Hydrodynamic Modeling: Hydrodynamic Equations. Prepared by Dragica Vasileska Professor Arizona State University
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1 Hyoyamc Moelg: Hyoyamc Equaos Peae by Dagca Vasleska Poesso Azoa Sae Uesy
2 D-Duso Aoaches Val he use aso omaes
3 Exeso o DD Aoaches Valy Iouco o he el-eee mobly
4 Velocy Sauao Imlcaos
5 Velocy Sauao
6 Dece Scalg
7 Challeges Scale Deces
8 Legh Scales Oe Whch These Eecs Occu
9 Moelg Heachy Semcouco Bolzma Equao Hyoyamc Moel D Duso Moel Ccu Sce Moel S E M I C L A S S I C A L QUANTUM Schöge Posso Sysem Quaum Bolzma/Wge Equao Quaum Hyoyamc Moel Quaum D Duso Moel Ccu Sce Moel?
10 Velocy Oeshoo Bulk S D elocy [cm/s].5x10 7 x x10 7 1x10 7 5x x kv/cm 5 kv/cm 10 kv/cm 50 kv/cm me [s] Eegy [ev] Tme eoluo o mea elocy. Elecc el s: a 1.0 kv/cm b kv/cm c 10 kv/cm a 50 kv/cm esecely Tme eoluo o he aeage eleco kec eegy. The elecc el equals: a 1 kv/cm b 5 kv/cm c 10 kv/cm a 50 kv/cm esecely. 1 kv/cm 5 kv/cm 10 kv/cm 50 kv/cm me [s]
11 Velocy Oeshoo
12 Velocy Oeshoo Co
13 Velocy Oeshoo Co
14 Hyoyamc Smulao Basc hyoyamc equaos Esemble elaxao aes a he calculao Dscezao o he balace o hyoyamc equaos
15 To ee he Balace Equaos oe sas h he BTE mulles h a aoae uco k a egaes oe k o ge: hee: R G J coll R e G J E Basc Hyoyamc Equaos
16 Balace equao o he cae esy s obae by assumg ha =1: Usg = +c oe ges he al om o he balace equao o he cae esy: coll R G e J 0 J coll Cae Desy
17 D s. Themal Eegy FD SOI 1 x INPUT VARIABLES Eleco esy Eleco hemal eegy Eleco eegy Chael Legh = 5 m Ts = 10 m
18 Momeum Balace equao s obae by assumg ha =: Ths leas o he ollog momeum balace equao: coll R e G E J J P coll e E J Momeum
19 Momeum Co Fo he lo o he momeum comoe x e hae ha: J xx J yx J zx J x x x y z Usg aga = +c oe ges: J m * j j m * j c c j Assumg agoal emeaue eso he aboe equao smles o: J x x x x y y The al om o he momeum balace equao s: x z z x kbt z kbt ee coll
20 Eegy The Eegy Balace equao s obae by assumg ha =E: The eegy balace equao s he o he om: coll B R e G T k E W E q J J coll B e T k E q
21 Closue To hae a close se o equaos oe ehe: a goes he hea lux alogehe b uses a smle ece o he calculao o he hea lux: * 5 m T k T B q Subsug T h he esy o he cae eegy he momeum a eegy balace equaos become: coll B coll e m k e m * 1 3 * 1 3 E E
22 Balace Equaos Moe coee se o balace equaos s ems o a : coll B coll coll e m k m e m m m m * 3 * * 1 * 3 * * E E
23 Reuco o D-Duso Moel The DD moel s obae by smlyg he momeum balace equao hch 1D s o he om: I seay-sae oe ges o he momeum balace equao ha: z m coll z B z P ee z P ee T k z z P 1 z ed E E e J z ' e D z e E ' hee:
24 Smlcaos The DD moel s he obae by makg he ollog assumos: The sbuo uco s close o he equlbum alue The eegy gae el s zeo. I he exee DD moel E a DE ae assume o ee uo he local el alues oly.
