Description of the plasma delay effect in silicon detectors

1.1 MB·English

by  Zbigniew Sosin

Description of the plasma delay effect in silicon
detectors
2
1 Z. Sosin
0
2
January 12, 2012
n
a
J

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1 Institute of Physics, Jagellonian University

]
h
t
- Abstract
l
c
u Anewmethodofmodelingofthecurrentsignalinducedbycharged
n
particle in silicon detectors is presented. The approach is based on
[
the Ramo-Shockley theorem for which the charge carrier velocities

1
are determined by taking into account not only the external electric
v
8 field generated by the electrodes, but also the Coulomb interaction
8 between the electron and hole clouds as well as their diffusion.
1
2
. keywords:
1
Plasma delay
0
2 Current pulse
1
Pulse shape analysis
:
v
Particles identification
i
X
Silicon detector
r
a
1 Introduction
Itisobviousthatidentificationofparticlesandfragmentsproducedinnuclear
reactions is crucial for any experimental or technical work in nuclear physics.
Among different ways of identifying charged particles the classical ∆E −E
telescope method remains still the flagship. Recently, an alternative method
based on the Pulse Shape Discrimination (PSD) technique applied for silicon
1
detectors is being developed and is increasingly drawing attention. Recent
results demonstrate that the method can offer charge and isotopic identifi-
cation comparable to that obtained with the classical ∆E−E method. The
main advantage of the PSD method comes from the fact that it requires
only one electronic channel for detection and identification. It is thus an
important point for designing and constructing multi-detector systems.
A significant difference between the ∆E − E and the PSD techniques
results from the fact that the former is governed basically by the energy loss
process (Bragg curve), while the latter is primarily related to the Plasma
Delay Effect (PDE) [1-6]. In silicon detectors, this effect manifests itself with
shortening of the pulse rise time with decreasing Z for low and intermediate
mass fragments, for which the generated charge is practically completely
collected by the detector electrodes. The experimental data demonstrates
that the PDE concerns particles with small Z, for which the Pulse Hight
Defect (PHD) is still of little importance.
For better understanding of the identification idea associated with PSD
technique, and for its future development, it is crucial to have at ones com-
mand a perfect simulation of the time dependence of the experimental signal
produced by an ion with a given charge, Z, atomic mass, A, and energy, E.
Themaingoalofsuchasimulationistodescribetheextractionandcollection
of the generated charge carriers moving in the external electric field distorted
due to the presence of a highly ionized track and due to the diffusion process
of the carriers.
As presented in [7], an approach in which the distortion of the electric
fieldcausedbythegeneratedcarriersisneglected, isabletocorrectlydescribe
the current signals for light charged particles (LCP), e.g. protons. However,
this simplified approach completely fails in case of heavy ions (HI) for which
the collection time of the generated carriers gets longer (τHI > τLCP). His-
torically, this difference, associated to a slower carrier collection for HI, was
quantified as a plasma delay (PD) effect. Since this effect influences the
current signal rise-time, it appears to be crucial for the PSD technique.
An attempt to describe phenomenologically the delayed carrier collection
time in silicon detectors has been recently proposed in [8]. The proposed
description took into account the polarization of the electron-hole pairs gen-
erated by the HI and connected it to the relative dielectric permittivity.
Another important assumption was that the dissociation of pairs in time oc-
curred with a constant probability and the modified electric field, inside and
outside of the ion range, was given by the Maxwell equation for the electric
2
field in the inhomogeneous medium. With these assumptions, the model was
indeed able to describe the experimental pulse shapes quite accurately.
In the present paper we propose another, more microscopic approach.
The main model assumptions are the following:
i. Propagation of the electric charges (electrons, holes) gener-
ated in the detector is represented by evolution of the Gaussian
clouds for which the centroids and variances are treated as inde-
pendent variables.
ii. Position of the centroid of each Gaussian is governed by
the drift process, while its variance undergoes both, diffusion and
the drift process.
