Scarce Documentation. Release David-Leon Pohl

Transcription

1 Scarce Documentation Release David-Leon Pohl Nov 17, 2017

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3 Contents 1 Installation Linux Windows Latest comits 5 3 Silicon 7 4 Potentials and fields Decription class Methods Solver Methods Examples Weighting field of a planar sensor Potential of a planar silicon sensor Weighting field of a 3D sensor Potential of a 3D sensor D potential in a planar silicon sensor Drifting e-h pairs in planar sensor Drifting e-h pairs in 3D sensor Silicon properties Indices and tables 43 Python Module Index 45 i

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5 Contents: Scarce stands for silicon charge collection efficiency and is a software to calculate the charge collection-efficiency of irradiated and segmented silicon sensors. Planar and 3D electrode configurations are supported. Additionally a collection of formulars is provided to calculate silicon properties. Contents 1

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7 CHAPTER 1 Installation The installation works with Linux and Windows. Mac OS might also work. 1.1 Linux This installation has been tested with Ubuntu LTS 64-bit and Anaconda Python bit. 1. Install the mesh creator gmsh: sudo apt-get install gmsh 2. Install Anaconda Python distribution: wget -O miniconda. sh For more information visit: 3. Install precompiled dependencies: conda install numpy pytables scipy matplotlib 4. Install sparse matrix solver (optional, increases speed): pip install -e git://pysparse.git.sourceforge.net/gitroot/pysparse/pysparse #egg=pysparse ez_setup 5. Download Scarce: git checkout 6. Install Scarce in development mode by typing: 3

8 cd Scarce && python setup.py develop 1.2 Windows This installation has been tested with Windows 7 64-bit and Anaconda Python bit. 1. Install the mesh creator gmsh that can be donwloaded here: 2. Install 64-bit Anaconda Python 2.7 distribution that can be donwloaded here: downloads#windows 3. Install precompiled dependencies by typing into the command prompt: conda install numpy pytables scipy matplotlib 4. Download Scarce here and unpack to a folder of your choise: zip 5. Install Scarce in development mode by typing: python setup.py develop 4 Chapter 1. Installation

9 CHAPTER 2 Latest comits REG by DavidLP at :17:39 MAINT by DavidLP at :51:26 ENH: simplify by DavidLP at :37:10 REG by DavidLP at :36:00 ENH: simplify more by DavidLP at :08:22 MAINT: formatting by DavidLP at :04:42 ENH: simplify example by DavidLP at :48:26 MAINT: formatting by DavidLP at :47:36 BUG: version string by DavidLP at :55:28 ENH: reduce pdf plot size and set contour by DavidLP at :19:27 5

10 6 Chapter 2. Latest comits

11 CHAPTER 3 Silicon Collection of (semi-) empiric functions describing silicon properties. scarce.silicon.get_depletion_depth(v_bias, n_eff, temperature) Depletion depth [um] of silcon with an effective doping concentration n_eff [10^12 /cm^3] at a temperature [K] and reverse bias V_bias [V]. Check/citetation of formulars needed! 7

12 Depletion depth in silicon N eff = /cm 3, T = 300 N eff = /cm 3, T = 300 N eff = /cm 3, T = 300 Depletion depth [um] Bias voltage [V] 8 Chapter 3. Silicon

13 400 Depletion depth in silicon Depletion depth [um] p type, V bias = 100 V, T = V bias [V] ρ [Ω cm] n type, V bias = 100 V, T = Resistivity [Ω cm] scarce.silicon.get_depletion_voltage(n_eff, distance) This function returns the full depletion Voltage [V] as a function of the effective doping concentration Neff [10^12 /cm^-3] and the distance between electrodes [um]. Formular is standard and for example used in: G. Kramberger et al., Nucl. Inst. And. Meth. A 476 (2002) Determination of effective trapping times for electrons and holes in irradiated silicon. 9

14 Full depletion voltage in silicon Electrode distance = 100 um Electrode distance = 150 um Electrode distance = 200 um Electrode distance = 250 um Depletion Voltage [V] Effective doping concentration [10 12 /cm 3 ] scarce.silicon.get_diffusion_potential() Diffusion potential [V] as a function of the effective doping concentration n_eff [10^12 / cm^3] and the temperature [K]. Check/citetation of formulars needed! 10 Chapter 3. Silicon

