Numerical Analysis of Coupled Circuit and Device Models Caren Tischendorf Humboldt University of Berlin ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003
Overview 1. motivation 2. network modeling 3. device modeling 4. coupling of both systems 5. formulation as abstract differential-algebraic system 6. index f¨ ur abstract DAEs 7. Galerkin approach for abstract DAEs ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 1
Circuit Modeling • Kirchhoff’s current law (KCL): Ai = 0 • Kirchhoff’s voltage law (KVL): A T e = u • circuit elements: g ( d q ( u,t ) , d φ ( i,t ) , u, i, t ) = 0 , e.g.: d t d t i = d q C ( u,t ) – capacitors: i = C d u d t , d t u = d φ L ( i,t ) – inductors: u = L d i d t , d t – voltage sources: u = v ( t ) , u = v ( i, ˆ u, t ) � i f ( d q ( x,t ) � ⇒ differential-algebraic equation (DAE) , x, t ) = 0 with x = e d t u ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 2
Replacement Circuit Models for More Complex Elements Gate Source SiO Drain 2 − − Device- n n − → Simulation p Si Bulk Advantages: • resulting system is a differential-algebraic system • fast simulation of the circuit is possible • circuits with many transistors ( > 10 6 ) can be simulated ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 3
Replacement Circuit Models for More Complex Elements Gate Source SiO Drain 2 − − Device- n n − → Simulation p Si Bulk Disadvantages: • interaction between circuit element and surrounding circuit might be insuffi- ciently regarded (essential for high frequency circuits) • more detailed models need a multitude of parameters ( > 500 per transistor) ⇒ parameter extraction is very time consuming ⇒ parameter adjustment becomes problematic for optimal circuit design ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 4
Wish: Coupling of Circuit and Device Simulation Gate Source SiO Drain 2 − − n n p + Si Bulk PDE DAE ⇒ PDAE ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 5
Network Equations by Modified Nodal Analysis d q ( A T C e, t ) + A R g ( A T R e, t ) + A L j L + A V j V + A S j S = − A I i s A C d t d φ ( j L , t ) − A T L e = 0 d t A T = V e v s A = ( A C , A R , A L , A V , A I , A S ) • e - nodal potentials • j L , j V - currents of inductances and voltage sources • j S - currents of semiconductors ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 6
Index of Network DAEs • DAE index is always ≤ 2 . [G¨ unther/Feldmann 96, T. 97, Reissig 98, Est´ evez Schwarz/T. 00] ( A C , A R , A V ) has not full row rank and Q T • DAE-Index = 2 ⇔ C A V has not full column rank ( Q C projector onto ker A T C ). ⇔ The network has an LI-cutset or a CV-loop with at least one VS. CV-loop LI-cutset ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 7
Problems of the Simulation of DAEs with Higher Index • Solution does not depend continuously on the initial data. • Initial values have to fulfill (hidden) constraints. • Simulation methods like BDF and trapezoidal rule can collapse. Example: Integration with inconsistent initial value 5 j exact Trapez V 4 3 2 2sin(t) 1 j V 0 −1 i(j ,t) −2 V −3 −4 −5 0 5 10 15 t ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 8
Problems of the Simulation of DAEs with Higher Index • Solution does not depend continuously on the initial data. • Initial values have to fulfill (hidden) constraints. • Simulation methods like BDF and trapezoidal rule can collapse. Example: Integration with inconsistent initial value 5 j exact Trapez V 4 3 2 2sin(t) 1 j V 0 −1 i(j ,t) −2 V −3 −4 −5 0 5 10 15 t ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 9
Problems of the Simulation of DAEs with Higher Index • Solution does not depend continuously on the initial data. • Initial values have to fulfill (hidden) constraints. • Simulation methods like BDF and trapezoidal rule can collapse. Example: Integration with consistent initial value 5 j exact Trapez V 4 3 2 2sin(t) 1 j V 0 −1 i(j ,t) −2 V −3 −4 −5 0 5 10 15 t ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 10
