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Charge collection in silicon detectors for strongly ionizing particles
NUCLEAR INSTRUMENTS AND METHODS II 3 (I973) 317-324;
© NORTH-HOLLAND PUBLISHING CO.
CHARGE COLLECTION IN SILICON DETECTORS FOR STRONGLY IONIZING PARTICLES W. SEIBT, K. E. SUNDSTROM and P. A. TOVE
Electronics Dept., Institute of Technology, University of Uppsala, Uppsala, Sweden
Received 13 April 1973 Experimental results for the plasma time in a semiconductor detector are given for alpha-particles, 160 ions and fission fragments. The plasma time tp is defined as the time needed to bring all carriers under the influence of the applied field. A simple theoretical model based on diffusion and space charge limited current erosion of the track is described and results in the expression tp = 1.32 x 10 10 (mE)~/F for a silicon detector where F is the electric field in the detector at the position of the
track, nl is the linear carrier density in the track and E is the particle energy. This gives good agreement with the experimental data for tracks longer than ---15/~m (alphas and 160 ions of energies 5.3-8.8 MeV and 22-42 MeV, respectively). The shorter tracks of fission fragments (90 MeV from e52cf) give shorter tp values then predicted by the formula above. This is explained by transfer from cylindrical to spherical erosion geometry.
1. Introduction
experimental investigations. The first formula obtained for the plasma erosion time was based on simple diffusion of the track and gave values orders of magnitude too long to explain the experimental resultsS). It was first the concept of space charge limited erosion currents 6) that b r o u g h t the calculated values into the region o f measured pulse rise times. It must, however, be pointed out that the measured plasma times and the calculated plasma erosion times were not really comparable, due to the difference in the erosion geometry. Also the experimental plasma times were obtained from the assumption that the measured rise times were quadratic superpositions of several time constants, including the rise time of the measuring amplifier and the RC-time from the undepleted detector material. Into this quadratic superposition was then also included the charge collection time, which in most cases 3'4'7) was further split into a calculated transport time and a plasma time. This means that the time delay introduced by the plasma effect is treated as equivalent to an integrating time constant acting on the voltage pulse, that should be obtained from the detector without plasma effect. It is obvious that the plasma time obtained in this way is difficult to relate to any physical process in the detector.
F o r heavily ionizing particles incident on a nuclear radiation semiconductor detector, the charge transport is delayed because of the " p l a s m a effect" which causes the charge collection time to be longer than that calculated from simple theory1). This has not only influence on the timing properties o f the detectors, but may also lead to inefficient charge collection, due to carrier trapping and recombination in the dense plasma or at the point where the plasma is in contact with the detector surface2'a9). An understanding of the above p h e n o m e n o n is thus o f central importance whenever semiconductor detectors are employed in heavy particle spectrometry. A l t h o u g h these detectors have been used for this purpose for over ten years, a general theoretical treatment of the plasma effect is almost made impossible by the three dimensional nature of the charge collection geometry, which is extremely difficult to treat mathematically without simplifying assumptions. Facing this problem, the earliest investigators o f the plasma effect confined themselves to the study o f voltage pulse rise times as a function of the electric field strength inside the detector depletion region at the position o f the ionized particle track. F r o m these studies empirical expressions for the plasma time were obtained3.4). Along with these studies attempts were made to calculate the time needed to erode the ionized track to the point where all charges are under the influence of the electric field. To make the problem tractable a geometry was chosen where the track is perpendicular to the electric field, which was not the case in the
317
A linear superposition of the plasma erosion time and the simple transit time seems more reasonable from a physical point o f view. A n uncertainty in this a p p r o a c h lies in the calculation o f the transit time since the last carriers to be collected move in an electric field which is modified by the carriers in drift in front o f them. A method to determine plasma erosion times according to the above definition, from
318
w. SEIBT et al.
measured current pulse shapes in totally depleted* detectors, was suggested by the authorsa). From a phenomenological discussion it was proposed that the plasma erosion time could be obtained from the decay of the current pulses, when the detectors were irradiated from the low field back side. Results thus obtained showed a relatively weak dependence of plasma time on the field. From a theoretical point of view the most accurate approach is of course to calculate the total charge collection times, although these times are less suited for a discussion of recombination effects in the plasma column. A simple theoretical model, yielding this time for particles incident parallel to the field, is discussed in ref. 10. The difficulty is to account for the transverse diffusion of the particle track. Several possibilities are discussed, but no clear choice could be made to fit the experimental results. Good agreement between experimental results and theoretical predictions has so far only been obtained through the use of computer methods H) and for comparatively light particles. In this case the total charge collection time in overdepleted detectors was computed from the general equations for charge transport in semiconductors, assuming one-dimensional geometry but also taking into account transverse diffusion. However, in spite of the simplified geometry, no closed form solution for the collection time could be found in this case. From the above discussion it seems that an expression for the plasma erosion time defined in 6) and for the case of tracks parallel to the field as a function of electric field strength and particle energy, could be very useful for the prediction of charge collection times and may also contribute to an explanation of that part of the pulse height defect 12'~3) which cannot be explained by nuclear collisions. For this case we will derive a theoretical expression for the plasma erosion time, defined as the time needed to bring all the carriers under the influence of the electric field. After a description of the experimental methods, measured charge collection times for e-particles, ~60 ions and fission fragments will be discussed in the light of the theoretical predictions.
