# Determination of the electric field in highly-irradiated silicon sensors
using edge-TCT measurements

R. Klanner1 G. Kramberger2 I. Mandić2 M. Mikuž2 M. Milovanović3 J. Schwandt1
1Institute for Experimental Physics, University of Hamburg, Luruper Chaussee
149, D 22761 Hamburg, Germany  2 Jozef Stefan Institute, Jamova 39, Sl 1000
Ljubljana, Slovenia  3 Jozef Stefan Institute, now at DESY Zeuthen,
Platanenallee 6, D 15738 Zeuthen, Germany

###### Abstract

A method is presented which allows to obtain the position-dependent electric
field and charge density by fits to velocity profiles from edge-TCT data from
silicon strip-detectors. The validity and the limitations of the method are
investigated by simulations of non-irradiated $n^{+}p$ pad sensors and by the
analysis of edge-TCT data from non-irradiated $n^{+}p$ strip-detectors. The
method is then used to determine the position dependent electric field and
charge density in $n^{+}p$ strip detectors irradiated by reactor neutrons to
fluences between 1 and $10\times 10^{15}$ cm-2 for forward-bias voltages
between 25 V and up to 550 V and for reverse-bias voltages between 50 V and
800 V. In all cases the velocity profiles are well described. The electric
fields and charge densities determined provide quantitative insights into the
effects of radiation damage for silicon sensors by reactor neutrons.

###### keywords:

Silicon detectors , radiation damage , edge-TCT , electric field , effective
doping density.

††journal: Elsevier

###### Contents

  1. 1 Introduction
  2. 2 Sensors, irradiation and measurements
  3. 3 Model and fit procedure
  4. 4 Electric field of the non-irradiated sensor for reverse bias
  5. 5 Electric field of the irradiated sensors
    1. 5.1 Forward bias
    2. 5.2 Reverse bias
    3. 5.3 Discussion of the results
  6. 6 Summary and conclusions
  7. 7 Appendix A: Effective weighting field for edge-TCT
  8. 8 Appendix B: Influence of electronics and charge carrier lifetimes on the velocity profiles

##  1 Introduction

The knowledge of the electric field and of the effective doping distribution
in silicon sensors is essential to predict and understand their performance.
Although qualitatively the physics of the change of the electric field with
hadron irradiation was understood already about 25 years ago [1, 2], a
reliable quantitative determination is still lacking. The attempts to
calculate the electric field in irradiated sensors using TCAD simulations with
trap parameters from spectroscopic measurements have been only partially
successful so far [3, 4, 5, 6, 7]. Difficulties are the large number of
radiation-induced states [8] with frequently only poorly known properties, the
problem of implementing cluster defects in the simulation [9] and the
influence of high electric fields on the defect properties. Thus, a method to
determine experimentally the electric field is highly desirable.

In Ref. [10] $n^{+}p$ strip sensors irradiated with hadrons up to fluences of
$10^{16}$ cm-2 were investigated using the edge-Transient Current Technique
(edge-TCT). In edge-TCT [11] the sensor is illuminated by a focussed near-
infrared laser beam, which is parallel to the sensor surface and normal to the
strips, and the current transients are recorded as a function of the distance
of the laser beam from the readout plane. From the initial part of the induced
current pulse, which is proportional to the sum of the hole and electron
velocities, the electric field at the position of the laser beam can be
determined. In Ref. [10] a parametrisation of the electric field (Fig. 5 of
[10]) has been assumed to extract the parameters of the model from the data.
As a cross-check the electric field thus determined has been used to simulate
the position dependence of the initial currents. Although an approximate
description has been achieved, significant differences are observed. In the
present paper a method is proposed, which allows determining the electric
field by directly fitting the initial currents without assumptions on the
shape of the electric field. The method is applied to the data of Ref. [10] as
well as to reference data for the non-irradiated strip-sensor.

The following section discusses the sensors, their irradiation and the edge-
TCT measurements. Then, the method used to extract the electric field from the
data is derived, the assumptions made discussed, and the fitting procedure
presented. To demonstrate the validity of the model, as a first step data from
a non-irradiated sensor are analysed and the results compared to the
expectations for a non-irradiated pad detector. Then the model is used to
determine the position dependence of the electric field in the sensor as a
function of forward and reverse bias voltage and irradiation fluence. Finally
the results are discussed and suggestions for further studies made.

##  2 Sensors, irradiation and measurements

The sensors investigated are $p$-type AC-coupled micro-strip detectors
fabricated on float-zone silicon by Hamamatsu Photonics [12]. The main
properties of the sensors are listed in Table 1. More information is given in
Refs. [10, 11, 13]. The isolation between the $n^{+}$-implants is achieved by
a $p$-spray implant with $2\times 10^{12}$ B-ions/cm2, and a single $p$-stop
implant with $8\times 10^{12}$ B-ions/cm2. This information is relevant for
estimating the electric field in-between the strips close to the Si-SiO2
interface.

thickness | crystal | pitch |  $n^{+}$ width | strip length | Area | B-doping | $U_{fd}$
---|---|---|---|---|---|---|---
$\upmu$m | orientation |  $\upmu$m |  $\upmu$m | cm | cm2 | cm-3 | V
300 | $\langle 100\rangle$ | 100 | 20 | 0.8 | 0.62 | $2.8\times 10^{12}$ | 180
Table 1: Sensor parameters. For the B-doping and the full depletion voltage,
$U_{fd}$, the values obtained from the fits for the non-irradiated strip
detector (Sect. 4) are given.

One strip of the sensor was connected via a current amplifier (10 kHz–1 GHz)
to a 1.5 GHz sampling scope. The remaining strips were connected via 50
$\Omega$ resistors to a fixed potential (virtual ground). For each measurement
the average of 400 transients was recorded.

The sensor was illuminated through a carefully polished edge with the light
from a laser with a wavelength of 1064 nm and a pulse width of 50 ps. At this
wavelength the light absorption length in non-irradiated silicon is about 1 mm
at room temperature [14]. It decreases with increasing temperature and with
increasing fluence, when irradiated by hadrons, [15]. The laser light was
focussed to a full-width-half-maximum of about $7\leavevmode\nobreak\ \upmu$m
at the position of the readout strip, and its position was controlled with
sub-micron precision by a computer-controlled moving stage.

For the measurements analysed in this paper, the sensors were irradiated with
neutrons in the TRIGA reactor of JSI in Ljubljana to 1 MeV neutron-equivalent
fluences, $\Phi_{eq}$, of (1, 2, 5, 10)$\leavevmode\nobreak\ \times 10^{15}$
cm-2 [16, 17]. The fluence uncertainty is estimated to be below 10 %. After
irradiation, the sensors were annealed for 80 min at $60^{\leavevmode\nobreak\
\circ}$C. In addition, data from a non-irradiated sensor were analysed. For
the irradiated sensors the measurements were taken at
$-20^{\leavevmode\nobreak\ \circ}$C, and for the non-irradiated one at
$+20^{\leavevmode\nobreak\ \circ}$C .

