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Noise of the induced signal pulses in semiconductor drift detectors
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
Noise of the induced signal pulses in semiconductor drift detectors Massimo Lazzaronia,*, Fabio E. Zocchib a
Dipartimento di Tecnologie dell’Informazione, Universita" degli Studi di Milano, Via Bramante, 65, Crema (CR) 26013, Italy b Media Lario S.r.l, Localita" Pascolo, Bosisio Parini (LC) 23842, Italy Received 10 October 2003; received in revised form 5 December 2003; accepted 9 December 2003
Abstract The noise associated with the signal current at the anode of a semiconductor drift detector is evaluated when the electrostatic induction giving rise to the signal is fully taken into account and the correct boundary condition at the anode for the electron density is considered. The consequent signal-to-noise ratio for both time and amplitude measurements is calculated as a function of the filter width and compared with previous results based on more simplified treatments. r 2003 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 85.30.z Keywords: Electrostatic induction; Detectors; Radiation detectors; Semiconductor drift detector; Signal-to-noise ratio
1. Introduction Since 1984 Gatti, Rehak and Walton have developed the theory of time resolution [1] for output signals from semiconductor drift detectors [2]. On the other hand when the output signal is approximately a Dirac current pulse, the theory of amplitude resolution is also well known. If the signal is supplied in a limited time interval Dt with unknown shape instead of being a pure Dirac pulse, then optimum filtering is achieved by a weighting function with a flat top region of duration Dt to avoid ballistic deficit. *Corresponding author. Tel.: +39-02-503-30073; fax: +3902-502-30010. E-mail addresses: [email protected] (M. Lazzaroni), [email protected] (F.E. Zocchi).
However, in semiconductor drift detectors excited by minimum ionizing particles or X-rays incident on the wafer, the width of the electron pulse at the anode is large enough to make the use of a simple flat top filter a non-optimum choice. Indeed in presence of parallel noise a better choice of the weighting function is possible if the shape of the signal and its noise statistics is taken into account [3]. In the early treatments, the statistics of input electrons was considered to be a Poisson distribution, i.e. a time-dependent white noise associated to the signal. In this case, if N is the total number of released electrons, the variance in the total number of collected electrons is again N [4]. Clearly this approach is oversimplified since it does not take into account that, if generation and recombination processes in the semiconductor can
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ARTICLE IN PRESS M. Lazzaroni, F.E. Zocchi / Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
be neglected, then the total number of collected electrons is fixed to N: This condition introduces a constraint that makes the electron statistical distribution sub-Poissonian. In their paper published in 1993, Gatti and Fusillo [3] developed a more rigorous statistical treatment of the shape fluctuations of the signal from a semiconductor drift detector, taking into consideration the rigid constraint of a fixed number of electrons collected at the anode and evaluating the consequences of this constraint to both time and amplitude measurements. The main result of their paper was that the covariance of the noise associated to the signal current in semiconductor detectors is reduced from its shot Poisson noise value due to the constraint on the total number of collected electrons. Indeed, they showed that the noise current covariance is given by [3] CI ðt1 ; t2 Þ ¼ dðt1 t2 ÞIðt1 Þ
Iðt1 ÞIðt2 Þ N
ð1Þ
provided the N electrons are considered statistically independent. The first term in Eq. (1) is the Poisson contribution, whereas, the second is the above-mentioned reduction that takes into account the constraint imposed by the fixed electron number. Clearly, the term Iðt1 ÞIðt2 Þ=N introduces a negative correlation between signal values at different times. Indeed, if the signal amplitude at some instant is greater than its mean value, in some other instant the amplitude must be smaller in order to comply with the constraint of a fixed pulse area. In deriving their result (1), Gatti and Fusillo did not take into account any information about the actual shape of the electron current pulse. On one side this gives to their treatment a wide generality but, on the other side, it makes their statistical model insensitive to the physics of the current pulse formation. In particular the current in a detector is not due to a pure charge collection at the anode but to the electrostatic induction of the drifting and diffusing electrons approaching the anode. When the width of the electron density distribution in the semiconductor is comparable or larger than the length of the induction region close to the anode, it is expected that the induction
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process introduces a strong correlation between signal values at different times. In this paper the statistics of the current signal at the anode is studied taking into account the actual inductive process of signal formation. As in Ref. [3] we assume that the total number of electrons is fixed. In addition the correct boundary condition for the electron density at the anode is considered. Indeed the presence of the anode forces the electron density to vanish, thus modifying the electron distribution near the anode. Since this is also the induction region, the anode boundary condition is essential to obtain meaningful results. The consequences of considering the effects of the induction process on the statistics of the current signal are also studied for both time and amplitude measurements. In particular, the expected signal-to-noise ratio after a simple Gaussian filter is calculated as a function of the filter width and compared with both the Poisson prediction and the results following the Gatti– Fusillo approach.
2. Noise of the signal current without induction In Ref. [3] the derivation of the result represented by Eq. (1) is rather lengthy. To better understand the assumption behind it, it is here derived again in a more straightforward way. Suppose that the anode current is due to a pure charge collection process. We consider just one electron and we assume that it is collected at the anode at the instant t ¼ t with a probability density function pðtÞ: Thus the current is given by dðt tÞ and its average value is Z þN IðtÞ dðt tÞpðtÞ dt ¼ pðtÞ: ð2Þ N
The autocorrelation function of the current is given by Z þN Z þN RI ðt1 ; t2 Þ dt1 dðt1 t1 Þdðt2 t2 Þ N
N
pðt1 ; t2 Þ dt2 ¼ pðt1 ; t2 Þ
ð3Þ
where pðt1 ; t2 Þ is the joint probability that t ¼ t1 and t ¼ t2 : Obviously, the joint probability
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pðt1 ; t2 Þ is zero if t1 at2 and is proportional to pðt1 Þ if t1 ¼ t2 : Thus, to assure normalization of pðt1 ; t2 Þ; we must have pðt1 ; t2 Þ ¼ dðt1 t2 Þpðt1 Þ: Finally, taking into account Eq. (2) the current covariance is given by CI ðt1 ; t2 Þ RI ðt1 ; t2 Þ Iðt1 ÞIðt2 Þ ¼ dðt1 t2 ÞIðt1 Þ Iðt1 ÞIðt2 Þ:
ð4Þ
The result for N electrons follows immediately in the assumption that the N electrons are independent. Indeed, under this hypothesis both the current and the autocorrelation and covariance functions are multiplied by N so that Eq. (1) is recovered.
