ElectronicMaterials - Lecture - Brillouin Zones, Lecture notes of Engineering Physics

Download ElectronicMaterials - Lecture - Brillouin Zones and more Lecture notes Engineering Physics in PDF only on Docsity! Brillouin zones Above are a number of views demonstrating the lifting of the degeneracy on the Bragg plane, and the creation of the energy gap. The upper left picture represents the parabolic energy due to a perfectly-free electron around the k-space origin. The next picture to the right shows two intersecting parabolae about two subsequent atoms in k-space. Notice the parabolae intersect halfway between the two atoms. The degeneracy at the Bragg plane halfway-point is lifted because the electrons are MOSTLY free but not ALL free. Redrawn to the right, getting rid of the duplicate information, the original parabola, as shown in the left half, gets a discontinuity on the right half at the Bragg plane. note that nearby the plane the energy curves into the Bragg plane. This is similar to the Fermi Surface from Slide #17. The bottom three pictures show three ways of presenting the same information. The Extended-Zone Scheme shows the parabolic shape of electron states extending out over many one-dimensional Brillouin Zones, with the appropriate gaps at the Bragg planes. The Reduced-Zone Scheme gives the same information, but with all the higher- order Brillouin zones folded over into the first zone. This portrayal of the band structure is usually the one most often used. Finally, the Repeated-Zone Scheme shows the same information of the Reduced- Zone Scheme repeated over several Brillouin zones. It is important to note that these three views all display the same information equally, they are just useful in different circumstances for presenting the band structure of the material. Quick and Dirty Preview of Solid State Physies The Fermi Energy and the Fermi Surface Fermi energy represents the sharp ocoupancy cutoff al T=0 for Ferm-Dirac syslems Ferm: surface is the locus of points in reciprocal space where the k- dependent energy is equal to lhe Fermi energy » Ferri surfaces become essential for study ind-D cases toremain sane = eee JOHNS HOPKINS Electrical Conduction + Ohm's Law AVEIR voltage drop (voltsy” A N. esista nce (ohms) current (amps) A - (CrOSS]~> sect. ¢——_ | area} | ay “ —\__—— * Resistivity, pand Conductivity, « -Geometry independent forms of Ohm's law AV UI L A Material Parameters: A “resistivity conductivity . J RA 1 -4 elect current —— (Ohmm) gg =F (Ohmm) - 4 densi intensity y Ande rson-205419.3 Conduction Requires Electron Transport + Metals hermal energy _ 4+ Puts many e intoa higher energy, it mobile state. net a” flow Energy, Energy, emp nearby, = accessible GAP higher empty =e band partly =o energy ede |e states filled valence | “- — le band | 3 band =e — filled | —3- = filled band | = > band Ande rson-205-19.6 Effect of T, Imperfections on p metal Imperfections increase resistivity] -grain boundarjesO ‘dislocations Act to scatter (deflect) e- ; “impurity atoms @ take less direct path Vacances * Data for Cu 6 yt : st ee wicloan Resistivity Tas ££: os gaa TT 2 at te 74 Ab wt% impurity 7 ZOoo © Ge : ae" aa WCW T a S2p Sea? p=p + Pimpurity + P — ou Tou the rinal impunty ~ def 47 Tee Fig. ial git 1_1_1__ Callister -200 -100 «0 OT (°C) Ande rson-205-19.8 Solid Electrolytes ‘Electrolyte - A substance that conducts electricity through the movement of ions. Most electrolytes are solutions or molten salts, but some electrolytes are solids and some of those are crystalline solids. Different names are given to such materials: -Solid Electrolyte -Fast Ion Conductor -Superionic Conductor Tonic vs. Electronic Conductivity Let's begin by comparing the properties of ionic conductors with the conventional electronic conductivity of metals. Metals -Conductivity Range = 10 S/cm< o< 10® S/cm -Electrons carry the current -Conductivity Increases linearly as temperature decreases (phonon scattering decreases as T 1) Solid Electrolytes -Conductivity Range = 10-3 S/em< o< 10 S/em -Ions carry the current -Conductivity decreases exponentially as temperature decreases (activated transport) _ Defects In order for an ion to move through a crystal it must hop from an occupied site to a vacant site. Thus ionic conductivity can only occur if defects are present. The two simplest types of point defects are Schottky and Frenkel defects, oo a ae ow a ae a eas aS a aa ana P+ NOS Schottky Defect Frenkel Defect (i.e. NaCl) (i.e. AgCl) Na*+Cl- ad ViatVei Ag’ = Viagt AQ" interstitial Applications of Ionic Conductors There are numerous practical applications, all based on electochemical cells, where ionic conductivity is needed and it is advantageous/necessary to use solids for all components. -Batteries eo -Fuel Cells -Gas Sensors Electrolyte Anode Cathode In such cells ionic conductors are needed for either the electrodes, the electrolyte or both. -Electrolyte (Material needs to be an electrical insulator to prevent short circuit) -Electrode (Mixed ionic and electronic conductivity is needed to avoid open circuit) Schematic of a Solid Oxide Fuel Cell FUEL CO, As SNS En CO, Hat O23 COs, HO + Be PRODUCTS: COs, H20 T= 700 é Op+ de > 20% EXCESS LO Le AIR Schematic of Rechargable Li Battery Di ischarge Eharge Load £ © g SSS WY NS Lit \ . Electrolyte , UT Current —_LixGg Collector Anode Liy.CoO2 = Current Cathode Collector Figure 6. Schematic illustration of the discharge and charge processes in a rechargeable lithium ion battery. In the Li,- CoO, cathode, the solid and open circles refer, respectively, to Co and O atoms (adapted from ref 14). Taken from A. Manthiram & J. Kim - “Low Temperature Synthesis of Insertion Oxides for Lithium Batteries", Chem. Mater. 10, 2895-2909 (1998).