Figure 1: Measured current transients from the non-irradiated sensor at
$20^{\leavevmode\nobreak\ \circ}$C with the laser beam $50\leavevmode\nobreak\
\upmu$m from the strips (from [10]).

Fig. 1 shows the measured current transients for the non-irradiated sensor
with the laser positioned $50\leavevmode\nobreak\ \upmu$m from the readout
strip for reverse bias voltages between 0 and 500 V. The initial peak is
dominated by the current induced by electrons, which move in the high field
towards the readout electrode. The tail at later times is caused by the holes,
which move in the decreasing field towards the $p^{+}$ back side of the
sensor. The rise time of the transient reflects the electronics response
function of the system sensor and readout.

As discussed in more detail in Sect. 8, there are different ways of obtaining
_velocity profile_ s, $vel$, from the data. Two examples are: The integration
of the transient up to the time $t_{int}$, and the maximum of the time
derivative of the initial pulse. As shown in Ref. [10], up to $t_{int}=800$ ps
the shape of $vel$ is independent of $t_{int}$. Shorter values of $t_{int}$
decrease the sensitivity of $vel$ to charge trapping but increase the relative
fluctuations. For the analysis of this paper $vel$ is obtained from the
maximal slopes of the current transients. For the sensors irradiated to the
fluences $\Phi_{eq}$ the velocity profiles $vel(y;U,\Phi_{eq})$, have been
measured as a function of the distance $y$ from the strip plane in
$5\leavevmode\nobreak\ \upmu$m steps and of the applied voltage $U$ up to 800
V for the irradiated sensors, and up to 500 V for the non-irradiated sensor.
In addition, the dark current $I_{dark}(U,\Phi_{eq})$ has been recorded.

##  3 Model and fit procedure

In this section the method used to extract the electric field from $vel(y)$ is
presented and the assumptions discussed. Fig. 2 shows the sketch of the cross
section of the sensor and defines the coordinate system.

Figure 2: Schematic drawing of the cross section of the strip sensor and the
coordinate system used. The region between the centers of two readout strips
is shown. The strip pitch is $p$, the sensor thickness $d$, and the length of
the strips in the $z$-direction is 8 mm.

A charge $Q$ moving with the velocity $\vec{v}\big{(}\vec{r}(t)\big{)}$ along
the trajectory $\vec{r}(t)$ induces in the readout electrode the current

$I(t)=Q\cdot\vec{v}\big{(}\vec{r}(t)\big{)}\cdot\overrightarrow{E_{w}}\big{(}\vec{r}(t)\big{)},$
(1)

where $\overrightarrow{E_{w}}(\vec{r})$ denotes the weighting field [18],
which has the units 1/cm. In general, a time-dependent weighting vector is
required to describe $I(t)$. However, as demonstrated in Ref. [19], time-
independent weighting fields can be used to describe the signal of fully
depleted non-irradiated sensors and for sensors irradiated by hadrons to
$\Phi_{eq}\gtrsim 10^{13}$ cm-2. The weighting field is obtained as the
difference of the electric field of the biased sensor plus $\Delta U=1$ V on
the readout electrode minus the electric field of the biased sensor divided by
$\Delta U$. For a light beam traveling at constant $y$ in the $x$-direction,
the current is induced by all charges generated along the light path. If the
light attenuation length is long compared to the strip pitch and the strip
length long compared to the sensor thickness, $d$, Eq. 1 can be used to
describe the induced current. In this case
$\overrightarrow{E_{w}}(\vec{r})=\hat{e}_{y}/d$, with the unit vector in the
y-direction $\hat{e}_{y}$, and $Q$ the moving charge per unit length
multiplied with the strip pitch, $p$.

This can be seen in the following way: If the dielectric constant in the
sensor is uniform and independent of the electric field, the effective
weighting field for the charge distribution
$\mathrm{d}^{3}Q/\mathrm{d}x\mathrm{d}y\mathrm{d}z=q_{0}\cdot\delta(y-y_{0})\cdot\delta(z)/p$
is
$\overrightarrow{\mathcal{E}_{w}}(\vec{r})=1/p\cdot\int_{-\infty}^{+\infty}\overrightarrow{E}_{w}(\vec{r})\leavevmode\nobreak\
\mathrm{d}x$. From symmetry arguments follows that for a sensor with an
infinite extension in the $x$\- and $z$-directions the integral does not
depend on $x$ and $z$. Symmetry arguments also require that the $x$\- and
$z$-components of $\mathcal{E}_{w}$ cancel, and that only the $y$-component is
finite. Changing the voltage on the readout electrode by 1 V does not change
the charge density inside the sensor, and from Gauss’ law follows that
$\overrightarrow{\mathcal{E}_{w}}$ is independent of $y$. From
$\int_{0}^{d}\mathcal{E}_{w}(y)\leavevmode\nobreak\ \mathrm{d}y=1$ follows the
result $\overrightarrow{\mathcal{E}_{w}}(\vec{r})=\hat{e}_{y}/d$.

The change of $\overrightarrow{\mathcal{E}_{w}}$ due to the finite attenuation
length of the near-infrared light used for the measurements is discussed in
Sect. 7.

The drift velocities of the charge carriers are
$\vec{v}_{i}(\overrightarrow{E},T)=\mu_{i}(E,T)\cdot\overrightarrow{E}$, where
the parametrisation of the hole mobility, $\mu_{h}$, and of the electron
mobility $\mu_{e}$ for electrons for $\langle 100\rangle$ silicon from Ref.
[20] are used; $T$ is the temperature. Although there is some evidence that
the low-field mobilities decrease after hadron irradiations for fluences
$\Phi_{eq}\gtrsim 10^{15}$ cm-2 [21, 22], fluence-independent $\mu_{i}$ values
are assumed.

Figure 3: Simulated potential, $\Phi$, in a strip detector with the parameters
given in Table 1 close to the readout strips biased to $-400$ V. Between the
strips there is a $2\leavevmode\nobreak\ \upmu$m thick SiO2 layer. The results
for Dirichlet boundary conditions (same $\Phi$ on top of the SiO2 as on the
strips) and for Neumann boundary conditions ($\partial\Phi/\partial y=0$ on
top of the SiO2) are shown. (Colour on-line)

For the electric field
$\overrightarrow{E}\big{(}\vec{r}\big{)}=E(y)\cdot\hat{e}_{y}$ is assumed. The
electric field close to the strip plane critically depends on many parameters,
some of which are only poorly known. It depends on the dark current, the
$p$-spray and $p$-stop doping, the charge densities at the Si-SiO2 interface
and the electrical boundary condition on the outer SiO2 surface. Simulations
using the software of Ref. [23] for two extreme conditions, Dirichlet and
Neumann boundary conditions, no dark current and no Si-SiO2 interface charges
are shown in Fig. 3. For the Dirichlet boundary condition (potential on the
outer Si-SiO2 equal to the strip potential), the electric field is similar to
the field of a pad diode, and the assumption that it has only a y-component
which is independent of $x$ is a good approximation. However, it is a very
poor description for the Neumann boundary condition (zero of the derivative of
the potential normal to the boundary). In Refs. [24, 25, 26] the electric
field for a $p^{+}n$ strip detector in the region close to the plane of the
readout strips has been studied as a function of ionising dose, biasing
conditions and time after switching on the voltage. It has been found, that
the boundary condition changes with time because of the finite surface
resistivity of the SiO2, which is a strong function of humidity and
temperature. Depending on the humidity, time constants varying between one
hour and several days have been measured at room temperature. In Ref. [26]
simulations of the changes of the electric field as a function of the time
after biasing a segmented $n^{+}p$-sensor are given. In the data analysis
presented in this paper, the effect of the lack of knowledge of the field
close to the strip plane is investigated by performing fits starting at
different minimal $y$-values.