3. Noise of the signal current in presence of induction As mentioned in the Introduction, the above treatment is very general but oversimplified. Indeed, the current in a detector is not due to a pure charge collection process but to the electrostatic induction of the drifting and diffusing electrons approaching the anode. The induction introduces a correlation between the elementary currents induced by the electron at two different positions and this in turn correlates the signal values at different times, thus modifying the prediction of Eq. (1). The covariance of the current signal is calculated in Section 3.1 when the electrostatic induction is considered as the signal formation mechanism. Then in Section 3.2 it is shown that the Gatti–Fusillo result (1) can be recovered from the general result in the limit of vanishing induction length. Finally, in Section 3.3 the expression of the current covariance is simplified in a form more suitable to numeric calculations.
to junctions at the surface of the structure. Fig. 1a depicts a simplified scheme of a semiconductor drift detector. The array of pþ strips on both sides of the wafer are reverse biased with respect to the bulk up to the complete depletion of the device leading to a parabolic potential channel. Applying a reverse bias decreasing as the strips get closer to the nþ anode(s) a drift field conveying the electrons towards the anodes is obtained. These devices are used as detectors when majority carriers are created by a ionizing particle. The interaction position is determined by the measurement of the drift time of the carriers whereas the total energy released within the detector is measured by the charge collected at the anode of the device. We now calculate both the average value and the covariance of the current induced at the anode of a semiconductor drift detector. To this end a specific model is required for the device. With reference to Fig. 1b, the following assumptions [5] will be made: (1) A one-dimensional scheme is assumed in which a delta-like charge of N electrons is generated at x0 > 0 at t ¼ 0: Then the n+
p+
p+
p+
Y W X p+
p+
p+
p+
p+ Z
(a)
3.1. Current covariance in presence of induction p+
The approach described in this Section is of general validity but it is here applied to semiconductor drift detectors [2]. Such detectors are fully depleted high resistivity silicon devices in which a static electric field controls the drift of the majority carriers for long distances in a buried channel parallel to the surface. The static electric field profile is produced by proper voltages applied
p+
p+
p+
p+
n+ Anode
x0 p+
p+
p+
p+
Y
p+ W
X
p+
(b) Fig. 1. Simplified pictorial view of a semiconductor drift detector: (a) conceptual diagram and (b) geometry of the simplified one-dimensional scheme used in the paper.
ARTICLE IN PRESS M. Lazzaroni, F.E. Zocchi / Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
electrons diffuse and drift toward the anode at x ¼ 0 owing to a time-independent drifting field EðxÞ: Due to the parabolic potential channel in the bulk of the device, the electron trajectories starting at different points along the thickness of the detector at a given x ¼ x0 are isochronous and this justify the onedimensional analysis. (2) The point x0 at which the electron–hole charge is generated is assumed to be far from the induction area and the hole charged is assumed to be collected by nearby field electrodes [2]. Thus the holes do not contribute to the induced signal at the anode. (3) The presence of the anode modifies the charge distribution in the detector by forcing the charge density to be zero at the anode [5]. (4) The electrons are assumed to be independent from each other. The Coulomb interaction, relevant at very high charge density only, is neglected and the time required for the separation of the initial electron–hole plasma is also neglected [6]. Under the drift-diffusion approximation the electron density nðx; tÞ satisfies the classical continuity equation [7] @nðx; tÞ @Jðx; tÞ ¼ @t @x @½EðxÞnðx; tÞ @2 nðx; tÞ þD ¼m ; @x @x2
ð5Þ
where Jðx; tÞ ¼ mEðxÞnðx; tÞ D
@nðx; tÞ @x
ð6Þ
is the current density, m is the mobility and D the diffusion coefficient both independent from x: In Eqs. (5) and (6) the first term is the drift component whereas the second term represents the diffusion component of the total current. Eq. (5) must be solved with the initial condition nðx; 0Þ ¼ Ndðx x0 Þ and the boundary condition nð0; tÞ ¼ 0 due to the anode. For a drift detector the boundary condition limx-þN nðx; tÞ ¼ 0 must be also assumed. However, if just one electron is considered, then nðx; tÞ looses its macroscopic meaning of average
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electron density. On the other hand, from a microscopic point of view nðx; tÞdx can be interpreted as the probability to find the electron at instant t between x and x þ dx: Such interpretation is also supported by a microscopic statistical analysis of diffusion [8] that, like the Brownian movement, is normally described by Wiener’s stochastic process [9]. According to this model the position of the electron is given by Wiener’s process sðtÞ such that, for any pair of instants t1 and t2 ; the position increment sðt2 Þ sðt1 Þ is independent from sðt1 Þ and is Gaussian distributed with zero average and variance 2Dðt2 t1 Þ: Taking into account that at t ¼ 0 the electron is at position sðt ¼ 0Þ ¼ x0 with certainty, it follows that the probability density that sðtÞ ¼ x is given by 2 1 psðtÞ ðx j sð0Þ ¼ x0 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 Þ =4Dt 4pDt
ð7Þ
which is also the solution of Eq. (5) when the electric field is zero and the anode is not present. Thus Wiener’s process description of diffusion gives a probability density function in agreement with the macroscopic result for infinite motion, even if this is just no more than a pure definition. When the presence of the anode at x ¼ 0 is taken into account, the above stochastic description must be modified since the random process stops as soon as the electron hits the anode, that is whenever sðtÞp0: Thus probability (7) must be substituted by the joint probability that sðtÞ ¼ x and s0 ¼ sðt0 ÞX0 for 0pt0 pt: The latter condition means that the process has never hit the anode before t: It is a standard result of the theory of Wiener’s process that such probability can be obtained by the Reflection Principle for Brownian motion [9] that gives, for xX0 and t > 0; psðtÞ ðx; s0 X0 j sð0Þ ¼ x0 Þ 2 2 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 Þ =4Dt pffiffiffiffiffiffiffiffiffiffiffi eðxþx0 Þ =4Dt 4pDt 4pDt ð8Þ which is exactly the solution of Eq. (5) when the drifting electric field is zero and the anode is present. In the general case the solution of Eq. (5) can be viewed as the extension of Eq. (8) to the
