Monte Carlo Simulation of Impact ionization processes in silicon

Transcription

1 Mnte Carl Simulatin f Impat inizatin presses in silin K. Taniguhi, M. Yamaji, T. Kunikiy, Y. Kamakura, H. Takenaka, M. Mrifuji, K. Snda and C. Hamaguhi Department f Eletrni Engineering, saka University Suita, saka, 565 Japan bstrat The energy-dependent impat inizatin rate in silin is derived by a first-rder perturbatin thery. The sattering rate is lse t that btained experimentally with a sft x-ray pht-emissin spetrspy. The reasnableness f the alulated results is als supprted by the lse agreement f simulatin results t available experimental data. Key features f the derived impat inizatin rate are (1) impat inizatin rate fr initial eletrns with energy belw 3 ev shws strng anistrpy, and (2) energy-dependent impat inizatin rate has a large pwer expnent. Fr pratial devie simulatin, we als derived a simple analytial expressin f impat inizatin rate under expnentially varying eletri field nditins. I. Intrdutin The dwn-saling f MSFETs indues higher eletri fields in the hannel beause the pwer supply vltage has been saled less aggressively than devie gemetries. In deep submirn MSFETs, the number f ht arriers is expeted t inrease quite rapidly, leading t the degradatin f devie harateristis. In rder t ensure lng term peratin f MSFETs, it is essential t mdel the behavirs f high energy eletrns. Devie simulatrs develped in the last deade nw make it pssible t reprdue devie harateristis but nne f these an aurately estimate the distributin f high energy arriers. Nte that high energy arriers diretly affet lng term reliability f devies as well as eletrn injetin effiieny in FLSH memry. Fr mre preditable T-CD, high energy arriers have t be simulated mre aurately. This requires the use f preise physial mdels instead f a traditinal parameter fitting apprah. mng several physial mdels, an impat inizatin rate mdel is far mre imprtant in rder t estimate the distributin f high energy arriers. The aims f this paper are twfld (1) t derive impat inizatin rate theretially and a simple impat inizatin mdel appliable fr pratial devie simulatin and (2) t verify the reasnableness f the derived impat inizatin rate thrugh the mparisn with available experimental data. II. Thery f impat inizatin Impat inizatin press is a kind f eletrn-eletrn interatin taking plae at the presene f high eletri field. When a high energy eletrn llides with a valene eletrn, it gives up its kineti energy t the valene eletrn whih is inized ver the band gap. Thus tw ndutin eletrns and a hle are left. 19

2 Thery f impat inizatin press in silin has been well established[l]. The impat inizatin rate is btained frm the Fermi's glden rule. SuXh 2 -» l', 2') = y [\M a \ 2 + M \M a - M b \ 2 ] x S(ei + e 2 - e v - e v ) Where 1 and 2 dente the ndutin and valene eletrn states befre sattering, while 1' and 2' the ndutin eletrn states after sattering. We used the Culmb ptential as an interatin Hamiltnian. The diret matrix element M a is given in the frms as M a = l^{r 1 )4> 2 {r 2 ) e 2 4we(q,u)\ri r 2 \ <f>i(ri)<h(r 2) T alulate the matrix elements, we need (1) wave funtins f eletrns invlving impat inizatin press and (2) a frequeny- and wave vetr-dependent dieletri funtin. fter expanding the Culmb ptential int Furier series, the diret matrix element is given by M a = J2 Tvl MGv^k^^G^k^^Guki) G l,g 2,G 1 G 2, q Ve W> u > x 2 (G 2, k 2 )S(-k v + k x - k 2 > + k 2 + G) In the alulatin, bth mmentum and energy amng the partiles are nserved thrugh the delta funtins. ls, bth the nrmal and umklapp presses were tk int aunt. Final wave vetr-dependent impat inizatin rate was derived frm the integratin ver an eightdimensinal k spae numerially. In ur alulatins, we used 113 plane waves and 15 reipral lattie vetrs. III. Calulated impat inizatin rate Figure 1 shws the alulated impahnizatin rate as a funtin f initial eletrn energy. The alulated results satter in a rather wide range fr initial eletrns with kineti energy belw 3 ev beause f its strng anistrpi nature. The anistrpy diminishes with inreasing the eletrn energy. In Fig. 1, several reprted analytial impat inizatin rates[2-5] are als pltted. lthugh they differ in three rders f magnitudes, they all reprted that the alulated impat inizatin effiients agree with experimental data. This means that nne f these impat inizatin rates have nt been well verified. In ther wrds, simulated results uld be fitted t any experimental data by simply adjusting phnn sattering rates. Reently Cartier et al. reprted a new mbined experimental and theretial effrt t find the eletrn-hle pair prdutin rate[6]. This was ahieved by using sft X-ray phtemissin spetrspy and by perfrming Mnte Carl simulatins. The thin slid urves shwn in Fig. 1 is the results f their experiment. lthugh there exist sme undulatins in the urves due t an artifiial fitting t three sets f Keldysh frmula[7], ur alulated data are essentially the same as theirs. Fr mre pratial use, we derived analytial frm f impat inizatin rate averaged ver all initial eletrn states with a given energy. Su.(e) = 1-0 x 10 n (e - l.uv) 4 * Cmpared with the Keldysh frm with pwer expnent f tw, the new istrpi impat inizatin rate has muh larger pwer expnent f 4.6, indiating sft impat inizatin threshld. Figure 2 shws the impat inizatin rates based n ther first priniple alulatins[8-11]. The verall trend f the alulated results shws similar harateristis sine the impat 20