Next the fit procedure is described, which is used to extract the electric
field, $E(y)$, from the measured velocity profile. For the fit $n_{E}$
electric field values, $E_{i}$, at the equidistant positions $y_{i}$ are
assumed, which are free parameters of the fit. For $E$ at other $y$ positions
a linear inter- and extrapolation is used, and the constraint
$\int_{0}^{d}E(y)\leavevmode\nobreak\ \mathrm{d}y=U$ is applied. The bias
voltage is denoted $U$. The values of the velocity profile measured at the
positions $y_{k}$ are denoted $vel_{k}$. The corresponding values of the model
are $u_{k}=\big{(}\mu_{h}(E_{k})+\mu_{e}(E_{k})\big{)}\cdot E_{k}$. For the
fit a $y$-range with $n_{k}$ values of $vel_{k}$ is selected, and

$\chi^{2}=\frac{1}{\sigma_{vel}^{2}}\sum_{k=1}^{n_{k}}\bigg{(}1-\frac{vel_{k}}{vscale\cdot
u_{k}}\bigg{)}^{2}+w_{pen}\sum_{i=2}^{n_{E}-1}\bigg{(}\frac{0.5\cdot(E_{i-1}+E_{i+1})-E_{i}}{E_{i}}\bigg{)}^{2}$
(2)

as a function of the parameters $E_{i}$ and _vscale_ is minimised. The scale
factor $vscale$ is introduced to normalise the $u_{k}$ values to the $vel_{k}$
values. The relative uncertainty of the $vel_{k}$ values is denoted
$\sigma_{vel}$, for which 2 % is assumed, and $w_{pen}$ is a penalty term,
which prevents excessive fluctuations of adjacent $E_{i}$ values. The choice
of the value of $w_{pen}$ is not too critical. A value was chosen, which
increases the $\chi^{2}$ of the fit by $\approx 50$ % compared to the value
for $w_{pen}=0$.

##  4 Electric field of the non-irradiated sensor for reverse bias

To test the analysis method, as a first step the velocity data of the non-
irradiated strip-sensor from Ref. [10] are analysed. The sensor parameters are
given in Table 1. The measurements were taken at $+20\leavevmode\nobreak\
^{\circ}$C for 25 reverse voltages between 25 and 500 V. The measured velocity
profiles are shown in Fig. 4.

Figure 4: Velocity profiles, $vel$, from Ref. [10] of the non-irradiated strip
sensor measured at $+20\leavevmode\nobreak\ ^{\circ}$C for reverse voltages
between 25 and 500 V in 25 V steps. For a given position $y$, $vel$ increases
with increasing bias voltage $U$.

The velocity profiles, $vel$, directly reflect the electric field and the
field dependence of the hole and electron mobilities. The maximum field is at
the $n^{+}p$ junction, which is close to $y=0$. This is also the region where
$vel$ has its maximum. Below full depletion, signals are only generated by the
charges produced in the depletion region, which expands to higher and higher
$y$ values with increasing bias voltage $U$. At low voltages the slope in the
falling part of $vel$, $\mathrm{d}vel/\mathrm{d}y$, is approximately
independent of $U$. This is expected as the slope of the electric field,
$\mathrm{d}E/\mathrm{d}y$, is constant for a uniform doping and at low fields
the carrier velocities are approximately proportional to the field. For
$U\gtrsim 180$ V the full depletion of the sensor is reached, and charges are
collected from the entire sensor. With increasing $U$ the velocity profiles
flatten, which reflects the decrease of the mobilities with electric field. In
the region of the $p^{+}$ backside implant close to $y=d\approx
300\leavevmode\nobreak\ \upmu$m, a spike in $vel$ is observed, in particular
for values of $U$ in the vicinity of the full-depletion voltage. It is caused
by the electric field from the diffusion of holes from the $p^{+}$ region into
the $p$ region. The effect decreases at low voltages, because of the finite
resistivity of the non-depleted silicon, which results in a time dependent
weighting field as discussed in Ref. [18]. The narrow width of the spike as
well as the sharpness of the decrease of $vel$ at the two sides of the sensor,
directly reflects the width of $7\leavevmode\nobreak\ \upmu$m FWHM of the
laser beam focus.

(a)

(b)

Figure 5:  (a) Results of the fits to the velocity profiles for the non-
irradiated sensor for different voltages. For clarity, the individual data are
shifted vertically by steps of 0.1. The symbols are the experimental data
$vel_{k}$ at the position $y_{k}$ and the lines the results of the model fit
using Eq. 2. (b) Electrical field values $E_{i}$ at the positions $y_{i}$ from
the fits (symbols). The dotted lines indicate the linear interpolation used to
calculate the velocity profiles of the model shown as lines in (a). The solid
lines are the electric fields expected for a pad diode with uniform doping.

Next the fits of the $vel$ data by the model discussed in Sect. 3 are
presented. The number of $E$ values to be fitted is chosen to be $n_{E}=14$,
and the $y$ positions of the $E_{i}$ values are $y_{i}=i\cdot
20\leavevmode\nobreak\ \upmu$m, with $i=1,\leavevmode\nobreak\
2,...,\leavevmode\nobreak\ n_{E}$ i. e. between $20\leavevmode\nobreak\
\upmu$m and $280\leavevmode\nobreak\ \upmu$m. Fig. 5 shows the results of the
fits to the data in the interval $y=11.6$ to $283.5\leavevmode\nobreak\
\upmu$m. It is seen that the model provides a good description of the
measurements, with a value of the $\chi^{2}$ as expected from the statistical
fluctuations of the data. Fig. 5(b) shows for selected $U$ values the results
for $E(y)$ (symbols) and the corresponding electric fields of a pad diode with
similar doping (solid straight lines). Typical differences are below $\pm
10$%. As expected, the biggest differences are observed for $y\lesssim
50\leavevmode\nobreak\ \upmu$m, where the influence of the strips, the surface
boundary conditions and the electronics response function are important.
Actually, the differences are smaller than expected from the discussions of
Sections. 3 and 8.

At low voltages, where the sensor is only partially depleted, the electric
field determined by the fit reproduces the linear $y$ dependence and the
expected depletion depth. At high voltages and small $y$ values the electric
fields are high and the drift velocities approach saturation. As a result the
accuracy of the field determination degrades and larger differences are
observed. Nevertheless, the results are satisfactory and demonstrate the
validity of the method.