15 0.8 Diffusion potential at thermal equilibrium in silicon Diffusion potential [V] T = 200 T = 250 T = 300 T = Effective doping concentration [10 12 /cm 3 ] scarce.silicon.get_eff_acceptor_concentration() Effective acceptor concentration [10^12 cm^-3] of irradiated n- and p- type silicon with and without oxygen enriched as a function of the fluence [10^12 cm^-2]. The data can be desribed by different line fits. The parameters were extracted from a plot taken for n-type silicon from: CERN/LHCC LEB Status Report/RD48 31 Dec and for p-type silicon from: RD50 Status Report 2006 CERN-LHCC , p. 4-6 Due to the difference in the data for different technologies a rather large error on the propotionality factor of 10% is assumed. 11

16 Acceptor concentration Neff [10 12 cm 3 ] Effective acceptor concentration p-type oxygenated p-type n-type oxygenated n-type Fluence [10 12 N eq /cm 2 ] scarce.silicon.get_free_path() Calculate mean free path [cm] of charge carriers from trapping propability and the velocity. The trapping propability is a function of the fluence and the velocity is a function of the electric field and the temperature. The electric field itself depends on the electrode geometry and the bias voltage. velocity = get_mobility(e_field, temperature, is_electron) * e_field trapping_time = get_trapping(fluence, is_electron, paper=1) 12 Chapter 3. Silicon

17 10 3 Charge carrier mean free path in irradiated silicon at saturation velocity (E = 10 6 V/cm) Electrons, T = 250 Holes, T = Trapping time [ns] Fluence [N eq /cm 2 ] scarce.silicon.get_mobility() Calculate the mobility [cm^2/vs] of charge carriers in silicon from the electrical field (E [V/cm]) the temperature (T [K]) and the charge carrier type (iselectron [0/1] otherwise hole). Formular derived from measured data of high purity silicon and the corresponding fit function parameters are used here. From: C. Jacononi et al., Solid state electronics, 1977, vol 20., p. 87 A review of some charge transport properties of silicon Note: The doping concentration is irrelevant for n_eff < 10^16/cm^3 13

18 Electron/-hole mobility [cm 2 /Vs] Charge carrier mobility in silicon Electrons, T = 250K Electrons, T = 300K Holes, T = 250K Holes, T = 300K Electric field [V/cm] 14 Chapter 3. Silicon

19 Electron/-hole velocity [cm/s] 1.2 1e V 200µm Charge carrier velocity in silicon Electrons, T = 250K Electrons, T = 300K Holes, T = 250K Holes, T = 300K 1000 V 200µm Electric field [V/cm] scarce.silicon.get_resistivity() Calculate the resitivity from: The effective doping concentration n_eff [10^12 / cm^3] the mobility [cm^2/vs] for n- and p-type silicon. The mobility istself is a function of the temperature [K] and the electric field [V/cm]. From edu/~bart/book/mobility.htm TODO: If you take the mobility[e_field] equation seriously, then there is no constant resitivity since the mobility depends also on the electric field. For low E-Fields <= 1000 V/cm the mobility is independent of the E flied and thus the resistivity. Likely this parameter is always given in low field approximation?! Source needed! 15

20 Resistivity of silicon (low e-field approximation) n-type p-type Resistivity [Ω cm] Effective doping concentration [cm 3 ] scarce.silicon.get_trapping() Calculate the trapping time tr (e^-(tr) in ns) of charge carriers in silicon as a function of the fluence [Neq/cm^2]. There was also a dependence on the temperature measured, that is omitted here! 16 Chapter 3. Silicon

21 10 4 Charge carrier trapping time in irradiated silicon Electrons Holes 10 3 Trapping time [ns] Fluence [N eq /cm 2 ] 17