3 inizatin rate simply reflets the energy band struture f silin. Hwever, ne pint t nte is that there still exists abut ne rder f magnitude disrepany amng the reprted values. The reasns fr the disrepany are nt larified yet. There may be several reasns hies f the integratin methd, pseudptential frm fatrs used, dieletri funtin, energy allwane used in the numerial alulatin. IV. Cmparisn with experimental results (1) Transient impat inizatin using anistrpi and istrpi I.I. rates In rder t study the reasnableness f the derived istrpi impat inizatin rate, we simulated the number f impat inizatin events under nn-steady state nditins by using istrpi and anistrpi sattering rates. N appreiable differenes between the tw ases indiate that it is reasnable t use istrpi impat inizatin rate instead f mre elabrate anistrpi nes withut lsing any physial meaning. In additin, the use f the istrpi impat inizatin rate greatly imprves the effiieny f Mnte Carl simulatin in terms f memry apaity as well as mputatinal time. (2) Impat inizatin effiient The number f impat inizatin events prdued by ne arrier per unit length is defined as impat inizatin effiient, whih varyies with the eletri field as exp(-b/e). The pen irles shwn Fig. 3 are the simulated impat inizatin effiient under nstant eletri field. Gd agreement with the experimental data demnstrates the validity f the impat inizatin rate derived. (3) Nn-lal impat inizatin effiient in expnentially varying eletri field In MSFETs eletri field in the velity saturatin regin hanges s rapidly that the hannel eletrns are n mre equilibrium. rding t simulatin studies, the eletri field in MSFETs was fund t vary expnentially with distane frm the pinh-ff pint tward the drain. The data pints in Fig. 4 shw alulated impat inizatin effiients under expnentially varying eletri field nditins. ll the pints belw the slid line are the data alulated under inreasing eletri field nditin, while the slid pints abve the slid line are the data fr dereasing eletri field nditins. Simulated harateristis length,, f the velity saturatin regin is simply expressed with the gate xide thikness and juntin depth. Figure 4 shws tw interesting features (l)under the inreasing eletri field, nn-lal impat inizatin effiient is signifiantly smaller than that in equilibrium state and (2) at lwer eletri field, the derease f the impat inizatin rate bemes mre prnuned. (4) Simulatin f substrate urrent Figure 5 shws the substrate urrents alulated using tw different impat inizatin mdels. The slid line shws the experimental data. nventinal drift-diffusin simulatin using the lal impat inizatin effiient verestimates substrate urrent. Hwever, the use f the nn-lal impat inizatin effiient shwn in Fig. 4 results in a reasnable agreement with experimental data. Nte that even quite simple devie simulatr based n the drift-diffusin mdel an predit impat inizatin urrent rretly ne the nn-lal impat inizatin effiient is implemented in a devie simulatr. (5) Quantum yield We simulated quantum yield t further verify the derived impat inizatin rate. By using p-h MSFET, eletrns are injeted frm the gate eletrde t the silin substrate thrugh the gate xide. If the injeted eletrns has enugh energy t make impat inizatin, sme f them reate eletrn-hle pairs in the silin substrate. The generated hles are lleted int the inversin layer while eletrns flw t the substrate eletrde. By measuring the eletrde urrents, the average number f generated eletrn-hle pairs per injeted eletrn is alulated. This is the definitin f quantum yield. 21

