THUAT TOAN BiCGSTAB(3) TIEN DIEU KIEN DE TIM N G H I E M CUA P I O r O N G TRINH POISSON BA CHIEU
TRAN QUdC TUAN Trudng THPT Le True, Tuyen Hoa, Qudng Binh
DINH NHU THAO Trudng Dai hgc Suphgm - Dgi hgc Hue
Tom tat: Chuong trinh dua tren thuat toan BiCGSTAB(3) vdi ti^n didu kien Jaeobi dupe xay dirog de tim nghiem cua phirong trinh Poisson ba chieu.
Chuong trinh nay sau do dupc ticb hop vao chuang trinh mo phong Monte- Carlo tap hpp tu hpp va ap dung vao viec mo phong cac di-6t p-i-n ban din GaAs de kiem tra hieu nlttig cua no. Hon the nura, de so sanh toe dp hpi tu va tinh on djnh ciia thuat toan chiing toi da tten hanh khao sat su phu thupc ciia chuan Euclid ciia vecto thang du vao so vong lap cua chuong trinh con Poisson. Cac ket qua chi ra rang chuong trinh giai phuang trinh Poisson dua tren thuat toan BiCGSTAB(3) tien dieu ki^n co cac dSc trung giong nhu chuang trinh sii dung thuat toan BiCGSTAB(3), ham y rang vai tro cua tien dieu kien Jaeobi trong tnrcmg hpp nay la khong dang ke.
I.GI6ITHIEU
Cdc linh kien ban dSn na-nd dang thu hut su quan tam manh me ciia gidi khoa hpc do tinh ling dpng cao ciia nd va hi?n dang dupc nghien ciiu rpng rai ca v^ mat ly thuydt va thvc nghi?m [I], [2]. Cac nghien cuu thuc nghiem ndi chung la rat ton kem va mat nhieu didi gian, khong nhiing th^ con doi hoi phai sir dung cong nghe cao ma chi mpt so nude cd the thuc hi6n dupc. Cac phuong phap nghien cuu ly thuyet hay dupc lira chpn 6.k khac phyc cac han che neu tren, dac biet trong giai do^n dau ciia vifc nghien cuu [3], [4], [5]. Vdi cac linh kien na-no ban din phuong phap mo phdng Monte-Carlo tap hop tf hpp la lira chpn hang dau vdi cac im diem noi trpi la tinh chinh xdc vd tinh dn dinh.
Trong qua trinh mo phong linh kien phuang phap Monte-Carlo tap hpp tu hpp cJin cap nhat phan bo cua dien the trong linh kien thong qua vific giai phuang trinh Poisson.
Phuang trinh nay thudng duoc giai bang phuong phap sai phan hiiu ban [3], khi do phuong trinh gdc chuyen thanh mpt he phucmg trinh tuyen tinh thua eye Idn vdi hang trieu phuong trinh va hang trieu In. He phuong trinh tren thudng duoc giai bSng cac phuang phap so chay tren cac sieu may tinh vdi bp nhd cue ldn ma Viet Nam hien nay chua CO [3], [4], [5], [6], [7]. Cac phuong phap khong gian con Krylov cd uu diem la cd the thuc hien tinh toan ma khong cSn luu trO cac so lieu tmng gian vi vay cac phuong phap nay co the chay tren mot may vi tinh ca nhan vdi ciu hinh tdi thieu [6], [7], [8], [9]. Cac thuat toan BiCGSTAB, BiCGSTAB t i k dilu kien, BiCGSTAB2, BiCGSTAB(3) va GPBiCG da dupc ap dung de giai phuong trinh Poisson ba chieu.
Cdc k6t qua nghien ciiu cho thay cac phuong phdp nay cd dp chinh xac cao va then gian tinh toan ngan [10], [11], [12]. Do la dpng luc de chiing toi tign hanh tim nghiem cua Tap chi Khoa hoc va Giao due, Tmong Dai hoc Su pham Hue
ISSN 1859-1612. S6 02(26)/2013:tr. 25-32
TRAN QU6C TU.^N - DINH N H L T T H A O
phuang trinh Poisson ba chidu bang thuat toan BiCGSTAB(3) tiln dieu kien dua tren tien dieu kien Jaeobi [10], [11], [12] vdi muc dich tim ra nhung phuang phap toi uu, boat dpng on dinh hon va cho ket qua nhanh han.
