# Determination of the p-spray profile for n+p silicon sensors using a MOSFET

E. Fretwurst  E. Garutti  R. Klanner  I. Kopsalis  J. Schwandt  M. Weberpals
Institute for Experimental Physics, University of Hamburg,
Luruper Chaussee 147, D 22761, Hamburg, Germany.

###### Abstract

The standard technique to electrically isolate the $n^{+}$ implants of
segmented silicon sensors fabricated on high-ohmic $p$-type silicon are
$p^{+}$-implants. Although the knowledge of the $p^{+}$-implant dose and of
the doping profile is highly relevant for the understanding and optimisation
of sensors, this information is usually not available from the vendors, and
methods to obtain it are highly welcome. The paper presents methods to obtain
this information from circular MOSFETs fabricated as test structures on the
same wafer as the sensors. Two circular MOSFETs, one with and one without a
$p^{+}$-implant under the gate, are used for this study. They were produced on
Magnetic Czochralski silicon doped with $\approx 3.5\,10^{12}$ cm-2 of boron
and $\langle 100\,\rangle$ crystal orientation. The drain-source current as
function of gate voltage for different back-side voltages is measured at a
drain-source voltage of 50 mV in the linear MOSFET region, and the values of
threshold voltage and mobility extracted using the standard MOSFET formulae.
To determine the bulk doping, the implantation dose and profile from the data,
two methods are used, which give compatible results. The doping profile, which
varies between $3.5\,10^{12}$ cm-3 and $2\,10^{15}$ cm-3 for the MOSFET with
$p^{+}$-implant, is determined down to a distance of a fraction of a $\upmu$m
from the Si-SiO2 interface. The method of extracting the doping profiles is
verified using data from a TCAD simulation of the two MOSFETs. The details of
the methods and of the problems encountered are discussed.

###### keywords:

Silicon pixel sensor , p-type silicon , p-spray , doping profile , MOSFET ,
TCAD simulations.

††journal: Elsevier

###### Contents

  1. 1 Introduction
  2. 2 MOSFETs investigated and measurement setup
  3. 3 Data analysis and results
    1. 3.1 MOSFET parameters extracted from the $I_{ds}(V_{gate})$ measurements
    2. 3.2 Doping determination: _Method 1_
    3. 3.3 Doping determination: _Method 2_
    4. 3.4 Comparison to TCAD simulations
  4. 4 Summary and conclusions
  5. 5 List of References

##  1 Introduction

In segmented $n^{+}p$ silicon sensors positive charges in the SiO2 close to
the Si-SiO2 interface can cause an electron accumulation layer, which
essentially shortens the $n^{+}$ implants of the electrodes. The positive
oxide charges are the result of the growing of the SiO2 on the Si. Radiation
damage due to ionising radiation typically further increases the density of
positive oxide charges. A $p^{+}$ implantation, either over the entire wafer
($p$-spray) or as strips ($p$-stop) or a combination of both is frequently
used to isolate the $n^{+}$ electrodes [1, 2, 3, 4]. In most cases the
implantation dose and the following thermal activation process is not
communicated by the vendor. However, the knowledge of the value and of the
density profile of active acceptors is required to understand and simulate the
performance of the sensors. This is particularly relevant if the sensors are
operated in a high radiation field, like at the CERN LHC or the European X-ray
Free-Electron Laser, EuXFEL. Therefore, reliable methods for determining the
profile of active acceptors are highly desirable. For electronics applications
a number of methods, both destructive and non-destructive, are readily
available. An overview can be found in Ref. [5]. Given the high resistivity of
several k$\Omega$ cm of the silicon used for detector fabrication, the
applicability and accuracy of the different methods has to be evaluated.

In this paper we use current-voltage measurements in the linear region of one
circular $n$ MOSFET with and a second one without a $p$-spray implant, to
determine the value and the profile of the $p$-spray implants. In addition,
the electron mobilities in the inversion layer at the Si-SiO2 interface as
function of the electric field normal to the interface for the two MOSFETs are
determined. The MOSFETs have been fabricated by Hamamatsu [6] on $\thicksim 4$
k$\Omega$ cm $p$-type silicon together with test sensors for the _CMS HPK
Campaign_ [7, 8, 9] of the CMS Collaboration working at the CERN LHC. For a
verification of the method, data from TCAD simulations of the two MOSFETs are
analysed with the same software as the experimental data, and input and
results compared. The paper presents the problems encountered using the
standard methods of the MOSFET analysis developed for electronics and how some
of them could be overcome. More information on the measurements and the
analysis can be found in [10].

##  2 MOSFETs investigated and measurement setup

The MOSFETs were fabricated on Magnetic Czochralski $p$-type silicon with the
crystal orientation $\langle 100\,\rangle$. Fig. 1(a) shows a cross section of
the circular MOSFET without $p$-spray implant. The thickness of the Si is
approximately $200\,\upmu$m. The Si-bulk dopant density, derived from the
$C-V$ measurement of pad diodes is $CN_{bulk}=(3.3\pm 0.3)\,10^{12}$ cm-3,
where the spread of the measured depletion voltage from different samples and
the uncertainty of the effective silicon thickness contribute about equally to
the uncertainty. Here and in the following we use $CN$ for the volume dopant
concentration with units [cm-3] and $N$ for the area dopant concentration with
units [cm-2]. The maximum dopant densities of the $n^{+}$ implants of Source
and Drain and of the $p^{+}$ back contact are approximately $10^{19}$ cm-3,
and the junction depths are about $2\,\upmu$m. The oxide thickness, determined
using capacitance measurements on MOS capacitors, is $t_{ox}=700\pm 5$ nm. The
metal overlaps of the gate over the $n^{+}$ implants are estimated to be about
$4\,\upmu$m.

Following the nomenclature of the _CMS HPK Campaign_ the MOSFET without
$p$-spray implant is called M200P, and the MOSFET with $p$-spray implant
M200Y.

(a)

(b)

Figure 1:  (a) Schematic cross section of the MOSFET. The dimensions are taken
from the GDS files of the photomask and (b) measurement setup.

Fig. 1(b) shows the biasing scheme for the MOSFET measurements, which were
made on a prober station at approximately $20^{\circ}$C in ambient atmosphere.
The Source was put on ground potential. The Drain was biased at $V_{ds}=50$
mV, and the Drain-Source current $I_{ds}$ was measured using a Keithley 6487
PicoAmmeter/Voltage Source. The backside voltage $V_{back}$ was set manually
in the range 0 to $-30$ V for the M200P, and from $+0.5$ V to $-30$ V for the
M200Y. As the extracted value of the doping concentration is very sensitive to
the exact value of $V_{back}$, this voltage has been recorded with an accuracy
at the 1 mV level, which is more precise than the setting accuracy of the
voltage source. For a given value of $V_{back}$, $V_{gate}$ was ramped from
$-6$ V to $+16$ V and $I_{ds}$ recorded. It was verified that the results for
ramping $V_{gate}$ up and down are compatible.