4 Figure 6 shws the alulated quantum yields using the MC simulatin tgether with experimental data[13-15]. The alulated data agree quite well with the data reprted by Takagi and Triumi. The disrepany amng the reprted quantum yields may riginate frm the alibratin errr f the injeted eletrn energy. (6) Transient impat inizatin We investigated the anistrpi impat inizatin in Si MSFETs reprted by Takagi[16]. The devies used fr their experiment were n-h Si MSFETs with single drain strutures n the (100) surfae. The measured data shws the gate vltage dependene f the substrate urrent with different hannel diretins. They fund that the substrate urrent alng the 45 ff the (il) diretin, meaning (001) diretin, bemes larger than that alng 0 r 90. The anistrpy dereases with inreasing substrate urrent. We simulated impat inizatin press with similar devie strutures. The alulated impat inizatin agrees well with the experimental data; the anistrpy diminishes with inreasing inizatin rate. V. Cnlusins We theretially derived impat inizatin rate. The energy dependene f the rate was fund t be the same as the pair generatin rate btained frm the sft x-ray pht-emissin spetrspy. The reasnableness f the mdels is als supprted by the lse agreement f the simulatin results t available experimental data. Key features f the derived impat inizatin rate are (1) impat inizatin rate fr initial eletrns with the energy belw 3 ev shws strng anistrpy whih diretly reflets energy band struture f silin, (2) energy dependent impat inizatin rate shws large pwer expnent, indiating the sft threshld f impat inizatin, (3) we demnstrated that bth istrpi and anistrpi impat inizatin rates lead t the same results. This means that fr mre pratial Mnte Carl simulatin we an use the energy dependent impat inizatin rate instead f mre mpliated wave-vetr dependent impat inizatin. Using the Mnte Carl simulatin, we derived a simple analytial expressin f nn-lal impat inizatin effiient under expnentially varying eletri field nditins. We demnstrated that the newly derived impat inizatin effiient well reprdues substrate urrent f MSFETs even in a drift-diffusin devie simulatin. Referene [I] E.. Kane, Phys. Rev., 159, 624 (1967). [2] J. Y. Tang and K. Hess, J. ppl. Phys., 54, 5139 (1983). [3] M. V. Fishetti and S. E. Laux, Phys. Rev., B38, 9721 (1988). [4] R. Thma, H. J. Peifer, W. L. Engl W. Quade, R. Brunetti and C. Jabni, J. ppl. Phys., 69, 2300 (1991). [5] Th. Vlgelsang and W. Hansen, J. ppl. Phys., 70, 1493 (1991). [6] E. Cartier, M. V. Fishetti, E.. Eklund, and F. R. MFeely, ppl. Phys. Lett, 62, 3339 (1993). [7] L. V. Keldysh, Sv. Phys., JETP 37, 509 (1960). [8] N. San and. Yshii, Phys. Rev., B45, 4171 (1992). [9] J. Bude, K. Hess and G. J. Iafrate, Phys. Rev., B45, (1992). [10] J. Bude and K. Hess, J. ppl. Phys., 72, 3554 (1992). [II] J. Wang and K. F. Brennan, J. ppl. Phys., 75, 313 (1994). [12] Y. Kamakura, H. Mizun, M. Yamaji, M. Mrifuji, K. Taniguhi, C. Hamaguhi, T. Kunikiy and M. Takenaka, J. ppl. Phys., 75, 3500 (1994). 22

5 [13] C. Chang, C. Hu and R. W. Brdersen, J. ppl. Phys., 57, 302 (1985). [14] D. J. DiMaria, T. N. Theis, J. R. Kirtley, F. L. Pesavent, D. W. Dng, and S. D. Brrsn, J. ppl. Phys., 57, 1214 (1985). [15] S. Takagi and. Triumi, Extended bstrats f the Japan Siety f pplied Phyis, Spring Meeting, 1993, bstrat N.31p-ZX-10. [16] S. Takagi and. Triumi, Teh. Dig. f Internatinal Eletrn Devies Meeting, p.711, S (8)=1.0X10 11 (S-1.1)4.6 1/se E 10' I ' I ' i i j. Thma ef a/. ^ r Tang and Hess L V = // 11,.7 r.7.7 ;' fj I r w. r 'ifj. r,. Jl. ** m s Vge sang and Haensh i i i i i -,, n i -''ys^} P***~~ Carfier etal Eletrn Energy (ev) Fishetti and Laux -» 1st band 2nd band, * 3rd band j i. il <n CD +-» C DC g " t k this wrk Wang etal. San etal. Kane - Bude etal. - Bude etal. (CB) Eletrn Energy (ev) Tl Figure 1. Calulated impat inizatin rates. Bld slid line represents a best fitted urve t the alulated impat inizatin rates. The ther fur lines exept the slid lines represent impat inizatin rate expressed by Keldysh frmula. The thin slid urves shw the impat inizatin rate fitted t a set f three Keldysh frmula. Figure 2. Cmparisn f reprted impat inizatin rates averaged ver all initial eletrn states as a funtin f energy measured frm the bttm f the ndutin band. 23

6 f~ 10 5 E 0) r 3= N "E 10* Mnte Carl "^ Mll etal. Lee etal. X Grant \ \ Van verstraeten et al / F (m/mv) Figure 3. Calulated impat-inizatin effiient as a funtin f reipral eletri field with reprted experimental data.,, 1 E 10" T ' ' CD ^ Q C N ~~ 10 2 L m L._ «?. B ; ~ ^ * i D * '". 4 Cnstant Field ; X =0.05um X = -0.05um- X=0.1um X = -0.1 um ' 4 X=0.2um * X=--0.2um - 1 ' 1 1. * * N. *» a D ^"v^ ^""""N^ Eletri Field -1 [m/mv] *> *'. " ^ ^-J 1 Figure 4. Calulated impat inizatin effiients under expnntially varying eletri field nditins. 10 r Vg(V) L/W= 1.0/10.0 xm) V d = 3V t^lgnm Xj=0.18nni Figure 5. Simulated substrate urrent tgether with experimental result at Vp=3.0V fr an n-h MSFET with the hannel length f 1.0 fim. Gate xide thikness f 10 nm and hannel dping nentratin f 1.2xl0 17 /m CD >- 10' -1. E C0 3 es-i Mnte Carl Chang etal. DiMaria fa/. Takagi etal Eletrn Energy (ev) Figure 6. Quantum yield at rm temperature as a funtin f eletrn energy. pen irles shw the alulated results. Curves are the experimental results. 24