25 Esemble Relaxao Raes To see ho oe ca calculae he momeum a he eegy esemble elaxao aes s ecessay o go back o he eo o R hee he elaxao me s: 0 R coll ' ' 1 1 ' S
26 Esemble Relaxao Raes The esemble elaxao ae hch aeas he cae esy balace equao s elae o he ealley ase may-alley semcoucos. I equals o: coll Oe usually calculaes hs esemble elaxao ae by usg Moe Calo smulao: 0 j j N T N T j j j
27 Momeum Relaxao Rae The momeum ae s eeme by a seay-sae MC calculao a bulk semcouco ue a uom bas elecc el o hch: coll m ee m e m e * 0 * * E E
28 Eegy Relaxao Rae The emsemble eegy elaxao ae s also eeme by a seay-sae MC calculao a bulk semcouco ue a uom bas elecc el o hch: e e e coll E E E
29 Smulao Examle Imoace o Velocy Oeshoo 5m FD-SOI MOSFET 5 m F D S O I M O S F E T. 5 x chael x m x m F D S O I M O S F E T. 5 x Aeage Eleco Velocy Alog The Chael 100m FD-SOI MOSFET T g a e = K T g a e = K T g a e = K 140m FD-SOI MOSFET. 5 x T g a e = K T g a e = K T g a e = K x m x Fo 80 m a lage eces smulae caes ae o he elocy oeshoo egme he lage oo o he chael x m x
30 Dscezao o he Balace Equaos Fo smlcy he equaos ll be sceze o a equally sace meshes. I 1D a e-eece oeao oao he RHS s o he equaos ha ee o be sole ae: * 3 * 3 * * 1 * 3 0 D xx xx B x x x x x x N e L G ee m L k m L L G m ee m L m L G L L G
31 Dscezao o he Balace Equaos Co The aous exlc schemes ha oe ca use ae lse belo: a Foa-me ceee-sace scheme FTCS: b U scheme: c Lax-Weo scheme: x L x L xx x ] [ 1 1 u x x L x ] [ ] [ ] [ xx x LW L L L x
32 Dscezao o he Balace Equaos co DuFo-Fakel scheme: e Leaog scheme: x L xx ] [ 1 1 x L l x ] [ 1 1
33 Eegy Balace Moel Imlemeao Slaco Fo a eale esco o he eegy balace/hyoyamc moel lease ee he lecue oes o hyoyamc moelg. Imoa eaues o hs examle ae: - The MATERIAL saeme s use o assg he eegy elaxao mes auel.el aumob.el - The MODELS saeme s use o selec he hyscal moels use hce - The IMPACT saeme s use o assg he eegy elaxao legh o he Selbehe moel Oly as o he lsg ha ae elea o he ece smulao a ae exace he ex e sles.
34 Moe Deals E.TAUR.VAR seces ha eleco emeaue eee eegy elaxao me s use. Use aamees TRE.T1 TRE.T TRE.T3 TRE.W1 TRE.W a TRE.W3 o maeal saeme o sece he eegy elaxao me. H.TAUR.VAR seces ha hole emeaue eee eegy elaxao me s use. Use aamees TRH.T1 TRH.T TRH.T3 TRH.W1 TRH.W a TRH.W3 o maeal saeme o sece he eegy elaxao me. HCTE seces ha boh eleco a hole emeaue ll be sole. HCTE.EL seces ha eleco emeaue ll be sole. HCTE.HO seces ha hole emeaue ll be sole. F.KSN seces he ame o a le coag a C eee uco secyg he eleco Pele coece as a uco o eleco eegy. F.KSP seces he ame o a le coag a C eee uco secyg he hole Pele coece as a uco o hole eegy. KSN seces hch ho cae aso moel ll be use o elecos. KSN=0 selecs he hyoyamc moel a KSN=-1 selecs he eegy balace moel. KSP seces hch ho cae aso moel ll be use o holes. KSP=0 selecs he hyoyamc moel a KSP=-1 selecs he eegy balace moel.
35 Moe Deals TAUMOB seces he eeece o elaxao mes h cae emeaue he mobly eo. I TAUMOB s sece he alues o MATERIAL saeme aamees TAUMOB.EL a TAUMOB.HO ae eee o he cae emeaue. TAUTEM seces he eeece o elaxao mes h cae emeaue. I TAUTEM s sece he alues o MATERIAL saeme aamees TAUREL.EL a TAUREL.HO ae eee o cae emeaue.. N.TEMP o HCTE.EL seces ha he eleco emeaue equao ll be sole. P.TEMP o HCTE.HO seces ha he hole emeaue equao ll be sole.
36 Sucue Beg Smulae
37 Smulao esuls: DD s. Eegy Balace Velocy sauao elocy oeshoo
38 FD Dece Geeaos Exame eaue 14 m 5 m 90 m Tox 1 m 1. m 1.5 m VDD 1V 1. V 1.4 V Souce/a og = 10 0 cm -3 Chael og = cm -3
39 Wha haes a 14 m FD SOI Dece?
40 Resuls o 14 m FD SOI Dece V G =1 V =0. s Hyoyamc D-uso
41 Resuls o 5 m FD SOI Dece V G =1. V Hyoyamc D-uso
42 Resuls o 90 m FD SOI Dece V G =1.4 V hyoyamc D-uso
43 Resuls om Hyoyamc Smulaos Dece Sucues Beg Smulae eaue 14 m 5 m 90 m Tox 1 m 1. m 1.5 m VDD 1V 1. V 1.4 V oeshoo 40% 136% % Souce/a og = 1E0 cm -3 Chael og = 1E18 cm -3 Oeshoo= ID HD -ID DD /ID DD % a o-sae
44 Hyoyamc Smulaos I summay e ee o use a leas hyoyamc moel o escbe he elocy oeshoo aoscale eces sa 14 m We hae o-saoay aso houghou he hole chael legh almos ballsc aso
45 Poblems h Hyoyamc Smulaos V G =1 V L G = 14 m 0.3 s 0. s =0.1 s
Continuous Time Markov Chains
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