TIM NGHIEM CUA P H l / O N G TRINH POISSON BA CHIEU BANG PHU'ONG PHAP CGS
LUONG ViNHGIANG - DINH NHU THAO Trudng Dgi hgc Suphgm - Dgi hgc Hue Tom tit: Trong bai bao nay chung toi trinh bay viec sii dung thuat toan CGS de giai phuong trinh Poisson ba chilu trong ehucmg trinh mo phdng linh kien di-ot p-i-n bdn din GaAs bing phuong pbap Monte - Carlo tdp hpp tu hpp, Cac ket qua thu dupc cho thay chuong trinh giai phumig trinh Poisson dua tren thu^t toan CGS tuy co dp chinh xac khong cao nhung thdi gian mo phong dugc cai thien dang kl so vdi thuat loan BICGSTAB(3).
l.GlOl THIEU
Trong nghien ciiu khoa hpc, viec nghien ciiu thyc nghi?m va ly thuyet cimg phdt trien song song vdi nhau. Trong mdt sd tmdng hop, vi?c nghien cira ly thuyet cd phan thudn lpi hon, chdng han ttong vdn de nghien cira cdc linh kien nano, budc dau nghien ciiu thuc nghiem se gap nhieu khd khdn vd tdn kem nen ngudi ta chpn phuong phdp nghien Cliu 1^ thuylt [1], [2], [3]. Mdt ttong sd cac phuang phdp dupc chpn d day Id phuong phdp md phdng Monte - Carlo tap hop ty hpp vi nd cd nhilu uu dilra ndi trdi, dac biSt la tinh chinh xac va tinh dn dinh sd. Day la phuong phdp ban cd dien vdi tdc dp tdn xa dupc tinh todn dya tt8n quy tic vdng Fermi vd vi$c khdo sdt dpng lyc hpc cua hat tai dya tten cdc phuong trinh chuyen ddng Newton [3].
Mpt ttong nhung bai todn co ban can gidi quydt Id bdi todn hat tai chuydn ddng ttong dien tmdng. De biet di?n trudng ta cdn biet dien the, phdn bd dien the ttong linh kien dupc tim bang cdch gidi phucmg trinh Poisson. Ddy Id rapt he phucmg trinh tuyen tinh rdt ldn, do dd cin diing cac giai thuat ddc biet de gidi [3].
Din nay dd cd nhieu phucmg phdp dupc xdy dyng nhu: Jaeobi, Gauss - Seidel, SOR, da d ludi (raultigrid), iLU [4]. Cac phuang phdp ndy cd the cho cdc ket qua chinh xdc tuy nhien dp on dinh sd khdng cao vd dp hdi tu thap. Gdn day, cdc phuang phdp khdng gian con Krylov nhu CG, GMRES, CGS, QMR, BiCG, BICGSTAB, BICGSTAB(3), BICGSTAB tien dieu kipn da dupc phdt trien va sii dung hi?u qud ttong viec gidi cdc he phuang ttinh tuyen tinh thua loai ldn [4], [5]. Mpt sd tdc gia da sii dung cdc phuang phdp BICGSTAB, BICGSTAB(3), BICGSTAB tiln dilu kien dl gidi phuang trinh Poisson va da thu dupc cdc ket qua chinh xdc vdi thdi gian tinh todn dupc riit ngin iihieu lan [6], [7], [8], [9]. Dl khdo sat rad rpng hudng nghien ciiu tten chiing tdi tien hanh tira nghiem ciia phuong ttinh Poisson bdng phuong phap CGS vdi muc dich tim ra nhiing phuong phdp tdi uu, ho^t ddng dn dinh, eho kit qud nhanh hon.
Chiing tdi da xdy dyng chuong trinh md phdng radi va thyc hien tinh toan tten mdy tinh cd ciu hinh Intel(R) Pentium(R) D CPU 2.80 GHz, DDRII 1GB. Kit qua chi ra ring chuang trinh md phdng sir dung thuat toan CGS tdi uu hon.