##  3 Data analysis and results

###  3.1 MOSFET parameters extracted from the $I_{ds}(V_{gate})$ measurements

Fig. 2 shows a selection of the $I_{ds}(V_{gate},V_{back})$ results. For the
M200Y measurements and $V_{back}>0.3$ V, the $p^{+}n$ junctions of Source and
Drain approach forward biasing and the diffusion current contributes
significantly to $I_{ds}$. Therefore for these data the $I_{ds}$ current
measured at $V_{gate}=-6$ V has been subtracted. Comparing the results of
M200Y, the MOSFET with $p$-spray implant, to the ones of M200P, the MOSFET
without $p$-spray implant, one notices: For $V_{back}\lesssim-2$ V, apart from
a shift of $V_{gate}$ by about 7 V, the curves and their spacings with
$V_{back}$ are similar, and for $V_{back}\gtrsim-2$ V, the spacings remain
approximately constant for M200P, but increase rapidly for M200Y. These
differences are caused by the $p$-spray implant, as will be shown in Sect.
3.2. In addition, the shapes of all curves are similar with the exception of
the M200Y measurement at $V_{back}=0.5$ V. This difference can be described by
a change of the electron mobility at the Si-SiO2 interface.

(a)

(b)

Figure 2:  Measured $I_{ds}(V_{gate})$ at $V_{ds}=50$ mV for the MOSFET (a)
M200P and (b) M200Y.

To extract the MOSFET parameters, the standard formula for an $n$-MOSFET in
the linear region, adapted for the circular geometry, is used [11, 12, 13]:

$I_{ds}\approx\frac{W}{L}\cdot\mu_{e}\cdot C_{ox}\cdot(V_{gate}-V_{th})\cdot
V_{ds}.$ (1)

The width-over-length ratio for the circular MOSFET is given by
$W/L=2\pi/\ln(r_{2}/r_{1})=4.964$, with $r_{1}$ the outer radius of the
Source-implant, and $r_{2}$ the inner radius of the Drain-implant. The value
of the oxide capacitance $C_{ox}=4.933$ nF/cm2. The mobility of the electrons
is denoted by $\mu_{e}$, where the following parametrisation of its dependence
on $V_{gate}$ and $V_{th}$ has been used [5]:

$\mu_{e}=\mu_{0}\cdot\frac{1}{1+\frac{V_{gate}-V_{th}}{V_{1/2}}},$ (2)

with $\mu_{0}$ the electron mobility at the Si-SiO2 interface for
$V_{gate}-V_{th}=0$, and $V_{1/2}$ the value of $V_{gate}-V_{th}$ at which the
mobility has decreased by a factor 2 relative to $\mu_{0}$, and the threshold
voltage $V_{th}$. The measurements were taken in the linear MOSFET region at
$V_{ds}=50$ mV. For $V_{back}=0$ it has been verified that in the range 25 mV
to 200 mV the results do not depend on the choice of $V_{ds}$ [14].

To determine the free parameters of the model, $V_{th}$, $\mu_{0}$ and
$V_{1/2}$, Eq. 1 was fitted to the data shown in Fig. 2. Figs. 3 and 4 show
the dependence on $V_{back}$ of the parameters determined. For M200P the
$V_{gate}$ voltage range selected for the fit was $\approx 2.5$ V above
$V_{th}$; the model describes the data within $\approx 0.2$ % and the
statistical errors obtained from the fit are $\delta V_{th}\approx 3.5$ mV,
$\delta\mu_{0}\approx 1.5$ cm2/V s and $\delta V_{1/2}\approx 0.2$ V if an
uncertainty of the $I_{ds}$ measurement of 0.1 % is assumed. For M200Y the
$V_{gate}$ voltage range selected for the fit was $\approx 5$ V above $V_{th}$
for positive $V_{back}$ values decreasing to $\approx 2.5$ V for the higher
negative $V_{back}$ values; the data are described within about 0.05 %, and
the uncertainties are for $\delta V_{th}$ between 5 and 10 mV, for
$\delta\mu_{0}\approx 1.5$ cm2/V$\cdot$s, and for $\delta V_{1/2}\approx 0.2$
V for an assumed 0.1 % $I_{ds}$ uncertainty.

Figure 3: Dependence of the threshold voltage $V_{th}$ on $V_{back}$ for
M200P, the MOSFET without $p$-spray implant and for M200Y, the MOSFET with
$p$-spray implant.

(a)

(b)

Figure 4:  Dependence on $V_{back}$ of (a) $\mu_{0}$, the electron mobility in
the inversion layer at the threshold voltage $V_{th}$, and of (b) $V_{1/2}$,
the value of $V_{gate}-V_{th}$ at which the mobility has decreased by a factor
2 relative to $\mu_{0}$.

Fig. 5 shows the dependence of the mobility on $E_{S}$, the electric field in
the Si at the Si-SiO2 interface. Using Gauss’s law it can be obtained from the
charge density of the inversion layer, $q_{0}\cdot N_{inv}$, and the charge
per unit area of the depleted silicon, $q_{0}\cdot N_{Si}$ (Eq. 14):

$E_{S}=\frac{q_{0}\cdot(N_{Si}+N_{inv})}{\varepsilon_{Si}}.$ (3)

Figure 5: Dependence of the electron mobility in the inversion layer of the
MOSFETs with and without $p$-spray implant as function of the electric field
component at the Si-SiO2 pointing from the SiO2 to the Si. The points are the
measurement results, and the lines the fit by Eq. 4.

In Fig. 5 the results of two different ways of determining the electron
mobility, $\mu_{e}$, are shown: For open symbols the mobility is obtained from
$\mu_{0}=\mu_{e}(V_{gate}=V_{th})$ of the $I_{ds}(V_{gate})$ fits for
different values of $V_{back}$, and for the filled symbols the values of
$\mu_{e}$ from the fits at the constant $V_{back}$ values of 0 and $-30$ V.
The two methods cover different regions of $E_{S}$, but agree within a few
percent in the regions of overlap.