Bai bdo dupc trinh bay nhu sau. Muc 2 trinh bay ve phucmg phdp tmh toan. Muc 3 trinh bay ve kit qua thu dupc va thao ludn vl tinh hieu qua ciia phuong phap. Trong phan nay chiing toi khdo sat dpng luc hpc ciia hat tai trong cdc di-6t p-i-n ban dan GaAs va chuan Euclid ciia vecta thang du de kiem tra hieu nang cua phuong phap. Muc 4 trinh bay kit luan.
2. PHLTONG PHAP
Gia sur vat lieu la dong nhat thi phuang trinh Poisson ba chieu co the dupc viet dudi dang nhu sau
aV tiV ^V
p
(1) dx- dy^ dz^ " s^'d day q) la dien the, f ^la hang s6 dien moi tinh trong linh kien va p{r,t) la mat dp dien tich do dong gdp cua cac dien tii va 16 trong dupc kich thich quang va cdc donor va acceptor trong di-6t. Dl co thi de dang thuc hien sai phan hihi ban ta chia mo Wnh Unh kien thanh cac 6 ludi va gia sii rang khoang each giiia cdc niit ludi theo cac chilu khong gian la bang nhau, Ax = Ay = Az. Tien hanh liy sai phdn hiru ban phucmg trinh (1) ta thu duoc be phuang trinh sau
'P.-l.,.t + %j-l.i + 'Pt.j.A-l - ^<P,,J,k + 'P,*\.j.k + <P. ;^!.i + V,.jM\ = - - ^ ^ Ax- , (2) dday i=\.N^, j = hN^ \kk = lN^ v&i N^, N^ va A^. ldn luot id s6 mit ludi theo cac chieu khong gian Ox, Oy vaOz. He phuang trinh (2) cd thi dupc vilt lai dudi dang mpt phuong trinh ma tran nhu sau
A<p = b, (3) d day b la ma tran cpt vdi cac ylu t6 la s6 hang b^n vl phai ciia he phuang trinh (2)
v a ^ Id mpt ma tran co dang
fa,) ('••) M fa.) k)
0 0 0 0 0
{PN,. \'«,-l,
u
(4)
THUAT TOAN BICGSTAB{3) TIEN DIEU KIEN DE TIM NGHIEM CUA PHUONG TRINH...
vdi [pj va [r j la cdc ma tran cheo cd dang
ip.)
(5)[g^} la cac ma tr^n ba dudng cheo cd dang (-2(1 +O 1
1 -2(1+ c) 1 2(1+ 0
1 2(1 + C)j
(6)
d day c = (Ax/ Ar)- = 1. Nhu da de cap, A la mot ma tran vudng doi xirng, ban the nua Id mpt ma tran thua lo^i ldn vdi cac ylu t6 khac khong chi ndm tren bay dudng cheo bao gom sdu dudng cheo nam ddi xung phia tren va phia dudi dudng cheo chinh. Ma tran A co the gom hang trieu hang vd hang trieu cot, phu thuoc vao dieu kien gidn doan hda. Trong tnrdng hpp nay A Id mot ma tran vdi kich thudc 39249 >t 39249.
Ci day thuat toan P-BiCGSTAB(3) - thuat todn BiCGSTAB(3) tiln dilu kien vdi tiln dieu kien Jaeobi dupc sii dung de xay dung mdt chuang trinh gidi phuang trinh Poisson dudi dang phuong trinh (3) [6], [7], Thu|t toan nay dupc chpn do tinh nang uu viet ciia n6. Thudt toan dupc trinh bay trong Bang 1.
Bang I. Thugt loan P-BiCGSTAB(3)
\ ^,',::,',";;".„
,( lmh' ffo- ii 1 Ukv
(•_ 1 = t) 0 .Vo = J„
^ Dm dAii I..11^
I'l' ™ ' '.-,.' I^ Ttnh ''• ';• I'o-"') 1 - ^ ",'^['
'>n' " ' i " ' " ; i j ' ' / * ' ' " ' ,,
•E *'•" l n > ; , T } ' ^ • f
_ 1 m p
> P
• " !
J 3 J ' '^T
t
U 4 i i C
'Z
•^l
;;_, j , ;^,, '
' . - ^r-i ' '' Pi ~ '^TfC'i™
^'"",'.""'
4,.j
" = ,-o"-'
;:lij;-?