Tap chi Khoa hgc va Giao due, Tmcmg Dai hoc Su pham Hui ISSN 1859-1612.36 01{21)/2012:tr 5-11
LLrONG \tN-H GIANG - DINH NHT'' THAO
(I) 2. GL\I PHUONG TRINH POISSON BA CHIEU BANG THUAT TOAN CGS Phuong trinh Poisson ba chilu trong tmdng hgp vat lieu ddng nhit cd dang
dx' dy~ dz' e^
a ddy q? la dien thi, p la mat dp dien tt'ch, e^ Id hing sd dien mdi fiiih trong linh kien;
X, v, - la ba bien khong gian. Thyc hien phep sai phan hiru ban va chia md hinh linh kien thanh cac d ludi bdng nhau Ax = A>" = Ar thi phuong trinh (1) dupc viet lai nhu sau
_ ^
V,-x.j.k +9',.,-u +9^^; i-1 - 6 ? ' , M + ^ - U . * +^.-j*U +^Lj.k.x = — ^ ^ A x % (2) £.
d d a y / = 1..V^. y = L.V, , k = l.X_ vdi ,\\. A \ , .V_ ldn lugt la sd mit ludi tiieo cdc chieu khdng gian Ch, O) . Oz. Day chinh la he phuong trinh myen tinh thua eye ldn md ta cdn gidi. Tbuat toan CGS de tim nghidra phuong ttinh Poisson dugc khai trien trong Bang 1,
Bang 1. Thugt todn CGS de tim nghiem cOa phucmg trinh Poisson Thuat toan CGS
Chpn Xa, To bdt ky Tinh Tfl = b- Ax^
Ldy r^=r^
A = 9o = 0 Bat diu vdng ldp chinh
6 = r, r,.,
p,= w+ fe[?,.,+ ii/?,.,)
v= Ap^
a='',/[ro,v)
q^ = u- av u= ii+ q^
X, = Jt,-1 + oa Neu JT, du chinh xdc thi thoat
r,= r^^- a Ad Ket thiic vdng lap chinh Ket thiic thudt toan
TIM NGHIEM CUA PHU'ONG TRINH POISSON BA CHIEU BANG THUAT TOAN CGS
3. KET QUA M O PHONG VA THAO LUAN
Md hinh cdu tnic ciia di-dt p-i-n ban dan GaAs gdm mot Idp bdn ddn thuin i kep giua hai ldp bdn dan pha tap loaip vd loai n nhu dupc chi ra ttong Hinh 1, ttong dd radi ldp cd dp day tuong iing Id d_, d^ vd d„. Mat dp pha tap acceptor vd donor tuong ling la
^A ^^ Nfy, cdc tap dugc phdn bd tir b l mat cua cdc ldp p va n vdo sdu ben ttong linh ki?n theo hdm phan bd Gauss. Trang thai cdn bdng nhiet cua linh kien dupc xdc lap bdng md phdng thdi gian thuc tmdc khi chieu xung laser vao linh kien.
Chung tdi da su dung phuong phap Monte - Carlo tap hpp ty hpp dl md phdng dpng lyc hpc ciia hat tdi ttong lmh
ki?n ttong tmdng hpp chieu rapt xung laser vdi chilu ddi cua xung la 1 2 ^ va ndng lupng photon la \.49eV.
Cdc tham sd cau tnic vimg ndng lupng dupc sii dung nhu sau: £ ' J ^ = I . 4 2 e K ,
ml^ = 0.063 niu -
"'r/i = 0 . 4 5 / t t g ,
ml^ = 0.222 ntj,, va dp chenh ISch ndng lupng giiia Tvd L
^„ = ^„ = 50n/", d, = 340«m
V^
Hinh 1. Mo hinh di-dt p-i-n GaAs Id Affj. = 0.29 eV. Chiing tdi gid thiet rang
A^^ =0.5x10" cm"', Nj^=2.5xi0"cm-', A^„ =5x10"' cm~^ sau thdi gian 1 ps. Kich thudc theo ba chilu khdng gian cua di-dt la L^yf-L^.y-L. =440nmxI00KmxI00«m, gid sii di-dt dupc nudi cdy theo phuong Ox.