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case of a non-vanishing arbitrary drifting electric field even if its derivation from the basic assumptions of Wiener’s process does not seem straightforward. With the above microscopic interpretation of the function nðx; tÞ for one electron, we can now calculate the average induced current and its covariance. This is done by means of Ramo’s theorem [10,11], according to which the current induced at the anode by a current density Jðx; tÞ in the detector is given by Z N * IðtÞ ¼ EðxÞJðx; tÞ dx ð9Þ 0
* * where EðxÞ ¼ dVðxÞ=dx is the on-axis weighting field obtained by solving Laplace’s equation for * the weighting potential VðxÞ inside the detector with the boundary condition that the voltage at the anode is 1 and the voltage at all other electrodes is 0. Substituting expression (6) for Jðx; tÞ in Eq. (9), integrating by part the second term and taking * into account that EðN; tÞ ¼ 0 and nð0; tÞ ¼ 0; Eq. (9) can be written as Z N * IðtÞ ¼ ½DE* 0 ðxÞ mEðxÞEðxÞ nðx; tÞ dx 0 Z N ¼ IðxÞnðx; tÞ dx; ð10Þ 0
* where IðxÞ DE* 0 ðxÞ mEðxÞEðxÞ is a known function of x: Following the probabilistic interpretation of nðx; tÞ; the quantity IðxÞ in Eq. (10) can be interpreted as the elementary current induced at the anode when the electron is between x and x þ dx at the instant t with probability nðx; tÞ dx: If N independent electrons are generated at x0 ; expression (10) of the current must be simply multiplied by N: To determine the autocorrelation function of the induced current, an expression for the joint probability psðt1 Þsðt2 Þ ðx1 ; x2 j sð0Þ ¼ x0 Þ of finding the electron between x1 and x1 þ dx1 at the instant t1 and between x2 and x2 þ dx2 at the instant t2 is required. This can be done taking into account that, according to Wiener’s model, successive increments of the electron positions are independent from the previous ones. Assuming t2 > t1 and
conditioning to the event sðt1 Þ ¼ x1 ; we find psðt1 Þsðt2 Þ ðx1 ; x2 j sð0Þ ¼ x0 Þ ¼ psðt2 Þ ðx2 j sðt1 Þ ¼ x1 Þpsðt1 Þ ðx1 j sð0Þ ¼ x0 Þ ¼ nx1 ðx2 ; t2 t1 Þnx0 ðx1 ; t1 Þ;
ð11Þ
where the index in nðx; tÞ denotes the point at which the electron is ‘‘generated’’. Thus the autocorrelation function of the current is Z NZ N RI ðt1 ; t2 Þ ¼ Iðx1 ÞIðx2 Þnx0 ðx1 ; t1 Þ 0
0
nx1 ðx2 ; t2 t1 Þ dx2 dx1
ð12Þ
and the covariance is CI ðt1 ; t2 Þ RI ðt1 ; t2 Þ Iðt1 ÞIðt2 Þ:
ð13Þ
For t2 ot1 it is enough to exchange t1 and t2 in the above relations. In the following this exchange will be tacitly assumed. Finally, for N independent electrons generated at x0 ; the autocorrelation must be multiplied by N so that the covariance becomes Iðt1 ÞIðt2 Þ : ð14Þ CI ðt1 ; t2 Þ ¼ RI ðt1 ; t2 Þ N 3.2. Recovering of the covariance function without induction Let us first consider the case when induction is * switched off. In this case EðxÞ ¼ dðxÞ and thus 0 IðxÞ ¼ Dd ðxÞ þ mEðxÞdðxÞ: Since nð0; tÞ ¼ 0; after integration by parts the current is, as expected, Z N IðtÞ ¼ D d0 ðxÞnðx; tÞ dx 0 @nðx; tÞ ¼D ¼ Jð0; tÞ ð15Þ @x x¼0 that is just the current due to a pure collection process. Substituting the expression for Iðx1 Þ in Eq. (12), the autocorrelation function after integration by parts becomes, for t2 > t1 ; @nx0 ðx1 ; t1 Þ RI ðt1 ; t2 Þ ¼ D @x1 x1 ¼0 Z N Iðx2 Þnx1 ¼0 ðx2 ; t2 t1 Þ dx2 : ð16Þ 0
The factor outside the integral sign is Iðt1 Þ whereas, the integral is the current due to an
ARTICLE IN PRESS M. Lazzaroni, F.E. Zocchi / Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
electron generated at the anode at t2 ¼ t1 and immediately collected and it is thus equal to dðt1 t2 Þ: A similar result applies also to the case t2 ot1 and consequently relation (4) is recovered. 3.3. Approximation of the covariance function in drift detectors Assuming, as it happens in drift detectors, that the induction region has a spatial extent d near the anode much less than the co-ordinate x0 ; then the autocorrelation function can be approximated in order to have a simplified expression more suitable for comparisons and numerical evaluations. For a uniform electric field EðxÞ E and assuming d5x0 ; the autocorrelation function (12) is different from zero only for t1 Ex0 =mE and jt2 t1 j Ex1 =mEpd=mE: It is concluded that jt2 t1 j5t1 and consequently that the width of the distribution nx0 ðx1 ; t1 Þ is much larger than the width of nx1 ðx2 ; t2 t1 Þ: The latter in particular is approxipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mately equal to 2 Djt2 t1 jp2 Dd=mE : By exploiting Einsten’s relation D ¼ kTm=q; it is shown that even at room temperature and at low drifting field of 100 V=cm; the width of pffiffiffi nx1 ðx2 ; t2 t1 Þ is about 0:031 d cm and thus it is also much less than d: Consequently, nx1 ðx2 ; t2 t1 Þ can be approximated in Eq. (12) by the delta distribution dðx2 x1 þ ðt2 t1 ÞmEÞ: In this case autocorrelation (12) for t2 > t1 becomes Z N Iðx1 Þ RI ðt1 ; t2 ÞE ðt2 t1 ÞmE
Iðx1 ðt2 t1 ÞmEÞnx0 ðx1 ; t1 Þ dx1 : ð17Þ
4. Signal filtering In all applications, the signal current at the detector anode is processed by a suitable filtering. In this section the theory of the signal filtering for both amplitude and time measurement is briefly resumed for further reference. If hðtÞ is the impulse response function of the filter, the output signal uðtÞ is the convolution of hðtÞ with the input
current, Z uðtÞ ¼
131
N
IðtÞhðt tÞ dt
ð18Þ
N
whereas the output covariance is [3] Z NZ N Cu ðt1 ; t2 Þ ¼ CI ðt1 ; t2 Þ N
N
hðt1 t1 Þhðt2 t2 Þ dt1 dt2 Z NZ N ¼ RI ðt1 ; t2 Þhðt1 t1 Þ N
N
hðt2 t2 Þ dt1 dt2 uðt1 Þuðt2 Þ:
ð19Þ
For amplitude measurement, the filter impulse response is chosen to be a unipolar function of time and the output signal is measured at its maximum at the instant tm : The squared noise-tosignal ratio is thus e2 ¼ Cðtm ; tm Þ=u2 ðtm Þ: For time measurement, the filter impulse response is an odd function of time and the measured time is at the instant tz at which uðtÞ crosses zero. The variance of the time noise is given by [1] t2 ¼ 0 Cðtz ; tz Þ=u 2 ðtz Þ; where u0 ðtz Þ is the slope of the output signal at the measurement time.
