Free carriers in materials

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Ernest Farmer
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1 Lctu / F cais in matials Mtals n ~ cm -3 Smiconductos n ~ cm -3 Insulatos n < 8 cm -3 φ isolatd atoms a >> a B a B.59-8 cm 3 ϕ ( Zq) q atom spacing a

2 Lctu / "Two atoms two lvls" φ a T splitting of vy lvl on two lvls du to intaction of atoms. T intaction is tunnling of lctons toug potntial bai. T tunnling bcoms considabl wn a a B

3 Lctu /3 "N atoms N lvls" pmittd ngy gap 3 fobiddn ngy gap 3 a B Intatom spacing vy lvl is split on N lvls. vy lcton in pmittd ngy blong to all atoms.

4 Lctu /4 Mtals, Smiconductos, Insulatos n ~ cm -3 n ~ 8-9 cm -3 n < 8 cm -3 Mtal Smiconducto Insulato Conduction g ngy gap Valnc g >3V

5 Lctu /5 π λ Dispsion law fo f lcton in vacuum d Boil 95 : p mv p m lcton wav vcto li on fo ligt wav π λ mv π p π p λ () m m Accoding to wav-paticl dualism concpt t lcton possss simultanously bot paticl and wav poptis and can b caactizd by wavlngt λ, wav vcto and impuls p

6 Lctu /6 lcton in smiconducto flctd lcton wav lcton wav pmittd ngy stats g ngy gap pmittd ngy stats As soon as lcton possss bot paticl and wav poptis it can b flctd and diffactd xactly li ligt wav. In a piodic lattic wit piod, lcton wav wit wavlngt satisfying Bagg condition (mλ, m-intg) can not popagat du to stong flction fom t ngy bai, i.. lcton wit t ngy in ctain intval (ngy gap) can not fly mov in cystal.

7 Lctu /7 lcton in smiconducto () a λ Condition of stong flction of lcton wav m, m,, 3... π λ Fobiddn gaps av to xist at dfinit wavvctos of lcton. π a m 3π π π π π fobiddn gap 3π

8 Billouin zons Lctu /8 () () F g F' g M' M S g P' S' g P 3π π π π π 3π π π T wavfunctions and lcton stats a quivalnt fo and ± mg (gπ/, m- intg): t stats M and M ; P and P, S and S', F and F' a quivalnt. It is not ncssay to us valus of lag tan π/a in dscibing any stats. Tis mans, it is noug to plot t lcton ngy () vs only witin fist Billouin zon.

9 Lctu /9 T ffctiv mass of lcton g π/ Fobiddn gap Conduction Valnc π/ () ), ( () () F lcton: m lcton in cystal: fo conduction m m Lt's ma an oigin in nw coodinats: ) ( Fo << π/

10 Lctu / Hols in smiconductos Hol concpt Ral spac ngativ lcton positiv ol Hol in -spac T collctiv motion of ngativ lctons in -spac in valnc is qual to movmnt of fw positivly cag ols.

11 Lctu / Hols in smiconducto Wn valnc is filld by lctons Conduction N i i ( cunt ) ν ngy gap If on lcton wit j is movd, t maining N- lctons a quivalnt to t on positivly cagd paticl wit quasimomntum: j Valnc N i i, i j + ( ) j j N i i j j

12 Hols in smiconducto Lctu / ' ' 3' 4' 5' Fo lctons in conduction wit paabolic dispsion Fo lctons in valnc wit paabolic dispsion mvb Fo N lctons (fully filld valnc ) t cunt is zo. Fo N- t is on lcton ' wic can contibut to t cunt dnsity j. j q m Stat fom : υ, ' υ, υ ' ( m T cunt of N- lctons ' j q in valnc mvb ' m lcton ffctiv mass in valnc (VB) is ngativ (-m VB ), i.. lcton ngy dcass wit quasimomntum ( ) incas. VB, ) υ m p m ' VB <

13 Hols in smiconducto Lctu /3 It is mo convnint to duc motion of t lctons wit ngativ ffctiv mass to motion of t quasipaticls wit positiv ffctiv mass m -m VB > and positiv cag. Cunt is qual to j + q m ' ' Tis cunt is qual to t on poducd by N- lctons ( ' ', m -m VB ) Wn m lctons a xitd fom valnc to impuity lvls o conduction t st of N-m lctons bav as m ols

14 Lctu /4 lctons and Hols Conduction ngy gap Valnc F -q fo lctons fo ols F +q F t p m m p m m Smiconducto m m GaAs InAs InP.63 m.3 m.77 m.5 m.4 m.85 m

Solid state physics. Lecture 3: chemical bonding. Prof. Dr. U. Pietsch

Solid state physics. Lecture 3: chemical bonding. Prof. Dr. U. Pietsch Solid stat physics Lctu 3: chmical bonding Pof. D. U. Pitsch Elcton chag dnsity distibution fom -ay diffaction data F kp ik dk h k l i Fi H p H; H hkl V a h k l Elctonic chag dnsity of silicon Valnc chag