2. Theoreticalmodel
normally to the detector surface. The charge in this track is polarized in the way previously described 6) so that the electric field becomes zero at the border between the polarized sheet and the rest of the plasma. The carriers in the polarized sheet start to drift in the electric field towards the other electrode and at the same time the plasma track is broadened by transverse diffusion. The diffusion which is visualized by the distribution of nc (t) in fig. 1 produces a decrease in the charge density in the track and this, together with the space charge limited extraction of carriers, causes the field to penetrate deeper and deeper into the plasma (see x ( t ) in fig. 1). We will now direct our attention to the central part of the track and study how the border x ( t ) between the polarized sheet and the rest of the plasma changes its position with time. For this purpose we consider a plasma column with a top surface A so small, that we may assume that the charge density in this column as a function of time is described by the solution to the diffusion equation in cylindrical coordinates for r = 0, yielding no(t) = nl/4ZrD . t.
(1)
Here n~(t) is the concentration of charge carriers in the middle of the column as a function of time, n~ is the total initial number of carriers per unit length in the track and D, is the ambipolar diffusion constant. At time t a small segment dx of the column (top area A) will thus contain an amount of charge equal to dQ
= qAnldX/47rD
(2)
a t.
It will now be assumed that charges are extracted from the top of the column by a space charge limited
J
to
M
t2
~
//
yx,,o, /t/xIt,I
The physical situation to be treated is a dense plasma track created by a heavily ionizing particle incident * Using overdepleted detectors eliminates the uncertainty introduced by the RC-time constant from undepleted detector material cf. ref. 9,7.
--'/-;'J.'¢\"'>'-
=
,o iol istonce from
centre of track
Fig. 1. Diffusion and current erosion o f plasma track.
319
C H A R G E C O L L E C T I O N IN S I L I C O N DETECTORS
current j ( t ) per unit area. Thus dQ/dt = j(t)A.
(3)
From (2) and (3) we get ( qnl/4 nO, t) dx = j ( t) dt, or, after integrating along the track and over the time needed to expose all carriers to the field: Qo =
qnldx = 4riD a o
tj(t)dt,
(4)
Jo
where Q0 is the total amount of charge created by the incident particle and tp is the plasma time. It remains now to find an appropriate expression for j(t). An attempt to use an extraction current of the order of the one-dimensional space charge limited transient current obtained in ~4), gives far too long values for the plasma time, so obviously the field acting on the cylindrical surface of the plasma column cannot be neglected. That this effect is of great importance has previously been recognized 6'1°) and especially at the end of the plasma column it should not be neglected,
pulse amp HP 462A
while it is smaller towards the detector surface. Due to the shielding effect of the carriers extracted from the end, the influence of the field on other parts of the plasma column is expected to be less important. We will assume that only those carriers at the end which are under the influence of the longitudinal field are also influenced by a transverse field component. The longitudinal field penetrates into the plasma a distance 6) 6 = eeoF/qnc. Although the carriers in this sheet are eventually spread along the whole detector thickness, a very high charge concentration is only maintained within a small distance from the plasmalS). If we assume that the transverse outflow of carriers is given by the space charge limited current value in cylindrical geometry6'16): .lscLc =-2~llee'o F2, the total current is given by iT
~
=
21rll(s%)ZF3/qnc.
For nc we use the value given by eq. (1) since we assume that ambipolar diffusion continues undisturbed except at the end of the plasma column. The current ix has to be supplied by the core of the plasma column
cathode follower
bias Ca)
2rClleeoF25
=
Tektronix sampling ocs type 661 external trig analog output
tt
X-Y recorder
signal input
pulse amp HP 35002A bias
+
Tektronix sampling osc type 661
averaging computer signat trig
external I trig I
(b)
detector
cathode follower
,,
analog J
"
signal input
w
~gate trig
signalinputoutput I
I
Fig. 2a. Experimental arrangement for measuring current pulse using a sampling oscilloscope preceded by a fast preamplifier. Fig. 2b. Set-up for current pulse measurement using a combination of sampling and averaging techniques.
320
w.
S E I B T et al.
which we assumed to have a top area A when deriving eq. (4). Thus we have to insert j(t) = iT/A into eq. (4) to get an expression for the plasma time obtaining
1 ( 3QoqnlA tv
=
~
F\32~DZj
.
(5a)
It is now convenient to introduce the particle energy E instead of the total created charge Qo. Since 3.5 eV are required to form an electron hole pair in silicon we finally get the expression tp
=
1.32
x
10-~0
(n]E)~/F,
(5b)
where we also have inserted # = 1400 cm2/V s obtained from 18) and Da = 16 cmZ/s. The area A was given the value 4.19 x 10 -8 cm 2 which will be discussed later in connection with the experimental results. That good agreement with measured plasma times could be obtained from eq. (5), if A is given a value of the order of the initial top area (radius about 1/Q of the plasma column, is a support for the model.
3. Measuring equipment The current pulses obtained when surface barrier detectors were exposed to a-particles, ]60 ions of different energies and fission fragments, were recorded. Olzservation of the current pulse shape gives more direct information about the carrier transport than does the voltage pulse (i.e., the current pulse integrated on a capacitor). The total collected charge, which is
I
directly obtained from the voltage pulse, can also be easily obtained from the current pulse by planimetric integration. The following electronic measuring set ups were used. In fig. 2a is shown the first set up, which had a rise time of 3.2 ns. The equivalent noise signal was of the order of 5 to 10 mV, referred to the input of the sampling oscilloscope. In fig. 2b is shown a second set up, the principle of which involved a combination of sampling and averaging techniques, in order to get a good signal-to-noise ratio (also independence of variability in signal amplitude)IT). In a third set up the same arrangement as in fig. 2a was used but the HP462A amplifier was replaced by two cascaded amplifiers of the later model HP35002A. This decreased the system rise time to l ns. The noise was also appreciably reduced.
