Notes on Fermi-Dirac Integrals 3 rd Edition

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1 Nots on rmi-dirac Intgrals 3 rd Edition Rasong Kim and Mark Lundstrom Ntwork for Computational Nanotchnology Purdu Univrsity Dcmbr 1, 8 (Last rvisd on August 4, 11) 1. Introduction rmi-dirac intgrals appar frquntly in smiconductor problms, so a basic undrstanding of thir proprtis is ssntial. Th purpos of ths nots is to collct in on plac, som basic information about rmi-dirac intgrals and thir proprtis. W also prsnt Matlab functions (s Appndix and [1]) that calculat rmi-dirac intgrals (th script dfind by Dingl [] and rviwd by Blakmor [3]) in thr diffrnt ways. To s how thy aris, considr computing th quilibrium lctron concntration pr unit volum in a thr-dimnsional (3D) smiconductor with a parabolic conduction band from th xprssion, g(e)de n = g(e) f (E)dE = ( 1 + E E ), (1) k B T E C whr g(e) is th dnsity of stats, f (E) is th rmi function, and dg. or 3D lctrons with a parabolic band structur, E C E C is th conduction band ( m ) 3/ g3d( E) = E E 3 π ħ C, () which can b usd in Eq. (1) to writ n = ( ) 3/ m E EC de 3 ( E E ) kbt π ħ E 1+ C. (3) By making th substitution, Eq. (3) bcoms ε = ( E E C ) k B T, (4) 1

2 n = whr w hav dfind 3/ m kbt 3 1/ ε dε, (5) π 1 ε ħ + C B E E k T. (6) By collcting up paramtrs, w can xprss th lctron concntration as n = N3D 1/ ( ), (7) π whr N π m k T = h B 3D 3/ (8) is th so-calld ffctiv dnsity-of-stats and 1/ 1/ ε dε ( ) (9) 1+ xp( ε ) is th rmi-dirac intgral of ordr 1/. This intgral can only b valuatd numrically. Not that its valu dpnds on, which masurs th location of th rmi lvl with rspct to th conduction band dg. It is mor convnint to dfin a rlatd intgral, 1/ ε dε 1/ ( ) π, (1) 1+ xp( ε ) so that Eq. (7) can b writtn as n = N. (11) 3D 1/ It is important to rcogniz whthr you ar daling with th Roman rmi-dirac intgral or th script rmi-dirac intgral. Thr ar many kinds of rmi-dirac intgrals. or xampl, in two dimnsional (D) smiconductors with a singl parabolic band, th dnsity-of-stats is g m ( E) =, (1) π ħ D

3 and by following a procdur lik that on w usd in thr dimnsions, on can show that th lctron dnsity pr unit ara is S D n = N, (13) whr and N m kbt D = πħ, (14) ε dε ( ) = = ln ( 1+ ε ) (15) 1+ is th rmi-dirac intgral of ordr, which can b intgratd analytically. inally, in on-dimnsional (1D) smiconductors with a parabolic band, th dnsity-of-stats is g 1D ( E) = m 1 πħ E E C, (16) and th quilibrium lctron dnsity pr unit lngth is whr L 1D 1/ n = N, (17) N 1D 1 m kbt =, (18) ħ π and 1/ 1 ε dε 1/ ( ) = π (19) + 1 ε is th rmi-dirac intgral of ordr 1/, which must b intgratd numrically.. Gnral Dfinition In th prvious sction, w saw thr xampls of rmi-dirac intgrals. Mor gnrally, w dfin 3

4 1 ε dε ( ) Γ ( + 1), () 1 + xp( ε ) whr Γ is th gamma function. Th Γ function is ust th factorial whn its argumnt is a positiv intgr, Also and Γ ( n) = n 1! (for n a positiv intgr). (1a) Γ(1 / ) = π, (1b) Γ( p + 1) = pγ( p). (1c) As an xampl, lt s valuat 1/ from Eq. (): 1/ 1 ε dε 1/ ( ) Γ (1/ + 1), (a) 1 + ε so w nd to valuat Γ(3 / ). Using Eqs. (1b-c), w find, so 1/ is valuatd as 1 π Γ (3/ ) = Γ (1/ + 1) = Γ (1/ ) =, (b) 1/ ε dε 1/ ( ) π, (c) 1 ε + which agrs with Eq. (1). or mor practic, us th gnral dfinition, Eq. () and Eqs. (1ac) to show that th rsults for and 1/ agr with Eqs. (15) and (19). 3. Drivativs of rmi-dirac Intgrals rmi-dirac intgrals hav th proprty that d d =, (3) 1 4

