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# URBAN GOVT & POL POL S 162

UCSB

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This 21 page Class Notes was uploaded by Johnathan McKenzie on Thursday October 22, 2015. The Class Notes belongs to POL S 162 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/226998/pol-s-162-university-of-california-santa-barbara in Political Science at University of California Santa Barbara.

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Date Created: 10/22/15

Lecture 9 ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 1 Reading Material f f Reading Finish Chapter 7 Solymar and Walsh ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 2 Effective Mas s gt In a one dimensional lattice the effective 39 1 w mass can be written as 2 62E m h ak2 hz 072E1 2Acoska 71 T h2 msecka gt Plots of E vs Vg and m are shown to the quot right Note that as an electron is moved from rest to higher and higher velocities its mass increases reaching in nite mass at krc2a 7m u A ECE162B Winter 2009 Professor Blumen nal Lecture 9 Slide 3 Effective Mass gt For a 3D crystal we can extend these results to propagation in X y or z hz 2A 12 seckxa x hxz sec k b 1 2Ayb2 y 2 sec kzc 2 gt An even in arbitrary directions eg the xy plane ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 4 gt In order to derive a formula for the effective number of free electrons lets take the one dimensional case and use an approach similar to What we used in the effective mass case The acceleration of the electron can be given by e q e dt hz 362 gt Summing over all electrons in occupied states and considering the rate of change of electric current in response to an applied electric eld E 2 dqvg dt dt 2 1 quez a E and considering the limit Where the density of states in the range dk is 1 W 3k 1 2 1 WE F dk 712 enfakZ ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 5 Effective Number of Free Electrons gt Assuming N non interacting electrons 2 Q N dt m gt Solving for N and using the electron acceleration result at TO we can de ne the effective number of free electrons E m e dt qZFe 2 l m d E dk n h2 dk2 ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 6 Effective Number of Free Electrons gt At T20 all states up to some energy Ea are lled and all states above are empty If Ea is inside the energy band the we integrate frorn k ka to kzka l m dE dE Ne 2 n h dk kkd dk kka 3 dE m2 dk M gt Which tells us that the effective number of electrons that can contribute to electrical conduction depends on the slope of the E k curve At high energy levels in the banbd dEdk tends to zero and the number of effective electrons for a full band is ZCFO ECE162B Winter 2009 Professor Blumenthal Lecture 9 Slide 7 Origin of SC Bandgap gt The discrete energy states of Discrete eleCtI OnS In an atOm are SEESEY energy levels E broadened from the A 1nd1V1dual atom states 1nto energy bands as the atoms are brought closer together to form a crystal gt When the bandgap energy is on the order of 2eV room temperature thermal energy can break covalent bonds and accelerate ionized electrons into empty states in the crystal Where they can conduct electrically gt Increasing atomic separation Semiconductors ECE162C Spring 2006 Professor Blumenthal Lecture 2 Slide 8 Lattice mismatch to silicon 5 0 5 IO 15 25 40 I I l I I o 39 ZnS 04 E 3390 ZnSe E a AIP O CdS Z T 05 3 g AIAS 0 r56 5 E 20 Gap CdSe 39 g 3 Q Ale CdTe a 3 O 39 gt E r GHAQ quot A n E 0 IU s m 10 5qu I 39 15 ISLE EIGe 20 I InSb 30 GB 4 HgTe39 50 O I I l J I 55 50 55 Lattice constant A ECE162B Winter 2009 Professor Blumenthal Lecture 1 Slide 9 Equilibrium gtEquilibrium gt Where is the minimum energy distance for two oppositely charged particles Attractive forces V H a Repuls1ve forces ECE162B Winter 2009 Professor Blumenthal E r Lecture 2 Slide 10 B ulk Elast1c Constant of a Sol1d II I H I 7 r gt If each atom distance compresses Ar then the total change in linear dimension is aro Ar and each face compresses by 12 aro Ar and the total work done to compress the material is 6 a Total Work Done on Material E T112 0 V From the Taylor expansion we can equate the total energy increase and the work done to solve for T T i 3 52E rz y gt The bulk