INTERMED MODRN PHYSC
INTERMED MODRN PHYSC PHY 3101
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This 20 page Class Notes was uploaded by Garett Kovacek on Thursday September 17, 2015. The Class Notes belongs to PHY 3101 at Florida State University taught by Staff in Fall. Since its upload, it has received 59 views. For similar materials see /class/205527/phy-3101-florida-state-university in Physics 2 at Florida State University.
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Date Created: 09/17/15
Solid State Physics 3 Sec rion 1046 Topics 0 Hea r Capaci ry of Elec rr39on Gas 0 Band Theory of Solids o Conduc ror39s Insula ror39s and Semiconduc ror39s Summary Special Eme CrediT As can be seen from The graph The pr39edicTion RREQOK 0 0C T RRO0002r2r02 fails aT very low Temper39aTur39es This is due in par39T To The failure of The equipar39TiTion Theor39em aT low Temper39aTur39es Challenge cr39eaTe a 40 x 103 30 x 103 20 gtlt 10 3 10x103 quot 2 ll la 2 1h 1392 1l4 116 1398 2390 beTTer39 model 3 TK 102003 by WH F eeeeee and Company Special ExTra CrediT Derive The TemperaTure 2 2 RRo ppo 7 H0 dependence of RRO by compuTing The average poTenTial energy ltEgt of a laTTice ion J assuming ThaT The energy 40X103 U lezrz E level of The nTh 2 VlbraTIonal sTaTe IS En n 8 raTher Than En ng as EinsTein had assumed lllllllllDue 2468101214161820 4 m before classes end 102003 by WH F eeeeee and Company 30 x 103 20 gtlt 10 3 10x103 quot HeaT CapaciTy of ElecTron Gas By definiTion The heaT capaciTy aT consTanT volume of The elecTr39on gas is given by dU V dT where U is The ToTaI energy of The gas For39 a gas of N elecTr39ons each wiTh average energy ltEgt The ToTaI energy is given by UNltEgt 5 Heat Capacity of Electron Gas Total energy U NltEgt f EnEdE 7r 8m 3 E32dE 2 hZ 6E EFkT In general this integral must be done numerically However39 for39 T ltlt TF we can use a reasonable approximation HeaT CapaciTy of Electron Gas AT T 0 The ToTa energy of The elecTron gas is UNltEgt NEEFJ For 0 lt T lt lt TF only a small fracTion kTEF of The elecTrons can be exciTed To higher energy STOTeS we Moreover The energy of r each is increased by roughly kT quot5 Hea r Capacity of Electron Gas Therefore The To ral energy can be wri r ren as F U NEF 05 E NkT 5 E where OL 724 as first shown by Sommerfeld The hea r capaci ry of The elec rron gas is predic red To be 2 gt C dU Nk T quot5 V 2 a HeaT CapaciTy of ElecTron Gas Consider 1 mole of copper In This case Nk R 2 CV LRL 2 T F For copper TF 89000 K Therefore even aT room TemperaTure T 300 K The contribution wk of The eecTron gas To The heaT capaciTy of copper is M small CV 0018 R quot5 Band Theory of Solids 50 for we have neglecTed The IoTTice of posiTively charged ions Moreover we have ignored The Coulomb repulsion beTween The elecTrons and The oTTrocTion beTween The IoTTice and The elecTrons The band Theory of solids Takes inTo occounT The inTerocTion beTween The elecTrons and The IoTTice ions Bond Theory of Solids Consider The po ren riol energy of o 1dimensionol solid which we approximate by The KronigPenney Model b Band Theory of Solids The Task is To compu re The quan rum s ra res and associated energy levels of This simplified model by solving The Schr b dinger equation UX ii Uo xii ab b 0 a ab m d 5 Hwyx Ewe 12 Band Theory of Solids For39 periodic poTenTials Felix Bloch showed ThaT The soluTion of The Schr39b39dinger39 equaTion musT be of The form kx and The wavefuncTion musT W06 16006 r39eflecT The periodiciTy of The IaTTice 0 lC na b Wxeiknab 13 Band Theory of Solids By requiring The wavefunc rion and i rs deriva rive To be con rinuous everywhere one finds energy levels Tha r are grouped in ro bands separa red by energy gaps The gaps occur at ka 2 in The energy gaps W are basically energy levels Tha r canno r occur in The solid Band Theory of Solids a E 2 2 2 Completely free E Z P 2 h k elec rr on 2m 2m k b E elec rr on in a n quot7 Ia r rice Allowed quot 7 1 E ergy gap bands g 39 1 m7g 3 15 WW quot I 1 0 na 2na 31151 41Ia k Band Theory of Solids When The wovefuncTions become sTonding waves One wove peaks oT The IoTTice siTes and anoTher peaks beTween Them W2 has lower energy Probability density W22 O O T a O Than W1 Moreover There is a jump in energy beTween These sToTes hence The energy gap lbl Bond Theory of Solids The allowed ranges of The wave vector k are called Brillouin zones zone 1 na lt k lt na zone 2 21a lt na zone 31ra lt k lt 21la etc The Theory can explain why some Alluwed m substances are conductors some insulators and Energy gap 557 l na Ena fixa Arra 2008 by w H Freeman and Campany o rhe rs semi conductors ConducTors InsulaTors SemiconducTors Sodium Na has one elecTron in The 35 5TaTe so The 35 energy level is halffilled Consequenle The a 35 band The valence band of solid sodium is also halffilled Moreover The 3p band which for Na i5 35 The conducTion band overlaps wiTh The 35 band 2 So valence elecTrons can easily be 23 raised To higher energy 5TaTe5 Therefore sodium is a good 15 conducTor 18 2003 by 39WH Freeman and Company ConducTors InsulaTors SemiconducTors NaCI is an in3uaTor wiTh a band gap of 2 eV which is much larger Than The Thermal energy aT T 300K a Forbidden 39 Allowed empty Allowed occupied l W M Key b C d k 31 3392393 3 1393 93 3 2394 W Conductor Insulator Semiconductor Conductor Therefore only a Tiny fracTion of elecTrons are in The conducTion band 19 ConducTors InsulaTors SemiconducTors Silicon and germanium have band gaps of 1 eV and 07 eV respecTively AT room TemperaTure a small a d Forbidden l Allowed m m m Allowed m occupied Key Conductor Conductor Insulator fracTion of The elecTrons are in The conducTion band Si and Ge are inTrinsic semiconducTors 20 Semiconductor
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