# Origin of Irradiation Synergistic Effects in Silicon Bipolar Transistors: a
Review

Yu Song<EMAIL_ADDRESS>College of Physics and Electronic Information
Engineering, Neijiang Normal University, Neijiang 641112, China  Su-Huai Wei
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100193, China

###### Abstract

The practical damage of silicon bipolar devices subjected to mixed ionization
and displacement irradiations is usually evaluated by the sum of separated
ionization and displacement damages. However, recent experiments show clear
difference between the practical and summed damages, indicating significant
irradiation synergistic effects (ISEs). Understanding the behaviors and
mechanisms of ISEs is essential to predict the practical damages. In this
work, we first make a brief review on the state of the art, critically
emphasizing on the difficulty encountered in previous models to understand the
dose rate dependence of the ISEs. We then introduce in detail our models
explaining this basic phenomenon, which can be described as follows. Firstly,
we show our experimental works on PNP and NPN transistors. A variable neutron
fluence and $\gamma$-ray dose setup is adopted. Fluence-dependent ‘tick’-like
and sublinear dose profiles are observed for PNP and NPN transistors,
respectively. Secondly, we describe our theoretical investigations on the
positive ISE in NPN transistors. We propose an atomistic model of
transformation and annihilation of displacement-irradiation-induced $\rm
V_{2}$ defects in p-type silicon under ionization irradiation, which is
totally different from the traditional picture of Coulomb interaction of oxide
trapped charges in silica on charge carriers in irradiated silicon. The
predicted novel dose and fluence dependences are fully verified by the
experimental data. Thirdly, the mechanism of the observed negative ISE in PNP
transistors is investigated in a similar way as in the NPN transistor case.
The difference is that in n-type silicon, displacement-irradiation-induced VO
defects also undergo an ionization-induced transformation and annihilation
process. Our results show that, the evolution of displacement defects due to
carrier-enhanced defect diffusion and reaction is the dominating mechanism of
the ISEs. Finally, we give a perspective on future investigations on the ISEs
when the displacement and ionization irradiations are present simultaneously.

Silicon bipolar transistor; irradiation synergistic effect; carrier-enhanced
defect diffusion; recombination-enhanced defect reaction; divacancy;
divacancy-oxygen complex; Coulomb interaction; defect modification

##  I Introduction

Irradiation synergistic effect. Silicon (Si)-based NPN and PNP transistors are
building blocks of modern microelectronics, such as operational amplifiers and
voltage comparators. When these devices are used in outer space and other
extreme environments such as nuclear reactors, both charge carriers and atomic
displacements will be generated in the silica and silicon materials, which
result in ionization and displacement damages in the devices. Oldham and
McLean (2003); Srour  _et al._ (2003) Conventionally, the practical damage is
usually assumed as a simple sum of the ionization damage (ID) and displacement
damage (DD), which then can be evaluated individually at ground by carrying
out $\gamma$-ray and neutron irradiation experiments, respectively. The
primary reason for the separability of the damages is that, the ID is caused
by the ionization defects, oxide trapped charges (OTs) and interface traps
(ITs), generated in the silica and near interface, Dressendorfer (1998) while
the DD is caused by the displacement defects, such as divacancies (V2) and
vacancy-oxygen complex (VO), generated in Si. Fleming  _et al._ (2010)
Accordingly, it was widely and always believed that there is no direct
interaction between these two irradiation-induced defect dynamics and the
practical damage can be separated.

However, in recent experiments it has been clearly demonstrated that, when
these bipolar transistors are simultaneously or subsequently irradiated by
displacement and ionization irradiations, the practical damage is usually
smaller or larger than the simply summed damage. Barnaby  _et al._ (2001,
2002); Gorelick  _et al._ (2004); Li  _et al._ (2012a, b, 2015); Wang  _et
al._ (2016); Song  _et al._ (2019, 2020a) These phenomena can be called as
irradiation synergistic effects (ISEs), which are negative in PNP transistors
and positive in NPN transistors, respectively. The presence of ISEs imply that
one cannot evaluate the practical damage by carrying out separated neutron and
$\gamma$-ray irradiations and simply summing the damages up. To obtain a
relatively true damage, one must know the behaviors of the ISEs and understand
the underlying mechanisms.

Figure 1: Schematic diagram of the Coulomb interaction mechanisms for NPN (a)
and PNP (b) transistors. The short arrows indicate the drift of the charge
carriers due to the influence of OTs in the oxide layer.

Existing model of the ISEs. Currently, the ISEs are explained in a point of
view of interfacial interaction in the device scale, see, Fig. 1. Especially,
the effect is believed to stem from the Coulomb interaction of OTs in the
silica layer on the charge distributions in the displacement-damaged silicon
base region, where the non-radiative Shockley-Read-Hall (SRH) carrier
recombination occurs. Barnaby  _et al._ (2001, 2002) To be fair, this point of
view is rather natural: the outcomes of the ID and DD are the ionization and
displacement defects in the silica layer and silicon region, respectively, so,
intuitively, the synergistic effect of the ID and DD should come from the
Coulomb interaction between the charged defects and carriers in these two
different regions. Actually, the opposite signs of the ISEs in NPN (which is
$+$) and PNP (which is $-$) transistors can be consistently explained by this
mechanism, as followings. A positive ISE arises in NPN transistors because the
positively charged OTs repel the positively charged holes as majority carriers
near the surface of the p-type silicon base, which reduces the difference of
hole and electron densities and enhances the SRH recombination current in the
base region, see Fig. 1(a). On the contrary, a negative ISE would arise in PNP
transistors because the positive OTs attract the negatively charged electrons
as majority carriers near the surface of the n-type silicon base, which widens
the difference of carrier densities and thus suppresses the SRH recombination
in the base region, see Fig. 1 (b).

Difficulty of the existing model in understanding the ISE. However, we notice
that some of the experimental observations of the ISEs cannot be explained by
the existing mechanisms. The most remarkable feature is the dose-rate
dependence of the effect. According to a reduced high-dose-rate sensitivity,
Pease  _et al._ (2008) the OTs will become fewer for decreasing dose rate
irradiation at a fixed total dose. Song  _et al._ (2020b) According to the
Coulomb interaction mechanism, the ISE should become weaker as the dose rate
decreases. However, this deduction is totally opposite to the experimental
results, where the synergistic effects between DD and ID are found to be
enhanced at lower dose rate $\gamma$-ray irradiations. Song  _et al._ (2019,
2020a) On the other hand, the ionization dose (dose rate) and displacement
fluence/flux dependencies of the ISEs are essential for the understanding and
modeling. However, existing experimental works usually use proton and electron
irradiations, where the simultaneously generated DD and ID are hard to
separate. Barnaby  _et al._ (2001); Li  _et al._ (2010) Meanwhile, existing
investigations usually consider fixed ionization dose or displacement fluence.
So, till now, the behavior of the ISEs as functions of the ionization dose,
dose rate, and displacement fluence is still not clear.

To sum up, the ISEs are important to evaluate practical damages of silicon
bipolar transistors in irradiation environments. However, there are still huge
gaps in the investigations of the behaviors and mechanisms of the effects. The
behavior of the ISEs is far from clear due to the fixed displacement fluence
and/or ionization dose in existing experiments. The understanding of the
effect is based on a point of view of an interfacial interaction, which seems
natural but cannot explain the dose rate dependence of the effect. Thus, new
experimental studies and mechanisms that can self-consistently describe the
experimental observations are urgently needed.

Figure 2: Setup of the neutron/$\gamma$-ray irradiation experiments for PNP
(a) and NPN (b) transistors. For each of the irradiation conditions, 8 PNP
transistors or 3 NPN transistors are used. The color stripes schematically
indicate the strength of damages.

##  II Behaviors of the ISEs

###  II.1 Experimental setup to explore the behaviors

To explore the behavior of the ISEs in PNP and NPN transistors, Song  _et al._
(2019, 2020a) LM324N chips with 4 PNP-type input-stage transistors in each
chip and 3DK9D NPN-type transistors were selected as the research object,
respectively. These devices were chosen because they are sensitive to both ID
and DD. The neutron irradiation hardly influences the dynamics of OTs in the
oxide layer, Raymond and Petersen (1987) hence, we adopt a sequential
neutron/$\gamma$-ray irradiation configuration. The processes of the
sequential irradiation experiments for PNP transistors is shown in Fig. 2 (a).
The transistors are first irradiated by neutron to the fluence as indicated in
the figure along the vertical direction; then they are irradiated by
$\gamma$-ray to the dose as indicated in the figure along the horizontal
direction. By these irradiations the practical damages can be obtained. The
summed damages as a reference are obtained by another two pure $\gamma$-ray
irradiations. For each neutron-$\gamma$-ray condition, 8 PNP transistors
(within 2 chips) are used to avoid possible misjudgment due to sample
variability. Note, existing experiments Barnaby  _et al._ (2001); Gorelick
_et al._ (2004); Li  _et al._ (2012a); Wang  _et al._ (2016) usually adopt
only a fixed displacement fluence and ionization dose, hence is very different
from the present experimental setup. The radiation sources, irradiation bias,
and testing systems are described in detail in Ref. Song  _et al._ , 2019. The
ISE of NPN transistors is investigated by a very similar sequential
neurton/$\gamma$-ray irradiation experiment, which is shown in Fig. 2 (b) and
has been described in detail in Ref. Song  _et al._ , 2020a.

Figure 3: The neutron fluence dependence of the input bias current of LM324N
chips. For each fluence, the two chips are shown by red and blue,
respectively; in each chip, the four transistors are shown by different symbol
shapes. Reproduced from Song, Y. _et al._ , _ACS Applied Electronic Materials_
2019, _1_ , 538–547. Copyright 2019 American Chemical Society.
Figure 4: The $\gamma$-ray dose dependence of the input bias current of LM324N
chips for initial neutron fluence of $0$ (a), $2\times
10^{13}~{}\text{cm}^{-2}$ (b), $3\times 10^{13}~{}\text{cm}^{-2}$ (c), and
$5\times 10^{13}~{}\text{cm}^{-2}$ (d), respectively. The dose rate is
$2.2~{}\text{mrad(Si)/s}$. In each split, the eight transistors are shown by
different colors and symbol shapes as in Fig. 3. Reproduced from Song, Y. _et
al._ , _ACS Applied Electronic Materials_ 2019, _1_ , 538–547. Copyright 2019
American Chemical Society.

###  II.2 Fluence-dependent ‘tick’-like dose curves of PNP transistors

Artificial summed damages. The individual DDs of the PNP transistors are shown
in Fig. 3, from which a sub-linear dependence of the input bias current
($I_{iB}$) on the neutron fluence is clearly seen. It is also seen that, the
DDs of the 8 transistors under a same fluence are different due to sample-to-
sample variability. To obtain the artificial summed damages, Nos. 1-2 and 9-10
chips were irradiated by a low and high dose rate $\gamma$ rays, respectively.
The results for the former are displayed in Fig. 4 (a). An almost linear
increase of the ID is seen and an average slope of $k_{0}^{L}=15.6$
nA/krad(Si) and $k_{0}^{H}=8.0$ nA/krad(Si) is extracted for the low and high
dose rate case, respectively. The summed damage reads:

$\Delta I_{iB}^{D+I}=D_{0}+k_{0}^{L(H)}x,$ (1)

where $D_{0}$ is the initial DD before $\gamma$-ray irradiations. A typical
result is plotted in Fig. 4 (c) as the dashed line.