For both M200P and M200Y the electron mobility at the Si-SiO2 interface
decreases with electric field. The mobility for the $p$-spray MOSFET is always
lower than for the non-$p$-spray MOSFET. The reason could be the additional
scattering of the electrons on the higher density of dopant atoms, however,
the decrease by up to $\approx 15$ % is larger than the $\approx 5$ % mobility
decrease at a doping of $2\,10^{15}$ cm-3 reported in Ref. [15]. Carrier-
carrier scattering in the inversion layer [16] also reduces the mobility. In
order to make the results available for simulations, the mobility has been
fitted by the function

$\mu_{e}^{E}=\frac{\mu_{0,e}^{E}}{1+\frac{E_{S}}{E_{1/2}}}.$ (4)

The results of the fits are shown as lines in Fig. 5, and the parameters
obtained in Table 1. The chosen parametrisation provides an adequate
description of the measurements.

|  $\mu_{0,e}^{E}$ [cm2/(V$\cdot$s)] |  $E_{1/2}$ [kV/cm]
---|---|---
M200P (data) | $1267\pm 15$ | $190\pm 15$
M200P (TCAD) | $1498\pm 15$ | $181\pm 15$
M200Y (data) | $1063\pm 20$ | $240\pm 20$
M200Y (TCAD) | $1259\pm 40$ | $259\pm 40$
Table 1: Parameters obtained by fitting the data of Fig. 5 by Eq. 4, and
similar for the TCAD simulations discussed in Sect. 3.4.

###  3.2 Doping determination: _Method 1_

In this section a simplified analysis is used to determine the bulk doping,
$CN_{bulk}$, the integrated $p$-spray dose, $N_{imp}$, and an estimate of the
maximal $p$-spray dopant density, $CN_{imp}$. Here and in the following we
call $CN$ the dopant density with units cm-3, and its integral with $N$ and
units cm-2. The method used is simpler than the way the dopant profile is
determined in Sect. 3.3 and less affected by measurement errors, as it does
not require a differentiation of experimental measurements. However,
assumptions have to be made on the surface potential, $\Phi_{S}$, and its
validity is limited to regions of constant doping density.

Figure 6: Schematic representation of the location of charges in a MOSFET in
inversion conditions, and qualitative dependence of the charge density,
$\rho$, transverse electric field, $E$, in the Si at the Si-SiO2 interface and
electric potential $\Phi$. For the charge densities $Q_{imp}$, $Q_{inv}$ and
$Q_{ox}$ $\delta$ function distributions at $x=0$ are assumed for the sketch
of $E$ and $\Phi$.

The method is explained with the help of Fig. 6. From the measurements we know
$V_{back}$, $V_{gate}$ and $I_{ds}$. From the physics of MOS structures we
know that the potential at the interface at inversion $\Phi_{S}\approx
2\,\psi_{B}+f_{ds}\cdot V_{ds}$, where
$\psi_{B}=\frac{k_{B}\,T}{q_{0}}\cdot\ln(CN_{A}/n_{i})$ is the distance of the
Fermi level from the middle of the band gap. The doping density at the Si-SiO2
is denoted $CN_{A}$, the Boltzmann constant, $k_{B}$, the absolute temperature
$T$, and the intrinsic charge carrier density at room temperature
$n_{i}\approx 10^{10}$ cm-3. The term $f_{ds}\cdot V_{ds}$ is the difference
of the average potential of the conducting channel of the MOSFET to the
potential at the $n^{+}p$ junction of the source. For a linear MOSFET
$f_{ds}=0.5$, and for a circular MOSFET, where the potential depends on the
logarithm of the radius, $f_{ds}=0.691$. The oxide charge density,
$Q_{ox}=q_{0}\cdot N_{ox}$, can be estimated by extrapolating $V_{th}$, shown
in Fig. 3, to $V_{back}=\Phi_{S}$, and using the relation
$Q_{ox}=C_{ox}\cdot\big{[}V_{th}(V_{back}=\Phi_{S})-\Phi_{S}\big{]}$. These
relations can be understood in the following way: For the threshold condition
$Q_{inv}=0$, and for $V_{back}=\Phi_{S}$, $Q_{Si}=Q_{dep}+Q_{imp}=0$. Thus
$Q_{ox}$ is the only relevant charge density in the MOSFET, and the biasing of
the MOSFET just corresponds to a SiO2 capacitor of thickness $t_{ox}$ charged
to a charge density $Q_{ox}$. For both M200P and M200Y a value of
$N_{ox}\approx 5\,10^{10}$ cm-2 is found.

From $I_{ds}$ and the electron mobility $\mu_{e}$, determined using Eq. 2, the
charge density of the inversion layer

$Q_{inv}=-q_{0}\cdot
N_{inv}=-\frac{L}{W}\cdot\frac{I_{ds}}{\mu_{e}(V_{gate},V_{back})\cdot
V_{ds}}$ (5)

is obtained. The negative sign takes into account that electrons make up the
inversion layer in $p$-type Si. Assuming an implantation depth, which is so
narrow that it can be approximated by a charge sheet at $x=0$, and a uniform
doping $CN_{A}(x)=CN_{bulk}$ in the Si, and taking into account that
$E_{Si}(w)=0$, the electric field in the Si bulk is

$E_{Si}(x)=-\int_{w}^{x}\frac{q_{0}\cdot
CN_{A}(\xi)}{\varepsilon_{Si}}\,\mathrm{d}\xi=\frac{q_{0}\cdot
CN_{bulk}}{\varepsilon_{Si}}\,\Big{(}w-x\Big{)},$ (6)

and the potential

$\Phi_{Si}(x)=V_{back}-\int_{-w}^{x}E_{Si}(\xi)\,\mathrm{d}\xi=V_{back}+\frac{q_{0}\cdot
CN_{bulk}}{2\,\varepsilon_{Si}}\,\Big{(}x-w\Big{)}^{2}.$ (7)

From $\Phi_{Si}(0)=\Phi_{S}$ follows

$w=\sqrt{\frac{2\,\varepsilon_{Si}}{q_{0}\cdot
CN_{bulk}}\,\Big{(}\Phi_{S}-V_{back}\Big{)}}\hskip
14.22636pt\mathrm{and}\hskip 14.22636ptE_{Si}(0)=\sqrt{\frac{2\,q_{0}\cdot
CN_{bulk}}{\varepsilon_{Si}}\,\Big{(}\Phi_{S}-V_{back}\Big{)}}.$ (8)

Taking into account the charge densities $Q_{imp}$, $Q_{inv}$, which are
negative, the positive oxide charge density $Q_{ox}$, and the boundary
conditions at the Si-SiO2 interface, we obtain the relation between $V_{gate}$
and $V_{back}$

$V_{gate}=\Phi_{S}+E_{ox}\cdot t_{ox}=\Phi_{S}+(\sqrt{2\,\varepsilon_{Si}\cdot
CN_{bulk}(\Phi_{S}-V_{back})/q_{0}}+N_{imp}+N_{inv}-N_{ox})\cdot
q_{0}/C_{ox}.$ (9)

Thus for regions of uniform doping, a linear relationship for $V_{gate}$
versus $\sqrt{2\,\Phi_{S}-V_{back}}$ is expected with the slope
$\sqrt{2\,\varepsilon_{Si}\cdot CN_{bulk}\cdot q_{0}}/C_{ox}$ and the
intercept $\Phi_{S}+(N_{imp}+N_{inv}-N_{ox})\cdot q_{0}/C_{ox}$.