5. Numerical results in silicon drift detectors The results of the previous sections are now applied to a specific example in order to better evaluate the effect of induction on the measured noise-to-signal ratio and to compare the results with both the Poisson and the Gatti–Fusillo approach. The solution of Eq. (5) in a drift detector for a uniform drifting field E and in presence of an anode at x ¼ 0 is given, for xX0 and tX0; by 2 1 nx0 ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 þmEtÞ =4Dt 4pDt 2 1 pffiffiffiffiffiffiffiffiffiffiffi emEx0 =D eðxþx0 þmEtÞ =4Dt ð20Þ 4pDt when a unit delta-like charge distribution is generated at x ¼ x0 at t ¼ 0: The first term is the drift diffusion in the absence of the anode. The second is the correction due to the anode. Relation (20) is indeed the required solution since both terms are separately solutions of the linear
ARTICLE IN PRESS M. Lazzaroni, F.E. Zocchi / Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133 6 5 4
6
equation (5). In addition both the initial and the boundary conditions are satisfied at t ¼ 0 and x ¼ 0; respectively. * For drift detectors the weighting function VðxÞ along the detector axis can be approximated by x=d * VðxÞEe for xX0; where d ¼ w=p; w being the detector thickness [5]. The above expression for * VðxÞ is the first leading term of a series expansion. Thus for drift detectors at constant drifting field IðxÞ ¼ g ex=d ; where g ¼ mE=d D=d 2 : The induced current (10) and the autocorrelation function (17), for t2 > t1 ; become Z N ex=d nx0 ðx; tÞ dx IðtÞ ¼ g
Current [10 e−/s]
132
3 2 1 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time [µs]
Fig. 2. Induced current at the detector anode.
0
g 2 ¼ eðDtdx0 þdmEtÞ=d 2" ! # 2Dt dx0 þ dmEt pffiffiffiffiffiffi erf 1 2d Dt g 2 eðDtþdx0 þdmEtÞ=d þmEx0 =D 2 " ! # 2Dt þ dx0 þ dmEt pffiffiffiffiffiffi erf 1 2d Dt
ð21Þ
RI ðt1 ; t2 ÞE g2 eðt2 t1 ÞmE=d Z N e2x1 =d nx0 ðx1 ; t1 Þ dx1 ðt2 t1 ÞmE 2
¼
g ð4Dt1 2dx0 þdmEðt2 þt1 ÞÞ=d 2 e 2" ! # 4Dt1 dx0 þ dmEt2 Þ pffiffiffiffiffiffiffiffi erf 1 2d Dt1 g2 2 eð4Dt1 þ2dx0 þdmEðt2 þt1 ÞÞ=d þmEx0 =D 2 " ! # 4Dt1 þ dx0 þ dmEt2 pffiffiffiffiffiffiffiffi erf 1 : 2d Dt1 ð22Þ
For the numerical calculation it has been assumed that E ¼ 100 V=cm; d ¼ 100 mm; x0 ¼ 0:1 cm [5] and m ¼ 1200 cm2 =V s for silicon at room temperature [7]. The diffusion coefficient D is related to the mobility m by Einstein’s relation D ¼ kTm=q at room temperature, T ¼ 300 K: Fig. 2 shows the current induced at the anode under these conditions.
The numerical filtering of the current at the output of the front-end electronics for the amplitude measurement has been done with a noncausal Gaussian filter with impulse response hðtÞ gðtÞ ¼ expðt2 =2s2 Þ: For time measurement the filter response function has been chosen as the time derivative of the above, hðtÞ g0 ðtÞ: In both cases different values of the filter time width s have been considered in the range between 20 ns and 1 ms: As a reference, the time width of the current signal at the anode is about 159 ns rms: Fig. 3 shows the squared noise-to-signal ratio, normalized to one electron, of the amplitude measurement as a function of the filter width. The results of the approach described in this paper that take into account the induction process are compared with both the Poisson and the Gatti–Fusillo treatments. A reduction of the squared noise-to-signal radio is evident up to a filter width s of about 0:8 ms: For shorter values of s the effect of the induction process is to introduce a higher negative correlation in the noise associated with the signal current, thus improving the performance of the measuring process. Apart from very low values of the time width of the filter, the noise-to-signal ratio is always less than the contribution associated with the fluctuation of the number of electron–hole pairs generated by the primary particle and described by Fano’s factor [12], equal to about 0.11 per electron in silicon [13]. A factor three improvement is also evident in Fig. 4 showing the noise time variance, normalized to one electron, of the time measurement as a function of the filter width. For time measurement,
ARTICLE IN PRESS
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1
Poisson 0.1
0.01
0.001
0.0001 0.0
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1
10
Time variance, τ 2 N [µs ]
Squared Noise to Signal Ratio, ε 2Ν
M. Lazzaroni, F.E. Zocchi / Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
Gatti Induction
0.1
Poisson - Gatti 0.01
Induction 0.2
0.4
0.6
0.8
1.0
Filter Width, σ [µs] Fig. 3. Squared noise-to-signal ratio, normalized to one electron, of the amplitude measurement as a function of the filter width when the Poisson, the Gatti–Fusillo and the full induction approach is applied.
the Poisson and the Gatti–Fusillo approaches give the same results [3] since at measurement time tz the filter output uðtz Þ in Eq. (19) is zero.
6. Conclusion The analysis of the statistics of the signal at the anode of a semiconductor drift detectors, when the electrostatic induction is fully taken into account, has shown that the negative correlation introduced between the signal values at different time instants is large enough for improving the signal-to-noise ratio with respect to the previous published estimation, in silicon at room temperature and for typical applications. It is expected that the contribution of induction in the statistics of signals from semiconductor particle detectors should be relevant whenever the width of the carrier density distribution is comparable or larger than the extension of the p induction This ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiregion. ffi condition is verified if 2 kTx0 =qE Xd which requires a low drifting field and a long drifting path.
0.001 0.0
0.2
0.4
0.6
0.8
1.0
Filter Width, σ [µs] Fig. 4. Noise time variance, normalized to one electron, of the time measurement as a function of the filter width when the Poisson/Gatti–Fusillo and the full induction approach is applied.
References [1] E. Gatti, P. Rehak, J.T. Walton, Nucl. Instr. and Meth. 226 (1984) 128. [2] E. Gatti, P. Rehak, Nucl. Instr. and Meth. 225 (1984) 608. [3] E. Gatti, G. Fusillo, Nucl. Instr. and Meth. A 236 (1993) 279. [4] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Company, New York, 1965. [5] E. Gatti, M. Sampietro, F.E. Zocchi, J. Appl. Phys. 74 (1993) 5940. [6] E. Gatti, A. Longoni, P. Rehak, M. Sampietro, Nucl. Instr. and Meth. A 253 (1987) 393. [7] S.M. Sze, Physics of Semiconductor Devices, 2nd Edition, Wiley, New York, 1981. [8] A. Einstein, Ann. D. Phys. 17 (1905) 549. [9] I. Karatsas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd Edition, Springer, New York, 1997. [10] S. Ramo, Proc. I.R.E. 27 (1939) 584. [11] C.K. Jen, Proc. I.R.E. 29 (1941) 345. [12] U. Fano, Phys. Rev. 72 (1947) 26. [13] H.J. Langmann, O. Meyer, Nucl. Instr. and Meth. 30 (1964) 209.