Md hinh linh kien dupc chia thanh cac d ludi khdng gian vdi khodng cdch giua cdc niit ludi Id Ax = A;' = A2 = 50xl0""'/K. Nhu vay theo phuong Ox ta se cd A'^^=89 niit ludi, theo phuong Oj'va Oz cd A^^ = 7^. = 21 niit ludi. Dien tmdng ngoai dugc dat vdo linh ki^n dpc theo phuong Ox vd di-dt dugc phdn eye nghich, xem Hinh I.
Hinh 2 rad ta sy thay ddi vdn tdc ttdi dat toan phdn vd van tdc ttdi dat ciia dien tir theo cdc phuong Ox, Oy, Oz iing vdi dien tmdng ngoai E^.^^ = 100kV/cm, dugc tinh todn vdi ldi gidi phuang trinh Poisson bdng thudt toan CGS. Tii dd thj ta thay rang dien tu chii yeu chuyen ddng ttdi dat theo phuang Ox cdn cdc thdnh phdn van tdc theo phuang Oy, Oz ddng gdp khdng ddng ke. DSc bidt, sau khi chieu xung laser, van tdc ttdi dat todn phdn cua dien tir (van tdc ttdi dat ciia dien tir theo phuang Ox) tdng nhanh vugt xa gid tt] bdo hda sau dd giam nhanh ve gid tri bao hda, hien tupng ndy dupc gpi Id sy vugt qud van tdc [1], [3].
LUONG ViNH GIANG - BINH NHl^r THAO
/%.
Hinh 2- Van tdc trdi dat todn phdn ctia dien Hinh 3. Vdn ldc trdi dat todn phdn aia dien lu tir va van tdc trdi dgt theo cdc phuang khdc nhu Id hdm cua thdi gian irng vdi cdc dien
nhau nhu Id hdm cua ihdi gian irng vdi trudng ngodi khdc nhau E„, = 100 kV/cm
Hinh 3 md td sy phu thupc cua van tde ttdi d^t ciia dien tii theo thdi gian iing vdi cac gia tt; dien tmdng ngoai E^ = lOkV I cm vd E^, =\00kVjcm. KSt qua cho thdy dien tmdng ngoai cdng cao thi sy vupt qud van tdc xdy ra cang sdm, dinh vupt qua van tdc cang cao va van tdc cang nhanh chdng vl tiem can gid tri bao hda. Dieu ndy dupc \^
giai, khi didn tmdng ngoai cdng cao thi sd di8n tOt ndm ttong edc trang thai cd the tham gia vao qua trinh tdn xa lien thimg lung cang Idn. Khi dien tii bi tan xa tir thung liing F sang thung liing L van tdc ciia dien tii bi giara nhieu do khdi lupng hi?u dung ciia di?n tir ttong thung lung L ldn han nhilu ldn khdi lupng hieu dyng ciia difn tir ttong thung lung r. He qua la vdn tdc cua diSn tii cdng giara nhanh ve gid tri bao hda.
Hinh 4 cho kit qua so sanh vdn tdc ciia di8n tii thu dupc bang hai chuong ttmh mfi phdng ba chilu su dung tiiuat todn BICGSTAB(3) va tiiudt todn CGS [7]. Hai dd tiii gdn nhu trimg nhau hodn todn. Dieu dd ndi len sy thanh cdng ttong viec sii dyng thudt todn dan gidn CGS de giai bai toan ehuyen dpng ciia hat tai ttong linh kidn di-dt p-i-n ban din GaAs (chuong trinh sii dung thuat todn BICGSTAB(3) da dupc chiing minh la cho kit qua phu hpp vdi thyc nghi?m [7]).
Hinh 5 rad td sy phan bd dien thi khdng gian ttong di-dt p-i-n ban dan GaAs theo edc phuong Ox va Oy tai mat cit z = lOnm iing vdi dien ttndng ngoai E^ = iOOkV/cm.
D6 thi cho thdy dien tiie phdn bd diu ttong linh kien va ehi biln thien nhd xung quanh mdt sd cac diem ludi theo phuong Ox vd gdn nhu khdng ddi tiieo hai phuong Oy,Oz.
Kdt qua hodn toan hpp ly do dien tmdng ngoai dupc dat vao linh kien theo phuang Ox vd cung phii hpp vdi cac kit qud da dupc cdng bd tmdc day [1], [7].