22 18 Chapter 3. Silicon

23 CHAPTER 4 Potentials and fields The field module calculates the potentials and fields of silicon sensors. The potential is determined numerically by solving these equations on a mesh: 2 Φ = 0 (4.1) 2 Φ = ρ ε (4.2) For the weighting potential equation (4.1) is solved with the boundary conditions: to Φ = 0 Φ r = 1(4.3) = = 0Φ r 1 19

24 The pixel readout electrode(s) are at a potential 1 and all other equipotential pixel parts (backside, bias columns, etc.) at 0. For the electric potential the equation (4.2) is solved with the boundary conditions: to Φ = V bias Φ r = V readout (4.3) = = V bias Φ r V readout The pixel readout electrode(s) are at V readout potential and the bias parts (backside, bias columns, etc.) are at V bias. The field is then derived via: E = φ Note: For simple cases (e.g. planar sensor with 100% fill factor) also analytical solutions are provided. The analytical results are also used to benchmark the numerical results in the automated unit tests. 4.1 Decription class The challenge for the field determination is that numerical differentiation of the potential amplifies numerical instabilities, thus the potential has to be smoothed before differentiation. A convinient interface is provided by the field description class. class scarce.fields.description(potential, min_x, max_x, min_y, max_y, nx, ny, smoothing=0.1) Class to describe potential and field at any point in space. The numerical potential estimation is used and interpolated. The field is derived from a smoothed potential interpolation to minimize numerical instabilities. 4.2 Methods scarce.fields.calculate_planar_sensor_w_potential(mesh, width, pitch, n_pixel, thickness) Calculates the weighting field of a planar sensor. 20 Chapter 4. Potentials and fields

25 scarce.fields.calculate_3d_sensor_w_potential(mesh, width_x, width_y, n_pixel_x, n_pixel_y, radius, nd=2) scarce.fields.get_weighting_potential_analytic(x, y, D, S, is_planar=true) Planar sensor: From Nuclear Instruments and Methods in Physics Research A 535 (2004) , with correction from wbar = pi*w/2/d to wbar = pi*w/d with: x [um] is the offset from the middle of the electrode y [um] the position in the sensor D [um] the sensor thickness S [um] the pixel pitch 3D sensor: Weighting potential for two cylinders with: D [um] distance between columns S [um] is the radius scarce.fields.get_weighting_field_analytic(x, y, D, S, is_planar=true) Calculates the analytical weighting field. Parameters x (number) X position in sensor in um y (number) Y position in sensor in um D (number) The sensor thickness in um S (number) 3D sensor only: electrode radius in um is_planar ({True, False}) To select 3D or planar geometry Returns Weighting field in x/y in V/um Return type array_like Notes From Nuclear Instruments and Methods in Physics Research A 535 (2004) , with correction w = The field is calculated from the drivation of the potential. π * w π * w > w = 2D D scarce.fields.get_potential_planar_analytic_1d(x, V_bias, V_readout, n_eff, D) Calculates the potential in the depletion zone of a planar sensor. Parameters x (array_like) Position in the sensor between 0 and D in μm V_bias (number) Bias voltage in volt. V_readout (number) Readout voltage in volt. n_eff (number) Effective doping concetration in cm 3 D (number) Thickness of the sensor in μm Notes The formular can be derived from the 1D Poisson equation (4.2), wich has the following general solution for x <= x dep with the full depletion assumption: Φ p = ρ 2ε x2 + const p,1 x + const p, Methods 21

26 For the undepleted region x > x dep there is no spacecharge (ρ = 0). Thus the generel solution of the 1D Laplace equation (4.1) can be used here: Φ l = const l,1 x + const l,2 For an underdepleted sensor (x dep <= D) these boundary conditions have to be satisfied: 1. Φ p (0) = V readout 2. Φ l (D) = V bias 3. Φ p (x dep ) = Φ l (x dep ) x Φ p(x dep ) = 0 x Φ l(x dep ) = 0 The following simultaneous equations follow: 1. Φ p = ρ 2ε x2 + const p,1 x + V readout 2. Φ l = (x D) const l,1 + V bias ρ 2ε x2 dep + const p,1 x dep + V readout = (x dep D) const l,1 + V bias ρ ε x dep + const p,1 = 0 5. const l,1 = 0 With the solution: 22 Chapter 4. Potentials and fields

