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Silicon drift chambers — first results and optimum processing of signals
Nuclear Instruments and Methods in Physics Research 226 (1984) 129-141 North-Holland, Amsterdam
129
Section IV. Electronics and readout methods SILICON DRIFT CHAMBERS - FIRST RESULTS AND OPTIMUM PROCESSING OF SIGNALS *
Emilio GATTI Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy
Pavel R E H A K Brookhaven National Laboratory, Upton, New York 11973, USA
Jack T. W A L T O N Lawrence Berkeley Laboratory, Berkeley, California 94720, USA
A 1) 2) 3)
Semiconductor Drift Chamber was produced and tested. The paper describes: principle of the chamber operation; the performance of the test chamber; the signal processing to obtain the best position resolution.
1. Introduction
Recently a new charge transport scheme in fully depleted semiconductors was proposed [1]. In the new scheme the electric field responsible for the charge transport was independent of the depletion field. The application of the new charge transport scheme led to several new semiconductor detector concepts: 1) semiconductor drift chambers; 2) ultralow capacitance, large area semiconductor spectrometers and photodiodes; 3) fully depleted thick CCDs. During the past several months, the semiconductor (silicon) drift chamber was realized and partially tested. The purpose of this contribution is twofold. First to report the observation of the signal from the chamber and second to present the optimization of the signal processing chain which provides the best position resolution. In sect. 2 we will briefly describe the principle of the semiconductor drift chamber. Sect. 3 describes the realization of the chamber and presents some test results. In Sect. 4 the signal and the noise analysis of the semiconductor drift chamber are presented.
2. Principles of the semiconductor drift chamber 11]
We are going to show how to fully deplete a thin, large silicon (or more generally semiconductor) wafer through a small contact at the edge of the wafer. After a complete depletion of the wafer we will superpose the second field to transport carriers towards the anode in such a way that the carrier transport time within the wafer is linearly related to the carrier path. Let us consider two conventional semiconductor junction ( d i o d e ) - detectors. Fig. l a shows two n-type silicon detectors with a p+ rectifying and an n + back contact. Two detectors are placed parallel to each other in such a way that the n + electrodes are in contact. We start to apply the same value of the reverse bias voltage to both detectors. * This research was supported by the US Department of Energy, Contract No. DE-AC02-76CH00016 and partially supported by the Italian INFN.
0168-9002/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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E. Gatti et aL / Silicon drift chambers
Fig. lb shows depleted regions (upper part) and the negative potential (lower part) of partially depleted detectors, we see that the formation of two depleted regions (one within each detector) is a consequence of the reverse biased rectifying p+n junctions. Uncovered fixed positive charge of the ionized donors in the depleted regions are compensated with an equal total negative charge "sitting" in the p+n junctions (one sided step junction approximation). The n + contacts do not play any role in the mechanism of the depletion. Their only function is to conduct electrons removed from the depleted regions to the external contact. Hence fig. lc shows the identical electrostatic situation to fig. lb, but achieved with a single semiconductor wafer twice as thick and with p+n rectifying junctions at two sides of the wafer. The undepleted semiconductor in the middle of the wafer acts as a conductor and thus replaces the function of two n + contacts in fig, lb. The connection to an external battery is at the edge of the wafer as shown in fig. la. With increasing bias voltage U, the thickness of the two depleted regions increases at the expense of the thickness of the undepleted conductive channel. Fig. l d shows the limiting case of a fully depleted silicon wafer; the central conductive channel has disappeared. The instability of the fully depleted situation against generated electron-hole pairs is not a problem. We can incline the potential shown in fig. l d in the direction perpendicular to the paper plane to carry away generated electrons, Fig. 2 shows the resulting potential function (negative) in two dimensions. A section of the surface shown here parallel to the plane z = 0 gives the parabola of fig. ld. Fig. 2 illustrates the operation of the semiconductor drift chamber. Electrons generated during the passage of a fast (minimum ionizing) particle are transported in a uniform electric field applied along the y axis in the middle of the valley towards an anode located further downstream (not shown). While electrons
p*JUNCTIONS
®
~
n*JUNCTIONS L _ . ~ ~i~)~'"
(a)
BELOW UNDEPLETED
UNDEPLETED
REGION
( REGO ,I N
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(
,JUNCTION
JUNCTIONS
(b)
(e)
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(d) Fig. 1. Princ]pie of the detector depletion from a virtual electrode: (a) Two standard n-type junction detectors with n + surfaces in contact. (b) Depleted and undepleted regions of two detectors when a reverse bias voltage was applied (top). The potential profile across two detectors (bottom). (c) Depleted and undepleted regions of a single n-type semiconductor wafer with p÷ n junctions on both sides (top). The potential profile of the wafer (bottom). (d) Fully depleted semiconductor wafer. The potential across the wafer (left). The central conductive channel has disappeared (right).
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E. Gatti et al. / Silicon drift chambers
are moving in the center of the wafer, their electric field is screened by surrounding p + n junctions which also act as electrodes. The signal at the anode is induced only when electrons arrive close to the anode. The time delay between the passage of the fast particle and the appearance of the signal at the anode is due to the drift of electrons within the wafer. For a uniform electric field in the y direction, the time delay is proportional to the distance between the passage of the fast particle and the anode, as in a standard gas drift chamber, Fig. 3 shows schematically a possible practical realization of a semiconductor drift chamber. It is produced from a thin ( = 300 # m thick, a few cm 2 in front area) high purity n-type silicon. The drift field is achieved by applying a linear potential to a strip array of rectifying p + n junctions on both faces of the detector. The only readout electrode (small capacitative anode) is also on the front face to facilitate the lithographic process during detector fabrication. The signal electrons thus have to arrive at the front face of the wafer. The bottom of the valley has to be moved from the center of the wafer towards the front face. The electrostatic configuration responsible for the turn of the bottom of the electron energy valley is shown in fig. 4. In the anode region of the drift chamber, the lower potentials are applied to the junctions on the anode side of the wafer rather than on the opposite side. The largest difference is in the anode plane. In this plane, the drift chamber looks like a standard junction detector depleted from one side of the wafer. A more detailed description of the principles of the semiconductor drift chamber and the related devices can be found in ref. [1].
3. Experimental test of the drift chamber A silicon semiconductor drift chamber similar to the one shown in fig. 3 was fabricated in the silicon laboratory at LBL and tested at BNL. The single crystal silicon used was n-type 10 kI2 cm ( N D = 5 x 1 0 U / c m 3) silicon 280 # m thick, supplied by the Komatsu Corporation. The chamber was produced by the planar process technique [2]. The wafer was cut to its final size, lapped, polished, cleaned and passivated by thermal oxidation at about 1030°C.
DRIFT FIELDELECTRODES ANODE
/ / / ~
]LJ Ill DRIFTDISTANCEI
~oI
/ ll I l ll ~0.0
~50.~0 T
i (MICRI]RSI
-
Iso.~
-
~ ~ 100.0
-200.0-
~ 300.0
400.0
L /
/2 T , LO
,o°°
o¢°
y (MICRONS1
Fig. 2. The negative potential of a fully depleted semiconductor detector when an additional linear lateral field along the y-axis is superpose& The field stabilizes the full depletion by sweeping away all generated charges. Fig. 3. Silicon drift chamber. The wafer is about 0.3 mm thick and has a front area of a few cm2. The surface is covered by a strip array of p+n junctions which provides the depletion and the lateral drift field. (Only junctions at the extremes of the wafer are shown.) Electrons produced by the passage of a fast particle drift towards the anode, which is the only readout contact on the wafer. IV. ELECTRONICS / READOUT
E. Gatti et al. / Silicon drift chambers
132
Using photolithographic equipment and etching techniques, windows were opened in the protective oxide to enable the doping by ion implantation techniques at these places while screening the rest of the wafer. First an arsenic (n +) implant was performed to produce an n + back contact for the anodes. (We have also two small guard anodes, not shown in fig. 3, below and above the collecting anode.) The arsenic-implanted dose was 2 x 1015/cm 2 at 25 keV. The preannealing was performed at 600°C for 30 min. The second implant of boron (p+) was performed on both sides of the wafer to produce p+n rectifying junctions. The boron implant was also performed at 25 keV; the dose was 4 x 10t4/cm 2. The wafer was thermally annealed at 800°C for 30 min. The aluminium pattern covering the implanted regions was also performed by photolithography. The detector was glued onto an especially designed thin printed circuit (PC) board and the detector junctions were connected to the PC lines by ultrasonic wire bonding. We did not have the full process completely under control. A few detectors had several rectifying junctions with a rather high leakage current. The chamber under test had to have only the first 2.5 mm of the drift distance active, because several following junctions showed a high leakage current and were disconnected. With the anode length 1 cm, the active area of the chamber under test was about 0.3 cm 2. (The drift length was 2.5 mm on one side and 0.5 mm on the other side of the anode.) The voltage on each (p÷n) junction was given as a sum of the "depletion" voltage and the "drift" voltage. The drift voltage was a linear function of the junction distance from the anode. The depletion voltage was a constant value for the majority of the junctions. Only junctions in the anode regions required
_~L
~'~'J
~0.0
× rl',CR@NSi
0
V
I~ 158.1/'~
100.0
200.0 y (MICRONS)
~uu,u
ANODE
Fig. 4. The negattve potential within the semiconductor drift chamber close to the readout anode. The p + n junctions on the back side of the wafer (opposite the anode) are b i a s e d to a higher negative potential, while the junctions on the anode side are biased less negatively. Consequently the bottom of the channel turns and electrons move towards the anode on the surface.