Figure 5: The typical experimental data (dots) and fitting curves (solid) of
the dose dependence of the input bias current of PNP-type LM324N for low (a)
and high (b) dose rate, respectively. The dashed lines show the summed
damages. Reproduced from Song, Y. _et al._ , _ACS Applied Electronic
Materials_ 2019, _1_ , 538–547. Copyright 2019 American Chemical Society.

Figure 6: The $\gamma$-ray dose and initial base current dependence of the
practical damages (dot) and fitting curves (solid curves) of 3DK9D transistors
for 10 rad/s (a) and 10 mrad/s (b) dose rates. The summed damages are shown by
the dashed lines. Reproduced from Song, Y. _et al._ , _ACS Applied Material &
Interfaces_ 2020, _12_ , 29993-29998. Copyright 2020 American Chemical
Society.

Practical damages and behavior of the negative ISE. The $\gamma$-ray response
of the $3^{rd}$-$8^{th}$ chips at low dose rate with different $D_{0}$’s are
displayed in Figs. 4 (b)-(d). It is seen that, the practical damages of all 24
transistors are smaller than their corresponding summed damages. This means
that we obtain a clear negative ISE in PNP transistors. Remarkably, the
practical damages display ‘tick’-like curves: the input-bias currents first
exponentially decrease for relatively small $\gamma$-ray dose and then
linearly increase for relatively large $\gamma$-ray dose. Moreover, the slope
of the linear increase is obviously smaller than that of the pure ID,
$k_{0}^{L}$. The $11^{th}$-$16^{th}$ chips display similar negative ISEs at
high dose rate $\gamma$-ray irradiations, see Ref. Song  _et al._ , 2019. To
further obtain the $D_{0}$ and $\gamma$-ray dose rate dependences of the ISE,
we plot six typical curves in Fig. 5, from which we can see that, the
exponential decay of the practical damage at small $\gamma$-ray dose becomes
stronger when the dose rate decreases; while the decrease of the slope at
large $\gamma$-ray dose becomes bigger when the initial DD increases. These
regular behaviors, especially the dose rate dependence of the ISE, cannot be
explained by the traditional Coulomb interaction mechanism.

###  II.3 Fluence-dependent sublinear dose curves of NPN transistors

The practical neutron/$\gamma$-ray damage (the base currents) of the 3DK9D NPN
transistors are shown in Fig. 6 (a) as a function of the $\gamma$-ray dose at
a dose rate of 10 rad/s (see the dots). Besides, the summed damages for three
typical samples are obtained from a direct sum of their DDs and the average ID
of five extra samples (see the dashed curves). From the results it is clear
that a positive ISE arises for the NPN transistors. From Fig. 6 (a) it is also
seen that, the practical damages increase sublinearly as the ionization dose
increases; meanwhile, the positive ISE becomes larger when the initial DD
increases. The practical and summed damages are shown in Fig. 6 (b) for the
low dose rate (10mrad/s) case. A second stage arises when the dose exceeds 2
krad. However, the two different stages are found to show similar dependences
on the $\gamma$-ray dose and neutron fluence as the high dose rate case.

To sum up, we have designed a variable neutron-fluence and $\gamma$-ray-dose
irradiation experiment to explore the behaviors of the negative and positive
ISEs in PNP and NPN transistors. The results show that both ISEs display
regular dependences on the $\gamma$-ray dose, dose rate, and neutron fluence.
The obtained features are quite different for PNP and NPN transistors.

##  III Mechanism and analytical model of the positive ISE in NPN transistors

In all previous works, Barnaby  _et al._ (2001, 2002); Li  _et al._ (2012a, b,
2015) the ISEs were attributed to the influence of the positively charged OTs
accumulated in silica on the charge density in irradiation damaged silicon,
see Fig. 1 (a) for NPN transistors and (b) for PNP transistors, respectively.
The different polarity of the ISE in PNP and NPN transistors could be
explained by this mechanism. However, as indicated above, the dose rate
dependence of the effect cannot be explained by the Coulomb interaction
mechanism. Besides, the above obtained ‘tick’-like and sublinear dose
dependence of PNP and NPN transistors, respectively, and their strong fluence
dependence are also hard to understand from the theory. These behaviors imply
that a new underlying mechanism for the ISEs in silicon bipolar transistors
should be developed.

We have systematically investigated this basic phenomena from the point of
view of _ionization-irradiation-induced evolution of displacement defects in
Si_. Song  _et al._ (2020a, 2019) The primary consideration is that, the
amplitude of the SRH recombination current in the bipolar transistors is in
direct proportion to the concentration of the displacement defects in Si;
while these defects can evolve under ionization irradiation, due to carrier-
enhanced defect diffusion Kimerling  _et al._ (1975); Kimerling (1976); Bar-
Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984); Car  _et al._
(1984) and recombination-enhanced defect reactions. Lang and Kimerling (1974);
Weeks  _et al._ (1975); Kimerling (1978) The diffusion of displacement defects
in Si can be enhanced by alternating capture and lose the ionization-
irradiation-induced electrons. Bar-Yam and Joannopoulos (1984a, b); Baraff and
Schlüter (1984); Car  _et al._ (1984) On the other hand, the defect reactions
can be enhanced by energy liberated upon non-radiative recombination. Weeks
_et al._ (1975) In p-type silicon, $\rm V_{2}$ is the only neutron-induced
active defect, while in n-type silicon, both $\rm V_{2}$ and VO are neutron-
induced active defects, see Fig. 7 (a). In this section, we will first
introduce the simpler case, i.e., the understanding and modeling of the ISE in
NPN transistors with p-type silicon as the channel. Song  _et al._ (2020a)
Then in the next section we further introduce the case of PNP transistors with
n-type silicon as the channel. Song  _et al._ (2019)

Figure 7: Schematic diagram of the proposed defect evolution mechanisms of the
ISE. (a) The energy levels of the displacement defects in Si. Bottom-half: in
p-type silicon the $\gamma$-ray irradiation can impel the transformation (with
a rate of $k_{1}$) and annihilation (with a rate of $k_{2}$) of the neutron-
irradiation-induced $\rm V_{2}$ defects. Upper-half: in n-type silicon the
evolutions can also happen for VO defects (with a rate of $k_{3}$). (b) The
predicted dose and fluence dependence of the base current in NPN transistors
that can be obtained in experiments, together with its compositions due to the
evolutions of $\rm V_{2}^{0/+}$, $\rm V_{2}O^{0/+}$, and $\rm V_{2}O^{+/+2}$
in p-type Si. (c) The predicted dose and fluence dependence of the base
current in PNP transistors, together with its compositions due to the
evolutions of $\rm V_{2}^{0/-}$ ($\rm V_{2}^{-/-2}$), $\rm V_{2}O^{0/-}$ ($\rm
V_{2}O^{-/-2}$), and $\rm VO^{0/-}$ in n-type Si.

The defect dynamics in p-type silicon under consideration is a $\gamma$-ray
induced transformation reaction from $\rm V_{2}$ to divacancy-oxygen complex
($\rm V_{2}O$) and a $\gamma$-ray induced annihilation reaction between $\rm
V_{2}$ and self-interstitials ($\rm Si_{i}$), Song  _et al._ (2020a) see the
bottom half of Fig. 7 (a). Based on these mechanisms, we first derive an
analytical model of the defect concentrations as a function of the
$\gamma$-ray dose and neutron fluence (Sec. III A). Then we relate the defect
concentrations to the base current of the NPN transistors to obtain a model
for the ISE (Sec. III B). We at last test the proposed model through the
variable-dose and -fluence data as shown in Sec. II C (Sec. III C).

###  III.1 Defect evolution model in p-type silicon

As identified by Deep Level Transient Spectroscopy (DLTS), Lang (1974a, b) the
main defect type in neutron irradiated p-type silicon is donor-like
divacancies ($\rm V_{2}^{0/+}$), Cheng  _et al._ (1966); Lee and Corbett
(1976); Lindström  _et al._ (1999) see Fig. 7 (a). We notice that, the
diffusion of $\rm V_{2}$ will be greatly enhanced when the damaged Si samples
are further subjected to a $\gamma$-ray irradiation. Song  _et al._ (2020a)
The microscopic picture of such a carrier-enhanced defect diffusion is as
followings. The $\gamma$-ray-induced charge carriers will shift the quasi-
Fermi energy in the Si samples. Accordingly, the $\rm V_{2}$ defects can
change from one charge state at its equilibrium site to another that is not at
equilibrium. Bar-Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984);
Car  _et al._ (1984) The latter will easily diffuse to its equilibrium site
without having to overcome a diffusion barrier. These behaviors largely
enhance the diffusion of $\rm V_{2}$ defect. Besides, many oxygen
interstitials ($\rm O_{i}$), which are electrically-inactive, have been
introduced in the Si sample in its growth process. The moving $\rm V_{2}$
defects will be trapped by $\rm O_{i}$ where they react to form another defect
$\rm V_{2}O$ that is electrically-active, Mikelsen  _et al._ (2005); Markevich
_et al._ (2014)

$\rm V_{2}+O_{i}\rightarrow V_{2}O,$ (2a) see Fig. 7 (a).

On the other hand, the energy liberated upon nonradiative recombination will
be converted into vibrational energy that can be utilized to promote the
defect reactions, which include the dissociation of clusters and re-emission
of self-interstitials in damaged Si. Lang and Kimerling (1974); Weeks  _et
al._ (1975); Kimerling (1978) The charge-enhanced defect diffusion also
applies to the emitted self-interstitials. The moving divacancies and self-
interstitials can collide and annihilate each other Song  _et al._ (2020a)

$\rm V_{2}+2Si_{i}\rightarrow 0.$ (2b)

The rate equation of $\rm V_{2}$ can be derived from Eq. (2). The result is
$\rm\partial V_{2}(t)/\partial t=-k_{1}O_{i}V_{2}(t)-k_{2}Si_{i}V_{2}(t)$,
where $k_{1}$ and $k_{2}$ are the transformation and annihilation reaction
velocities in Eqs. (2a) and Eq. (2b), respectively, see Fig. 7 (a). To
simplify the calculations, we replace $V_{2}(t)$ and $\rm Si_{i}$ in the last
term by their initial concentrations, which are directly related to the
irradiation fluence of neutron, $F=V_{2}(0)$. This replacement can be done
because the annihilation rate $k_{2}$ is relatively small Mikelsen  _et al._
(2005); Markevich  _et al._ (2014) and the concentration of $\rm Si_{i}$ is
directly proportion to the total amount of vacancies. The above equation
becomes

$\partial V_{2}(t)/\partial t=-\kappa_{1}V_{2}(t)-\kappa_{2}F^{2},$ (3a) where
$\kappa_{1}=k_{1}O_{i}$ and $\kappa_{2}=k_{2}Si_{i}/V_{2}$ are effective
transformation and annihilation velocities. The accompanying rate equation for
$\rm V_{2}O$ defects reads $\partial V_{2}O(t)/\partial t=\kappa_{1}V_{2}(t).$
(3b)

