The Single-Electron Model 1
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1 R The Singe-Eetron Moe P M R q q (L) 8th May , Mihae Marer
2 Approximations P M q q R R uei treate as assia potentias. Many eetrons ombine with nuei in ose shes to orm ions. A eetroni egrees o reeom eiminate, turne to eetive potentia or ions. Phonon ynamis. Ions treate as stati potentia. Born-Oppenheimer approximation. Ions treate assiay. Stuies o ohesive energy, moeuar ynamis. Couomb interation inorporate se-onsistenty into attie potentia. Singe eetron approximation, partiay justiie by Fermi iqui theory Ioni potentia eiminate, interating eetrons move in uniorm positive potentia. Jeium, starting point or ie theoreti wor. Ions arrange in a attie, orming perioi potentia or singe eetrons. Starting point or irstprinipes auations. Wea ioni potentias justiie by pseuopotentias. Ioni potentia eiminate. Free Fermi gas, surprisingy eetive with aai metas. 8th May , Mihae Marer
3 Approximations 3 Main physia iea is the Paui prinipe, whih has two onsequenes:. It popuates sois with eetrons whose energies wou assiay signiy temperatures on the orer o 0,000 K.. It prevents a eetrons but those whose energy iers sighty rom the highest oupie state rom partiipating in transport proesses Important terms: Oupation number Fermi energy Fermi surae Density o states Sommere expansion Eetive mass 8th May , Mihae Marer
4 003, Mihae Marer The Basi Hamitonian 4 h m U r r r rr r Singe eetron probem: h m U r r r (L3) Free eetron gas h m r r r r (L4) 8th May 003 (L)
5 x y L L x x y z Perioi Bounary Conitions 5 x y z z y z z z z z (L5) 8th May , Mihae Marer
6 e i Densities o States 6 r (L6) with o the orm L ẍ ÿ z (L7) The eigenvaue orresponing to the eigenuntion (6) is 0 h m (L8) Oupation number o a state inexe by groun state, an 0 otherwise. is i this one-eetron state is part o the 8th May , Mihae Marer
7 Densities o States 7 F states esribe by Eq. (L7) oupy a ubi attie in neighboring points separate by istanes o L, or reiproa spae, with spae voume per state L is 3. 8th May , Mihae Marer
8 F Densities o States 8 (L9) F L 3 F (L0) F 3 F (L) orrespons to q 3 q (L) 8th May , Mihae Marer
9 Deinition o Density o States D 9 Density o eetroni states or ensity o eves D 3 (L3) eine so that F D F (L4) To avoi perpetuay writing D, aopt the notation D 3 (L5) 8th May , Mihae Marer
10 D F F D F F F Energy Density o States 0 From here onwars, re question mars braet areas eiberatey et ban so that stuents an i them in! (L6) To in D, note that (L7) (L8) (L9) 8th May , Mihae Marer
11 D Resuts or Free Eetrons 0 (L0) 4 h 3 m m h (L) (L) m (L3) 0 ev ev 3 m (L4) 8th May , Mihae Marer
12 Resuts or Free Eetrons Eetrons in sphere o raius F is (L5) (L6) F (L7) F 3F 3 (L8) F 3 n n Å3 3Å (L9) 8th May , Mihae Marer
13 D Resuts or Free Eetrons 3 raius parameter r s 4 3 r3 s r s (L30) Fermi energy, F, or Fermi eve F h F m 3646 n Å3 3 ev (L3) Fermi surae, eetrons with energy F. F h F m 358 n Å m s (L3) F 3 n F 4 0 n Å3 ev 3 Å(L33) 8th May , Mihae Marer
14 D D One- an Two-Dimensiona Formuae 4 One imension: D (L34) m h (L35) Two imensions: m h (L36) 8th May , Mihae Marer
15 Statistia mehanis o noninterating eetrons 5 Z gr e states n 0 Using the mathematia at that n 0 n 3 0 e n (L37) (L38) M M A n A n (L39) n 0 n 0 n M 0 n 0 one has that Z gr e n (L40) n 0 e (L4) 8th May , Mihae Marer
16 Statistia mehanis o noninterating eetrons 6 D Thereore the gran potentia is given by B T nz gr (L4) B T n e (L43) B T n e (L44) 8th May , Mihae Marer
17 where... n 003, Mihae Marer Fermi Funtion 7 (L45) D D e e (L46) (L47) 8th May 003
18 , Mihae Marer Fermi Funtion 8 exp : BT : BT : BT : BT e e (L48) (L49) D (L50) D 8th May 003 (L5) 5 05
19 B Cassia Limit 9 Botzmann statistis Ce (L5) when e (L53) At ow temperatures (L54) Fermi temperature: T F F (L55) 8th May , Mihae Marer
20 F TF Eements as ree eetron gases 0 Eement Z n F F r s a 0 (0 m 3 ) (0 8 m ) (ev) (0 4 K) (0 8 ms ) Li Ag Be A Sn Sb Mn Fe Co i th May , Mihae Marer
21 TD Sommere Expansion Paraox that ensity o states too sma sove by F (L56) 30 0 B T B T 05 B T 5 8th May , Mihae Marer
22 with n n n H an H 003, Mihae Marer (L57) Sommere Expansion H H H n n an H n n BT H n n n H n (L58) (L59) x b x n e x b sinb x n b 0 (L60) (L6) H 6 BT H BT 4 H (L6) 8th May 003
23 003, Mihae Marer 0 D Speii Heat 3 T (L63) D (L64) 6 BT (L65) T T T (L66) D 0 D 6 BT D (L67) T 3 BT D D (L68) 8th May 003 D
24 D D D D Speiaize to Free Fermi Gas 4 F 6 B T D D F F (L69) To orer T F 0 F F D F 6 B T 6 B T F D F (L70) 0 F 3 BT D F 6 B T F D (L7) (L7) As preite, F T. Linear oeiient, Sommere parameter T 8th May , Mihae Marer
25 Speiaize to Free Fermi Gas 5 T B F n B(L73) Speii heat eetive mass o the eetron. 8th May , Mihae Marer
26 Meta Z (mj moe Speiaize to Free Fermi Gas 6 K ) Meta Z (mj moe Expt. Eq. (L73) Expt. Eq. (L73 Li A a Ga K In Rb Sn Cs Pb Cu Sb Ag Bi Au Mn.8.0 Be Fe Mg UPt Ca.73.5 UBe 3 00 Sr Ba.7.9 Zn C Stewart (983), an Stewart (984). 8th May , Mihae Marer K )
27 Speiaize to Free Fermi Gas 7 Heavy Fermions 8th May , Mihae Marer
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