5 which oftn coms in usful. or xampl, w hav an analytical xprssion for, which mans that w hav an analytical xprssion for 1, d 1 1 = = d 1 +. (4) Similarly, w can show that thr is an analytic xprssion for any rmi-dirac intgral of intgr ordr,, for, = P, (5) ( ) ( 1+ ) ( ) whr P k is a polynomial of dgr k, and th cofficints p k, i ar gnratd from a rcurrnc rlation [4] (not that th rlation in Eq. (6c) is missing in p. of [4]) p k, = 1, (6a) p = 1+ i p k + 1 i p i = 1,, k 1, k, i k 1, i k 1, i 1 (6b) pk, k = pk 1, k 1. (6c) 4 or xampl, to valuat 4 ( ) = ( 1+ ) P ( ) gnratd from Eqs. (6a-c) as [4], polynomial cofficints ar p, = 1, p = 1, p = p = 1, (7) 1, 1,1, p = 1, p = p p = 4, p = p = 1,,,1 1,1 1,, 1,1 and w find = = +. (8) ( ) i ( 1 4 ) 4 p i 4 1 i + = 1+ 4, 4. Asymptotic Expansions for rmi-dirac Intgrals It is usful to xamin rmi-dirac intgrals in th non-dgnrat ( << ) and dgnrat ( >> ) limits. or th non-dgnrat limit, th rsult is particularly simpl, 5

6 ( ), (9) which mans that for all ordrs,, th rmi-dirac intgral approachs th xponntial in th non-dgnrat limit. To xamin rmi-dirac intgrals in th dgnrat limit, w considr th complt xpansion for th rmi-dirac intgral for > 1 and > [, 5, 6] whr 1 n ( 1) 1, (3) n + 1 tn ( ) = + cos n ( π ) 1 n ( n) + = Γ + n= 1 n µ ζ, and ( n) µ = 1 µ t =, 1 n 1 ( 1 1 n t = = ) ( n) n ζ is th Rimann zta function. Th xprssions for th rmi-dirac intgrals in th dgnrat limit ( >> ) com from Eq. 1 (3) as ( ) + Γ ( + ) blow. [7]. Spcific rsults for svral rmi-dirac intgrals ar shown 1/ 1/ ( ), π (31a) 3/ 4 1/ ( ), 3 π (31b) 1 1( ), (31c) 5/ 8 3/ ( ), 15 π (31d) 1 3 ( ). 6 (31) Th complt xpansion in Eq. (3) can b rlatd to th wll-known Sommrfld xpansion [8, 9]. irst, not that th intgrals to calculat carrir dnsitis in Eqs. (1) and (3) ar all of th form H ( E) f ( E) de. (3) - If H ( E ) dos not vary rapidly in th rang of a fw kbt about Taylor xpansion of H ( E ) about H ( E) H E E as [9] ( E E ) n n E= E n= de n! 6 n E, thn w can writ th d =. (33) Using this Taylor sris xpansion, th intgral in Eq. (3) can b writtn as (s [9] for a dtaild drivation)

7 E n 1 d H E f E de H E de k T a H E n = +, (34) B n n 1 n 1 de = E= E whr = , (35) a n n n n and it is notd that an = tn. Equation (34) is known as th Sommrfld xpansion [8, 9]. Typically, th first trm in th sum in Eq. (34) is all that is ndd, and th rsult is E π H ( E) f ( E) de H ( E) de + ( kbt ) H ( E ). (36) 6 If w scal E by kbt in Eq. (34), ε E kbt, thn Eq. (34) bcoms d H f d H d a H n 1 = + n n 1 n 1 dε = ε = ( ε ) ( ε ) ε ( ε ) ε ( ε ). (37) Thn th Sommrfld xpansion for th rmi-dirac intgral of ordr can b valuatd by H ε = ε Γ + 1 in Eq. (37), and th rsult is ltting t. (38) + 1 n ( ) = n n= Γ ( + n) Equation (38) is th sam as Eq. (3) xcpt that th scond trm in Eq. (3) is omittd [5]. In th dgnrat limit, howvr, th scond trm in Eq. (3) vanishs, so th Eqs. (3) and (38) giv th sam rsults as Eqs. (31a-). 5. Approximat Exprssions for Common rmi-dirac Intgrals rmi-dirac intgrals can b quickly valuatd by tabulation [, 7, 1, 11] or analytic approximation [1-14]. W brifly mntion som of th analytic approximations and rfr th radr to a Matlab function. Bdnarczyk t al. [1] proposd a singl analytic approximation that valuats th rmi-dirac intgral of ordr = 1/ with rrors lss than.4 [3]. Aymrich- Humt t al. [13, 14] introducd an analytic approximation for a gnral, and it givs an rror of 1. for 1/ < < 1/ and.7 for 1/ < < 5/, and th rror incrass with largr. Th Matlab fuction, D_int_approx.m, [1] calculats th rmi-dirac intgral dfind in Eq. (1) with ordrs 1/ using ths analytic approximations. Th sourc cod of this rlativly short function is listd in th Appndix. 7

