A Primer on Semiconductor Device Simulation Mark Lundstrom Purdue University Network for Computational Nanotechnology 1) The Semiconductor Equations 2) Discretization 3) Numerical Solution 4) Physical Models 5) Examples www.nanoHUB.org NCN
1) A Continuity Equation Wabash River Rate of increase of water level in lake = (in flow - outflow) + rain - evaporation r � p ( ) � R J p q + G = �� • � t www.nanoHUB.org NCN
1) The Semiconductor Equations Conservation Laws: Constitutive Relations: r r r r D = �� 0 E = � �� 0 V � D = � � • + � N A ( ) r � � = q p � n + N D ( ) = G � R ( ) r J n � q r r � • J n = nq µ n E + qD n n � r r r r ( ) = G � R J p = pq µ p E � qD p p ( ) � J q � • p R = f ( n , p ) (steady-state) etc. www.nanoHUB.org NCN
1) The Mathematical Problem The “Semiconductor Equations” r 3 coupled, nonlinear, D = � � • second order PDE ’ s r ( ) = G � R ( ) J n � q for the 3 unknowns: � • V ( r n ( r p ( r r r ) r ) r ) ( ) = G � R ( ) J q � • p Conservations laws: exact Transport eqs. (drift-diffusion): approximate www.nanoHUB.org NCN
1) The Depletion Approximation (i) analytical solutions (e.g. depletion approximation) 0 < V G < V T dD SiO 2 ( ) dx = q p � n � N A P-Si −ρ d 2 V 2 = qN A ( ) x < W dx � S � 0 y W www.nanoHUB.org NCN
1) The DA vs. Numerical Solution + qN D � qN A PN Junction Educational Tool www.nanoHUB.org NCN
1) Asymmetric Junction + qN D = + 0.016 C/cm 3 ? � qN A = � 0.8 C/cm 3 PN Junction Educational Tool www.nanoHUB.org NCN
1) Asymmetric Junction inversion layer in a PN junction! E F E I www.nanoHUB.org NCN
1) The Minority Carrier Diffusion Equation (i) analytical solutions (e.g. minority carrier diffusion eq) dp X J p = pq µ p E � qD p dy V > 0 P + N -Si ( ) d J p q = � R � � � p dy � p e � y / L P Δ p L P = D p � p d 2 � p 2 � � p D p = 0 dy � p y www.nanoHUB.org NCN
2) The Grid N nodes (ii) “exact” numerical solutions 3N unknowns V i , j n i , j p i , j www.nanoHUB.org NCN
2) Discretization df = f i + 1 � f i f(x) 2 ) + O ( h dx h ( ) x i + 1/ 2 “centered h difference” x i-1 x i x i+1 x Local truncation error (LTE) www.nanoHUB.org NCN
2) Nonuniform Grid Example: MOS problem h = 1 Ang = 0.1 nm −ρ N � 100,000! V G > V T y 1 µ m 10 µ m 10 nm Nonuniform mesh: N ~ 100 LTE is O(h) www.nanoHUB.org NCN
2) Numerical Errors: finite word length df � f i + 1 � f i LTE --> 0 as h --> 0 dx x i + 1/ 2 h f i+1 ---> f i as h ---> 0 significance errors: f i + 1 = 0.1234567890 � 10 7 10 significant digits 7 10 significant digits f i = 0.1234567889 � 10 f i + 1 � f i = 0.1 � 10 � 2 1 significant digit! www.nanoHUB.org NCN
2) Numerical Error vs. Grid Spacing Numerical Error LTE significance error h For numerical solution of PDE ’ s, LTE typically dominates, make h as small as possible (but small h increases N , solution time, and memory!) www.nanoHUB.org NCN
2) Numerical Error: Example L J = q R ( x ) dx A/cm 2 � 0 J � 1.3 � 10 � 24 A/cm 2 let A = 10 µ m x 10 µ m I � 1.3 � 10 � 30 A 1 electron every 15M years PN Junction Educational Tool www.nanoHUB.org NCN
2) Discretization: Example Gridding: P-Si 1) resolve variations in the unknowns 2) minimize LTE N-Si 3) minimize N (solution time) (from Mark Pinto) www.nanoHUB.org NCN
2) Discretization: Example Gridding examples Uniform rectangular grid General tensor product 9409 points 1156 points Terminating line- rectangular General triangular 387 points 264 www.nanoHUB.org NCN
2) Discretization: Example Uniform General rectangular grid tensor product 9409 points 1156 points Terminating line- General rectangular triangular 387 points 264 www.nanoHUB.org NCN
2) Discretization: Tips Gridding tips • place nodes where V, p, and n are expected to vary • avoid abrupt changes in h • verify the accuracy of the grid by re-solving with a finer grid NOTE: for simple MOS geometries, gridding can be automated e.g. MINIMOS automatically defines a grid and redefines it when the bias changes www.nanoHUB.org NCN
2) Discretizing a PDE Poisson Current Continuity r r ( ) J n = � q G � R D = � � • � • r r � � � � D d � = � d � � • J d � = � q ( G � R ) d � � • n � � � � r r r r J � � D • d S � d � = � � n � q • d S ( G � R ) d � = S � S � www.nanoHUB.org NCN