The initial conditions for the two defects are $\rm V_{2}(0)$ $=F$ and $\rm
V_{2}O(0)$$=0$. Solving the resulting coupled equations with these conditions,
we obtain the $\gamma$-ray dose dependence of the concentration of $\rm V_{2}$
and $\rm V_{2}O$

$V_{2}(D)=(F+\kappa_{1}^{-1}\kappa_{2}F^{2})(e^{-\kappa_{1}R^{-1}D}-1)+F,$
(4a)
$V_{2}O(D)=(F+\kappa_{1}^{-1}\kappa_{2}F^{2})(1-e^{-\kappa_{1}R^{-1}D})-\kappa_{2}F^{2}R^{-1}D.$
(4b)

Here we have changed $\partial/\partial t$ to $R\partial/\partial D$, where
$R$ is the dose rate. From the pair of solutions it is clear that, the
concentration of $\rm V_{2}$ is an exponential decay function of the
$\gamma$-ray dose, while the concentration of $\rm V_{2}O$ is both an
asymptotic growth function and a linear decay behavior of the $\gamma$-ray
dose. Song  _et al._ (2020a) The neutron fluence dependence of the defect
concentrations is indicated by the factor $F$ in Eq. (4). Song  _et al._
(2020a)

Comparison with the annealing experiments. Evolution of displacement defects
also happens in traditional annealing process induced by elevated temperature.
However, the self-interstitial emission mechanism described in Eq. 2 (b) is
absent for this case, Kimerling (1978); Pellegrino  _et al._ (2001) as the
energy is not enough to decompose the clusters. As a result, the defect
evolution is only due to defect diffusion and transformation induced by
elevated temperatures, as described by Eq. (2a). Then, the defect
concentrations as a function of the annealing time ($t$) reduces to

$V_{2}(t)=Fe^{-\kappa_{1}R^{-1}t},$ (5a)
$V_{2}O(t)=F(1-e^{-\kappa_{1}R^{-1}t}).$ (5b)

These results are the same as those in Ref. Mikelsen  _et al._ , 2005. It is
noticed that, here the increase of $\rm V_{2}O$ equals exactly to the decrease
of $\rm V_{2}$, and we will not observe a linear decay term of the defect
concentration as described in Eq. (4b).

###  III.2 Recombination current model of NPN transistors

To show the behavior of the ISE in NPN transistors, we relate the defect
concentrations in p-type Si to the SRH base current in the NPN transistor,
which are found to be in direct proportion to each other. Pierret and Neudeck
(1987); Adell and Boch (2014) The donor levels of the defects in p-type Si
have been identified by DLTS. Mikelsen  _et al._ (2005); Markevich  _et al._
(2014) As shown in Fig. 7 (a), there is one active donor level for the $\rm
V_{2}$ defect, $\rm V_{2}^{0/+}$ at $E_{v}+0.19$ eV, and two active donor
levels for the $\gamma$-ray induced $\rm V_{2}O$ defect, $\rm V_{2}O^{0/+}$ at
$E_{v}+0.24$ eV and $\rm V_{2}O^{+/+2}$ at $E_{v}+0.09$ eV. The contribution
factors of each defect charge state to the base current can be denoted as
$\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$, see Fig. 7 (a). Note, the
same contribution factor of $\lambda_{1}=\lambda_{3}$ is often assumed
Ganagona  _et al._ (2014) as $\rm V_{2}^{0/+}$ and $\rm V_{2}O^{0/+}$ have
almost the same donor levels and hole capture cross sections. Srour  _et al._
(1973); Myers  _et al._ (2008); Liu  _et al._ (2015) The $\gamma$-ray-induced
changes of the base current corresponding to the evolution of defects in Eq.
(4) read Song  _et al._ (2020a)

$\Delta I_{B}^{V_{2}^{0/+}}(D,f)=(f+\eta f^{2})(e^{-D/D_{c}}-1),$ (6a) $\Delta
I_{B}^{V_{2}O^{0/+}}(D,f)=(f+\eta f^{2})(1-e^{-D/D_{c}})-\eta f^{2}D/D_{c},$
(6b) $\Delta I_{B}^{V_{2}O^{+/+2}}(D,f)=\lambda_{21}\Delta
I_{B}^{V_{2}O^{0/+}}(D,f).$ (6c)

Here the measurable initial DD due to the neutron-irradiation-induced $V_{2}$
is denoted as $f=\lambda_{1}F$, the ratio between the $\gamma$-ray-
irradiation-induced annihilation and transformation velocities is denoted as
$\eta=\kappa_{1}^{-1}\kappa_{2}/\lambda_{1}$, $D_{c}=\kappa_{1}^{-1}R$ is a
characteristic dose of the transformation from $\rm V_{2}$ to $\rm V_{2}O$,
and $\lambda_{21}=\lambda_{2}/\lambda_{1}$ is a ratio between the contribution
factors of the first and second donor levels of $\rm V_{2}O$. The total change
of the base current is the sum of the above three components, which reads Song
_et al._ (2020a)

$\Delta I_{B}(D,f)=\lambda_{21}(f+\eta
f^{2})(1-e^{-D/D_{c}})-(1+\lambda_{21})\eta f^{2}D/D_{c}.$ (7)

This is the desired analytical model of the positive ISE in NPN transistors,
which is based on the ionization-irradiation-induced dynamics of displacement
defects in p-type silicon and shows the explicit dependence of the applied
$\gamma$-ray dose and initial DD (neutron fluence). These four current
profiles are plotted in Fig. 7(b) by using different colors.

The derived model predicts that, the positive ISE of NPN transistors would
possess one asymptotic growth term and one linear decay term as a function of
the $\gamma$-ray dose, Song  _et al._ (2020a) see the black curves in Fig. 7
(b). The asymptotic growth term of the ISE stems from the ionization-
irradiation-induced asymptotic growth of the $\rm V_{2}O^{+/+2}$ defect, see
the magenta solid curves in Fig. 7 (b). The linear decay term of the ISE
results from the two ionization-irradiation-induced linear decaying components
of $\rm V_{2}O^{0/+}$ and $\rm V_{2}O^{+/+2}$, see the blue and magenta
straight solid lines in Fig. 7 (b). As shown by the red and blue dashed curves
in Fig. 7 (b), the remaining asymptotic growth of $\rm V_{2}O^{0/+}$ and the
exponential decay of $\rm V_{2}^{0/+}$ cancel each other. The reason is the
one-to-one transformation between $\rm V_{2}$ and $\rm V_{2}O$ defects and the
same contribution factors of their first donor levels. The proposed model also
predicts an almost linear dependence of the magnitude of the asymptotic growth
term on the initial DD ($f$), and a quadratic dependence of the slope of the
linear decay term on the initial DD. Song  _et al._ (2020a) The reasons are
the one-to-one correspondence between $\rm V_{2}$ and $\rm V_{2}O$ in the
transformation and the binary behavior of interstitials and vacancies in the
annihilation. Song  _et al._ (2020a) The proposed defect dynamics model also
predicts a dose-rate dependence of the two independent synergistic terms.

Figure 8: The linear (blue dashed) and exponential (red dashed) terms of the
practical damage (black dot and solid) for a typical NPN (a) and PNP (b)
transistor. For both cases, the linear term is smaller than the average pure
ID (black dashed), indicating the presence of a linear decay term in the ISE.
Reproduced from Song, Y. _et al._ , _ACS Applied Material & Interfaces_ 2020,
_12_ , 29993-29998. Copyright 2020 American Chemical Society. Reproduced from
Song, Y. _et al._ , _ACS Applied Electronic Materials_ 2019, _1_ , 538–547.
Copyright 2019 American Chemical Society.

###  III.3 Experimental verification and parameter extraction of the model of
NPN transistors

To testify the above proposed model for NPN transistors, we try to fit the
obtained data in Sec. II.C. by using a fitting model

$\Delta I_{B}^{j}=k_{j}D+A_{j}(1-e^{-D/D_{c}^{j}}),$ (8)

where $j$ denotes the sample index. The first term in the fitting model
contains both the linear decay term in the proposed physical model and the IDs
of the samples. As the ID varies among different samples, the physical model,
Eq. (7), cannot be directly used. In stead, the presence of the linear decay
term can be identified by the statistical difference between the extracted
$k_{j}$ and $k_{0}$ (the slopes of pure ID). The correspondence between the
fitting and physical parameters is $k_{j}-k_{0}=-(1+\lambda_{21})\eta
f^{2}/D_{c}$. The second term in the fitting model is exactly the asymptotic
growth term in the proposed physical model. The correspondence between the
parameters is $A_{j}=\lambda_{21}(f+\eta f^{2})$ and $D_{c}^{j}=D_{c}$. The
fitting curves for the high dose rate case are plotted in Fig. 6 (a), which
are found to work very well for all 9 samples. The separated linear component
(blue dashed) and asymptotic growth term (red dashed) of a typical sample are
shown in Fig. 8 (a). A significant decrease is found in the linear component
relative to the average slope of pure ID (black dashed). For all 9 samples’
statistical results, see $k_{j}$ plotted in Fig. 9 (a). Similar perfect
fitting can be done for the low dose rate case within the first stage, see the
fitting curves plotted in Fig. 6 (b). These facts verify the predicted
asymptotic growth and linear decay terms in the ISE of NPN transistors,
reflecting the proper underlying defect evolution physics in the prediction of
dose dependence of the ISE.

To further testify the initial DD dependence of the proposed model, we plot
the fitting parameters of Eq. (8) ($k_{j}$, $A_{j}$, and $D_{c}^{j}$) in Fig.
9 as a function of the base current. A decreasing trend is found in Fig. 9 (a)
for the slope of the linear term $k_{j}$ as a function of the initial DD,
$D_{0}$, which starts from the pre-irradiation base current, $\rm
I_{B}^{0}\approx 2mA$. Further investigation shows that a quadratic dependence
is satisfied, i.e., $k_{j}=k_{0}-\alpha D_{0}^{2}$. This behavior is
consistent with the model’s prediction. Calculations show that $\alpha=5\times
10^{-3}\rm mA^{-1}krad^{-1}$ and $17\times 10^{-3}\rm mA^{-1}krad^{-1}$ for
the high and low dose rate $\gamma$-ray irradiation, respectively. A linearly
increasing trend is found in Fig. 9 (b) for the amplitude of the asymptotic
growth term $A_{j}$ as a function of the initial DD. This feature is also the
same as the prediction of the model. The characteristic dose of the asymptotic
growth term $D_{c}^{j}$ shown in Fig. 9 (c) is insensitive to the DD. These
results further confirm the novel initial DD dependence predicted by the
proposed model, reflecting the proper defect evolution physics underlying the
model.

The significant difference between the red and blue dots in Fig. 9 also
reflect the predicted dose-rate dependence of the ISE. For a fixed total dose,
the carrier-enhanced defect diffusion and recombination-enhanced defect
reactions become stronger for decreasing dose rate, which means dramatically
increased action time. This fact readily explains the dose rate dependence of
the ISE in NPN transistors, which, however, cannot be explained by the Coulomb
interaction mechanism.

