Charge cloud simulations, overview and very first results

Size: px

Start display at page:

Download "Charge cloud simulations, overview and very first results"

Joseph Dennis
5 years ago
Views:

1 W eierstraß -Institut für Angew andte Analysis und Stochastik K. Gärtner Charge cloud simulations, overview and very first results SRH generation 6M pairs, 7.5ns Berlin, July 28, 2008 charge cloud sim. meeting

2 Outline - Van Roosbroeck system, analytic/discrete properties - Goals and present status - Time scales cloud movement, example CCD charge handling - Status work and Summary charge cloud sim. meeting 2

3 Van Roosbroeck s Equations ɛ w = C n + p, (1) n t + µ nn φ n = R, (2) p t µ pp φ p = R, (3) in S Ω, S = (0, T ), Ω IR N, 2 N 3, a bounded polyhedral domain, Ω = Γ D Γ N, Γ D closed, positive surface measure. Boundary conditions: hom. Neumann on insulating parts, Dirichlet on Ohmic contacts, and gates: hom./inhom. Neumann φ/w ( w/ ν + α(w w Γ ) = 0, ν outer normal vector). charge cloud sim. meeting 3

4 Free energy, dissipation rate (ψ, n, p ): thermal equilibrium solution of ɛ ψ = f + e ψ e ψ in Ω, ψ = ψd on Γ D, n ψ + α(ψ ψ Γ ) = 0 on Γ gate, n = e ψ, p = e ψ. Free energy: F (ψ, n, p) = [n(log n n 1) + n + p(log p p 1) + p ] dω ψ ψ 2, with h 2 = ɛ h 2 dω + αh 2 dγ. Dissipation rate: d(ψ, n, p) = [nµ n φ n 2 + pµ p φ p 2 + r(x, n, p)(np 1) log(np)] dω 0. charge cloud sim. meeting 4

5 Very short summary: analytic results 1) thermodynamic equilibrium boundary conditions: the free energy decays for any initial data along trajectories (exponentially) to its equilibrium value, the thermodynamic equilibrium is unique; 2) existence of bounded steady state solutions (smoothness assumptions on data, models and domains, space dimensions), uniqueness close to equilibrium; 3) time dependent problems: existence and uniqueness for finite time intervals (avalanche blow up), in special cases global in time (no easy bounds far from steady state, non trivial attractor dimensions); 4) 1d unipolar problem: unique, bounded steady state solution. Discrete counterparts: On boundary conforming Delaunay grids 1)... 4) (up to exponential decay) can be proved, too! Byproduct of proofs: possible averages for µ(x, n, p, φ i ),... resulting in positive solutions. charge cloud sim. meeting 5

6 General approach a) Discretization with proven properties (price to pay for general domains: Delaunay grids); b) Implicit, dissipative time discretization, time step control based on free energy, dissipation rate, source integrals...; c) Newton s method and implicit damping; d) linear systems solved by a combination of direct and iterative methods. If a),b),c) are solved, solving the linear systems is the challenge! Complexity estimates direct methods (spd case, d space dimension, n unknowns per space dimension, grids of size N = n d ): Operations for d = 2 d = 3 Factorization N 3 2 N 2 Solution Nlog(N) N 4 3 charge cloud sim. meeting 6

7 Goals and present status Numerics goals: - complete understanding, fast and reliable algorithms; - reduced complexity for linear systems (huge condition numbers); - general anisotropic boundary conforming Delaunay meshes; - extended mobility, recombination models; - a limited number of applications, not entirely covered by commercial tools; Status: - bottleneck is solving the lin. systems (limits problem sizes 10 6 nodes, 128GB); - essential parts are SMP-parallel; - precise current evaluation (typical rel. balance error ) - formal and a few HLL stationary and dynamic examples solved; Dream: 10 7 nodes, mainly by algorithmic improvements. charge cloud sim. meeting 7

8 Goals and present status Main steps to get things going: a) construction of a 2d Delaunay grid b) extension (transl., rotation) to 3d with boundary, material assignment c) doping profiles (analytic, 1d, 2d DIOS interpolation) d) description source distribution (MC results), charge integrals e) material, contact assignment, time functions... f) IV, I(t), V(t), charge integrals, graphics General detector design problems: huge, sometimes complex domains, very large or small charges, small differences; charge cloud sim. meeting 8

pin Diode, 3D 10 o sector, ToSCA grid in the r z plane, doping: top N +, bottom P +, volume N = 2 10 12 /cm 3, top

9 pin Diode, 3D 10 o sector, ToSCA grid in the r z plane, doping: top N +, bottom P +, volume N = /cm 3, top 200V, depletion 120V, R = 75µm, Z = 300µm, nodes, tets, 1.6M pairs full cylinder charge cloud sim. meeting 9

10 Time scales cloud movement deposition time (4ps), no changes in the electrostatic pot. charge cloud sim. meeting 10

11 Time scales cloud movement local dd time (100ps), dipole moment developed, pot. flattened charge cloud sim. meeting 11

12 Time scales cloud movement drift time (1ns), particle current in the volume charge cloud sim. meeting 12

13 Time scales cloud movement cloud separation time ns, pair number and shape dependent charge cloud sim. meeting 13

14 CCD charge cloud sim. meeting 14

15 CCD Minimal configuration: total 3 registers l. blue: channel stop red stoarge regions X = 75µm, Y = 75µm, Z = 150µm, nodes, tets, BACK -50V, R2=-10V, R1=R3=R4= -15 / -18V, MOS1=MOS2=MOS3=5V Doping in thermal voltages charge cloud sim. meeting 15

16 CCD Steps in the computation: - Depletion, - reduce n artificially (multiply by 10 7, time integration 0.01as..1ns), 5.8 electrons are still in the domain; - charge cloud close to BACK beneath R2 and let it distribute; - shift the electrons: R2 R3 R2; - check the charge integrals in the lower 99.5% of the domain. charge cloud sim. meeting 16

17 CCD Total charge balance charge cloud sim. meeting 17

18 Planned work E1) generalization of the initial cloud distribution E2) mobility models µ(n, p, φ n, φ p ) M0) comparison of 2d (TeSCA cylindrical symmetry) and 3d simulations M1) simulation of typical diode pixel detectors with the present code version D1) libraries, images of the code (min. twice a year) linux, AIX D2) support for the design groups (getting started with a typical example, prototype computations at WIAS) more and details see contract and appendix charge cloud sim. meeting 18

19 comparison TeSCA cylindrical symmetry charge cloud sim. meeting 19

20 comparison TeSCA cylindrical symmetry charge cloud sim. meeting 20

21 comparison TeSCA cylindrical symmetry charge cloud sim. meeting 21

22 Status contract and Summary Contract signed in June (with date April 1). Jens Griepentrog will work on the related position; Th. Koprucki will use the code at WIAS for VCSEL simulations; Work on user documentation started; Begin of October: 1st delivery to HLL, Uni HH, RAL; Next steps: mobility models, larger clouds; Thank you for your attention charge cloud sim. meeting 22

Charge cloud simulations, status report

W eierstraß-institut für Angew andte Analysis und Stochastik K. Gärtner Charge cloud simulations, status report SRH generation 6M pairs, 7.5ns Hamburg, Oct. 14, 2008 XDAC meeting Outline - Short recapitulation