elastic modulus c is de ned as Tamale ECE162B Winter 2009 ProfessorBlumenthal Lecture 2 Slide 11 Ionic Bonds gt How much energy would it take to take this crystal apart Cub1c Latt1ce Crystal gt This is the Cohesive Energy Ec gt Summing the electrostatic forces of the crystal gt 6 Cl ions bonded to each Na ion at distance a 6 q l 4n 0a gt 12 Na ions bonded to each Na ion at a distance a sqrt2 12 1 12 77 47120 L115 gt 8 Cl ions bonded to each Na ion at a distance a sqrt3 2 2 2 gt Keepgoing E 6Li 12 1 8L 47120 a 47120 am 47120 5113 47150 12 6 ii a 015 015 2 EC Mai ECE162B Winter 2009 Professor Blumenthal 47550 Lecture 2 Slide 12 Metallic Bonds gt A metal can be thought of as a lattice of thermally Vibrating atoms With a sea of free conducting electrons scattering off each other and the xed atoms gt But these electrons are con ned for the most part to the metal and we have to do some work to remove them from the metal gt The metallic bond is similar to the ionic bond in that electrostatic forces generated by the moving electrons play a key role in keeping the lattice together ECE162B Winter 2009 Professor Blumenthal Lecture 2 Slide 13 Covalent Bond gtIn general the Covalent bond is very strong gt Electrons are not easily removed bond broken or free to conduct gt Insulator gtIn some material systems compound the covalent bond is more easily broken and the electrons freed by breaking this bond can conduct gt Semiconductor ECE162B Winter 2009 Professor Blumenthal Lecture 2 Slide 14 Van der Waals Forces gt What happens When we have an electrically neutral atom not ionized and there is no attraction to neighboring electrons gt How do neutral atoms form bonds gt The electron position density function uctuates in time forming a displacement between the electron and nucleus gt This is called an oscillating dipole gt One dipole Will generate a eld static or dynamic that Will distort nearby atoms into becoming dipoles like oppose opposites attract gt Tendency on average is for attraction or the Van der Waals Force ECE162B Winter 2009 Professor Blumenthal Lecture 2 Slide 15 Feynman s Coupled Mode Approach gt Richard Feynman Physicist Nobel Laureate see Feynman Lectures on Physics Addison Wesley Longman ISBN 13 978 0201021158 gt Developed an approach that can be used to describe all physical systems that are coupled gt Sometimes called coupled mode theory gt Other examples besides pendulums include electrical oscillators atoms acoustics and Vibrational systems etc gt Step 1 Write a model for each part of the system by itself gt Step 2 Introduce a coupling factor in each equation that describes how each part of the system interacts With the other parts of the system gt This couple can be in time space time and space or other manner ECE162B Winter 2009 Professor Blumenthal Lecture 3 Slide 16 Coupled Schrodinger s Equations gtLets consider the simplest case Where there are two solutions jl and j2 gt Our two Diff equations Will look like m d6010 dt m d6020 dt H11w1t Er quot 74 H22602t ECE162B Winter 2009 Professor Blumenthal Lecture 3 Slide 17 COUPICdStateS o gt As the two are brought together from in nity only one solution has attractive and repulsive forces balance out to a stable solution gt In the time dynamic solution the electron can jump from one nucleus to the other and a sharing can occur forming a bond E r S gt Only A stable minimum gt E0 A ECE162B Winter 2009 Professor Blumenthal Lecture 3 Slide 18 The Free Electron Model of rMetals W gt Now lets look at a cube With sides L that contains the electrons The electron energy can be written as E2 2 2 2 2 K x Ky KZ 2 2 2 2 8 L2nxnynz m nxnynZ l23 ECE162B Winter 2009 Professor Blum enthal Lecture 4 Slide 19 gt Assume that all electrons With energy EF 22kBTs E s EF contribute to the speci c heat and can be treated as classical electrons ie there are plenty of available states to excite them into gt Assume each classical electron has an average energy 32kBT then the average energy of the electrons that can contribute too the SH and the resulting speci c heat of these electrons is ivenb g y ltEgt3KBT2KBT F d E CV 1 EMMA 669231 dT dT 2 EF E F ECE162B Winter 2009 Professor Blumenthal Lecture 5 Slide 21

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