Exam Study Practice

1. 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 on your own. • Equations given: See Formula Posted

2. Exam Study Questions for Crystals • What is a crystal? • What is a lattice? • What are the lattice primitive translation vectors? • What is an atomic basis? • What is a unit cell? • What is the primitive unit cell? • Be able to point out a unit cell, give the atomic basis, generate a lattice with translation vectors, write the translation vectors for a lattice, give a primitive unit cell.

3. Exam Study Questions for Quantum Effects • What is the photo-electric effect, why is it important? • What is a photon? • What is the energy of a photon? • What increases when light intensity? (brightness) increases? • Why do atoms emit light? • Why do atoms emit light at very specific frequencies? • What is the Bohr Atom, what did it explain? • What two forces are equated to analyze the Bohr atom? • What physical quantity did Bohr quantize (limit to specific values)? • Why did the Bohr atom not predict? • What fundamental equation of QM can predict the hydrogen atom fine emission spectrum.

4. Exam Study Questions for Quantum Effects • Be able to separate the original time-space partial differential SWE into two separate Eigenvalue equations, one for space and one for time (just like we did in class)? • What must be true about the potential in the SWE to use separation of variables. • What is the solution to the SWE? • What is the relationship between the wave function and the probability of finding a particle? • What are the momentum and position operators in QM? • Be able to calculate the expected values of position and momentum. • What does the Heisenberg Uncertainty Principle say? How is it related to the ideas of probability and the wave-function? • Explain particle tunneling. Can it be predicted using classical or QM, why?

5. Important Concepts on Energy Bands for Crystalline Solids: Including Semconductors • Semiconductors and Metals are crystalline solids. • Electrons exist only in allowed bands of energy. • Bands are composed of many quantum states. • Each band has approximately the same number of states as there are atoms in crystal. • Bands that are completely filled with electrons cannot conduct electricity • In semiconductors, at T=0oK, all bands are either totally full or totally empty. • The band with highest energy that is totally full is call the valence band (VB). • The next highest band in energy is the conduction band (CB). • In semiconductors there is an energy gap between the valence and conduction bands • At T > 0oT, some electrons obtain enough energy to jump the gap and exist in the conduction band. • This process makes the conduction band partially full, and the valence band partially empty. Therefore, both of these bands can conduct at T > 0oT

6. Important Concepts on Energy Bands and Conduction for Crystalline Solids: Including Semconductors • States in the VB where electrons have left from are now empty. These empty states are called holes. • Mathematically, holes act like positively charged electrons and are treated as such. • The most popular semiconductor material is Silicon which composes more than 95% of our chips. • Pure silicon is also called intrinsic silicon. • In 1 cubic cm of intrinsic silicon at room temperature there are about 1010 electrons in the conduction band and 1010 holes in the valence band. • In a metal at room temperature the CB is typically ½ full meaning that there are about 1023 conduction electrons/cm3 • Therefore, the conductivity of a metal is approximately 1013 greater than that of a semiconductor at 300oK (recall current density: J=qnv).

7. Important Concepts on Energy Bands and Conduction for Crystalline Solids: Including Semconductors • We usually want to increase and control the conductivity in semiconductors. We achieve this with doping. • With doping we substitute impurities for silicon atoms into the lattice that give rise to mobile electron in the CB or mobile holes in the VB. • In N-type material, the impurities give rise to electrons in the CB • In P-type material, the impurities give rise to holes in the VB. • N-type doping works as follows in Si: Silicon has 4 valence electrons. Phosphorous, which has 5 valence electrons, is substituted for a Si atom, leaving one of the 5 electrons essentially unbound. This electron enters the CB and can now be conducting. • P-type dopants have 3 valence electrons. They accept electrons from the VB, leaving mobile holes in VB, which can now be conducting. • Doping levels in semiconductors typically range from 1014 to 1019/cm3 • Adding 1017 phosphorous atoms to Si will make it 107 times more conductive than intrinsic Si. • Doped semiconductors are called extrinsic semiconductors. • The fact that we can dope semiconductors to selectively control their conductivity allows for microelectronics to exist as we know it.

8. Important Concepts on Energy Bands and Conduction for Crystalline Solids: Including Semconductors • By studying the band structure (QM states) we understand the doping and conduction possibilities of a semiconductors. • The band structure also tells us the instantaneous (group) velocity of an electron (by the slope Vg=(1/hbar)(dE/dk) • Band structure tells us effective mass , 1/m* = (1/hbar)2(d2E/d2k), by the curvature of the band near the minium. • The effective mass and group velocity help to account for the quantum effects of crystal without having to solve the Schrodinger equation for every application. • Mobility is another very important concept in semiconductors. It tells you the average velocity of a conduction electron (or hole) in the presence of an electric field. Vavg= mobility x field (mobility = μ= qτ/m*) • The mean free time between collisions is τ, which is also obtained using the band structure.

9. Important Concepts on Energy Bands and Conduction for Crystalline Solids: Including Semconductors • Another important concept in semiconductors is the use of Fermi-Dirac Statistics (Fermi statistics for short). • The Fermi function tells the probability that a quantum state is occupied by an electron. • Fermi statistics are often used to calculate the number of electrons and holes in the CB and VB, respectively. • Fermi statistics are strickly only applicable during equilibrium conditions (when no current is flowing). • The Fermi Level is an important parameter in Fermi statistics. The Fermi Level is the energy where the probability that is state is occupied = ½.