To study the effects of the non-uniform fields at small $y$ values (see Fig.
3) and of the electronics response function, fits for different $y$ ranges are
performed. The results obtained when varying the lower $y$ value between
$11.6\leavevmode\nobreak\ \upmu$m and $51.3\leavevmode\nobreak\ \upmu$m, and
the upper $y$ value between $273.5\leavevmode\nobreak\ \upmu$m and
$283.5\leavevmode\nobreak\ \upmu$m, give equally good descriptions of the data
and compatible $E$ values.

Figure 6: The scaling factor $vscale(U)$ for the non-irradiated detector.

Fig. 6 shows the voltage dependence of $vscale(U)$, with which the signal from
the model calculation has to be multiplied in order to describe the velocity
profiles. It has to be introduced, as the relation between the initial slope
of the pulse, which is used to determine the velocity profiles, and the actual
velocities of the charge carriers, is not known with sufficient accuracy. An
increase of _vscale_ with decreasing $U$ is expected from the finite rise
time, $t_{r}$, of the initial current transient. As discussed in Sect. 8, a
finite $t_{r}$ causes a decrease of $vel$ compared to the value for $t_{r}=0$
close to the boundary of the sensor. For the measurements of this paper, the
effects become significant for $y\lesssim 40\leavevmode\nobreak\ \upmu$m and
for $y\gtrsim 270\leavevmode\nobreak\ \upmu$m. The absolute decrease is
approximately independent of $U$, however, the relative decrease is biggest at
low $U$. As a result _vscale_ has to increase with decreasing $U$ in order to
satisfy the condition $U=\int_{0}^{d}E\leavevmode\nobreak\ dy$.

To summarise this section: The model proposed in Sect. 3 provides an excellent
description of the velocity data from the edge-TCT measurements of the non-
irradiated silicon strip detectors in the 25 V to 500 V range of the
measurements. The extracted electric field distributions agree approximately
with the expectation for non-irradiated silicon strip detectors. In particular
the voltage dependence of the depletion depth and the linear decrease of the
electric field with the distance from the $n^{+}p$-junction are well
described. The scale factor between the model and the velocity, $vscale$,
depends on the voltage, which is understood only qualitatively. The results
obtained for the non-irradiated silicon strip detectors are sufficiently
encouraging, so that the method is used in the next section to extract the
electric fields in highly-irradiated silicon strip detectors.

##  5 Electric field of the irradiated sensors

###  5.1 Forward bias

In this section the analysis of the $vel$ data Ref. [10] for forward voltages
and three neutron fluences are presented. The voltage ranges are $25-300$ V
for $\Phi_{eq}=2\times 10^{15}$ cm-2, $25-500$ V for $\Phi_{eq}=5\times
10^{15}$ cm-2 and $25-400$ V for $\Phi_{eq}=10^{16}$ cm-2. The data were
recorded at $-20\leavevmode\nobreak\ ^{\circ}$C.

Figs. 7, 8 and 9 show the results of the fits to the velocity profiles. For
all $U$ values the velocity profiles are described by the fit within their
statistical uncertainties. Varying the $y$ range of the fit has no significant
effect on the $E$ values. At the lowest voltages $E$ is independent of $y$, as
expected for an ohmic resistor. At higher voltages a transition to a
$y$-dependent electric field is observed. The voltage at which the transition
takes place increases with $\Phi_{eq}$. In Sect. 5.3 these results are
discussed, and compared to the ones for reverse bias.

(a)

(b)

Figure 7:  (a) Fits to the velocity profiles for the sensor irradiated to
$\Phi_{eq}=10^{16}$ cm-2 for forward bias. For clarity, the individual data
are shifted vertically by steps of 0.1. The symbols are the experimental data
and the solid lines, which are hardly visible because of the good description
of the data, are the results of the model fit. (b) Electrical fields from the
fits to the velocity profiles. The lines indicate the linear interpolation
used to calculate the velocity profiles of the model shown as lines in (a).

(a)

(b)

Figure 8:  Same as Fig. 7 for $\Phi_{eq}=5\times 10^{15}$ cm-2.

(a)

(b)

Figure 9:  Same as Fig. 7 for $\Phi_{eq}=2\times 10^{15}$ cm-2.

###  5.2 Reverse bias

In this section the analysis of the $vel$ data for reverse voltages and four
neutron fluences from Ref. [10] are presented. The neutron fluences are
$\Phi_{eq}=(1,\leavevmode\nobreak\ 2,\leavevmode\nobreak\
5\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ 10)\times 10^{15}$
cm-2, and the voltage range $U=50\leavevmode\nobreak\
\mathrm{to}\leavevmode\nobreak\ 800$ V except for the $5\times 10^{15}$ cm-2
data, where the maximum voltage is 750 V. The data were taken at
$-20\leavevmode\nobreak\ ^{\circ}$C.

(a)

(b)

(c)

(d)

Figure 10:  Fits to the velocity profiles for reverse bias for the $\Phi_{eq}$
values (a) $10^{15}$ cm-2, (b) $2\times 10^{15}$ cm-2, (c) $5\times 10^{15}$
cm-2 and (d) $10^{16}$ cm-2. For clarity, the individual data are shifted
vertically by steps of 0.1. The symbols are the experimental data and the
solid lines, which are hardly visible because of the good description of the
data, are the results of the fit.

Fig. 10 compares the measured $vel$ profiles to the fit results for different
reverse voltages and $\Phi_{eq}$. The fits are performed in the range
$21\leavevmode\nobreak\ \upmu$m $\leqslant y\leqslant 278\leavevmode\nobreak\
\upmu$m. Changing the fit range makes little changes to the overall results,
however if the lower $y$ value is reduced or the upper $y$ value increased,
differences are observed in these $y$ ranges. Again, the model describes the
data within their statistical uncertainties.

(a)

(b)

(c)

(d)

Figure 11:  Electric fields determined by the fits to the velocity profiles
for reverse bias for the $\Phi_{eq}$ values (a) $10^{15}$ cm-2, (b) $2\times
10^{15}$ cm-2, (c) $5\times 10^{15}$ cm-2, and (d) $10^{16}$ cm-2.

Fig. 11 shows $E(y)$ obtained from the fits to the $vel$ profiles. The $y$
dependence $E(y)$ is opposite to what is observed for forward bias: maximal
fields towards $y=0$ and $y=d$ and a low-field region in-between. The
flattening of $E(y)$ towards $y=0$ at $\Phi_{eq}=10^{15}$ and $2\times
10^{15}$ cm-2 is ascribed to the effects of the $x$-dependence of $E$ in strip
sensors discussed in Sect. 3, the response function of the readout discussed
in Sect. 8 and the finite width of the light beam. The values of $E$ in the
central low-field region are compatible with the expectations from the field
produced by the dark current with the resistivity of intrinsic silicon similar
to the situation at low forward bias voltages. At $U=50$ V and
$\Phi_{eq}=10^{16}$ cm-2, $E(y)$ is constant over the entire sensor. With
increasing voltage and fluence $E$ increases at low $y$ values. At $U=800$ V
and $\Phi_{eq}=10^{16}$ cm-2, a value $E>100$ kV/cm is reached, and the regime
of charge multiplication is approached [27, 28, 29]. With increasing fluence
and voltage the extension of the high-field region at low $y$ decreases. Also
for $y$ values approaching $y=d$ an increase of $E(y)$ is observed, which is
called _double junction_ in the literature [2]. However, compared to the
electric field close to $y=0$, the value towards $y=d$ is much smaller and
does not exceed 10 kV/cm.