4. Experimental results The physical properties of the silicon surface barrier detectors used in the experiments are given in table I. Measurements were done for a-particles, 160 ions and fission fragments. Alphas were obtained from ThB and Po sources, 160 ions from a van-de-Graaff generator and fission fragments from a 252Cf source. All particles had well defined energies, except the zs2Cf fission fragments for which an average energy of 90 MeV was assumed. This corresponds to the lowmass, high-energy peak in the energy spectrum. The triggering level was set to exclude lower energies. Examples of the current pulse shapes obtained, for
vo= ,56v W= 5 6 0 ~ m
l~,lAs t~
ns 33,) 2~
h
!T00
z6.s ~,6 2Lg ~2
,79
,,,
'OLT
/~2s
,~
"~
,
l~,~
!2"~
~s.s
ns
~2.~
to
n~
3,
3.0
-.
~
;
11"
Fig. 3. C u r r e n t pulse shapes obtained with detector El for front-injected 21°po c~-particles, at different bias voltages. Original s h o w n in one instance, other curves redrawn s m o o t h e d , for clarity.
CHARGE COLLECTION
DIODE
/ 1 1 i
1 I
I
///A
fltfl/i
l/filt'i/
F 5
o.
t ///,A~k 1tl//~ t 1t 1 1 ~ . Illfl, e//-
'
I " I 37~ I s~l ~9 I ~ i ~,~
19~1 !?.~. 8< I ' ° 1 ~7 i ~
\ \ \\\~t,.
20
30
o.
~ -~ ,3., 2,8 ,o~ ~o~
° ::
\\-"%'q~
t0
321
IN S I L I C O N DETECTORS
~7.8 ,so
~sec
Fig. 4. Current pulse shapes obtained with detector F5, for front-injected 42 MeV 160 ions, at different bias voltages.
TABLE 1 Detector
pa k-Q cm
Slice thickness w mm
VoW
F5 F7 F3 E4 E1
9.0 8.8 7.0 14.4 21.0
0.29 0.41 0.27 0.43 0.56
38 75 36 42 56
a
Calculated from the measured value of capacitance (at depletion voltage V0) and the slice thickness. b Determined from the plot of measured capacitance versus voltage. For V > V0 the capacitance becomes constant.
5.3 MeV alphas from 21°Po and 42 MeV 160 ions, front injection and their variation with applied bias are shown in figs. 3 and 4. In order to make a comparison with the theoretical predictions possible, the experimental data were treated in the following way. From the total length of the current pulse, which can be obtained with much better accuracy than when voltage (integrated) pulses are recorded, we subtract the calculated transit time tt=kw2//2 V to obtain the "plasma time". In the formula for tt the variation of the mobility /2 with field was taken into account, using information from ref. 18, k is a correction factor. This assumption of a linear superposition of plasma and transit time is in agreement with our previous suggestions (cf. e.g., fig. 4 in ref. 6). Actually the
transit time under space charge free conditions in a totally depleted detector is t t' = (t~z/2/2V0) In (V+ Vo)/ ( V - V o ) . Further modification of the transit time is caused by field perturbations connected with space charge limited (SCL) carrier transport. The presence of space charge lengthens the transport time of those carriers travelling behind the space charge (the last carriers leaving the plasma). For the case of an ideal plane parallel geometry the transport time has been shown to be lengthened a factor 1.3 to 1.5, depending on the ratio of applied voltage and the depletion voltageS4). In our case this effect should be smaller, because of sidewise spreading of the moving charge in front of the last carriers. These effects are taken into account by choosing k = 1.1 in the expression for t t. Experimental total collection times obtained as described above for alphas, and ~60 ions are shown in figs. 5 and 6. The times corresponding to the 10-90% pulse rise times of the corresponding voltage pulses were also obtained by planimetric integration of the current pulses. In all cases this time was very nearly half the total collection time, so previously published 1°'11) 10-90% voltage pulse rise times can be compared with half the values of tm presented in this work. Since the plasma times for a certain field strength predicted by eq. (5) are independent of the particular detector and only depend on the type of particle and its energy, plasma times obtained with different detectors but for a certain type of particles should follow the same theoretical line. Experimental results for
322
w. SEIBT et al.
different detectors are therefore given in the same plots. Results for 5.3 and 8.8 MeV alphas are given in fig. 7 and for 42 MeV 160 ions in fig. 8. The separation between the theoretical lines for 5.3 and 8.8 MeV in ltmo 3O _
ns,
A Detector E1
& O 0
a o
Detector EZ,
o
Detector F7
o
O a 0
20
0
a° L~
o
A O OA A A O A O
10
I
I
I
I
I
I .~.F
3
4
5
6
7
8
fig. 7 is small compared with the experimental error (cf. eq. 5). There is, however, a clear tendency among the experimental points to give shorter plasma times for the particles of the lower energy. In order to compare these results with the theoretical predictions, a value had to be given to the area A in eq. (5). If this value is chosen to be A =4.19 x 10 -8 cm 2 (corresponding to a radius of ~ 1 . 1 5 x 10 - 4 c m ) good agreement is obtained with the experimental results, both for 5.3 and 8.8 MeV alphas and 42 MeV 160 ions. These particle types have penetration depths into silicon of 55 #m for 8.8 MeV and ~-25 p m for both 5.3 MeV alphas and 42 MeV 160 ions. In these cases a track which is much longer than the width is ns
kV/cm
,tp o •
3O
Fig. 5. Experimental total collection times in nanoseconds (full length of current pulse) for 5.3 MeV (detectors E1 and E4) and 8.8 MeV (detector F7) c~-particles. F is the calculated field strength at the front contact of the detector.
o
Detector F3 Detector F5 Detector F7
20 ns
tm
4O
•=
o • o
o o
mid
30
Detector F3 Detector F5 Detector F7
•
°o
10
zllmm II o
0
I
I
I
I
t
t
3
4
5
6
?