The first results of the model calculations indicate some binding effects be-
tween the holes and electrons, in a region similar to that predicted by the
phenomenological model of Ref. [8]. In this region also the electric field
shows similar behavior to that presented in [8].
The detailed description of the new model is presented in the following
section. The preliminary results of the calculations and comparison to the
experimental data are presented in Section 3; Conclusions and possible ex-
tensions of the model applicability are given in Section 4.
2 Description of the model
A particle entering the silicon detector is assumed to degrade its energy
according to the Bragg curve which relates the generated ionization B(x) to
the particle position x. We assume that the X direction is perpendicular to
the detector surface. In order to describe the initial, local, density of the
electrons ρ (r,t = 0) and holes ρ (r,t = 0) we assume that the ionization is
e h
proportional to the local stopping power B(x) = 1 dE(x), where w = 3.62
w dx
eV is the energy for an electron-hole pair production and dE(x) is the local
dx
stopping power [9]. Just after stopping of the impinging ion, the carrier
density can be described as:
ˆ
ρ (r,t = 0) = −ρ (r,t = 0) = − B(x(cid:48))δ(x−x(cid:48))δ(y)δ(z)dx(cid:48) (1)
e h
where r = [x,y,z]. This assumption states that for t = 0 the ionization is
localized along the X axis only and disappears elsewhere.
3
In order to describe the time evolution of the generated ionization we
assume its distribution in the following form:
ˆ
ρ (r,t) = B (x(cid:48),t)G (x−x(cid:48),y,z,t)dx(cid:48) (2)
e e e
ˆ
ρ (r,t) = B (x(cid:48),t)G (x−x(cid:48),y,z,t)dx(cid:48) (3)
h h h
which is analog to (1) and we set:
−B (x,t = 0) = B (x,t = 0) = B(x) (4)
e h
Functions G and G are assumed to be Gaussians:
e h
(cid:32) (cid:33)
1 (x−x(cid:48))2 +y2 +z2
G (x−x(cid:48),y,z,t) = exp − (5)
e (cid:113) 2σ2(x,t)
(2π)3σ3(x,t) e
e
and
(cid:32) (cid:33)
1 (x−x(cid:48))2 +y2 +z2
G (x−x(cid:48),y,z,t) = exp − (6)
h (cid:113) 2σ2(x,t)
(2π)3σ3(x,t) h
h
If σ → 0 and σ → 0 for t → 0 then the functions G and G can be
e h e h
regarded as representations of the δ function, thus:
G (x−x(cid:48),y,z,t = 0) = G (x−x(cid:48),y,z,t = 0) = δ(x−x(cid:48))δ(y)δ(z)dx (7)
e h
Now, the goal is to describe the time evolution of the functions B , B and
e h
G , G (fordeterminationofG , G itissufficienttoderivethetimeevolution
e e e e
of their variances σ2 and σ2). In order to do it we define the one dimensional
e h
densities associated with the variable x as:
ˆ ˆ
∞ ∞
η (x,t) = dy dzρ (r,t) (8)
e e
−∞ −∞
ˆ ˆ
∞ ∞
η (x,t) = dy dzρ (r,t) (9)
h h
−∞ −∞
4
and we divide the thickness of the detector d into N intervals ∆x = dSi.