8 If a bttr accuracy is rquird and a longr CPU tim is allowd, thn th approximations proposd by Haln and Pulfry [15, 16] may b usd. In this modl, svral approximat xprssions ar introducd basd on th sris xpansion in Eq. (3), and th rror is lss than 1-5 for 1/ 7 / [15]. Th Matlab function, Dx.m, [1] is th main function that calculats th rmi-dirac intgrals using this modl. This function includs tabls of cofficints, so it is not simpl nough to b shown in th Appndix, but it can b downloadd from [1]. Thr also hav bn discussions on th simpl analytic calculation of th invrs rmi- Dirac intgrals of ordr = 1/ [3]. This has bn of particular intrst bcaus it can b usd to 1 calculat th rmi lvl from th known bulk charg dnsity in Eq. (11), as ( n N ) =. 1/ 3D Joyc and Dixon [17] xamind a sris approach that givs.1 for max 5.5 [3], and a simplr xprssion from Joyc [18] givs.3 for max 5 [3]. Nilsson proposd two diffrnt full-rang ( 1 ) xprssions [19] with.1 and.5 [3]. Nilsson latr prsntd two mpirical approximations [] that giv.1 for and max, rspctivly [3]. max Numrical Evaluation of rmi-dirac Intgrals rmi-dirac intgrals can b valuatd accuratly by numrical intgration. Hr w brifly rviw th approach by Prss t al. for gnralizd rmi-dirac intgrals with ordr > 1 [1]. t In this approach, th composit trapzoidal rul with variabl transformation ε xp( t ) = is usd for 15, and th doubl xponntial (DE) rul is usd for largr. Doubl prcision (ps, ~ ) can b achivd aftr 6 to 5 itrations [1]. Th Matlab function, D_int_num.m, [1] valuats th rmi-dirac intgral numrically using th composit trapzoidal rul following th approach in [1]. Th sourc cod is listd in th Appndix. This approach provids vry high accuracy, but th CPU tim is considrably longr. An onlin simulation tool that calculats th rmi-dirac intgrals using this sourc cod has bn dployd at nanohub.org []. Not that th numrical approach w considr in this not is rlativly simpl, and thr ar othr advancd numrical intgration algorithms [3] suggstd to improv th calculation spd. In ig. 1, w compar th accuracy and th timing of th thr approachs that calculat. Th rmi-dirac intgral of ordr 1 ) is calculatd for 1 1 = ( 1/ with spacing =.1 using approximat xprssions ( D_int_approx.m and Dx.m ) and th rigorous numrical intgration ( D_int_num.m ) with doubl-prcision. Th rlativ rrors, whr 1/,approx of th approximat xprssions ar calculatd as 1/, approx 1/, num 1/, num and 1/,num rprsnt th rsults from th approximat xprssion and th numrical intgration rspctivly. Th lapsd tim masurd for ach approach (using Matlab commands tic/toc for Pntium 4 CPU 3.4 GHz and. GB RAM) clarly shows th compromis btwn th accuracy and th CPU tim. 8

9 ig. 1. (a) Rlativ rrors from th approximat xprssions for ( ) with rspct to th 1/ numrical intgration ( D_int_num.m ). (A) Rlativ rror from D_int_approx.m. (B) Rlativ rror from Dx.m. All Matlab functions ar availabl in [1].(b) Th absolut valus of th rlativ rrors in th log scal. Th lapsd tim masurd for th thr approachs clarly shows th trad-off btwn th accuracy and th CPU tim. 9

