- Email: [email protected]
Charge collection characterization of a 3D silicon radiation detector by using 3D simulations
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296 www.elsevier.com/locate/nima
Charge collection characterization of a 3D silicon radiation detector by using 3D simulations Juha Kalliopuskaa,b,1,, Simo Era¨nenb, Risto Oravac,d a
Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland VTT Technical Research Center of Finland, Micro- and Nanoelectronics, Tietotie 3, P.O. Box 1000, FI-02044 VTT, Finland c High Energy Physics Division, University of Helsinki, P.O. Box 64, FI-00014, Finland d CERN, Helsinki Institute of Physics, CERN/PH, CH-1211 Geneva, Switzerland
b
Available online 29 November 2006
Abstract In 3D detectors, the electrodes are processed within the bulk of the sensor material. Therefore, the signal charge is collected independently of the wafer thickness and the collection process is faster due to shorter distances between the charge collection electrodes as compared to a planar detector structure. In this paper, 3D simulations are used to assess the performance of a 3D detector structure in terms of charge sharing, efficiency and speed of charge collection, surface charge, location of the primary interaction and the bias voltage. The measured current pulse is proposed to be delayed due to the resistance–capacitance (RC) product induced by the variation of the serial resistance of the pixel electrode depending on the depth of the primary interaction. Extensive simulations are carried out to characterize the 3D detector structures and to verify the proposed explanation for the delay of the current pulse. A method for testing the hypothesis experimentally is suggested. r 2006 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 29.40.Gx; 28.52.Av Keywords: Silicon radiation detectors; 3D detectors; Charge collection characteristics; Charge sharing
1. Introduction In this paper, characterization of the performance of a 3D detector pixel element [1] is reported in terms of its charge collection (CC) efficiency and time. The CC time dependence on the surface charge concentration at the silicon–oxide interface, bias voltage, charge sharing and the location of the primary interaction within the detector bulk are studied by using 3D simulations. The delay in the CC process, proposed to be caused by an induced resistance– capacitance (RC) product, is analyzed by simulating the Corresponding author. Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland. Tel.: +358 414 313 212; fax: +359 9 456 7012. E-mail addresses: [email protected].fi (J. Kalliopuska), simo.eranen@vtt.fi (S. Era¨nen), [email protected] (R. Orava). 1 The Ph.D. studies of the corresponding author has been financed by Helsingin Sanomain 100-vuotissa¨a¨tio¨.
0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.10.370
detector structures with varying thicknesses. A cylindrical capacitor structure, for which the RC product can be calculated in a straightforward manner, is used as a convenient reference. A delay in the simulated current pulse is shown to occur depending on the depth of the primary interaction of an incident particle in the detector bulk. A way to measure the CC pulse delay is suggested. The 2D and 3D simulations were carried out by using the ICE-TCAD simulation software for semiconductor devices [2]. A rectangular 3D pixel element was defined as a reference detector structure for the characterization. The detector structures and procedures used in the simulations are further described in Appendix A. 2. Results Dependence of the CC time and charge sharing on the impact location of an incident minimum ionizing particle
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