Si N
Let us assume that in the interval ∆x (i = 1,N) the associated values of σ
i ei
and σ do not change substantially within the radius of a few sigma around
hi
∆x , and that a linear approximation can be used for the functions B (x(cid:48),t)
i e
and B (x(cid:48),t) within ∆x :
h i
B (x(cid:48),t) = p (t)x(cid:48) +q (t) (10)
e ei ei
B (x(cid:48),t) = p (t)x(cid:48) +q (t) (11)
h hi hi
With the above assumption for x within an interval ∆x one can approximate
i
the densities:
ˆ ˆ ˆ
∞ ∞ ∞
η (x,t) (cid:119) dx(cid:48) dy dz(p (t)x(cid:48) +q (t))G (x−x(cid:48),y,z,t) = p (t)x+q (t)
e ei ei e ei ei
−∞ −∞ −∞
(12)
ˆ ˆ ˆ
∞ ∞ ∞
η (x,t) (cid:119) dx(cid:48) dy dz(p (t)x(cid:48) +q (t))G (x−x(cid:48),y,z,t) = p (t)x+q (t)
h ei ei h hi hi
−∞ −∞ −∞
(13)
which means that, in practice, one can use the same coefficients for linear
expansion of both, the densities η , η and of the functions B , B .
e h e h
We introduce also the x (t) and x (t) coordinates, which denote the
e0 eN
beginning and end of the B (x,t) distributions for electrons. Similar coor-
e
dinates x (t) and x (t) are introduced for holes (see right-upper panel on
h0 hN
Fig. 2).
2.1 Electric field determination
In order to determine the drift velocity associated with the centers of Gaus-
sians G , G we have to calculate the respective effective electric field acting
e h
on the carriers described by above distributions. Such a field is determined
by a static voltage applied to the detector electrodes and by the Coulomb
interaction between the Gaussian charge clouds. The detector static field at
position x, considered from the rear side of the detector (order n − p from
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the point of view of the particle entering the detector, see e.g. [8]), is given
as:
2V x V −V
d d
E (x) = + (14)
stat d2 d
Si Si
where the bias voltage V is assumed to be higher than the depletion voltage
V which for the bulk concentration of donors N and permittivity ε = ε ε
d D r 0
reads as:
eN d2
D
V = (15)
d
2ε ε
r 0
In order to find the modification of the electric field caused by the gen-
erated plasma, let us consider two Gaussians describing the distribution of
the charges Z , Z , centered at a relative distance r =|r −r | and char-
1 2 12 1 2
acterized by variances σ2, σ2 respectively. The mutual interaction potential
1 2
of the clouds can then be expressed in the form which can be often found in
quantum molecular dynamics calculations (see e.g. [10])
v(r , r , σ , σ ) = e2Z1Z2 ¨ exp(cid:16)−(r12−σ12r01)2(cid:17)exp(cid:16)−(r22−σ22r02)2(cid:17)d3r d3r =
01 02 1 2 (2πσ σ )3 |r −r | 1 2
1 2 1 2
(cid:16) (cid:17)
erf √r12
= e2Z Z 2σ (16)
1 2
r
12
(cid:112)
where σ = σ2 +σ2.
1 2
Let us now assume that the intervals ∆x are small enough to enable the
i
linear approximation for the charge densities ρ , ρ and the functions B ,B
e h e h
with the use of the coefficients p and q. For simplicity, we introduce variables
p , q in lieu of the p (t), p (t) and q (t), q (t). If we denote the endpoints
ei hi ei hi
of the ∆x interval by c and d then for a Gaussian centered at a point a
i
and representing the charge Z , its interaction with the charge located in the
a
interval ∆x = (c, d) given by (12) and (13), can be formulated as:
i
ˆ (cid:16) (cid:17)
e2 d erf |√x−a|
V (a,c,d,p,q,σ ,σ ) = Z dx (px+q) 2σs (17)
C a i a
ε |x−a|
c
where the σ and σ above denote the standard deviations of the Gaussian
a i
describing the charge Z and of the Gaussian from the interval ∆x , respec-
a i
(cid:112)
tively, and σ = σ2 +σ2.