E. Gatti et aL / Silicon drift chambers
133
special values of the depletion voltage (see fig. 4). To facilitate electron collection by anodes, a constant overdepletion voltage was imposed on all anodes. Fig. 5 shows the capacitance-voltage (C-V) characteristics of the silicon drift chamber, that is, the dependence of the total anode capacitance on the depletion voltage. The potential on the horizontal axis in fig. 5 is the potential on the electrode opposite the anode. Other potentials during the measurements were applied through an appropriate voltage divider. (The majority of junctions had an applied voltage equal to 1 / 4 of the voltage of the junction opposite to the anode.) The measurement shown in fig. 5 was performed with zero drift voltage and a 4 V overdepletion voltage imposed on the anode. We see an abrupt drop in anode capacitance at the voltage corresponding to the total depletion, when the conductive channel retracts. The remaining capacitance is the geometrical capacitance between the anode and surrounding electrodes. Its calculated value is 2 pF. The measured plateau value is about 6 pF. The difference is due to stray capacitances in connection at the holding PC board and at connections to the bridge terminals.
100
CANODE(PF)
DRIFT CHAMBER C- V CHARACTERISTICS
50
°,o° °o o°
20
•,,
VANOD E= OV
10 5
2
-VovER = 4V -VoPPos,rE (v) .......
5
10 '
20 . .
. . . . 50 . . .
100 v
Fig. 5. Capacitance of the silicon drift chamber anode as a function of the depletion voltage. The potential on the horizontal axis was applied on the junction opposite the anode. The potential on other junctions was applied through a voltage divider. The divider was such that, at the voltage corresponding to the full depletion, the electrostatic configuration was as shown in fig. 4. We can see an abrupt drop in anode capacitance at the depletion
~TO
ANODE
THEGAUSSIAN SHAPPINGAMPL.
1~-'~713-Sr go SOURCE
f (3)
-- -
/ Si DRIFTCHAMBER/
FAST SCINTILLA -TOR /p+nJUNCTIONS
I
I '
I
I p÷nJUNCTIONS ~ L~_ ELECTRODES)
..~ I ~ANODE I
DRIFTING • ELECTRONS
~ ~-CROSSINO° t ":~--~ii~ ;'~O~-RES~-~1
~ 2.5 rnm
b)
Fig. 6. (a) Experimental setup for observation of signals produced by fast fl- crossing the silicon drift chamber. Fast electrons emitted by a fl- source were collimated and after crossing the drift chamber they stopped in a fast scintillator. The scintillator signal was used to define the crossing time. The anode of the chamber was connected to a charge sensitive preamplifier followed by a Gaussiar. shaping amplifier. (b) The geometry of the detection process when the collimator slit is 2.5 mm from the drift chamber anode. Electrons produced by the ionization are focused onto the center plane of the chamber and drift towards the anode. IV. ELECTRONICS / R E A D O U T
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E. Gatti et al. / Silicon drift chambers
Fig. 6a shows the experimental setup used for the observation of signals produced by fast f l - s crossing the chamber at different distances from the anode. Electrons from a 9°Sr f l - source were collimated by a movable aluminum collimator. After crossing the silicon drift chamber, the electrons entered a fast scintillator. The scintillator threshold was adjusted in such a way that f l - electrons were close to the ionization minimum when crossing the drift chamber. The anode of the drift chamber was connected to a charge sensitive preamplifier which was followed by a Gaussian shaping amplifier. Fig. 7 shows waveforms at the output of the Gaussian shaping amplifier on the scope triggered by signals from the scintillator. Waveforms are shown for four different positions of the collimator. The waveforms in fig. 7a were obtained when the collimator slit was just above the anode of the drift chamber. We see several traces peaking on average at the second vertical line on the scope screen. Fig. 7b shows waveforms when the slit was moved by 0.5 mm. The waveforms now peak about 200 ns later than in fig. 7a. Figs. 7c and d correspond to anode-slit distances of 1.5 and 2.5 mm, respectively. Each displacement of 1 m m corresponds roughly to a 200 ns delay in the average waveform peaking time. The lower efficiency (empty traces) of the detection at a distance of 2.5 m m was probably due to the fact that part of the f l - s crossed the chambers outside the active volume.
Fig. 7. Wavelengths at the output of the Gaussian shaping amplifier for the four positions of the collimator slit: (a) slit just above the anode; (b) slit 0.5 mm away from the anode; (c) slit 1.5 mm away from the anode; (d) slit 2.5 mm away from the anode. The scope sweep was triggered by the signal from the fast scintillator. We can see the shift of the average peak position of about 200 ns for 1 mm of the slit displacement ( E D -~ 200 V/cm).
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E. Gatti et al. / Silicon drift chambers
Fig. 6b shows to scale the geometry of the detection process when the collimator slit was 2.5 mm from the drift chamber anode. Electrons created by the ionization of a fast fl- are focused onto the center plane of the chamber and drift towards the anode. We would like to stress the fact that the collimator slit had more than 6 mm in the direction parallel to the anode. The fact that the detection efficiency was high suggests the uniformity of the detector performance along the anode direction. The noise visible on waveforms in fig. 7 was due to a rather high value of the anode leakage current ( -- 2 /~A). This is not the state of the art [3] and we plan to improve the chamber in this respect.
4. Signal processing for optimum position resolution of semiconductor drift chambers In this section we will derive the form of the signal filter which optimizes the position resolution of the drift chamber. The filter minimizes the effect of random fluctuations due to the following factors: 1) detector leakage current; 2) series amplifier noise; 3) electron diffusion (and mutual repulsion of electrons). Let us start our analysis by considering the hypothetical case when all three above-mentioned sources of fluctuations are switched off. For the potential shown in fig. 2, the component of the electron velocity towards the anode (y-direction) is independent of the position of the electron. Electrons produced by a fast particle which crossed the chamber perpendicular to its face would reach the anode at the same time. The form of the current signal induced at the anode in this hypothetical case would be a sharp peak (fig. 10 of ref. [1]). The height would fluctuate according to the ionization loss of a fast particle; however, the peaking time would define the arrival time of the electrons. Knowing the time of passage of the fast particle and the drift velocity of electrons in the drift chamber, the measurement of the peak time gives the position of the particle. The problem of defining the time of arrival of a signal is well known in nuclear spectroscopy. Generally, the anode signal drives a linear filter (which includes the preamplifier) and the arrival time is defined when the filter output waveform crosses a given threshold. The optimum signal processing for the drift chamber thus consists in the realization of the linear filter which minimizes the variance of the threshold crossing time. Let us call i(t) the signal waveform at the anode, that is, at the input of the linear filter. We introduce the normalized form of the input signal f(t) with the relation:
i(t) = qNf(t), where
f~ f(t)dt = 1,
(1)
where q is the electron charge and N is the number of electrons in the signal. The linear filter (time invariant) is defined by its impulse response h(t). The signal at the output of the linear filter due to the input i(t) is given by the convolution integral: s(,)=
qN f _ L f ( t -
"r) .h ('r )d~" = qN f'_Lf(t - ~ ) -w(-'r )dr.
