# 9. Si Detectors

"*All composite things decay, strive diligently*."\....Buddha Gauthama

"*To see the world in a grain of sand*."\....William Blake

The gas detectors which we discussed in Section 8 are intrinsically
limited due to diffusion, where the limit on the localization of the
drifting ionization is of order 100![](media/image1.emf) for drift
distances \~ 1 cm. For the study of heavy quark decays, (see Section
13), the characteristic decay length ranges from
100![](media/image2.emf) to 400![](media/image3.emf). The length scale
is set by ![](media/image4.emf) ,the lifetime of the particle in its
rest frame. Therefore, ![](media/image5.emf) is the decay length. Hence,
if we want to resolve the production vertex (Fig. I.1) as separated from
the subsequent quark decay vertices, we need a detection element with
better spatial resolution than the proportional wire chamber. (See the
discussion of "Vertex Detectors" in Section 13.)

## 9.1 Impact Parameter and Secondary Vertex

A bubble chamber photograph of the production of a pair of charmed
mesons by an incident photon is shown in Fig. 9.1. Note that the
laboratory decay lengths are time dilated by a factor
![](media/image6.emf) and are of order 1000![](media/image7.emf) or 1
mm. If we draw the daughter momentum vectors back to the production
vertex we can see that the impact parameter, or the transverse distance
between the path of the decay particles and the production point are of
the same order as the lifetime, ![](media/image8.emf). Note that the
bubble chamber technique, although it contains an enormous amount of
information per event, is much too slow a technology to use in the study
of the relatively rare production of heavy quarks. It has, in fact, not
been discussed in this text. It can, in principle however, resolve the
production and decay vertices as illustrated in Fig. 9.1.

![](media/image9.png){width="4.2in" height="5.811111111111111in"}

Fig. 9.1: Bubble chamber event showing pair production and subsequent
decay of charmed D mesons. Note that the decay lengths are \~1000 μm =
1mm. (From Ref. 10, with permission.)

Silicon detectors are now the high rate devices of choice in the study
of heavy quarks at both fixed target and colliding beam machines. Let\'s
consider the resolution which is required of such a detector. A particle
which is produced and subsequently decays into a secondary track which
has an impact parameter b is shown in Fig. 9.2. The distance along the
track is increased by a factor of ![](media/image10.emf) over the
intrinsic "lifetime" ![](media/image11.emf). The impact parameter is
proportional to that distance, D, and the angle ![](media/image12.emf)
which the decaying track makes with respect to the heavy quark parent.

![](media/image13.wmf)

> Fig. 9.2: Geometry of a short-lived particle D decaying into a
> secondary track a with impact parameter b.

$\begin{array}{r}
z_{D}\ \text{\textasciitilde}\ \gamma_{D}\ (c\tau)_{D}\ \text{\textasciitilde}\ D \\
b\ \text{\textasciitilde}\ \theta\ z_{D}
\end{array}$ (9.1)

The decay angle ![](media/image14.emf) is approximately the transverse
momentum divided by the longitudinal momentum. (See Appendix A.)

$\begin{array}{r}
\theta\ \text{\textasciitilde}\ \frac{p_{T}^{a}}{p^{a}} \\
b\ \text{\textasciitilde}\ \frac{p_{T}^{a}}{p^{a}}\ (c\tau)_{D}
\end{array}$ (9.2)

For two body decays the transverse momentum impulse is again set by the
physics (see Appendix A). The intrinsic physics in this case is fixed by
the mass of the parent quark, M~D~, decaying into daughters, which are
here, assumed to be essentially massless. The longitudinal component is
set by the sharing of the momentum, p~D~ , of the heavy quark parent by
the two daughters.

$\begin{array}{r}
D\  \rightarrow \ a\  + \ b \\
p_{T}^{a}\ \text{\textasciitilde}\ \frac{M_{D}}{2} \\
p^{a}\ \text{\textasciitilde}\ \frac{p_{D}}{2}
\end{array}$ (9.3)

Therefore the impact parameter b is of the same order of magnitude as
the lifetime, (c![](media/image15.emf)~D~), independent of the γ factor
of the heavy quark.

$\begin{array}{r}
\theta\ \text{\textasciitilde}\ \frac{M_{D}}{p_{D}}\ \text{\textasciitilde}\ \frac{1}{\gamma_{D}} \\
b\ \text{\textasciitilde}\ (c\tau)_{D}
\end{array}$ (9.4)

Therefore, the required resolution of the detector, ignoring multiple
scattering, does not depend on the momentum of the heavy quark. Multiple
scattering considerations (see Section 5) would bias us towards studying
higher momentum quarks. The detector should have a resolution \<\<
![](media/image16.emf) or \~ (10-50) ![](media/image17.emf).

Typical values for the "pitch", P, or the spacing of electrode strips
which collect the signals in Si detectors are in the range
20-100![](media/image18.emf), which satisfies this criterion. Assuming
uniform illumination, if only which strip is hit is recorded, the
resolution is the pitch, P, divided by $\sqrt{\text{12}}$. (See Appendix
J.)

$\frac{P}{2}\ y^{2}\frac{\text{dy}}{\int_{}^{}\text{dy}} = \ \frac{P^{2}}{\text{12}}$
(9.5)

If the charge on the strips is recorded, we may do better (see Section 8
on cathode strip charge measurements). For pitch P = 50μm, the
resolution, P/$\sqrt{\text{12}}$, is 15 μm. A basic limitation is again
set by diffusion. With a drift voltage of 50 Volts and a drift distance
of 300 ![](media/image19.emf) (these values will be justified later in
this Section), we find ![](media/image20.emf) for the diffusion error,
(see Section 8).