s a i
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The above form allows us to describe the respective effective electric field
E acting on the Gaussian located at a point a as
x
ˆ (cid:16) (cid:17)
1 ∂V e2 d ∂ erf |√x−a|
E (a,c,d,p,q,σ ,σ ) = − C = − dx (px+q) 2σs (18)
x a i
Z ∂a ε ∂a |x−a|
a c
Since ∂ = − ∂ , the above formula can be expressed as:
∂a ∂x
ˆ (cid:16) (cid:17)
e2 d ∂ erf |√x−a|
E (a,c,d,p,q,σ ,σ ) = dx (px+q) 2σs =
x a i
ε ∂x |x−a|
c
 (cid:16) (cid:17) (cid:16) (cid:17) ˆ (cid:16) (cid:17)
e2 erf |√d−a| erf √|c−a| e2 d ∂ erf |√x−a|
= q 2σs − 2σs + p dxx 2σs (19)
ε |d−a| |c−a| ε ∂x |x−a|
c
After integrating by parts one obtains:
 (cid:16) (cid:17) (cid:16) (cid:17)  (cid:16) (cid:17) (cid:16) (cid:17)
e2 erf |√d−a| erf √|c−a| e2 erf |√d−a| erf √|c−a|
Ex = q 2σs − 2σs + pd 2σs −c 2σs +
ε |d−a| |c−a| ε |d−a| |c−a|
ˆ (cid:16) (cid:17)
e2 d erf |√x−a|
− p dx 2σs (20)
ε |x−a|
c
The last integral can be easily evaluated by expanding the error function.
Thetotaleffectiveelectricfieldmodifiedduetothepresenceofplasmacan
be obtained by summing up contributions of charges located in all intervals
(c ,d ) and of their mirror charges induced in the detector electrodes (see
i i
Fig. 1).
Inordertodeterminethetimeevolutionofthechargedistributiononehas
to know, in addition, the generalized force associated with the σ variable.
a
Similarly as for the effective electric field, the net value of the force acting in
the σ direction can be obtained by summing over all ingredients associated
a
with the charge distribution. This leads to the following expression for the
interaction with the charge located in the ∆x interval:
i
ˆ (cid:16) (cid:17)
∂V e2 d ∂ erf |√x−a|
F = − C = − Z dx (px+q) 2σs =
σa,a,cd ∂σ ε a ∂σ |x−a|
a c a
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(cid:32) (cid:32) (cid:33) (cid:32) (cid:33)(cid:33)
e2pσ (a−d)2 (a−c)2
a
= −Z exp − −exp − +
a(cid:112)πεσ 2σ2 2σ2
2 s s s
e2σ (ap+q) (cid:18) (cid:18)(a−d)(cid:19) (cid:18)(a−c)(cid:19)(cid:19)
+Z a erf √ −erf √ (21)
a εσ2 2σ 2σ
s s s
Now we are in position to calculate the time evolution of the charge
generated in the detector. This process is determined by the drift and by the
diffusion of the interacting clouds of electrons and holes. We will consider
the evolution of the centroids and of the variances of Gaussians representing
a fraction of the charge distribution located in the middle of the intervals
∆x and at the start- and end-points of the distributions of electrons and
i
holes (points x (t), x (t) and x (t), x (t)) .
e0 eN h0 hN
2.2 Evolution of the function B
In the present subsection we will describe the numerical method used to
determine the time evolution of the ionization clouds. In the following we
assume that the evolution of the functions B , B is determined by the time
e h
evolutions of the coefficients p , q and p , q . In order to find how the
ei ei hi hi
expansion coefficients p , q and p , q propagate in time, we have to
ei ei hi hi
investigate the time evolution of the functions η (x,t) and η (x,t).
e h
Below we consider the formulas for electrons only, keeping in mind that
the formulas for holes are analogical.