The physical parameters in Eq. (7) can be calculated from their correspondence
to the fitting parameters. The results are shown in Table 1 and some valuable
facts can be inferred. The values of $\lambda_{21}$ at different dose rates
indicate that the second donor level of $\rm V_{2}O$ contributes 5 or 10 times
less than the first donor level of $\rm V_{2}O$ and $\rm V_{2}$. The values of
$\eta$ implies that $\eta f\sim 1\times 10^{-3}\ll 1$, which supports $\eta
f^{2}\ll f$ in Eq. (7) hence the linear dependence of the growth term on the
initial DD. The dose rate sensitivity of both the dynamic parameters
($D_{c}=\kappa_{1}^{-1}R$ and $\eta=\kappa_{1}^{-1}\kappa_{2}/\lambda_{1}$)
and the contribution factor ($\lambda_{21}=\lambda_{2}/\lambda_{1}$) of the
defects result in the dose rate dependence of the ISE.

Table 1: Calculated physical parameters for the ISE in NPN transistors.  dose rate | $\lambda_{21}$ | $D_{c}~{}\rm(krad)$ | $\eta~{}\rm(mA^{-1})$
---|---|---|---
10 mrad/s | 0.23 | 0.26 | $\rm 3.9\times 10^{-3}$
10 rad/s | 0.11 | 0.38 | $\rm 1.5\times 10^{-3}$

##  IV Mechanism and analytical model of the negative ISE in PNP transistors

As discussed in Sec. II B, a negative ISE is observed for PNP transistors. The
practical neutron-$\gamma$-ray damage displays an interesting fluence-
dependent ‘tick’-like dose curve. In this section this effect will be
understood and modeled in a similar way as that for the NPN transistors. Note
that, in n-type silicon, both $\rm V_{2}$ and VO are neutron-irradiation-
induced active defects.

###  IV.1 ISE due to the evolution of $\rm V_{2}$ in n-type silicon

Similar to the p-type silicon case, $\rm V_{2}$ in n-type silicon as acceptor
levels will transform to $\rm V_{2}O$ due to the above mentioned carrier-
enhanced diffusion of $\rm V_{2}$ under the $\gamma$-ray irradiation,
Kimerling  _et al._ (1975); Kimerling (1976); Bar-Yam and Joannopoulos (1984a,
b); Baraff and Schlüter (1984); Car  _et al._ (1984) see Fig. 7 (a). $\rm
V_{2}$ will also be annihilated by self-interstitials which are emitted due to
recombination-enhanced defect reactions. Lang and Kimerling (1974); Weeks  _et
al._ (1975); Kimerling (1978) The ionization dose and neutron fluence
dependence of the concentrations of $\rm V_{2}$ and $\rm V_{2}O$ are given by
Eq. (4).

Different from the p-type silicon case, both $\rm V_{2}$ and $\rm V_{2}O$ in
n-type silicon have two acceptor levels: $\rm V_{2}^{-/-2}$ at $\rm
E_{c}-0.23eV$, $\rm V_{2}^{0/-}$ at $\rm E_{c}-0.42eV$, $\rm V_{2}O^{-/-2}$ at
$\rm E_{c}-0.20eV$, and $\rm V_{2}O^{0/-}$ at $\rm E_{c}-0.46eV$, Mikelsen
_et al._ (2005) see the upper half in Fig. 7 (a). Denoting the contribution
factors of these acceptor charge states to the base current of PNP transistors
as $\lambda_{4}$, $\lambda_{5}$, $\lambda_{6}$, and $\lambda_{7}$,
respectively. Note, the same contribution factor of $\lambda_{4}=\lambda_{6}$
($\lambda_{5}=\lambda_{7}$) can be assumed, because $\rm V_{2}^{0/-}$ and $\rm
V_{2}O^{0/-}$ ($\rm V_{2}^{-/-2}$ and $\rm V_{2}O^{-/-2}$) have almost the
same acceptor levels and electron capture cross sections. Mikelsen  _et al._
(2005) Multiplying the defect concentrations, Eq. (4), by these factors, we
obtain the corresponding components of the base current in PNP transistors,
which are plotted as the red and blue profiles in Fig. 7 (c). As the
exponential decay curve of $\rm V_{2}^{0/-}$ ($\rm V_{2}^{-/-2}$) and the
asymptotic growth curve of $\rm V_{2}O^{0/-}$ ($\rm V_{2}O^{-/-2}$) cancel
each other, we can find that the net result due to the ionization-induced
evolution of $\rm V_{2}$ displacement defects in n-type Si is a linear decay
term of the ISE in PNP transistors

$\Delta I_{B}(D,f_{1};V_{2}\rightarrow V_{2}O)=-\eta_{1}f_{1}^{2}D/D_{c}.$ (9)

Here $f_{1}=(\lambda_{4}+\lambda_{5})F$ is the measurable initial DD due to
$\rm V_{2}$ in PNP transistors and
$\eta_{1}=\kappa_{1}^{-1}\kappa_{2}/(\lambda_{4}+\lambda_{5})$ is the ratio
between the annihilation and transformation reactions for $\rm V_{2}$ in
n-type silicon; these factors are different from the $f$ and $\eta$ factors in
Eq. (7) for NPN transistors.

###  IV.2 ISE due to the evolution of VO in n-type silicon

Different from p-type silicon, neutron-irradiation-induced VO is also an
effective recombination center in n-type silicon, see the upper half in Fig. 7
(a). So, besides $\rm V_{2}$ the $\gamma$-ray-irradiation-induced evolution of
VO should also be considered for PNP transistors. Due to the above mentioned
carrier-enhanced diffusion of $\rm VO$ Kimerling  _et al._ (1975); Kimerling
(1976); Bar-Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984); Car
_et al._ (1984) and recombination-enhanced emission of self-interstitial, Lang
and Kimerling (1974); Weeks  _et al._ (1975); Kimerling (1978) the
transformation and annihilation reactions of VO will happen

$\rm VO+O_{i}\rightarrow VO_{2},$ (10a) $\rm VO+Si_{i}\rightarrow O_{i}.$
(10b)

According to Ref. Pellegrino  _et al._ , 2001, the transformation reaction
happens because the moving VO will be trapped by electrically-inactive $\rm
O_{i}$, which forms an electrically non-active defect $\rm VO_{2}$, see Fig.
7(a). The annihilation reaction happens because the moving vacancy-oxygen
complex and self-interstitials can collide and generates electrically-inactive
$\rm O_{i}$. Eqs. (2) and (10) make up the full defect dynamics accounting for
the negative ISE in PNP transistors.

Using the same modeling approach as $\rm V_{2}$ and $\rm V_{2}O$ in Sec. III
B, the concentrations of VO and $\rm VO_{2}$ can be obtained as functions of
the $\gamma$-ray dose and initial DD

$VO(D)=(G+\kappa_{3}^{-1}\kappa_{4}G^{2})(e^{-\kappa_{3}R^{-1}D}-1)+G,$ (11a)
$VO_{2}(D)=(G+\kappa_{3}^{-1}\kappa_{4}G^{2})(1-e^{-\kappa_{3}R^{-1}D})-\kappa_{4}G^{2}R^{-1}D.$
(11b)

Here $\kappa_{3}=k_{3}\rm O_{i}$ and $\kappa_{4}=k_{4}\rm Si_{i}/VO$ are
effective reaction velocities of the transformation and annihilation processes
of the defects in n-type Si, respectively, see Fig. 7 (a). The used initial
conditions are $\rm VO(0)$ $=G$ and $\rm VO_{2}(0)=0$. The results indicate
that, similar to the $\rm V_{2}$ defect, the concentration of VO will
exponentially decay under ionization irradiation, and similar to $\rm V_{2}O$
defect, the concentration of $\rm VO_{2}$ will asymptotically increase and
linearly decay as a function of the $\gamma$-ray dose.

The faded VO defect contributes an acceptor level $\rm E_{c}-0.17eV$ Svensson
_et al._ (1991) and the generated $\rm VO_{2}$ defect is non-active in n-type
silicon. Denoting the contributing factor of VO to the recombination current
as $\lambda_{8}$, see Fig. 7 (a). The base current in PNP transistors due to
VO and $\rm VO_{2}$ can be solved as a function of the $\gamma$-ray dose and
initial DD. The net effect is found as an exponential decay of the initial
base current ($f_{2}$) contributed by VO

$\Delta I_{B}(D,f_{2};VO\rightarrow
VO_{2})=(f_{2}+\eta_{2}f_{2}^{2})(e^{-D/D_{c}^{\prime}}-1),$ (12)

see the magenta curve in Fig. 7 (c). Here $f_{2}=\lambda_{8}G$ is the
measurable initial DD due to VO defect,
$\eta_{2}=\kappa_{3}^{-1}\kappa_{4}/\lambda_{8}$ is the rate ratio between the
annihilation and transformation reactions of VO in Si, and
$D_{c}^{\prime}=\kappa_{3}^{-1}R$ is a characteristic dose of the
transformation reaction of VO in Si.

Summing up Eqs. (9) and (12), we at last obtain the defect dynamics model for
the negative ISE of the base current in PNP transistors

$\Delta
I_{B}(D,f)=-\eta_{1}f_{1}^{2}D/D_{c}-(f_{2}+\eta_{2}f_{2}^{2})(1-e^{-D/D_{c}^{\prime}}),$
(13)

see the black curves in Fig. 7 (c).

Eq. (13) predicts that, there would be two independent terms in the ISEs of
the PNP transistors: one linear decay term depending on the $\gamma$-ray dose,
which stems from the ionization-irradiation-induced transformation and
annihilation of $\rm V_{2}$ in n-type silicon; and one exponential decay term
depending on the $\gamma$-ray dose, which stems from the ionization-
irradiation-induced transformation and annihilation of VO in n-type silicon.
Eq. (13) also predicts an almost linear dependence of the amplitude of the
exponential decay on the initial DD due to VO ($f_{2}$) and a quadratic
dependence of the slope of the linear decay term on the initial DD due to $\rm
V_{2}$ ($f_{1}$). Eq. (13) also predicts a dose rate dependence of both terms
of the ISE.

Figure 10:  The initial DD dependence of the fitting parameters $k_{j}$ (a)
and $A_{j}$ (b) in Eq. (14) for the curves in Fig. 4 at the low (red) and high
(blue) dose rate.

###  IV.3 Experimental verification of the model of PNP transistors

Following a similar way as the NPN transistor case (Sec. III C), the proposed
model, Eq. (13), is testified by the data of PNP devices displayed in Sec.
II.B. The fitting model is

$\Delta I_{B}^{j}=k_{j}D-A_{j}(1-e^{-D/D_{c}^{j}}).$ (14)

Again, the first term in the fitting model contains both the linear decay term
in the proposed physical model and the unknown linear ID; the presence of the
linear decay term can be identified by the statistical difference between
$k_{j}$ and $k_{0}$. The correspondence between the fitting parameters in Eq.
(14) and physical parameters in Eq. (13) is
$k_{j}-k_{0}=-\eta_{1}f_{1}^{2}/D_{c}^{j}$. The second term in Eq. (14) is
exactly the exponential decay term in Eq. (13). The correspondence between the
parameters are $A_{j}=(f_{2}+\eta_{2}f_{2}^{2})\sim f_{2}$ and
$D_{c}^{j}=D_{c}\textquoteright$. It is noted that $A_{j}$ is exactly the
initial base current due to the VO defects. The fitting profiles are plotted
in Fig. 5 as solid curves, which are found to work very well for all samples.
The dashed curves in Fig. 8 (b) show the linear component and the exponential
decay term of a typical fitting curve of the PNP device. A significant
decrease is found for the linear component (blue dashed) with respect to the
average slope of pure ID (black dashed). These facts verify the predicted
linear and exponential decay terms in the ISE of PNP transistors, reflecting
the proper defect evolution physics in the prediction of the dose dependence
of the ISE in PNP transistors.