(MIP) traversing the detector surface was simulated in 2D at 20 V bias voltage by using the pixel structure described in Fig. A.1(a). The incident MIPs hit the detector surface in 10 different locations as shown in Fig. A.1(a) by a dashed line extending from inside to the outer edge of the pixel element. The CC is considered complete when 90% of the over-all signal charge is collected, i.e. possible tail of the current response distribution is ignored. As depicted in Fig. 1, the signal charge is collected in less than 10 ns at 20 V within the main pixel volume. The time interval between recording the peak of the charge pulse and in collecting the over-all charge signal is due to spreading of charge carriers during the CC process. The charge is shared within a few microns from the pixel edge, only, and the signal charge—induced by an incident MIP—is collected by the pixel electrode with an efficiency exceeding 95%. Similar results are obtained for somewhat different 3D detector structure, compatible with the Medipix2 read-out chip [3,4]. The detector pixel is fully depleted at about 20 V bias voltage. It is, however, of advantage to use a higher reverse bias voltage in order to increase the electric field for the benefit of a faster CC. Fig. 2(a) depicts the dependence of CC on the applied bias voltage. The maximum bias voltage is defined by the saturation limit of the velocity of the charge carriers; doubling the reverse bias from 40 to 80 V does not significantly speed up CC. At 10 V, the detector is not yet fully depleted, and the CC process is slowed down appreciably. The 3D CC efficiencies are depicted in Fig. 2(b) for a MIP entering the center of a quarter of a 3D detector pixel element (see Fig. A.1(c)). The bias voltage is set to 20 V; Charge sharing and charge collection time dependence on interaction location 100 20
Peak time Charge collection time Charge collected correctly Charge shared
10
60
40
Relative charge [%]
Time [ns]
80
20
0
surface recombination and different uniform surface charge concentrations at the silicon–oxide interface are included in the simulation. The charge collected by the pþ electrode was calculated by integrating over the current response. With the surface charge of 6 1011 cm2 , a high leakage current in excess of 10 mA was observed. This is due to the breakdown of the pn-junction and no results for surface charge above 5 1011 cm2 could be considered in the analysis. Introduction of a surface charge creates an undepleted accumulation layer below the oxide–silicon interface and modifies the electric field [5]. The charge carriers created within the accumulation layer have to diffuse outside this volume in order to get collected by the electric field. Due to this, the CC time increases slightly as seen in Fig. 2(b), where the CC efficiency curves are seen to saturate later than the ones depicting the situation without adding the surface charge. The CC efficiency falls with an increased surface charge concentration due to the holes recombining with electrons in the accumulation region. 2.1. Dependence of charge collection time on RC constant The CC time of a 3D detector pixel is observed to depend on the RC product of the pixel and on the depth of the primary interaction by the incident particle. These correlations have been analyzed by simulating the CC time as a function of detector thickness and by comparing the simulation results with theoretical calculations for a simplified reference structure and by demonstrating the dependence of the CC time delay on the depth of the primary interaction by an incident MIP. The RC product of a fully depleted 3D detector pixel can be approximated by a cylindrical capacitor in combination with a resistance represented by a pþ electrode. For this combination, the RC product can be calculated in a straightforward way by using Eqs. (1) and (2), where denotes the electrical permittivity (for silicon Si ¼ 11:8 0 ), h gives the wafer thickness, r1 and r2 the inner and outer radii of the cylinder, rmin the minimum resistance of the doping profile of the pþ electrode, and rmin the radius of the region with the minimum resistance. The approximation used for the resistance of the pþ electrode is valid since most of the current concentrates into the region with the minimal resistance [6]. C cyl ¼ 2p
h lnðr2 =r1 Þ
(1)
Rel ¼ rmin
h . pr2min
(2)
0 0
20 40 Distance from the edge [um]
60
Fig. 1. Dependence of CC time and charge sharing on the interaction location of the incident radiation. The X -axis shows the entrance locations of the MIP from the pixel edge in microns (mm); the peak time of the current response signal and the CC time are given on the left Y -axis and the charge collected correctly by the pixel and the charge shared by the neighboring pixels are given on the right Y -axis in percentages ð%Þ.