###  5.3 Discussion of the results

This section illustrates how the results of the analysis of the precise edge-
TCT data presented in this paper together with the dark current measurements
provides an insight into the effects of radiation damage in silicon detectors.
The formulae used for the following discussion can be found in standard
textbooks on semiconductor devices [3, 30]. From the electric field values,
$E_{i}$, the total charge carrier density, $N_{tot}(y)$, is obtained

$N_{tot}(y)=\frac{\rho(y)}{q_{0}}=\frac{\mathrm{d}E(y)}{\mathrm{d}y}\cdot\frac{\varepsilon_{Si}}{q_{0}}=N^{+}(y)-N^{-}(y)+n_{h}(y)-n_{e}(y).$
(3)

$N_{tot}(y)$ can be positive and negative. The densities of fixed positive and
negative space-charge carriers are denoted $N^{+}$ and $N^{-}$, respectively.
They are given by the sum over the densities of the individual states in the
silicon band gap multiplied with the probabilities that they are charged.
Whereas the densities of the individual states for a given fluence $\Phi_{eq}$
are expected to be independent of position and voltage, the probabilities that
the states are charged are not. They depend on the densities of free
electrons, $n_{e}(y)$, and holes, $n_{h}(y)$, which are related to the current
density, $\vec{j}$, and the electric field by

$\vec{j}=\vec{j}_{e}+\vec{j}_{h}=\mu_{e}\cdot(n_{e}\cdot
q_{0}\cdot\vec{E}+k\cdot T\cdot\vec{\nabla}n_{e})+\mu_{h}\cdot(n_{h}\cdot
q_{0}\cdot\vec{E}-k\cdot T\cdot\vec{\nabla}n_{h}).$ (4)

For the steady state situation, $\partial n_{e}/\partial t=\partial
n_{h}/\partial t=0$ and from the current-continuity equation follows
$\vec{\nabla}\vec{j}=0$. From the additional model assumption that the
electric field depends only on $y$ follows that $\vec{j}$ has only a
$y$-component and that $j$ is constant in the entire detector. Its value is
given by the dark current divided by the sensor area. Under the same
assumptions the separate current-continuity equations for holes and electrons
are

$\mathrm{d}j_{h}/\mathrm{d}y=q_{0}\cdot(G_{h}-U_{h})\hskip
14.22636pt\mathrm{and}\hskip
14.22636pt\mathrm{d}j_{e}/\mathrm{d}y=-q_{0}\cdot(G_{e}-U_{e}),$ (5)

with the generation rates, $G_{h}$ and $G_{e}$, and the recombination rates,
$U_{h}$ and $U_{e}$, for holes and electrons, respectively. In the absence of
an external source which generates electron-hole pairs in the detector,
$G_{h}=G_{e}$ and $U_{h}=U_{e}$. In addition, $E(y)$ is constrained by
$\int_{0}^{d}E(y)\leavevmode\nobreak\ \mathrm{d}y=U$, and the sign of $E(y)$
is the same in the entire detector.

It has to be noted that taking derivatives can result in significant
fluctuations. To avoid such problems, in Eq. 2 the number $n_{E}$ and the
positions $y_{i}$, where the $E_{i}$ values are determined by the fit, as well
as $w_{pen}$ have to be optimised for every voltage and fluence. Such a study
has not been done, and therefore just two examples, one for forward and one
for reverse bias are shown, for which the fluctuations after differentiation
appear acceptable. Fig. 12 shows the two examples for $N_{tot}(y)$ for
selected bias voltages $U$.

(a)

(b)

Figure 12:  Total charge-carrier density distributions, $N_{tot}(y)$, obtained
from the derivative $\mathrm{d}E/\mathrm{d}y$ for (a) forward bias and
$\Phi_{eq}=2\times 10^{15}$ cm-2, and (b) reverse bias and $\Phi_{eq}=10^{16}$
cm-2. The dashed lines show $+N_{tot}$ and the solid lines $-N_{tot}$. Note
the different scales in (b) for $-N_{tot}$ (left) and $+N_{tot}$ (right).

For forward bias, Fig. 12(a) shows that $N_{tot}(y)$ is positive at low $y$
and negative at high $y$. This is expected, because electrons move in the
$+y$-direction and holes in the $-y$-direction, resulting in an excess of
holes at low $y$ and an excess of electrons at high $y$, which changes the
occupation of the states in the silicon band gap accordingly. For reverse bias
the situation is opposite: holes move in the $+y$-direction and electrons in
the $-y$-direction, and the sign of $N_{tot}(y)$ is reversed, as can be seen
in Fig. 12(b). In both cases the sign of $N_{tot}$ changes approximately in
the middle of the detector, where extended low $|N_{tot}|$-regions are
observed. Towards the electrodes $|N_{tot}|$ has maxima.

First the results for forward bias and low voltages ($U\lesssim 50$ V for
$\Phi_{eq}=2\times 10^{15}$, and $\lesssim 100$ V for $10^{16}$ cm-2) are
discussed. Electrons are injected through the forward-biased $n^{+}p$-junction
at $y=0$. They drift in the $+y$-direction and along their path recombine
until the recombination-generation equilibrium is reached. Similarly, holes
are injected through the $p^{+}p$-junction at $y=d$, recombine on their path
in the $-y$-direction until the recombination-generation equilibrium is
reached. In these transition regions, a small increase of $|N_{tot}|$,
typically below $10^{12}$, is observed. In the central equilibrium region, the
density of electrons is expected to be
$n_{e}=\sqrt{\mu_{h}(0)/\mu_{e}(0)}\cdot n_{i}$, and the density of holes
$n_{h}=\sqrt{\mu_{e}(0)/\mu_{h}(0)}\cdot n_{i}$ [19] The low-field mobilities
are $\mu_{e}(0)$ and $\mu_{h}(0)$, and the intrinsic charge carrier density
$n_{i}\approx 1.2\times 10^{8}$ cm-3 at $-20\leavevmode\nobreak\ ^{\circ}$C.
Fig. 12(a) shows that in this region $|N_{tot}|<10^{11}$ cm-3, and the
electric field is the result of the ohmic resistance of the current given by
Eq. 4: $E=j/\big{(}q_{0}\cdot(n_{e}\cdot\mu_{e}+n_{h}\cdot\mu_{h})\big{)}$.
For low voltages, the electric field shown in Fig. 9(b) agrees within $\pm 20$
% with this prediction. At small $y$-values, where $n_{e}\gg n_{i}$ the value
of $E$ has to decrease in order to satisfy the requirement of a constant $j$.
The situation is analogous for $y$ approaching $d$, where holes are injected
through the $p^{+}p$-junction and $n_{h}\gg n_{i}$. Fig. 9(b) shows that this
decrease is actually observed at both sides of the detector. The slower
increase of $E(y)$ for low $y$-values compared to the decrease at high
$y$-values is expected because $\mu_{e}>\mu_{h}$. The observation from Figs.
7(b), 8(b) and 9(b) that at higher $\Phi_{eq}$-values the equilibrium
condition is reached at shorter distances and also extends to higher voltages,
is explained by the higher density of radiation-induced states and therefore
higher recombination rates. For higher forward voltages, regions with positive
$N_{tot}$ develop for small values of $y$, and with negative $N_{tot}$ at
larger $y$-values. In-between, $N_{tot}$ remains low, and the electric field
is high and approximately constant with a value, well in excess of $U/d$,
which can be seen in Fig. 9(b). This region is similar to a depletion region
with low doping concentration.

