Chem 53, UC, Bereley Chem 53A Crystl Structure Chem 53B Electroic Structure Chem 53, UC, Bereley 1
Chem 53, UC, Bereley Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-3, 8-10 Kittel: chpter 6-9 Gerste: Chpter 7, 11 Burdett: chpter 1-3 Hoffm: p1-0 Chem 53, UC, Bereley
Chem 53, UC, Bereley Si NW Chem 53, UC, Bereley 3
Chem 53, UC, Bereley Chem 53, UC, Bereley Crrier Mobility Mometum gied durig the me free flight ee v d * m vd ee m* Mometum lost i collisio Drift velocity Mobility: the rtio of the drift velocity over the pplied electric field v d E e m* 4
Chem 53, UC, Bereley Chem 53, UC, Bereley Idepedet Electros Free Electro Approimtio 5
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7 Chem 53, UC, Bereley ( m E From 1D to 3D: z z y y z y A r si si si ( ] ( ( [( ( z z y y m E Chem 53, UC, Bereley Periodic Boudry Coditio ( ( E F K F K ( ( r r m Solutio: trvelig ple wve ep( i A Where: m E (
Chem 53, UC, Bereley Normliztio: 3D Periodic Boudry Coditio * Aep( i r dr 1 A ep( i rep( i r dr A V V: uit cell volume V 1/ ep( i r de Broglie wvelegth Chem 53, UC, Bereley Eergy Eigevlue: Momet Opertor: E( m pˆ i r pˆ ( r i ( m y z ( r ( r r Aep( i r E F Mometum Eigevlue: K F K p 8
9 Chem 53, UC, Bereley With periodic boudry coditios: 1 z z y y i i i e e e z z z y y y D spce: Are per poit: y 3D spce: Are per poit: V z y 3 8 A regio of spce of volume will coti: llowed vlues. 3 3 8 8 ( V V Chem 53, UC, Bereley Reciprocl ttice ( c b c b ( c b c b ( c b b c Reciprocl lttice is lwys oe of 14 Brvis ttice.
Chem 53, UC, Bereley K spce desity of level: V 3 8 No-iterctig electros: Puli eclusio priciple Ech wve vector two electroic level (spi up/dow Fermi wve vector: F Volume eclosed by the Fermi surfce: 4 3 3 F Chem 53, UC, Bereley # of llowed sttes withi: 4 3 3 3 F V 3 8 F 6 V # of electros N: N 3 F 3 V Electroic desity: N V 3 F 3 F ( 3 1/3 10
Chem 53, UC, Bereley Free & idepedet electro groud stte: Fermi wve vector Eclosed Fermi sphere F ( 3 1/3 Fermi Surfce Fermi Mometum Fermi eergy Fermi velocity p E v F F F F m p F F / m* Chem 53, UC, Bereley Estimtio bsed o coductio electro desity: V N 1 4 r 3 3 3 s 3 F E F F m F (9 / 4 r s E 1/3 F 1.9 r s 50.1eV ( rs / 0 Rdius of sphere where volume Equls to the volume per Coductio electro -3, for my metl Fermi eergy for metllic elemets: 1.5 15 ev Fermi temperture: EF 58. 4 TF 10 K ( r / B s 0 11
Chem 53, UC, Bereley Chem 53, UC, Bereley Desity of Sttes The umber of orbitls/sttes per uit eergy rge D( E dn de E N 3 N ( m m V V E 3 3/ ( /3 dn V m 3/ D( E ( E de 1/ 1
Chem 53, UC, Bereley Qutum Cofiemet d Dimesiolity Chem 53, UC, Bereley Fermi-Dirc distributio: f ( E ep[( E 1 E / F B T ] 1 13
Chem 53, UC, Bereley Chem 53, UC, Bereley 14
Chem 53, UC, Bereley Chem 53, UC, Bereley 15
Chem 53, UC, Bereley Nerly Free Electro Model Addig smll perturbtio by the periodic potetil of the ioic cores E F K F K E( m Chem 53, UC, Bereley Periodic Boudry Coditio m ( r ( r ( ( Solutio: trvelig ple wve Aep( i Where: E F K F K E( m 16
Chem 53, UC, Bereley 3D Periodic Boudry Coditio Normliztio: Aep( i r * dr 1 A ep( i rep( i r dr A V V: uit cell volume V 1/ ep( i r de Broglie wvelegth Chem 53, UC, Bereley Periodic Potetils d Bloch's Theorem V ( r V ( r R R ttice vector Bloch s theorem: the eigesttes of the Hmiltoi bove c be chose to hve the form of ple wve times fuctio with the periodicity of the Brvis ttice. Bloch Wvefuctio: ir e ( r u( r V u( r u( r R periodic prt of Bloch fuctio 17
Chem 53, UC, Bereley Brgg reflectio of electro wves i crystl is the cuse of the eergy gp. First Brgg reflectio: Other gp: Chem 53, UC, Bereley 18
Chem 53, UC, Bereley Reciprocl ttice d R ' ' R 1 b 3 c e ikr 1 i e ' R( K ' 1 ue Coditio Reciprocl lttice vector For ll R i the Brvis ttice Chem 53, UC, Bereley For 1D ttice: Reciprocl lttice vector: ' K Diffrctio Coditio: 1 K C be eteded to 3D 19
Chem 53, UC, Bereley Brgg reflectio of electro wves i crystl is the cuse of the eergy gp. First Brgg reflectio: First Brilloui Zoe Other gp: Chem 53, UC, Bereley Wiger-Seitz cell 0
Chem 53, UC, Bereley The wvefuctio t re ot trvelig wve of free electros: ep( i ep( i Isted: equl prts of the wves trvelig to the left d right A wve trvels either to the left or to the right is stdig wve. Chem 53, UC, Bereley Two differet stdig wves: ( ep( i ep( i cos ( ep( i ep( i si Probbility desity: 1
Chem 53, UC, Bereley Pile electro betwee the core ioshigher eergy Pile electro o the core ioslower eergy Chem 53, UC, Bereley Eteded zoe scheme reduced zoe scheme
Chem 53, UC, Bereley Fermi Surfce E( m E F K F K Chem 53, UC, Bereley For divlet elemets: free electro model 3
Chem 53, UC, Bereley Chem 53, UC, Bereley For erly free electro: 1. Iterctio of electro with periodic potetil opes gp t zoe boudry. Almost lwys Fermi surfce will itersect zoe Boudries perpediculrly. 3. The totl volume eclosed by the Fermi surfce depeds oly o totl electro cocetrtio, ot o iterctio 4
Chem 53, UC, Bereley Alli Metl N, Cs: sphericl Fermi surfce r r 0.4 Al. Erth metl: Be, Mg:: erly sphericl Fermi surfce D cse r r 0.56 Chem 53, UC, Bereley 5
Chem 53, UC, Bereley Chem 53, UC, Bereley 6
Chem 53, UC, Bereley Chem 53, UC, Bereley 7
Chem 53, UC, Bereley Chem 53, UC, Bereley 8
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