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By combining Eqs. (1) and (2), an approximation for the RC product of the 3D detector structure is obtained as t ¼ Rel C cyl ¼ 2rmin
h2 . r2min lnðr2 =r1 Þ
(3)
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
294
From Eq. (3) it is explicitly seen that the RC product, depends on the detector thickness, h, as: th2 . For comparing the analytical calculation (Eq. (3)) with the simulation results, detector thickness is varied while the distance of CC by the pixel electrode remains the same. By using Eq. (3) with the parameter values of Si , rmin ¼ 2 mm, rmin ¼ 0:04 O cm ð1018 cm3 Þ, r1 ¼ 10 mm and r2 ¼ 60 mm, the following approximate CC time delays are obtained: t1 10 fs, t300 1 ns, t600 4 ns and t900 10 ns. The external resistor of R ¼ 100 O dominates the over-all resistance of the 1 mm thick detector structure. Figs. 3(a) and (b) depict the current response and integrated charge of the detector structure as a function of the wafer
a
thickness. The distributions for a 1 mm thick detector are included for reference with no delay time in the CC process; for these the time constant is t1 10 fs. Both the peak current and tail increase as a function of the wafer thickness (Fig. 3(a)). The delay in CC time depends on the wafer thickness (th2 ) and causes the tails of the simulated distributions. 2.2. Charge collection time dependence on the interaction depth of the incident radiation An ionizing radiation creates charge carriers (electron– hole pairs) in the detector bulk. When moving towards the
b
Charge collection of the MIP at different reverse bias voltages
Charge collection efficiency and dependence on surface effects at 20 V. 1
50
10 20 40 80
0 0
5e-09
1e-08
1.5e-08
V V V V
Charge collection efficiency
Percentage of charge collected
100
1
0.8
0.8
0.6
0.6
0.4
0.4
with with with with
0.2
0
2e-08
0
surface surface surface surface
5e-09
charge, c=5e11 charge, c=1e11 charge, c=5e10 recombination
1e-08
0.2
0 1.5e-08
Time [s]
Time [s]
Fig. 2. (a) Dependence of charge collection time on the reverse bias voltage. An MIP hitting into the center of the 3D quarter pixel at 1 ns (the structure shown in Fig. A.1(c)). (b) Dependence of charge collection efficiency at 20 V on the surface recombination and surface charge concentrations of C ¼ 5 1010 cm2 , C ¼ 1 1011 cm2 and C ¼ 5 1011 cm2 .
Current response with a varied wafer thickness
Charge collection with a varied wafer thickness 1
0.8 5e11 Charge collected
Charge scaled current [arbitrary units]
0
1e10
1.5e10
900 microns 600 microns 300 microns 1 micron
2e10
0.6
0.4
1 micron 300 microns 600 microns 900 microns
0.2
0 0
5e09
1e08 Time [s]
1.5e08
0
1e08 Time [s]
2e08
Fig. 3. (a) Current response pulses with detector thicknesses of 1, 300, 600 and 900 mm. (b) Charge collection time dependence on the wafer thickness. The curves are integrals of (a).
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
detector electrodes, the charge carriers induce a current flow into the electrodes. The opposite sign electric charge is drawn from the circuit through contacts until the full depth of the interaction is reached, i.e. the current flows into the electrodes up the interaction distance. Movement of charge within the electrodes induces a serial resistance, created by the detector structure, to depend on the interaction depth. The time delay caused by the interaction depth is demonstrated in Fig. 4(a), where a MIP traverses the detector structure described in Fig. A.1(c) up to a depth of 30 mm from the top and bottom sides of the 300 mm thick detector. The MIP entering the detector from the bottom side simulates particle interactions deep in the bulk. In Fig. 4(a) the electric current densities induced by the shallow (a MIP entering the detector from top) and deep (a MIP entering the detector from bottom) interactions are depicted for different time intervals with respect the arrival time of the incident MIP. In the uppermost panels in Fig. 4(a), the MIPs have just entered the detector bulk. In the lower panels, diagonal crosssections after 1 ns of the MIP arrivals are shown. The current is seen to be drawn deeper into the bulk for the MIPs entering the detector from the bottom side. A higher serial resistance is created in this case and, therefore, a delay in CC follows. In Fig. 4(b) the CC efficiency and current response pulses depicted in Fig. 4(a) are shown. The CC time for the particles interacting deeper in the bulk is longer when compared to the CC time of the shallow interactions. The time difference between the two example cases appears to be shorter than predicted by directly applying Eq. (3). The
295
discrepancy is due to the charge clouds that, in the case of the structures simulated in Fig. 4(a), can spread towards the center of the bulk (perpendicular to the electric field) due to diffusion and repulsion of the charge carriers. The delay can be increased by changing the detector parameters (see Eq. (3)) to yield a larger RC product.