Nuclear Instruments and Methods in Physics Research A 523 (2004) 126–133
Noise of the induced signal pulses in semiconductor drift detectors Massimo Lazzaronia,*, Fabio E. Zocchib a
Dipartimento di Tecnologie dell’Informazione, Universita" degli Studi di Milano, Via Bramante, 65, Crema (CR) 26013, Italy b Media Lario S.r.l, Localita" Pascolo, Bosisio Parini (LC) 23842, Italy Received 10 October 2003; received in revised form 5 December 2003; accepted 9 December 2003
Abstract The noise associated with the signal current at the anode of a semiconductor drift detector is evaluated when the electrostatic induction giving rise to the signal is fully taken into account and the correct boundary condition at the anode for the electron density is considered. The consequent signal-to-noise ratio for both time and amplitude measurements is calculated as a function of the filter width and compared with previous results based on more simplified treatments. r 2003 Elsevier B.V. All rights reserved. PACS: 29.40.Wk; 85.30.z Keywords: Electrostatic induction; Detectors; Radiation detectors; Semiconductor drift detector; Signal-to-noise ratio
1. Introduction Since 1984 Gatti, Rehak and Walton have developed the theory of time resolution [1] for output signals from semiconductor drift detectors [2]. On the other hand when the output signal is approximately a Dirac current pulse, the theory of amplitude resolution is also well known. If the signal is supplied in a limited time interval Dt with unknown shape instead of being a pure Dirac pulse, then optimum filtering is achieved by a weighting function with a flat top region of duration Dt to avoid ballistic deficit. *Corresponding author. Tel.: +39-02-503-30073; fax: +3902-502-30010. E-mail addresses: [email protected] (M. Lazzaroni), [email protected] (F.E. Zocchi).
However, in semiconductor drift detectors excited by minimum ionizing particles or X-rays incident on the wafer, the width of the electron pulse at the anode is large enough to make the use of a simple flat top filter a non-optimum choice. Indeed in presence of parallel noise a better choice of the weighting function is possible if the shape of the signal and its noise statistics is taken into account [3]. In the early treatments, the statistics of input electrons was considered to be a Poisson distribution, i.e. a time-dependent white noise associated to the signal. In this case, if N is the total number of released electrons, the variance in the total number of collected electrons is again N [4]. Clearly this approach is oversimplified since it does not take into account that, if generation and recombination processes in the semiconductor can
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be neglected, then the total number of collected electrons is fixed to N: This condition introduces a constraint that makes the electron statistical distribution sub-Poissonian. In their paper published in 1993, Gatti and Fusillo [3] developed a more rigorous statistical treatment of the shape fluctuations of the signal from a semiconductor drift detector, taking into consideration the rigid constraint of a fixed number of electrons collected at the anode and evaluating the consequences of this constraint to both time and amplitude measurements. The main result of their paper was that the covariance of the noise associated to the signal current in semiconductor detectors is reduced from its shot Poisson noise value due to the constraint on the total number of collected electrons. Indeed, they showed that the noise current covariance is given by [3] CI ðt1 ; t2 Þ ¼ dðt1 t2 ÞIðt1 Þ
Iðt1 ÞIðt2 Þ N
ð1Þ
provided the N electrons are considered statistically independent. The first term in Eq. (1) is the Poisson contribution, whereas, the second is the above-mentioned reduction that takes into account the constraint imposed by the fixed electron number. Clearly, the term Iðt1 ÞIðt2 Þ=N introduces a negative correlation between signal values at different times. Indeed, if the signal amplitude at some instant is greater than its mean value, in some other instant the amplitude must be smaller in order to comply with the constraint of a fixed pulse area. In deriving their result (1), Gatti and Fusillo did not take into account any information about the actual shape of the electron current pulse. On one side this gives to their treatment a wide generality but, on the other side, it makes their statistical model insensitive to the physics of the current pulse formation. In particular the current in a detector is not due to a pure charge collection at the anode but to the electrostatic induction of the drifting and diffusing electrons approaching the anode. When the width of the electron density distribution in the semiconductor is comparable or larger than the length of the induction region close to the anode, it is expected that the induction
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process introduces a strong correlation between signal values at different times. In this paper the statistics of the current signal at the anode is studied taking into account the actual inductive process of signal formation. As in Ref. [3] we assume that the total number of electrons is fixed. In addition the correct boundary condition for the electron density at the anode is considered. Indeed the presence of the anode forces the electron density to vanish, thus modifying the electron distribution near the anode. Since this is also the induction region, the anode boundary condition is essential to obtain meaningful results. The consequences of considering the effects of the induction process on the statistics of the current signal are also studied for both time and amplitude measurements. In particular, the expected signal-to-noise ratio after a simple Gaussian filter is calculated as a function of the filter width and compared with both the Poisson prediction and the results following the Gatti– Fusillo approach.
2. Noise of the signal current without induction In Ref. [3] the derivation of the result represented by Eq. (1) is rather lengthy. To better understand the assumption behind it, it is here derived again in a more straightforward way. Suppose that the anode current is due to a pure charge collection process. We consider just one electron and we assume that it is collected at the anode at the instant t ¼ t with a probability density function pðtÞ: Thus the current is given by dðt tÞ and its average value is Z þN IðtÞ dðt tÞpðtÞ dt ¼ pðtÞ: ð2Þ N
The autocorrelation function of the current is given by Z þN Z þN RI ðt1 ; t2 Þ dt1 dðt1 t1 Þdðt2 t2 Þ N
N
pðt1 ; t2 Þ dt2 ¼ pðt1 ; t2 Þ
ð3Þ
where pðt1 ; t2 Þ is the joint probability that t ¼ t1 and t ¼ t2 : Obviously, the joint probability
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pðt1 ; t2 Þ is zero if t1 at2 and is proportional to pðt1 Þ if t1 ¼ t2 : Thus, to assure normalization of pðt1 ; t2 Þ; we must have pðt1 ; t2 Þ ¼ dðt1 t2 Þpðt1 Þ: Finally, taking into account Eq. (2) the current covariance is given by CI ðt1 ; t2 Þ RI ðt1 ; t2 Þ Iðt1 ÞIðt2 Þ ¼ dðt1 t2 ÞIðt1 Þ Iðt1 ÞIðt2 Þ:
ð4Þ
The result for N electrons follows immediately in the assumption that the N electrons are independent. Indeed, under this hypothesis both the current and the autocorrelation and covariance functions are multiplied by N so that Eq. (1) is recovered.