As an aside, not all measurements are resolution limited. Consider the
measurement of heavy quark decay with lifetime ![](media/image21.emf).
If the decay time t is measured with an error ![](media/image22.emf),
the probability to observe time ![](media/image23.emf) is
![](media/image24.emf) \~ ![](media/image25.emf) where
$P(t,t^{'})\text{}\text{\textasciitilde}\text{}\text{exp}( - \ \frac{\left( t - t^{'} \right)^{2}}{2}\sigma^{2})$,
(see Appendix J). The shape of ![](media/image26.emf) has the same time
structure with the same lifetime, ![](media/image27.emf), but the
overall rate at the same time value t′ as t is increased by a factor
exp![](media/image28.emf). Therefore, Gaussian measurement errors of the
decay time do not change the shape of the decay distribution, at least
for ![](media/image29.emf). The proof of this is left to the reader.

$N(t) = e^{\frac{- t}{t}},N(t^{'})\text{\textasciitilde}e^{\frac{- t^{'}}{t}}e^{\frac{\left( \frac{s}{t} \right)^{2}}{2}}$
(9.6)

## 9.2 Band Gap, Intrinsic Semiconductors and Ionization

The band gap, E~g~, is defined to be the minimum energy needed to excite
an electron into the conduction band. In silicon it has a value of 1.12
eV. In a solid the sharp atomic energy levels are smeared into "bands"
by the mutual interactions among the electrons (see Section 2). In a
semiconductor such as silicon the "valence band" is full, and it is
separated from the allowed states in the empty "conduction band" by a
"band gap" E~g~. This gap can be thought of as a residue of the sharp
energy difference between atomic states.

The intrinsic conductivity, ![](media/image30.emf) of pure silicon is
due to the thermal excitation of charge carriers across the gap, E~g~,
into the conduction band. The intrinsic carrier density, n~i~ =
![](media/image31.emf), at room temperature is about 10^11^/cm^3^ (see
Ref. 9.1). The n~i~ refers to negative electrons in the conduction band
while p~i~ refers to the mobile "holes" remaining in the valence band.
Note that this carrier density is rather small in that the full electron
density is
$n_{\text{si}} = \frac{N_{o}\rho}{A}\text{\textasciitilde}5\text{.}\text{01}´\frac{\text{10}^{\text{22}}}{\text{cm}^{3}},\frac{n_{i}}{n_{\text{si}}}\text{\textasciitilde}\text{10}^{- \text{12}}$.
The small value of the ratio is due to the fact that E~g~/kT \~ 40, so
the Boltzmann factor, ![](media/image32.emf) is very small, \~
4x10^-18^. Therefore, silicon has relatively few mobile charge carriers
and is a poor conductor.

The intrinsic resistivity, ![](media/image33.emf), is the reciprocal of
the intrinsic conductivity, ![](media/image34.emf). A high conductivity
metal such as copper has a resistivity of \~10^-6^
![](media/image35.emf) which is about 10^11^ times less than pure
silicon, reflecting n~i~/n~si~. In a metal there are allowed unfilled
states in the valence band so that the valence electrons can participate
in conduction without paying a big penalty due to a small Boltzmann
factor.

The conductivity of semiconductor material is controlled by "doping" the
crystals with small levels of impurities. Typical doping levels are
chosen, (e.g. parts per million), such that the doped conductivity of
the majority carriers, (set by impurity levels) is \>100 times the
intrinsic conductivity while that for the minority carriers is
consequently \<100 times less since the product n~i~ p~i~ = np remains
constant under doping. For example, n \~ 10^13^/cm^3^ and p \~
10^9^/cm^3^ for a lightly doped n type semiconductor. Thus doping with
parts per million impurities will dominate the intrinsic charge carriers
while still keeping the resistivity high with respect to good
conductors. The unit of charge, the electron charge, is taken to be q in
this Section to avoid confusion with the exponential function. The
carrier mobility is $\mu$ (see Section 8). Electrons have a mobility of
\~1400 cm^2^ /$(V \cdot \text{sec})$.

$\begin{array}{r}
n_{i} = p_{i} = \frac{\text{10}^{\text{11}}}{\text{cm}^{3}}|_{T = \text{300}Kalignl\begin{array}{r}
o
\end{array}^{}}
\end{array}\rho_{i} = \frac{1}{{qmn}_{i}} = \text{200}k\Omega\text{cm} = \frac{1}{s_{i}}$
(9.7)

When we consider the detection of ionization energy, the energy needed
to create an electron-hole pair in Si is 3.6 eV. That's greater than the
band gap energy E~g~ because of the existence of competing dissipative
processes such as phonon, or lattice vibration, excitation. The silicon
detector acts as a "unity gain" ionization detector operating similarly
to those discussed in Section 8. In a detector of active depth
300![](media/image36.emf), the energy loss in silicon due to ionization
is about 116 keV which liberates a source charge, q~s~, of 5.1 fC.

$\begin{array}{r}
E_{\text{pair}}\ \text{\textasciitilde}\ 3\text{.}6\text{eV} \\
d\ \text{\textasciitilde}\ \text{300}\mu m \\
q_{s}\text{\textasciitilde}q\left( \frac{{\Delta E}_{\text{ion}}}{E_{\text{pair}}} \right) \\
q_{s}\ \text{\textasciitilde}\ 5\text{fC}\ \text{\textasciitilde}\ \text{32},\text{000}q
\end{array}$ (9.8)

Silicon detectors are "unity gain" devices. Compared to gas filled
detectors the yield per unit length of ionization is increased by the
density ratio and the ionization energy ratio, I~o~/E~g~ for a total
factor of \~ 10,000. This factor approximately offsets the "gas gain",
G, available in PWC devices. The drift time and diffusion width both
depend on the drift distance, d, which means that we expect Si devices
to be fast and capable of precise position measurements (d \~ 300 μm).