As we will show later
∂ ∂
η (x,t) = − (η (x,t)v (x,t)) (22)
e e xe
∂t ∂x
thus, the differential dη (x,t) can be written as
e
(cid:18) (cid:19)
∂ ∂
dη (x,t) = − v (x,t) η (x,t)+η (x,t) v (x,t) dt (23)
e xei e e xe
∂x ∂x
If the x denotes the center of the interval ∆x and q and p are the
i i ei ei
coefficients of linear expansion
η (x,t) = p (t)(x−x )+q (24)
e ei i ei
8
and, if one denotes the average velocity and the average linear density asso-
ciated with the interval ∆x by v (t) and η (t), respectively, then:
i xei xei
dη (t) = −(v (t)p (t)+η (t)ϕ (t))dt (25)
ei xei ei ei xei
Here, ϕ (t) is the differential coefficient of v (x,t) at the point x . As one
xei xei i
can see, in order to calculate the above increment, we have to trace the time
dependent tables of η (t), v (t). The tables of p (t), q (t) and ϕ (t) are
ei xei ei ei xei
obtained by fitting the smooth curves to the distributions η (t), v (t) in
ei xei
every time step. For t = 0 the η (t = 0) is given by the Bragg curve. In
ei
order to make use of the formula (25) we need to construct the respective
tables for velocities. Knowing the effective electric field for electrons in the
x direction, E , one can assume that the respective average velocity of the
xei
center of Gaussian located at a point x is proportional to the strength of the
i
field:
v = µ E (26)
xei xe xei
where µ and µ are the electron and hole mobilities, respectively.
xe xh
Knowing the drift velocity, one can calculate the evolution of the charge
deposited in every interval ∆x including the edge intervals with variable
i
ends x (t), x (t) and x (t), x (t).
e0 eN h0 hN
2.3 Charge propagation in the perpendicular direction
The diffusion and transport processes in the electric field influence also the
widths of the charge distributions located in every ∆x interval. Extending
i
the above reasoning we can assume that the velocity, v , describing the
σei
rate of the standard deviation expansion in perpendicular direction has three
components:
v = vE +vD +vT (27)
σei σei σei σei
The first term, vE , results from the field described by (21). In analogy
σei
to the charge drift in x direction, one can assume that this component is
proportionaltothefieldactingonthechargeZ associatedwiththeGaussian
a
with a standard deviation σ
a
F
E = σa (28)
σei
Z
a
9
where F is the net force given by interaction (21). Thus, the respective
σa
velocity can be expressed as
vE = µ E (29)
σei σe σei
where, the µ parameter is the only free parameter of the model. It seems,
σe
however, that it can be determined theoretically in the future. A similar
parameter for description of the hole propagation can be calculated assuming
the following proportion:
µ µ
σe xe
= (30)
µ µ
σh xh
The velocity vD follows from the solution of the second Fick’s law for diffu-
σei
sions of Gaussian density distributions:
∂σ D
vD = ei | = e (31)
σei ∂t vσEei=0,vσTei=0 σ
ei
where D is the diffusion coefficient for electrons.
e
Another process which affects, on average, the widths of the respective
Gaussiandistributionsusedtodescribethechargelocatedintheinterval∆x ,
i
is related to the transport of the carriers. In order to describe this process
we consider an increase of the variance, σ2, of the Gaussian distribution of
the charge of Z particles contained in an interval ∆x, with linear density
η(x) = Z . Let the (cid:80)r2 denote the sum of squares of deviations of particle
∆x i
positions from the average. In this consideration we neglect the influence of
the diffusion and of mutual interactions of clouds on the propagation of the
variance.
(cid:80)r2
The change of σ2 ≡ i , resulting from the flow of particles into and
Z
out of the cell ∆x (with the net value of dZ) is, in general, equal to
d(cid:0)(cid:80)r2(cid:1) σ2(x)dZ
dσ2 = i − (32)
Z Z
As one can see, in order to find the increment dσ2 we have to find the
increments dZ and d(cid:0)(cid:80)r2(cid:1).
i
Let us begin with the description of d(cid:0)(cid:80)r2(cid:1). If the accretion of particles
i
in an interval ∆x across the points x = x − ∆x/2 and x = x + ∆x/2 is
1 2
denoted by dZ and dZ , respectively, and the variances at these points are
1 2
10