towards simulation of electric fields in silicon detectors using the robin hood method

Towards simulation of electric fields in silicon detectors using the Robin Hood method

Hrvoje Štefančić

Theoretical Physics Division Ruđer Bošković Institute

(based on joint work with Hrvoje Abraham and Predrag Lazić)

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

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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


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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


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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.


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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


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The result – a point charge close to an insulated sphere


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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– ...


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The metallic plate at a fixed potential


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Water in connected pits – the redistribution problem


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Linear memory consumption• The required memory scales linearly with N (number of

triangles) – other BEM have a memory requirement ~ N2

• 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 ε0


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Complex geometries• Corrugated plane (P.Lazić, B. N. J. Persson, Surface-roughness–

induced electric-field enhancement and triboluminescence, Europhys. Lett. 91 (2010) 46003)


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The same convergence at many scales• Point charge close to the sphere kept at a fixed

potential


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The Robin Hood method and the particle detectors and accelerators

• Micro-pattern detectors (Micromegas)• Katrin collaboration experimental setup• IEC fusor


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Micro-pattern detectorsPredrag Lazić, Denis Dujmić, Joseph A. Formaggio, Hrvoje Abraham, Hrvoje Štefančić, New approach to 3D electrostatic calculations for micro-pattern detectors, JINST 6 (2011) P12003


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Types of micro-mesh• Dependence of the electronic transparency on the type of

micro-mesh (rectangular, cylindrical, woven, calendered)


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Electric field at the micro-mesh


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Electric potential at the micro-mesh


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Ez at the micro-mesh


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Exy at the micro-mesh


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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


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Electronic transparency vs. optical transparency


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Dielectric spacer

• Full cylinder or hollow (capillary), vertical or horizontal

No spacer Full spacer Hollow spacer


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Detector system of the KATRIN experimental setup

J. A. Formaggio, P. Lazić, T. J. Corona, H. Štefančić, H. Abraham, and F. Gluck, Solving for Micro- and Macro-Scale Electrostatic Configurations Using theRobin Hood Algorithm, Progress In Electromagnetics Research B, 39 (2012) 1.


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IEC fusor


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Si microstrip detectors – from the viewpoint of electric field modeling

• Periodic structures with many details • Various dielectric layers (SiO2, Si with various dopants)• The geometry of dielectric layers is not precisely known

(especially for SiO2)• Particular elements have orders of magnitude different

dimensions (e.g. Al strips have length ~ cm, width ~ mm, thickness ~ mm) – potential problems with sharp-angled triangles

• The separation of strips (~ 50 mm) is much smaller than their width


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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 SiO2 layer


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No SiO2 – total field strength


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No SiO2 – field strength in the x direction


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No SiO2 – field strength in the y direction


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No SiO2 – field strength in the z direction


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“Thick” SiO2 layer – start from the Si block


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“Thick” SiO2 layer – add a Si cylinder


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“Thick” SiO2 layer – make a Boolean difference of the Si plate and the cylinder


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“Thick” SiO2 layer – add a SiO2 cylinder


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“Thick” SiO2 layer – make a Boolean difference of the SiO2 cylinder and a cube


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“Thick” SiO2 layer – add Al plates


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What next?

• Realistic geometry of the SiO2 layer• Dependence on the thickness of the Si

(decoupling of top and bottom strips)• Spatial charge accumulated at SiO2

• Full system analysis (interference of adjacent strips)

• Simulation of charge transport (big step)


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THANK YOU FOR YOUR ATTENTION

towards simulation of electric fields in silicon detectors using the robin hood method

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