The fitting parameters $k_{j}$ and $A_{j}$ are plotted in Figure 10 as a
function of the base current, to further testify the initial DD dependence of
the proposed model. containing the initial DD ($f_{1}+f_{2}$). The fitting
characteristic dose ($D_{c}\textquoteright$) is not sensitive to the initial
DD, and is 0.47 krad and 0.50 krad for the low and high dose rates,
respectively. Note, these values are larger than those for the $V_{2}$ defects
as shown in tab. 1, reflecting a slower transformation process for VO defects
in Si. The data of high dose rate with the neutron fluence of $1\times
10^{12}\rm cm^{-2}$ is of low quality and are excluded from the fitting
process. It is clear from Fig. 10 (b) that, the amplitude of the exponential
decay term, $A=f_{2}$, shows an almost linear dependence on the initial DD,
which is started from the pre-irradiation base currents, $\rm I_{B}^{0}\approx
14nA$, see the dashed guide lines. This dependence is consistent with the
prediction of the proposed model. It is also found that for the low (high)
dose rate $\gamma$-ray irradiation, the value of $A=f_{2}$ is only 1/38
(1/170) of the initial DD, $f_{1}+f_{2}$. This fact implies that the neutron-
induced (annealable) VO defects are much fewer than the neutron-induced $\rm
V_{2}$ defects. As can be seen in Fig. 10 (a), the slope of the linear term
$k_{j}$ decreases as the initial DD increases. A quadratic dependence is found
for the overall trend, $k_{j}=k_{0}-\alpha(f_{1}+f_{2})^{2}$. Since $f_{2}$ is
much smaller than $f_{1}$, this result means $k_{j}=k_{0}-\alpha f_{1}^{2}$,
which is also consistent with the model’s prediction. The value of $\alpha$ is
$2\times 10^{-5}\rm nA^{-1}krad^{-1}$ for both low and high dose rates. The
above results further verify the predicted initial DD dependence of the ISE in
the proposed model, reflecting the proper physics of defect evolution
underlying the model.

The significant difference between the red and blue dots in Fig. 10 reflects
the dose-rate dependence of the ISE predicted by the model. The reason is that
the ionization-induced defect transformation and annihilation becomes stronger
for lower dose rate irradiation. This fact readily explains the dose rate
dependence of the ISE in PNP transistors, which cannot be explained by the
Coulomb interaction mechanism. Different from the NPN transistor case, the
physical parameters cannot be calculated from the fitting parameters.

##  V Perspective

Several important works should be carried out in future investigations on the
ISEs of silicon bipolar transistors. The first is the DLTS characterization of
the ionization-irradiation-induced evolution of the displacement defects in
n-type and p-type silicon, which can provide direct evidences for the
atomistic mechanisms of the ISEs. The second is the explicit dose rate
dependence of the ISEs, from which the nature of the two underlying
mechanisms, i.e., carrier-enhanced defect diffusion and recombination-enhanced
defect reactions, can be further explored. The third and last is the behavior
and mechanism of ISE due to _simultaneous_ displacement and ionization
irradiations. The displacement defect dynamics contains processes of fast
generation, short-term annealing, and long-term annealing of displacement
defects. Srour  _et al._ (2003) The above results show that the displacement
defects during or after the long-term annealing process can be strongly
modified by ionization irradiations. When the transistors are subjected to
simultaneous displacement and ionization irradiations, which is the usual
cases, the ionization irradiation can further dramatically alter the short-
term annealing of displacement defects. Gregory and Sander (1967) The behavior
and mechanism of such a _simultaneous ISE_ should be an important research
subject for both basic researches and practical applications.

##  VI Conclusion

In summary, the irradiation synergistic effect (ISE) is an important physical
phenomenon to understand and predict the practical damages of semiconductor
devices from the individual displacement and ionization irradiation
experiments. In this work, we have made a concise review on the current
investigation status of the effect in silicon bipolar transistors. We want to
emphasize that, the existing experimental studies of the behavior of the
effect are usually limited to fixed displacement fluence and/or fixed
ionization dose, while the previous theoretical model encounters physical
troubles to explain the dose rate dependence of the ISE.

Here, we describe in detail our recent investigations of the behavior of the
effect and introduce our new mechanism of the effect. Our experimental
investigation uses a variable-fluence and dose neutron/$\gamma$-ray
irradiation setup as well as a large sample size to explore the behavior of
the ISE. Our theoretical investigation adopts a point of view of ionization-
irradiation-induced evolution of displacement defects in Si, that is totally
different from the Coulomb interaction mechanism in previous works. We
demonstrate that both the positive and negative ISEs in NPN and PNP
transistors can be fully and self-consistently explained by the same basic
mechanism, i.e., the ionization-irradiation-induced transformation and
annihilation of the displacement-irradiation-induced defects in silicon.
However, as the electrically-active displacement defects and their
transformation products are different for p-type and n-type silicon, the ISEs
of NPN and PNP transistors show quite different behaviors. Specifically, the
ionization-driven transformation and annihilation of active $\rm V_{2}$ defect
in p-type silicon give arises to a positive ISE in NPN transistors containing
an asymptotic growth and a linear decay term; the same processes of both
active $\rm V_{2}$ and VO defects in n-type silicon give arises to a negative
ISE in PNP transistors containing a linear and exponential decay terms. Both
ISEs show a regular dependence on the initial displacement damage of the
bipolar transistors, which can also be self-consistently explained by the
defect evolution mechanism. The dose rate problem encountered by the previous
Coulomb interaction mechanism can be naturally explained by the present model.

We believe that the uncovered behavior and mechanism of the ISEs will be
helpful to predict the practical damage of silicon bipolar transistors. The
developed experimental and modeling methods in our works can be extended to
the investigation of ISEs in other semiconducting materials and devices.

## Acknowledgments

This work was supported by the Science Challenge Project under Grant No.
TZ2016003-1 and National Natural Science Foundation of China (NSFC) under
Grant Nos. 51672023; 11634003; U1930402; 11404300.

## References

  * Oldham and McLean (2003) T. R. Oldham and F. McLean, IEEE transactions on nuclear science 50, 483 (2003).
  * Srour  _et al._ (2003) J. Srour, C. J. Marshall, and P. W. Marshall, IEEE Transactions on Nuclear Science 50, 653 (2003).
  * Dressendorfer (1998) P. V. Dressendorfer,  _Basic Mechanisms for the New Millennium_ , Tech. Rep. (Sandia National Labs., Albuquerque, NM (United States), 1998).
  * Fleming  _et al._ (2010) R. M. Fleming, C. Seager, D. Lang, and J. Campbell, Journal of Applied Physics 108, 063716 (2010).
  * Barnaby  _et al._ (2001) H. J. Barnaby, R. D. Schrimpf, A. L. Sternberg, V. Berthe, C. R. Cirba, and R. L. Pease, IEEE Transactions on Nuclear Science 48, 2074 (2001).
  * Barnaby  _et al._ (2002) H. J. Barnaby, S. K. Smith, R. D. Schrimpf, D. M. Fleetwood, and R. L. Pease, IEEE Transactions on Nuclear Science 49, 2643 (2002).
  * Gorelick  _et al._ (2004) J. L. Gorelick, R. Ladbury, and L. Kanchawa, IEEE transactions on nuclear science 51, 3679 (2004).
  * Li  _et al._ (2012a) X. Li, C. Liu, H. Geng, E. Rui, D. Yang, and S. He, IEEE Transactions on Nuclear Science 59, 439 (2012a).
  * Li  _et al._ (2012b) X. Li, C. Liu, E. Rui, H. Geng, and J. Yang, IEEE Transactions on Nuclear Science 59, 625 (2012b).
  * Li  _et al._ (2015) X. Li, C. Liu, and J. Yang, IEEE Transactions on Nuclear Science 62, 1375 (2015).
  * Wang  _et al._ (2016) C. Wang, W. Chen, Z. Yao, X. Jin, Y. Liu, S. Yang, and Z. Wang, Nuclear Instruments and Methods in Physics Research A 831, 322 (2016).
  * Song  _et al._ (2019) Y. Song, Y. Zhang, Y. Liu, J. Zhao, D. Meng, H. Zhou, X. Wang, M. Lan, and S.-H. Wei, ACS Applied Electronic Materials 1, 538 (2019).
  * Song  _et al._ (2020a) Y. Song, H. Zhou, X.-F. Cai, Y. Liu, P. Yang, G.-H. Zhang, Y. Zhang, M. Lan, and S.-H. Wei, ACS Applied Materials & Interfaces 12, 29993 (2020a).
  * Pease  _et al._ (2008) R. L. Pease, R. D. Schrimpf, and D. M. Fleetwood, in  _Radiation and Its Effects on Components and Systems (RADECS), 2008 European Conference on_ (IEEE, 2008) pp. 18–32.
  * Song  _et al._ (2020b) Y. Song, G. Zhang, Y. Liu, H. Zhou, L. Zhong, G. Dai, X. Zuo, and S.-H. Wei, arXiv preprint arXiv:2008.04486 (2020b).
  * Li  _et al._ (2010) X. Li, H. Geng, C. Liu, Z. Zhao, D. Yang, and S. He, IEEE Transactions on Nuclear Science 57, 831 (2010).
  * Raymond and Petersen (1987) J. Raymond and E. Petersen, IEEE Transactions on Nuclear Science 34, 1621 (1987).
  * Kimerling  _et al._ (1975) L. Kimerling, H. DeAngelis, and J. Diebold, Solid State Communications 16, 171 (1975).
  * Kimerling (1976) L. Kimerling, IEEE Transactions on Nuclear Science 23, 1497 (1976).
  * Bar-Yam and Joannopoulos (1984a) Y. Bar-Yam and J. Joannopoulos, Physical Review B 30, 1844 (1984a).
  * Bar-Yam and Joannopoulos (1984b) Y. Bar-Yam and J. Joannopoulos, Physical Review B 30, 2216 (1984b).
  * Baraff and Schlüter (1984) G. A. Baraff and M. Schlüter, Physical Review B 30, 358 (1984).
  * Car  _et al._ (1984) R. Car, P. J. Kelly, A. Oshiyama, and S. T. Pantelides, Physical Review Letters 52, 1814 (1984).
  * Lang and Kimerling (1974) D. Lang and L. Kimerling, Physical Review Letters 33, 489 (1974).
  * Weeks  _et al._ (1975) J. D. Weeks, J. C. Tully, and L. Kimerling, Physical Review B 12, 3286 (1975).
  * Kimerling (1978) L. Kimerling, Solid-State Electronics 21, 1391 (1978).
  * Lang (1974a) D. Lang, Journal of Applied Physics 45, 3014 (1974a).
  * Lang (1974b) D. Lang, Journal of applied physics 45, 3023 (1974b).
  * Cheng  _et al._ (1966) L. Cheng, J. C. Corelli, J. Corbett, and G. Watkins, Physical Review 152, 761 (1966).
  * Lee and Corbett (1976) Y.-H. Lee and J. W. Corbett, Physical Review B 13, 2653 (1976).
  * Lindström  _et al._ (1999) J. Lindström, L. Murin, V. Markevich, T. Hallberg, and B. Svensson, Physica B: Condensed Matter 273, 291 (1999).
  * Mikelsen  _et al._ (2005) M. Mikelsen, E. Monakhov, G. Alfieri, B. Avset, and B. Svensson, Physical Review B 72, 195207 (2005).
  * Markevich  _et al._ (2014) V. Markevich, A. Peaker, B. Hamilton, S. Lastovskii, and L. Murin, Journal of Applied Physics 115, 012004 (2014).
  * Pellegrino  _et al._ (2001) P. Pellegrino, P. Lévêque, J. Lalita, A. Hallén, C. Jagadish, and B. G. Svensson, Physical Review B 64, 195211 (2001).
  * Pierret and Neudeck (1987) R. F. Pierret and G. W. Neudeck,  _Advanced Semiconductor Fundamentals_ , Vol. 6 (Addison-Wesley Reading, MA, 1987).
  * Adell and Boch (2014) P. Adell and J. Boch, in  _IEEE Nuclear and Space Radiation Effects Conference (NSREC), Short Course 2014_ (Paris, France, 2014).
  * Ganagona  _et al._ (2014) N. Ganagona, L. Vines, E. Monakhov, and B. G. Svensson, Journal of Applied Physics 115, 034514 (2014).
  * Srour  _et al._ (1973) J. R. Srour, S. Othmer, and O. L. Curtis Jr,  _Radiation Effects on Semiconductor Materials and Devices._ , Tech. Rep. (NORTHROP RESEARCH AND TECHNOLOGY CENTER HAWTHORNE CALIF, 1973).
  * Myers  _et al._ (2008) S. M. Myers, P. J. Cooper, and W. R. Wampler, Journal of Applied Physics 104, 044507 (2008).
  * Liu  _et al._ (2015) C. Liu, X. Li, J. Yang, G. Ma, and Z. Sun, IEEE Transactions on Nuclear Science 62, 3381 (2015).
  * Svensson  _et al._ (1991) B. Svensson, B. Mohadjeri, A. Hallén, J. Svensson, and J. Corbett, Physical Review B 43, 2292 (1991).
  * Gregory and Sander (1967) B. Gregory and H. Sander, IEEE Transactions on Nuclear Science 14, 116 (1967).