27 { ρ Φ(x) = 2ε x2 ρ ε x depx + V readout x x dep V bias x > x dep with x dep = 2ε ρ (V readout V bias) If the sensor is fully depleted (x dep > D) only Φ p has to be solved with the following boundary conditions: 1. Φ p (0) = V readout 2. Φ p (D) = V bias The following simultaneous equations follow: 1. Φ p = ρ 2ε x2 + const p,1 x + V readout 2. Φ p (D) = ρ 2ε D2 + const p,1 D + V readout = V bias With the solution: Φ(x) = ρ ( Vbias V readout 2ε x2 + ρ ) D 2ε D x + V readout For the generell solution follows: ρ 2ε x2 ρ ε x depx + V readout Φ(x) = V bias ( ρ 2ε x2 ρ 2ε D x 2 ) dep D + 1 x + V 2 readout x dep D, x x dep x dep D, x > x dep x dep > D with x dep = 2ε ρ (V readout V bias) scarce.fields.get_electric_field_analytic(x, y, V_bias, n_eff, D, V_readout=0, S=None, is_planar=true) Calculates the electric field. Planar sensor: Calculates the field E_y[V/um], E_x = 0 as a function of the position x between the electrodes [um], the bias voltage V_bias [V], the effective doping concentration n_eff [10^12 /cm^-3] and the sensor Width D [um]. The analytical function from the detector book p. 93 is used. The field is derived from the potential: E = Φ 4.2. Methods 23

28 From the solution of the potential follows in 1D ( E( x) = E(x)): ρ ε x dep ρ ε x x dep D, x x dep E(x) = 0 ( x dep D, x > x dep ρ 2ε D x 2 ) dep D + 1 ρ 2 ε x x dep > D with x dep = 2ε ρ (V readout V bias) 3D sensor: Calculates the field E_x/E_y [V/um] in a 3d sensor as a function of the position x,y between the electrodes [um], the bias Voltage V_bias [V], the effective doping concentration n_eff [cm^-3], the electrode distance D [um] and radius R [um]. So far the same field like the weighting field is used > space charge is ignored. 24 Chapter 4. Potentials and fields

29 CHAPTER 5 Solver The solver module provides numerical ODE solver functions. These functions are either implemented here or provide an interface to other solvers. class scarce.solver.driftdiffusionsolver(pot_descr, pot_w_descr, T=300, geom_descr=none, diffusion=true, t_e_trapping=0.0, t_h_trapping=0.0, t_e_t1=0.0, t_h_t1=0.0, t_r=0.0, save_frac=20) Solve the drift-diffusion equation for pseudo particles. Solving via euler forward difference. 5.1 Methods scarce.solver.solve(var, equation, **kwargs) Interface to the fipy solver used for the 2d poisson equation. Parameters var (fipy.cellvariable) Fipy meshed cell variables to solve the equation for equation (fipy equation) e.g. fipy.diffusionterm() = 0 kwargs (kwargs) Arguments of fipy equation.solve(kwargs) Notes A linear LU solver is forced here, since otherwise pysparse based solver do not converge properly. 25

30 26 Chapter 5. Solver

31 CHAPTER 6 Examples 6.1 Weighting field of a planar sensor Example that creates a planar silicon sensor with a given geometry. Calculates the weighting potential and fields. For comparison also the analytical result of a planar sensor with 100% fill factor (width = pitch) is created. Note: With increasing distance from the center pixel the numerical result deviates from the analytical one. This shows that is is important to use several pixels (> 5) to get a proper field description in the center pixel. 27

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33 Scarce Documentation, Release The mesh size determines the quality of the numerical result and can be changed in the example. 6.2 Potential of a planar silicon sensor Example that creates a planar silicon sensor with a given geometry. Calculates the electrical potential and fields. For comparison also the analytical result of a planar sensor with 100% fill factor (width = pitch) is created. Warning: The calculation of the depletion region is simplified. If the depletion is not at a contant y position in the sensor (e.g. for pixels with very small fill factor) it deviates from the correct solution Potential of a planar silicon sensor 29

