Transport Coefficients For A GaAs Hydro dynamic Model Extracted From Inhomogeneous Monte Carlo Calculations

Transcription

1 Tanspot Coeffcents Fo A GaAs Hydo dynamc Model Extacted Fom Inhomogeneous Monte Calo Calculatons MeKe Ieong and Tngwe Tang Depatment of Electcal and Compute Engneeng Unvesty of Massachusetts, Amhest MA 0003, USA Abstact The valdty of enegydependent elaxaton tmes often used n a GaAs hydodynamc model has been caefully examned usng the selfconsstent Monte Calo smulaton We have found that those tanspot coeffcents assocated wth the ntevalley tansfe fom the lowe to the uppe valley ae not snglevalued functons of the aveaged electon enegy n the valley If, nstead, the valley populaton ato o the aveage enegy weghted by the valley populaton ato s used a substantal mpovement n accuacy can be acheved I Intoducton Conventonally, macoscopc (aveaged) elaxaton tmes such as, p and TW appeang n the hydodynamc (HD) tanspot equatons ae assumed enegydependent and detemned by pefomng Monte Calo (MC) calculatons unde steady state and homogeneous feld condtons These expessons ae often extended to the case of nhomogeneous felds wthout any justfcaton Sandbon et al [], usng the MC smulaton, found that both the enegy and momentum elaxaton tmes n an equvalent snglevalley model unde the tansent condton dffe vey much fom the steady state and homogeneous feld values Yamada [2] also obseved the dscepancy n the elaxaton tmes between the homogeneous and nhomogeneous feld condtons He suggested that the elaxaton tmes should depend not only on the enegy but also on the valley populaton In ths study, nstead of a snglevalley model, a theevalley HD model fo GaAs has been developed The tanspot coeffcents appeang n the HD model fo homogeneous and nhomogeneos feld condtons ae evaluated by a sngle patcle MC smulaton pogam and a multpatcle selfconsstent MC smulaton pogam, espectvely A one dmensonal N + N N + GaAs ballstc dode was used as a test devce Ths appoach allows us to goously examne the valdty of the enegy dependence of each elaxaton tme as well as povdes us valuable nfomaton fo a moe appopate descpton of the elaxaton tmes II Moments of the Boltzmann Tanspot Equaton The HD tanspot equatons can be obtaned by takng vaous moments of the Boltzmann tanspot equaton (BTE) [3] Extendng the wok of [4], [5] to a multvalley system, we obtan the followng steadystate consevaton equatons fo the tth valley vm)[ + + JS], a) 'nj 'nk T"njt "^nk 65

2 Lv(n U l )F=^^ q A (2) n pa fjj Pk lv (^)«!# = Il^!»5 5 + a]& + =B, (3) n TW Ttyj T~Wk ^ T e j Tl T e k («) (Wj + &) F = ^ ^ *, (4) n /*«M«J M«fe whee F; = (t>),7 = ( & ), W; = fa), St = (ve) and R = (e^k) In the conventonal HD tanspot model, the tanspot coeffcents ( n, f,tw,etc) ae usually assumed to depend on the aveage enegy,w{) Ths appoach gnoes the dependence of the tanspot coeffcents on the shape of the dstbuton functon III SelfConsstent Monte Calo Smulaton To examne the accuacy of enegydependent tanspot coeffcents, we begn wth a gous soluton of the steadystate BTE by the MC method The smulaton pogam uses an analytcal multvalley, nonpaabolc band The followng types of scatteng ae taken nto account acoustc phonon scatteng, optcal phonon scatteng, pola optcal phonon scatteng, onzed mputy scatteng, equvalent and nonequvalent ntevalley scatteng The vaous scatteng paametes ae smla to those used n [6] In ths wok, a onedmensonal N + N N + GaAs stuctue wth a 05 fm Negon was examned The dopng denstes of the thee layes wee N d = X 0 7 cm' 3, x 0 6 cm" 3 and x 0 7 cm~ 3, espectvely The appled bas was 20 volts Fg dsplays the dopng densty and electc feld pofles wthn the devce as obtaned fom the selfconsstent MC (SCMC) smulaton Fg 2 shows the Y valley velocty and enegy pofles I Results and Dscussons At each poston wthn the devce we evaluated the tanspot coeffcents and the aveage enegy n each valley by the SCMC pogam We found that these coeffcents ae geneally a functon of the local aveage enegy n the valley except fo those due to the ntevalley tansfe fom the lowe to the uppe valley (e T > L, * X, and L» X) Fgs3,5 and 7 espectvely dsplay the Y * L ntevalley tansfe coeffcents, n x,, and TWTL vesus the aveage T valley enegy, W, fo both the homogeneous and the nhomogeneous feld calculaton The "hysteess" loops clealy ndcate that none of them can be descbed as a snglevalued functon of W Fo these cases, the enegydependent tanspot coeffcents whch wee obtaned fom the homogeneous feld calculaton always undeestmate the actual one n the nceasng feld egon and oveestmate the same n the deceasng feld egon If, nstead, W s weghted by the valley populaton ato (e, ^W) then the hystess loop fo ntl can be sgnfcantly educed (see Fg 4) Ths s motvated by the fact that the valley populaton ato moe o less eflects the facton of electon populaton whch have suffcent enegy to tansfe fom the lowe to the uppe valley We also found that usng the valley populaton ato alone (e, = ) the hystess a seen loop can be consdeably educed fo /pl and TWTL * n Fgs6 and 8,espectvely 66

