Lecture 1: Semiconductor Physics I. Fermi surface of a cubic semiconductor

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1 Leture 1: Semiodutor Physis I Fermi surfae of a ubi semiodutor 1

2 Leture 1: Semiodutor Physis I Cotet: Eergy bads, Fermi-Dira distributio, Desity of States, Dopig Readig guide: Ludstrom

3 3D Eergy Bads - Effetive Mass Effetive mass approximatio: Kieti Eergy: E E = ħ m k x + k y + k z Effetive Mass: m = ħ d E dk Group Veloity: v x = 1 ħ de k dk x = ħ m k x IAs Desity of states: D 3D = m 1.5 π ħ 3 E E 3

4 Eergy Bads E(k) ad E(x) E Mometum spae E Regular spae E k Spatial desriptio E k E C k x Badgap E g (ev) Badgap E g (ev) E v x The bottom of the odutio bad (E C ) orrespods to the potetial eergy of a eletro. E k =E-E orrespods to the kieti eergy 4

5 D Eergy Bads Quatum Wells For a very thi slab of semiodutor material the eletros are ofied i the z diretio z y W Ifiite quatum well eergies E ħ π m W x Sub-bad eergy dispersio E k = E ħ m k x + k y + E + E C E Desity of states: D D = m πħ Θ(E E ) k x x Carriers are free to move i the x-y plae! 5

6 Semiodutor Badgaps Semiodutors with differet badgaps. Biarys (III-V): GaAs, IP, IAs.. Teraries: I 1-x Ga x As, I 1- xal x As, GaAs 1-x Sb x.. Materials with similar lattie ostats a easily be ombied. GaAs-Al x Ga 1-x As IP-I 0.53 Ga 0.47 As-I 0.5 Al 0.48 As IAs-AlSb-GaSb 6

7 Semiodutor heterostrutures Differet materials have differet affiities ad differet bad gaps. χ Vauum level Whe formig heterostrutures differet materials obtai odutio ad valee bad offsets to eah other DE C odutio bad offsets DE V valee bad offsets χ 1 ΔE C E G1 E G χ 3 ΔE v E G3 Note: This is a simplisti explaatio! Tabulated affiities ad badgaps aot be used to aurately alulate DE C ad DE V betwee materials. 7

8 Semiodutor Badgaps Commo (almost) lattie mathed semiodutors: GaAs Al 1-x Ga x As IP-I 0.53 Ga 0.47 As- I 0.5 Ga 0.48 As IAs-GaSb-Alsb The materials have differet E g ad m* DE C ad DE V haraterize the heterojutios Not lattie mathed, but tehologially importat: Si (SiGe) GaN-AlN-IGaN-AlGaN IP-IAs 8

9 Crystal Growth: Lattie Mathig III - Strai a 5.95Å I 0.7 Ga 0.3 As I 1-x Ga x As o IP Metamorphi I 0.7 Ga 0.3 As Pseudomorphi I 0.7 Ga 0.3 As IP a=5.87å Lattie Mismath: f = Δa a a=5.87å Disloatios poor material quality! Small lattie-mismath: Large ritial thikess before relaxatio Large lattie mismath: Small ritial thikess before relaxatio 9

10 Fermi Level ad F/D Statistis Eletros obey Fermi- Dira statistis: f 0 (E, E F ) = e E E F kt E f E Total oetratio of free eletros: = න E D 3D (E)f 0 E E F de -type material E g E v D 3D = m 1.5 π ħ 3 E E For IAs: N = m -3 N v = m -3 Effetive desity of states: N C = πm kt h 1.5 η F N 0 dx 1 exp x x E f E / kt F 1/ (( E f E ) / kt ) 10

11 Fermi Level ad F/D Statistis Itegral for eletro oetratio: N C = π න 0 η 1 + e η η F dη = F 1/(η F ) E f E g E N C = πm kt h 1.5 -type material E v N e (E f E C )/kt Maxwell-Boltzma statistis, valid if < 0.05N, or E -E f >> 3kT For IAs: N = m -3 N v = m -3 E f E kt l N C + N N 1/4 Other approximatios for alulatio of E F -E C if is kow! E f E kt l N 1 8 N N... (Joye-Dixo approximatio) 11

12 3D Approximatios - Comparisos E f 10 E kt l N 1 8 N N N F ( 1/ E f E ) Exat! (withi the EMA) /N C E f E kt l N C E f E kt l N C + N N 1/ (E F -E C )/kt Very easy to alulate E f -E if is kow! (reverse is usually ot true umerial alulatios eeded.) 1

13 Itrisi Semiodutor Thermally exited harges Itrisi semiodutor (Boltzma approx.) has thermally exited harges: E C = p = i = N V N C e E G kt E F E F E C E G + ktl N C N V Oly strogly valid if <N C ad p<n V! (Boltzma approx.) E v A itrisi semiodutor has E F i lose to the middle of the badgap Ex: IAs: i = m - GaAs: i = m - 13

14 Semiodutor Crystals - dopig Itrodutio of door atoms ito the semiodutor reates free eletros E d e e e e E Eah ioized door atom oe free eletro (e-) ad oe fixed door io. (N D+ ) E g E v Door: -type material Itrodutio of aeptor atoms ito the semiodutor aptures a eletro from the valee bad E g E Eah ioized aeptor atom oe free hole (h+) ad oe fixed aeptor io. (N A- ) E a e e e E v Hole: p-type material Dopig Rage m -3 14

15 Carrier Coetratio Doped Semiodutors N-dopig = p-dopig N d + 4 i + N d N d N d i E F E C E v p = N a + 4 i + N a N a p = i Mass atio law (Boltzma approx) N-type material: N d i p i N d i Majority Carriers Miority Carriers 15

16 Semiodutor Crystals high dopig Low dopig isolated states High dopig ioized dopat atoms iterat with bad edge door atoms iside above E. No freezeout! E E v This is usually the ase for most small badgap III-Vs 16

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