1. of APS (Full Depleted FDAPS, MAPS and Hybrid Technology HPD)
A Theoretical Investigation of the Influence of High Irradiation on Signal to Noise Resolution (S/N) of different types
of APS (Full Depleted FDAPS, MAPS and Hybrid Technology HPD)
Vasilij Kushpil, Svetlana Kushpil
Nuclear Physics Institute , ASCR, Řež near Prague
We use a simplified model for the pixel diode based on the differential equation to describe a process of diffusion and drift. This model allows to estimate the S/N resolution depended on the radiation absorbed dose. The estimations were done for the HPD, FDAPS, MAPS.We also studied the dependence the S/N resolution on the geometrical proportions of the diode's area. The results can be useful for developing of new PD with high radiation hardness.
Detector technology and geometry
The Monolitic Active Pixel Sensor (MAPS)
Full depleted active pixel sensor (FDAPS) and hybrid pixel detector (HPD)
have the simular structure.
We calculate a ratio of the active area (Sactive) of the APS to total area (Stotal) of the APS (see formula 1). The figure 2,b shows the dependence Stotal /Sactive
from parameter k, where k=b/a (see Fig.2,a).
𝑆 𝑎𝑐𝑡𝑖𝑣𝑒 𝑆 𝑡𝑜𝑡𝑎𝑙 =1+ 𝑙 𝑎 ⋅{ 𝑀−1 𝑘⋅𝑀 + 𝑁−1 𝑁 }+ 𝑙 𝑎 2 ⋅{ 𝑁−1 ⋅ 𝑀−1 𝑘⋅𝑁⋅𝑀 } a) b)
Figure 1: Principle of operation of the MAPS based on the PN diode.
Figure 2: a) Geometry of the typical pixel detector consisted of a group of the elements (diodes), where A,B are sizes of the pixel detector; a,b are sizes of the active area; l is dead space between pixel elements,N/M are numbers of the elements in row/column;
b) The dependence of the ratio Stotal /Sactive from parameter k
Figure 3: Structure of the FDAPS/HPD based on the PIN diode.
Physical modeles
Let's calculate the drift current (Idrift ) generated by a MIP in a PIN diode . We use the equations [1]:
The noise of PIN diode is defined by leakage currents [2]:
The solution is:
𝐼 𝑃𝐼𝑁 𝑉 =𝑒⋅𝑆⋅ 𝐷 𝑛 ⋅ 𝑁 𝑖 2 𝐿 𝑛 ⋅ 𝑁 𝑑 + 𝑊𝑛 𝑉 ⋅ 𝑁 𝑖 2⋅ 𝐿 𝑛 2 𝐷 𝑛 +𝐺⋅𝑊𝑛 𝑉 ⋅ 𝑁 𝑖 ⋅ 1+exp − 𝑉 φ 𝑇
𝑑𝑛 𝑑𝑡 = ∂𝑛 ∂𝑡 + μ 𝑒 ⋅𝐸⋅ ∂𝑛 ∂𝑥
𝐼 𝑡 𝑑𝑟𝑖𝑓𝑡 = 𝑒 𝑊 𝑛 ⋅ μ 𝑒 ⋅exp 𝑒⋅ μ 𝑒 ⋅ 𝑁 𝑑 ε 𝑜 ⋅ ε 𝑆𝑖 ⋅[ 𝐸 𝑜 ⋅ 𝑊 𝑛 −ξ 𝑡 + 𝑒⋅ 𝑁 𝑑 ε 𝑜 ⋅ ε 𝑆𝑖 ⋅ 𝑊 𝑛 2 −ξ 𝑡 2 ]
∂𝐸 ∂𝑥 = −𝑒⋅ 𝑁 𝑑 ε 𝑜 ⋅ ε 𝑆𝑖
(1)
The noise of PN diode is defined by leakage currents [2]:
𝐼 𝑡 𝑒 = 𝑒 𝑊 𝑛 ⋅ μ 𝑒 ⋅∫ 𝑛 𝑥,𝑡 ⋅𝐸 𝑥 𝑑𝑡
where:
ξ 𝑡 − 𝐸 𝑜 ⋅ ε 𝑜 ⋅ ε 𝑆𝑖 𝑒⋅ 𝑁 𝑑 1−exp −𝑒⋅ μ 𝑒 ⋅ 𝑁 𝑑 ε 𝑜 ⋅ ε 𝑆𝑖 1−exp −𝑒⋅ μ 𝑒 ⋅ 𝑁 𝑑 ε 𝑜 ⋅ ε 𝑆𝑖
𝐼 𝑃𝑁 𝑉 =𝑒⋅𝑆⋅ 𝐷 𝑛 ⋅ 𝑁 𝑖 2 𝐿 𝑛 ⋅ 𝑁 𝑑 +𝐺⋅𝑊𝑛 𝑉 ⋅ 𝑁 𝑖 ⋅ 1+exp − 𝑉 φ 𝑇
The diffusion current (Idiff ) generated by a MIP in a PN diode can be obtained :
∂𝑛 ∂𝑡 ≈ 𝐷 𝑛 ⋅ ∂𝑛 ∂𝑥
The signal to noise for Johnson noise of resistor Req [3]:
Figure 4: Dependence of the leakage current on voltage for the PIN diode (red line) and the PN diode (blue line)
The solution is:
(2)
𝑆 𝑁 = 𝐼 𝑠𝑖𝑔𝑛𝑎𝑙 2 ⋅ 𝑅 𝑒𝑞 〈 𝐼 𝑙𝑒𝑎𝑘 〉 2 ⋅ 𝑅 𝑒𝑞
𝐼 𝑑𝑖𝑓𝑓 𝑡 ≈ 𝑒⋅ 𝑁 𝑒 𝐷 𝑛 𝐿 𝑛 ⋅δ 𝑡
δ 𝑡 ≈ 1−𝐾⋅ 𝑡 𝑇 𝑐𝑜𝑙𝑙 1−𝐾⋅ 𝑡 𝑇 𝑐𝑜𝑙𝑙
where
Tcoll – collection time, and K≤1
Signal and noise
Formula (3) gives us a trap concentration (Fig.7)
for different k and a, when Isurf= Ibulk
Ni - own concentration electrons in Si
Nt - trap concentration in bulk near surface
W(Vbias) – depth of depletion region
λ(Vbias) — depth of surface potential
Τi — life time of electrons
Τt — life time of electron traps
Vbias — bias voltage
Figure 6: Scheme of the leakage current thermally generated in a diode near surface.
Ratio of the leakage currents gene-
rated in a bulk and near surface
can be estimated:
𝐼 𝑏𝑢𝑙𝑘 𝐼 𝑠𝑢𝑟𝑓 ≈ τ 𝑡 τ 𝑖 ⋅ 𝑁 𝑖 𝑁 𝑡 ⋅ 𝑊 𝑉 𝑏𝑖𝑎𝑠 λ 𝑉 𝑏𝑖𝑎𝑠 ⋅𝑙 ⋅𝐹 𝑘,𝑎
Figure 5: Signals as a dependence of the current on time for the PIN and the PN diodes. The values were calculated from (1) and (2).
(3)
Figure 7: The dependence of the trap concentration on parameter k=b/a for different length of the diffusion(Lp)
Figure 8: Dependence of the ratio signal to noise (S/N) for the PIN diode and the PN diode on bias voltage for different length of diffusion
where:
𝐹 𝑘,𝑎 = 𝑎⋅𝑏 2⋅ 𝑎+𝑏
Conclusion
Radiation influence
The theoretical investigation allowed us to conclude:
1. Active area of an APS could be more effectively used if b/a>2;
2. To minimize the leakage current thermally generated in a diode
near surface one can choose the ratio b/a if the diffusion length
is known ;
3. Signal-to-noise ratio for the APS increases with increasing bias voltage,
while for the MAPS it decreases with increasing bias voltage;
4. Absolute value of signal-to-noise ratio for the MAPS is larger than for
signal-to-noise ratio for APS (in frame of considered model);
5. For the irradiated the APS and the MAPS we studied a behavior
of signal-to-noise ratio depending on absorbed dose.
Signal-to-noise ratio decreases with increasing dose. Moreover, for
the APS signal-to-noise ratio decreases more faster than signal-to-noise
ratio for the MAPS.
Figure 9: Dependence of the S/N ratio on an absorbed dose of 1-MeV electrons for the PIN(red,blue) and PN (green,pink)
Figure 10: Dependence of the S/N ratio on an absorbed dose of 1-MeV electrons for the PIN(blue) and PN (red). The value of the S/N ratio was normalized to maximal value.
References
[1] Eremin, V. Chen, W. Li, Z "Analytical Solutions Of Minimum Ionization Particle Induced Current Shapes Of Silicon Detectors And Simulation Of
Charge Collection Properties".Nuclear Science Symposium and Medical Imaging Conference, 31 Oct-6 Nov 1993, pp
[2] by Rolf Enderlein, " FUNDAMENTALS OF SEMICONDUCTOR
PHYSICS AND DEVICES" 1997
[3] John C. Russ (2007). The image processing handbook. CRC Press. ISBN