3 The esult fo T C TL s vey smla to TWTL an d that fo f t L s smla to fl Once the hystess loop s educed, these tanspot coeffcents can now be modelled empcally as snglevalued functons of the valley populaton ato o the enegy weghted by the valley populaton ato Conclusons A SCMC smulaton pogam was used to examne the conventonal assumpton of enegydependent tanspot coeffcents n a multvalley system We found that the tanspot coeffcents elated to the ntevalley tansfe fom the lowe to the uppe valley (e, + L, T X and L + X) ae not a snglevalued functon of the aveage enegy n the valley A substantal mpovement n the accuacy can be acheved f the valley populaton ato o the aveage enegy weghted by the valley populaton ato s used Acknowledgement The authos acknowledge Pof D Navon fo caefully eadng the manuscpt Ths wok was suppoted n pat by NSF Gant ECS Refeences [] PA Sandbon, A Rao and P A Blakey, IEEE, Tans Electon Devces, ol36, No7, pp244253, Jul, 989 [2] Yoshno Yamada, IEICE Tans, ole73, No2, pp255259, Feb, 990 [3] Blotekjag, K, IEEE Tans Electon Devces,ol ED7, Nol, pp3847, Jan, 970 [4] SC Lee and Tw Tang, Sold State Elec, ol35, P56569, Ap, 992 [5] Tw Tang,S Ramaswamy and J Nam,to appea n IEEE Tans Electon Devces [6] MA Lttlejohn, JR Hause, TH Glsson, J Appl Phys, ol48, pp , Nov,

4 E j f / / IA 6 x(um) l 0'» Fgue Imputy dopng pofle and ts selfconsstent electc feld fo an abupt N + N N + stuctue wth an appled bas of 20 volts 0" A ' I II A IA A x(jun) * 0 * ST Fgue 2 Aveage electon enegy and velocty n the T valley, 0 0 H m " ';' ' ' ^ ' I 0 0 lllll lllll 0 bomot ldx u s^ A 03 ^ nthomo t^^ homo %n 2 3 (njn T )Wf(ev) l"l Fgue 3 Cae elaxaton tme, vesus the aveage T valley enegy,^ Fgue 4 Cae elaxaton tme, nl, vesus 68

5 00 0 'T~ " S " ' ^ "* ^ txllll '!! 4,» 'A '"x >, ^ N ">v N»»»»»* "^^ 0 nhomo fa 0 oa OJ W (ev) *" 03 0 nhomo ^fl homo fl 4 («d*) ~z Fgue 5 Cae moblty, fl, vesus the aveage T valley enegy, Wp Fgue 6 Cae moblty, f^l, vesus (j*) 0 p [ t ;" j IOOJO IOJO ' 3 jj IJO J* ^ K " ** 4 3 ~z W g IJO 0 nhon W (cv) 0^ ] OA OJ 0 OJO nhomo "C^YL ~ homo t^yl 4 ' ^ ^ t, c,,,,, Fgue 8 Enegy elaxaton tme, TWTL, vesus Fgue 7 Enegy elaxaton tme, TWTL, vesus the 6J * aveage T valley enegy, Wp ^'' 69