(2)
We have introduced the filter weighting function w(t) as the time mirror image of the filter impulse response h ( t ). The time variance of the threshold crossing can be obtained to first order by geometrical consideration of the output waveform. The time variance is given by the ratio of the signal amplitude variance and the squared slope of the signal at the crossing time. To simplify the notation without loss of generality, we will assume that the undisturbed (noiseless) output crossing occurs at time t = 0. The time variance to minimize can be written: c 2-- [var(1) + var(2) + v a r ( 3 ) ] /
qN.
j'(,).w(~')d,
,
(3)
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136
where the total amplitude variance is the sum of the amplitude variances due to the three factors mentioned above. (All three sources of fluctuations have different origins and are therefore uncorrelated.) We have to find the dependence of all three amplitude variances on the weighting function w(t). The first variance in eq. (3) is due to a detector leakage current, or more exactly to the shot noise of this current. We can calculate the variance directly from Campbell's theorem [4] which states: if a linear instrument responds to an indefinitely short pulse of charge q at t = 0 by a deflection s(t) = q. h(t), then its mean response to a random series of pulses occurring at a mean rate v is:
g= vqL2h( t )dt, and the variance:
(4)
var(s)=uq2L%h2(t)dt.
(5)
The detector leakage current is due to a thermal generation of electron-hole pairs in the bulk of the detector. The arrival of the electrons at the anode is random as required by the theorem and the variance can be written: var(1) =
vq2fw2( r)d r,
(6)
where the leakage current equals vq. The second source of the amplitude fluctuations is the series noise of the preamplifier which is usually given by its power spectrum density e n2 (V2/Hz). We can deriv'e the formula of the amplitude variance due to the voltage series noise of the amplifier using Campbell's theorem again. A white noise with the power density spectrum e~ is equivalent to a random sequence of voltage pulses. In order to use eq. (5), we have to take into account that this noise source is in series with the detector anode. We substitute this series voltage noise source with its transformed current equivalent noise source, that is, a source which feeds the circuit in the same way as the input signal and produces the same noise at the output of the filter. This transformation in the time domain is the product of the total input capacitance C and the time derivative. The overall impulse response for the equivalent shot noise source is C. d / d t[h (t)]. The variance due to the series amplifier noise can be written: var(2) =
(C2e2n/2)L%(w'(r))adr.
(7)
Eqs. (6) and (7) are the standard expressions for stationary parallel and series noise, respectively [5]. Fluctuations due to the electron diffusion and their mutual repulsion are fluctuations in the form of f ( t ) and therefore non-stationary. The input signal is formed by N electrons, each having probability f ( t ) . d t of arriving at the anode between time t and t + dt. The mean number of electrons within that time interval is N .fit). dt and the number fluctuates with variance N .f(t). dt (variance of a Poisson distribution). The action of the filter is deterministic. At the time of interest, t = 0, the filter output value is the sum (integral) of input values at different times r multiplied by the weighting function w0- ), eq. (2). If the fluctuations in the number of arriving electrons are uncorrelated, the variance of the filter output at time t = 0 is the sum (integral) of variances with the appropriate weight. Hence: var(3) =
q2Nf'_%f( "r). w2( "r) d r .
(8)
The ratio to be minimized eq. (3) can now be written:
q2v£%w2(r ,2 _
)dr+ (C2e:fl2)f~w'2(,)dr+ -
q2N£%f(r ). w2(~") d r
q2N2[ f'_%f'( z)w('r)d,] 2 Eq. (9) was obtained by substituting eqs. (6), (7) and (8) into eq. (3).
(9)
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137
By standard variational calculus, we find that the optimum weighting function has to satisfy the following differential equation:
_ C2e 2 2 2~ .f'(t). _~ )w(~-)d~-, q2N2.W"(t)+ f ( t ) . w ( t ) + - ~ w ( t ) = 2 , ~ i n f f'('r where train2 is the minimum value of the time variance. We will solve eq. (10) assuming a Gaussian shape of the function
f(t)
1 = 2~o'exp-
t2
1
202
2~o
f(t).
X2
exp
where we have introduced a new variable x =
2qZNo . e _ X 2 / 2 . v ( x ) v"(x) - C2e2 272~
(10)
2 '
t/o.
(11) Eq. (10) can be rewritten:
2q2~'o2u(x) C~en
2 0 q2N2 --(min 02
¢r
2 f
C2e,2 x e -x /2 -~Y e-y2/2v(y)dy,
or v"( x ) - A e-X2/2 . v( x ) - Bv( x ) = - k . x . e-X2/2 f ~ y e-y2/2v( y )d y,
(12)
where v(x) = w(t); derivations are relative to the new variable x and new constants A, B and k were used to simplify the notation. We expand the unknown function v(x) in a series of Hermite polynomials multiplied by a common weighting factor, e -x2/2.
v(x) = e -~/2. ~ a , n , ( x ) .
(13)
n=0
Using the recurrence property of Hermite polynomials, the second derivative of
v"(x)=e-X2/2 ~ a ' [ a H ' + 2 ( x ) + n ( n -
2n+12
v(x)
can be written:
H,(x)
(14)
where H, ( x ) - 0 for n < 0. The integral on the right hand side of eq. (12) can be evaluated with the help of the orthogonality of Hermite polynomials.
f~ y e -y2/2. v ( y ) d y = f~%[Hi(y)/21
.e -~2/2. e -y%/2. ~ a , H , ( y ) . d y = ½ff%e-Y2alH2(y)dy n=0
= a , . v~-~, where we have used the explicit form of
(15)
Hl(y ) =
2y. Using eqs. (14) and (15), eq. (12) can be rewritten:
~-"=o ¼II'+2(x)+n(n-1)I-I~-2(x)-2n+12 - A e -x2/2 ~ a,H,(x)
, 1t1(x) + K T a i v ~
H~(x)-BH~(x)
~- =- O.
(16)
n=0
From eq. (16) we already see that all a, with n even have to be zero, that is, v(x) is an odd function. In order to calculate a,s, we have to eliminate the exponential factor from eq. (16). We can write:
e -x~/2. ~ a,,H,,(x)= ~ b.H.(x). n~O
(17)
n=0 IV. E L E C T R O N I C S
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E. Gatti et al. / Silicon drift chambers
H,,(x). e - x 2
Multiplying eq. (17) by
b,,,.v/~.2".m! = ~_,
a°f
n~O
and integrating with respect to x we obtain
e-3X2/2.H,(x)H,,,(x)dx,
--O0
or bm = (1/v/~2 m. m ! ) . ~
a,X . . . .
(18)
n=0
where ~,,,, are calculable numbers. The left hand side of eq. (17) is therefore
e -~2/2. ~ a,H,,(x)= ~ H,,(x) oo n=0
n=0
v~- 2"n m~_-0a ' x ' ' ' " !
=
(19)
Substituting eq. (19) into eq. (16) we obtain
a,(~H,+2(x)+n(n-a)H n 2 ( x ) 2n+lH,(x)-B.Hn(x)) n=O H,,(x) ~ , Hl(x ) ~=oamXn,,,+ K - - - ~ alv/~ = 0. -
2
~ A
n~0
V~-2",!
(20)
Equating to zero the coefficients of Hermite polynomials of all degrees, we obtain a linear system of equations for the a,s. To illustrate the method, we will find the first 3 non-zero coefficients, al, a 3 and a 5, in eq. (18). The problem is reduced to a system of three linear homogeneous equations. The system has a non-trivial solution only if the determinant of the system matrix is equal to zero. This condition gives us the value of the factor k and from it directly the value of the optimum (minimal) time variance ( m: i n [eq. (12)]. Now we can assign an arbitrary value to one of three unknown a i (a 1 = 1, for example). This freedom reflects the fact that the amplitude (gain) of the optimum filter is arbitrary. The values of the two remaining a ' s ( a 2 , a 3 ) follow directly from the system. Let us consider a silicon drift chamber with the following parameters: C = 5 pF,
e~ = 3 × 1018 VZ/Hz,
u = 3 × 1011 e l / s ( I = 50 nA)
N = 20000 el. (thickness 1 = 0.3 mm), and
o = 140 ns (sanft = 0.7 cm, Edrift = 100 V / c m ) .
Dimensionless constants A and B in eq. (12) are then 0.762 and 4.01, respectively. The determinant of the system matrix became zero for k equal to 6.35. The minimal time jitter is 4.5 ns. This time jitter at the given electron drift velocity corresponds to the position resolution of 6 btm (rms). The value of a ' s are aI = 1
a 3 = 0.393
and
a 5 --- 0.936 × 10 -3.
The weighting function of the optimum filter is therefore (fig. 8):
v(x)
= [ H l ( x ) + 0.0393H3(x ) + 0.936 × 10-3//5 ( x ) ] " e -x~/2,
(21)
which is an odd function of x. The output waveform of the optimum filter due to the input of the form f ( t ) (eq. (11) is the convolution of this even function f(t) with the odd function of the filter impulse response (eq. 2) and is therefore an odd function of time t. The optimum system thus turns out to be a zero crossing timing circuit with the important antiwalk property. We can also see that the weighting function v(x) has for all practical purposes a finite duration and is therefore realizable. We are now going to study the dependence of the obtainable resolution on the drift field, Ed, for a given drift chamber geometry. In the optimization process we have assumed a certain time width o of the electron swarm arriving at the anode [eq. (11)]. This time width o is due to the diffusion and the mutual repulsion of electrons and depends on the total drift time. To gain insight into the dependence of the obtainable
E. Gatti et al. / Silicon drift chambers
139
resolution on the drift field, Eo, we will study the dependence of the position resolution due to individual limiting factors separately. We will further simplify the calculation assuming that only the limiting process studied contributes to the position resolution.