"~ J'-
j j , " , ' ; / ; \
"-"'" ^'
>p '•
;:-
I '
-'
>p - - Ta y-, •••> " j
: • " ' ' ' • " ' ' ' -to -= -yj - • ' m i T. = l • ; - C ^ m I - 7 • , - ^ , ia --- -^
7 - . . . • . >p
ro = I „ + - , > , >p A . - A , + :•;•.; >p
-Vo = j'a O i l kill 1','.'. ri..'
kl! ;i;;;: ;,';;,":,';'.';,„
TRAN QUOC TUAN - DINH NHU THAO
Thuat toan nay CO th6 giai he phuong trinh (2) theo mot each khong can liru trii so lieu trung gian boi viec sit dung dang cong thiic ttrong minh da cho dai vol moi phutmg trinh trong he phuong trinh (2) thay vi sit dung dang tuong minh cua cac ma tran A va b d6i voi m6i buoc tinh toan. BS ki6m tra hieu nang ciia thuat toan, chuang trinh mm nay duoc tich hop vao chuong trinh mo phong Monte-Carlotap hop tu hop va ap dung dl mo phong dpng lire hoc cua hat tai trong diot p-i-n ban din GaAs [1]. Tien dieu kien Jaeobi duoc thuc hien tai nhung noi co kj' hieu (>P) trong Bang 1.
3. KET QUA MO PHONG VA THAO LUAN
Mo hinh cSu tnic cua di-6t p-i-n ban din GaAs g6m mpt lop ban din thuan (i) kep gifta hai lop ban din pha lap loai p va loai n nhu dupc chi ra trong Hinh 1, dp day ciia cac l*p nay, lop i, lop p va lop n tuong ung la d,, d^ va d,. Mat dp pha t^p acceptor va donor tuong ung la Af, va Afj,, cac tap duoc phan b6 theo ham Gauss tit bS mat cua cac lop p va n. Cac dien ttr va 16 tr6ng dupc kich thich quang bed cac xung laser cue ngin voi mjt do cao.
^.^-.
I^
Hinh 1. Mo hinh di-6t p-i-n GaAs
Cdc budc tinh todn chi tilt duac thuc hien dung theo phucmg phap Monte-Carlo tap hpp tu hap dp dung cho ciu tnic linh kien co kich thudc na-n6 [1], [3]. Cac hgt tai kich thich quang dupc tao ra trong linh kien bSng cdch chieu m6t xung laser vdi chilu dai xung la 12 fs va nang lupng photon la 1.49 eV. Cdc tham s6 khac dupc gia thilt cd gid trj nhu sau (/^=£/„= 50 nm, (/,= 340 nmva A^^= 0.5x10" cm•\ TV^ = 2 . 5 x 1 0 " cm'^ Mat dp hat tai quang N^, bang 5xlO'^ cm'"'. Cac tham s6 khac co the duac dm thay trong tai lieu tham khdo [1], Budc thdi gian va kich thudc 6 ludi ldn lupt la 0.25 fs vd 10 A.
Kich thudc theo ba chilu khong gian cua di-6t la L^xL xL. = 440nmx 100nmx 100nm, gid su di-6t dupc nuoi cay theo phuang Ox.
Linh kien dupc chia thdnh cdc 6 ludi khdng gian vdi Ax = Ay = Az = 5 nm. Nhu v§y ta se cd 89 nut ludi theo phuang Ox, 21 niit ludi theo phucmg Oy vd 21 mit ludi theo phuang O:. tuc la N^ = 89, A', = 21 va N. = 21. Dien trudng ngoai dupc dat vao linh kien dpc theo phuang Ox vd di-6t dupc phan cue nghjch.
Hinb 2 mo td su phu thupc cua van t6c trdi dat ciia dien tii theo thdi gian ung vdi dien trudng ngoai f^, = 100 kV/cm, duac tinh toan bdi phuong phdp Monte-Carlo tap hpp tu hop tich hop vdi hai thuat toan giai phuang trinh Poisson khac nhau P-
THU.AT TOAN B1CGSTAB(3) TI^K DIEU KIEN DE TTM NGHIEM CUA PHU'ONG TRINH... 29
BiCGSTAB(3) va BiCGSTAB(3). Kit qua cho thiy hai b6 mo phdng nay cd dp chinh xac nhu nhau, cho ket qua gdn nhu trung khdp vdi nhau.