3. Conclusions The CC efficiency is shown to decrease significantly when the surface charge at the silicon–oxide interface reaches the value causing the pn-junction to break down. In addition, it is demonstrated that a high surface charge concentration increases the CC time. In a rectangular 3D pixel element, the electric charge is substantially shared only within the volume in close neighborhood of the pixel; within about 5 mm distance from the edge of the pixel element. Within the detector volume outside the edge region, most of the charge is collected within 10 ns at 20 V. The CC process is shown to depend on the interaction depth of the incident ionizing particles. For the incident MIPs, the CC time increases with the detector thickness. For the 3D detectors manufactured with a very high aspect ratios, this delay in CC time has to be accounted for. An experimental method to test the CC time delay (see Figs. 4(a) and (b)) could be based on a low energy laser and fast read-out system. Since the observed CC time delay could be used to define the interaction depth of incident radiation, interesting new applications may become available with the 3D detectors.
MIP entering 30 microns from the front and back sides of the detector at 1 ns MIP entrance from top
1
MIP entrance from bottom
0.8
Diagonal cut of (a) 1 ns after MIP entrance
Diagonal cut of (b) ns after MIP entrance
Current [A]
0.6 -4e09 0.4
-6e09
current, MIP to front
0.2
Relative charge collected
-2e09
current, MIP to back charge, MIP to back charge, MIP to front
0
2e-09
4e-09
6e-09
8e-09
0 1e-08
Time [s]
Fig. 4. Demonstration of charge collection time dependence on the interaction depth of the incident ionizing radiation. An MIP hitting into the center of the quarter pixel from the top and bottom and penetrating 30 mm. (a) Induced current densities at the entrance of the MIP (top) and 1 ns after (bottom). (b) The current response pulses and charge collection curves of (a).
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Fig. A.1. Total doping concentrations (p- and n-type) and the pnjunctions (full lines) of the simulated structures. (a) A 2D simulation of an array of neighboring pixels, (b) a 2D simulation of a quarter pixel (dotted region in (a)), (c) a 3D simulation of the quarter pixel and (d) a 3D simulation of the quarter pixel with thickness of 1 mm.
Appendix A. Description of the simulated detector structures The simulated detector pixel has a 70 mm pitch between nþ - and pþ -type, or nþ - and nþ -type, poly-silicon electrodes that penetrate through varied thicknesses of n -type silicon bulk. Each pþ electrode is surrounded by eight nþ electrodes such that the nþ electrodes form a square with the pþ electrode in the middle. Resistivity of the n -type bulk is about 4500 O cm ð1 1012 cm3 Þ and lifetimes of the charge carriers are 1 105 s. The maximum concentrations in the error function profiles of the p- and n-type
electrodes are C max ¼ 1 1018 cm3 . The diameter of the electrodes is about 20 mm. The dimensions of the simulated detector pixels and the physical models used in the simulations are introduced in closer detail in Ref. [5]. Four different structures were simulated to fully characterize the performance of the detector structure and to study the current response delay phenomena. These structures are shown in Fig. A.1. Each of these structures is connected into an external circuit with a 100 O resistor— the pþ electrode is connected to the ground via the 100 O resistor, and the nþ electrodes are connected directly to the ground in parallel. Uniform surface charge and surface recombination model at the silicon/oxide interface of structure in Fig. A.1(c) were used only when studying the charge collection efficiency. In other simulations no interface charge was present. Charge collection is studied by introducing a MIP that penetrates through the silicon wafer and creates 80 electron–hole-pairs/mm uniformly along its path in the silicon with a lateral spread of 5 mm defined by a complementary error function. A denser grid was manually inserted around the penetration path of the MIP. References [1] S. Parker, C. Kenney, J. Segal, IEEE Trans. Nucl. Sci. A NS-395 (1997) 328. [2] Integrated Systems Engineering (ISE), Release 6.1, ISE TCAD Manual, 2000. [3] X. Llopart, M. Campbell, R. Dinapoli, D.S. Segundo, E. Pernigotti, IEEE Trans. Nucl. Sci. NS-49 (5) (2002) 2279. [4] V.A. Wright, D. Davidson, V. O’Shea, L. Donnohue, L. Lea, K. Robb, K. Smith, S. Nenonen, H. Sipila¨, 3d medipix—a new generation of X-ray detectors, in: Nuclear Science Symposium (NSS) Conference Record, vol. 2, Rome, Italy, 2004, pp. 1336–1343. [5] J. Kalliopuska, S. Era¨nen, R. Orava, 3d simulations of 3d silicon radiation detector structures, Nucl. Instrum. and Meth. A 568 (2006) 27. [6] R.S. Muller, T.I. Kamins, Device electronics for integrated circuits, second ed., Wiley, New York, 1977, pp. 110–113.
Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296 www.elsevier.com/locate/nima
Charge collection characterization of a 3D silicon radiation detector by using 3D simulations Juha Kalliopuskaa,b,1,, Simo Era¨nenb, Risto Oravac,d a
Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland VTT Technical Research Center of Finland, Micro- and Nanoelectronics, Tietotie 3, P.O. Box 1000, FI-02044 VTT, Finland c High Energy Physics Division, University of Helsinki, P.O. Box 64, FI-00014, Finland d CERN, Helsinki Institute of Physics, CERN/PH, CH-1211 Geneva, Switzerland
b
Available online 29 November 2006
Abstract In 3D detectors, the electrodes are processed within the bulk of the sensor material. Therefore, the signal charge is collected independently of the wafer thickness and the collection process is faster due to shorter distances between the charge collection electrodes as compared to a planar detector structure. In this paper, 3D simulations are used to assess the performance of a 3D detector structure in terms of charge sharing, efficiency and speed of charge collection, surface charge, location of the primary interaction and the bias voltage. The measured current pulse is proposed to be delayed due to the resistance–capacitance (RC) product induced by the variation of the serial resistance of the pixel electrode depending on the depth of the primary interaction. Extensive simulations are carried out to characterize the 3D detector structures and to verify the proposed explanation for the delay of the current pulse. A method for testing the hypothesis experimentally is suggested. r 2006 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 29.40.Gx; 28.52.Av Keywords: Silicon radiation detectors; 3D detectors; Charge collection characteristics; Charge sharing
1. Introduction In this paper, characterization of the performance of a 3D detector pixel element [1] is reported in terms of its charge collection (CC) efficiency and time. The CC time dependence on the surface charge concentration at the silicon–oxide interface, bias voltage, charge sharing and the location of the primary interaction within the detector bulk are studied by using 3D simulations. The delay in the CC process, proposed to be caused by an induced resistance– capacitance (RC) product, is analyzed by simulating the Corresponding author. Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland. Tel.: +358 414 313 212; fax: +359 9 456 7012. E-mail addresses: [email protected].fi (J. Kalliopuska), simo.eranen@vtt.fi (S. Era¨nen), [email protected] (R. Orava). 1 The Ph.D. studies of the corresponding author has been financed by Helsingin Sanomain 100-vuotissa¨a¨tio¨.
0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.10.370
detector structures with varying thicknesses. A cylindrical capacitor structure, for which the RC product can be calculated in a straightforward manner, is used as a convenient reference. A delay in the simulated current pulse is shown to occur depending on the depth of the primary interaction of an incident particle in the detector bulk. A way to measure the CC pulse delay is suggested. The 2D and 3D simulations were carried out by using the ICE-TCAD simulation software for semiconductor devices [2]. A rectangular 3D pixel element was defined as a reference detector structure for the characterization. The detector structures and procedures used in the simulations are further described in Appendix A. 2. Results Dependence of the CC time and charge sharing on the impact location of an incident minimum ionizing particle
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