## 9.3 The Si Diode Fields 

The charge quoted in Eq. 9.8 is a rather small signal charge, and
consequently it is fairly difficult to detect. The idea is to set up a
situation where the other currents are small, so that the ionization
current pulse can be detected. Consider a p-n junction diode consisting
of silicon doped with p type impurities on one side and n type
impurities on the other. The dielectric constant for silicon is
![](media/image37.emf) \~ 1 pF/cm (recall ![](media/image38.emf) = 0.089
pF/cm and ![](media/image39.emf) \~ 11.9 ![](media/image40.emf), Table
1.1). The current, I, voltage, V, relationship for a p-n junction is
given below.

$\begin{array}{r}
I(V) = I_{o}\left\lbrack e^{\frac{\text{qV}}{\text{kT}}} - 1 \right\rbrack \\
R_{F}º\left( \frac{\text{dI}}{\text{dV}} \right)^{- 1} = \frac{\text{kT}}{\text{qI}} = \text{25}\Omega|_{\text{300}^{o}K,1\text{mA}}
\end{array}$ (9.9)

Here ![](media/image41.emf) represents the thermal "reverse" current due
to the thermal excitation of the minority carriers in reverse biased
operation, V\< 0. The relevant scales are the applied energy qV with
respect to the thermal energy kT. If ![](media/image42.emf) and
![](media/image43.emf) then ![](media/image44.emf) and we have a small
reverse current. If ![](media/image45.emf), then
![](media/image46.emf)and we have large forward currents. The diode acts
as a "switch" with "position - on/off" controlled by V.

The forward biased resistance, R~F~, is estimated using a differential
form of Ohm\'s law which gives us a 25 Ω resistance for a forward biased
diode at room temperature with a 1 mA standing current. Fluctuations in
the forward current (shot noise is discussed later in this Section)
would swamp the ionization signal. Therefore, we need to operate the
diode as a reversed biased p-n junction. As we will see, the reverse
current at room temperature is \~ 10 nA.

A schematic of the location of the charge carriers in reverse mode is
shown in Fig. 9.3 where the n type and p type are shown with an applied
voltage V\< 0 leading to a "depletion" region centered on the junction.
The applied voltage sweeps the mobile charge carriers out of the
junction region leaving a uniform ion number density n. As the applied
voltage is increased, the mobile charge carriers are swept further out
of the junction region leaving the fixed ion charge which is positive
for the n type region and negative for the p type region.

                                                                                                                           ←
d →

![](media/image47.png){width="5.291666666666667in"
height="2.2083333333333335in"}

Fig. 9.3: Schematic of a reverse biased p-n semiconductor diode showing
the depletion region. (From Ref. 11, with permission.)

For an applied reverse potential V~D~, the geometry at "full depletion",
where all the mobile charge carriers are just swept out, is shown in
Fig. 9.4a. We have assumed a very thin n side of the junction diode. The
number density of doping impurities is n~A~ on the p side and n~D~ on
the n side. Charge conservation requires that the n type is highly doped
as it is thin.

![](media/image48.emf) (9.10)

This static situation can be treated using Maxwell's equations. The
electric field vanishes at x = A and x = -D.

$\begin{array}{r}
\overrightarrow{\nabla} \cdot \overrightarrow{D}\  = \text{}\rho \\
\overrightarrow{\nabla} \cdot \ \overrightarrow{E}\  = \ \lbrack\text{qn}\frac{\rbrack}{\underline{C}}\text{},\ \underline{C}\ \text{}\text{or}\ \text{}\varepsilon\text{}(\text{CGS})
\end{array}$ (9.11)

In Eq. 9.11 ρ is the charge density and not the resistivity and n is the
ion number density. The charge in the depletion region is assumed to be
static and uniformly distributed. Thus, the electric field is linear in
x.

$\begin{array}{r}
E(x > 0) = - {qn}_{A}(x - A\frac{)}{\underline{C}} \\
E(x < 0) = {qn}_{D}(x + D\frac{)}{\underline{C}} \\
q\left\lbrack \frac{n_{A}A}{D} \right\rbrack(x + D\frac{)}{\underline{C}}
\end{array}$ (9.12)

The slope discontinuity in E is due to the discontinuity of charge at
the junction seen in Fig. 9.4a. We integrate a second time to get the
potential applying the boundary conditions V(x = A) = -V~D~, V(x = -D) =
0.

$\begin{array}{r}
\overrightarrow{E}\  = \  - \text{}\overset{\rightarrow}{\nabla V} \\
V(x < 0)\ \text{} = \ \text{} - \frac{qn_{A}A(x + D)^{2}}{2D\underline{C}} \\
(V(x > 0) + V_{D})\text{}\  = \text{}\ \frac{qn_{A}(x - A)^{2}}{2\underline{C}}
\end{array}$ (9.13)

Note that we have ignored the "built in potential" due to thermal
excitation of the charge carriers. It is V~bi~ \~ kT/q = 25 mV @ 300^o^C
(Eq. 9.9) times a factor of order one, ln (n/n~i~). For a heavy doping,
10^6^ times the intrinsic level, we have a \~ 0.36 V drop due to thermal
excitation. Compared to a depletion voltage, \~ 50 V, this "diode drop"
is negligible.

![](media/image49.wmf)

Fig. 9.4: a) Geometry of a p-n junction. The static charge number
density is n~D~ and n~A~. The full depletion region is d = A +D \~A. b)
Electric field for a p-n junction. c) Electric potential for a p-n
junction.