TIM NGHIEM CUA PHUONG TRINH POISSON BA CHIEU BANG THU.4T TOAN CGS
Hinh 4. Van tdc trdi dgt todn phdn cua dien tu theo thai gian img vdi hai thugl lodn B1CGSTAB(3) va CGS dudi dien trucmg
ngodi E^^ = 70 kV/cm
Hinh 5. Phdn bd dien the khdng gian trong di- dt p-i-n bdn ddn GaAs iheo hai phucmg Ox. Oy
tai mat cdi 2 = 1 Onm ung vdi E^,=\00kV/cm
Hinb 6 chi ra viec so sdnh sy phdn bd dien the ttong khdng gian theo true Oxt&iy = z = 10 nm ling vdi hai thuat toan BICGSTAB(3) vd CGS dudi dien ttudng ngodi E^ = 70 kV/cm. Ta thiy hai dudng nay triing khit nhau, qua dd mpt ldn niia ndi len sy thanh cdng trong vi$c sir dung thudt todn CGS de gidi phuang trinh Poisson ba chieu.
Cdc ket qud thu dupc ndi len sy thdnh cdng trong viec sii dung thudt todn CGS, den day ta nhdn thay vdi bdi todn tim nghiem ciia phuong trinh Poisson ba chieu thi viec su dung thudt todn CGS cd uu diem hon cdc thudt toan khdc bdi tinh don gidn ciia nd. Tuy nhien, mdt vdn de khdng kem phdn quan trpng cdn dugc thdo li4n Id dp hpi tp, tinh dn dinh sd vd chi phi thdi gian tinh toan.
Hinh 7 so sdnh dp hdi tu ciia thuat todn BICGSTAB(3) vd fliudt todn CGS. Kit qua cho thay my dp hdi tu ciia thudt toan BICGSTAB(3) nhanh hon nhung dd thi tuong ung vdi thudt toan CGS tron hon dieu ndy ndi len tinh dn dinh sd cua thudt todn CGS. Khi tien hanh chay chuong trinh md phdng tten rady tinh cd can hinh Intel(R) Pentium(R) D CPU 2.80 GHz, DDRII I GB vdi sai sd 10 "' tiii chuong trinh md phdng vdi tiiudt todn BICGSTAB(3) chay ttong khodng thdi gian 61 phiit ttong khi chuang trinh rad phdng vdi thudt toan CGS chay ttong khoang thdi gian 42 phiit. Qua dd chiing ta thiy viec su dung thudt todn CGS ddi vdi bai todn tim nghiem cua phuong trinh Poisson ba chieu Id tdi uu hon.
LUONG ViNH GIANG - DINH NHU THAO
— V s i lAi-li«i dii won phin
'V
/\ \
E^^WkV/am
\ \
Hinh2. Vgn ldc trdi dgt lodn phdn cua dien Hinh 3. Van tdc trdi dgt todn phdn cua diin lu td va vdn tdc trdi dgt theo cdc phucmg khdc nhu Id hdm cua thdi gian ung vdi cdc dien
nhau nhu Id hdm cOa thdi gian ung vdi trudng ngoai khdc nhau E^ = 100 kV/cm
Hmh 3 rad td sy phu thudc cua van toe ttdi dat ciia di$n tir theo thdi gian iing vdi cdc gid tti dien tmdng ngoai E^ = 70kV I cm vd E^, =lOOkV/cm. Kit qua cho thdy dien tmdng ngoai cdng cao thi su vugt qud vdn tdc xay ra cdng sdm, dinh vugt qua van tdc cdng cao vd van tdc cdng nhanh chdng vl tiera can gia trj bao hda. Dieu nay dupc ly gidi, khi di?n tmdng ngoai cdng cao thi so di?n tir ndm ttong cac ttang thai cd thi tham gia vdo qud trinh tan xa lien thung lung cdng ldn. Khi dien tir bi tan xa tir thung lung F sang thtmg lung L van tdc ciia dldn tii hi giam nhieu do khdi lupng hi^u dung eiia dien tir ttong thung liing L ldn hon nhilu lin khdi lupng hieu dung ciia di$n tu ttong thung lung F. H? qua la v^n tdc cua didn tu cdng gidm nhanh vl gid tti bao hda.