The situation for reverse bias, where the electric field points in the
$+y$-direction, is very different. The reverse-biased $n^{+}p$-junction
prevents the injection of holes at $y=0$, and the $p^{+}p$-junction at $y=d$
the injection of electrons, and the dark current is caused by the generation
of free charge carriers. As can be seen in Fig. 12(b) for reverse voltages as
low as 50 V, $|N_{tot}|\lesssim 10^{11}$ cm-3, and the electric field is
approximately uniform and given by the ohmic resistivity of the damaged
silicon and the current density (see Fig. 11(d)). Thus the situation is
similar to the one for low forward voltages. With increasing reverse voltage,
regions of high $|N_{tot}|$ develop towards both electrodes. Their sign is
negative for small $y$, where the dark current is dominated by electrons, and
positive at high $y$-values, where it is dominated by holes. As can be seen
from Fig. 12(b) the value of $|N_{tot}|$ at low $y$ is approximately a factor
20 higher than at high $y$. It can also be seen that with increasing voltage,
the two boundaries of the high-$|N_{tot}|$ region move towards the centre of
the detector and the low-field region shrinks. However, for
$\Phi_{eq}=10^{16}$ cm-2 even at a voltage as high as 800 V, an ohmic low-
field region remains in the centre.

In the region of high $|N_{tot}|$ the contribution of $n_{e}$ and $n_{h}$ is
negligible, and $N_{tot}\approx N^{+}-N^{-}$. As for a given $\Phi_{eq}$ the
density of radiation-induced levels does not depend on position nor on applied
voltage, the value of $N_{tot}$ directly reflects the fraction of these levels
which are in a charged state. It is also noted that typical values for the sum
of the introduction rates of all defects is of order 1 cm-1 [31] resulting in
a density of approximately $10^{16}$ cm-3 defect states for
$\Phi_{eq}=10^{16}$ cm-2. Thus the observation of $|N_{tot}|\approx 10^{14}$
cm-3 indicates that only a small fraction these states are charged. It should
also be noted that, as discussed in Refs. [8, 31], the introduction rates for
the different damage states and the removal of dopants depend on the type of
particles used for the irradiation, and the results for irradiation with high-
and low-energy charged hadrons can well be quite different.

Comparing $E(y)$ for forward and reverse bias reveals that for forward bias
the region of high electric fields extends over a significantly larger region
of the detector. This explains why for high $\Phi_{eq}$ the measured charge
collection for forward bias is higher than for reverse bias [21, 32, 33].

##  6 Summary and conclusions

A straight-forward method is proposed and used to determine the position-
dependent electric field and the total charge density of radiation-damaged
silicon detectors from velocity profiles measured by edge-TCT (Transient
Current Technique). The velocity profiles are extracted from the initial
slopes of the current transients from charges produced by the light of a near-
infrared laser beam injected into a strip sensor parallel to its surface.
Simulations of the current transients for a non-irradiated sensor convolved
with a simplified response function are used to investigate the validity and
the limitations of the method. The main difficulties arise from the finite
bandwidth of the readout and the poor knowledge of the electric field in the
region close to sensor surface at which the strip electrodes are located. For
the strip sensors investigated the effect is particularly important because of
the small ratio of implant-width to strip-pitch of $20\leavevmode\nobreak\
\upmu$m to $100\leavevmode\nobreak\ \upmu$m.

The method is first applied to data from a non-irradiated $n^{+}p$ sensor of
$300\leavevmode\nobreak\ \upmu$m thickness for reverse voltages below and
above the full-depletion voltage. The model provides a description of the
measured velocity profiles within their statistical uncertainty and the
electric fields approximately agree with the expectations of a non-irradiated
pad sensor. In particular the expected voltage dependence of the depletion
depth is well reproduced.

Next the velocity profiles of $300\leavevmode\nobreak\ \upmu$m thick $n^{+}p$
strip sensors irradiated by reactor neutrons to neutron-equivalent fluences
$\Phi_{eq}=(2,\leavevmode\nobreak\ 5\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ 10)\times 10^{15}$ cm-2 for forward voltages
between 25 V and up to 500 V are analysed. Again, the model describes the
experimental data within their statistical uncertainties. In agreement with
expectations it is found that at low voltages the electric field in the sensor
is the result of the ohmic voltage drop of the dark current in silicon with
intrinsic resistivity. The intrinsic resistivity is the result of the
generation-recombination equilibrium reached in highly radiation-damaged
silicon at low electric fields. At higher voltages a high field region
develops in the centre of the detector, which extends over most of the sensor
depth. This explains the observation that above a certain radiation dose the
charge collection efficiency is higher for forward than for reverse bias.

Finally the method is used to analyse the velocity profiles for reverse
voltages between 50 V and 800 V of a $300\leavevmode\nobreak\ \upmu$m thick
$n^{+}p$ strip sensor irradiated by reactor neutrons to
$\Phi_{eq}=(1,\leavevmode\nobreak\ 2,\leavevmode\nobreak\
5\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ 10)\times 10^{15}$
cm-2. Again, the model describes the experimental data within their
statistical uncertainties. The electric field has its maximum at the
$n^{+}p$-junctions of the readout strips. The maximum field increases with
reverse voltage, and also with $\Phi_{eq}$ at higher voltages. At 800 V its
value exceeds 100 kV/cm, which is close to the onset of charge multiplication
for electrons. The extension of the high-field region decreases with
increasing fluence. At 800 V and $\Phi_{eq}=10^{16}$ cm-2 the high-field
region is less than $100\leavevmode\nobreak\ \upmu$m deep, which explains the
observation that for high radiation damage the charge collection of
$150\leavevmode\nobreak\ \upmu$m and $300\leavevmode\nobreak\ \upmu$m thick
sensors is similar, i. e. that the charge collection efficiency of thin
silicon sensors is superior. With increasing $\Phi_{eq}$ and increasing
reverse voltage an increase of the electric field at the $p^{+}p$ back side of
the sensor is observed – a phenomenon known as double junction. However, the
maximum field is below 10 kV/cm, thus more than an order of magnitude smaller
than at the $n^{+}p$-junction. From the derivative of the electric field with
respect to the depth in the sensor the position-dependent total charge
density, $N_{tot}$, is estimated and its sign determined. For
$\Phi_{eq}=10^{16}$ cm-2 and reverse voltages above 300 V $N_{tot}\approx
10^{14}$ cm-3 and the charge sign is negative. In the region towards the
$p^{+}p$-junction $N_{tot}$ increases steadily with increasing reverse voltage
and a value of $N_{tot}\approx 7\times 10^{12}$ cm-3 is reached at 800 V. The
charge sign is positive in this region.