Figure 7: $V_{gate}$ as function of $\sqrt{\Phi_{S}-V_{back}}$ for different
$N_{inv}$ values for the MOSFETs M200P and M200Y.

Fig. 7 shows as examples the results for
$N_{inv}=0,\,5\,10^{10}\,$cm${}^{-2},\,\mathrm{and}\,10^{11}\,\mathrm{cm}^{-2}$.
For $N_{inv}=0$ the value of $V_{th}$ has been used for $V_{gate}$, and for
$N_{inv}>0$ a linear interpolation of $V_{gate}$ between the two closest
$N_{inv}$ values using Eq. 5. Table 2 presents the MOSFET doping parameters
extracted by this analysis.

For M200P straight lines are observed for all values of $N_{inv}$, from which
we conclude that the doping density is uniform throughout the silicon. As
expected, the curves are shifted by steps of $q_{0}\cdot N_{inv}/C_{ox}=1.63$
V for the chosen $N_{inv}$ steps of $5\,10^{10}$ cm-3. Using Eq. 9, from the
slope a value of the doping of $CN_{bulk}=(3.81\pm 0.15)\,10^{12}$ cm-3 is
obtained.

For M200Y two linear regions are observed, which is the result of the
$p$-spray doping. As function of $N_{inv}$, the curves are shifted by the same
amount as the M200P curves. From the value of the slope, a bulk doping of
$CN_{bulk}=(3.69\pm 0.15)\,10^{12}$ cm-3 is obtained, which is similar to the
value from the M200P and the bulk doping from $C-V$ measurements of pad diodes
reported in Sect. 2. The second linear region at low $V_{back}$ voltages has a
slope, which corresponds to a $p$-implant doping density $CN_{imp}=(1.9\pm
0.15)\,10^{15}$ cm-3. Using Eq. 9, the integrated $p$-spray implant of
$N_{imp}=(2.17\pm 0.05)\,10^{11}$ cm-2 is derived from the differences of the
intercepts of the straight lines for high and low $V_{back}$ voltages. The
ratio $N_{imp}/CN_{imp}=(1.14\pm 0.12)\,\upmu$m yields an estimate of the
implantation depth.

|  $\Phi_{S}$ [V] |  $CN_{bulk}$ [cm${}^{-3}]$ |  $CN_{imp}$ [cm${}^{-3}]$ |  $N_{imp}$ [cm${}^{-2}]$
---|---|---|---|---
M200P | 0.33 | $(3.81\pm 0.15)\,10^{12}$ | – | –
M200Y | 0.65 | $(3.69\pm 0.15)\,10^{12}$ | $(1.9\pm 0.2)\,10^{15}$ | $(2.17\pm 0.05)\,10^{11}$
Table 2: Parameters obtained for the bulk and the $p$-implant doping for M200P
and M200Y.

The analysis presented is quite similar to methods used for the analysis of
the doping profiles in MOSFETs for electronics [5]: the _Threshold Voltage
Method_ [17], which corresponds to the analysis with $N_{inv}=0$, and the
_Constant Drain-Source Current Method_ [18, 19], which uses the dependence of
$V_{gate}$ on $V_{bulk}$ for constant $I_{ds}$. We found it necessary to
correct $I_{ds}$ for the change in mobility with electric field, in order to
have a constant $N_{inv}$. Fig. 8 compares the results for $CN_{bulk}$ for
M200P and M200Y and for $CN_{imp}$ using the requirements of constant
$N_{inv}$ to constant $I_{ds}$ in the analysis. Whereas for the constant
$N_{inv}$ requirement the extracted doping is constant within a few percent
for $N_{inv}$ values between 0 and $4\,10^{11}$ cm-2, the constant $I_{ds}$
requirement results in a systematic increase. We conclude that for the
determination of the doping densities from MOSFETs on high-ohmic Si the
_Constant $N_{inv}$ Method_ should be used instead of the _Constant Current
Method_.

(a)

(b)

Figure 8:  Values of the doping density determined from the straight-line fits
to the data as shown in Fig. 7 as function of $N_{inv}$ assuming constant
$I_{ds}$ or constant $N_{inv}$ (a) for the bulk doping, $CN_{bulk}$, of M200P
and M200Y, and (b) for the $p$-spray doping, $CN_{imp}$, of M200Y.

###  3.3 Doping determination: _Method 2_

In this section an attempt is made to determine the doping profiles as well as
the integrals of the doping profile of the M200Y and M200P MOSFETs. We first
note that, like for the $C-V$ method used for doping-profile determinations,
also for MOSFETs the majority carrier concentration, $p(x)$, (holes for an
$n$-MOSFET), and not the doping profile, $CN(x)$ is determined. If $CN(x)$
changes rapidly compared to the Debye length
$L_{D}=\sqrt{(\varepsilon_{Si}\cdot k_{B}\cdot T)/(q_{0}^{2}\cdot CN)}$, the
diffusion of holes causes a difference between $CN(x)$ and $p(x)$ [20, 18, 21,
22]:

$CN(x)=p(x)-\frac{\varepsilon_{Si}\cdot k_{B}\cdot
T}{q_{0}^{2}}\cdot\frac{\mathrm{d}^{2}\ln\big{(}p(x)\big{)}}{\mathrm{d}x^{2}}.$
(10)

For a doping $CN=10^{13}$ cm-3 the Debye length $L_{D}=1.3\,\upmu$m at room
temperature, and the difference between the doping profile and the majority-
charge carrier distribution, the _Debye correction_ , can be significant.

Eq. 10 can be derived under the assumption that the influence of minority
charge carriers can be ignored. In this case Gauss’s law reads

$\frac{\mathrm{d}E_{Si}(x)}{\mathrm{d}x}=\frac{q_{0}}{\varepsilon_{Si}}\cdot\Big{(}p(x)-CN(x)\Big{)},$
(11)

and the steady-state current-continuity equation for zero current flow is:

$j(x)=q_{0}\cdot\Big{(}D_{h}\cdot\frac{\mathrm{d}p(x)}{\mathrm{d}x}-p(x)\cdot\mu_{h}\cdot
E(x)\Big{)}=0\,,$ (12)

with the hole mobility $\mu_{h}$ and the hole diffusion constant
$D_{h}=\mu_{h}\cdot(k_{B}\cdot T)/q_{0}$. Inserting the derivative of Eq. 12
into Eq. 11 results in Eq. 10.