(MIP) traversing the detector surface was simulated in 2D at 20 V bias voltage by using the pixel structure described in Fig. A.1(a). The incident MIPs hit the detector surface in 10 different locations as shown in Fig. A.1(a) by a dashed line extending from inside to the outer edge of the pixel element. The CC is considered complete when 90% of the over-all signal charge is collected, i.e. possible tail of the current response distribution is ignored. As depicted in Fig. 1, the signal charge is collected in less than 10 ns at 20 V within the main pixel volume. The time interval between recording the peak of the charge pulse and in collecting the over-all charge signal is due to spreading of charge carriers during the CC process. The charge is shared within a few microns from the pixel edge, only, and the signal charge—induced by an incident MIP—is collected by the pixel electrode with an efficiency exceeding 95%. Similar results are obtained for somewhat different 3D detector structure, compatible with the Medipix2 read-out chip [3,4]. The detector pixel is fully depleted at about 20 V bias voltage. It is, however, of advantage to use a higher reverse bias voltage in order to increase the electric field for the benefit of a faster CC. Fig. 2(a) depicts the dependence of CC on the applied bias voltage. The maximum bias voltage is defined by the saturation limit of the velocity of the charge carriers; doubling the reverse bias from 40 to 80 V does not significantly speed up CC. At 10 V, the detector is not yet fully depleted, and the CC process is slowed down appreciably. The 3D CC efficiencies are depicted in Fig. 2(b) for a MIP entering the center of a quarter of a 3D detector pixel element (see Fig. A.1(c)). The bias voltage is set to 20 V; Charge sharing and charge collection time dependence on interaction location 100 20
Peak time Charge collection time Charge collected correctly Charge shared
10
60
40
Relative charge [%]
Time [ns]
80
20
0
surface recombination and different uniform surface charge concentrations at the silicon–oxide interface are included in the simulation. The charge collected by the pþ electrode was calculated by integrating over the current response. With the surface charge of 6 1011 cm2 , a high leakage current in excess of 10 mA was observed. This is due to the breakdown of the pn-junction and no results for surface charge above 5 1011 cm2 could be considered in the analysis. Introduction of a surface charge creates an undepleted accumulation layer below the oxide–silicon interface and modifies the electric field [5]. The charge carriers created within the accumulation layer have to diffuse outside this volume in order to get collected by the electric field. Due to this, the CC time increases slightly as seen in Fig. 2(b), where the CC efficiency curves are seen to saturate later than the ones depicting the situation without adding the surface charge. The CC efficiency falls with an increased surface charge concentration due to the holes recombining with electrons in the accumulation region. 2.1. Dependence of charge collection time on RC constant The CC time of a 3D detector pixel is observed to depend on the RC product of the pixel and on the depth of the primary interaction by the incident particle. These correlations have been analyzed by simulating the CC time as a function of detector thickness and by comparing the simulation results with theoretical calculations for a simplified reference structure and by demonstrating the dependence of the CC time delay on the depth of the primary interaction by an incident MIP. The RC product of a fully depleted 3D detector pixel can be approximated by a cylindrical capacitor in combination with a resistance represented by a pþ electrode. For this combination, the RC product can be calculated in a straightforward way by using Eqs. (1) and (2), where denotes the electrical permittivity (for silicon Si ¼ 11:8 0 ), h gives the wafer thickness, r1 and r2 the inner and outer radii of the cylinder, rmin the minimum resistance of the doping profile of the pþ electrode, and rmin the radius of the region with the minimum resistance. The approximation used for the resistance of the pþ electrode is valid since most of the current concentrates into the region with the minimal resistance [6]. C cyl ¼ 2p
h lnðr2 =r1 Þ
(1)
Rel ¼ rmin
h . pr2min
(2)
0 0
20 40 Distance from the edge [um]
60
Fig. 1. Dependence of CC time and charge sharing on the interaction location of the incident radiation. The X -axis shows the entrance locations of the MIP from the pixel edge in microns (mm); the peak time of the current response signal and the CC time are given on the left Y -axis and the charge collected correctly by the pixel and the charge shared by the neighboring pixels are given on the right Y -axis in percentages ð%Þ.