# The Name of the Title Is Hope

Anonymous Author(s)

(2018)

###### Abstract.

While high-resolution time series has allowed us to develop useful pattern-
based time series data mining algorithms, there can be more serious privacy
infringement compared to low resolution data. Previous studies show that high
resolution time series such as hourly recorded household power demand sensor
data could be easily used to infer a wide range of sensitive information such
as the number of residents, age, marital status, long-term illness, and
employment status. Moreover, large time series patterns, due to their less
frequent occurrence, could leak more sensitive information. We take the first
step to consider the problem of sensitive large pattern could leak sensitive
information and sharing perturbed raw data is very hard to protect while
maintain pattern-based data mining task utility. To resolve the problem of
information leak from large patterns, we investigate a naive solution of
directly sharing matrix profile of short subsequence length computed from
sensitive time series data. It turns out that this simple solution prevents
recover the data from matrix profile distance. However, we find the location
of the sensitive large pattern could leak from the matrix profile index using
our propose location and entropy based attack methods. We further propose to
share a perturbed matrix profile via random k-nearest neighbor, which can
maintain the utility of downstream tasks, but prohibit the location leakage of
sensitive pattern. We evaluate our proposed method against baseline methods
through quantitive analysis and multiple real-world case studies to show the
effectiveness of the proposed method while protecting the sensitive
information.

datasets, neural networks, gaze detection, text tagging

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Computer systems organization
Embedded systems††ccs: Computer systems organization Redundancy††ccs: Computer
systems organization Robotics††ccs: Networks Network reliability

##  1\. Introduction

With the advancement of sensor technology, large volumes of high-resolution
time series are collected in a variety of domains such as cardiology
(adami2019electrocardiographic), manufacturing (jardine2006review), remote
sensing (gao2017trajviz) and home automation (dahmen2017smart). While high-
resolution continuously-collected time series has allowed us to develop useful
pattern-based algorithms, there can be more serious privacy infringement
compare to low resolution data. Previous studies show that time series of less
than per hour recorded sensor could be easily used to infer a wide range of
sensitive information such as number of residents, residents age, marital
status, long-term illness, and employment status (teng2022balancing).
Moreover, large time series patterns, due to the less frequently occurring,
can be used to identify behavior of users. For example,

  * •

In manufacturing process, while local short-length pattern is corresponding to
the machine’s local functionality and health condition for parts, but longer
repeated pattern are much more rare and might be used to identify more about
the product information and production activity if with other side information
available.

  * •

In agricultural data, sensors are often installed to monitor the growing of
crops or monitor behavior of farm animals such as cows and chicken. While
small utility length of pattern is used to identify animal activities such as
milking or egg laying, large frequent pattern detected could be potentially
used to identify a farmer’s routine, work hours, and infer their farm working
hours, hence farm income, and other more sensitive information that small
pattern is useful for designing model for time series data mining tasks.

While there is an urgent need to extract valuable information and insights
from the data, it can be very costly or nearly impossible for domain experts
to manually inspect and analyze large scale data. In the last two decades,
researchers have defined and explored new time series primitives that can help
extract meaningful patterns in large scale time series (keogh2005hot;
chiu2003probabilistic; mueen2009exact; mercer2021matrix). It has been shown
that these patterns have the potential to provide great insights and further
our knowledge in many application domains.For example,

  * •

time series motifs (mueen2009exact; yeh2016matrix) are temporally frequent
subsequences. They can reveal common patterns or regularities such as electric
power usage patterns, and be used for data summarization or even anomaly
detection (gao2017trajviz).

  * •

time series discord (keogh2005hot) is defined as the most unusual subsequence
in a time series. They are useful in detecting abnormal or rare patterns in a
time series, and have a wide range of applications such as cardiac event
detection, machine failure detection, or sudden fall detection for elderly
patients.

  * •

time series shapelet (ye2009time; rakthanmanon2013fast) learns the features
maximally in classification has great interpretation in health care, activity
recognition, disaggregation.

Matrix profile (MP) (yeh2016matrix; zhumatrix) is a recently proposed data
structure which outputs the distance and subsequence index to the most similar
subsequence for every time series subsequence, and is proven to effectively
work as a reliable and informative subsequence representation to support many
down-streaming tasks such as motif discovery, anomalous detection, shapelet
discovery, data summarization. As shown in Fig. 1(a), data holders (e.g.,
factories with smart sensors, e-commerce platforms and hospitals) may capture
clients’ time series and send to cloud servers/third party modelers for data
modeling. The third party modelers will run matrix profile and then design
model for improving the quality of data holders’ service or production related
decision making. For example, a data holder can be a factory with smart
sensors installed on their facilities but do not have the domain knowledge to
model the data. They are interested in finding mechanical faults related to
time series anomaly, hence they have to send the data to the third party
modeler, who might design a model based on matrix profile for the desired
downstream task of anomaly detection. This scenario is called public as the
modeler can access the data and build the model.

It is actually a non-trivial problem to reach the goal, because we do not
assume that we know what location of meaningful pattern of small utility
length could be embedded in the sensitive large pattern, so we cannot just
remove some random segment of time series. One might wonder, can we add some
noises on the raw data to perturb the data and publish it, as similar idea has
achieved success in the statistics such as aggregation (dwork2006calibrating).
Unfortunately, these approaches would not suit our purpose, it needs more
noises to disrupt a large pattern than a small utility length pattern. As a
result, the perturbed time series would be indifferent in similarity on small
utility length, making the time series unusable for potential downstream
tasks.

(a) Traditional pipeline (b) Sharing Matrix Profile  (c) Sharing perturbed
Matrix Profile

Figure 1. Brief comparisons of matrix profile (MP) based data mining pipeline
under different scenarios.

Since it is risky to share raw data or the perturbed raw data directly, in
this paper, we would like to take the first step to address the information
leakage in time series pattern-based data mining algorithms. A naive solution
would be directly sharing matrix profile computed from sensitive time series
data as shown in Fig. 1(b) since matrix profile has the ability to identify
various patterns such as motifs, anomalies, while only recording the distance
and location to the nearest neighbor for each subsequence of a distinct
length, hiding information that would have been revealed by the raw time
series. To broaden the application and usability for matrix profile, we
investigate the private concern introduced by matrix profile from recovering
raw time series data and large sensitive patterns. We find that although
recovering raw time series data and large sensitive patterns from matrix
profile distance is difficult, however, the matrix profile index could leak
the location of large sensitive patterns in some scenarios. To overcome the
limitation of naive solution, as shown in Fig. 1(c), instead of directly
sharing the original matrix profile, we propose to share a perturbed matrix
profile via random k-nearest neighbor, which can maintain the utility
downstreaming tasks, but prohibit sensitive information leakage from larger
and relatively rare time series pattern discovery. In summary, our
contribution is as follows:

  * •

We take the first step to consider the problem of sensitive large pattern
could leak sensitive information and sharing perturbed raw data is very hard
to protect while maintain pattern-based data mining task utility.

  * •

To resolve the information leakage from large pattern, we investigate a naive
solution of directly sharing matrix profile of short subsequence length
computed from sensitive time series data. It turns out that this simple
solution prevents recover the data from matrix profile distance.

  * •

However, we find the location of the sensitive large pattern could leak from
the matrix profile index using our propose location and entropy based attack
methods.

  * •

We further propose to share a perturbed matrix profile via random k-nearest
neighbor, which can maintain the utility of downstreaming tasks, but prohibit
the location leakage of sensitive pattern.

  * •

We evaluate our proposed method against baseline methods through quantitive
analysis and multiple real-world case studies to show the effectiveness of the
proposed method while protecting the sensitive information.

##  2\. RELATED WORK

In the last two decades, vast of research efforts have been put into time
series data mining tasks such as time series classification (bagnall2018uea),
discord (keogh2005hot), motif discovery (mueen2009exact), and segmentation
(gharghabi2017matrix). Different from point-based tasks, pattern-based time
series approaches are based on similarity on subsequence level of instead of
point values to capture the notion of shape information, which . Pattern-based
time series methods handle large scale time series well with excellent
performance and interpretation. Recently, Matrix profile (yeh2016matrix;
zhumatrix) is proposed as an efficient and effective data representation on
subsequence level and support major fundamental tasks for downstream
applications in a broad range of domains applications (zhu2020swiss). Existing
work such as (zymbler2021matrix) typically use the raw data to compute matrix
profile as the first step, and then design algorithms or models based on the
computed matrix profile. None of existing work consider the problem of
protecting any privacy threat of using matrix profile, especially the
information leak caused by patterns. There are a number of data sanitizing
approaches proposed for solving the problem of releasing time series
aggregated statistics under differential privacy (dwork2006calibrating), which
has become the gold standard in the privacy-research community. According to
the granularity of privacy protection, these studies can be broadly classified
into three categories: event-level, user-level and $w$-event privacy. For
example, Dwork et al. (dwork2010differential) proposed a binary tree technique
based event-level DP algorithm for finite streams. Fan et al.
(fan2013adaptive) presented FAST for realizing user-level DP on finite streams
with a framework of sampling-and-filtering. Event-level tries to hide a single
event in streams while user-level aims to hide all events of a single user. To
balance event-level and user-level privacy, Kellaris et al.
(kellaris2014differentially) proposed $w$-event privacy for infinite streams,
which aims to guarantee DP for any time window consisting of $w$ consecutive
time instances (or time interval or timestamps for simplicity). In summary,
these time series DP solutions cannot be directly applied to our setting,
since these work mainly focus on computing the aggregated estimates (e.g.,
prefix-sum and moving average) of the values under differential privacy, while
our goal is to release subsequence level representation that benefit different
downstream tasks while prevent leaking the information of large patterns. From
a first glance, this problem is similar with variable motif discovery problem
given a matrix profile with a short subsequence length and the goal is to
detect a larger length motif. However, we would like to point out that
existing matrix profile-based variable length motif discovery algorithms such
as (mueen2009exact; linardi2018matrix; hime) cannot fulfill the adversary
goal. This is because these algorithms heavily rely on the mean and standard
deviation of each subsequences to accurately estimate the similarity
relationship between long subsequences via small pattern, whereas in our
setting, the attacker could only access to the given matrix profile and matrix
profile index.