3. Noise of the signal current in presence of induction As mentioned in the Introduction, the above treatment is very general but oversimplified. Indeed, the current in a detector is not due to a pure charge collection process but to the electrostatic induction of the drifting and diffusing electrons approaching the anode. The induction introduces a correlation between the elementary currents induced by the electron at two different positions and this in turn correlates the signal values at different times, thus modifying the prediction of Eq. (1). The covariance of the current signal is calculated in Section 3.1 when the electrostatic induction is considered as the signal formation mechanism. Then in Section 3.2 it is shown that the Gatti–Fusillo result (1) can be recovered from the general result in the limit of vanishing induction length. Finally, in Section 3.3 the expression of the current covariance is simplified in a form more suitable to numeric calculations.
to junctions at the surface of the structure. Fig. 1a depicts a simplified scheme of a semiconductor drift detector. The array of pþ strips on both sides of the wafer are reverse biased with respect to the bulk up to the complete depletion of the device leading to a parabolic potential channel. Applying a reverse bias decreasing as the strips get closer to the nþ anode(s) a drift field conveying the electrons towards the anodes is obtained. These devices are used as detectors when majority carriers are created by a ionizing particle. The interaction position is determined by the measurement of the drift time of the carriers whereas the total energy released within the detector is measured by the charge collected at the anode of the device. We now calculate both the average value and the covariance of the current induced at the anode of a semiconductor drift detector. To this end a specific model is required for the device. With reference to Fig. 1b, the following assumptions [5] will be made: (1) A one-dimensional scheme is assumed in which a delta-like charge of N electrons is generated at x0 > 0 at t ¼ 0: Then the n+
p+
p+
p+
Y W X p+
p+
p+
p+
p+ Z
(a)
3.1. Current covariance in presence of induction p+
The approach described in this Section is of general validity but it is here applied to semiconductor drift detectors [2]. Such detectors are fully depleted high resistivity silicon devices in which a static electric field controls the drift of the majority carriers for long distances in a buried channel parallel to the surface. The static electric field profile is produced by proper voltages applied
p+
p+
p+
p+
n+ Anode
x0 p+
p+
p+
p+
Y
p+ W
X
p+
(b) Fig. 1. Simplified pictorial view of a semiconductor drift detector: (a) conceptual diagram and (b) geometry of the simplified one-dimensional scheme used in the paper.
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electrons diffuse and drift toward the anode at x ¼ 0 owing to a time-independent drifting field EðxÞ: Due to the parabolic potential channel in the bulk of the device, the electron trajectories starting at different points along the thickness of the detector at a given x ¼ x0 are isochronous and this justify the onedimensional analysis. (2) The point x0 at which the electron–hole charge is generated is assumed to be far from the induction area and the hole charged is assumed to be collected by nearby field electrodes [2]. Thus the holes do not contribute to the induced signal at the anode. (3) The presence of the anode modifies the charge distribution in the detector by forcing the charge density to be zero at the anode [5]. (4) The electrons are assumed to be independent from each other. The Coulomb interaction, relevant at very high charge density only, is neglected and the time required for the separation of the initial electron–hole plasma is also neglected [6]. Under the drift-diffusion approximation the electron density nðx; tÞ satisfies the classical continuity equation [7] @nðx; tÞ @Jðx; tÞ ¼ @t @x @½EðxÞnðx; tÞ @2 nðx; tÞ þD ¼m ; @x @x2
ð5Þ
where Jðx; tÞ ¼ mEðxÞnðx; tÞ D
@nðx; tÞ @x
ð6Þ
is the current density, m is the mobility and D the diffusion coefficient both independent from x: In Eqs. (5) and (6) the first term is the drift component whereas the second term represents the diffusion component of the total current. Eq. (5) must be solved with the initial condition nðx; 0Þ ¼ Ndðx x0 Þ and the boundary condition nð0; tÞ ¼ 0 due to the anode. For a drift detector the boundary condition limx-þN nðx; tÞ ¼ 0 must be also assumed. However, if just one electron is considered, then nðx; tÞ looses its macroscopic meaning of average
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electron density. On the other hand, from a microscopic point of view nðx; tÞdx can be interpreted as the probability to find the electron at instant t between x and x þ dx: Such interpretation is also supported by a microscopic statistical analysis of diffusion [8] that, like the Brownian movement, is normally described by Wiener’s stochastic process [9]. According to this model the position of the electron is given by Wiener’s process sðtÞ such that, for any pair of instants t1 and t2 ; the position increment sðt2 Þ sðt1 Þ is independent from sðt1 Þ and is Gaussian distributed with zero average and variance 2Dðt2 t1 Þ: Taking into account that at t ¼ 0 the electron is at position sðt ¼ 0Þ ¼ x0 with certainty, it follows that the probability density that sðtÞ ¼ x is given by 2 1 psðtÞ ðx j sð0Þ ¼ x0 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 Þ =4Dt 4pDt
ð7Þ
which is also the solution of Eq. (5) when the electric field is zero and the anode is not present. Thus Wiener’s process description of diffusion gives a probability density function in agreement with the macroscopic result for infinite motion, even if this is just no more than a pure definition. When the presence of the anode at x ¼ 0 is taken into account, the above stochastic description must be modified since the random process stops as soon as the electron hits the anode, that is whenever sðtÞp0: Thus probability (7) must be substituted by the joint probability that sðtÞ ¼ x and s0 ¼ sðt0 ÞX0 for 0pt0 pt: The latter condition means that the process has never hit the anode before t: It is a standard result of the theory of Wiener’s process that such probability can be obtained by the Reflection Principle for Brownian motion [9] that gives, for xX0 and t > 0; psðtÞ ðx; s0 X0 j sð0Þ ¼ x0 Þ 2 2 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 Þ =4Dt pffiffiffiffiffiffiffiffiffiffiffi eðxþx0 Þ =4Dt 4pDt 4pDt ð8Þ which is exactly the solution of Eq. (5) when the drifting electric field is zero and the anode is present. In the general case the solution of Eq. (5) can be viewed as the extension of Eq. (8) to the