In Refs. [18, 19] the following formulae for the majority-charge carrier
density $p(x)$ as function of the distance $x$ from the Si-SiO2 for a MOSFET
with arbitrary doping distribution $CN(x)$ are derived

$x=\frac{\varepsilon_{Si}}{C_{ox}}\cdot\frac{\mathrm{d}V_{back}}{\mathrm{d}V_{gate}}\hskip
14.22636pt\mathrm{and}\hskip
14.22636ptp(x)=\frac{C_{ox}^{2}}{q_{0}\cdot\varepsilon_{Si}}\cdot\Bigg{(}\frac{\mathrm{d}^{2}V_{back}}{\mathrm{d}V_{gate}^{2}}\Bigg{)}^{-1},$
(13)

where $x=w(V_{back},N_{inv})$ 111This equality follows from the depletion
approximation, which states that the charge density is probed at the edge of
the depletion region. For the determination of doping profiles using $C-V$
measurements, the corresponding relation is $x=w=\varepsilon_{Si}/C_{ox}$ is
the depletion depth for a given value of $V_{back}$ and $N_{inv}$, and the
values of $V_{gate}$ are obtained by interpolating the
$I_{ds}(V_{gate},V_{back})$ measurements for a given $N_{inv}$ value using Eq.
5. Anticipating that the second derivatives from the experimental data have
large uncertainties and that the Debye correction to the integral of the
dopant density, $N_{Si}$, is small compared to the measurement uncertainties,
we also give the formula for the integral of $p(x)$ over the interval $x_{0}$
to $x$, which we call $N_{Si}^{\ast}$

$\int_{x_{0}}^{x}p(x)\,\mathrm{d}x=\frac{C_{ox}}{q_{0}}\cdot\Big{(}V_{gate}(x)-V_{gate}(x_{0})\Big{)}\approx
N_{Si}^{\ast}(x,x_{0}).$ (14)

This equation, which can be derived from Eq. 13, directly follows from charge
neutrality for the entire MOSFET. We note that $N_{Si}^{\ast}$ includes both
$p$-spray implant and bulk-dopant densities (see also Fig. 6).

Fig. 9 shows for the MOSFETs M200P and M200Y the depletion depth $w(V_{back})$
for the threshold voltage, and for $N_{inv}=5\,10^{10}$ cm-2 and $10^{11}$
cm-2. For the derivative $\mathrm{d}V_{back}/\mathrm{d}V_{gate}$ in Eq. 13, a
second order polynomial is put through the $V_{back}$ points below, at and
above the corresponding $V_{gate}$ value. For the derivative of the first and
last $V_{gate}$ value, the selected $V_{gate}$ points are shifted up and down
by one, respectively. Using second order polynomial fits through 5 points or
smoothing of the measured points gives compatible results.

For M200P a square root dependence of $w\big{(}|V_{back}|\big{)}$ is observed,
as expected for a uniform doping. For M200Y the change of doping for small
values of $-V_{back}$ is much weaker, because of the $p$-spray implant. This
is apparent from Fig. 9(b), where the $y$-axis for the M200P is scaled by a
factor 10 compared to M200Y.

Fig. 10 shows the dependence of $w$ on $V_{gate}-(q_{0}\cdot N_{inv})/C_{ox}$
for a limited $w$ range. For $w=0$ the Si is non depleted and acts like a
conductor connected to the gate by the capacitance $C_{ox}$, which is charged
up by the oxide-charge density $q_{0}\cdot N_{ox}$. For M200P, with its
uniform doping density, the potential in the Si is constant and
$\Phi_{S}\approx V_{back}$ and $|V_{gate}-V_{back}|\approx q_{0}\cdot
N_{ox}/C_{ox}$. Therefore, extrapolating the $w(V_{back})$ and
$w(V_{gate}-(q_{0}\cdot N_{inv})/C_{ox})$ curves to $w=0$ allows to determine
$\Phi_{S}$ and $N_{ox}$. The results are shown in Table 3. For M200Y, the
doping is very nonuniform due to the $p$-spray implant, the potential in the
Si depends on $x$, even for $w=0$, and the situation is significantly more
complicated.

(a)

(b)

Figure 9:  Depletion depth $w$ as function of $-V_{back}$ for M200P and M200Y for the threshold voltage, for $N_{inv}=5\,10^{10}$ cm-2 and 1$0^{11}$ cm-2, for (a) the entire, and (b) the $-V_{back}$ range between -0.75 and 1.75 V. As discussed in the text, the extrapolation to $w=0$ allows to estimate the potential at the interface $\Phi_{S}$. Note that in (b) the $y$ scale for M200Y is expanded by a factor 10.  Figure 10: Depletion depth $w$ as function of $V_{gate}-(q_{0}\cdot N_{inv})/C_{ox}$ for M200P and M200Y for the threshold voltage, corresponding to $N_{inv}=0$, for $N_{inv}=5\,10^{10}$ cm-2 and 1$0^{11}$ cm-2. As discussed in the text, the extrapolation to $w=0$ allows to estimate the oxide-charge density $N_{ox}$. Note that the $y$ scale for M200Y is expanded by a factor 20.  |  $V_{back}(w=0)$ [mV] |  $V_{gate}(w=0)$ [V] |  $N_{ox}$ [cm${}^{-2}]$
---|---|---|---
M200P (data) | $150\pm 80$ | $-1.82\pm 0.10$ | $(6.0\pm 0.3)10^{10}$
M200P (TCAD) | $350\pm 80$ | $-2.0^{+1.0}_{-0.5}$ | $(6.2^{+3.0}_{-1.5})10^{10}$
M200Y (data) | $510\pm 25$ | $-0.61\pm 0.05$ | –
M200Y (TCAD) | $800^{+500}_{-250}$ | $-2.0\pm 1.0$ | –
Table 3: Values of $V_{back}$ and $V_{gate}$ for zero depletion depth, $w$ for
the experimental data and the data simulated with TCAD discussed in Sect. 3.4.

Fig. 11 shows $N_{Si}^{\ast}(x,0)$, the integral over the free charge carrier
density $p(x)$ between zero depletion width, $w=x=0$ and $x$, which to a good
approximation is equal to the integral of the dopant density from the Si-SiO2
interface to $x$ (see Fig. 6). The values obtained for the three values of
$N_{inv}$ presented in the figure, zero for $V_{th}$, $5\,10^{10}$ cm-2 and
$10^{11}$ cm-2 are on top of each other in the figure and thus agree. The
agreement for the other $N_{inv}$ values, which are not shown, is similar. For
$x\gtrsim 10\,\upmu$m the slope of the linear increase of $N_{Si}^{\ast}(x)$
is the same within 1 % for M200P and M200Y, from which we conclude that they
have the same constant bulk doping. For $x\lesssim 2\,\upmu$m the M200Y data
show a rapid increase, which reflects the $p$-spray implant.

(a)

(b)

Figure 11:  Approximate dopant density $N_{Si}^{\ast}$ as function of $x$, the
distance from the Si-SiO2 interface for M200P and M200Y for the threshold
voltage, $N_{inv}=5\,10^{10}$ cm-2 and 1$0^{11}$ cm-2, for (a) the entire
studied $x$ range, and (b) the $x$ range up to 2.5 $\upmu$m.