A plot of V(x) is provided in Fig. 9.4c. Note that V and its' slope, E ,
are continuous across the x = 0 junction. Let us make a simplifying
assumption. Continuity of V(x\<0) and V(x\>0) at x = 0 requires that
V~D~ = q n~A~ A(A+D)/2[C]{.underline}. If the doping levels are much
different, n~D~ \>\> n~A~ then, A \>\> D. In that case, d ≡ A+D → A and

![](media/image50.emf) (9.14)

The n layer, in this approximation, is very thin. Defining n~A~ = p for
simplicity, we find that E(x) \~ -q p(x-d)/[C]{.underline}, V(x) \~
-V~D~ + qp(x-d)^2^/2[C]{.underline}. The relationship of the applied
voltage and the depletion thickness, is:

$V_{D}\text{} = \text{}\left\lbrack \text{qp} \right\rbrack \frac{d^{2}}{2}\underline{C},\text{}\left( \frac{2\pi}{\varepsilon}\left\lbrack \text{qp} \right\rbrack d^{2},\text{MKS} \right)$
(9.15)

For example, a 300![](media/image51.emf) depletion region, d, with a
doped resistivity of ![](media/image52.emf) = 6 kΩcm has a depletion
voltage V~D~ of 51 V and a capacity per unit area, [C]{.underline}/d of
35 pF/cm2. Note that in comparison with the intrinsic values there are
\~30 times more majority charge carriers in this example, so that the
doping is fairly light. A detector of area 25 mm^2^ has a source
capacity, C~s~, of \~ 8 pF.

## 9.4 The Si Diode; Signal Formation at Depletion

A plot of data taken with a silicon strip detector, Fig. 9.5, shows the
coincidence rate of a Si strip, S~1~, with an exterior scintillator
array of multiple counters or "telescope", T, triggered on minimum
ionizing particles as a function of the applied voltage. It\'s clear
that as the voltage increases the coincidence rate rises because the
depleted region is expanding rapidly and more charge is being collected,
V~D~ \~ d^2^. At a fixed detection threshold, V~T~, more charge means
higher efficiency. The amount of charge rises until a voltage is reached
where the detector is "fully depleted". In that case the efficiency
comes to a "plateau" without further increase. This occurs at \~50V
applied voltage as estimated from Eq. 9.15.

![](media/image53.wmf)

Fig. 9.5: Data taken with a 50 μm pitch Si microstrip plane on the
coincidence ratio of strip S1 with external telescope T as a function of
applied voltage V.

In Fig. 9.6 is shown a photograph of a real detector used in an
experiment, 50![](media/image54.emf) pitch Si detector mounted as a
planar detector for a "fixed target" geometry. Note that transverse
diffusion and ![](media/image55.emf) ray emission limit the "pitch"
size. In Fig. 9.7 the silicon detectors are mounted as azimuthal
coordinate detectors in a cylindrical geometry more appropriate to the
solenoidal field used by colliding beam detectors. (See Section 13.)

![](media/image56.png){width="4.0055555555555555in"
height="4.977777777777778in"}

Fig. 9.6: Photograph of a 50 μm strip spacing Si detector mounted as a
planar detector. Note the "fanout" of silicon strips to high density
connectors which accommodate the electronics. (Photo - Fermilab.)

What about the time for pulse formation? First we find the electric
field as a function of depth just at depletion, ![](media/image57.emf).
For $V_{D}\text{} = \text{}\text{50}V$
$\text{and}\ \text{}d\text{} = \text{}\text{300}\mu m\text{}\ \text{is}\text{}E(0)\text{} = \text{}3\frac{\text{kV}}{\text{cm}}\text{.}$

![](media/image58.png){width="4.55625in" height="3.629861111111111in"}

Fig. 9.7: Photograph of a Si detector mounted as an azimuthal coordinate
detector in a cylindrical geometry. (Photo - Fermilab.)

We use the methods developed in Section 8 to find the charge position as
a function of time. For the special case of point ionization created at
x = 0 at t = 0 (see Fig. 9.4b) the charge will reach
$\text{x=d}\text{at}t \rightarrow ¥$.

$\begin{array}{r}
\text{dx}\  = \ \mu\text{Edt}\text{}\  = \  - \mu\left( \frac{2V_{D}}{d^{2}} \right)(x - d)\text{dt} \\
x(t)\  = \ d\left\lbrack 1 - e^{\frac{- t}{\tau_{D}}} \right\rbrack,\text{}\tau_{D}\  = \ \frac{\underline{C}}{\mu}\left\lbrack \text{qp} \right\rbrack\text{},\text{} \\
x(0)\  = \ 0,\text{}\ x(\infty) = d
\end{array}$ (9.16)

See Appendix H for more general results and more details. Note that τ~D~
= [C]{.underline}ρ so that high speed operation favors low resistivity,
highly doped Si while a desire for low leakage currents favors high
resistivity.

The current is found using the formalism developed in Section 8

$\begin{array}{r}
I(t)\  = \frac{\text{dQ}}{\text{dt}}\  = \ \mu\ q_{s}\frac{E^{2}}{V_{D}}, \\
\text{}\  = \frac{4\mu\ q_{s}\ V_{D}}{d^{2}}e^{- 2\frac{t}{\tau_{D}}}\  = \ 2\ q_{s}\tau_{D}\left\lbrack e^{- 2\frac{t}{\tau_{D}}} \right\rbrack
\end{array}$ (9.17)

The characteristic charge collection time τ~D~ can be thought of as the
charge going a distance d in a field E(0) with charge carriers having a
mobility ![](media/image59.emf).

