory of Electronic Structure in Solids Read: Chapter 10 (statistical physics) and Chapter 11 (solid-state physics) Homework due Wednesday Nov. 11th Chapter 10: 1, 3, 6, 9, 11, 20, 23, 25, 29, 35 Reminder: Exam 2 on Monday, November 16th

Band Theory of Solids To understand the electronic states of a solid we must consider the presence of many N~1023 atoms The band theory of solids describes the interaction between the electrons and the lattice ions that comprise a solid

Band Theory: “Bound” Electron Approach For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E. Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level). Two atoms Six atoms Solid of N atoms Electrons must occupy different energies due to Pauli Exclusion principle. Phys 320 - Baski

Band Theory of Solids Consider the potential energy of a 1-dimensional solid which we approximate by the Kronig-Penney Model

Band Theory of Solids The task is to compute the quantum states and associated energy levels of this simplified model by solving the Schrödinger equation 1 2 3

Band Theory of Solids For periodic potentials, Felix Bloch showed that the solution of the Schrödinger equation must be of the form and the wavefunction must reflect the periodicity of the lattice: 1 2 3

Band Theory of Solids By requiring the wavefunction and its derivative to be continuous everywhere, one finds energy levels that are grouped into bands separated by energy gaps. The gaps occur at The energy gaps are basically energy levels that cannot occur in the solid 1 2 3

Band Theory of Solids Completely free electron electron in a lattice

Band Theory of Solids When, the wavefunctions become standing waves. One wave peaks at the lattice sites, and another peaks between them. Ψ2, has lower energy than Ψ1. Moreover, there is a jump in energy between these states, hence the energy gap

Band Theory of Solids The allowed ranges of the wave vector k are called Brillouin zones. zone 1: -p/a < k < p/a The theory can explain why some substances are conductors, some insulators and others semi conductors

Fermi-Dirac “Filling” Function Probability of electrons to be found at various energy levels. Important factoid: 300K (room temp) = 25.86meV Temperature dependence of Fermi-Dirac function shown as follows:

Conductors, Insulators, Semiconductors Sodium (Na) has one electron in the 3s state, so the 3s energy level is half-filled. Consequently, the 3s band, the valence band, of solid sodium is also half-filled. Moreover, the 3p band, which for Na is the conduction band, overlaps with the 3s band. So valence electrons can easily be raised to higher energy states. Therefore, sodium is a good conductor

Conductors, Insulators, Semiconductors NaCl is an insulator, with a band gap of 2 eV, which is much larger than the thermal energy at T=300K Therefore, only a tiny fraction of electrons are in the conduction band

Conductors, Insulators, Semiconductors Silicon and germanium have band gaps of 1 eV and 0.7 eV, respectively. At room temperature, a small fraction of the electrons are in the conduction band. Si and Ge are intrinsic semiconductors

Summary In solids, the discrete energy levels of the individual atoms merge to form energy bands Energy gaps arise in solids because they contain standing wave states The size of the energy gap between the valence and conduction bands determines whether a substance is a conductor, an insulator or a semiconductor