Semiconductor Equations (Drift Diffusion Model) Gate div ( ε grad V ) = q ( n − p − N ) − ∂ t n + 1 Source q div J n = R ( n, p, J n , J p ) SiO Drain 2 − − n n ∂ t p + 1 q div J p = − R ( n, p, J n , J p ) p Si = q ( D n grad n − µ n n grad V ) J n = q ( − D p grad p − µ p p grad V ) J p Bulk • V - electrostatic potential • n , p - electron and hole concentration • J n , J p - current density of electrons and holes ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 11
Boundary and Coupling Conditions e l + c · A T = S e + W on Γ O ∪ Γ S V αV − α ( e l + c · A T grad V · ν = S e ) + β on Γ MI grad V · ν = 0 on Γ I = = on Γ O n n 0 , p p 0 J n · ν = − qv n ( n − n 0 ) , J p · ν = qv p ( p − p 0 ) on Γ S J n · ν = − qR surf ( n, p ) , J p · ν = qR surf ( n, p ) on Γ MI J n · ν = 0 , J p · ν = 0 on Γ I � j S k = ( J n + J p − ε grad ∂ t V ) · ν d σ Γ k • Γ O , Γ S - Ohmic and Schottky contacts • Γ MI - metal-insulator contacts • Γ I - insulator contacts • Γ k - contacts at the k -th terminal of the semiconductor ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 12
Homogenization Let f ( x ) = ( f 1 ( x ) , ..., f b S − 1 ( x )) T and g ( x ) be smooth functions on Ω with � 1 if x ∈ Γ k ⊆ (Γ O ∪ Γ S ∪ Γ MI ) , f k ( x ) = grad f k · ν = 0 on Γ 0 if x ∈ (Γ O ∪ Γ S ∪ Γ MI ) \ Γ k , and g = W on Γ O ∪ Γ S , grad g · ν = 0 on Γ MI ∪ Γ I . ˜ V ( x, t ) := V ( x, t ) − e l ( t ) − f ( x ) · A T S e ( t ) − g ( x ) ⇒ ˜ ε grad ˜ V · ν + α ˜ V = ˜ grad ˜ V = 0 on Γ O ∪ Γ S , β on Γ MI , V · ν = 0 on Γ I with ˜ β := β − αg − ε grad g · ν . ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 13
Complete Coupled System d q C ( A T C e, t ) + A R g ( A T R e, t ) + A L j L + A V j V + A S j S + A I i s = 0 A C d t d φ L ( j L , t ) − A T L e = 0 d t A T V e − v s = 0 div ( ε grad ˜ q ( n − p − N ) − div ( ε grad ( f · A T V ) = S e + g )) − ∂ t n + 1 q div J n = R ( n, p, J n , J p ) ∂ t p + 1 q div J p = − R ( n, p, J n , J p ) q ( D n grad n − µ n n grad ( ˜ V + f · A T = S e + g )) J n q ( − D p grad p − µ p p grad ( ˜ V + f · A T J p = S e + g )) � [( J n + J p ) · ν χ 1 − ε ∂ t grad ˜ j S = V · ν χ 2 ] d σ Γ ˜ ε grad ˜ V · ν + α ˜ V = ˜ grad ˜ V = 0 on Γ O ∪ Γ S , β on Γ MI , V · ν = 0 on Γ I + boundary conditions for n and p as well as J n and J p ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 14
Coupled System as Abstract Differential-Algebraic System (I) qC ( A T Cu 1 ,t ) A d φL ( u 2 ,t ) D ( u, t ) = with , dt D ( u ( t ) , t ) + B ( u ( t ) , t ) = 0 − r 1 u 5 u 6 u 7 ARg ( A T Ru 1 ,t )+ ALu 2+ AV u 3+ ASu 4+ AIis ( t ) − A T Lu 1 A T AC 0 0 0 0 V u 1 − vs ( t ) 0 I 0 0 0 div ( ε grad u 5) − q ( u 6 − u 7 − N )+div ( ε grad ( f · A T Su 1+ g )) 0 0 0 0 0 0 0 0 0 0 − 1 A = B ( u, t ) = , , q div u 8+ R ( u 6 ,u 7 ,u 8 ,u 9) 0 0 0 I 0 0 0 0 0 I 1 0 0 0 0 0 q div u 9+ R ( u 6 ,u 7 ,u 8 ,u 9) 0 0 0 0 0 0 0 I 0 0 u 8 − q ( Dn grad u 6 − µnu 7grad ( u 5+ f · A T Su 1+ g )) u 9 − q ( − Dp grad u 7 − µpu 7grad ( u 5+ f · A T Su 1+ g )) u 4 − r 2 ( u 8+ u 9) � � where r 1 v := Γ ε grad v · ν χ 2 d σ, r 2 v := Γ v · ν χ 1 d σ and u ( t ) = ( e ( t ) , j L ( t ) , j V ( t ) , j S ( t ) , ˜ V ( · , t ) , n ( · , t ) , p ( · , t ) , J n ( · , t ) , J p ( · , t )) ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 15
Coupled System as Abstract Differential-Algebraic System (I) A D ( · , t ) A d dt D ( u ( t ) , t ) + B ( u ( t ) , t ) = 0 B ( · , t ) 9 ns X 1 = R n − 1 , l =1 R kl − 1 X 2 = R nL , X 3 = R nV , X := X with i =1 X i X 4 = X ns l =1 H 2 (Ω l ) : v l = 0 on Γ l O ∪ Γ l S } , X 5 = { v ∈ X ns ns l =1 H 1 (Ω l ) , X 6 = X 7 = X X 8 = X 9 = l =1 H (div ; Ω l ) . X ns l =1 L 2 (Ω l )) 5 × X 4 := X 1 × X 2 × X 3 × ( X Y ns R nC × X 2 × X 4 × ( l =1 H 1 (Ω l )) 2 Z := X ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 16
Coupled System as Abstract Differential-Algebraic System (I) A D ( · , t ) A d dt D ( u ( t ) , t ) + B ( u ( t ) , t ) = 0 B ( · , t ) • X , Y , Z - real Hilbert spaces • A , D ( · , t ) continuous operators • B ( · , t ) is an unbounded operator! ETH Z¨ urich, MACSI-NET Workshop, 2.-3. May 2003 C. Tischendorf 17
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