8
...._F
kV/cm
OO
•
Fig. 8. Plasma times derived f r o m fig. 6.
r 3
20~
I 4
I 5
I Ol 6 7
I 8
F kV/cm ns
Fig. 6. Experimental total collection times for 42 MeV 160 ions.
t
Detector
El
a tm
70
•
tp
6O
a
ns ~tp 10 9
Detector
E1
¢, Detector o Detector
E4 F7
50
o "
40
¢, o
7 30
5 a
a
A
~
A
A
3
tm
o
tp
E4
/,,
~' •
o
4
o zx •
6
2
Detector
O
20
~
a a ~ ao a
I 3
I 4
I 5
I 6
I 7
I 8
vF kV/cm
Fig. 7. Plasma times derived from fig. 5 as the difference between tm and the calculated transit time (see text). The line is the calculated plasma time according to eq. (5).
10 2
i 3
i 4
t 5
i 6
n 7
n 8
~ 9
~'F kV/cm
Fig. 9. Experimental total collection times and corresponding experimental plasma times for light fission fragments from 252Cf.
CHARGE
ns
Detector
O
O 3O
COLLECTION
F7
trn
® tp
Oo
% Oo
20
*%% 10
9
,4
5
6
I,
I
I ~
7
8
9
kV/cm
Fig. 10. Experimental total collection times and corresponding experimental plasma times for 22 MeV 160 ions. Detector F7. Also included is the theoretical line calculated from eq. (5). produced. For fission fragments, on the other hand, this is not the case, since the penetration depth is only a few microns. It is therefore difficult to compare the results for fission fragments with the theory which is based on a cylindrical geometry. For fission fragments a spherical geometry is probably a better choice6). This would lead to faster charge collection than that predicted by the theory in this paper. Experimentally this is also what is observed in fig. 9, where the plasma times for the fission fragments are only a factor 1.25 higher than for the 42 MeV 160 ions in fig. 8. Applying eq. (5) to the fission fragments would lead to a much higher factor, due to the high linear density n~ in combination with the higher energy of the fission fragments. An intermediate example are the data obtained with 22 MeV 160 ions in fig. 10. Here the points for the measured plasma times fall slightly below the theoretical line calculated from eq. (5), indicating that the penetration depth of the particles has reached the critical lower value, where the assumption of a cylindrical erosion geometry becomes invalid.
IN
SILICON
DETECTORS
323
not possible to design a theoretical model covering all penetration depths. The model is therefore based on a cylindrical geometry and describes the experimental data well for penetration depths exceeding "-~15 pm. A spherical erosion geometry is more effective and so the experimental data for fission fragments are significantly shorter than what the cylindrical model predicts. Nevertheless, all experimental data follow the 1IFdependence in eq. (5) obtained for cylindrical geometry and the experimental plasma times are independent of the particular detector as predicted by eq. (5). An important result is that, according to this equation, the plasma time is not only dependent on the particle energy but, through the linear charge density nl, also on the type of particles. This is experimentally confirmed by the data for 5.3-8.8 MeV alphas and 42 MeV ~60 ions, where the penetration depths should be long enough for eq. (5) to be valid. The evaluation of the plasma time in this work follows the previous suggestion by the authors 6'8) to consider the total charge collection time as a linear superposition of the plasma time and a simple space charge free transit time. This way of evaluation has the advantage, that the plasma time obtained can be related to a physical process in the detector and thus can be used to discuss e.g., trapping effects. Experimentally, however, the method is somewhat more difficult than methods involving the measurement of the voltage pulse from the detector, which probably is a reason that it has not been used previously. Some of the measurements were done in the Tandem Van-de-Graaff accelerators of the Danish Atomic Energy Commission at Ris6 and at the University of Utrecht. We owe our sincere thanks to Dr B. Elbek and Professor A. M. H o o g e n b o o m of these laboratories as well as to their coworkers for the research facilities put at our disposal. The travels were made possible by grants from " K n u t och Alice Wallenbergs Stiftelse". Some stimulating discussions with Professor J. W. Mayer are also greatfully acknowledged.
5. Conclusion and discussion
The time needed to erode the plasma, created by heavily ionizing particles, to the point where all charges are under the influence of the electric field, was studied experimentally for different particles and compared with a simple theoretical model. The particles were 5.3 and 8.8 MeV alphas, 22 and 42 MeV 160 ions and 90 MeV fission fragments, thus covering a large range of energies and particle penetration depths. Since the erosion geometry is changed from cylindrical to spherical when the particle range decreases, it was
References 1) G. L. Miller, W. L. Brown, P. F. Donovan and I. M. Mackintosh, IRE Trans. Nucl. Sci. NS-7 no. 2-3 (1960) 185. 2) j. M. Caywood, C. A. Mead and J. W. Mayer, Nucl. Instr. and Meth. 79 (1970) 329. 3) O. Meyer and H. J. Langmann, Nucl. Instr. and Meth. 34 (1965) 77. 4) H. Meyer, IEEE Trans. Nucl. Sci. NS-13 no. 3 (1966) 180. 5) G. Dearnaley and D. C. Northrop, Semiconductor counters for nuclear radiation (Spon, London, 1963) ch. 2, p. 68. 6) p. A. Tove and W. Seibt, Nucl. Instr. and Meth. 51 (1967) 261.
324
w . SEIBT et al.