Hinh 4 cho ket qua so sdnh v|n tdc ciia didn tii thu dupc bdng hai chuong trinh rad phdng ba chilu sii dpng tiiudt todn BICGSTAB(3) va tiiudt todn CGS [7]. Hai dd tiij gdn nhu trung nhau hoan toan. Dilu dd ndi len sy thanh cdng ttong viee sii dyng thuat toan don gian CGS de gidi bai todn chuyin dpng cua hat tdi ttong llnh kien di-dt p-i-n bdn din GaAs (chuang ttinh su dung tiiuat todn BICGSTAB(3) da dupc chimg rainh la cho kit qud phu hpp vdi thyc nghiem [7]).
Hinh 5 md ta sy phan bd didn thi khdng gian ttong di-dt p-i-n bdn ddn GaAs tiieo cac pbuong Ox vd Oy tai mat cdt z = I Onm ung vdi di?n tiudng ngodi E^, = iOOkV/cm.
Dd till cho thiy dien thi phan bd diu ttong linh ki$n va chi biln tiiien nhd xung quanh mpt sd cac diem ludi tiieo phuang Ox vd gin nhu khdng ddi tiieo hai phuang Oy,Oz.
Kit qua hoan toan hpp ly do di?n tmdng ngodi dupc dat vao linh kien theo phuong Ox vd ciing phii hgp vdi cdc ket qua dd dupc cdng bd tmdc ddy [1], [7],
TIM NGHIEM CUA PHUONG TRINH POISSON BA CHIEU BANG THUAT TOAN CGS
Hinh 4. Van Idc trdi dgt todn phdn ciia dien Ur theo thdi gian irng vdi hai thugt lodn BICGSTAB(3) vd CGS dudi dien tnrdng
ngodi E^, = 70kV/cm
Hinh 5- Phdn bo dien the khdng gian trong di- dt p-i-n bdn ddn GaAs theo hai phuang Ox, Oy
tgi mat cat z= \ Onm irng vdi E^=\00kV/cm
Hinh 6 chi ra viec so sdnh sy phdn bd dien thd ttong khdng gian theo true Oxtaiy=z = 10 nm img vdi hai thugt todn BICGSTAB(3) vd CGS dudi dien tmdng ngodi E^, = 70kV/cm. Ta thay hai dudng ndy trimg khit nhau, qua dd mpt ldn niia ndi ldn sy thdnh cdng ttong vi^c sii dyng thudt todn CGS dd giai phuong trinh Poisson ba chieu.
Cdc kit qud thu dupe ndi len sy thanh cdng ttong viec su dung thudt todn CGS, den ddy ta nhdn thdy vdi bai todn tim nghiem ciia phuang ttinh Poisson ba chidu thi vide sir dung thudt todn CGS cd uu diem hon cdc thuat todn khac bdi tinh don gidn ciia nd. Tuy nhidn, rapt vdn dd khdng kdm phdn quan trpng cdn dupc thdo ludn la dp hdi tu, tinh dn dinh sd vd chi phi thdi gian tinh toan.
Hinh 7 so sdnh dp hdi tu ciia thuat todn BICGSTAB(3) vd thudt todn CGS. Kdt qud cho thdy tuy do hdi tu ciia thudt todn BICGSTAB(3) nhanh han nhirag do thi tuong ling vdi thuat todn CGS tton hon dilu nay ndi len tinh dn dinh sd ciia thudt toan CGS. Khi tien hdnh chay chuong trinh md phdng ttdn mdy tinh cd cdu hmh lntel(R) Pentium(R) D CPU 2.80 GHz, DDRII I GB vdi sai sd 10" thi chuong ttinh md phdng vdi tiiuat toan BICGSTAB(3) chay ttong khoang thdi gian 61 phiit ttong khi chuong trinh md phdng vdi thuat todn CGS chay ttong khodng thdi gian 42 phut. Qua do chiing ta thdy vide sir dung thudt toan CGS ddi vdi bdi todn tim nghiem ciia phuang trinh Poisson ba chidu la tdi uu hon.
