Noise

The noise in a silicon detector system plays an essential role, since the signals are very low. Especially with strip detectors, it is important to know each contribution to optimize the design. This is easier for pixel detectors, since their active area is very small and there is virtually no readout line impedance, which reduces many noise components down to negligible values compared to strip detectors.

The electronic noise in silicon detector systems is given in terms of equivalent noise charge (ENC) referred to the input. The main noise source is the input transistor in the amplifier, with a noise figure depending on geometry and electrical parameters [24,31,34,35]. Noise contributions of further electronic stages are usually neglected. Due to its integrating nature, the load capacitance plays an important role for the amplifier noise. In a simple approach, the amplifier noise can be described by the sum of a constant value (parallel noise) and a part which scales with the load capacitance $C$ (series noise),

$${\rm ENC_{C}=ENC_{C,p}+ENC_{C,s}}\,C\quad.$$ (2.35)

With the capacitive load which is typical for a strip detector, the amplifier noise can be as small as $250\,\rm e$ with a peaking time of a few microseconds. When faster shaping is required, the noise increases. The CMS front-end amplifier APV25 with a peaking time of $T_p=50\,\rm ns$ has a noise figure of about $970\,\rm e$ with a typical capacitive load.

Figure: Noise sources in a silicon strip detector.

Apart from the amplifier, there are other noise sources in the system. Fig.  shows noise related components in a typical AC coupled strip detector configuration with a polysilicon bias resistor. The AC coupling capacitor can be neglected in these considerations. The current source $I_{\rm Leak}$ is the fraction of the detector leakage current which is seen by one strip, $R_P$ is the polysilicon resistor, $C$ is the detector strip capacitance and $R_S$ is the line resistance of the strip. In reality, the line resistance and the strip capacitance are distributed along the strip as in a transmission line, so the effective impedance differs from the concentrated values. Nevertheless, its influence is limited, so we will use the concentrated values as an approximation.

Parallel noise sources are the constant part of the amplifier $\rm ENC_{C,p}$, leakage current fluctuations $\rm ENC_{ILeak}$ and the polysilicon resistor noise $\rm ENC_{RP}$. The capacitive fraction of the amplifier noise $\rm ENC_{C,s}$ and the readout line resistor noise $\rm ENC_{RS}$ are series noise sources. As expected, the peaking time $T_p$ plays an key role in the noise functions. Numerical noise equations, in which the physical constants are already expressed by numbers, can be written as

$${\rm ENC_{Ileak}} = 106\,\sqrt{I_{\rm Leak}\,T_p}\quad,$$ (2.36)

$${\rm ENC_{RP}} = 758\,\sqrt{\frac{T_p}{R_P}}\quad{\rm and}$$ (2.37)

$${\rm ENC_{RS}} = 0.395\,C\,\sqrt{\frac{R_S}{T_p}}$$ (2.38)

with $\rm ENC\,[e]$, $I_{\rm Leak}\,\rm [nA]$, $T_p\,\rm [\mu s]$, $R_P\,\rm [M\Omega]$, $R_S\,\rm [\Omega]$ and $C\,\rm [pF]$. Parallel noise contributions rise with increasing peaking time, while series noise behaves opposite. The total noise figure is the square sum of the individual contributions, since the individual sources are uncorrelated,

$${\rm ENC^2=\sum ENC_i^2}\quad.$$ (2.39)

The deconvolution method (see section , p. ) compromises the noise. Both intrinsic amplifier noise components increase due to the signal processing and the external series noise is amplified, while the parallel noise is reduced. It has been shown [32,33] that the ratio between peak and deconvolution mode noise can be expressed for parallel and series terms as

$$\frac{{\rm ENC_{p,d}}}{{\rm ENC_p}} = \frac{e^{-2}}{x^2}\left( e^{2x}-4x-e^{-2x} \right)\quad{\rm and}$$ (2.40)

$$\frac{{\rm ENC_{s,d}}}{{\rm ENC_s}} = \frac{e^{-2}}{x^2}\left( e^{2x}+4x-e^{-2x} \right)\quad,$$ (2.41)

where $x=T/T_p$ is the ratio between sampling time and peaking time.

To get a feeling for the magnitude of individual noise components, these figures will be calculated and compared for the DELPHI Very Forward Tracker (VFT) [23] and an average CMS silicon detector. The VFT uses the MX6 readout chip, while the CMS strip detectors will be instrumented with the APV25 described in section , p. . The APV25 noise will be shown for both peak and deconvolution modes.

Table: Noise related numbers of the DELPHI Very Forward Tracker (VFT) and an average CMS strip detector. The VFT uses FOXFET bias resistors with a dynamic resistance at operating conditions as shown and the leakage current of the CMS detector corresponds to a moderately irradiated state.

	DELPHI VFT	CMS
Amplifier	MX6	APV25
$I_{\rm Leak}\,\rm [nA]$	0.3	100
$R_P\,\rm [M\Omega]$	36	1.5
$C\,\rm [pF]$	9	18
$R_S\,\rm [\Omega]$	25	50
$T_p\,\rm [\mu s]$	1.8	0.05
$\rm ENC_{C}\,[e]$	$325+23\,\rm pF^{-1}$	$250+36\,\rm pF^{-1}$ (peak)
		$400+60\,\rm pF^{-1}$ (deconvolution)

Table: Noise numbers resulting from the typical values given in tab. . The total noise in the second last row is the square sum of the above contributions, and the signal-to-noise ratio with a MIP charge of $22500\,\rm e$ is shown below.

	DELPHI VFT	CMS
		peak	deconvolution
${\rm ENC_{C}}\,\rm [e]$	$532$	$898$	$1480$
${\rm ENC_{Ileak}}\,\rm [e]$	$\:\;78$	$237$	$\:\;103$
${\rm ENC_{RP}}\,\rm [e]$	$169$	$138$	$\:\;\:\;60$
${\rm ENC_{RS}}\,\rm [e]$	$\:\;13$	$225$	$\:\;345$
${\rm ENC}\,\rm [e]$	$564$	$966$	$1524$
${\rm SNR_{MIP}}$	$39.9$	$23.3$	$14.8$

Tab.  gives an overview of typical detector and readout parameters, while the resulting noise contributions are shown in tab. . The dominant noise source of both detector systems is the amplifier chip, whose contribution principally depends on the peaking time. It is obvious that the signal-to-noise-ratio ($\rm SNR$) was not an important issue in DELPHI, while it is a crucial figure for CMS especially with the deconvolution method, which will be the default mode of operation.

The noise figures given here include the detector together with the input transistor of the front-end amplifier. In reality, other components in the read-out chain beyond this point also contribute to the total observed noise. Line drivers and receivers, the transmission line and the digitization typically add a few hundred electrons of noise. However, since this contribution is uncorrelated as well, the total square sum is still dominated by the front-end noise.