4.1. Resolution limited by the Poisson noise of the detector leakage current We have var(2)= var(3)= 0, and eq. (10) with the function f ( t ) given by eq. (11) has the solution
w ( t ) = ( t / o ) e -t2/2°2
and
Emin2= 4V/~vo3/N2.
(22)
Assuming that the time spread o of the signal is dominated by the diffusion process, that is, o
Ud
U3/2
~3/2
(23)
E3/2 '
where D is the diffusion constant, s is the drift distance and ~ is the electron mobility, the position resolution 31 can be written (24)
~ l = Ud ' f m i n ~ E d 1"25.
4.2. Resolution limited by the series amplifier noise Putting var(1) = var(3) = 0 into eq. (10) we immediately find the solution [5].
]_l__rt
w(t)
1
v~--oovc~-o J
e_~2/2O2.dt "
-~
1
and
2n Emi
_
~/~C2.eZo q2N2
(25)
Using eq. (23) for the position resolution we obtain
3l oc E °'25.
(26)
4.3. Resolution limited by Poisson fluctuations in the pulse shape Here var(1) = var(2)= 0 and from eq. (10) we see the solution [6,7]
w( t )
t/o
and
o2 E2 mim = -N- •
(27)
/L
1.0
0.5
-0.5
t
-0.1
cr
I
-6
-t.
-2
I
2
I
i
t.
I
I
6
Fig. 8. The Weighting function of the filter to optimize the position resolution of the silicon drift chamber. The timing signal is obtained when the convolution of the input signal with this function crosses zero. IV. ELECTRONICS / READOUT
140
E. Gatti et al. / Silicon drift chambers
For the position resolution we obtain
*z e °5.
(28)
For the considered value of the drift chamber parameters, the main contribution to the chamber position resolution comes from the detector leakage current (3.8 out of a total of 4.5 ns, corresponding to 5 l = 6 ~tm). Increasing the drift field E a from 100 to 300 V / c m , we obtain a limiting resolution of 3.5 t~m for the same value of the detector leakage current. If the detector leakage current is decreased by a factor of 100, the obtainable position resolution becomes about 2.8 btm for the drift field of 100 V / c m . A position resolution of 3/~m with the drift distance 7 m m corresponds to a relative resolution better than 0.1%. It is not unlikely that the final resolution will be limited by other imperfections not yet considered. A primary candidate here for the semiconductor drift chamber is the non-uniformity in the detector bulk impurity concentration. The electron transport within the chamber uses the electric field generated by ionized donors within the chamber. The non-uniformity of the donors produces a non-uniform depth of the potential valley (fig. 2). Electrons are transported in the bottom of the valley and the donor non-uniformity creates an additional component of the electric field there. It can be easily shown that the value of this field is E = ( q 1 2 / 8 % E ~ ) " grad No,
(29)
where l is the chamber thickness, N O is the density of electrically active donors and q, ~0, cr retain their previous meaning. Eq. (24) shows that the importance of the doping uniformity N D increases with the square of the chamber thickness. A thinner chamber is less sensitive to the impurity concentration non-uniformity; however, the charge produced by a fast particle in a thinner chamber is smaller and the resolution may deteriorate due to the increase importance of the series noise of the amplifier (eq. 25). To decrease the electronic noise, we can incorporate the first transistor into the anode structure of the chamber [8]. Moreover, the decrease of the series noise due to the transistor action would improve the performance of the chamber at higher values of the drift velocity. High drift velocity chambers are required in many high multiplicity and (or) high rate experiments.
5. Conclusions A new charge transport mechanism in fully depleted semiconductors was described. The application of the new mechanism leads to several new detectors. The simplest device, which we have called the semiconductor drift chamber, a very unique device, was produced by planar technology at LBL. The test results confirmed the validity of the principles. The optimum processing of the signal from the chamber is described. The position resolution of the semiconductor drift chamber is a few ~tm. We wish to thank V. Radeka for the fruitful and stimulating discussions and in particular, for the kind hospitality given to E. Gatti in the Instrumentation Division of BNL. We are indebted to A. Longoni for help during the measurements. Discussions with G. Soncini and S. Solmi are acknowledged. And finally, we appreciate the interest and encouragement of F. Goulding in this work.
References [1] E. Gatti and P. Rehak, in Proc. 2nd Pisa Meeting on Advanced Detectors, Grosseto, Italy (1983) Nucl. Instr. and Meth. 225 (1984) 608. F. Cappasso, IEEE Trans Electron Devices ED-29 (1982) 1388. [2] We were guided by the process described in the followingpaper: J. Kemmer et al., IEEE Trans. Nucl. Sci. NS-29 (1982) 733. [3] J. Kemmer, these Proceedings (Semiconductor Detectors '83), p. 89. [4] F.N.H. Robinson, Noise and fluctuations in electronic devices and circuits (Clarendon, Oxford, 1974) p. 15.
E. Gatti et al. / Silicon drift chambers
[5] [6] [7] [8]
141
V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1) (1974) 51. E. Gatti and V. Svelto, Nucl. Instr. and Meth. 39 (1966) 309. J. Llacer, IEEE Trans. Nucl. Sci. NS-28 (1981) 630. T.C. Madden and G.L. Miller, Proc. Gatlinburg Conf. on Semiconductor Nuclear Particle Detectors and Circuits, Nucl. Sci. Series, Rept No. 44, Publ. 1693 (Nat. Acad. Sci., Washington DC, 1969) p. 314.
Discussion Damerell: Regarding your proposed C C D option, I am not clear that you avoid the problems encountered in fabricating standard C C D on high resistivity silicon, namely resistivity drops and problems with the operation of the on-chip MOSFET, which need islands of low resistivity. In the case where these problems were solved on prototype devices (by Westinghouse), full charge collection was obtained over the full depletion depth ( - 200 ~m). Rehak: The question regards technology and I do not understand why the silicon resistivity drops during the fabrication process. If reproducible, this could be taken into account. The low resistivity islands are not required if one can avoid having MOSFET on the wafer. Hyams: Do you have any knowledge of the variation of drift velocity with doping concentration? Rehak: Yes, the electron scattering on impurities in very pure silicon is much less than the electron scattering on phonons. This means that we have to control the temperature very accurately. Drukier: Do you expect that temperature inhomogeneity inside the silicon chip can introduce jitter in the drift velocity and therefore affect the spatial resolution? Rehak: The silicon crystal is a good thermal conductor and the crystal heating by the energy deposit of the particles is very small. However, instabilities in the detector leakage current could produce temperature differences which are difficult to calibrate and which decrease the spatial resolution. Lutz: You discussed the influence of doping variations on the drift field and mentioned that the sensitivity to these variations is reduced by a factor of four if the wafer is half as thick. It seems to me that you can obtain the same effect by moving the potential valley out of the centre towards one electrode, by putting unequal voltages on the drift electrodes. Rehak: You are right, however, the improvement is slow. By moving the potential valley from the centre towards one side (e.g. to 1 / 4 depth) one improves by 25% only. The electron transport in this non-symmetric configuration requires two times finer lithography for the junction array on the rear side of the wafer. Parker: Could you reduce noise due to leakage current and obtained second coordinate information by subdividing the anode and using multichannel readout? (the binary code or grey code schemes use anodes, covering about half the length and so do not have a large leakage current reduction). Rehak: Yes, in fact we are designing a multiple anode readout, which still may have a large number of elements. The grey code scheme is one attempt to reduce the amount of electronics. Clearly, one does not have the benefit of leakage current reduction. Tore: 1) Could you quote performance of your detector in terms of spatial resolution (in percentage of the m a x i m u m covered length), counting rate and m a x i m u m coverable area? 2) One of the 3 noise sources which you mentioned (the leakage current) gave a dominant contribution to the time dispersion ( 3 - 4 times larger than the others). If this is the case, moderate cooling (Peltier-element) could improve the performance. Rehak: 1) Possible spatial resolution 1 - 0 . 1 % ; counting rate < 1 MHz. 2) Primary interest is to make simple apparatus and avoid cooling. By adjustment of design, the three noise sources could be equalized.