$\begin{array}{r}
I(o)\text{}\  = \ 2\frac{q_{s}}{\tau_{D}} \\
I(\infty)\  = \ 0 \\
\tau_{D}\  = \ \left\lbrack \frac{d^{2}}{2}V_{D}\mu \right\rbrack \\
\text{} = \ \frac{d}{\left\lbrack \mu E(0) \right\rbrack}\text{}
\end{array}$ (9.18)

With an electron mobility, ![](media/image60.emf)~e~, of 1400
cm2/![](media/image61.emf), the drift velocity is of order 4.2 x 10^6^
cm/sec or = 42 ![](media/image62.emf)m/nsec. This velocity should be
compared to a gas where the electron drift velocity (see Fig. 8.2) is
comparable, i.e. a few cm/μsec. However, the drift distances are much
smaller so that the charge collection time is shorter for Si devices.
For a drift distance of 300 ![](media/image63.emf) we get τ~D~ \~ 7
nsec. The holes have a mobility \~3 times less and therefore the hole
signal is about 3 times slower.

Note that τ~D~ was derived just at depletion. Clearly if we can raise
the applied voltage it will drive the electric field into the diode and
reduce the collection time. (See Appendix H). A limitation to this
method of speeding up the detector comes because there exists a reverse
breakdown voltage. Normally, we do not operate far above depletion
$|V|\  \leq \ \text{100}V$ in order to avoid breakdown.

A plot of the coincidence rate between a Si strip S~1~ and a
scintillation counter "telescope", T, as a function of the relative
delay between them is shown in Fig. 9.8. The basic falloff time is \~ 12
nsec. From Section 2, a coincidence rate will persist on a time scale
equal to the intrinsic "time jitter" of the device. One component of
that time jitter is the resolving time of the Si detector. Clearly, the
Si strip is experimentally confirmed to be a high-speed detector.

![](media/image64.wmf)

Fig. 9.8: Plot of coincidence rate between a Si strip S1 and an external
scintillation counter T array as a function of the time delay inserted
in the coincidence logic.

Recalling that the source charge (q~s~) is 5 fC, we find a peak electron
(hole) current 710 nA (240 nA).

$\begin{array}{r}
q_{s} = 5\text{fC} \\
t_{D}\text{\textasciitilde}7n\text{sec}(\text{20}n\text{sec}) \\
I(o)\text{\textasciitilde}\frac{q_{s}}{t_{D}} = \text{710}\text{nA}(\text{240}\text{nA})
\end{array}$ (9.19)

Having established the size of the current signal, let us turn to issues
of noise.

## 9.5 Noise Sources - Thermal and Shot Noise

Typical reverse currents for junctions as a function of temperature are
shown in Fig. 9.9. Note the exponential relationship between the
operating temperature and the reverse leakage current, I~o~. Since this
current is due to the thermal excitation of minority carriers, such
behavior is not unexpected. The scale for reverse currents at room
temperature for the lowest leakage junctions is 2-20 nA. Therefore the
signal to noise ratio in such detectors is adequate if high quality high
resistivity detectors are used (see Eq. 9.7, high resistivity means
small intrinsic current). Note that "bulk damage" of the crystal
lattice, say from irradiation due to detector operation at a high
luminosity accelerator, induces defects which act as carriers, thus
increasing the reverse current. For example, a neutron "fluence" of
\~10^14^ n/cm^2^ may cause discernible degradation in performance.
Looking at Fig. 9.9, cooling the Si devices is an option to restore low
noise operation, and indeed, solid state detectors are routinely run at
low temperature in many applications.

![](media/image65.png){width="3.459722222222222in"
height="3.473611111111111in"}

Fig. 9.9: Reverse leakage current as a function of temperature. (From
Ref. 12, with permission.)

Now let\'s consider briefly methods of detection for these small pulses.
It is the smallness of the signal compared to a PWC which dictates a
less cavalier treatment of front end electronics in this case. A
schematic of an operational amplifier realization of a charge integrator
is shown in Fig. 9.10 (see Appendix I). The ionization detectors we have
discussed are all a capacitive source of charge; the signal can be
thought of as a current source. Using Q = CV, I = CdV/dt, we find that
the output voltage V~o~ is proportional to the source charge q~s~. If
the amplifier gain G is \>\> Cs/C then the device acts as an ideal
operational amplifier (G→∞), V~o~ = qs/C. A typical output pulse of such
a device is seen in Fig. 8.4. A more detailed schematic showing the
noise sources, e~n~, for a signal with source charge q~s~, driven by
source capacity Cs and source resistance Rs into a generic amplifier
with gain G and output filter with "transfer function"
![](media/image66.emf) is shown in Fig. 9.11.

![](media/image67.wmf){width="5.027083333333334in"
height="3.229861111111111in"}

Fig. 9.10: Charge sensitive preamplifier.

Let us now look in more detail at the behavior of the noise sources.
There are dissipative elements such as resistors which are a source of
intrinsic "thermal noise". The size of the thermal currents is set by
the thermal energy, kT, in the resistor. We assume that the frequency
spectrum of this noise is "white"; it is uniform over all frequencies.
That means, physically, that the noise occurs impulsively, or as a delta
function, in time. \[Recall that the Fourier transform of a plane wave
with fixed momentum, ![](media/image68.emf), is e^ikx^ whose intensity
is spatially uniform.\]

![](media/image69.wmf){width="6.069444444444445in"
height="1.2638888888888888in"}

Fig. 9.11: Amplifier and bandwidth limiting filter, f(ω) with source
capacity, C~S~, source resistance, R~S~, source charge, q~S~, and input
noise voltage, e~n~.

The noise power in a resistor is the thermal energy kT spread uniformly
over all frequencies, ![](media/image70.emf). This thermal noise power
leads to a thermal resistor current I~T~, since power = [P]{.underline}
= VI = I^2^R.

$\text{dI}_{{}_{T}}^{2} = \frac{2\text{kT}}{R}\left( \frac{dw}{p} \right)$
(9.20)

Additionally there is "shot noise" which occurs because standing
currents, I, are in fact due to the motion of discrete charge carriers
since charge is quantized in multiples of q. Fluctuations in the numbers
of these charge carriers lead to the "shot noise" of the standing
current, $\text{dI}_{s}\text{\textasciitilde}qdw$. If charge were not
quantized, q → 0 and dI~s~ → 0.

