Amplifier

While it is possible to build a discrete amplifier for a single or few channels, the huge number of readout channels in high energy physics experiments demands high integration. Present front-end chips usually include 128 separate amplifier channels together with analog bias generators, sample/hold circuits, the associated control logic and a multiplexing output stage. In some cases, a pipeline storage or a digitization circuit are included as well. All of these functions are built into an integrated circuit (IC) with a die size of less than $1\,\rm cm^2$.

Since a semiconductor detector produces a current signal, the amplifier must have a low-resistance current input. The shape of the current pulse depends on the bias voltage (see section , p. ). There are specialized fast low-noise amplifiers which can visualize the current waveform [30], but usually it is more convenient to measure the integrated current, which corresponds to the collected charge. Thus, the first stage of the amplifier is an integrator. As the MIP charge is only 22500 electrons, special attention must be paid to noise minimization, which is done by a special filter (``shaper'') in the second stage of the amplifier.

Figure: The most widely used amplifier principle for silicon detectors: An integrating preamplifier followed by a CR-RC shaper.

The CR-RC shaping method employs a semi-Gaussian filter which allows easy implementation. Fig.  shows the principal schematics of an integrating preamplifier together with a CR-RC shaper. With equal resistors, capacitors and thus time constants $T_p=R_i C_i$, the transfer function of such an amplifier in the Laplace domain is given by

$$\frac{V_{\rm out}}{I_{\rm in}}=\frac{A\, T_p}{(1+s\, T_p)^2}$$ (2.31)

with the factor $A$ determined by the preamplifier. The practical implementation of this circuitry often includes transconductance amplifiers [31], resulting in a more complicated transfer function. Nevertheless, eq.  gives a good approximation for that case.

Since the input current pulse is always much shorter than the shaping time constant, it can be approximated by a Dirac-$\delta$ pulse weighted with the collected charge $Q_c$. The amplifier response to such an input is

$$v_{\rm out}=A\,Q_c\,\frac{t}{T_p}\,e^{-\frac{t}{T_p}}$$ (2.32)

in the time domain. Since the maximum output voltage is reached at $T_p$, the time constant is also known as ``peaking time''.

The simulation discussed in section , p. , has been used to feed an integrating preamplifier with CR-RC shaper with the calculated detector currents and compare the output to the one obtained with an idealized Dirac-$\delta$ input pulse (eq. ). A shaping time of $50\,\rm ns$, which is also used in the APV amplifier for CMS, was chosen. The simulated amplifier output waveforms with silicon detectors of $300$, $400$ and $500\,\rm\mu m$ thickness are compared to a Dirac-$\delta$ input pulse in fig. .

Figure: APV shaper ( $T_p=50\,\rm ns$) output voltage with MIP signals coming from $300, 400$ and $500\rm\, \mu m$ thick detectors and an ideal Dirac-$\delta$ pulse.

In practice, the preamplifier has a resistor feedback in addition to the integrating capacitor so its output smoothly returns to zero. Thus, drift and saturation are avoided without disturbing the principal function. However, the time constant of feedback capacitor and resistor must be small enough to avoid pile-up effects with frequent signals.

The preamplifier input, as seen from the detector, is capacitive when neglecting the feedback resistor. Its value is given by the feedback capacitor (typically around $1\,\rm pF$) multiplied by the open-loop gain (typically $1000$), leading to a typical input capacitance of $1\,\rm nF$. The detector signal current is divided between the strip (or pixel) capacitance and the amplifier input capacitance, so their ratio should be as high as possible towards the amplifier. Taking the above values with a typical strip detector ($20\,\rm pF$), the charge loss is $2\%$ and thus negligible.