Hinh 2. Van tdc iroi dgt cua dien lir theo phuang Ox nhu la ham ciia thai gian ung vdi E^i = 100 kV/cm, dugc tinh loan bdi phuang phdp Monle-Carlo lap hgp lu hgp lich hgp vdi
hai Ihugi loan P-BiCGSTAB(3) va BiCGSTAB(3)
Hmh 3 mo td sy phan bd dien the khdng gian trong di-6t p-i-n ban ddn GaAs theo hai phuang Ox vd Oy tgi mat cdt z = 50 nm dng vdi E^, = 100 kV/cm, dupc tinh toan bdi phuang phdp Monte-Carlo tap hpp tu hap tich hap vdi hai thuat todn P-BiCGSTAB(3) vd BiCGSTAB(3). Hai do thi trong hinh 3 gdn nhu trung nhau hoan toan. Ket qud nay cung phii hop vdi cdc kit qua da dupc cdng b6 trudc day [1], [11]. Dilu do hdm y rdng thugt todn P-BiCGSTAB(3) cung cd the giai hieu qud phuong trinh Poisson. Hinh 4 mo ta su phdn bd dien the khong gian trong di-6t p-i-n ban ddn GaAs theo phuang Ox tai mat cat y = z = 50 nm irng vdi E^, = 100 kV/cm. Hai do thi trong hinb 4 ciing gan nhu trung nhau hoan toan. Ket qua nay mot ldn niia khdng dinh tmh hieu qua cua thuat toan P-BiCGSTAB(3).
Hinh 3. Phdn bo diin thi khong gian trong di-dt p-i-n bdn ddn GaAs theo hai phucmg Ox vd Oy lai mat cdi z = 50 mn img vdiE^, = 100 kV/cm. dugc linh loan bdi phuang phdp Monle-
Carlo lap hgp lu hgp tich hgp vdi hai ihudi loan P-BiCGSTAB(3) vd BiCGSTAB(3)
TRAN QUOC TUAN - DINH NHU' THAO
Hmh 4. Phdn bd diin Ihi khong gtan Irong di-ol p-i-n bdn ddn GaAs theo phuang Ox lgi mat cat y = z = SO nm ung vdi E^, = 100 kV/cm, dugc linh todn bdi phuang phdp Monte-Carlo
igp hgp lu hgp lich hgp vdi hai thugl loan P-BiCGSTAB(3) vd BiCGSTAB(3) De so sdnh tdc dp hpi tu vd tinh 6n djnh cua thuat toan P-BiCGSTAB(3) vdi cdc dac tnmg tuong ung ciia thudt toan BiCGSTAB(3), chiing toi da tien hdnh khao sat sir phu thupc ciia chudn Euclid cua vecto thdng du vao so vong lap ciia chuong trinh con Poisson, Hinb 5. Chudn Euclid ciia vecto thdng du dupc tinh theo cong thiic [6]
woi r = b - A<p \k vecto thang du.
Sotofi):l3p[vunK]
Hinh 5. Si^phu thuoc cita logarithm oia chudn Euclid CMO vecta thang du vdo s6 vong lap cua chttcmg trinh con Poisson Irong ehucmg trinh mo phong Monte-Carlo tip hap nr hap tich vai
hai thugt todn P-BiCGSTAB(3) vd BiCGSTA3(3)
Tir do thi ta thiy ring hai thuat toan P-BiCGSTAB(3) va BiCGSTAB(3) CO cac die trung gan nhu gi6og nhau hoan toan, ham y ting hai thuat toan nay co ciing t6c dp hpi tu va cho ket qua 6n dinh nhu nhau bit chip vai tro cua ti6n dilu kien Jaeobi. Noi each khac, tien dieu kien Jaeobi khong hoat ddng hi€u qua trong trucmg hpp nay. Kit luan
THUAT TOAN BICGSTAB(3) TIEN DIEU KIEN DE TIM NGHIEM CL'A PHU'ONG TRINH...
nay trdi ngupc vdi kit luan phd biln rdng khi tich hop tien dilu kien vdo bat ky phuong phap nao thi phuang phap do ciing deu cai thien toe dp hpi tu vd dp dn dinh.