$\text{dI}_{s}^{2} = \text{qI}\left( \frac{dw}{p} \right)$ (9.21)

We need a filter to limit the bandwidth. Otherwise we will have an
infinite amount of noise since all frequencies are hypothesized to be
equally probable. We assume a simple shaping network which cuts off
linearly at low frequencies and falls as ![](media/image71.emf) at high
frequencies with a single "shaping time", *τ*. A schematic plot of
G![](media/image72.emf) as a function of ![](media/image73.emf) is given
in Fig. 9.12.

![](media/image74.wmf)

Fig. 9.12: Plot of the transfer function Gf(ω) which → 0 as ω → 0 and as
ω → ∞ and peaks at ω = 1/τ.

$f(w) = G\left\lbrack \frac{\text{wt}}{1 + (\text{wt})^{2}} \right\rbrack$
(9.22)

We define the transconductance, g~m~, to be the inverse of the base
resistance of a hypothetical front end transistor in the amplifier. We
assume a front end transistor with a base resistance characteristic of a
forward biased diode. The relationship between standing emitter current
I~E~ and base resistance R~B~ is one we have already seen in Eq. 9.9.

$R_{B} = \frac{\text{kT}}{\text{qI}_{E}} = \frac{1}{g_{m}},(\text{Eq}\text{.}9\text{.}9)$
(9.23)

To set the scale for these quantities, we evaluate the coefficients
numerically (at room temperature);

$\begin{array}{r}
\sqrt{\text{qI}} = 0\text{.}4\text{nA}\sqrt{I(A) \cdot \text{Hz}}, \\
\sqrt{\frac{2\text{kT}}{R}} = 0\text{.}\text{09}\text{nA}\sqrt{\frac{\text{Hz}}{R(\Omega)}}
\end{array}$ (9.24)

Clearly we are dealing with nA scale noise currents per unit frequency
range. For example, with a 100 MHz frequency range, dω, at 1 mA standing
current, we have 126 nA shot noise - a level which becomes uncomfortably
close to the signal level, Eq. 9.19.

We assume three sources of noise, (Fig. 9.11); the thermal noise in the
source resistor R~S~, shot noise due to the standing front end emitter
current I~E~, (amplifier) and thermal noise in the base resistor R~B~.
These noise currents flow into the source capacity, C~S~, and by
assumption not into source resistance, R~S~, because the source
resistance is assumed to be much greater than the impedance, \~ 1/ω C~S~
of the source capacitor at the relevant peaking frequencies,
![](media/image75.emf). Note that I = CdV/dt = C$iw$V means that the
capacitor impedence Z~C~ is equal to = V/I = 1/$iw$C.

These assumptions lead to an expression for the total square summed
input noise voltage ![](media/image76.emf). Since the noise sources are
statistically independent we add them in quadrature (see Appendix J).
The first 2 terms go as ![](media/image77.emf) while the last goes as
![](media/image78.emf) The source capacity is important at low
frequencies.

$\begin{array}{r}
d{\overline{V}}^{2} = d{\overline{I}}_{T}^{2}Z_{\text{CS}}^{2} + d{\overline{I}}_{S}^{2}Z_{\text{CS}}^{2} + d{\overline{I}}_{T}^{2}R_{B}^{2} \\
\left\lbrack \frac{2\text{kT}}{R_{S}({wC}_{S})^{2}} + \frac{\text{qI}_{B}}{\left( {wC}_{S} \right)^{2}} + \frac{2\text{kT}}{g_{m}} \right\rbrack\frac{dw}{p}
\end{array}$ (9.25)

## 9.6 Filtering and the "Equivalent Noise Charge"

The noise voltages are referred to the amplifier input. They go through
the amplifier and through the output low and high pass filter network,
f(![](media/image79.emf)), to produce a mean square voltage at the
output terminals, ![](media/image80.emf).

$\begin{array}{r}
\left\langle V^{2} \right\rangle = \int_{}^{}{|f(w)}|{}^{2}(d{\overline{V}}^{2}dw)dw \\
 = G^{2}\left\lbrack \left( \frac{\text{kT}}{2R_{S}} + \frac{\text{qI}_{B}}{4} \right)\frac{t}{C_{S}^{2}} + \frac{\text{kT}}{2g_{m}t} \right\rbrack
\end{array}$ (9.26)

Note that without the shaping network this integral would diverge. The
noise going into C~S~ diverges at low frequencies because the C~S~
impedence diverges at low frequencies. In comparison, the base thermal
noise needs to be cut off at high frequencies. Thus
![](media/image81.emf) has terms that goes as ![](media/image82.emf) and
as 1/![](media/image83.emf).

For the signal charge itself, q~s~, there is an optimized output from
the filter if the frequency distribution of the source is well matched
to the shaping time ![](media/image84.emf) of the filter.

$\text{input}q_{s} \rightarrow \frac{\left( q_{s}\frac{G}{C_{s}} \right)}{e} = V,\text{at}w\text{\textasciitilde}\frac{1}{t}\text{at}\text{the}\text{output}$
(9.27)

In Eq. 9.27 and below e is the base of the natural log not the
electronic charge which we have defined to be q in this Section in order
to avoid confusion.

The behavior of the signal allows us to define an equivalent noise
charge, ENC, referred to the input, since q~s~ appears on C~S~.

$(\frac{\text{ENC} \cdot G}{C_{s}}e)^{2}º\left\langle V^{2} \right\rangle$
(9.28)

The ENC is the noise charge that occurs due to the noise sources
referred to the input with respect to the output signal that arises for
optimal shaping, V \~ (q~s~ G/C~S~e) \~ G V~s~/e. It is convenient to
further define "parallel" and "series" equivalent noise charges. Those
noise charges are proportional to the square root of the shaping time
![](media/image85.emf) (parallel) and 1/τ (series).