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Section IV. Electronics and readout methods SILICON DRIFT CHAMBERS - FIRST RESULTS AND OPTIMUM PROCESSING OF SIGNALS *
Emilio GATTI Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, Italy
Pavel R E H A K Brookhaven National Laboratory, Upton, New York 11973, USA
Jack T. W A L T O N Lawrence Berkeley Laboratory, Berkeley, California 94720, USA
A 1) 2) 3)
Semiconductor Drift Chamber was produced and tested. The paper describes: principle of the chamber operation; the performance of the test chamber; the signal processing to obtain the best position resolution.
1. Introduction
Recently a new charge transport scheme in fully depleted semiconductors was proposed [1]. In the new scheme the electric field responsible for the charge transport was independent of the depletion field. The application of the new charge transport scheme led to several new semiconductor detector concepts: 1) semiconductor drift chambers; 2) ultralow capacitance, large area semiconductor spectrometers and photodiodes; 3) fully depleted thick CCDs. During the past several months, the semiconductor (silicon) drift chamber was realized and partially tested. The purpose of this contribution is twofold. First to report the observation of the signal from the chamber and second to present the optimization of the signal processing chain which provides the best position resolution. In sect. 2 we will briefly describe the principle of the semiconductor drift chamber. Sect. 3 describes the realization of the chamber and presents some test results. In Sect. 4 the signal and the noise analysis of the semiconductor drift chamber are presented.
2. Principles of the semiconductor drift chamber 11]
We are going to show how to fully deplete a thin, large silicon (or more generally semiconductor) wafer through a small contact at the edge of the wafer. After a complete depletion of the wafer we will superpose the second field to transport carriers towards the anode in such a way that the carrier transport time within the wafer is linearly related to the carrier path. Let us consider two conventional semiconductor junction ( d i o d e ) - detectors. Fig. l a shows two n-type silicon detectors with a p+ rectifying and an n + back contact. Two detectors are placed parallel to each other in such a way that the n + electrodes are in contact. We start to apply the same value of the reverse bias voltage to both detectors. * This research was supported by the US Department of Energy, Contract No. DE-AC02-76CH00016 and partially supported by the Italian INFN.
0168-9002/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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E. Gatti et aL / Silicon drift chambers
Fig. lb shows depleted regions (upper part) and the negative potential (lower part) of partially depleted detectors, we see that the formation of two depleted regions (one within each detector) is a consequence of the reverse biased rectifying p+n junctions. Uncovered fixed positive charge of the ionized donors in the depleted regions are compensated with an equal total negative charge "sitting" in the p+n junctions (one sided step junction approximation). The n + contacts do not play any role in the mechanism of the depletion. Their only function is to conduct electrons removed from the depleted regions to the external contact. Hence fig. lc shows the identical electrostatic situation to fig. lb, but achieved with a single semiconductor wafer twice as thick and with p+n rectifying junctions at two sides of the wafer. The undepleted semiconductor in the middle of the wafer acts as a conductor and thus replaces the function of two n + contacts in fig, lb. The connection to an external battery is at the edge of the wafer as shown in fig. la. With increasing bias voltage U, the thickness of the two depleted regions increases at the expense of the thickness of the undepleted conductive channel. Fig. l d shows the limiting case of a fully depleted silicon wafer; the central conductive channel has disappeared. The instability of the fully depleted situation against generated electron-hole pairs is not a problem. We can incline the potential shown in fig. l d in the direction perpendicular to the paper plane to carry away generated electrons, Fig. 2 shows the resulting potential function (negative) in two dimensions. A section of the surface shown here parallel to the plane z = 0 gives the parabola of fig. ld. Fig. 2 illustrates the operation of the semiconductor drift chamber. Electrons generated during the passage of a fast (minimum ionizing) particle are transported in a uniform electric field applied along the y axis in the middle of the valley towards an anode located further downstream (not shown). While electrons
p*JUNCTIONS
®
~
n*JUNCTIONS L _ . ~ ~i~)~'"
(a)
BELOW UNDEPLETED
UNDEPLETED
REGION
( REGO ,I N
~EP~ETEO~ ® F~,~ d i e(~ ) REGION® (3
(
,JUNCTION
JUNCTIONS
(b)
(e)
t-u
® ® ®
l
(Z) ® ® i
L FULLYDEPLETED
(d) Fig. 1. Princ]pie of the detector depletion from a virtual electrode: (a) Two standard n-type junction detectors with n + surfaces in contact. (b) Depleted and undepleted regions of two detectors when a reverse bias voltage was applied (top). The potential profile across two detectors (bottom). (c) Depleted and undepleted regions of a single n-type semiconductor wafer with p÷ n junctions on both sides (top). The potential profile of the wafer (bottom). (d) Fully depleted semiconductor wafer. The potential across the wafer (left). The central conductive channel has disappeared (right).
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E. Gatti et al. / Silicon drift chambers
are moving in the center of the wafer, their electric field is screened by surrounding p + n junctions which also act as electrodes. The signal at the anode is induced only when electrons arrive close to the anode. The time delay between the passage of the fast particle and the appearance of the signal at the anode is due to the drift of electrons within the wafer. For a uniform electric field in the y direction, the time delay is proportional to the distance between the passage of the fast particle and the anode, as in a standard gas drift chamber, Fig. 3 shows schematically a possible practical realization of a semiconductor drift chamber. It is produced from a thin ( = 300 # m thick, a few cm 2 in front area) high purity n-type silicon. The drift field is achieved by applying a linear potential to a strip array of rectifying p + n junctions on both faces of the detector. The only readout electrode (small capacitative anode) is also on the front face to facilitate the lithographic process during detector fabrication. The signal electrons thus have to arrive at the front face of the wafer. The bottom of the valley has to be moved from the center of the wafer towards the front face. The electrostatic configuration responsible for the turn of the bottom of the electron energy valley is shown in fig. 4. In the anode region of the drift chamber, the lower potentials are applied to the junctions on the anode side of the wafer rather than on the opposite side. The largest difference is in the anode plane. In this plane, the drift chamber looks like a standard junction detector depleted from one side of the wafer. A more detailed description of the principles of the semiconductor drift chamber and the related devices can be found in ref. [1].
3. Experimental test of the drift chamber A silicon semiconductor drift chamber similar to the one shown in fig. 3 was fabricated in the silicon laboratory at LBL and tested at BNL. The single crystal silicon used was n-type 10 kI2 cm ( N D = 5 x 1 0 U / c m 3) silicon 280 # m thick, supplied by the Komatsu Corporation. The chamber was produced by the planar process technique [2]. The wafer was cut to its final size, lapped, polished, cleaned and passivated by thermal oxidation at about 1030°C.
DRIFT FIELDELECTRODES ANODE
/ / / ~
]LJ Ill DRIFTDISTANCEI
~oI
/ ll I l ll ~0.0
~50.~0 T
i (MICRI]RSI
-
Iso.~
-
~ ~ 100.0
-200.0-
~ 300.0
400.0
L /
/2 T , LO
,o°°
o¢°
y (MICRONS1
Fig. 2. The negative potential of a fully depleted semiconductor detector when an additional linear lateral field along the y-axis is superpose& The field stabilizes the full depletion by sweeping away all generated charges. Fig. 3. Silicon drift chamber. The wafer is about 0.3 mm thick and has a front area of a few cm2. The surface is covered by a strip array of p+n junctions which provides the depletion and the lateral drift field. (Only junctions at the extremes of the wafer are shown.) Electrons produced by the passage of a fast particle drift towards the anode, which is the only readout contact on the wafer. IV. ELECTRONICS / READOUT
E. Gatti et al. / Silicon drift chambers
132
Using photolithographic equipment and etching techniques, windows were opened in the protective oxide to enable the doping by ion implantation techniques at these places while screening the rest of the wafer. First an arsenic (n +) implant was performed to produce an n + back contact for the anodes. (We have also two small guard anodes, not shown in fig. 3, below and above the collecting anode.) The arsenic-implanted dose was 2 x 1015/cm 2 at 25 keV. The preannealing was performed at 600°C for 30 min. The second implant of boron (p+) was performed on both sides of the wafer to produce p+n rectifying junctions. The boron implant was also performed at 25 keV; the dose was 4 x 10t4/cm 2. The wafer was thermally annealed at 800°C for 30 min. The aluminium pattern covering the implanted regions was also performed by photolithography. The detector was glued onto an especially designed thin printed circuit (PC) board and the detector junctions were connected to the PC lines by ultrasonic wire bonding. We did not have the full process completely under control. A few detectors had several rectifying junctions with a rather high leakage current. The chamber under test had to have only the first 2.5 mm of the drift distance active, because several following junctions showed a high leakage current and were disconnected. With the anode length 1 cm, the active area of the chamber under test was about 0.3 cm 2. (The drift length was 2.5 mm on one side and 0.5 mm on the other side of the anode.) The voltage on each (p÷n) junction was given as a sum of the "depletion" voltage and the "drift" voltage. The drift voltage was a linear function of the junction distance from the anode. The depletion voltage was a constant value for the majority of the junctions. Only junctions in the anode regions required
_~L
~'~'J
~0.0
× rl',CR@NSi
0
V
I~ 158.1/'~
100.0
200.0 y (MICRONS)
~uu,u
ANODE
Fig. 4. The negattve potential within the semiconductor drift chamber close to the readout anode. The p + n junctions on the back side of the wafer (opposite the anode) are b i a s e d to a higher negative potential, while the junctions on the anode side are biased less negatively. Consequently the bottom of the channel turns and electrons move towards the anode on the surface.