7) A. Alberighi Quaranta, A. Taroni and G. Zanarini, IEEE Trans. Nucl. Sci. NS-15 no. 3 (1968) 373. s) p. A. Tove, W. Seibt and K. E. SundstrSm, Proc. Ispra Syrup. Nuclear electronics (6-9 May 1969) p. 101; also UUIP-649 (May 1969). 9) p. A. Tove, W. Seibt and W. Leitz, Nucl. Instr. and Meth. 51 (1967) 304. 10) A. H. Krulisch and R. C. Axtmann, IEEE Trans. Nucl. Sci. NS-14 no. 4 (1967) 58. 11) A. Taroni and G. Zanarini, Nucl. Instr. and Meth. 67 (1969) 277. 12) B. D. Wilkins, M. J. Fluss, S. B. Kaufman, C. E. Gross and
E. P. Steinberg, Nucl. Instr. and Meth. 92 (1971) 381. la) E. P. Steinberg, S. B. Kaufman, B. D. Wilkins, C. E. Gross and M. J. Fluss, Nucl. Instr. and Meth. 99 (1972) 309. 14) W. Seibt, Solid-State Electron., 16 (1973) 1017. 15) A. Many and G. Rakavy, Phys. Rev. 126 (1962) 1980. 16) M. A. Nicolet, J. Appl. Phys. 37 (1966) 4224. 17) R. Janney, K. E. SundstrOm and P. A. Tove, UUIP-674 (Dec. 1969). 18) p. A. Tove, G. Andersson, G. Ericsson and R. Lidholt, IEEE Trans. Electron Devices ED-17 (1970) 407. 19) p. A. Tove and G. Andersson, J. Appl. Phys., 44 (1973) 2690.
© NORTH-HOLLAND PUBLISHING CO.
CHARGE COLLECTION IN SILICON DETECTORS FOR STRONGLY IONIZING PARTICLES W. SEIBT, K. E. SUNDSTROM and P. A. TOVE
Electronics Dept., Institute of Technology, University of Uppsala, Uppsala, Sweden
Received 13 April 1973 Experimental results for the plasma time in a semiconductor detector are given for alpha-particles, 160 ions and fission fragments. The plasma time tp is defined as the time needed to bring all carriers under the influence of the applied field. A simple theoretical model based on diffusion and space charge limited current erosion of the track is described and results in the expression tp = 1.32 x 10 10 (mE)~/F for a silicon detector where F is the electric field in the detector at the position of the
track, nl is the linear carrier density in the track and E is the particle energy. This gives good agreement with the experimental data for tracks longer than ---15/~m (alphas and 160 ions of energies 5.3-8.8 MeV and 22-42 MeV, respectively). The shorter tracks of fission fragments (90 MeV from e52cf) give shorter tp values then predicted by the formula above. This is explained by transfer from cylindrical to spherical erosion geometry.
1. Introduction
experimental investigations. The first formula obtained for the plasma erosion time was based on simple diffusion of the track and gave values orders of magnitude too long to explain the experimental resultsS). It was first the concept of space charge limited erosion currents 6) that b r o u g h t the calculated values into the region o f measured pulse rise times. It must, however, be pointed out that the measured plasma times and the calculated plasma erosion times were not really comparable, due to the difference in the erosion geometry. Also the experimental plasma times were obtained from the assumption that the measured rise times were quadratic superpositions of several time constants, including the rise time of the measuring amplifier and the RC-time from the undepleted detector material. Into this quadratic superposition was then also included the charge collection time, which in most cases 3'4'7) was further split into a calculated transport time and a plasma time. This means that the time delay introduced by the plasma effect is treated as equivalent to an integrating time constant acting on the voltage pulse, that should be obtained from the detector without plasma effect. It is obvious that the plasma time obtained in this way is difficult to relate to any physical process in the detector.
F o r heavily ionizing particles incident on a nuclear radiation semiconductor detector, the charge transport is delayed because of the " p l a s m a effect" which causes the charge collection time to be longer than that calculated from simple theory1). This has not only influence on the timing properties o f the detectors, but may also lead to inefficient charge collection, due to carrier trapping and recombination in the dense plasma or at the point where the plasma is in contact with the detector surface2'a9). An understanding of the above p h e n o m e n o n is thus o f central importance whenever semiconductor detectors are employed in heavy particle spectrometry. A l t h o u g h these detectors have been used for this purpose for over ten years, a general theoretical treatment of the plasma effect is almost made impossible by the three dimensional nature of the charge collection geometry, which is extremely difficult to treat mathematically without simplifying assumptions. Facing this problem, the earliest investigators o f the plasma effect confined themselves to the study o f voltage pulse rise times as a function of the electric field strength inside the detector depletion region at the position o f the ionized particle track. F r o m these studies empirical expressions for the plasma time were obtained3.4). Along with these studies attempts were made to calculate the time needed to erode the ionized track to the point where all charges are under the influence of the electric field. To make the problem tractable a geometry was chosen where the track is perpendicular to the electric field, which was not the case in the
317
A linear superposition of the plasma erosion time and the simple transit time seems more reasonable from a physical point o f view. A n uncertainty in this a p p r o a c h lies in the calculation o f the transit time since the last carriers to be collected move in an electric field which is modified by the carriers in drift in front o f them. A method to determine plasma erosion times according to the above definition, from
318
w. SEIBT et al.
measured current pulse shapes in totally depleted* detectors, was suggested by the authorsa). From a phenomenological discussion it was proposed that the plasma erosion time could be obtained from the decay of the current pulses, when the detectors were irradiated from the low field back side. Results thus obtained showed a relatively weak dependence of plasma time on the field. From a theoretical point of view the most accurate approach is of course to calculate the total charge collection times, although these times are less suited for a discussion of recombination effects in the plasma column. A simple theoretical model, yielding this time for particles incident parallel to the field, is discussed in ref. 10. The difficulty is to account for the transverse diffusion of the particle track. Several possibilities are discussed, but no clear choice could be made to fit the experimental results. Good agreement between experimental results and theoretical predictions has so far only been obtained through the use of computer methods H) and for comparatively light particles. In this case the total charge collection time in overdepleted detectors was computed from the general equations for charge transport in semiconductors, assuming one-dimensional geometry but also taking into account transverse diffusion. However, in spite of the simplified geometry, no closed form solution for the collection time could be found in this case. From the above discussion it seems that an expression for the plasma erosion time defined in 6) and for the case of tracks parallel to the field as a function of electric field strength and particle energy, could be very useful for the prediction of charge collection times and may also contribute to an explanation of that part of the pulse height defect 12'~3) which cannot be explained by nuclear collisions. For this case we will derive a theoretical expression for the plasma erosion time, defined as the time needed to bring all the carriers under the influence of the electric field. After a description of the experimental methods, measured charge collection times for e-particles, ~60 ions and fission fragments will be discussed in the light of the theoretical predictions.