2) Control Volume x (i, j -1) y 3 unknowns at each node: (i +1, j) (i -1, j) V ij , n ij , p ij (i,j) Need 3 equations (i, j +1) at each node “control volume” www.nanoHUB.org NCN
2) Discretizing Poisson ’ s Equation ( ) h = � i , j h 2 D R + D B � D L � D T (i, j -1) D T D L = � S � 0 E L D L � � S � 0 (i +1, j) (i -1, j) (i,j) ( ) V i � 1, j � V i , j h D L D R D B (i, j +1) i , j V i , j � 1, V ( ) = 0 F i � 1, j , V i , j , V i + 1, j , V i , j + 1 , n i , j , p i , j V www.nanoHUB.org NCN
2) The 3 Discretized Equations i , j = 0 x F V (i, j -1) i , j = 0 F y n i , j = 0 (i +1, j) (i -1, j) (i,j) F p (i, j +1) 3 unknowns at each node N nodes 3N unknowns and 3N equations (nonlinear!) www.nanoHUB.org NCN
2) Discretization: pitfalls (i, j -1) r ( ) J n = � q G � R � • (i -1, j) (i,j) dV dn J nL = � nq µ n dx + kT µ n J nL dx (i, j +1) The simplest approach….. � � � � � � = � n i � 1, j + n i , j V i , j � V i � 1, j � + n i , j � n i � 1, j J nL � � � � � ( ) kT µ n 2 h kT / q h � � � � � � www.nanoHUB.org NCN
2) Discretization: pitfalls ( ) � � V n i , j = 2 kT / q J nL = 0 ( ) + � V n i � 1, j 2 kT / q (equilibrium) ( ) � V = V i , j � V i � 1, j > 2 kT / q fails when: (use Scharfetter-Gummel discretization instead!) www.nanoHUB.org NCN
3) Numerical Solution • have a system of 3N nonlinear equations to solve • recall Poisson ’ s equation at node (i,j): i , j V i , j � 1, V ( ) = 0 F i � 1, j , V i , j , V i + 1, j , V i , j + 1 , n i , j , p i , j V r r [ ] A V = b linear if n ij and p ij are known V � � 1 r V � � 2 [A]: V = � � M � � V � � N www.nanoHUB.org NCN
3) Curse of Dimensionality Linear systems: 1D N ~ 100 nodes [A]: 100 x 100 2D N ~ 10,000 [A]: 10,000 x 10,000 3D N ~ 100,000 [A]: huge! Sparseness = # of non-zero elements / total number (~ 5 / N for 2D) Linear system solution methods: direct iterative www.nanoHUB.org NCN
3) Uncoupled Numerical Solution The semiconductor equations are nonlinear! (but they are linear individually) Guess V,n,p Uncoupled solution procedure Solve Poisson for new V repeat Solve electron until cont for new n satisfied Solve hole cont for new p www.nanoHUB.org NCN
3) Coupled vs. Uncoupled Numerical Solution 1) Uncoupled (sequential) method: basis of Gummel ’ s method memory efficient may converge rapidly at low bias; slowly at high bias 2) Coupled method: a generalization of Newton ’ s method requires more memory converges more quickly may require a careful initial guess (e.g. from a sequential method) www.nanoHUB.org NCN
3) Numerical Solution: Stopping How do we know when we ’ re done? 1) r || r || r r V k , r k , r � � F k ) � � F V ( n p r r V r r k , r k , r = r � � � k ) � F 0 � = F n ( V n p r Is a measure of the � � r n � r r k , r k , r k ) � � numerical error F p ( V n p F � � � � p 2) k = V k + 1 � V k � V Δ V k --> 0 as k --> oo www.nanoHUB.org NCN
3) Convergence diverging Convergence tips: change in n, p, or V Residual norm or • check problem definition converging slowly • take small steps in voltage • increase k max if converging tol • change convergence criterion Iteration # • try another method www.nanoHUB.org NCN
3) Numerical Solution: Summary Summary: Solving Partial Differential Equations 1) Begin with a set of equations and boundary conditions 2) Discretize the equations on a grid with N nodes to obtain 3N nonlinear equations in 3N unknowns 3) Solve the system of nonlinear equations by iteration www.nanoHUB.org NCN
4) Physical Models The physical parameters in the semiconductor equations need to be modeled. e.g. µ i µ = 1) doping dependent mobility � 1 + N D N µ o 2) field dependent mobility µ = 1 + E E cr 3) recombination 2 np � n i R = ( ) � po + p + p 1 ( ) � no n + n 1 4) etc. www.nanoHUB.org NCN
4) Physical Models: Example MINIMOS physical parameters (see Ch. 2 of manual) 1) doping, field, and temperature dependent mobility 2) SRH recombination 3) impact ionization 4) band-to-band tunneling 5) interface and traps 6) intrinsic carrier concentration 7) hot carrier transport model parameters 8) Monte Carlo transport model parameters www.nanoHUB.org NCN
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