293
By combining Eqs. (1) and (2), an approximation for the RC product of the 3D detector structure is obtained as t ¼ Rel C cyl ¼ 2rmin
h2 . r2min lnðr2 =r1 Þ
(3)
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
294
From Eq. (3) it is explicitly seen that the RC product, depends on the detector thickness, h, as: th2 . For comparing the analytical calculation (Eq. (3)) with the simulation results, detector thickness is varied while the distance of CC by the pixel electrode remains the same. By using Eq. (3) with the parameter values of Si , rmin ¼ 2 mm, rmin ¼ 0:04 O cm ð1018 cm3 Þ, r1 ¼ 10 mm and r2 ¼ 60 mm, the following approximate CC time delays are obtained: t1 10 fs, t300 1 ns, t600 4 ns and t900 10 ns. The external resistor of R ¼ 100 O dominates the over-all resistance of the 1 mm thick detector structure. Figs. 3(a) and (b) depict the current response and integrated charge of the detector structure as a function of the wafer
a
thickness. The distributions for a 1 mm thick detector are included for reference with no delay time in the CC process; for these the time constant is t1 10 fs. Both the peak current and tail increase as a function of the wafer thickness (Fig. 3(a)). The delay in CC time depends on the wafer thickness (th2 ) and causes the tails of the simulated distributions. 2.2. Charge collection time dependence on the interaction depth of the incident radiation An ionizing radiation creates charge carriers (electron– hole pairs) in the detector bulk. When moving towards the
b
Charge collection of the MIP at different reverse bias voltages
Charge collection efficiency and dependence on surface effects at 20 V. 1
50
10 20 40 80
0 0
5e-09
1e-08
1.5e-08
V V V V
Charge collection efficiency
Percentage of charge collected
100
1
0.8
0.8
0.6
0.6
0.4
0.4
with with with with
0.2
0
2e-08
0
surface surface surface surface
5e-09
charge, c=5e11 charge, c=1e11 charge, c=5e10 recombination
1e-08
0.2
0 1.5e-08
Time [s]
Time [s]
Fig. 2. (a) Dependence of charge collection time on the reverse bias voltage. An MIP hitting into the center of the 3D quarter pixel at 1 ns (the structure shown in Fig. A.1(c)). (b) Dependence of charge collection efficiency at 20 V on the surface recombination and surface charge concentrations of C ¼ 5 1010 cm2 , C ¼ 1 1011 cm2 and C ¼ 5 1011 cm2 .
Current response with a varied wafer thickness
Charge collection with a varied wafer thickness 1
0.8 5e11 Charge collected
Charge scaled current [arbitrary units]
0
1e10
1.5e10
900 microns 600 microns 300 microns 1 micron
2e10
0.6
0.4
1 micron 300 microns 600 microns 900 microns
0.2
0 0
5e09
1e08 Time [s]
1.5e08
0
1e08 Time [s]
2e08
Fig. 3. (a) Current response pulses with detector thicknesses of 1, 300, 600 and 900 mm. (b) Charge collection time dependence on the wafer thickness. The curves are integrals of (a).
ARTICLE IN PRESS J. Kalliopuska et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 292–296
detector electrodes, the charge carriers induce a current flow into the electrodes. The opposite sign electric charge is drawn from the circuit through contacts until the full depth of the interaction is reached, i.e. the current flows into the electrodes up the interaction distance. Movement of charge within the electrodes induces a serial resistance, created by the detector structure, to depend on the interaction depth. The time delay caused by the interaction depth is demonstrated in Fig. 4(a), where a MIP traverses the detector structure described in Fig. A.1(c) up to a depth of 30 mm from the top and bottom sides of the 300 mm thick detector. The MIP entering the detector from the bottom side simulates particle interactions deep in the bulk. In Fig. 4(a) the electric current densities induced by the shallow (a MIP entering the detector from top) and deep (a MIP entering the detector from bottom) interactions are depicted for different time intervals with respect the arrival time of the incident MIP. In the uppermost panels in Fig. 4(a), the MIPs have just entered the detector bulk. In the lower panels, diagonal crosssections after 1 ns of the MIP arrivals are shown. The current is seen to be drawn deeper into the bulk for the MIPs entering the detector from the bottom side. A higher serial resistance is created in this case and, therefore, a delay in CC follows. In Fig. 4(b) the CC efficiency and current response pulses depicted in Fig. 4(a) are shown. The CC time for the particles interacting deeper in the bulk is longer when compared to the CC time of the shallow interactions. The time difference between the two example cases appears to be shorter than predicted by directly applying Eq. (3). The
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discrepancy is due to the charge clouds that, in the case of the structures simulated in Fig. 4(a), can spread towards the center of the bulk (perpendicular to the electric field) due to diffusion and repulsion of the charge carriers. The delay can be increased by changing the detector parameters (see Eq. (3)) to yield a larger RC product.