2. Theoreticalmodel
normally to the detector surface. The charge in this track is polarized in the way previously described 6) so that the electric field becomes zero at the border between the polarized sheet and the rest of the plasma. The carriers in the polarized sheet start to drift in the electric field towards the other electrode and at the same time the plasma track is broadened by transverse diffusion. The diffusion which is visualized by the distribution of nc (t) in fig. 1 produces a decrease in the charge density in the track and this, together with the space charge limited extraction of carriers, causes the field to penetrate deeper and deeper into the plasma (see x ( t ) in fig. 1). We will now direct our attention to the central part of the track and study how the border x ( t ) between the polarized sheet and the rest of the plasma changes its position with time. For this purpose we consider a plasma column with a top surface A so small, that we may assume that the charge density in this column as a function of time is described by the solution to the diffusion equation in cylindrical coordinates for r = 0, yielding no(t) = nl/4ZrD . t.
(1)
Here n~(t) is the concentration of charge carriers in the middle of the column as a function of time, n~ is the total initial number of carriers per unit length in the track and D, is the ambipolar diffusion constant. At time t a small segment dx of the column (top area A) will thus contain an amount of charge equal to dQ
= qAnldX/47rD
(2)
a t.
It will now be assumed that charges are extracted from the top of the column by a space charge limited
J
to
M
t2
~
//
yx,,o, /t/xIt,I
The physical situation to be treated is a dense plasma track created by a heavily ionizing particle incident * Using overdepleted detectors eliminates the uncertainty introduced by the RC-time constant from undepleted detector material cf. ref. 9,7.
--'/-;'J.'¢\"'>'-
=
,o iol istonce from
centre of track
Fig. 1. Diffusion and current erosion o f plasma track.
319
C H A R G E C O L L E C T I O N IN S I L I C O N DETECTORS
current j ( t ) per unit area. Thus dQ/dt = j(t)A.
(3)
From (2) and (3) we get ( qnl/4 nO, t) dx = j ( t) dt, or, after integrating along the track and over the time needed to expose all carriers to the field: Qo =
qnldx = 4riD a o
tj(t)dt,
(4)
Jo
where Q0 is the total amount of charge created by the incident particle and tp is the plasma time. It remains now to find an appropriate expression for j(t). An attempt to use an extraction current of the order of the one-dimensional space charge limited transient current obtained in ~4), gives far too long values for the plasma time, so obviously the field acting on the cylindrical surface of the plasma column cannot be neglected. That this effect is of great importance has previously been recognized 6'1°) and especially at the end of the plasma column it should not be neglected,
pulse amp HP 462A
while it is smaller towards the detector surface. Due to the shielding effect of the carriers extracted from the end, the influence of the field on other parts of the plasma column is expected to be less important. We will assume that only those carriers at the end which are under the influence of the longitudinal field are also influenced by a transverse field component. The longitudinal field penetrates into the plasma a distance 6) 6 = eeoF/qnc. Although the carriers in this sheet are eventually spread along the whole detector thickness, a very high charge concentration is only maintained within a small distance from the plasmalS). If we assume that the transverse outflow of carriers is given by the space charge limited current value in cylindrical geometry6'16): .lscLc =-2~llee'o F2, the total current is given by iT
~
=
21rll(s%)ZF3/qnc.
For nc we use the value given by eq. (1) since we assume that ambipolar diffusion continues undisturbed except at the end of the plasma column. The current ix has to be supplied by the core of the plasma column
cathode follower
bias Ca)
2rClleeoF25
=
Tektronix sampling ocs type 661 external trig analog output
tt
X-Y recorder
signal input
pulse amp HP 35002A bias
+
Tektronix sampling osc type 661
averaging computer signat trig
external I trig I
(b)
detector
cathode follower
,,
analog J
"
signal input
w
~gate trig
signalinputoutput I
I
Fig. 2a. Experimental arrangement for measuring current pulse using a sampling oscilloscope preceded by a fast preamplifier. Fig. 2b. Set-up for current pulse measurement using a combination of sampling and averaging techniques.
320
w.
S E I B T et al.
which we assumed to have a top area A when deriving eq. (4). Thus we have to insert j(t) = iT/A into eq. (4) to get an expression for the plasma time obtaining
1 ( 3QoqnlA tv
=
~
F\32~DZj
.
(5a)
It is now convenient to introduce the particle energy E instead of the total created charge Qo. Since 3.5 eV are required to form an electron hole pair in silicon we finally get the expression tp
=
1.32
x
10-~0
(n]E)~/F,
(5b)
where we also have inserted # = 1400 cm2/V s obtained from 18) and Da = 16 cmZ/s. The area A was given the value 4.19 x 10 -8 cm 2 which will be discussed later in connection with the experimental results. That good agreement with measured plasma times could be obtained from eq. (5), if A is given a value of the order of the initial top area (radius about 1/Q of the plasma column, is a support for the model.