$\begin{array}{r}
\text{ENCP} = \left( C_{S}\frac{e}{G} \right)\sqrt{\left\langle V^{2} \right\rangle_{P}}ºe\sqrt{t\left( \frac{\text{kT}}{2\text{Rs}} + \frac{{qI}_{B}}{4} \right)} \\
\text{ENCS} = \left( C_{S}\frac{e}{G} \right)\sqrt{\left\langle V^{2} \right\rangle_{S}}{ºeC}_{s}\sqrt{\frac{\text{kT}}{2}g_{m}t}
\end{array}$ (9.29)

Clearly, for low noise operation we want low temperature, large source
resistance (ideal current source) and small standing base current in
order to minimize the parallel noise. Similarly, we try to obtain small
source capacity and small base resistance to reduce series noise. In
addition, high-speed operation argues for short shaping times,
![](media/image86.emf), but a compromise with enhanced series noise may
be required, ENCS \~ 1/$\sqrt{t}$. Short shaping time implies a large
bandwidth, ![](media/image87.emf) which inevitably puts more noise into
the system. We should then, if possible, operate at reduced temperature
because thermal noise in the resistors is reduced (see Eq. 9.29).

A plot of the noise as a function of source capacity for typical
preamplifier parameters is shown in Fig. 9.13. You can see the existence
of a "parallel" part of the equivalent noise which is independent of
source capacity and a "series" part which becomes important at large
source capacity and which increases linearly as C~s~. Clearly, we try to
minimize the capacitance of the source because it causes noise. For
example, long Si strips are economically desirable, but we may incur too
high a penalty due to increased source capacity if high speed (small
![](media/image88.emf)) operation is planned.

![](media/image89.png){width="3.4034722222222222in"
height="3.265277777777778in"}

Fig. 9.13: Noise in RMS electrons referred to the input as a function of
source capacitance for typical preamplifier parameters. The linear
dependence for series noise dominance is evident for \>100 pF capacity.
(From Ref. 12, with permission.)

## 9.7 Front End Transistor Noise

So far we have neglected the transistor as an active element and as a
noise source. For front-end transistors, the equivalent noise voltage
referred to the input, ![](media/image90.emf) (see Fig. 9.11), is often
quoted in the manufacturer's specification. For a typical transistor,
$e_{n}\text{\textasciitilde}1\frac{\text{nV}}{\sqrt{\text{Hz}}}$, and a
series noise charge can be defined.

$(\text{ENCS})\text{\textasciitilde}e_{n}\frac{C_{s}}{\sqrt{t}}$ (9.30)

As an example, a source capacity ![](media/image91.emf) = 25 pF using a
25 nsec shaping time ![](media/image92.emf), yields a front end
transistor noise of 1000 electrons, to be compared to a Si signal of
31,000 electrons.

The plot of the ENC in rms electrons for various types of front-end
transistors is shown as a function of shaping time in Fig. 9.14 for a 10
pF source capacity C~S~. For short shaping times the expected
$\frac{1}{\sqrt{t}}$ behavior is observed. At the point, $t$ \~ 10 nsec,
we find ENC \~ 800 e or e~n~ \~ 1.3 $\frac{\text{nV}}{\sqrt{\text{Hz}}}$
for bipolar transistors. For high speed applications bipolar transistors
are favored (GaAs if money is no object), while for slower operation
junction FETs are the best choice. Clearly, the devil is in the details.

![](media/image93.png){width="2.21875in" height="4.164583333333334in"}

Fig. 9.14: Equivalent noise charge in RMS electrons as a function of
shaping time τ for a 10 pF source capacity and a variety of front end
transistors. (From Ref. 13, with permission.)

Can transistors improve dramatically? Well, it is often remarked upon
that the speed of computer chips doubles every \~ year and this is
quoted as a "rule". However, all rules only work in limited regions of
the parameters. The speed of a transistor is determined by the "feature
size". That size is currently \~ 1 μm or \~ 10000 atoms. If the trend
toward shrinking feature size were to continue for 7 years, the size
would go to 0.01 μm or 100 atoms. At that size quantum effects are
crucial and the extrapolation collapses.

## 9.8 Total Noise Charge

As a numerical example, assume a source charge of 5.1 fC which is
appropriate to a silicon detector. Use a transconductance, g~m~, of
(25![](media/image94.emf))^-1^. Pick a source capacity of 30 pF which is
not unreasonable for the leads and connections of a typical silicon
strip detector. Use a 10 nsec shaping time, τ, which given the intrinsic
speed of the silicon detector, would be a match to τ~D~ and would not
compromise the Si high rate performance.

We find a series noise charge of 1125 electrons with this source
capacity. For a source resistance of 1 ![](media/image95.emf) and a base
current of 1 mA the parallel equivalent noise charge is 825 electrons,
which means a total signal to noise, folding the series and parallel
noise in quadrature, of 23. That value of S/N is typical of those
achieved in high quality silicon systems. Clearly, the S/N value must be
made to be \>\>1 if we are to operate efficiently and simultaneously run
with low accidental counting rates, (see Section 2).

