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 lakmor [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 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) kt π ħ E 1+ C. (3) y making th substitution, Eq. (3) bcoms ε = ( E E C ) k T, (4) 1

2 n = whr w hav dfind 3/ mkt 3 ε dε, (5) π 1 ε ħ + C E E k T. (6) y collcting up paramtrs, w can xprss th lctron concntration as n= N3D ( ), (7) π whr N π mkt = h 3D 3/ (8) is th so-calld ffctiv dnsity-of-stats and ε dε ( ) (9) 1+ xp( ε ) is th rmi-dirac intgral of ordr. 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, ε dε ( ) π, (1) 1+ xp( ε ) so that Eq. (7) can b writtn as n= N. (11) 3D 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 mkt D = π ħ, (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 π ħ E E C, (16) and th quilibrium lctron dnsity pr unit lngth is whr L 1D n = N, (17) N 1D 1 mkt =, (18) ħ π and 1 ε dε ( ) = π (19) 1 ε + is th rmi-dirac intgral of ordr, 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 j 1 ε dε j Γ ( j + 1), () 1 + xp( ε ) whr Γ is th gamma function. Th Γ function is just 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 from Eq. (): 1 ε dε ( ) Γ (1/ + 1), (a) 1 + ε so w nd to valuat Γ(3 / ). Using Eqs. (1b-c), w find, so is valuatd as 1 π Γ (3/ ) =Γ (1/ + 1) = Γ (1/ ) =, (b) ε dε ( ) 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 agr with Eqs. (15) and (19). 3. Drivativs of rmi-dirac Intgrals rmi-dirac intgrals hav th proprty that d d j =, (3) j 1 4

5 which oftn coms in usful. or xampl, w hav an analytical xprssion for mans that w hav an analytical xprssion for 1,, which d 1 1 = = d 1 +. (4) Similarly, w can show that thr is an analytic xprssion for any rmi-dirac intgral of intgr ordr, j, for j, = P, (5) ( ) ( 1+ ) j j j ( ) whr P k is a polynomial of dgr k, and th cofficints p ki, 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, (6b) ki, k 1, i k 1, i 1 pkk, = 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) j which mans that for all ordrs, j, 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 j > 1 and > [, 5, 6] whr 1 n ( 1) 1, (3) n j+ 1 tn j = + cos n ( π j) j 1 n ( j 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 ( ) j+ Γ ( + ) blow. [7]. Spcific rsults for svral rmi-dirac intgrals ar shown j j ( ), π (31a) 3/ 4 ( ), 3 π (31b) 1 1, (31c) 5/ 8 3/( ), 15 π (31d) (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 kt 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) 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+ ( kt) H ( E). (36) 6 If w scal E by kt in Eq. (34), ε EkT, 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 j can b valuatd by j H ε = ε Γ j+ 1 in Eq. (37), and th rsult is ltting t. (38) j+ 1 n j = n n= Γ ( j + 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. dnarczyk t al. [1] proposd a singl analytic approximation that valuats th rmi-dirac intgral of ordr j = with rrors lss than.4 [3]. Aymrich- Humt t al. [13, 14] introducd an analytic approximation for a gnral j, and it givs an rror of 1. for < j < and.7 for < j < 5/, and th rror incrass with largr j. Th Matlab fuction, D_int_approx.m, [1] calculats th rmi-dirac intgral dfind in Eq. (1) with ordrs j 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 j 7/ [15]. Th Matlab function, Djx.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 j = [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 ( nn ) =. 3D Joyc and Dixon [17] xamind a sris approach that givs Δ.1 for max 5.5 [3], and a simplr xprssion from Joyc [18] givs Δ.3 for max 5 [3]. Nilsson proposd two diffrnt full-rang ( 1 ) xprssions [19] with Δ.1 and Δ.5 [3]. Nilsson latr prsntd two mpirical approximations [] that giv Δ.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 j > 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 nanohu.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 j j = ( with spacing =.1 using approximat xprssions ( D_int_approx.m and Djx.m ) and th rigorous numrical intgration ( D_int_num.m ) with doubl-prcision. Th rlativ rrors, whr,approx of th approximat xprssions ar calculatd as, approx, num, 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. G 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 numrical intgration ( D_int_num.m ). (A) Rlativ rror from D_int_approx.m. () Rlativ rror from Djx.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. lakmor, "Approximations for rmi-dirac intgrals, spcially th function usd to dscrib lctron dnsity in a smiconductor," Solid-Stat Elctron., vol. 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.. 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. r, 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. dnarczyk and J. dnarczyk, "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, 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, 3/,, 5/, 3, and 7/" [J. Appl. Phys. 57, 571 (1985)]," J. Appl. Phys., vol. 59, no. 6, p. 64, [17] W.. 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.. 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. 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. Natarajan 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, j ) Analytic approximations for rmi-dirac intgrals of ordr j > - Dat: Sptmbr 9, 8 Author: Rasong Kim (Purdu Univrsity) Inputs ta: ta_ j: D intgral ordr Outputs y: valu of D intgral (th "script " dfind by lakmor (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. dnarczyk and J. dnarczyk, Phys. Ltt. A, 64, 49 (1978) []J. S. lakmor, 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 j < - rror( 'Th ordr should b qual to or largr than -.') ls x = ta; switch j cas y = log( 1 + xp( x ) ); analytic xprssion cas 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( j + 1 ) to mak it consistnt with [1] and []. a = 14.9; b =.64; c = 9 / 4; y = ( ( j + 1 ) ^ ( j + 1 )./ ( b + x + ( abs( x - b ).^ c + a ).^ ( 1 / c ) ).^ ( j + 1 )... + xp( -x )./ gamma( j + 1 ) ).^ -1./ gamma( j + 1 ); othrwis Modl proposd in [4] Exprssions from qs. (6)-(7) of [4] Th intgral is dividd by gamma( j + 1 ) to mak it consistnt with [1] and []. a = ( / 4 ( j + 1 ) + 1 / 4 ( j + 1 ) ^ ) ^ ( 1 / ); b = j; c = + ( - sqrt( ) ) ^ ( - j ); y = ( ( j + 1 ) ^ ( j + 1 )./ ( b + x + ( abs( x - b ).^ c + a ^ c ).^ ( 1 / c ) ).^ ( j + 1 )... + xp( -x )./ gamma( j + 1 ) ).^ -1./ gamma( j + 1 ); nd nd 1

13 D_int_num.m function [ y N rr ] = D_int_num( ta, j, tol, Nmax ) Numrical intgration of rmi-dirac intgrals for ordr j > -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_ j: 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 lakmor (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. 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.^ j./ ( 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( j + 1 ); 13

Notes on Fermi-Dirac Integrals 2 nd Edition. Raseong Kim and Mark Lundstrom Network for Computational Nanotechnology Purdue University

Notes on Fermi-Dirac Integrals 2 nd Edition. Raseong Kim and Mark Lundstrom Network for Computational Nanotechnology Purdue University Nots on rmi-dirac Intgrals nd Edition Rasong Kim and Mark Lundstrom Ntwork for Computational Nanotchnology Purdu Univrsity Sptmbr 3, 8 1. Introduction rmi-dirac intgrals appar frquntly in smiconductor