LUONG ViNH GLWG - DINH NHL' THAO
I f Irmk OB PlwHB Inefj
Hinh 7. Suphu thugc cua chudn Euclid cua veclo thang du vao so vdng lap ciia chuong Irinh
con Poisson ung vdi E^ = lOOkV/cm Hinh 6. Phdn bd dien thi khdng gian trong
di-dt p-i-n ban ddn GaAs theo phuang Ox iai mat cdt y = z= 1 Onm irng vdi hai thuat
lodn B1CGSTAB(3) vd CGS dudi dien irudng ngodi E^ = 70 kV/cm 4. KET LUAN
Chiing tdi da thanh cdng ttong vide sii dyng thuat todn CGS dd giai phuong trinh Poisson ba chilu trong md phdng linh kidn nano bdn din bdng phuang phdp Monte - Carlo tap hpp ty hpp. De khdo sat cdc ddc tnmg ciia phirong phap chiing tdi da dp dyng dd md phdng dpng lyc hpc ba chieu cua hat tdi ttong di-dt p-i-n ban din GaAs vd so sanh vdi cdc kdt qud md phdng da dugc cdng bd tmdc ddy. Ket qua cho thiy chuong trinh gidi phuong trinh Poisson sii dung thuat todn CGS cd dp hpi tu cham han chuong trinh su dung thuat todn BICGSTAB(3) nhung cho kdt qud dn dinh hon vd thdi gian md phdng dupc cdi thien ddng kd so vdi chuong trinh sii dyng thudt toan BICGSTAB(3).
LCfl CAM ON
Cdc tac gia xin giii ldi cam on chan thanh tdi PGS. TS. Nguyin H6og Quang, Ban Hpp tac Quic tl, Vien Khoa hoc va Cong nghe Viet Nam, chii nhiem de tai NAFOSTED (103.02.109.09) da ho trg mpi mat cho cac tac gia trong qua tiinh hoan tbanb bai bao nay.
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 electric fields.
Joumal oflhe Physica! Society of Japan 73,3177-3181.
[2] G. Klatt et al. (2011). Photo-Dember terahertz emitter excited with an Er: fiber laser.
Appl. Phys. Len. 98, 021114 - 021114-3.
[3] K. Tomizawa (1993). Numerical simulation of submicron semiconductor devices.
Arteeh House, Boston London.
TIM NGHIEM CUA PHUONG TRINH POISSON BA CHIEU BANG THUAT TOAN CGS
[4] H. A. Vorst (2003). Iterative Krylov methods for large linear systems. Cambridge University.
[5] Y. Saad (2000). Iterative Methods for Sparse Linear Systems. SLAM.
[6] 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.
[7] 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] 5, i9-26.
[8] D. N . Thao and N. T. Ngoc (2010). 3D sunulation of semiconductor devices using preconditioned BICGSTAB algorithm with Jaeobi preconditioner for the solution of Poisson's eqaalion. Journal of Science and Education, 16, 34-41.
[9] D. N. Thao, D. T. D. My, N, C. P, Thi and N, T. Thuy (2011)., Three-dimensional simulation of nano semiconductor devices using GPBICG algorithm for the solution of the Poisson's equation. Joumal of Science, 65,215-223.
Title: OBTAINING THE SOLUTION OF THE 3D POISSON'S EQUATION BY MEANS OF THE CGS METHOD
A b s t r a c t : In this article, we present the CGS algorithm used to solve Poisson's equation in three-dimensional simulation in GaAs p-i-n diodes semiconductor by the self-consitent ensemble Monte Carlo method. These results suggest that CGS algorithm solve 3D Poisson's equation less accuracy than BICGSTAB(3) algorithm but the simulation time is significantly improved.
TS. DINH N H U THAO
Khoa Vat ly, T m d n g Dai hpc Su pham - Dai hpc Hue DT: 099-686-7668. Email: [email protected] LUONG ViNH G L\ N G
Hoc vien Cao hpc, Tmcmg D?i hoc Su pham - Dai hoc Hue DT: 0974.193.736