E. Gatti et aL / Silicon drift chambers
133
special values of the depletion voltage (see fig. 4). To facilitate electron collection by anodes, a constant overdepletion voltage was imposed on all anodes. Fig. 5 shows the capacitance-voltage (C-V) characteristics of the silicon drift chamber, that is, the dependence of the total anode capacitance on the depletion voltage. The potential on the horizontal axis in fig. 5 is the potential on the electrode opposite the anode. Other potentials during the measurements were applied through an appropriate voltage divider. (The majority of junctions had an applied voltage equal to 1 / 4 of the voltage of the junction opposite to the anode.) The measurement shown in fig. 5 was performed with zero drift voltage and a 4 V overdepletion voltage imposed on the anode. We see an abrupt drop in anode capacitance at the voltage corresponding to the total depletion, when the conductive channel retracts. The remaining capacitance is the geometrical capacitance between the anode and surrounding electrodes. Its calculated value is 2 pF. The measured plateau value is about 6 pF. The difference is due to stray capacitances in connection at the holding PC board and at connections to the bridge terminals.
100
CANODE(PF)
DRIFT CHAMBER C- V CHARACTERISTICS
50
°,o° °o o°
20
•,,
VANOD E= OV
10 5
2
-VovER = 4V -VoPPos,rE (v) .......
5
10 '
20 . .
. . . . 50 . . .
100 v
Fig. 5. Capacitance of the silicon drift chamber anode as a function of the depletion voltage. The potential on the horizontal axis was applied on the junction opposite the anode. The potential on other junctions was applied through a voltage divider. The divider was such that, at the voltage corresponding to the full depletion, the electrostatic configuration was as shown in fig. 4. We can see an abrupt drop in anode capacitance at the depletion
~TO
ANODE
THEGAUSSIAN SHAPPINGAMPL.
1~-'~713-Sr go SOURCE
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~ ~-CROSSINO° t ":~--~ii~ ;'~O~-RES~-~1
~ 2.5 rnm
b)
Fig. 6. (a) Experimental setup for observation of signals produced by fast fl- crossing the silicon drift chamber. Fast electrons emitted by a fl- source were collimated and after crossing the drift chamber they stopped in a fast scintillator. The scintillator signal was used to define the crossing time. The anode of the chamber was connected to a charge sensitive preamplifier followed by a Gaussiar. shaping amplifier. (b) The geometry of the detection process when the collimator slit is 2.5 mm from the drift chamber anode. Electrons produced by the ionization are focused onto the center plane of the chamber and drift towards the anode. IV. ELECTRONICS / R E A D O U T
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E. Gatti et al. / Silicon drift chambers
Fig. 6a shows the experimental setup used for the observation of signals produced by fast f l - s crossing the chamber at different distances from the anode. Electrons from a 9°Sr f l - source were collimated by a movable aluminum collimator. After crossing the silicon drift chamber, the electrons entered a fast scintillator. The scintillator threshold was adjusted in such a way that f l - electrons were close to the ionization minimum when crossing the drift chamber. The anode of the drift chamber was connected to a charge sensitive preamplifier which was followed by a Gaussian shaping amplifier. Fig. 7 shows waveforms at the output of the Gaussian shaping amplifier on the scope triggered by signals from the scintillator. Waveforms are shown for four different positions of the collimator. The waveforms in fig. 7a were obtained when the collimator slit was just above the anode of the drift chamber. We see several traces peaking on average at the second vertical line on the scope screen. Fig. 7b shows waveforms when the slit was moved by 0.5 mm. The waveforms now peak about 200 ns later than in fig. 7a. Figs. 7c and d correspond to anode-slit distances of 1.5 and 2.5 mm, respectively. Each displacement of 1 m m corresponds roughly to a 200 ns delay in the average waveform peaking time. The lower efficiency (empty traces) of the detection at a distance of 2.5 m m was probably due to the fact that part of the f l - s crossed the chambers outside the active volume.
Fig. 7. Wavelengths at the output of the Gaussian shaping amplifier for the four positions of the collimator slit: (a) slit just above the anode; (b) slit 0.5 mm away from the anode; (c) slit 1.5 mm away from the anode; (d) slit 2.5 mm away from the anode. The scope sweep was triggered by the signal from the fast scintillator. We can see the shift of the average peak position of about 200 ns for 1 mm of the slit displacement ( E D -~ 200 V/cm).
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E. Gatti et al. / Silicon drift chambers
Fig. 6b shows to scale the geometry of the detection process when the collimator slit was 2.5 mm from the drift chamber anode. Electrons created by the ionization of a fast fl- are focused onto the center plane of the chamber and drift towards the anode. We would like to stress the fact that the collimator slit had more than 6 mm in the direction parallel to the anode. The fact that the detection efficiency was high suggests the uniformity of the detector performance along the anode direction. The noise visible on waveforms in fig. 7 was due to a rather high value of the anode leakage current ( -- 2 /~A). This is not the state of the art [3] and we plan to improve the chamber in this respect.
4. Signal processing for optimum position resolution of semiconductor drift chambers In this section we will derive the form of the signal filter which optimizes the position resolution of the drift chamber. The filter minimizes the effect of random fluctuations due to the following factors: 1) detector leakage current; 2) series amplifier noise; 3) electron diffusion (and mutual repulsion of electrons). Let us start our analysis by considering the hypothetical case when all three above-mentioned sources of fluctuations are switched off. For the potential shown in fig. 2, the component of the electron velocity towards the anode (y-direction) is independent of the position of the electron. Electrons produced by a fast particle which crossed the chamber perpendicular to its face would reach the anode at the same time. The form of the current signal induced at the anode in this hypothetical case would be a sharp peak (fig. 10 of ref. [1]). The height would fluctuate according to the ionization loss of a fast particle; however, the peaking time would define the arrival time of the electrons. Knowing the time of passage of the fast particle and the drift velocity of electrons in the drift chamber, the measurement of the peak time gives the position of the particle. The problem of defining the time of arrival of a signal is well known in nuclear spectroscopy. Generally, the anode signal drives a linear filter (which includes the preamplifier) and the arrival time is defined when the filter output waveform crosses a given threshold. The optimum signal processing for the drift chamber thus consists in the realization of the linear filter which minimizes the variance of the threshold crossing time. Let us call i(t) the signal waveform at the anode, that is, at the input of the linear filter. We introduce the normalized form of the input signal f(t) with the relation:
i(t) = qNf(t), where
f~ f(t)dt = 1,
(1)
where q is the electron charge and N is the number of electrons in the signal. The linear filter (time invariant) is defined by its impulse response h(t). The signal at the output of the linear filter due to the input i(t) is given by the convolution integral: s(,)=
qN f _ L f ( t -
"r) .h ('r )d~" = qN f'_Lf(t - ~ ) -w(-'r )dr.
(2)
We have introduced the filter weighting function w(t) as the time mirror image of the filter impulse response h ( t ). The time variance of the threshold crossing can be obtained to first order by geometrical consideration of the output waveform. The time variance is given by the ratio of the signal amplitude variance and the squared slope of the signal at the crossing time. To simplify the notation without loss of generality, we will assume that the undisturbed (noiseless) output crossing occurs at time t = 0. The time variance to minimize can be written: c 2-- [var(1) + var(2) + v a r ( 3 ) ] /
qN.
j'(,).w(~')d,
,
(3)
IV. ELECTRONICS / READOUT
E. Gatti et al. / Silicon drift chambers
136
where the total amplitude variance is the sum of the amplitude variances due to the three factors mentioned above. (All three sources of fluctuations have different origins and are therefore uncorrelated.) We have to find the dependence of all three amplitude variances on the weighting function w(t). The first variance in eq. (3) is due to a detector leakage current, or more exactly to the shot noise of this current. We can calculate the variance directly from Campbell's theorem [4] which states: if a linear instrument responds to an indefinitely short pulse of charge q at t = 0 by a deflection s(t) = q. h(t), then its mean response to a random series of pulses occurring at a mean rate v is:
g= vqL2h( t )dt, and the variance:
(4)
var(s)=uq2L%h2(t)dt.