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case of a non-vanishing arbitrary drifting electric field even if its derivation from the basic assumptions of Wiener’s process does not seem straightforward. With the above microscopic interpretation of the function nðx; tÞ for one electron, we can now calculate the average induced current and its covariance. This is done by means of Ramo’s theorem [10,11], according to which the current induced at the anode by a current density Jðx; tÞ in the detector is given by Z N * IðtÞ ¼ EðxÞJðx; tÞ dx ð9Þ 0
* * where EðxÞ ¼ dVðxÞ=dx is the on-axis weighting field obtained by solving Laplace’s equation for * the weighting potential VðxÞ inside the detector with the boundary condition that the voltage at the anode is 1 and the voltage at all other electrodes is 0. Substituting expression (6) for Jðx; tÞ in Eq. (9), integrating by part the second term and taking * into account that EðN; tÞ ¼ 0 and nð0; tÞ ¼ 0; Eq. (9) can be written as Z N * IðtÞ ¼ ½DE* 0 ðxÞ mEðxÞEðxÞ nðx; tÞ dx 0 Z N ¼ IðxÞnðx; tÞ dx; ð10Þ 0
* where IðxÞ DE* 0 ðxÞ mEðxÞEðxÞ is a known function of x: Following the probabilistic interpretation of nðx; tÞ; the quantity IðxÞ in Eq. (10) can be interpreted as the elementary current induced at the anode when the electron is between x and x þ dx at the instant t with probability nðx; tÞ dx: If N independent electrons are generated at x0 ; expression (10) of the current must be simply multiplied by N: To determine the autocorrelation function of the induced current, an expression for the joint probability psðt1 Þsðt2 Þ ðx1 ; x2 j sð0Þ ¼ x0 Þ of finding the electron between x1 and x1 þ dx1 at the instant t1 and between x2 and x2 þ dx2 at the instant t2 is required. This can be done taking into account that, according to Wiener’s model, successive increments of the electron positions are independent from the previous ones. Assuming t2 > t1 and
conditioning to the event sðt1 Þ ¼ x1 ; we find psðt1 Þsðt2 Þ ðx1 ; x2 j sð0Þ ¼ x0 Þ ¼ psðt2 Þ ðx2 j sðt1 Þ ¼ x1 Þpsðt1 Þ ðx1 j sð0Þ ¼ x0 Þ ¼ nx1 ðx2 ; t2 t1 Þnx0 ðx1 ; t1 Þ;
ð11Þ
where the index in nðx; tÞ denotes the point at which the electron is ‘‘generated’’. Thus the autocorrelation function of the current is Z NZ N RI ðt1 ; t2 Þ ¼ Iðx1 ÞIðx2 Þnx0 ðx1 ; t1 Þ 0
0
nx1 ðx2 ; t2 t1 Þ dx2 dx1
ð12Þ
and the covariance is CI ðt1 ; t2 Þ RI ðt1 ; t2 Þ Iðt1 ÞIðt2 Þ:
ð13Þ
For t2 ot1 it is enough to exchange t1 and t2 in the above relations. In the following this exchange will be tacitly assumed. Finally, for N independent electrons generated at x0 ; the autocorrelation must be multiplied by N so that the covariance becomes Iðt1 ÞIðt2 Þ : ð14Þ CI ðt1 ; t2 Þ ¼ RI ðt1 ; t2 Þ N 3.2. Recovering of the covariance function without induction Let us first consider the case when induction is * switched off. In this case EðxÞ ¼ dðxÞ and thus 0 IðxÞ ¼ Dd ðxÞ þ mEðxÞdðxÞ: Since nð0; tÞ ¼ 0; after integration by parts the current is, as expected, Z N IðtÞ ¼ D d0 ðxÞnðx; tÞ dx 0 @nðx; tÞ ¼D ¼ Jð0; tÞ ð15Þ @x x¼0 that is just the current due to a pure collection process. Substituting the expression for Iðx1 Þ in Eq. (12), the autocorrelation function after integration by parts becomes, for t2 > t1 ; @nx0 ðx1 ; t1 Þ RI ðt1 ; t2 Þ ¼ D @x1 x1 ¼0 Z N Iðx2 Þnx1 ¼0 ðx2 ; t2 t1 Þ dx2 : ð16Þ 0
The factor outside the integral sign is Iðt1 Þ whereas, the integral is the current due to an
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electron generated at the anode at t2 ¼ t1 and immediately collected and it is thus equal to dðt1 t2 Þ: A similar result applies also to the case t2 ot1 and consequently relation (4) is recovered. 3.3. Approximation of the covariance function in drift detectors Assuming, as it happens in drift detectors, that the induction region has a spatial extent d near the anode much less than the co-ordinate x0 ; then the autocorrelation function can be approximated in order to have a simplified expression more suitable for comparisons and numerical evaluations. For a uniform electric field EðxÞ E and assuming d5x0 ; the autocorrelation function (12) is different from zero only for t1 Ex0 =mE and jt2 t1 j Ex1 =mEpd=mE: It is concluded that jt2 t1 j5t1 and consequently that the width of the distribution nx0 ðx1 ; t1 Þ is much larger than the width of nx1 ðx2 ; t2 t1 Þ: The latter in particular is approxipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mately equal to 2 Djt2 t1 jp2 Dd=mE : By exploiting Einsten’s relation D ¼ kTm=q; it is shown that even at room temperature and at low drifting field of 100 V=cm; the width of pffiffiffi nx1 ðx2 ; t2 t1 Þ is about 0:031 d cm and thus it is also much less than d: Consequently, nx1 ðx2 ; t2 t1 Þ can be approximated in Eq. (12) by the delta distribution dðx2 x1 þ ðt2 t1 ÞmEÞ: In this case autocorrelation (12) for t2 > t1 becomes Z N Iðx1 Þ RI ðt1 ; t2 ÞE ðt2 t1 ÞmE
Iðx1 ðt2 t1 ÞmEÞnx0 ðx1 ; t1 Þ dx1 : ð17Þ
4. Signal filtering In all applications, the signal current at the detector anode is processed by a suitable filtering. In this section the theory of the signal filtering for both amplitude and time measurement is briefly resumed for further reference. If hðtÞ is the impulse response function of the filter, the output signal uðtÞ is the convolution of hðtÞ with the input
current, Z uðtÞ ¼
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N
IðtÞhðt tÞ dt
ð18Þ
N
whereas the output covariance is [3] Z NZ N Cu ðt1 ; t2 Þ ¼ CI ðt1 ; t2 Þ N
N
hðt1 t1 Þhðt2 t2 Þ dt1 dt2 Z NZ N ¼ RI ðt1 ; t2 Þhðt1 t1 Þ N
N
hðt2 t2 Þ dt1 dt2 uðt1 Þuðt2 Þ:
ð19Þ
For amplitude measurement, the filter impulse response is chosen to be a unipolar function of time and the output signal is measured at its maximum at the instant tm : The squared noise-tosignal ratio is thus e2 ¼ Cðtm ; tm Þ=u2 ðtm Þ: For time measurement, the filter impulse response is an odd function of time and the measured time is at the instant tz at which uðtÞ crosses zero. The variance of the time noise is given by [1] t2 ¼ 0 Cðtz ; tz Þ=u 2 ðtz Þ; where u0 ðtz Þ is the slope of the output signal at the measurement time.