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35 6.3 Weighting field of a 3D sensor Example creating the weighting field and potential of a 3D pixel array. Note: The weighting potential and field is only correct if the pixel is surrounded by other pixels, thus n_pixel_x = n_pixel_y = Weighting field of a 3D sensor 31

36 The mesh size determines the quality of the numerical result and can be changed in the example. 6.4 Potential of a 3D sensor Example that creates a 3D pixel array with a given geometry. 32 Chapter 6. Examples

37 Note: The calculation of a partially depleted 3D sensor is supported Potential of a 3D sensor 33

38 6.5 1D potential in a planar silicon sensor Example that shows the potential in a planar sensor in 1D. A not fully depleted is supported.the poisson equation is solved numerically and compared to the analytical solution. The 1D solution is also correct in the 2D case for a sensor with 100% fill factor. The numerical solving for underdepleted sensors is not straight forward, due to the a priory unknown depletion depth. This example shows also how to deal with that. 34 Chapter 6. Examples

39 6.6 Drifting e-h pairs in planar sensor Example that moves e-h pairs in a planar sensor. Calculates the induced charge from e-h pairs drifting through the silicon Drifting e-h pairs in planar sensor 35

40 36 Chapter 6. Examples

41 6.7 Drifting e-h pairs in 3D sensor Example that moves e-h pairs in a 3D sensor. Calculates the induced charge from e-h pairs drifting through the silicon. Warning: The 3D field is low between pixels. Thus diffusion should be activated to leave this field minima quickly to give reasonable results. The following shows the induced charge for e-h pairs at different start positions with and without diffusion. With diffusion: 6.7. Drifting e-h pairs in 3D sensor 37

42 Without diffusion: 38 Chapter 6. Examples

43 With diffusion: 6.7. Drifting e-h pairs in 3D sensor 39

44 Without diffusion: 40 Chapter 6. Examples

45 6.8 Silicon properties This example creates plots for all silicon properties available in Scarce. These plots are shown in Silicon Silicon properties 41

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47 CHAPTER 7 Indices and tables genindex modindex search 43

48 44 Chapter 7. Indices and tables

49 Python Module Index s scarce.examples.plot_properties, 41 scarce.examples.potential_1d, 34 scarce.examples.sensor_3d, 32 scarce.examples.sensor_3d_weighting, 31 scarce.examples.sensor_planar, 29 scarce.examples.sensor_planar_weighting, 27 scarce.examples.transient_3d, 37 scarce.examples.transient_planar, 35 scarce.fields, 19 scarce.silicon, 7 scarce.solver, 25 45

50 46 Python Module Index

51 Index C calculate_3d_sensor_w_potential() (in module scarce.fields), 21 calculate_planar_sensor_w_potential() (in module scarce.fields), 20 D Description (class in scarce.fields), 20 DriftDiffusionSolver (class in scarce.solver), 25 G get_depletion_depth() (in module scarce.silicon), 7 get_depletion_voltage() (in module scarce.silicon), 9 get_diffusion_potential() (in module scarce.silicon), 10 get_eff_acceptor_concentration() (in module scarce.silicon), 11 get_electric_field_analytic() (in module scarce.fields), 23 get_free_path() (in module scarce.silicon), 12 get_mobility() (in module scarce.silicon), 13 get_potential_planar_analytic_1d() (in module scarce.fields), 21 get_resistivity() (in module scarce.silicon), 15 get_trapping() (in module scarce.silicon), 16 get_weighting_field_analytic() (in module scarce.fields), 21 get_weighting_potential_analytic() (in module scarce.fields), 21 S scarce.examples.plot_properties (module), 41 scarce.examples.potential_1d (module), 34 scarce.examples.sensor_3d (module), 32 scarce.examples.sensor_3d_weighting (module), 31 scarce.examples.sensor_planar (module), 29 scarce.examples.sensor_planar_weighting (module), 27 scarce.examples.transient_3d (module), 37 scarce.examples.transient_planar (module), 35 scarce.fields (module), 19 scarce.silicon (module), 7 scarce.solver (module), 25 solve() (in module scarce.solver), 25 47