4. KET LUAN
Chiing tdi dd xdy dung thanh cdng chuang trinh giai phuang trinh Poisson ba chieu dua tren thuat todn P-BiCGSTAB(3). Chuang trinh gidi phuong trinh Poisson ba chieu ndy sau do duac tich hpp vao bp cdng cu md phong Monte-Carlo tap hop ty hpp de kiem tra hi?u ndng vd so sanh vdi b6 cdng cu mang iing dya tren thudt todn BiCGSTAB(3). Hon the nua, de so sanh t6c dp h6i tu va tinh 6n dinh cua thuat toan P-BiCGSTAB(3) vdi cdc ddc trung tuong iing ciia thudt toan BiCGSTAB(3) chiing tdi dd tiln hdnh khao sdt sy phu thupc cua chudn Euclid ciia vecto thdng du vao so vong lap ciia chuang trinh con Poisson. Cac ket qud chi ra rang chuong trinh gidi phuong trinh Poisson dua tren thudt todn P-BiCGSTAB(3) tiln dilu kien cd tdc dp hpi tu tuong duang chuang hinh sir dung thuat toan BiCGSTAB(3), dieu do cd ngbta la vai tro ciia tien dieu kifin Jaeobi trong trudng hap ndy Id khdng dang ke.
TAI LIEU THAM KHAO
[1] D. N Thao, S. Katayama, and K. Tomizawa (2004), Numerical simulation of THz radiation by coherent LO phonons in GaAs p-i-n diodes under high electnc fields, Joumal oflhe Physical Society of Japan 73,3177-3181.
[2] G. Klatt et al. (2011), Photo-Dember terahertz emitter excited with an Er: fiber laser, AppL Phys. Lett. 98, 021114 - 021114-3.
[3] K. Tomizawa (1993), Numerical simulation of submicron semiconductor devices, Arteeh House, Boston London, and references therein.
[4] C. Jacoboni and L. Reggiani (1983), The Monle Carlo method for the solution of charge transport in semiconductors with applications lo covalenl materials. Rev.
Mod. Phys. 55, 645 - 705.
[5] C. Jacoboni, P. Lugii (1989), The Monle Carlo method for semiconductor device simulation, Springer-Verlag, Wien New York.
[6] H. A. Vorst (2003), Iterative Krylov methods for large linear systems, Cambridge University.
[7] Y. Saad (2000), Iterative methods for sparse linear systems, Society for Industrial and Applied Mathematics - SIAM.
[8] C. Broyden and M. Vespucci (2004), Krylov solvers for linear algebraic systems, Elsevier Science, Amsterdam.
[9] Shao-Liang Zhang (1997), GPBi-CG: Generalized product-type methods based on Bi- CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput., Vol 18, No. 2, 537-551.
[10] G. Speyer, D. Vasileska and S.M. Goodnick (2001), Efficient Poisson equation solvers for large scale 3D simulations. Technical Proceedings of the 2001 International Conference on Modeling and Simulation of Microsystems. Nanolech 2001,Vol. 1,23-26.
TRAN QUOC TUAN - DINH NHU THAO
[11] D. N. Thao and L. H. Hai (2010), 3D simulation of semiconductor devices using BICGSTAB (3) for the solution of Poisson's equation, Joumal of Science and Education 15, 19-26.
[12] D. N. Thao and N. T. Ngoc (2010), 3D simulation of semiconductor devices using preconditioned BICGSTAB algorithm with Jaeobi preconditioner for the solution of Poisson's sc^Xiou., Journal of Science and Education 16, 34-41.
Title: THE PRECONDITIONAL BiCGSTAB(3) ALGORITHM FOR THE SOLUTION OF THE THREE-DIMENSIONAL POISSON'S EQUATION
Abstract: A solver based on the preconditional BiCGSTAB(3) algorithm with the Jaeobi preconditioner (P-BiCGSTAB(3))was built to find the solution of the three-dimensional Poisson equation. This solver was then integrated into the self-consistent ensemble Monte-Carlo simulator and was applied to simulate GaAs p-i-n diodes lo test its efficiency. Furthermore, in order to examine the convergent rate and stability of the algorithm, we also investigate the dependence of the logarithm of tbe Euclidean noim of the residual vector on the number of iterations of the Poisson solver. The obtained results showed thai the P-BiCGSTAB(3) based Poisson solver has the same characteristics as the BiCGSTAB(3) based Poisson solver, this means that the role of the Jaeobi preconditioner is negligible in this case.
TS. DINH NHU THAO
Tnrcmg Dai hpc Su ph^m - Dai hpc Hui, 34 Le Lpi, TP. Hue DT: 099-686-7668, Email: [email protected] ThS TRANQUOCTUAN
Tmirag THPT Le True, Tuyen Hoa, Quang Binh DT: 091-344-6567, Email: [email protected]