To the best of our knowledge, there has been no existing work have solution
for the data sharing issue or potential leakage of patterns from using matrix
profile, nor the task to protect from large pattern inference while
maintaining small pattern utility.

##  3\. Background and Preliminary

In this section, we first review necessary time series related notations.

###  3.1. Time series Notations

###### Definition 3.1.

A time series $T=[t_{1},t_{2},\ldots,t_{n}]$ is a set of observation ordered
by time, where $t_{i}$ is a finite real number and $n$ is the length of time
series $T$.

Time series subsequences can be extracted from time series $T$ by sliding a
window along time series.

###### Definition 3.2.

A Subsequence $S^{T}_{i,l}=[t_{i},t_{i+1},\ldots,t_{i+l-1}]$ of time series
$T$ is a contiguous set of points in time series $T$ starting from position
$i$ with length $l$. Typically $l\ll n$, and $1\leq i\leq n-l+1$.

Previous work such as (yeh2016matrix; keogh2005hot) requires subsequence
comparison be non-trivial match, which refer to the subsequence itself or
those that not overlapping more than 50% with the subsequence when comparing
the distances.

###### Definition 3.3.

Given a time series $T$, a subsequence $S_{i,l}$ with length $l$ is considered
a non-trivial match of another subsequence $S_{j,l}$ of length $l$ if
$|i-j|>l/2$.

In many applications, we are interested in finding similar “shapes” between
two subsequences. Z-normalized Euclidean distance is used to achieve scale and
offset invariance (lin2007experiencing)

###### Definition 3.4.

A Z-normalized Euclidean Distance $d(S_{p,l},S_{q,l})$ of subsequences
$S_{p,l}$, $S_{q,l}$ of length $l$ is computed as

$\tiny{\sqrt{\sum_{m=1}^{l}{(\frac{t_{p+m-1}-\mu_{p,l}}{\sigma_{p,l}}-\frac{t_{q+m-1}-\mu_{q,l}}{\sigma_{q,l}}})^{2}}},$

where $\mu_{p,l}$, $\sigma_{p,l}$ and $\mu_{q,l}$, $\sigma_{q,l}$ are the
means and standard deviations of subsequences $S_{p,l}$ and $S_{q,l}$
respectively.

Z-normalization step is very critical, as noted in previous work — “with- out
normalization time series similarity has essentially no meaning. More
concretely, very small changes in offset rapidly dwarf any information about
the shape of the two time series in question” (keogh2003need). There is
additional benefit of preventing data leaking brought by utilizing
Z-normalization distance, and we will discuss in details in Subsection C.

###### Definition 3.5 (Distance Profile).

Given a query subsequence $Q_{l}$ of length $l$ and a time series $T$, a
distance profile $D_{T}(Q)$ is a vector containing the Z-normalized Euclidean
distances between $Q$ and each subsequence of the same length in time series
$T$. Formally,
$D_{(}Q_{l},T)=[d(Q_{l},S^{T}_{1}),d(Q_{l},S^{T}_{2}),\cdots,d(Q_{l},S^{T}_{n-l+1})]\quad$.

###### Definition 3.6.

non-overlapping

###### Definition 3.7.

A matrix profile Distance($MPD_{l}$) of time series $T$ is a vector of the
Z-normalized Euclidean distances between every subsequence $S_{i,l}$ and its
nearest neighbor (most similar) subsequence in time series $T$. Formally,
$MP=[\min(D(S_{1,l},T)),\min(d(S_{2,l},T)),\cdots,\min(d(S_{n-l+1,l},T))].$

###### Definition 3.8.

A matrix profile Index ($MPI_{l}$) of time series $T$ is a vector of the
Z-normalized Euclidean distances between every subsequence $S_{i,l}$ and its
nearest neighbor (most similar) subsequence in time series $T$. Formally,
$MPI_{l}=[\arg\min(d(S_{1,l},T),\cdots,\arg\min(d(S_{n-l+1,l},T)].$

We use matrix profile (MP) to refer to the two vectors of matrix profile
distance and matrix profile index together. Formally, $MP(1,j)=$We share the
two dimensional vector MP for the downstream data mining task.

We next describe the join matrix profile or AB-join matrix profile
(gharghabi2018matrix) and one derivative called masked join matrix profile,
which will be used in our proposed method:

###### Definition 3.9 (AB/BA Join Matrix Profile).

Given two time series $A$ and $B$, an AB Join Matrix Profile, $MP_{AB}$, is a
matrix profile such that $MP_{AB}(1,i)=\min(D(S^{A}_{i},B))$ and
$MP_{AB}(2,i)=\arg\min(D(S^{A}_{i},B))$.

###### Definition 3.10 (Masked AB-Join Matrix Profile ($MP_{AB,I}$)).

Given two time series $A$ and $B$, a Masked AB Join Matrix Profile with a mask
index set $I$ of $B$ , $MP_{AB}$, is a matrix profile such that
$MP_{AB}(1,i)=\min_{i\not\in I}(D(S^{A}_{i},B))$ and
$MP_{AB}(2,i)=\arg\min_{i\not\in I}(D(S^{A}_{i},B))$.

Matrix profile has rich information about the data and the pattern location.
For example, time series motif can be found by exacting minimum value of
matrix profile and the matrix profile index.

The goal of our paper is closely related to patterns, specifically, we mean by
time series motifs, which is frequent set patterns repeated occurs in the
data. Thus, in this paper, we use the word motif and pattern interchangeably.

###### Definition 3.11.

A Time series set motifs is a set of subsequences $P_{l}$ of length $l$ in
time series $T$ such that the distance between each of the subsequences and a
seed subsequence $S_{seed}$ is less than a threshold $\epsilon$. Specifically,
$P_{l}=\\{S_{i,l}|\forall(S_{i,l},,S_{j,l})\in
P_{l},d(S_{i,l},S_{j,l})\leq\epsilon\\}$.

As previous work noted, once a pair of motif is detected, all instances of
motif can be found by with a similarity query (begum2014rare). Hence, we
define our pattern leakage to be specifically for the leak of a pair motif,
which can be found by matrix profile and matrix profile index.

###### Definition 3.12.

Time series pair motif ($S_{i,l},S_{j,l}$) of length $l$ is a pair of
subsequences $S_{i,l},S_{j,l}$ such that $i=\arg\min(MP_{l})$ and
$MPI_{l}[i]=j$.

As demonstrated in previous work, different lengths of subsequences could
carry different information . (otpional example). In our problem setting, we
assume there exists an appropriate subsequence length which can be easily
identified by the domain experts such as doctors or engineers, and is suitable
for the downstream task. On the other hand, a large length motif, potentially
unrelated to the downstream tasks, could carry irrelevant but sensitive
information and should not be shared.

For the sake of simplicity, we denote the small subsequence length related to
the downstream tasks as utility length $L_{util}$, and the sensitive pattern
length as sensitive length $L_{sens}$.

##  4\. Problem Statement

In practice, companies may utilize sensitive data for data mining in order to
provide better services. As shown in Fig. 1(a), data holders (e.g., factories
with smart sensors, e-commerce platforms and hospitals) may capture clients’
time series and send to cloud servers/third party modelers for data modeling.
The third party modelers will run matrix profile and then design model for
improving the quality of data holders’ service or production related decision
making. However, directly transmitting raw data to modelers would allow
malicious modelers to use the large patterns to infer sensitive information
such as demographic information.

Since it is risky to share data directly, a better protocol (Fig. 1(b)) is to
first learn knowledge from data by generating the matrix profile (MP) and
matrix profile index (MPI) with utility length $L_{util}$, and then send it to
the external cloud servers/modelers for downstream applications. Then two
natural questions arises:

  * •

Does sharing MP Distance and MP Index of small utility length protect the
values of raw time series?

  * •

Does sharing MP Distance and MP Index of small utility length protect the
locations for large patterns?

To answer these questions, in the following subsections, we conduct a
comprehensive study of matrix profile in terms of the privacy protection.

###  4.1. Benefit of Matrix Profile for Privacy Protection

####  4.1.1. Difficult to recover raw data from MPD and MPI

Z-normalization makes it extremely difficult to recover the data from the
release of matrix profile, since it requires every subsequence to extract its
mean and divide the value of standard deviation respectively. With
Z-normalization, if we would like to recover the data from the distance, we
have to solve the value of data from pairwise distance between subsequence
with the mean $\mu_{i,l}$ and standard deviation $\sigma_{i,l}$ as parameters
in every equation. Since $\sigma_{i}$ is a non-linear function of variable
$t_{i}$ to $t_{i+l-1}$, it makes the inverse problem ill-defined. One could
get infinitely many possible time series data as possible solutions satisfying
a given matrix profile, and hence is unable to retrieve the original data.

####  4.1.2. Difficulty to infer large motif location from MPD

In addition, Z-normalization naturally surpass the correlation between small
pattern and the large pattern. As pointed out by previous work, “the distance
between a pair of short subsequences does not necessarily share similar
behavior with the distance between long subsequences if the length difference
is large”(hime). Specifically, the distance of subsequences of sensitive
pattern length has a non-linear relationship with the distance of short
utility length. Thus, it is difficult to recover values from a small length
MPD to values of large senstive MPD, hence MPD does not leak sensitive motif
locations.

Figure 2. Although Large Pattern Existed in the Time Series, from Distance
Value in the Matrix Profile, it seems like there are three small patterns
separably occurred in the data

Nevertheless, remember that both matrix profile MPD and MPI has to be shared.
Although it is difficult to attack the raw data and sensitive pattern location
from MPD, can attacker derive the sensitive pattern location from MPI? We will
describe our attack and propose a defense in the next sections.

