After the initiation of an avalanche, the avalanche current must be quenched to avoid damaging the diode. This can be achieved by lowering the biasing voltage of the SPAD below its breakdown voltage preventing the occurrence of the multiplication process. The simplest way to accomplish this is by using a Passive Quenching Circuit (PQC) [43] consisting of a high- value ballast resistor connected in series with the SPAD (Fig. 2.8). In quiescent conditions, the SPAD is reverse-biased to the desired voltage (above breakdown voltage) through the ballast resistor. When an avalanche is triggered in response to an impinging photon, the diode current
(Id) swiftly rises to its peak value, given by the transient excess bias voltage (Vex = Vd - Vbd) divided by the diode resistance (Rd) as below :
Id (t)
=
𝑉𝑑(𝑡)−𝑉𝑏𝑑𝑅𝑑
=
𝑉𝑒𝑥(𝑡)
𝑅𝑑 (2.2)
where, Vd is the diode voltage and Vbd the breakdown voltage of the SPAD. The diode resistance Rd is given by the series of space–charge resistance of the avalanche junction and ohmic resistance of the neutral semiconductor crossed by the current. The value of Rd depends on the semiconductor device structure, which is typically lower than 500Ω for SPADs with a wide area and thick depletion layer and from a few hundred ohms to various kiloohms for devices with a small area and a thin junction [44]. The self-sustaining avalanche current develops a voltage drop across the high resistive load and the avalanche process simply quenches itself. The avalanche current discharges the parasitic capacitance at the SPAD cathode so that Id exponentially drops and reaches the steady state value of Ids.
Ids
=
𝑉𝑂𝑃−𝑉𝑏𝑑𝑅𝑑+𝑅𝑞
≈
𝑉𝐸𝑋
𝑅𝑞
(
since 𝑅𝑑 ≪ 𝑅𝑞) (2.3) The quenching time constant or the exponential decay of cathode voltage is set by the terminal capacitance Cd and the resistances Rq and Rd in parallel as demonstrated by the following equation.Tq
=
𝐶𝑑𝑅𝑑 𝑅𝑞
𝑅𝑑+𝑅𝑞
≈
𝐶𝑑𝑅𝑑 (since 𝑅𝑑 ≪ 𝑅𝑞) (2.4)
As Id declines, the diode voltage also reduces approaching Vbd which causes a decline in the number of carriers traversing the avalanche region. Since the avalanche process is statistical, so it may happen that after a random time none of the carriers crossing the high field region cause impact ionization. However, the probability of the occurrence of zero multiplied carriers rapidly increases when the diode current Id falls below the latching current level Iq (Fig. 2.9). The value of Iq is not sharply defined, although a value of about 100µA has often been used. If the steady state current Ids is set to a value much lower value than the latching current Iq, then the PQC works in the best possible way, quenching the avalanche within a well defined time with fairly small jitter. However, if Ids approaches Iq, quenching occurs with a progressively longer delay and wider time jitter and above Iq, the avalanche is no longer quenched and a steady current flows just like that of normal diodes used as voltage references in electronic circuits which may cause permanent damage due to excessive heating. The value of the ballast resistor must be
Figure 2.9. Timing diagram during passive quench and recharge operation
large enough to ensure that the asymptotic Ids value is sufficiently lower than the quenching level Iq. A good rule of thumb suggests that Ids should not exceed 20µA. The value of the quenching resistor Rq should be at least 50 kΩ for 1V of applied excess bias voltage Vex. Henceforth, lower values of Rq should be avoided and the minimum recommended values to be employed range from 50 to 500 kΩ for thin-junction SPAD devices that work with VEX from 1 to 10 V, and 200 kΩ to 2.5 MΩ for thick-junction SPADs that work with VEX from 4 to 50 V. The avalanche process is self-sustaining above the latching current level Iq (<100µA) and self- quenching below it. Jitter of the quenching time with respect to the avalanche onset influences the value of Iq and results in the corresponding jitter of diode voltage Vq at which quenching
occurs. Theoretically, Vq is slightly higher than Vbd as can be inferred from the equation below:
Vq
=
𝑉𝑏𝑑+ 𝐼𝑞𝑅𝑑 (2.5)The total charge Qav in the avalanche pulse is related to the steady state current Ids and can be evaluated using the equation :
Qav
=
𝑉𝑂𝑃−𝑉𝑞 𝐶𝑑≈
𝑉𝐸𝑋 𝐶𝑑≈ 𝐼
𝑓𝑇𝑟(2.6) Tr
=
𝑅𝑞𝐶𝑑(2.7)
Tr corresponds to the characteristic time constant of the voltage recovery. An output pulse can be obtained from a PQC as shown in Fig 2.8 (a). The voltage-mode output generates a peak amplitude of : Vu
=
(𝑉𝑂𝑃− 𝑉𝑏𝑑− 𝐼𝑞𝑅𝑑) 𝑅𝑑 𝑅𝑞+𝑅𝑑≈
𝑉𝐸𝑋 𝑅𝑑 𝑅𝑞(2.8)
One drawback of this configuration producing a voltage waveform from the fast avalanche current pulse is that the detector timing performance is not fully exploited owing to the time constant Tq, generated by the intrinsic low-pass filter, that acts on the fast current pulse to produce the voltage waveform. Such low-pass filtering has a detrimental effect on the photon timing accuracy but can be compensated by employing a very low threshold level in the subsequent fast timing electronics.