Resistive Read-out in Thin Silicon Sensors with Internal Gain

In the past few years, the performance capabilities of silicon sensors in terms of combined spatial and temporal resolutions have improved significantly. This evolution is due to the introduction of two innovations in the design of silicon sensors: (i) Low-Gain Avalanche Diode and (ii) Resistive Read-out.

The Low-Gain Avalanche Diode [1, 2] design has been first introduced to compensate for the loss of signal due to charge trapping in irradiated sensors. However, this design found its main application in the field of precision timing, with the introduction of Ultra-Fast Silicon Detector (UFSD) [3]. This R&D has spurred a strong evolution in the field of accurate timing using silicon detectors.

Resistive read-out was first proposed to achieve an LGAD design with 100% fill factor [4] (the so-called AC-LGAD). Then, it was subsequently recognized to lead to excellent spatial precision with a concurrent reduction of the number of read-out electrodes [5, 6]. Sensors based on resistive read-out are called RSDs (Resistive Silicon Detectors). In the productions completed so far  [7, 8, 9], RSDs have an AC-coupled design; recently, the design has been extended to DC-coupled read-out [10] (DC-RSD). The key feature of RSDs is that the signal is shared among the electrodes near the hit point analogously to a current divider: each pad $i$ sees a fraction $I_{i}$ of the total signal $I_{o}$ that depends on the impedance $Z_{j}$ between the impact point and the pads, $I_{i}=I_{o}(1/Z_{i})/\sum_{1}^{n}(1/Z_{j})$.

Figure 1 shows the sketch of a silicon sensor that incorporates both the LGAD and RSD innovations. The design is based on an n-in-p sensor, has a continuous gain implant just underneath the cathode, and the cathode is resistive to ensure electrode isolation and signal sharing. The presence of the gain implant, the signature feature of the LGAD design, creates a high electric field in the volume underneath the $n^{+}$ resistive layer and leads to signal multiplication. In this sketch, the metal electrodes are directly implanted in the resistive $n^{+}$ sheet to ensure a DC coupling between the sensor and the electronics (DC-RSD). The RSD design has uniform electric and weighting fields over the whole sensor volume, a requisite for good temporal resolution.

According to the ECFA roadmap [11], the next generation of large silicon trackers will be deployed at either circular (FCC) or linear (ILC, CLIC) lepton colliders. The physics aims of these future lepton colliders, revolving around very precise flavor physics, can only be achieved with silicon trackers with very small impact parameter resolutions. Table 1 reports a few important parameters of future silicon vertex trackers at e⁺e^- colliders.

The trackers need to have a superb position resolution (less than 5 $\upmu$m), be very thin, sustain a high hit particle rate, and use very little power. Two quantities need to be minimized to achieve these goals: (i) the single hit resolution $\sigma_{hit\;pos}$, and (ii) the multiple scattering resolution $\sigma_{MS}$. The two terms $\sigma_{hit\;pos}$ and $\sigma_{MS}$ are linked to each other and to the type of read-out architecture. The pixel size determines the spatial resolution if a tracker employs single-pixel read-out. Only very small pixels (25x25 $\upmu$m²) achieve the required precision of 5 $\upmu$m, and it is practically impossible to reach better resolutions. On the other hand, if a tracker uses the traditional design for charge sharing to improve $\sigma_{hit\;pos}$, the sensor needs to be quite thick (at least 200 $\upmu$m), leading to a large $\sigma_{MS}$ contribution. The very mechanism that optimizes $\sigma_{hit\;pos}$ is detrimental to $\sigma_{MS}$: thick sensors, necessary for signal sharing, cause significant multiple scattering and deteriorate the overall accuracy of the tracker system. Multiple scattering is further increased by the cooling infrastructures: for this reason, the levels of power consumption reported in Table 1 are such that air cooling can be employed. This request is particularly difficult to achieve when using single pixel read-out as the power used by millions of pixels is large. Although research and development in silicon detectors is very active in many fields, currently, no design can achieve the performance listed in Table 1 [12].

According to our present R&D studies, the key to meeting the demand of the next generation of lepton colliders is to use thin silicon sensors that combine resistive read-out and internal gain.

In the present silicon tracker paradigm, the targeted position resolution determines the pixel size. These systems reach an excellent spatial resolution using millions of pixels and amplifiers that, at any given time, are mostly empty; in many situations, less than 0.1 % of pixels see a signal. In a much more efficient design, the density of particles determines the pixel size. For example, the pixel size should be such that less than a few per mille of pixels are hit by 2 particles at the same time. Resistive read-out allows reaching excellent position resolution while using large pixels, so the pixel size is determined by occupancy and not by the needed position resolution. The presence of internal gain boosts the signal and allows using thin sensors.

Figure 2 illustrates why the combination of resistive read-out and low-gain amplification is so powerful. With this design, the signals are shared among a few pads, have large amplitudes, are uniform over the sensor surface, and are short.

The experimental features arising from the design innovations can be summarised as:

1. 1.

   Sharing allows using large pixels, i.e., having enough space for the read-out electronics.

2. 2.

   Sharing combined with internal amplification achieves excellent spatial precision.

3. 3.

   Sharing (fewer pixels) combined with internal amplification (less need for amplification in the electronics) reduces power consumption.

4. 4.

   Large and short signals combined with a uniform response yield excellent temporal precision.

5. 5.

   Large signals and a uniform response yield 100% fill factor.

6. 6.

   Large signals and a uniform response yield 100% efficiency.

7. 7.

   Short signals make it possible to work at a high rate.

8. 8.

   Thin sensors (and thin front-end electronics) enable a low material budget.

The second RSD production at FBK [9] (RSD2) was designed to test several optimizations of the first FBK RSD production. An in-depth study of the RSD2 spatial and temporal resolutions can be found in [15]. An important improvement has been the introduction of cross-shaped electrodes, which significantly increase the response uniformity. Figure 6 shows on the left pane the layout of a sensor composed of an array of 450 \micronspixels with cross-shaped electrodes while on the right, a photograph of the sensors. Wirebonders are visible for a group of 16 electrodes.

The combined spatial and temporal resolutions for several sensors with cross-shaped electrodes are presented in Figure 7. The presented results are obtained at a gain of about 30.

The two key points are: (i) The spatial resolution is about 3% of the pixel size, and it scales linearly with the pixel size, (i) the temporal resolution is fairly constant at about 35-40 ps as a function of the pixel size.

The development of resistive read-out in thin sensors with internal multiplication is driven by the needs of future experiments and offers a viable solution to meet the demanding requirements of future charged particle trackers. More generally, it will be relevant for all applications requiring accurate photons and charged particle localization. Our preliminary results show that a concurrent spatial resolution of about 3% of the pixel size and temporal resolution of about 35 - 40 ps is achievable. The development of the read-out electronics should be tailored to the need of the specific application: at low power consumption, a simple ToT system is a good choice, while for higher power consumption, a double ToT read-out or a waveform sampler can be considered.

We kindly acknowledge the following funding agencies and collaborations: INFN – FBK agreement on sensor production; Dipartimenti di Eccellenza, Univ. of Torino (ex L. 232/2016, art. 1, cc. 314-337); Ministero della Ricerca, Italia, PRIN 2017, Grant 2017L2XKTJ – 4DinSiDe; Ministero della Ricerca, Italia, FARE, Grant R165xr8frt_fare, Compagnia di San Paolo, Italia, Grant TRAPEZIO 2021.