It is concluded that the method proposed in this paper successfully describes
the velocity profiles from edge-TCT measurements and allows to extract the
position-dependent electric field and the charge density in highly radiation-
damaged silicon sensors. Although the main emphasis of the paper is on the
method and its limitations, its application to sensors damaged by reactor
neutrons already provides quite some insight into the changes of the detector
properties due to radiation damage. Such information can also be used to
verify the results of TCAD simulations based on radiation-induced states in
the silicon band gap. As a next step the method should be used to analyse
edge-TCT data for sensors irradiated by charged particles of different
energies to understand the impact on the sensor performance of the
experimentally observed differences in introduction rates of the different
damage states. As one of the uncertainties in the analysis is the small ratio
of strip implant to strip pitch, measurements with sensors with a smaller
inter-strip distance are desirable.

An extension of the method to analyse the entire current transient and not
only its initial rise appears feasible. At least in principle, this extension
would allow to determine the position-dependent lifetimes of holes and
electrons in addition to the position-dependent electric field. Such
information would enable a complete description of the performance of highly
radiation-damaged sensors and thus be an important element in the simulation
and data analysis of silicon sensors in high-radiation environments.

##  7 Appendix A: Effective weighting field for edge-TCT

In Sect. 3 it is shown that for a charge distribution uniformly distributed in
$x$ (see Fig.2) the effective weighting field required to evaluate the edge-
TCT data, $\overrightarrow{\mathcal{E}_{w}}=\hat{e}_{y}/d$, the weighting
field of a fully depleted pad detector. In this section numerical calculations
are used to calculate $\overrightarrow{\mathcal{E}_{w}}(y)$ for a charge
distribution $\propto e^{-x/\lambda_{abs}}$, where $\lambda_{abs}$ is the
light-absorption length. For the wavelength $\lambda=1064$ nm, the light used
for the measurements, $\lambda_{abs}\approx 1.7$ mm for non-irradiated silicon
at a temperature of $-20\leavevmode\nobreak\ ^{\circ}$C [34]. At this
wavelength $\lambda_{abs}$ decreases with temperature. In addition, a decrease
by approximately 15 % after irradiation to $\Phi_{eq}=10^{16}$ cm-2 has been
observed at $+20\leavevmode\nobreak\ ^{\circ}$C [15].

Using the program of Ref. [23] the weighting field of the strip sensor has
been simulated. The center of the central strip, to which the potential of +1
V was applied, is positioned at $x=y=0$ (see Fig. 2). A total of 13 strips, 6
for negative and 6 for positive $x$ were simulated. The potential of the
strips excluding the central one, of the rear electrode at $y=d$ and of the
silicon at the centers of the outer strips at $x=\pm\leavevmode\nobreak\
600\leavevmode\nobreak\ \upmu$m were set to zero. For the interface in-between
the strips at $y=0$ symmetric boundary conditions were assumed. The simulated
potential has a similar shape as the potential of half of the potential shown
on the right-hand side of Fig. 3 for the Neumann boundary conditions on the
SiO2 surface. Both $x$ and $y$-components of the weighting field,
$\overrightarrow{{E}_{w}}$, vary strongly with $x$ and $y$.

The effective weighting field is obtained from the integral
$\overrightarrow{\mathcal{E}_{w}}=\big{(}\int\overrightarrow{{E}_{w}}(\vec{r})\cdot
e^{-x/\lambda_{abs}}\leavevmode\nobreak\ \mathrm{d}x\big{)}/p$. As expected,
for $\lambda_{abs}\gg p$ the $x$-component of
$\overrightarrow{\mathcal{E}_{w}}$ is zero and the $y$-component is
independent of $y$. However, the value is $\approx 30$ cm-1 instead of
$1/d=33$ cm-1. The reason is that the region outside of the
$\pm\leavevmode\nobreak\ 600\leavevmode\nobreak\ \upmu$m simulated contributes
$\approx 10$ % to $\mathcal{E}_{w}$. It is concluded that edge-TCT
measurements should not be taken for strips too close to the edge of the
sensor, and that several of the strips adjacent to the readout strip have to
be connected to ground, ideally with the impedance of the readout electronics.

(a)

(b)

Figure 13: Influence of a finite light absorption length $\lambda_{abs}$ on
$y$-dependence of the effective weighting field ${\mathcal{\vec{E}}}_{w}$ for
a strip sensor with the thickness $d=300\leavevmode\nobreak\ \upmu$m and the
strip pitch $p=100\leavevmode\nobreak\ \upmu$m. (a) $R_{y}$, the ratio of the
$y$-component of $\mathcal{\vec{E}}_{w}$ for different values of
$\lambda_{abs}$ to the value of $\mathcal{E}_{w}$ for $\lambda_{abs}=1$ m. (b)
$x$-component of $\mathcal{\vec{E}}_{w}$ for different values of
$\lambda_{abs}$, which can be compared to the $y$-component of
$\mathcal{E}_{w}=1/d=33.3$ cm-1 for $\lambda_{abs}\gg p$.

Fig. 13(a) shows $R_{y}(y)$, the ratio of the $y$-component of
$\overrightarrow{\mathcal{E}_{w}}$ for different values of $\lambda_{abs}$ to
the one for $\lambda_{abs}=1\leavevmode\nobreak\ \mathrm{m}$, i. e.
$\lambda_{abs}\gg p$. It can be seen that for $\lambda_{abs}=2$ mm, which is
close to the value for the laser light used for the measurements, $R_{y}=1$
within $\pm\leavevmode\nobreak\ 1$ %, and the deviation from 1 is small
compared to other uncertainties of the method. Fig. 13(b) shows the
$x$-component of $\overrightarrow{\mathcal{E}_{w}}$ for different values of
$\lambda_{abs}$. For small $y$, values as high as $\approx 5$ cm-1 are found
for $\lambda_{abs}=2$ mm, which corresponds to $\approx 15$ % of the
$y$-component of $\mathcal{E}_{w}$. Its effect on the analysis depends on the
electric field distribution in the region of the readout plane, which is not
known. No further study has been made on this topic.

##  8 Appendix B: Influence of electronics and charge carrier lifetimes on the
velocity profiles

In this appendix simulations are performed with the aim to investigate the
influence of the finite bandwidth of the readout electronics and the finite
lifetimes of electrons and holes on the determination of the electric field
from the velocity profiles.

