Supplementary Material
Discovery of a Photoresponse Amplification Mechanism in
Compensated PN Junctions
Yuchun Zhou, 1 Yu-Hsin Liu, 2 Samia N. Rahman, 1 David Hall, 1 L. J. Sham3 and YuHwa Lo1, a)
1Department
of Electrical and Computer Engineering, University of California, San
Diego 9500 Gilman Drive, La Jolla, California 92093, USA
2Materials Science and Engineering Program, University of California, San Diego 9500
Gilman Drive, La Jolla, California 92093, USA
3Department of Physics, University of California, San Diego 9500 Gilman Drive, La
Jolla, California 92093, USA
a) To
whom correspondence should be addressed. Email address: ylo@ucsd.edu
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Supplementary Material
Analysis and simulation of the amplified photoresponse from the cycling excitation
process (CEP).
In the following, we derive the expression of the amplification factor of the cycling
excitation process (CEP).
Fig. S1: Carrier multiplication by cycling excitation process initiated from a single
photon incident on either the p-side or n-side of the heavily doped and compensated
junction.
Fig. S1 shows how the photogenerated primary (zeroth generation) electron-hole pair
initiates the cycling excitation process. In the figure, we label each electron and hole
according to its generation. For example, we use “0” to denote the primary electron and
hole generated by photon absorption. The number of electron-hole pairs produced by the
i -th generation hot electron and hot hole is assumed to be X i and Yi , respectively. X i
and Yi are considered to be independent random variables and all X i ’s and Yi ’s have
their mean values:
X i  x (i  1, 2,3,...)
(1)
Y j  y (i  1, 2,3,...)
(2)
The average number of electron-hole pairs produced by the absorption of a single photon
in the p-side over the series of cyclic excitations is given by:
Gpside  1  X 0  X 0Y1  X 0Y1 X 2  X 0Y1 X 2Y3  X 0Y1 X 2Y3 X 4  
(3)
Using the relations in Eq. (1) and Eq. (2), we simplify Eq. (3) into
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Supplementary Material
G p  side 
1 x
1  xy
(4)
Similarly, the average number of e-h pairs due to absorption of a single photon in the nside is derived as
Gnside  1  Y0  Y0 X1  Y0 X1Y2  Y0 X1Y2 X 3  Y0 X1Y2 X 3Y4  
(5)
Again, using the relations in Eq. (1) and Eq. (2), Eq. (5) is simplified as
Gn side 
1 y
1  xy
(6)
Finally, the total number of e-h pairs of the device is derived as
G junction 
Pp (1  x)  Pn (1  y )
( Pp  Pn )(1  xy )
(7)
where Pp and Pn in Eq. (7) are the probabilities for the incoming photon to be absorbed in
the p-side and n-side of the device.
Fig. S2: (a) Histograms of gain distribution with the e-h pair generation probability at
p  0.5 with a mean gain value of 2.0 and an excess noise factor of 1.26. (b) Histograms
of gain distribution for p  0.95 with a mean gain value of 20.0 and an excess noise
factor of 1.48.
The above analysis yields the average gain. Next we apply Monte Carlo simulations to
verify the analytical model by comparing the mean value of gain with the analytic
expression Eq. (7) and to obtain the gain distribution, which yields the noise
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Supplementary Material
characteristics of the amplification mechanism. Fig. S2 shows the histograms of the
simulated gain distribution for 10000 events assuming that ⟨𝑋𝑖 ⟩ = ⟨𝑌𝑖 ⟩ is set to 0.5 and
0.95 in parts (a) and (b), and 𝑃𝑝 = 𝑃𝑛 is also assumed for the calculations. In the
simulations, the value of 𝑋𝑖 or 𝑌𝑖 was set to be either “0” or “1” indicating if an excitation
event occurs or not. Under low bias condition, the probability for a carrier to create more
than 1 excitation event is assumed negligible.
The simulated mean gain values in the insets show excellent agreement with the results
from the analytical model (Eq. (7)). Moreover, based on the histograms, the excess noise
⟨𝐺 2
⟩
factor (𝑁. 𝐹. = ⟨𝐺 𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛⟩2) is found to be N.F. =1.26 for ⟨𝐺⟩ = 2 and 1.48 for ⟨𝐺⟩ = 20.
𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛
The excess noise factor for the CEP process appears to be lower than that of conventional
Si avalanche photodiodes having their excess noise factor greater than 2 at ⟨𝐺⟩ = 20.
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