1. Towards simulation of electric fields in silicon detectors using the Robin Hood method Hrvoje Štefančić Theoretical Physics Division Ru đ er Bošković Institute (based on joint work with Hrvoje Abraham and Predrag Lazić)

2. Electric field strength in a segment of a Si microstrip detector Peculiar behavior of the Si microstrip detector at the interstrip region observed by the IRB group (communicated by Soić and Grassi) The configuration of the electric field in the interstrip region might be relevant for the explanation Robin Hood as a method for the precise calculation of the field This workshop as a testing ground of the idea (with some preliminary results) Discuss and asses the potential of the method in Si detectors Learn about possible unpercieved opportunities or overlooked pitfalls In which direction to proceed 2 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

3. Outline of the presentation The description of the Robin Hood method – a novel tool Properties of the RH method (capacity of a unit cube, corrugated surfaces) – a powerful tool Applications in particle detectors (micro-pattern) – a tool relevant for the field of particle detectors Preliminary results for silicon detectors – a discussion point 3 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

4. The Robin Hood method How do we calculate the electric field in some electrostatic system with a complex geometry? – Solve the Poisson equation by the discretization of the 3D space - Finite Difference Methods (FDM), Finite Element Methods (FEM) – Determine the sources at surfaces (surface charge distributions at conductors, polarization of dielectrics) – Boundary Element Methods (BEM) – The Robin Hood method falls into the class of Boundary Element Methods Predrag Lazić, Hrvoje Štefančić, Hrvoje Abraham, J. Comp. Phys. 213 (2006) 117. Predrag Lazić, Hrvoje Štefančić, Hrvoje Abraham, Engineering Analysis with Boundary Elements, 32 (2008) 76. 4 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

5. The Robin Hood method – how does it work? Imagine and example of a point charge close to an insulated sphere – Divide the surface of the sphere into triangles (discretization) – Calculate the initial value of the electric potential at all triangles – Find the triangles with the maximal and the minimal value of the potential – Transfer charge from the triangle with the maximal potential to the triangle with the minimal potential so that after the transfer their potentials are equal (that is why the Robin Hood name – taking from the rich to give to the poor) – Update the value of the potential at all triangles – Iterate the procedure (find max and min, charge transfer, update) until the requirement on the precision is achieved 5 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

6. The result – a point charge close to an insulated sphere Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 6

7. The Robin Hood method The Robin Hood method is also applicable to – Conductors at a fixed potential – Dielectrics (the condition of equipotentiality is replaced by the condition on fields on both sides of the interface between the dielectrics) – Magnetostatics – Electromagnetism – Systems of linear equations –... 7 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

8. The metallic plate at a fixed potential 8 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

9. Water in connected pits – the redistribution problem 9 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

10. Linear memory consumption The required memory scales linearly with N (number of triangles) – other BEM have a memory requirement ~ N 2 The record (2008) in the precision of the capacity of the unit cube: C=0.66067786 ±8 x 10 -8 in units of 1/4 π ε 0 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 10

11. Complex geometries Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 11 Corrugated plane (P.Lazić, B. N. J. Persson, Surface-roughness– induced electric-field enhancement and triboluminescence, Europhys. Lett. 91 (2010) 46003)

12. The same convergence at many scales Point charge close to the sphere kept at a fixed potential Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 12

13. The Robin Hood method and the particle detectors and accelerators Micro-pattern detectors (Micromegas) Katrin collaboration experimental setup IEC fusor Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 13

14. Micro-pattern detectors 14 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 Predrag Lazić, Denis Dujmić, Joseph A. Formaggio, Hrvoje Abraham, Hrvoje Štefančić, New approach to 3D electrostatic calculations for micro-pattern detectors, JINST 6 (2011) P12003

15. Types of micro-mesh Dependence of the electronic transparency on the type of micro-mesh (rectangular, cylindrical, woven, calendered) Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 15

16. Electric field at the micro-mesh Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 16

17. Electric potential at the micro-mesh Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 17

18. E z at the micro-mesh Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 18

19. E xy at the micro-mesh Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 19

20. Transparency vs. field Calculations using the electric-field tracking (EFT) method and micro-tracking (MT) The less symmetrical the electric field at the micro- mesh, the larger the electronic transparency Transparency in general higher for EFT than for MT Transparency the best for cylindrical and calendered, intermediate for woven and the worst for rectangular Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 20

21. Electronic transparency vs. optical transparency Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 21

22. Dielectric spacer Full cylinder or hollow (capillary), vertical or horizontal Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 22 No spacerFull spacerHollow spacer

23. Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 23

24. Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 24

25. Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 25

26. Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 26

27. Detector system of the KATRIN experimental setup Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 27 J. A. Formaggio, P. Lazić, T. J. Corona, H. Štefančić, H. Abraham, and F. Gluck, Solving for Micro- and Macro-Scale Electrostatic Configurations Using the Robin Hood Algorithm, Progress In Electromagnetics Research B, 39 (2012) 1.

28. IEC fusor 28 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

29. Si microstrip detectors – from the viewpoint of electric field modeling Periodic structures with many details Various dielectric layers (SiO 2, Si with various dopants) The geometry of dielectric layers is not precisely known (especially for SiO 2 ) Particular elements have orders of magnitude different dimensions (e.g. Al strips have length ~ cm, width ~ mm, thickness ~  m) – potential problems with sharp-angled triangles The separation of strips (~ 50  m) is much smaller than their width 29 Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012

30. Si microstrip detectors – modelling assumptions Entire system could be analyzed in full (this lies within the capabilities of the Robin Hood Solver), but it is more instructive to focus on the interstrip region – the most interesting configuration of the electric fields – Observed reversed polarity signals for particles passing the interstrip region (Soić, Grassi) Define an “elementary cell” – centered at the interstrip region All layers of Si (differently doped) have the same dielectric constant (Capan) Al strips can be well described as conducting plates Various geometries for the SiO 2 layer Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 30

31. No SiO 2 – total field strength Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 31

32. No SiO 2 – field strength in the x direction Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 32

33. No SiO 2 – field strength in the y direction Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 33

34. No SiO 2 – field strength in the z direction Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 34

35. “Thick” SiO 2 layer – start from the Si block Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 35

36. “Thick” SiO 2 layer – add a Si cylinder Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 36

37. “Thick” SiO 2 layer – make a Boolean difference of the Si plate and the cylinder Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 37

38. “Thick” SiO 2 layer – add a SiO 2 cylinder Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 38

39. “Thick” SiO 2 layer – make a Boolean difference of the SiO 2 cylinder and a cube Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 39

40. “Thick” SiO 2 layer – add Al plates Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 40

41. What next? Realistic geometry of the SiO 2 layer Dependence on the thickness of the Si (decoupling of top and bottom strips) Spatial charge accumulated at SiO 2 Full system analysis (interference of adjacent strips) Simulation of charge transport (big step) Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 41

42. THANK YOU FOR YOUR ATTENTION Silicon Detector Workshop, Split, Croatia, 8- 10 October 2012 42