3. Measuring equipment The current pulses obtained when surface barrier detectors were exposed to a-particles, ]60 ions of different energies and fission fragments, were recorded. Olzservation of the current pulse shape gives more direct information about the carrier transport than does the voltage pulse (i.e., the current pulse integrated on a capacitor). The total collected charge, which is
I
directly obtained from the voltage pulse, can also be easily obtained from the current pulse by planimetric integration. The following electronic measuring set ups were used. In fig. 2a is shown the first set up, which had a rise time of 3.2 ns. The equivalent noise signal was of the order of 5 to 10 mV, referred to the input of the sampling oscilloscope. In fig. 2b is shown a second set up, the principle of which involved a combination of sampling and averaging techniques, in order to get a good signal-to-noise ratio (also independence of variability in signal amplitude)IT). In a third set up the same arrangement as in fig. 2a was used but the HP462A amplifier was replaced by two cascaded amplifiers of the later model HP35002A. This decreased the system rise time to l ns. The noise was also appreciably reduced.
4. Experimental results The physical properties of the silicon surface barrier detectors used in the experiments are given in table I. Measurements were done for a-particles, 160 ions and fission fragments. Alphas were obtained from ThB and Po sources, 160 ions from a van-de-Graaff generator and fission fragments from a 252Cf source. All particles had well defined energies, except the zs2Cf fission fragments for which an average energy of 90 MeV was assumed. This corresponds to the lowmass, high-energy peak in the energy spectrum. The triggering level was set to exclude lower energies. Examples of the current pulse shapes obtained, for
vo= ,56v W= 5 6 0 ~ m
l~,lAs t~
ns 33,) 2~
h
!T00
z6.s ~,6 2Lg ~2
,79
,,,
'OLT
/~2s
,~
"~
,
l~,~
!2"~
~s.s
ns
~2.~
to
n~
3,
3.0
-.
~
;
11"
Fig. 3. C u r r e n t pulse shapes obtained with detector El for front-injected 21°po c~-particles, at different bias voltages. Original s h o w n in one instance, other curves redrawn s m o o t h e d , for clarity.
CHARGE COLLECTION
DIODE
/ 1 1 i
1 I
I
///A
fltfl/i
l/filt'i/
F 5
o.
t ///,A~k 1tl//~ t 1t 1 1 ~ . Illfl, e//-
'
I " I 37~ I s~l ~9 I ~ i ~,~
19~1 !?.~. 8< I ' ° 1 ~7 i ~
\ \ \\\~t,.
20
30
o.
~ -~ ,3., 2,8 ,o~ ~o~
° ::
\\-"%'q~
t0
321
IN S I L I C O N DETECTORS
~7.8 ,so
~sec
Fig. 4. Current pulse shapes obtained with detector F5, for front-injected 42 MeV 160 ions, at different bias voltages.
TABLE 1 Detector
pa k-Q cm
Slice thickness w mm
VoW
F5 F7 F3 E4 E1
9.0 8.8 7.0 14.4 21.0
0.29 0.41 0.27 0.43 0.56
38 75 36 42 56
a
Calculated from the measured value of capacitance (at depletion voltage V0) and the slice thickness. b Determined from the plot of measured capacitance versus voltage. For V > V0 the capacitance becomes constant.
5.3 MeV alphas from 21°Po and 42 MeV 160 ions, front injection and their variation with applied bias are shown in figs. 3 and 4. In order to make a comparison with the theoretical predictions possible, the experimental data were treated in the following way. From the total length of the current pulse, which can be obtained with much better accuracy than when voltage (integrated) pulses are recorded, we subtract the calculated transit time tt=kw2//2 V to obtain the "plasma time". In the formula for tt the variation of the mobility /2 with field was taken into account, using information from ref. 18, k is a correction factor. This assumption of a linear superposition of plasma and transit time is in agreement with our previous suggestions (cf. e.g., fig. 4 in ref. 6). Actually the
transit time under space charge free conditions in a totally depleted detector is t t' = (t~z/2/2V0) In (V+ Vo)/ ( V - V o ) . Further modification of the transit time is caused by field perturbations connected with space charge limited (SCL) carrier transport. The presence of space charge lengthens the transport time of those carriers travelling behind the space charge (the last carriers leaving the plasma). For the case of an ideal plane parallel geometry the transport time has been shown to be lengthened a factor 1.3 to 1.5, depending on the ratio of applied voltage and the depletion voltageS4). In our case this effect should be smaller, because of sidewise spreading of the moving charge in front of the last carriers. These effects are taken into account by choosing k = 1.1 in the expression for t t. Experimental total collection times obtained as described above for alphas, and ~60 ions are shown in figs. 5 and 6. The times corresponding to the 10-90% pulse rise times of the corresponding voltage pulses were also obtained by planimetric integration of the current pulses. In all cases this time was very nearly half the total collection time, so previously published 1°'11) 10-90% voltage pulse rise times can be compared with half the values of tm presented in this work. Since the plasma times for a certain field strength predicted by eq. (5) are independent of the particular detector and only depend on the type of particle and its energy, plasma times obtained with different detectors but for a certain type of particles should follow the same theoretical line. Experimental results for
322
w. SEIBT et al.