5. Numerical results in silicon drift detectors The results of the previous sections are now applied to a specific example in order to better evaluate the effect of induction on the measured noise-to-signal ratio and to compare the results with both the Poisson and the Gatti–Fusillo approach. The solution of Eq. (5) in a drift detector for a uniform drifting field E and in presence of an anode at x ¼ 0 is given, for xX0 and tX0; by 2 1 nx0 ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi eðxx0 þmEtÞ =4Dt 4pDt 2 1 pffiffiffiffiffiffiffiffiffiffiffi emEx0 =D eðxþx0 þmEtÞ =4Dt ð20Þ 4pDt when a unit delta-like charge distribution is generated at x ¼ x0 at t ¼ 0: The first term is the drift diffusion in the absence of the anode. The second is the correction due to the anode. Relation (20) is indeed the required solution since both terms are separately solutions of the linear
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6
equation (5). In addition both the initial and the boundary conditions are satisfied at t ¼ 0 and x ¼ 0; respectively. * For drift detectors the weighting function VðxÞ along the detector axis can be approximated by x=d * VðxÞEe for xX0; where d ¼ w=p; w being the detector thickness [5]. The above expression for * VðxÞ is the first leading term of a series expansion. Thus for drift detectors at constant drifting field IðxÞ ¼ g ex=d ; where g ¼ mE=d D=d 2 : The induced current (10) and the autocorrelation function (17), for t2 > t1 ; become Z N ex=d nx0 ðx; tÞ dx IðtÞ ¼ g
Current [10 e−/s]
132
3 2 1 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Time [µs]
Fig. 2. Induced current at the detector anode.
0
g 2 ¼ eðDtdx0 þdmEtÞ=d 2" ! # 2Dt dx0 þ dmEt pffiffiffiffiffiffi erf 1 2d Dt g 2 eðDtþdx0 þdmEtÞ=d þmEx0 =D 2 " ! # 2Dt þ dx0 þ dmEt pffiffiffiffiffiffi erf 1 2d Dt
ð21Þ
RI ðt1 ; t2 ÞE g2 eðt2 t1 ÞmE=d Z N e2x1 =d nx0 ðx1 ; t1 Þ dx1 ðt2 t1 ÞmE 2
¼
g ð4Dt1 2dx0 þdmEðt2 þt1 ÞÞ=d 2 e 2" ! # 4Dt1 dx0 þ dmEt2 Þ pffiffiffiffiffiffiffiffi erf 1 2d Dt1 g2 2 eð4Dt1 þ2dx0 þdmEðt2 þt1 ÞÞ=d þmEx0 =D 2 " ! # 4Dt1 þ dx0 þ dmEt2 pffiffiffiffiffiffiffiffi erf 1 : 2d Dt1 ð22Þ
For the numerical calculation it has been assumed that E ¼ 100 V=cm; d ¼ 100 mm; x0 ¼ 0:1 cm [5] and m ¼ 1200 cm2 =V s for silicon at room temperature [7]. The diffusion coefficient D is related to the mobility m by Einstein’s relation D ¼ kTm=q at room temperature, T ¼ 300 K: Fig. 2 shows the current induced at the anode under these conditions.
The numerical filtering of the current at the output of the front-end electronics for the amplitude measurement has been done with a noncausal Gaussian filter with impulse response hðtÞ gðtÞ ¼ expðt2 =2s2 Þ: For time measurement the filter response function has been chosen as the time derivative of the above, hðtÞ g0 ðtÞ: In both cases different values of the filter time width s have been considered in the range between 20 ns and 1 ms: As a reference, the time width of the current signal at the anode is about 159 ns rms: Fig. 3 shows the squared noise-to-signal ratio, normalized to one electron, of the amplitude measurement as a function of the filter width. The results of the approach described in this paper that take into account the induction process are compared with both the Poisson and the Gatti–Fusillo treatments. A reduction of the squared noise-to-signal radio is evident up to a filter width s of about 0:8 ms: For shorter values of s the effect of the induction process is to introduce a higher negative correlation in the noise associated with the signal current, thus improving the performance of the measuring process. Apart from very low values of the time width of the filter, the noise-to-signal ratio is always less than the contribution associated with the fluctuation of the number of electron–hole pairs generated by the primary particle and described by Fano’s factor [12], equal to about 0.11 per electron in silicon [13]. A factor three improvement is also evident in Fig. 4 showing the noise time variance, normalized to one electron, of the time measurement as a function of the filter width. For time measurement,
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Time variance, τ 2 N [µs ]
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Gatti Induction
0.1
Poisson - Gatti 0.01
Induction 0.2
0.4
0.6
0.8
1.0
Filter Width, σ [µs] Fig. 3. Squared noise-to-signal ratio, normalized to one electron, of the amplitude measurement as a function of the filter width when the Poisson, the Gatti–Fusillo and the full induction approach is applied.
the Poisson and the Gatti–Fusillo approaches give the same results [3] since at measurement time tz the filter output uðtz Þ in Eq. (19) is zero.
6. Conclusion The analysis of the statistics of the signal at the anode of a semiconductor drift detectors, when the electrostatic induction is fully taken into account, has shown that the negative correlation introduced between the signal values at different time instants is large enough for improving the signal-to-noise ratio with respect to the previous published estimation, in silicon at room temperature and for typical applications. It is expected that the contribution of induction in the statistics of signals from semiconductor particle detectors should be relevant whenever the width of the carrier density distribution is comparable or larger than the extension of the p induction This ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiregion. ffi condition is verified if 2 kTx0 =qE Xd which requires a low drifting field and a long drifting path.
0.001 0.0
0.2
0.4
0.6
0.8
1.0
Filter Width, σ [µs] Fig. 4. Noise time variance, normalized to one electron, of the time measurement as a function of the filter width when the Poisson/Gatti–Fusillo and the full induction approach is applied.
References [1] E. Gatti, P. Rehak, J.T. Walton, Nucl. Instr. and Meth. 226 (1984) 128. [2] E. Gatti, P. Rehak, Nucl. Instr. and Meth. 225 (1984) 608. [3] E. Gatti, G. Fusillo, Nucl. Instr. and Meth. A 236 (1993) 279. [4] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Company, New York, 1965. [5] E. Gatti, M. Sampietro, F.E. Zocchi, J. Appl. Phys. 74 (1993) 5940. [6] E. Gatti, A. Longoni, P. Rehak, M. Sampietro, Nucl. Instr. and Meth. A 253 (1987) 393. [7] S.M. Sze, Physics of Semiconductor Devices, 2nd Edition, Wiley, New York, 1981. [8] A. Einstein, Ann. D. Phys. 17 (1905) 549. [9] I. Karatsas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd Edition, Springer, New York, 1997. [10] S. Ramo, Proc. I.R.E. 27 (1939) 584. [11] C.K. Jen, Proc. I.R.E. 29 (1941) 345. [12] U. Fano, Phys. Rev. 72 (1947) 26. [13] H.J. Langmann, O. Meyer, Nucl. Instr. and Meth. 30 (1964) 209.