Fr Carrir : Carrir onntrations as a funtion of tmpratur in intrinsi S/C s. o n = f(t) o p = f(t) W will find that: n = NN i v g W want to dtrmin how m

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1 MS 0-C 40 Intrinsi Smiondutors Bill Knowlton Fr Carrir find n and p for intrinsi (undopd) S/Cs Plots: o g() o f() o n( g ) & p() Arrhnius Bhavior

2 Fr Carrir : Carrir onntrations as a funtion of tmpratur in intrinsi S/C s. o n = f(t) o p = f(t) W will find that: n = NN i v g W want to dtrmin how many fr - s and h + s xist nar thir rsptiv band dgs as a funtion of: o Tmpratur Calld Fr Carrir, this is again prformd using: o Statistial Mhanis o Quantum Mhanis Knowlton

3 Lt n b dfind as th onntration of ltrons (#/volum) in th ondution band. = n C = T g b( ) f ( ) d [] = C B g b () is th dnsity of stats in D. It is obtaind by solving th partil in th box in dimnsions. Hn, it is obtaind using quantum mhanis. m ff b = π = g ( ) 8 N ; h m ff whr N = 8π h Th Frmi-Dira distribution funtion, F() is givn: F ( ) = ( f ) + Knowlton

4 Lt n b dfind as th onntration of ltrons (#/volum) in th ondution band. = C T n = N d = ( f ) C B + Lt: T Multiply by: n = N = = ( f ) C B + d Lt: f + f = = ε η Whr: f ε = and η = Knowlton 4

5 Substituting, w hav: Nd to: = ε n = N d Transform d to dε = Dtrmin th limits for ε : F j (η) is th Frmi-Dira Intgral of Ordr j: j x= x Fj ( η ) = dx Γ 0 x ( j + ) x= + η Γ(j+) is th Gamma Funtion of ordr j+ ( ) 0 x= j x Γ j + x dx Knowlton =x= 5 C B + ε η d d ε( ) = dε = 0 = ; So: d = dε ε( ) = ε( = ) = = 0; And: ε( = ) = = Thus: n N ( ε = ε = ) d ε [] ε = 0 ε η = N Γ F / + ( η ) ( ) N = N

6 Hr: F j (η) = F / (η) And: Γ(j+) = Γ(/) F j (η) was not analytially solvd until 995. Can now us Mathmatia to solv. Gamma funtions, whih ar lss ompliatd than F j (η), an b solvd using intgral tabls or Mathmatia. BOLZMANN S APPROXIMATION: F j (η) was not analytially solvd until 995, so historially th Boltzmann s Approximation was usd. Boltzmann s Approximation: f is wll within th nrgy band gap and not nar th band dgs of C and V. ε f η = >> Knowlton 6

7 BOLZMANN S APPROXIMATION: F j (η) was not analytially solvd until 995, so historially th Boltzmann s Approximation was usd. Boltzmann s Approximation: f is wll within th nrgy band gap and not nar th band dgs of C and V. ε f η = >> This lads to: ( ) = = ε η + ε η ε η η ε [] Knowlton 7

8 Not that th Frmi-Dira distribution funtion boms th Maxwll-Boltzmann distribution funtion. Thus, w hav Arrhnius bhavior rsulting in thrmal ativation of ltron onntration in th ondution band. Substituting quation [] into [], w hav: ε = ε n= N dε ε = 0 ε η = = ε = ε = 0 ε = 0 η ε N ε dε ε = η ε N ε dε Knowlton 8

9 Not that w hav: Γ(j+) = Γ(/) ε = 0 η ε n= N ε dε η π = NΓ Not: Γ = η = N ( f ) = N whr: N = N C π ε = C π Hn: n= N ( f ) * π m whr N = h N C is known as th fftiv Dnsity of Stats Knowlton 9

10 Compar bfor th Boltzmann Approximation: n N ( ε = ε = ) 0 d ε ε = ε η + / / = N Γ F F ( η ) ( η ) [] To aftr th Boltzmann Approximation: n= N d = NΓ = N η ε = η ε η ε ε ε = 0 π η Hn, ompar ths two funtions: ( ) n F / η n η Knowlton 0

11 Hn, ompar ths two funtions: MS 40-C 40 ( ) n F / η n η Robrt F. Pirrt, Advand Smiondutor Fundamntals, Modular Sris on Solid Stat Dvis, nd d., Volum VI (Prnti Hall, 00) p. -6. Not: η divrgs from F / (η) whn η, or f - or v - f ~. Knowlton