(5)
The detector leakage current is due to a thermal generation of electron-hole pairs in the bulk of the detector. The arrival of the electrons at the anode is random as required by the theorem and the variance can be written: var(1) =
vq2fw2( r)d r,
(6)
where the leakage current equals vq. The second source of the amplitude fluctuations is the series noise of the preamplifier which is usually given by its power spectrum density e n2 (V2/Hz). We can deriv'e the formula of the amplitude variance due to the voltage series noise of the amplifier using Campbell's theorem again. A white noise with the power density spectrum e~ is equivalent to a random sequence of voltage pulses. In order to use eq. (5), we have to take into account that this noise source is in series with the detector anode. We substitute this series voltage noise source with its transformed current equivalent noise source, that is, a source which feeds the circuit in the same way as the input signal and produces the same noise at the output of the filter. This transformation in the time domain is the product of the total input capacitance C and the time derivative. The overall impulse response for the equivalent shot noise source is C. d / d t[h (t)]. The variance due to the series amplifier noise can be written: var(2) =
(C2e2n/2)L%(w'(r))adr.
(7)
Eqs. (6) and (7) are the standard expressions for stationary parallel and series noise, respectively [5]. Fluctuations due to the electron diffusion and their mutual repulsion are fluctuations in the form of f ( t ) and therefore non-stationary. The input signal is formed by N electrons, each having probability f ( t ) . d t of arriving at the anode between time t and t + dt. The mean number of electrons within that time interval is N .fit). dt and the number fluctuates with variance N .f(t). dt (variance of a Poisson distribution). The action of the filter is deterministic. At the time of interest, t = 0, the filter output value is the sum (integral) of input values at different times r multiplied by the weighting function w0- ), eq. (2). If the fluctuations in the number of arriving electrons are uncorrelated, the variance of the filter output at time t = 0 is the sum (integral) of variances with the appropriate weight. Hence: var(3) =
q2Nf'_%f( "r). w2( "r) d r .
(8)
The ratio to be minimized eq. (3) can now be written:
q2v£%w2(r ,2 _
)dr+ (C2e:fl2)f~w'2(,)dr+ -
q2N£%f(r ). w2(~") d r
q2N2[ f'_%f'( z)w('r)d,] 2 Eq. (9) was obtained by substituting eqs. (6), (7) and (8) into eq. (3).
(9)
E. Gatti et al. / Silicon drift chambers
137
By standard variational calculus, we find that the optimum weighting function has to satisfy the following differential equation:
_ C2e 2 2 2~ .f'(t). _~ )w(~-)d~-, q2N2.W"(t)+ f ( t ) . w ( t ) + - ~ w ( t ) = 2 , ~ i n f f'('r where train2 is the minimum value of the time variance. We will solve eq. (10) assuming a Gaussian shape of the function
f(t)
1 = 2~o'exp-
t2
1
202
2~o
f(t).
X2
exp
where we have introduced a new variable x =
2qZNo . e _ X 2 / 2 . v ( x ) v"(x) - C2e2 272~
(10)
2 '
t/o.
(11) Eq. (10) can be rewritten:
2q2~'o2u(x) C~en
2 0 q2N2 --(min 02
¢r
2 f
C2e,2 x e -x /2 -~Y e-y2/2v(y)dy,
or v"( x ) - A e-X2/2 . v( x ) - Bv( x ) = - k . x . e-X2/2 f ~ y e-y2/2v( y )d y,
(12)
where v(x) = w(t); derivations are relative to the new variable x and new constants A, B and k were used to simplify the notation. We expand the unknown function v(x) in a series of Hermite polynomials multiplied by a common weighting factor, e -x2/2.
v(x) = e -~/2. ~ a , n , ( x ) .
(13)
n=0
Using the recurrence property of Hermite polynomials, the second derivative of
v"(x)=e-X2/2 ~ a ' [ a H ' + 2 ( x ) + n ( n -
2n+12
v(x)
can be written:
H,(x)
(14)
where H, ( x ) - 0 for n < 0. The integral on the right hand side of eq. (12) can be evaluated with the help of the orthogonality of Hermite polynomials.
f~ y e -y2/2. v ( y ) d y = f~%[Hi(y)/21
.e -~2/2. e -y%/2. ~ a , H , ( y ) . d y = ½ff%e-Y2alH2(y)dy n=0
= a , . v~-~, where we have used the explicit form of
(15)
Hl(y ) =
2y. Using eqs. (14) and (15), eq. (12) can be rewritten:
~-"=o ¼II'+2(x)+n(n-1)I-I~-2(x)-2n+12 - A e -x2/2 ~ a,H,(x)
, 1t1(x) + K T a i v ~
H~(x)-BH~(x)
~- =- O.
(16)
n=0
From eq. (16) we already see that all a, with n even have to be zero, that is, v(x) is an odd function. In order to calculate a,s, we have to eliminate the exponential factor from eq. (16). We can write:
e -x~/2. ~ a,,H,,(x)= ~ b.H.(x). n~O
(17)
n=0 IV. E L E C T R O N I C S
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E. Gatti et al. / Silicon drift chambers
H,,(x). e - x 2
Multiplying eq. (17) by
b,,,.v/~.2".m! = ~_,
a°f
n~O
and integrating with respect to x we obtain
e-3X2/2.H,(x)H,,,(x)dx,
--O0
or bm = (1/v/~2 m. m ! ) . ~
a,X . . . .
(18)
n=0
where ~,,,, are calculable numbers. The left hand side of eq. (17) is therefore
e -~2/2. ~ a,H,,(x)= ~ H,,(x) oo n=0
n=0
v~- 2"n m~_-0a ' x ' ' ' " !
=
(19)
Substituting eq. (19) into eq. (16) we obtain
a,(~H,+2(x)+n(n-a)H n 2 ( x ) 2n+lH,(x)-B.Hn(x)) n=O H,,(x) ~ , Hl(x ) ~=oamXn,,,+ K - - - ~ alv/~ = 0. -
2
~ A
n~0
V~-2",!
(20)
Equating to zero the coefficients of Hermite polynomials of all degrees, we obtain a linear system of equations for the a,s. To illustrate the method, we will find the first 3 non-zero coefficients, al, a 3 and a 5, in eq. (18). The problem is reduced to a system of three linear homogeneous equations. The system has a non-trivial solution only if the determinant of the system matrix is equal to zero. This condition gives us the value of the factor k and from it directly the value of the optimum (minimal) time variance ( m: i n [eq. (12)]. Now we can assign an arbitrary value to one of three unknown a i (a 1 = 1, for example). This freedom reflects the fact that the amplitude (gain) of the optimum filter is arbitrary. The values of the two remaining a ' s ( a 2 , a 3 ) follow directly from the system. Let us consider a silicon drift chamber with the following parameters: C = 5 pF,
e~ = 3 × 1018 VZ/Hz,
u = 3 × 1011 e l / s ( I = 50 nA)
N = 20000 el. (thickness 1 = 0.3 mm), and
o = 140 ns (sanft = 0.7 cm, Edrift = 100 V / c m ) .
Dimensionless constants A and B in eq. (12) are then 0.762 and 4.01, respectively. The determinant of the system matrix became zero for k equal to 6.35. The minimal time jitter is 4.5 ns. This time jitter at the given electron drift velocity corresponds to the position resolution of 6 btm (rms). The value of a ' s are aI = 1
a 3 = 0.393
and
a 5 --- 0.936 × 10 -3.
The weighting function of the optimum filter is therefore (fig. 8):
v(x)
= [ H l ( x ) + 0.0393H3(x ) + 0.936 × 10-3//5 ( x ) ] " e -x~/2,
(21)
which is an odd function of x. The output waveform of the optimum filter due to the input of the form f ( t ) (eq. (11) is the convolution of this even function f(t) with the odd function of the filter impulse response (eq. 2) and is therefore an odd function of time t. The optimum system thus turns out to be a zero crossing timing circuit with the important antiwalk property. We can also see that the weighting function v(x) has for all practical purposes a finite duration and is therefore realizable. We are now going to study the dependence of the obtainable resolution on the drift field, Ed, for a given drift chamber geometry. In the optimization process we have assumed a certain time width o of the electron swarm arriving at the anode [eq. (11)]. This time width o is due to the diffusion and the mutual repulsion of electrons and depends on the total drift time. To gain insight into the dependence of the obtainable
E. Gatti et al. / Silicon drift chambers
139
resolution on the drift field, Eo, we will study the dependence of the position resolution due to individual limiting factors separately. We will further simplify the calculation assuming that only the limiting process studied contributes to the position resolution.