In order to estimate the density of free charge carriers, $p(x)$,
$N_{Si}^{\ast}(x,x_{0})$ is differentiated with respect to $x$. The results
are shown in Fig. 12. It can be seen that, in particular around the maximum of
$p(x)$, major differences for the three $N_{inv}$ values shown are observed.
The reason is that the values of $p(x)$ are very sensitive to the exact values
of $V_{gate}$, which are obtained by interpolating the $I_{ds}(V_{gate})$
results, and of $V_{back}$, which has been recorded with an accuracy of
$\approx 1$ mV, significantly more precise than the setting uncertainty of the
Keithley 6487. In order to obtain a smooth result, the individual values of
$V_{back}$ had to be changed manually by up to $\pm 2$ mV in the analysis. The
change of a single $V_{back}$ value by 2 mV in the region of the maximum of
$p(x)$, results in an S-shaped deviation with an amplitude of $\approx 30$ %.
Thus the determination of $p(x)$ can only be considered an estimate, however
the integral $N_{Si}^{\ast}(x,0)$ is a reliable determination. Its uncertainty
is given by the uncertainty of determining the value of $x=0$. We note that
for understanding the effect of the $p$-spray doping on the isolation and
resistance between $n^{+}$ implants on $p$-Si, the integral
$N_{Si}^{\ast}(x,0)$ is the relevant quantity. Nevertheless, in order to
provide a doping profile, which can be used in TCAD simulations, and to
estimate the size of the Debye correction, $p(x)$ has been fitted by the
phenomenological function

$p(x)=A\cdot\exp\Bigg{(}\frac{-(x-\mu)^{2}}{2\cdot\big{(}\sigma_{0}^{2}+\sigma_{1}^{2}\cdot(x-\mu)^{2}\big{)}}\Bigg{)}+B,$
(15)

which is a Gaussian function with a width, which increases with the distance
from the mean value $\mu$, and finally approaches a constant plus the constant
$B$. The function only approximately describes the observed $p(x)$ dependence
and deviations of up to 20 % are observed. The best description is obtained
for low $N_{inv}$ values and in Fig. 13 data and fit for $N_{inv}=10^{10}$
cm-2 are shown. The Debye correction (Eq. 10) amounts to $+5$ % at the maximum
of $p(x)$, and to $-20$ % at $x=3\,\upmu$m, and thus is similar to the
uncertainties of the measurement results. Although this is only a crude
estimate, it is clear that that the Debye correction does not explain the non-
Gaussian tails of $p(x)$. The parameters from the fit are given in Table 4.
For large $x$ values the constant bulk doping of
$A\cdot\exp(-0.5/\sigma_{1}^{2})+B=3.8\,10^{12}$ cm-3 is obtained.

(a)

(b)

Figure 12:  Free charge carrier density $p$ as function of $x$, the distance
from the Si-SiO2 interface for M200P and M200Y for the threshold voltage,
$N_{inv}=5\,10^{10}$ cm-2 and 1$0^{11}$ cm-2, for (a) the entire $x$ range
studied, and (b) the $x$ range up to 2.5 $\upmu$m.

(a)

(b)

Figure 13: Free charge carrier density $p(x)$, fit by Eq. 15 and doping density after the Debye correction, $CN(x)$, as function of $x$, for M200Y for $N_{inv}=10^{10}$ cm-2 for (a) the entire studied $x$ range, and (b) the $x$ range up to 12 $\upmu$m; the right $y$ scale refers to the ratio of the Debye correction to $p(x)$.  $A$ [cm-3] |  $B$ [cm-3] |  $\mu\,[\upmu$m] |  $\sigma_{0}\,[\upmu$m] | $\sigma_{1}$
---|---|---|---|---
$1.96\,10^{15}$ | $-2.93\,10^{13}$ | 0.462 | 0.418 | 0.350
Table 4: Parameters of the fit of Eq. 15 to the data of Fig. 11. The function
describes the data with an estimated uncertainty of $\approx 20$ %.

###  3.4 Comparison to TCAD simulations

To verify the analysis methods used to extract the MOSFET parameters, in
particular the doping profile close to the Si-SiO2 interface, simulations
using SYNOPSYS TCAD [23] were performed. The MOSFET geometry used for the
simulation is given in Fig 1(a). For the M200P a constant $p$-doping density
of $3.5\,10^{12}$ cm-3 and for M200Y the $p$ doping profile derived from the
measurements in Sect. 3.3 with the values given in Table 4, are assumed. They
correspond to a maximal $p^{+}$ doping of $2\,10^{15}$ cm-3 at a distance of
$0.46\,\upmu$m from the Si-SiO2 interface, and a bulk doping of $3.8\,10^{12}$
cm-3. The back contact is simulated by an $p^{+}$ implant with a maximal
doping density of $10^{19}$ cm-3 and a depth of $2\,\upmu$m. For the oxide
charge density a value of $5\,10^{10}$ cm-2 is assumed for both MOSFETs. For
the doping dependence of the electron mobility the model of Masetti [24] with
the transverse electric field dependence of Lombardi [15] and the carrier-
carrier scattering model of Conwell-Weisskopf [16], are used.

For the M200Y the grid has 356 000 points and 709 000 elements, with grid
spacings in the Si close to the Si-SiO2 interface of 0.2 nm, 0.8 nm, 0.8 nm
and further spacings of 1.4 nm. Such small spacings are required for a
realistic simulation of the inversion layer. Fig. 14 shows the layout, the
grid and the doping distribution of the M200Y in the region of the corner of
the source $n^{+}$ implant implemented for the simulation. The simulation of
the complete data set for the M200Y takes 35 hours on 16 Intel XENON ES-2640v3
CPUs operating at 2.6 GHz. The requirements for the simulation of the M200P is
significantly less challenging and time consuming. We note that, apart from
the circular geometry, the basic structure of the M200Y is very similar to a
segmented $n^{+}p$ sensor, and we find that a fine grid close to the Si-SiO2
interface is required for obtaining reliable results.

Figure 14: Grid and doping distribution implemented in the TCAD simulation for
the MOSFET M200Y at the corner of the source $n^{+}$ implant. The distance
from the center of the circular MOSFET is denoted by $r$.

In the following, we present and discuss some of the results of the analysis
of the data from the TCAD simulation. The dependence of $I_{ds}$ on $V_{gate}$
for the different values of $V_{back}$ of the simulated data is very similar
to the experimental data shown in Fig. 2. An exception are the results for
M200Y for $V_{back}=0.5$: The $V_{back}=0.5$ curve is closer to the
$V_{back}=0.4$ V curve for the simulation than for the experimental data. The
fits of Eqs. 1 and 2 to $I_{ds}(V_{gate})$, which are used to determine the
free parameters of the model, are of similar quality as for the experimental
data, with deviations between fit results and data at the 0.1 % level.

