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

Size: px

Start display at page:

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

Frederick York
6 years ago
Views:

1 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 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. To s how thy aris, considr computing th quilibrium lctron concntration pr unit volum in a thr-dimnsional smiconductor with a parabolic conduction band from th xprssion, g(e)de n = g(e) f (E)dE = ( 1+ E E ) k B, (1) T E C E C whr g(e) is th dnsity of stats, f (E) is th rmi function, and E C is th conduction band dg. or thr dimnsional lctrons, ( m ) 3/ g3d ( E) = E E 3 π C () which can b usd in qn. (1) to writ n = ( ) 3/ m E EC de 3 ( E E ) kbt π E 1+ C. (3) By making th substitution, qn. (3) bcoms ε = ( E E C ) k B T (4) n = whr w hav dfind ( ) 3/ mkt B 3 ε dε, (5) π 1 ε +

2 ( ) E E k T. (6) C B By collcting up paramtrs, w can xprss th lctron concntration as whr n = N 3D π ( ) (7) N π mkt = h B 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 qn. (7) can b writtn as 3D ( ) n= N. (11) 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 dimnsions, th dnsity-ofstats is ( ) m gd E =, (1) π and by following a procdur lik that on w usd in thr dimnsions, on can show that th lctron dnsity pr unit ara is

3 ( ) n = N (13) S D whr and N mkt B D = π, (14) ε dε ( ) = = ln ( 1+ ε ) (15) 1+ is th rmi-dirac intgral of ordr, which can b intgratd analytically. inally, in on dimnsion, th dnsity-of-stats is g 1D ( E) = m 1 π E E C (16) and th quilibrium lctron dnsity pr unit lngth is whr ( ) n = N (17) L 1D N 1D 1 mkt B = (18) 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 j 1 ε dε j( ) Γ ( j + 1), () 1 + xp( ε )

4 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 qn. (): 1 ε dε ( ) Γ (1/ + 1), (a) 1 + ε so w nd to valuat Γ(3 / ). Using qns. (1b) and (1c), 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 qn. (1). or mor practic, us th gnral dfinition, qn. () and qns. (1a-c) to show that th rsults for ( ) and ( ) 1/ agr with qns. (15) and (19). 3. Drivativs of rmi-dirac Intgrals rmi-dirac intgrals hav th proprty that d d j =, (3) j 1 which oftn coms in usful. or xampl, w hav an analytical xprssion for ( ), which mans that w hav an analytical xprssion for ( ) 1,

5 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 [1] p k, = 1 (6a) ( ) ( ) p = 1+ i p k+ 1 i p i = 1,, k 1 (6b) k, i k 1, i k 1, i 1 pkk, = pk 1, k 1. (6c) 4 or xampl, to valuat 4( ) = ( 1+ ) P( ) gnratd from qns. (6a-c) as [1], polynomial cofficints ar p, = 1 p1, = 1 p1,1 = p, = 1 (7) 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, ( ) (9) J

6 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 > [, 3] 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 t =, ( ) 1 n 1 ( 1 1 n t = = ) ( n) n, and ( n) μ ζ μ= 1 ζ is th Rimann zta function. Th xprssions for th rmi-dirac intgrals in th dgnrat limit ( >> ) com from qn. 1 (3) as ( ) j+ Γ ( + ) blow. [4]. Spcific rsults for svral rmi-dirac intgrals ar shown j j ( ) (31a) π 3/ 4 ( ) 3 π (31b) 1 1( ) (31c) 5/ 8 3/( ) 15 π (31d) 1 3 ( ) 6 (31) Now w rlat th complt xpansion in qn. (3) to th Sommrfld xpansion [5, 6]. Th Sommrfld xpansion for a function H ( ε ) is xprssd as whr n 1 d H( ε) f ( ε) dε = H( ε) dε + an H n 1 ( ε ) n= 1 dε ε = (3) = , (33) a n n n n and it is notd that an = tn. Thn th Sommrfld xpansion for th rmi-dirac intgral of j H ε = ε Γ j+ 1, and th rsult is ordr j can b valuatd by ltting ( ) ( )

7 t. (34) j+ 1 n J( ) = n n= Γ ( j + n) Equation (34) is th sam as qn. (3) xcpt that th scond trm in qn. (3) is omittd [3]. In th dgnrat limit, howvr, th scond trm in qn. (3) vanishs, so th qns. (3) and (34) giv th sam rsults as qns. (31a-). 5. Approximat Exprssions for Common rmi-dirac Intgrals rmi-dirac intgrals can b quickly valuatd by tabulation [, 4, 7, 8] or analytic approximation [9-11]. W brifly mntion som of th analytic approximations and rfr th radr to a Matlab function. Bdnarczyk t al. [9] proposd a singl analytic approximation that valuats th rmi-dirac intgral of ordr j = 1/ with rrors lss than.4 [1]. Aymrich- Humt t al. [1, 11] introducd an analytic approximation for a gnral j, and it givs an rror of 1. for 1/ < j < 1/ and.7 for 1/ < j < 5/, and th rror incrass with largr j. Th Matlab fuction, D_int_approx.m [13], calculats th rmi-dirac intgral dfind in qn. (1) with ordrs j 1/ using ths analytic approximations. Th sourc cod of this rlativly short function is shown in th Appndix. If a bttr accuracy is rquird and a longr CPU tim is allowd, thn th approximations proposd by Haln and Pulfry [14, 15] may b usd. In this modl, svral approximat xprssions ar introducd basd on th sris xpansion in qn. (3), and th rror is lss than 1-5 for 1/ j 7 / [14]. Th Matlab function, Djx.m [13], 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. 6. 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 [16]. 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. Th doubl prcision (ps, ~ ) can b achivd aftr 6 to 5 itrations [16]. Th Matlab function, D_int_num.m [13], valuats th rmi-dirac intgral numrically using th composit trapzoidal rul following th approach in [16]. Th sourc cod is shown in th Appndix. This approach provids vry high accuracy, but th CPU tim is considrably longr.

8 Rfrncs [1] M. Goano, "Algorithm 745: computation of th complt and incomplt rmi-dirac intgral," ACM Trans. Math. Softw., vol. 1, no. 3, pp. 1-3, [] R. Dingl, "Th rmi-dirac intgrals ( ) ( ) 1 p ε! ( 1) 1 p = p ε + dε," Applid Scintific Rsarch, vol. 6, no. 1, pp. 5-39, [3] R. B. Dingl, Asymptotic Expansions: Thir Drivation and Intrprtation. London: Acadmic Prss, [4] 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 , [5] 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. [6] N. W. Ashcroft and N. D. Mrmin, Solid Stat Physics. Philadlphia: Saundrs Collg Publishing, [7] 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. 59-4, [8] 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. [9] D. Bdnarczyk and J. Bdnarczyk, "Th approximation of th rmi-dirac intgral ( )," Physics Lttrs A, vol. 64, no. 4, pp , [1] 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/ [11] X. Aymrich-Humt,. Srra-Mstrs, and J. Millan, "A gnralizd approximation of th rmi--dirac intgrals," J. Appl. Phys., vol. 54, no. 5, pp , [1] J. S. Blakmor, "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. [13] R. Kim and M. S. Lundstrom (8), "Nots on rmi-dirac Intgrals (nd Edition)," Availabl: [14] 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 , [15] 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, [16] 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.

9 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 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 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

10 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 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 N: numbr of itrations rr: rror 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.^ 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 );

Notes on Fermi-Dirac Integrals 3 rd Edition

Notes on Fermi-Dirac Integrals 3 rd Edition 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