10 Rfrncs [1] R. Kim and M. S. Lundstrom (8), "Nots on rmi-dirac Intgrals (3rd Edition)," Availabl: 1 p ε [] R. Dingl, "Th rmi-dirac intgrals = (!) ( + 1) 1 p p ε dε," Applid Scintific Rsarch, vol. 6, no. 1, pp. 5-39, [3] J. S. Blakmor, "Approximations for rmi-dirac intgrals, spcially th function usd to dscrib lctron dnsity in a smiconductor," Solid-Stat Elctron., vol. 1/ 5, no. 11, pp , 198. [4] M. Goano, "Algorithm 745: computation of th complt and incomplt rmi-dirac intgral," ACM Trans. Math. Softw., vol. 1, no. 3, pp. 1-3, [5] R. B. Dingl, Asymptotic Expansions: Thir Drivation and Intrprtation. London: Acadmic Prss, [6] T. M. Garoni, N. E. rankl, and M. L. Glassr, "Complt asymptotic xpansions of th rmi--dirac intgrals 1 ( 1) p ε = Γ + ( 1+ ) p p ε dε," J. Math. Phys., vol. 4, no. 4, pp , 1. [7] J. McDougall and E. C. Stonr, "Th computation of rmi-dirac functions," Philosophical Transactions of th Royal Socity of London. Sris A, Mathmatical and Physical Scincs, vol. 37, no. 773, pp , [8] A. Sommrfld, "Zur Elktronnthori dr Mtall auf Grund dr rmischn Statistik," Zitschrift für Physik A Hadrons and Nucli, vol. 47, no. 1, pp. 1-3, 198. [9] N. W. Ashcroft and N. D. Mrmin, Solid Stat Physics. Philadlphia: Saundrs Collg Publishing, [1] A. C. Br, M. N. Chas, and P.. Choquard, "Extnsion of McDougall-Stonr tabls of th rmi-dirac functions," Hlvtica Physica Acta, vol. 8, pp , [11] P. Rhods, "rmi-dirac functions of intgral ordr," Procdings of th Royal Socity of London. Sris A, Mathmatical and Physical Scincs, vol. 4, no. 178, pp , 195. [1] D. Bdnarczyk and J. Bdnarczyk, "Th approximation of th rmi-dirac intgral," Physics Lttrs A, vol. 64, no. 4, pp , / [13] X. Aymrich-Humt,. Srra-Mstrs, and J. Millán, "An analytical approximation for," Solid-Stat Elctron., vol. 4, no. 1, pp , th rmi-dirac intgral 3/ [14] X. Aymrich-Humt,. Srra-Mstrs, and J. Millan, "A gnralizd approximation of th rmi--dirac intgrals," J. Appl. Phys., vol. 54, no. 5, pp , [15] P. V. Haln and D. L. Pulfry, "Accurat, short sris approximations to rmi-dirac intgrals of ordr -1/, 1/, 1, 3/,, 5/, 3, and 7/," J. Appl. Phys., vol. 57, no. 1, pp , [16] P. Van Haln and D. L. Pulfry, "Erratum: "Accurat, short sris approximation to rmi-dirac intgrals of ordr -1/, 1/, 1, 3/,, 5/, 3, and 7/" [J. Appl. Phys. 57, 571 (1985)]," J. Appl. Phys., vol. 59, no. 6, p. 64, [17] W. B. Joyc and R. W. Dixon, "Analytic approximations for th rmi nrgy of an idal rmi gas," Appl. Phys. Ltt., vol. 31, no. 5, pp ,

11 [18] W. B. Joyc, "Analytic approximations for th rmi nrgy in (Al,Ga)As," Appl. Phys. Ltt., vol. 3, no. 1, pp , [19] N. G. Nilsson, "An accurat approximation of th gnralizd Einstin rlation for dgnrat smiconductors," Phys. Stat. Solidi (a), vol. 19, pp. K75-K78, [] N. G. Nilsson, "Empirical approximations for th rmi nrgy in a smiconductor with parabolic bands," Appl. Phys. Ltt., vol. 33, no. 7, pp , [1] W. H. Prss, S. A. Tukolsky, W. T. Vttrling, and B. P. lannry, Numrical Rcips: Th Art of Scintific Computing, 3rd d. Nw York: Cambridg Univrsity Prss, 7. [] X. Sun, M. Lundstrom, and R. Kim (11), "D intgral calculator," Availabl: [3] A. Nataraan and N. Mohankumar, "An accurat mthod for th gnralizd rmi-dirac intgral," Comput. Phys. Commun., vol. 137, no. 3, pp , 1. 11

