Semiconductor Statistical Mechanics (Read Kittel Ch. 8)

Transcription

1 EE30 - Solid State Electroics Semicoductor Statistical Mechaics (Read Kittel Ch. 8) Coductio bad occupatio desity: f( E)gE ( ) de f(e) - occupatio probability - Fermi-Dirac fuctio: g(e) - desity of states / uit volume. For a isotropic, parabolic bad, geeralize free-electro theory: ge ( ) m * e π ( E E _ c ) 1 h m * e π _ h ε 1 dε exp[ ( ε E F + ) kt] where ε E. Defie dimesioless variables: η ε E c η kt c μ kt E F kt m * e kt π _ h η 1 dη exp( η μ+ η c ) F 1 ( μ η c ) Fermi-Dirac itegrals (tabulated i Semicoductor Statistics, J.S. Blakemore, Pergamo, 196) F ( x) π 0 z dz 1 + exp( z x) Occupatio statistics Hadout.fm

2 EE30 - Solid State Electroics effective desity of states : Recall the discussio of degeerate / o-degeerate Fermi-gas. N C is desity for the degeerate case. Some umbers: * 19 For Si, m e 1.18m o ( desity of states mass);.8 10 cm 3 at 300K. 17 For GaAs, m e 0.067m o ; cm 3 at 300K cm 3 at 4K Aisotropic bads desity of states mass: ν degeeracy factor - # of equivalet CB valleys 6 i Si 1 i GaAs Maxwell-Boltzma approximatio If E F is well iside bad-gap (o-degeerate case): E F» kt, the the Fermi fuctio Boltzma factor F ( x) z e x z d π 0 z for e x 1 -- This expressio ca be iterpreted as if there are states all located at bad edge

3 EE30 - Solid State Electroics Holes: use the distributio for empty states: f p ( E) 1 f FD ( E) exp[ ( E F E) kt] p E v ge ( )[ 1 f FD ( E) ] de E V p N V F 1 ( η V μ) η V kt Maxwell-Boltzma approx: 1 N V -- m * h kt πh Itrisic case (pure semicoductor, o dopig) charge eutrality: p i F E F E c E V E F kt N V F kt The itrisic case is early always o-degeerate, so we ca write: Now, take the log of both sides, ad solve for : E F E + E V kt N F l V

4 EE30 - Solid State Electroics E F is ear midgap. E F is exactly at midgap at T0. E Badgap also decreases with T E F E v E F heads for low-mass For high eough T, large mass ratio, ca get high temperature degeeracy. Examples: ISb, IAs above ~ 400K. The itrisic carrier desities are idepedet of. T E F kt N V e E G E G : eergy gap i N V e E G kt 1 * * -- ( m 4 e mh ) 3 4 kt πh e E G kt i E G Measuremet of vs. T ca be used to determie

5 EE30 - Solid State Electroics Extrisic case (doped semicoductors) shallow impurities E D or E A close to bad edges. Easily ioized at RT E d E a E V II III IV V VI B C N O Al Si P S Z Ga Ge As Se Cd I S Sb Te GaAs dopats Si dopats E d (mev) E a (mev) E d (mev) E a (mev) S 6 Se 6 Te 6 Si 6 36 Ge 6 40 S C 6 6 Z 31 Be 8 P 44 As 49 Sb 39 B 45 Al 69 Ga 73 I

6 EE30 - Solid State Electroics Notice that these ioizatio eergies are very similar. This suggests a simple hydrogeic model: For Si, m e 0.74m 0 (mobility mass), ε 11.9ε 0. So i this model, E d 71 mev. For GaAs, m e 067m 0, ε 13.1ε o, so E d 5 mev. I geeral, we may have both doors & acceptors. Complete ioizatio case Charge eutrality: p N a For the o-degeerate case still holds. N a i Solve this quadratic equatio for : i N a ( ) N a Similarly: p N a ( ) 1 i N a For N a» i :, ad

7 EE30 - Solid State Electroics Statistical Mechaics for Doors & Acceptors Icomplete ioizatio Remove assumptio of complete ioizatio of the dopats. Fid the temperature depedece of pe,, F For simplicity, cosider -type case, doors oly N a 0. Ca geeralize later. i - desity of ioized doors - desity of eutral doors - total desity of doors Assume 1 electroic state per door atom exp[ ( E d E f ) kt] i exp[ ( E d E F ) kt] exp[ ( E d E F ) kt] the, If the door states have degeeracies, g i, g (i.e. spi), the this expressio is modified to: N di g i ---- exp[ ( E d E F ) kt] g g i exp[ ( E d E F ) kt] g For a simple moovalet door

8 EE30 - Solid State Electroics For acceptors, the aalogous expressio is: N a g N a exp[ ( E a E F ) kt] g i What we wat to do is determie the free carrier desity: (o degeerate statistics) let E d kt; η c kt; μ E F kt i p, (assumig -type: >>p) N d e μ + Elimiate μ by usig e μ η c. e μ e μ η d e μ η c e η c so e η c η + d which is a quadratic equatio for. The solutio is: -----e η c 4 ( ) e η c ± + - root is uphysical To gai physical isight we examie limitig behaviors of this relatio:

9 EE30 - Solid State Electroics Low temperature, η c» 1 ( kt «E d ) reserve regio -----e η c 4 ( ) e η c 1 N d 1 ( e η c ) Here, E F falls i betwee, E d. Sort of a mii-gap E F For moderate T, such that e η c E d < 1, or kt > , expad the : l( 8 ) E d -----e η c 4 ( ) e η c This is called the exhaustio regio ioizatio.) (Here s where we usually wat to be - complete For really high T, >>p is o loger true. How high does T have to be for this? E G kt > ~ l N v or fially: E G 1 N kt -- c N v < ~ l

10 EE30 - Solid State Electroics kt > ~ l( N v ) Whe this coditio is true, the we basically have itrisic: E g log itrisic kt e E g exhaustio reserve ( E d ) kt e 1/T