4.1. Resolution limited by the Poisson noise of the detector leakage current We have var(2)= var(3)= 0, and eq. (10) with the function f ( t ) given by eq. (11) has the solution
w ( t ) = ( t / o ) e -t2/2°2
and
Emin2= 4V/~vo3/N2.
(22)
Assuming that the time spread o of the signal is dominated by the diffusion process, that is, o
Ud
U3/2
~3/2
(23)
E3/2 '
where D is the diffusion constant, s is the drift distance and ~ is the electron mobility, the position resolution 31 can be written (24)
~ l = Ud ' f m i n ~ E d 1"25.
4.2. Resolution limited by the series amplifier noise Putting var(1) = var(3) = 0 into eq. (10) we immediately find the solution [5].
]_l__rt
w(t)
1
v~--oovc~-o J
e_~2/2O2.dt "
-~
1
and
2n Emi
_
~/~C2.eZo q2N2
(25)
Using eq. (23) for the position resolution we obtain
3l oc E °'25.
(26)
4.3. Resolution limited by Poisson fluctuations in the pulse shape Here var(1) = var(2)= 0 and from eq. (10) we see the solution [6,7]
w( t )
t/o
and
o2 E2 mim = -N- •
(27)
/L
1.0
0.5
-0.5
t
-0.1
cr
I
-6
-t.
-2
I
2
I
i
t.
I
I
6
Fig. 8. The Weighting function of the filter to optimize the position resolution of the silicon drift chamber. The timing signal is obtained when the convolution of the input signal with this function crosses zero. IV. ELECTRONICS / READOUT
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E. Gatti et al. / Silicon drift chambers
For the position resolution we obtain
*z e °5.
(28)
For the considered value of the drift chamber parameters, the main contribution to the chamber position resolution comes from the detector leakage current (3.8 out of a total of 4.5 ns, corresponding to 5 l = 6 ~tm). Increasing the drift field E a from 100 to 300 V / c m , we obtain a limiting resolution of 3.5 t~m for the same value of the detector leakage current. If the detector leakage current is decreased by a factor of 100, the obtainable position resolution becomes about 2.8 btm for the drift field of 100 V / c m . A position resolution of 3/~m with the drift distance 7 m m corresponds to a relative resolution better than 0.1%. It is not unlikely that the final resolution will be limited by other imperfections not yet considered. A primary candidate here for the semiconductor drift chamber is the non-uniformity in the detector bulk impurity concentration. The electron transport within the chamber uses the electric field generated by ionized donors within the chamber. The non-uniformity of the donors produces a non-uniform depth of the potential valley (fig. 2). Electrons are transported in the bottom of the valley and the donor non-uniformity creates an additional component of the electric field there. It can be easily shown that the value of this field is E = ( q 1 2 / 8 % E ~ ) " grad No,
(29)
where l is the chamber thickness, N O is the density of electrically active donors and q, ~0, cr retain their previous meaning. Eq. (24) shows that the importance of the doping uniformity N D increases with the square of the chamber thickness. A thinner chamber is less sensitive to the impurity concentration non-uniformity; however, the charge produced by a fast particle in a thinner chamber is smaller and the resolution may deteriorate due to the increase importance of the series noise of the amplifier (eq. 25). To decrease the electronic noise, we can incorporate the first transistor into the anode structure of the chamber [8]. Moreover, the decrease of the series noise due to the transistor action would improve the performance of the chamber at higher values of the drift velocity. High drift velocity chambers are required in many high multiplicity and (or) high rate experiments.
5. Conclusions A new charge transport mechanism in fully depleted semiconductors was described. The application of the new mechanism leads to several new detectors. The simplest device, which we have called the semiconductor drift chamber, a very unique device, was produced by planar technology at LBL. The test results confirmed the validity of the principles. The optimum processing of the signal from the chamber is described. The position resolution of the semiconductor drift chamber is a few ~tm. We wish to thank V. Radeka for the fruitful and stimulating discussions and in particular, for the kind hospitality given to E. Gatti in the Instrumentation Division of BNL. We are indebted to A. Longoni for help during the measurements. Discussions with G. Soncini and S. Solmi are acknowledged. And finally, we appreciate the interest and encouragement of F. Goulding in this work.
References [1] E. Gatti and P. Rehak, in Proc. 2nd Pisa Meeting on Advanced Detectors, Grosseto, Italy (1983) Nucl. Instr. and Meth. 225 (1984) 608. F. Cappasso, IEEE Trans Electron Devices ED-29 (1982) 1388. [2] We were guided by the process described in the followingpaper: J. Kemmer et al., IEEE Trans. Nucl. Sci. NS-29 (1982) 733. [3] J. Kemmer, these Proceedings (Semiconductor Detectors '83), p. 89. [4] F.N.H. Robinson, Noise and fluctuations in electronic devices and circuits (Clarendon, Oxford, 1974) p. 15.
E. Gatti et al. / Silicon drift chambers
[5] [6] [7] [8]
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V. Radeka, IEEE Trans. Nucl. Sci. NS-21 (1) (1974) 51. E. Gatti and V. Svelto, Nucl. Instr. and Meth. 39 (1966) 309. J. Llacer, IEEE Trans. Nucl. Sci. NS-28 (1981) 630. T.C. Madden and G.L. Miller, Proc. Gatlinburg Conf. on Semiconductor Nuclear Particle Detectors and Circuits, Nucl. Sci. Series, Rept No. 44, Publ. 1693 (Nat. Acad. Sci., Washington DC, 1969) p. 314.
Discussion Damerell: Regarding your proposed C C D option, I am not clear that you avoid the problems encountered in fabricating standard C C D on high resistivity silicon, namely resistivity drops and problems with the operation of the on-chip MOSFET, which need islands of low resistivity. In the case where these problems were solved on prototype devices (by Westinghouse), full charge collection was obtained over the full depletion depth ( - 200 ~m). Rehak: The question regards technology and I do not understand why the silicon resistivity drops during the fabrication process. If reproducible, this could be taken into account. The low resistivity islands are not required if one can avoid having MOSFET on the wafer. Hyams: Do you have any knowledge of the variation of drift velocity with doping concentration? Rehak: Yes, the electron scattering on impurities in very pure silicon is much less than the electron scattering on phonons. This means that we have to control the temperature very accurately. Drukier: Do you expect that temperature inhomogeneity inside the silicon chip can introduce jitter in the drift velocity and therefore affect the spatial resolution? Rehak: The silicon crystal is a good thermal conductor and the crystal heating by the energy deposit of the particles is very small. However, instabilities in the detector leakage current could produce temperature differences which are difficult to calibrate and which decrease the spatial resolution. Lutz: You discussed the influence of doping variations on the drift field and mentioned that the sensitivity to these variations is reduced by a factor of four if the wafer is half as thick. It seems to me that you can obtain the same effect by moving the potential valley out of the centre towards one electrode, by putting unequal voltages on the drift electrodes. Rehak: You are right, however, the improvement is slow. By moving the potential valley from the centre towards one side (e.g. to 1 / 4 depth) one improves by 25% only. The electron transport in this non-symmetric configuration requires two times finer lithography for the junction array on the rear side of the wafer. Parker: Could you reduce noise due to leakage current and obtained second coordinate information by subdividing the anode and using multichannel readout? (the binary code or grey code schemes use anodes, covering about half the length and so do not have a large leakage current reduction). Rehak: Yes, in fact we are designing a multiple anode readout, which still may have a large number of elements. The grey code scheme is one attempt to reduce the amount of electronics. Clearly, one does not have the benefit of leakage current reduction. Tore: 1) Could you quote performance of your detector in terms of spatial resolution (in percentage of the m a x i m u m covered length), counting rate and m a x i m u m coverable area? 2) One of the 3 noise sources which you mentioned (the leakage current) gave a dominant contribution to the time dispersion ( 3 - 4 times larger than the others). If this is the case, moderate cooling (Peltier-element) could improve the performance. Rehak: 1) Possible spatial resolution 1 - 0 . 1 % ; counting rate < 1 MHz. 2) Primary interest is to make simple apparatus and avoid cooling. By adjustment of design, the three noise sources could be equalized.
IV. E L E C T R O N I C S / R E A D O U T