ECE 595, Secto 0 Numercal Smulatos Lecture 9: FEM for Electroc Trasport Prof. Peter Bermel February, 03
Outle Recap from Wedesday Physcs-based devce modelg Electroc trasport theory FEM electroc trasport model Numercal results Error Aalyss
Recap from Wedesday Thermal trasfer overvew Covecto Coducto Radatve trasfer FEM Modelg Approach Numercal Results Error Evaluato
Physcs-Based Devce Modelg D. asleska ad S.M. Goodck, Computatoal Electrocs, publshed by Morga & Claypool, 006. //03 ECE 595, Prof. Bermel
Electroc Trasport Theory Wll assume electroc badstructureskow, ad take a semclasscal approach Electrostatcs modeled va Possso s equato: ) D A ε = p N N Charge coservato s requred: t p t = J q = J p q U U S. Selberherr: "Aalyss ad Smulato of Semcoductor Devces, Sprger, 984. p
Electroc Trasport Theory Both p-type ad -type currets gve by a sum of two terms: Drft term, derved from Ohm s law Dffuso term, derved from Secod Law of Thermodyamcs d J = q x) µ E x) qd dx d J p = qp x) µ pe x) qdp dx S. Selberherr: "Aalyss ad Smulato of Semcoductor Devces, Sprger, 984.
FEM Electroc Trasport Model Much lke earler work, wll employ the followg strategy: Specfy problem parameters, cludg bulk ad boudary codtos Costruct fte-elemet mesh over spatal doma Geerate a lear algebra problem Solve for key feld varables: φ x,y,z,t) p x,y,z,t) x,y,z,t)
FEM Electroc Trasport Model Regardg the grd set-up, there are several pots that eed to be made: I crtcal devce regos, where the charge desty vares very rapdly, the mesh spacg has to be smaller tha the extrsc Debye legth determed from the maxmum dopg cocetrato that locato of the devce εkbt N max e Cartesa grd s preferred for partcle-based smulatos It s always ecessary to mmze the umber of ode pots to acheve faster covergece A regular grd wth small mesh aspect ratos) s eeded for faster covergece L D = D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
Posso Solver The D Posso equato s of the form: d ϕ e = p N D N A dx ε ) EF E = exp = exp ϕ / T ) kbt E E F p = exp = exp ϕ / T ) kbt D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
Posso Solver Perturbg potetal by δ yelds: d ϕ e dx ε e δ ε d ϕ e dx ε ϕ / / / ) T ϕ T e e C = ϕ / / ) T ϕ T e e ϕ / / ) / / / ) T ϕ T ew ϕ T ϕ T e e ϕ e e C = e e e old ϕ / / ) T ϕ T ε ϕ ew old δ = ϕ ϕ D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State e ε
Posso Solver Reormalzed form d ϕ = p C) δ p ) dx d ϕ dx p ϕ = p C p = ) ew ) ) ew old δ ϕ ϕ ϕ old D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
Posso Solver Italze parameters: -Mesh sze -Dscretzato coeffcets -Dopg desty -Potetal based o charge eutralty Solve for the updated potetal gve the forcg fucto usg LU decomposto Update: - Cetral coeffcet of the learzed Posso Equato - Forcg fucto Equlbrum solver > tolerace Test maxmum absolute error update < tolerace D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
Curret Dscretzato The dscretzato of the cotuty equato coservatve form requres the kowledge of the curret destes J J x) = e x) µ E ed x) = ep x) µ E ed p o the md-pots of the mesh les coectg eghborg grd odes. Sce solutos are avalable oly o the grd odes, terpolato schemes are eeded to determe the solutos. There are two schemes that oe ca use: a)learzed scheme:,, p, µ ad D vary learly betwee eghborg mesh pots b) Scharfetter-Gummel scheme: electro ad hole destes follow expoetal varato betwee mesh pots p p D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State p
Naïve Learzato Scheme Wth the learzed scheme, oe has that / J e / µ / ed / a J = / / = eµ / eµ / ed a a ed / a a Ths scheme ca lead to substatal errors regos of hgh electrc felds ad hghly doped devces. / a D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
Scharfetter-GummelScheme Oe solves the electro curret desty equato for ), subject to the boudary codtos The soluto of ths frst-order dfferetal equato leads to x ed a e x ed a e J = = / / / / / µ µ ) ad ) = = [ ] = = = t B t B a ed J e e g g g t t / / ) / ) / ) ), ) ) ) = x e x x B s the Beroul fucto D. asleska, EEE533 Semcoductor Devce ad Process Smulato Lecture Notes, Arzoa State
ADEPT.0 Avalable o aohubfrom Prof. Gray s team: https://aohub.org/tools/adeptpt
ADEPT.0 Ca customze all the calculato detals:
ADEPT.0 Outputs clude electrostatc Posso) soluto:
Eergy bad dagram ADEPT.0
ADEPT.0 Carrer cocetratos:
ADEPT.0 Ad fally, realstc I- curves:
Next Class Is o Moday, Feb. 5 Next tme, we wll cover electroc badstructures