Fig. 15 shows the dependence of $V_{th}$ on $V_{back}$ derived from the TCAD
data. When compared to the experimental data of Fig. 3, only minor differences
are observed. Fig. 16 shows the dependence of the electron mobility $\mu_{e}$
on the electric field at the Si-SiO2 interface, $E_{S}$, derived from the TCAD
data, to be compared to Fig. 5 for the experimental data. Here, major
differences are observed. For both M200P and M200Y the mobility values from
the simulation are significantly higher. For M200Y the mismatch of the
mobility determined from the $V_{back}=0$ V and the $-30$ V simulations, is
larger than for the experimental data. Table 1 gives the parameters of the fit
of Eq. 4 to the simulated data, which is shown as solid line in the figure. In
spite of significant differences between experimental and simulated values, we
have not implemented the experimentally determined mobility parametrisation
into the TCAD simulation, as the value of the mobility should not influence
the determination of the doping profile, which is the main aim of the paper.

Figure 15: Dependence of the threshold voltage $V_{th}$ on $V_{back}$ derived
from the TCAD data for the M200P and the M200Y MOSFETs. Figure 16: Dependence
of the electron mobility in the inversion layer on the electric field
component normal to the Si-SiO2 interface derived from the TCAD data for the
M200P and the M200Y MOSFETs. The points are the simulation results, and the
lines the fit by Eq. 4.

Using _Method 1_ of Sect. 3.2, which assumes regions of constant doping
density and the knowledge of the potential at the Si-SiO2 interface,
$\Phi_{S}$, the bulk doping, $CN_{bulk}$, and the maximum and integral of the
$p$-spray doping, $CN_{imp}$ and $N_{imp}$, have been extracted from the
simulated data. In Table 5 the results are compared to the input values of the
simulation. For the M200P MOSFET the values for the bulk doping, which is
constant, agree. For the M200Y MOSFET the extracted bulk doping is 20 % higher
than the input value, where we note that the extracted value depends on range
of $V_{back}$ used in the analysis: For the range $-5$ V to $-30$ V the value
is $CN_{bulk}=4.75\,10^{12}$ cm-3, whereas for the range $-20$ V to $-30$ V
$CN_{bulk}=4.15\,10^{12}$ cm-3. As expected, the values for the integrated
dose, $N_{imp}$, which is the relevant parameter for understanding the
isolation of $p$ implants and which is determined with an accuracy of $\approx
5$ %, agree. The maximum of the $p$-implant doping, $N_{imp}$, and the
effective implantation depth, $d_{imp}=N_{imp}/CN_{imp}$, also agree within
their significantly larger uncertainties.

|  $CN_{bulk}$ [cm${}^{-3}]$ |  $CN_{imp}$ [cm${}^{-3}]$ |  $N_{imp}$ [cm${}^{-2}]$ |  $d_{imp}$ [$\upmu$m]
---|---|---|---|---
M200P (input) | $3.5\,10^{12}$ | – | – | –
M200P (results) | $(3.5\pm 0.1)\,10^{12}$ | – | – | –
M200Y (input) | $3.8\,10^{12}$ | $1.93\,10^{15}$ | $2.1\,10^{11}$ | 1.09
M200Y (results) | $(4.3\pm 0.5)\,10^{12}$ | $(1.6\pm 0.3)\,10^{15}$ | $(2.0\pm 0.1)\,10^{11}$ | $1.25\pm 0.20$
Table 5: Comparison of the input data to the analysis results using _Method 1_
for the TCAD simulations.

The further analysis of the simulated data follows _Method 2_ of Sect. 3.3.
Using Eq. 13 the depletion depth, $w$, is determined for constant values of
the charge density of the inversion layer, $N_{inv}$. The results on the
dependence of $w$ on $V_{back}$ and on $V_{gate}-(q_{0}\cdot N_{inv})/C_{ox}$
are shown in Fig. 17 for three values of $N_{inv}$. It can be seen that the
results are independent of $N_{inv}$: Except for the $V_{back}=0.5$ V results
for M200Y, the different $N_{inv}$ points are on top of each other. Comparing
to the experimental data, which are shown in Figs. 9 and 10, the shape of the
curves for both M200P and M200Y are compatible, but the M200Y curve is shifted
by $\approx+0.5\,\upmu$m and the M200P curve by $\approx+2\,\upmu$m relative
to the experimental curves. We do not understand the reason for this
difference. The values for $V_{back}$ and of $V_{gate}$ extrapolated to $w=0$
are reported in Table 3. Because of the larger extrapolation in $w$, the
uncertainties for the simulated data are larger than for the experimental
data. Within their uncertainties the results are compatible.

(a)

(b)

Figure 17:  Depletion depth $w$ as function of (a) $-V_{back}$, and (b) of
$V_{gate}-(q_{0}\cdot N_{inv})/C_{ox}$ for M200P and M200Y for the threshold
voltage, for $N_{inv}=5\,10^{10}$ cm-2 and 1$0^{11}$ cm-2, for the analysis of
the simulated data. Note that in the $y$ scale on the right side for M200Y is
expanded by a factor 10 for (a) and by a factor 20 for (b).

The majority charge carrier density, $p(x)$, is obtained by differentiating
$N_{Si}^{\ast}$ from Eq. 14 with respect to $x$ from Eq. 13. In Fig. 18 the
results are compared to the input doping profiles: $3.5\,10^{12}$ cm-3 for
M200P, and the function given in Eq. 15 with the parameters from Table 4 for
M200Y. It is found that the results do not depend on $N_{inv}$, and only the
values for $N_{inv}=5\,10^{10}$ cm-2 are shown. For $x\gtrsim 0.5\,\upmu$m the
reconstructed and input values agree within $\lesssim 10$ %. The values for
$x\lesssim 0.5\,\upmu$m show big fluctuations, which appear unphysical and are
not described by the smooth parametrisation. Investigating the simulated
charge density and field distributions reveals large fluctuations in this
region, which may cause the fluctuations in the results. We also note that for
the simulation the integrated $p$-spray dose is $\approx 5$ % higher than the
input value.

In spite of these differences, we consider the agreement between analysis
results and input data satisfactory, and conclude that the proposed method of
determining doping profiles using MOSFETs on high-ohmic silicon is valid.

(a)

(b)

Figure 18:  Comparison of the input to the reconstructed free charge carrier
density $p$ as function of $x$, the distance from the Si-SiO2 interface for
the simulated M200P and M200Y data, where $N_{inv}=5\,10^{10}$ cm-2 has been
selected; (a) the entire $x$ range studied, and (b) the $x$ range up to 12
$\upmu$m.