$\begin{array}{r}
\text{ENCS}\ \text{\textasciitilde}\ \text{1125}\ \text{electrons}\text{} \\
\text{ENCP}\ \text{\textasciitilde}\ \text{825}\ \text{electrons} \\
\frac{S}{N}\  = \ \frac{q_{s}}{\sqrt{\text{ENCS}^{2} + \text{ENCP}^{2}}}\ \text{\textasciitilde}\ \text{23} \\
\text{Front}\text{}\text{end}\text{}\text{noise}\ \text{} = \ \text{}(e_{n}C_{s}\frac{)}{\sqrt{\tau}}\text{}\ \text{\textasciitilde}\ \text{}\text{800}e
\end{array}$ (9.31)

These noise considerations are not simply an academic exercise. They are
symbols for what appears on your scope trace. For example, the noise due
to source capacity, C~S~, is evident in Fig. 8.4. In Fig. 9.15 are shown
oscilloscope traces in the case of a wire chamber. We see the raw pulses
which have a short rise time (5 nsec/division trace) due to the pulse
formation time, τ~o~, which we evaluated to be a few nsec (Section 8).
The pulses have a long 1/t tail as we mentioned in Section 8 which can
be electronically cancelled as shown in Fig. 9.15b and 9.15c. We see
that pulse shaping does indeed achieve a shorter pulse. However, the
"base line", which is the signal to noise is becoming worse. The
equivalent noise charge basically tells us what the level of "hash" is
on the base line of our scope trace. It\'s that level of "hash" which
defines the possible levels for pulse discrimination, V~T~ (Section 2).

![](media/image96.png){width="5.690277777777778in"
height="3.915277777777778in"}

Fig. 9.15: Oscilloscope traces for a wire chamber. a) raw pulses, b)
tail cancelled, c) pole/zero cancelled, d) impulse response. Note that
the width of the baseline is a direct measure of the ENC for the system.
(From Ref. 5, with permission.)

In Fig. 9.16 there is a photo of some Applications Specific Integrated
Circuit chips, or "ASICs", which are mounted directly on a detector. We
can see that the level of technology which is used in such applications
is rather high. The electronics is mounted very close to the detector in
order to minimize electromagnetic noise pickup as an additional noise
source and also in order to minimize the source (leads) capacity. We are
constantly attempting to hold down the sources of noise. These tight
requirements are an example of the fact that detector design is
something of an artform.

![](media/image97.png){width="4.156944444444444in"
height="3.298611111111111in"}

Fig. 9.16: Photograph of integrated electronics for detectors. (Photo -
Fermilab.)

## 9.9 Hybrid Si Devices

Recently "hybrid photodetectors" have become commercially available.
These devices combine aspects of photomultipliers and Si devices which
open up new applications. An example is the hybrid photodiode which has
a photocathode in a vacuum vessel along with a Si diode. The
photocathode, held at \~10 kV, emits photoelectrons which are
accelerated across a small gap, \~1 mm, in a short, \< 1 nsec, transit
time and strike the diode. The released charge, G \~\[(10,000)/3.6\] =
2778 electrons per photoelectron, is collected rapidly by the diode.
Thus the device acts like a compact, fast photomultiplier tube with gain
\~2800. In addition, the device can work in magnetic fields up to 5T if
the $\overrightarrow{E}$ and $\overrightarrow{B}$ axes are aligned,
since then the path is a helix with axis along the $\overrightarrow{E}$
direction and with small radius of curvature (see Appendix G). Finally,
the Si diode can be cut into many independent devices, allowing for
operation with multiple channels per device making for low cost per
channel. These devices are to be used in the next generation of high
energy physics experiments.

## References

Section 9: Si Detectors

\[1\] [Introduction to Solid State Physics]{.underline}, C. Kittel, John
Wiley and Sons (1953).

\[2\] [Physics of Semiconductor Devices]{.underline}, J.M. Sze, John
Wiley and Sons (1981).

\[3\] [Semiconductor Detectors]{.underline}, G. Bertolini, A. Coche,
North-Holland Publishing Co. (1968).

\[4\] [Semiconductor Detectors for Nuclear Radiation]{.underline}, G.
Dearnaley, D.C. Northrop, John Wiley & Sons (1966).

\[5\] Low Noise Techniques in Detectors, V. Radeka, Annual Review of
Nuclear and Particle Science, [38]{.underline} 217 (1988).

\[6\] P.G. Rancoita, A. Seidman, Rev Nuova, Cimento [7]{.underline} 5
(1982).

\[7\] H.M. Heijne, CERN Report, 83-06 (1983).

\[8\] G. Lutz, A.S. Schwarz, "Silicon Devices for Charged-Particle Track
and Vertex Detection" Annual Review of Nuclear and Particle Science 295
[45]{.underline} (1995).

\[9\] I.S. Hughes, [Elementary Particles]{.underline}, Cambridge
University Press, 1972.

\[10\] T. Ferbel, [Experimental Techniques in High Energy
Physics]{.underline}, Addison-Wesley Publishing Co., Inc. (1987).

\[11\] Silicon Photodiodes and Charge Sensitive Amplifiers for
Scintillation Counting and High Energy Physics, Hamamatsu Photonics
(1993).

\[12\] [Instrumentation in Elementary Particle Physics]{.underline},
C.W. Fabjan, J.E. Pilcher, Trieste 1987, World Scientific Publishing Co.
(1988).

Table of Contents

9\. Si Detectors [1](#si-detectors)

9.1 Impact Parameter and Secondary Vertex
[1](#impact-parameter-and-secondary-vertex)

9.2 Band Gap, Intrinsic Semiconductors and Ionization
[5](#band-gap-intrinsic-semiconductors-and-ionization)

9.3 The Si Diode Fields [6](#the-si-diode-fields)

9.4 The Si Diode; Signal Formation at Depletion
[10](#the-si-diode-signal-formation-at-depletion)

9.5 Noise Sources - Thermal and Shot Noise
[15](#noise-sources---thermal-and-shot-noise)

9.6 Filtering and the "Equivalent Noise Charge"
[19](#filtering-and-the-equivalent-noise-charge)

9.7 Front End Transistor Noise [21](#front-end-transistor-noise)

9.8 Total Noise Charge [22](#total-noise-charge)

9.9 Hybrid Si Devices [25](#hybrid-si-devices)

References [26](#references)