3. Conclusions The CC efficiency is shown to decrease significantly when the surface charge at the silicon–oxide interface reaches the value causing the pn-junction to break down. In addition, it is demonstrated that a high surface charge concentration increases the CC time. In a rectangular 3D pixel element, the electric charge is substantially shared only within the volume in close neighborhood of the pixel; within about 5 mm distance from the edge of the pixel element. Within the detector volume outside the edge region, most of the charge is collected within 10 ns at 20 V. The CC process is shown to depend on the interaction depth of the incident ionizing particles. For the incident MIPs, the CC time increases with the detector thickness. For the 3D detectors manufactured with a very high aspect ratios, this delay in CC time has to be accounted for. An experimental method to test the CC time delay (see Figs. 4(a) and (b)) could be based on a low energy laser and fast read-out system. Since the observed CC time delay could be used to define the interaction depth of incident radiation, interesting new applications may become available with the 3D detectors.
MIP entering 30 microns from the front and back sides of the detector at 1 ns MIP entrance from top
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MIP entrance from bottom
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Diagonal cut of (b) ns after MIP entrance
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0.6 -4e09 0.4
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Fig. 4. Demonstration of charge collection time dependence on the interaction depth of the incident ionizing radiation. An MIP hitting into the center of the quarter pixel from the top and bottom and penetrating 30 mm. (a) Induced current densities at the entrance of the MIP (top) and 1 ns after (bottom). (b) The current response pulses and charge collection curves of (a).
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Fig. A.1. Total doping concentrations (p- and n-type) and the pnjunctions (full lines) of the simulated structures. (a) A 2D simulation of an array of neighboring pixels, (b) a 2D simulation of a quarter pixel (dotted region in (a)), (c) a 3D simulation of the quarter pixel and (d) a 3D simulation of the quarter pixel with thickness of 1 mm.
Appendix A. Description of the simulated detector structures The simulated detector pixel has a 70 mm pitch between nþ - and pþ -type, or nþ - and nþ -type, poly-silicon electrodes that penetrate through varied thicknesses of n -type silicon bulk. Each pþ electrode is surrounded by eight nþ electrodes such that the nþ electrodes form a square with the pþ electrode in the middle. Resistivity of the n -type bulk is about 4500 O cm ð1 1012 cm3 Þ and lifetimes of the charge carriers are 1 105 s. The maximum concentrations in the error function profiles of the p- and n-type
electrodes are C max ¼ 1 1018 cm3 . The diameter of the electrodes is about 20 mm. The dimensions of the simulated detector pixels and the physical models used in the simulations are introduced in closer detail in Ref. [5]. Four different structures were simulated to fully characterize the performance of the detector structure and to study the current response delay phenomena. These structures are shown in Fig. A.1. Each of these structures is connected into an external circuit with a 100 O resistor— the pþ electrode is connected to the ground via the 100 O resistor, and the nþ electrodes are connected directly to the ground in parallel. Uniform surface charge and surface recombination model at the silicon/oxide interface of structure in Fig. A.1(c) were used only when studying the charge collection efficiency. In other simulations no interface charge was present. Charge collection is studied by introducing a MIP that penetrates through the silicon wafer and creates 80 electron–hole-pairs/mm uniformly along its path in the silicon with a lateral spread of 5 mm defined by a complementary error function. A denser grid was manually inserted around the penetration path of the MIP. References [1] S. Parker, C. Kenney, J. Segal, IEEE Trans. Nucl. Sci. A NS-395 (1997) 328. [2] Integrated Systems Engineering (ISE), Release 6.1, ISE TCAD Manual, 2000. [3] X. Llopart, M. Campbell, R. Dinapoli, D.S. Segundo, E. Pernigotti, IEEE Trans. Nucl. Sci. NS-49 (5) (2002) 2279. [4] V.A. Wright, D. Davidson, V. O’Shea, L. Donnohue, L. Lea, K. Robb, K. Smith, S. Nenonen, H. Sipila¨, 3d medipix—a new generation of X-ray detectors, in: Nuclear Science Symposium (NSS) Conference Record, vol. 2, Rome, Italy, 2004, pp. 1336–1343. [5] J. Kalliopuska, S. Era¨nen, R. Orava, 3d simulations of 3d silicon radiation detector structures, Nucl. Instrum. and Meth. A 568 (2006) 27. [6] R.S. Muller, T.I. Kamins, Device electronics for integrated circuits, second ed., Wiley, New York, 1977, pp. 110–113.