12 Dnsity of Stats funtion: g ( ) J.P. MKlvy, Solid Stat Physis for nginring & Matrials Sin (Krigr Publishing Company, 99). Knowlton

13 Frmi-Dira distribution funtion: f ( ) = ( ) + f b Robrt F. Pirrt, Advand Smiondutor Fundamntals, Modular Sris on Solid Stat Dvis, nd d., Volum VI (Prnti Hall, 00) p. -6. Dnsity of Stats funtion: John P. MKlvy, Solid Stat Physis for nginring & Matrials Sin (Krigr Publishing Company, 99). Knowlton

14 What happns whn you ombin th Frmi lvl distribution with th dnsity of stats distribution? Frmi-Dira distribution & Dnsity of Stats funtions: John P. MKlvy, Solid Stat Physis for nginring & Matrials Sin (Krigr Publishing Company, 99). Knowlton 4

15 What happns whn you ombin th Frmi lvl distribution with th dnsity of stats distribution? +χ CB g() (- ) / [- f()] For ltrons Ara = n n () F F v v p () VB For hols Ara = p 0 Fig 5.7 (a) g() (b) f() () n () or p () (d) (a) nrgy band diagram. (b) Dnsity of stats (numbr of stats pr unit nrgy pr unit volum). () Frmi-Dira probability funtion (probability of oupany of a stat). (d) Th produt of g() and f() is th nrgy dnsity of ltrons in th CB (numbr of ltrons pr unit nrgy pr unit volum). Th ara undr n () vs. is th ltron onntration in th ondution band. From Prinipls of ltroni Matrials and Dvis, Third dition, S.O. Kasap ( MGraw-Hill, 005) Knowlton 5

16 A similar approah is usd to find th onntration of hols, p, in th valn band. Lt p b dfind as th onntration of hols (#/volum) in th valn band. = = = V T V T V T g vb () is th dnsity of stats in D in th valan band. g vb () is obtaind by solving th partil in th box in dimnsions. Hn, g vb () is obtaind using quantum mhanis. + h m ff vb π v v g ( ) = 8 = P ; h m whr P 8 h h ff = π ( ) + ( ) p = g f d ( ) ( ) = gvb f d [] = V T vb + h Knowlton 6

17 valuation of p vntually rsults in th produt of a Frmi-Dira intgral of ordr ½ and Gamma funtion of ordr /. p * m ff ε 0 ε η = π + * m ff = Γ F π dε ( η ) [] Not - This may hlp th drivation: 0 f ( x) dx = f ( x) dx 0 Knowlton 7

18 Applying th BOLZMANN S APPROXIMATION: Boltzmann s Approximation: f is wll within th nrgy band gap and not nar th band dgs of C and V. That is: f Hn: ε η = >> ( ) = = ε η ε η + ε η η ε Substituting into quation [], w hav: p * m ff ε 0 ε η = π + * mff η ε 0 π dε ε dε Now w hav a Gamma Funtion that is asily found in intgral tabls: * mff p = Γ π f v p= N v Knowlton 8 η

19 n: onntration of ltrons in th Condution Band (CB) pr unit volum. MS 40-C 40 i.., ltron dnsity = p: onntration of hols in th Valn Band (VB) pr unit volum. i.., hol dnsity n = g ( ) f ( ) d = b whr g ( ) = = 0 b n= N Knowlton 9 [ ] υ p = g ( ) f ( ) d υ whr g ( ) υb ( f ) * π m whr N = h p = N υ ( f υ ) υ * h π m whr Nυ = h b

20 Intrinsi arrir onntration as a funtion of /T (Arrhnius plot). Aftr Prof..H. Hallr, UC Brkly Ltur nots Knowlton 0

21 n: onntration of ltrons in th Condution Band (CB) pr unit volum. i.., ltron dnsity np = n = = = i NN NN NN υ υ υ ( f ) ( f υ ) ( ) g υ * * πmh N υ πm whr N = & = h h b n i = = NN NN υ υ g g f g = + 4 ln m m ff, h ff, +

22 Rfrns: J.S. Blakmor, Smiondutor, (Dovr, 987) p John P. MKlvy, Solid Stat Physis for nginring & Matrials Sin (Krigr Publishing Company, 99) Ch. 9. Robrt F. Pirrt, Advand Smiondutor Fundamntals, Modular Sris on Solid Stat Dvis, nd d., Volum VI (Prnti Hall, 00) p. -6. David K. Frry & Jonathan P. Bird, ltroni Matrials and Dvis, (Aadmi Prss, 00) p S.O. Kasap, Prinipls of ltroni Matrials and Dvis, rd d. (MGraw-Hill, 005) p