##  4 Summary and conclusions

In this paper an attempt is made to determine the doping profile of the
$p$-spray implant, which is used to electrically isolate $n^{+}$ implants in
segmented $n^{+}p$ silicon sensors. For circular MOSFETs with and without
$p$-spray implant, produced as test structures together with silicon sensors,
the Drain-Source current, $I_{ds}$, has been measured as function of the gate
voltage, $V_{gate}$, for different values of the back-side voltage,
$V_{back}$. The measurements were performed at room temperature and ambient
atmosphere in the linear MOSFET region for a Drain-Source voltage $V_{ds}=50$
mV on a standard chuck. The value of $V_{back}$ was recorded with an accuracy
of $\lesssim 1$ mV, which is required for a precise determination of the
doping profile close to the Si-SiO2 interface. In order to determine the
doping density at sub-micron distances from the Si-SiO2 interface, the
measurements have to cover also positive values of $V_{back}$, for which the
$n^{+}p$ implants of Source and Drain approach forward biasing and a
significant forward current is observed.

To determine the MOSFET threshold voltage, $V_{th}(V_{back})$, and the
mobility of the electrons in the inversion layer,
$\mu_{e}(V_{gate},V_{back})$, the $I_{ds}(V_{gate},V_{back}$) data are fitted
using the standard MOSFET formula derived from the Brews charge-sheet model
and a parametrisation of the dependence of the mobility on $V_{gate}$. The
$I_{ds}(V_{gate},V_{back})$ data are also used to determine
$V_{gate}(V_{back},N_{inv})$, the dependence of the gate voltage on the back-
side voltage for constant area density of the electrons in the inversion
layer, $N_{inv}$. The value of $N_{inv}$ is derived from $I_{ds}$, taking into
account the dependence of the electron mobility on $V_{gate}$ and $V_{back}$.

Two methods are used to determine the doping densities for the MOSFETs with
and without $p$-spray implant:

  1. 1.

Assuming that the MOSFETs have regions of constant doping, a linear dependence
of $V_{th}$ and of $V_{gate}(N_{inv}=const.)$ on $\sqrt{\Phi_{S}-V_{back}}$ is
expected (Eq. 9), where $\Phi_{S}$ is the potential at the Si-SiO2 interface.
The dopant density is proportional to the square of the slope. The comparison
of the slopes for $V_{th}$ and $V_{gate}$ for different $N_{inv}$ values
allows to check the consistency of the doping determination.

  2. 2.

From the derivatives $\mathrm{d}V_{back}/\mathrm{d}V_{th}$ and
$\mathrm{d}V_{back}/\mathrm{d}V_{gate}$ at constant $N_{inv}$, the depletion
depth $w(V_{back})$ is derived (Eq. 13). Assuming the depletion approximation,
the distance $x$ from the Si-SiO2 interface where the doping is determined, is
equal to $w$. The integral of the density of holes $p(x)$ (the majority charge
carriers in $p$-type Si), up to the distance $x$,
$\int^{x}_{0}p(\xi)\,\mathrm{d}\xi$, is proportional to the voltage
differences $V_{th}(x)-V_{th}(0)$ and $V_{gate}(x)-V_{gate}(0)$ (Eq. 14). The
threshold voltage corresponding to the depletion depth $w$ is $V_{th}(w)$, the
threshold voltage extrapolated to zero depletion depth is $V_{th}(0)$, and
similar for $V_{gate}(x)$ for constant $N_{inv}$. The derivative of the
integral $\int_{0}^{x}p(\xi)\,\mathrm{d}\xi$ gives $p(x)$. Fitting $p(x)$ with
a phenomenological function allows us to estimate the Debye correction, the
difference between $p(x)$ and the doping density, and thus determine the
doping profile from a sub-micron distance from the Si-SiO2 interface to the
maximum depletion depth given by the maximum $|V_{back}|$ value of the
measurement. The comparison of the results obtained using $V_{th}$ and
$V_{gate}$ for different $N_{inv}$ values provides a check of the consistency
of the doping density determination.

_Method_ 1 is straight-forward and does not require to differentiate data
points, however the value of the potential at the interface has to be assumed
and the impact of the assumption of locally constant doping is not so clear.
For the MOSFET without $p$-spray implant it gives precise values for the
doping and the oxide-charge density. For the MOSFET with $p$-spray implant it
gives a precise value for the integrated implant dose, and approximate values
for the bulk doping beyond the implant region, for the maximal implant doping
density and its width.

_Method_ 2 is more involved, as data and distributions derived from data have
to be differentiated. However, it also gives more details on the doping
profile. Its main assumption is the depletion approximation, and no value for
the potential at the Si-SiO2 interface has to be assumed. The dependence of
$\int^{x}_{0}p(\xi)\,\mathrm{d}\xi$, the integral of the majority-charge
carrier (hole) density on the distance $x$ from the Si-SiO2 interface, is
precisely determined. The hole density, $p(x)$, which is obtained by
differentiation, is very sensitive to the exact knowledge of $V_{back}$:
Changes of a single $V_{back}$ value by 2 mV can result in changes of close-by
$p$ values by $\pm 30$ %. For M200Y, where positive $V_{back}$ values close to
forward biasing have been applied, the doping profiles at a fraction of a
$\upmu$m from the Si-SiO2 interface can be determined. The typical uncertainty
of $p(x)$ is $\approx 20$ %. Finally, a fit of a phenomenological
parametrisation to $p(x)$ allows to apply the Debye correction to $p(x)$ and
thus obtain a doping profile for the use in TCAD simulations and model
calculations.

The results from both methods are consistent. To verify the methods, two
MOSFETs with similar parameters as the investigated ones were simulated using
Synopsys TCAD, and data with the same drain, gate and source voltages as the
experimental ones have been generated. The same analysis software as for the
experimental data has been used, and the results compared to the input
parameters and the experimental values. For the field dependence of the
electron mobility in the inversion layer, significant differences between
experimental and simulated results have been found, which however should not
influence significantly the extraction of the doping profiles. For distances
exceeding $0.5\,\upmu$m from the Si-SiO2 interface, the extracted doping
profiles for both MOSFETs are consistent with the input values. For distances
smaller than $0.5\,\upmu$m, the doping profile extracted from the simulated
data shows large unphysical fluctuations, which reflect large fluctuations of
the simulated charge density distribution close to the Si-SiO2 interface.

The work presented in this paper demonstrates how circular MOSFETs, fabricated
as test structures together with sensors on high-ohmic $p$-type Si, can be
used to determine the bulk doping as well as the doping profile of the $p$
implants, which are required for isolating the $n^{+}$ electrodes of segmented
$n^{+}p$ sensors.

## Acknowledgements

We thank Peter Buhmann and Michael Matysek for maintaining the measurement
infrastructure of the Hamburg Detector Laboratory, where the measurements were
performed in excellent shape, which is a necessary condition for the precision
results presented in this paper. We thank Dr Frank Schluenzen from the DESY IT
Group for setting up and maintaining the IT infrastructure used for the TCAD
simulations. Ioannis Kopsalis acknowledges the fellowship by the DAAD (German
Academic Exchange Service), which allowed him to obtain a PhD from the
Department of Physics of the University of Hamburg.

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