12 Appndix D_int_approx.m function y = D_int_approx( ta, ) Analytic approximations for rmi-dirac intgrals of ordr > -1/ Dat: Sptmbr 9, 8 Author: Rasong Kim (Purdu Univrsity) Inputs ta: ta_ : D intgral ordr Outputs y: valu of D intgral (th "script " dfind by Blakmor (198)) or mor information in rmi-dirac intgrals, s: "Nots on rmi-dirac Intgrals (3rd Edition)" by Rasong Kim and Mark Lundstrom at Rfrncs [1]D. Bdnarczyk and J. Bdnarczyk, Phys. Ltt. A, 64, 49 (1978) []J. S. Blakmor, Solid-St. Elctron, 5, 167 (198) [3]X. Aymrich-Humt,. Srra-Mstrs, and J. Millan, Solid-St. Elctron, 4, 981 (1981) [4]X. Aymrich-Humt,. Srra-Mstrs, and J. Millan, J. Appl. Phys., 54, 85 (1983) if < -1/ rror( 'Th ordr should b qual to or largr than -1/.') ls x = ta; switch cas y = log( 1 + xp( x ) ); analytic xprssion cas 1/ Modl proposd in [1] Exprssions from qs. ()-(4) of [] mu = x.^ x. ( xp( -.17 ( x + 1 ).^ ) ); xi = 3 sqrt( pi )./ ( 4 mu.^ ( 3 / 8 ) ); y = ( xp( - x ) + xi ).^ -1; cas 3/ Modl proposd in [3] Exprssions from q. (5) of [3] Th intgral is dividd by gamma( + 1 ) to mak it consistnt with [1] and []. a = 14.9; b =.64; c = 9 / 4; y = ( ( + 1 ) ^ ( + 1 )./ ( b + x + ( abs( x - b ).^ c + a ).^ ( 1 / c ) ).^ ( + 1 )... + xp( -x )./ gamma( + 1 ) ).^ -1./ gamma( + 1 ); othrwis Modl proposd in [4] Exprssions from qs. (6)-(7) of [4] Th intgral is dividd by gamma( + 1 ) to mak it consistnt with [1] and []. a = ( / 4 ( + 1 ) + 1 / 4 ( + 1 ) ^ ) ^ ( 1 / ); b = ; c = + ( - sqrt( ) ) ^ ( - ); y = ( ( + 1 ) ^ ( + 1 )./ ( b + x + ( abs( x - b ).^ c + a ^ c ).^ ( 1 / c ) ).^ ( + 1 )... + xp( -x )./ gamma( + 1 ) ).^ -1./ gamma( + 1 ); nd nd 1

13 D_int_num.m function [ y N rr ] = D_int_num( ta,, tol, Nmax ) Numrical intgration of rmi-dirac intgrals for ordr > -1. Author: Rasong Kim (Purdu Univrsity) Dat: Sptmbr 9, 8 Extndd (composit) trapzoidal quadratur rul with variabl transformation, x = xp( t - xp( t ) ) Valid for ta ~< 15 with prcision ~ps with 6~5 valuations. Inputs ta: ta_ : D intgral ordr tol: tolranc Nmax: numbr of itrations limit Not: Whn "ta" is an array, this function should b xcutd rpatdly for ach componnt. Outputs y: valu of D intgral (th "script " dfind by Blakmor (198)) N: numbr of itrations rr: rror or mor information in rmi-dirac intgrals, s: "Nots on rmi-dirac Intgrals (3rd Edition)" by Rasong Kim and Mark Lundstrom at Rfrnc [1] W. H. Prss, S. A. Tukolsky, W. T. Vttrling, and B. P. lannry, Numrical rcipis: Th art of scintific computing, 3rd Ed., Cambridg Univrsity Prss, 7. for N = 1 : Nmax a = -4.5; limits for t b = 5.; t = linspac( a, b, N + 1 ); gnrat intrvals x = xp( t - xp( -t ) ); f = x. ( 1 + xp( -t ) ). x.^./ ( 1+ xp( x - ta ) ); y = trapz( t, f ); if N > 1 tst for convrgnc rr = abs( y - y_old ); if rr < tol brak; nd nd y_old = y; nd if N == Nmax rror( 'Incras th maximum numbr of itrations.') nd y = y./ gamma( + 1 ); 13