different detectors are therefore given in the same plots. Results for 5.3 and 8.8 MeV alphas are given in fig. 7 and for 42 MeV 160 ions in fig. 8. The separation between the theoretical lines for 5.3 and 8.8 MeV in ltmo 3O _
ns,
A Detector E1
& O 0
a o
Detector EZ,
o
Detector F7
o
O a 0
20
0
a° L~
o
A O OA A A O A O
10
I
I
I
I
I
I .~.F
3
4
5
6
7
8
fig. 7 is small compared with the experimental error (cf. eq. 5). There is, however, a clear tendency among the experimental points to give shorter plasma times for the particles of the lower energy. In order to compare these results with the theoretical predictions, a value had to be given to the area A in eq. (5). If this value is chosen to be A =4.19 x 10 -8 cm 2 (corresponding to a radius of ~ 1 . 1 5 x 10 - 4 c m ) good agreement is obtained with the experimental results, both for 5.3 and 8.8 MeV alphas and 42 MeV 160 ions. These particle types have penetration depths into silicon of 55 #m for 8.8 MeV and ~-25 p m for both 5.3 MeV alphas and 42 MeV 160 ions. In these cases a track which is much longer than the width is ns
kV/cm
,tp o •
3O
Fig. 5. Experimental total collection times in nanoseconds (full length of current pulse) for 5.3 MeV (detectors E1 and E4) and 8.8 MeV (detector F7) c~-particles. F is the calculated field strength at the front contact of the detector.
o
Detector F3 Detector F5 Detector F7
20 ns
tm
4O
•=
o • o
o o
mid
30
Detector F3 Detector F5 Detector F7
•
°o
10
zllmm II o
0
I
I
I
I
t
t
3
4
5
6
?
8
...._F
kV/cm
OO
•
Fig. 8. Plasma times derived f r o m fig. 6.
r 3
20~
I 4
I 5
I Ol 6 7
I 8
F kV/cm ns
Fig. 6. Experimental total collection times for 42 MeV 160 ions.
t
Detector
El
a tm
70
•
tp
6O
a
ns ~tp 10 9
Detector
E1
¢, Detector o Detector
E4 F7
50
o "
40
¢, o
7 30
5 a
a
A
~
A
A
3
tm
o
tp
E4
/,,
~' •
o
4
o zx •
6
2
Detector
O
20
~
a a ~ ao a
I 3
I 4
I 5
I 6
I 7
I 8
vF kV/cm
Fig. 7. Plasma times derived from fig. 5 as the difference between tm and the calculated transit time (see text). The line is the calculated plasma time according to eq. (5).
10 2
i 3
i 4
t 5
i 6
n 7
n 8
~ 9
~'F kV/cm
Fig. 9. Experimental total collection times and corresponding experimental plasma times for light fission fragments from 252Cf.
CHARGE
ns
Detector
O
O 3O
COLLECTION
F7
trn
® tp
Oo
% Oo
20
*%% 10
9
,4
5
6
I,
I
I ~
7
8
9
kV/cm
Fig. 10. Experimental total collection times and corresponding experimental plasma times for 22 MeV 160 ions. Detector F7. Also included is the theoretical line calculated from eq. (5). produced. For fission fragments, on the other hand, this is not the case, since the penetration depth is only a few microns. It is therefore difficult to compare the results for fission fragments with the theory which is based on a cylindrical geometry. For fission fragments a spherical geometry is probably a better choice6). This would lead to faster charge collection than that predicted by the theory in this paper. Experimentally this is also what is observed in fig. 9, where the plasma times for the fission fragments are only a factor 1.25 higher than for the 42 MeV 160 ions in fig. 8. Applying eq. (5) to the fission fragments would lead to a much higher factor, due to the high linear density n~ in combination with the higher energy of the fission fragments. An intermediate example are the data obtained with 22 MeV 160 ions in fig. 10. Here the points for the measured plasma times fall slightly below the theoretical line calculated from eq. (5), indicating that the penetration depth of the particles has reached the critical lower value, where the assumption of a cylindrical erosion geometry becomes invalid.
IN
SILICON
DETECTORS
323
not possible to design a theoretical model covering all penetration depths. The model is therefore based on a cylindrical geometry and describes the experimental data well for penetration depths exceeding "-~15 pm. A spherical erosion geometry is more effective and so the experimental data for fission fragments are significantly shorter than what the cylindrical model predicts. Nevertheless, all experimental data follow the 1IFdependence in eq. (5) obtained for cylindrical geometry and the experimental plasma times are independent of the particular detector as predicted by eq. (5). An important result is that, according to this equation, the plasma time is not only dependent on the particle energy but, through the linear charge density nl, also on the type of particles. This is experimentally confirmed by the data for 5.3-8.8 MeV alphas and 42 MeV ~60 ions, where the penetration depths should be long enough for eq. (5) to be valid. The evaluation of the plasma time in this work follows the previous suggestion by the authors 6'8) to consider the total charge collection time as a linear superposition of the plasma time and a simple space charge free transit time. This way of evaluation has the advantage, that the plasma time obtained can be related to a physical process in the detector and thus can be used to discuss e.g., trapping effects. Experimentally, however, the method is somewhat more difficult than methods involving the measurement of the voltage pulse from the detector, which probably is a reason that it has not been used previously. Some of the measurements were done in the Tandem Van-de-Graaff accelerators of the Danish Atomic Energy Commission at Ris6 and at the University of Utrecht. We owe our sincere thanks to Dr B. Elbek and Professor A. M. H o o g e n b o o m of these laboratories as well as to their coworkers for the research facilities put at our disposal. The travels were made possible by grants from " K n u t och Alice Wallenbergs Stiftelse". Some stimulating discussions with Professor J. W. Mayer are also greatfully acknowledged.
5. Conclusion and discussion
The time needed to erode the plasma, created by heavily ionizing particles, to the point where all charges are under the influence of the electric field, was studied experimentally for different particles and compared with a simple theoretical model. The particles were 5.3 and 8.8 MeV alphas, 22 and 42 MeV 160 ions and 90 MeV fission fragments, thus covering a large range of energies and particle penetration depths. Since the erosion geometry is changed from cylindrical to spherical when the particle range decreases, it was
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