In order to use electric fields in the simulations, which are similar to the
ones shown in Fig. 11(d), a pad sensor with a doping density of $2\times
10^{13}$ cm-3 has been simulated. The corresponding depletion voltage is
$\approx 1400$ V for a $300\leavevmode\nobreak\ \upmu$m thick sensor. In the
non-depleted region of the sensor the electric field is assumed to be 1 kV/cm
because of the resistivity of the radiation-damaged silicon and the dark
current. No increase of the field towards $y=d$, the _double junction_ , is
implemented, as for the irradiated detectors studied the effect is minor. In
the following results for two voltages, 200 V and 1000 V, are shown, for which
the maximum fields at $y=0$ are 35 kV/cm and 80 kV/cm, respectively. The
depletions depths for these two voltages are $115\leavevmode\nobreak\ \upmu$m
and $275\leavevmode\nobreak\ \upmu$m. It has been checked that the conclusions
from these simulation are the same if these quite arbitrary assumptions are
changed.

For the effective weighting field $\hat{e}_{y}/d$ is assumed. For the drift of
the charge carriers the mobility parametrisation of Ref. [20] is used and
effects of charge diffusion are neglected. With the help of Eq. 1 the time
dependence of the induced current $I(t;y_{0})$ for a uniform chain of
electron-hole pairs produced at $y_{0}$ is calculated. Examples for the
generated current transients are shown in Fig. 14 as dashed lines.

(a)

(b)

(c)

(d)

Figure 14:  Current transients for the simulations discussed in the text for
electron-hole pairs generated uniformly in $x$ at $t=0$ and different $y$
values. The transients before (dashed lines) and after (solid lines) the
convolution with a Gaussian with $\sigma_{t}=250$ ps for the voltages $U=200$
V and $U=1000$ V are shown for (a) $y=25\leavevmode\nobreak\ \upmu$m, (b)
$y=50\leavevmode\nobreak\ \upmu$m, (c) $y=100\leavevmode\nobreak\ \upmu$m, and
(d) $y=200\leavevmode\nobreak\ \upmu$m.

To investigate the influence of the electronic response function on the
initial rise of the measured transient, the simulated current transient,
$I(t)$, is convolved with a Gaussian response function with an rms time spread
$\sigma_{t}$. Both functions are evaluated at 512 time steps in the interval
between $-3$ ns to $+15$ ns, resulting in a time step of $\approx 35$ ps. The
electron-hole pairs are generated at $t=0$, and the Gaussian response function
is also centred at $t=0$. The method of the Fast-Fourier-Transform (FTT) is
used for the convolution. For the estimation of $\sigma_{t}$ the convolved
current simulated for the non-irradiated sensor at $y=50\leavevmode\nobreak\
\upmu$m is compared to the measured current shown in Fig. 1. The value of
$\sigma_{t}$ is varied until the rise time of the convolved current agrees
with the measured value. The value found is $\sigma_{t}\approx 250$ ps, which
corresponds to a bandwidth $BW=0.0935\cdot\sqrt{2}/\sigma_{t}\approx 530$ MHz
and a rise time $t_{r}=0.3394/BW\approx 640$ ps.

Fig. 14 shows several examples of current transients before and after
convolution. It is noted that for small $y$ values the electrons which drift
to $y=0$ produce a narrow flat region, which is strongly suppressed by the
convolution. As a result the maximum of the convolved transient is much lower
than the original transient. For larger values of $y$ the width of the
electron signal becomes wider than the full width, $\Gamma_{t}$, of the
Gaussian response function, and the maxima of the transients before and after
convolution become similar. We thus expect that the determination of the
electric field from the velocity profiles will only be reliable for
$y>v_{e}\cdot\Gamma_{t}$, where $v_{e}$ is the electron velocity close to
$y=0$. A similar effect from holes is expected at $y=d$. However, for the
neutron-irradiated sensors studied in this paper, the electric field is much
lower in this region than at $y=0$, and the effects are much smaller. In
principle this effect can be taken into account in the analysis, which,
however, has not been done.

Two methods of determining the velocity profiles from the convolved transients
are investigated for the simulated data:

  1. 1.

The slope method, where it is found that taking the maximum slope or the slope
at $t=0$ makes hardly a difference.

  2. 2.

The time integral method, for which the transient is integrated up to the time
$t_{int}$.

(a)

(b)

Figure 15: Comparison of velocity profiles before (dotted lines) and after
(solid lines) the convolution of the simulated current transients for $U=200$
V and $U=1000$ V for (a) the slope method, and (b) the charge integration
method for $t_{int}=-250$ ps.

Fig. 15 compares the results of the two methods. For the slope methods
significant deviations between the simulated velocity profile and the velocity
profile derived after the convolution appear below $y=50\leavevmode\nobreak\
\upmu$m. Differences are also observed at the transition between the depleted
and the non-depleted region ($y=120\leavevmode\nobreak\ \upmu$m for 200 V, and
$y=260\leavevmode\nobreak\ \upmu$m for 1000 V). For the charge integration
method, the integration time $t_{int}$ can be optimised. It is found that for
$t_{int}\gtrsim 250$ ps the results are identical to method 1. However, for
$t_{int}=-250$ ps, i. e. the integration of the convolved transient starts
$\sigma_{t}$ before the pulse without convolution, the agreement can be
extended down to $\approx 30\leavevmode\nobreak\ \upmu$m and the deviations
when approaching the non-depleted region disappears. However such a short
integration time results in significant experimental errors, and thus cannot
be used in the analysis.

For the study of the effects of the finite lifetimes of the free charge
carriers, $\tau_{i}$, the numbers of the drifting charges are reduced
following an exponential in time, and the resulting velocity profiles are
compared to the velocity profiles for $\tau_{i}$ values large compared to
$t_{trans}$, the transit time of the charge carriers. For both methods, where
$t_{int}=+250$ ps is used for method (2), the results are similar and,
assuming $\tau=\tau_{e}=\tau_{h}$, it is found from the simulations that the
relative reduction of $vel$ is approximately
$\sigma_{t}/\tau=0.25\leavevmode\nobreak\ \mathrm{ns}/\tau$, with $\sigma_{t}$
the rms width of the Gaussian transfer function introduced before. As the
$\tau_{i}$ values of silicon irradiated to fluences of $\Phi_{eq}=10^{16}$
cm-2 are only a few ns, the effect is significant. However, in particular if
the $\tau_{i}$ values are independent of position, the reduction of $vel$ is
similar in the entire sensor and partially compensated by the requirement
$\int_{0}^{d}E(y)\leavevmode\nobreak\ \mathrm{d}y=U$.

To summarise this Appendix: The simulations show that both, the finite
bandwidth of the readout and free charge-carrier lifetimes of a few
nanoseconds and less, significantly influence the determination of the
electric field from velocity profiles. Whereas the effect of the finite
bandwidth is limited to the edges of the sensor, short lifetimes affect the
electric field determination in the entire sensor. To avoid these
uncertainties the entire transients and not only its initial rise should be
used in the analysis. However, this requires the precise understanding of the
electronic transfer function, which could be possibly achieved using the
method described in Refs. [35, 36].

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