###  4.2. Threat Model

Before we get into our attack methods, we define the threat model as follows:

Adversary Goal. Given matrix profile $MP_{l}$ and matrix profile index
$MP_{l}$ of subsequence length $l$, the adversary aims to locate two senstive
motifs of length $L_{sens}$. We assume $L_{sens}$ is much larger than utlity
length $L_{util}$ for the downstream tasks. The success rate for detecting the
location of large motif is defined as the ov

Adversary Knowledge. We assume a strong adversary (e.g., honest-but-curious
modeler), who has no access to the sensitive time series data, but has the
white-box access to the matrix profile $MP_{l}$ and matrix profile index
$MPI_{l}$ with short subsequence length $l$ pre-generated from the sensitive
data and the downstream tasks computed over the matrix profile $MP_{l}$ and
$MPI_{l}$.

##  5\. Attack Methods

The goal of the attack method is to locate large motifs with subsequence
length $L_{sens}$ given a smaller length matrix profile. A diligent reader may
wonder whether our problem can be solved by existing variable motif discovery
techniques such as (mueen2013enumeration; linardi2018matrix; hime;
mercer2021matrix). Unfortunately, under our setting, the attacker could only
use a matrix profile of a single length to infer a much larger pattern in the
data, the data is unknown, making it impossible to fulfill the adversary goal.
Therefore, we need to design the attack algorithm from a brand-new
perspective.

###### Definition 5.1 (Consecutive Index Block (CIB)).

Given a matrix profile index $MPI_{l}$, a Consecutive Index Block (CIB)
consist of a set of index where $|MPI_{l}(i)-MPI_{l}(i+1)|<l$ for every $i$.

Since we only care about the CIB under the location of sensitive region, for
the sake of clarity and simplicity, we refer this type of CIB as sensitive
CIB.

Our proposed attack algorithms are based on common behaviors in matrix
profile: large motifs often occurs when there are many consecutive indices
matching each other in small utility length matrix profile.

Figure 3. Consecutive neighborhood Preserving Property could leak the location
of large patterns

We use a simple time series contains only one large pattern to illustrate the
behavior of $MPI$ when a potential sensitive pattern presented. Figure 3.top
shows the time series contains a large motif with two identical instances of
length 600. Figure 3.bottom shows the corresponding $MPI$ given a query length
100. On the noisy area in the time series, the index changes frequently. This
is because the nearest neighbor of the consecutive subsequences could appear
in arbitrary locations. In contrast, at the motif location, the index
evolution is smooth and gradual. Furthermore, the indexes stored in $MPI$
under the motif region are coincided with the location of the second instance
of the motif. For example, $MPI(5000)$ is 7000, which is exactly the location
of the second instance. This example shows that $MPI$ could potentially leak
large motif location.

The distinguishable characteristic of sensitive pattern region in $MPI$ also
present when noisy real world data. Figure 4. Figure 4(a) shows a snippet of a
dishwasher time series. Figure 4.bottom shows that the index corresponding to
the two motif highlighted in the raw time series computed via matrix profile
of half of the motif length. Although it is not as clean as the previous
example, by simply compute the histogram of the index, we obtained the
histogram shown in Fig. 5, it is not very difficulty to find that the index
corresponding to the highest frequent appears area is still match the location
of another instance highlighted.

Figure 4. MPI index of a dishwasher time series with two time series Figure 5.
MPI index of a dishwasher time series with two time series Algorithm 1
Proposed Attack Algorithm

1:Input: $MP$, $MPI$, $L_{attack}$

2:Output: large pair motif indices $idx1$, $idx2$

3: /* Calculate sum of distances in each sliding window of length
$L_{attack}$*/

4:$SumDist=$Score($MP$,$L$)

5:$idx1=\arg\min$($SumDist$)

6: /* Identify second motif index based 1st motif index*/

7:$idxPool=MPI[idx1:idx1+L-1]$

8:$idx2$=$\arg\max($GetIntervalFrequency($idxPool$))

9:return $idx1$, $idx2$

Inspired by the above intuition, we introduce our basic framework of the
attack method. It is shown in Algorithm 1. Given a attack length $L_{attack}$,
the attack algorithm will first scan through the time series with a sliding
window equal to $L_{attack}$ and assign a significant score for each
subsequence of length $L_{attack}$ (Line 4). The subsequence of with the most
outstanding score will be identified as the first instance of the motif. Then,
based on the observation we obtained through previous example, we could
identify the location of the second instance of the motif based on the the
highest frequency indexes appeared in the first identified area (Line 7-8).

We next introduce three different significant score that could potentially
used in the basic framework to reveal the location of large motif. Given a
index subsequence from $MPI$ of length $L_{attack}$, we propose two scores
based on $MPI$:

  * •

Location-based Score the length of the consecutive index change not greater
than $L_{attack}$.

  * •

Entropy-based Score The negative entropy value given all indices in
$S_{attack}$.

In the experiment, we shows that all three attack method could retrieve some
degree of large patten information if $MP$ is not protected.

##  6\. Defense Strategy

Before introducing our proposed defense method, we hope that our defense will
be compatible and allow the development for the current and future matrix
profile algorithms and models for the downstream tasks.

Since the attack models rely on detecting the Consecutive Index Block (CIB) in
the matrix profile index, we propose a defense strategy to perturb CIB of
matrix profile with a random replaced with masked matrix profile as shown in
Fig. 1(c). This strategy can both prevent the attacker recover CIB under the
sensitive motif region, at the same time, preserve the functionality of matrix
profile and hence benefit the downstream tasks. We hope that our defense will
be compatible and allow the development for the current and future matrix
profile algorithms and models for the downstream tasks.

###  6.1. Overall Defense Algorithm

The overall algorithm is described in Algorithm 1. The algorithm consists of
two main components. The first component aims at generating “fake” 1-NN to
makes the sensitive CIB become similar to other parts of the data. The second
component aims at manipulate the permuted $MPD$ so that the modification
become difficult to be detected by the attacker.

###  6.2. Randomly Perturb MP in CIB with Masked AB-Join Matrix Profile

In this component, the goal is to make the indexes in the long consecutive
intervals behaving more distributed and similar to other parts of the data,
but still meaningful for downstream task. To achieve this goal, we perturb the
MP index with AB-Join matrix profile value and index between CIB and the
remaining of the time series after masked the 1-NN location.

Intuitively, we observed that CIB behavior of the mask matrix profile often
follows similar behavior the actual one, and in the most cases, the distance
fluctuation caused by replacing is relatively small, as there are other
similar patterns in data. So the replacing could generate indistinguishable
CIB switch behavior. So in this step, we aims at perturb the original .

Concretely, for any subsequence of length $L_{sens}$ which is overlapped with
the long CIB (greater than $L_{perm}$), we trigger the permutation function to
replace 1-NN value and index stored in $MP$ with the masked AB join matrix
profile. Specifically, for any value and index falls in the CIB, starting from
$K=2$, we replace the original sequence with $Top-K$ sequence. If the $Top-K$
sequence hit a CIB with length greater than a random length $L_{r}and$, the
algorithm increase the $K$ and performs the same process until all index in
the CIB is examined. The randomness adds further protection. even the
adversary doubt there were a protection, he/she does not know whether it is a
2-NN or 3-NN, and bring more difficult if one tries to get an lower bound to
guess about any location for large motif.

In addition, we exclude the original $MPI(start)$ from all indexes $[start-
L_{sens}start]$ when computing matrix profile, and the original $MPI(end)$
from all indexes $[end,end+L_{sens}]$ when computing matrix profile. Finally,
we put

Figure shows an example…

###  6.3. Remove Potential Conflicted Indices

A smart attacker might examine some properties in perturbed matrix profile and
detecting the motif locations by identifying the locations containing
conflicted matrix profile index. To make it consistent, we perform two post-
processing strategy for matrix profile index to avoid potential conflicts
detected by adversary.

Adjust ‘Cycle’ Link Distance Sometimes, there are ‘cycles’ in the matrix
profile index – given two subsequence $S_{i,l}$ and $S_{j,l}$, $MPI_{l}[i]=j$
and $MPI_{l}[j]=i$. We call $S_{i,l}$ and $S_{j,l}$ symmetric nearest
neighbors. While this is not generally hold in matrix profile index, there can
be a portion of subsequences satisfying the symmetric condition depending on
data. After replacing with K-NN index, we identify ‘cycles’ in matrix profile
index and check their distance in $MP_{l}$. If the distances are not equal,
adjust their $MP$ distances to be the minimum of the two.

Adjust minimum/maximum value We want to preserve the value of maximum and
minimum value or original $MP_{l}$ for downstream tasks. Hence, we adjust any
value for the perturbed matrix profile exceeding the range back to the range.

Algorithm 2 Defense Algorithm

1:Input: $T$,$MP$, $MPI$

2:Output: Perturbed $MP$,$MPI$

3:$SegSet=$GetConsecutiveInterval($MPI$)

4:for Each $Segment>L_{perm}$ do

5: /* When the interval length is sufficient large, indicates we need to
perturb indices */ $idx$ =MajorIndex($Segment$) $mask\\_idx\\_set$.add($idx$)

6: for interval i overlap with $Segment$ do

7: $dist^{\prime}=$MaskDistanceProfile($T$,$i$,$mask\\_idx\\_set$)

8: $MP^{\prime}_{i}$=min($dist_{new}$);$MPI^{\prime}_{i}$=argmin($dist_{new}$)

9: end for

10: for $\ \textbf{do}i$ in $[Segment.StartSegment.End]$

11: $dist^{\prime}=$MaskDistanceProfile($T$,$i$,$mask\\_idx\\_set$)

12:
$MP^{\prime}_{i}$=min($dist_{new}$);$MPI^{\prime}_{i}$=argmin($dist_{new}$)

13: if LastConsecutiveIdxCount(i)$>T$ then

14: $mask\\_idx\\_set$.add(MPI’(i))

15: end if

16: if HitRandomIdx(i) then

17: $mask\\_idx\\_set$.removefirst()

18: $T=median(lens)$

19: else

20: $T=RandomGenerate(L/4)$

21: end if

22: end for

23:end for

24:$MP^{\prime}$,$MPI^{\prime}$ = FakeCycleLink($MP^{\prime}$,$MPI^{\prime}$)

25:$MP^{\prime}$,$MPI^{\prime}$ = ReScaleValue($MP^{\prime}$,$MPI^{\prime}$)

26:return $MP$,$MPI$

###  6.4. Advantage of Proposed Algorithm

Our defense algorithm is carefully designed and has several advantages
attribute to the kth nearest neighbor replacing. First, our algorithm plays a
specifically defending strategy to the attack method on protecting the
location of large motif index, while keeping the downstream utility in mind.
The replaced values are still meaningful as k-nearest-neighbor value can be
seen as an approximation to 1-near-neighbor value. Second, our strategy only
perturb a small mount of index, so there is less information loss compare with
the original matrix profile and matrix profile index. Third, our defense
strategy from visually, generate a relatively uniform distribution and the
defense is consist between new generated MP and MPI. As a result, it is
difficult for the adversary to find a shortcut to achieve their attack goal.

##  7\. Experimental Evaluation

###  7.1. Detecting Planted Motif while Protected Pattern Leakage

We first test our MP in two planted motif experiment to demonstrate its
ability to prevent attackers to infer large motifs while still maintain good
performance on detecting small motifs. We choose this task as it is a very
challenging task due to the fact the utility of small motif and large motif is
almost aligning together, but the adversarial goal is opposite to the utility
of the downstream task. In this subsection, we conducts two experiment to
evaluate the attack and protection performance in two scenario: 1) when small
motif is at a different location from large sensitive motif 2) when small
motif is part of the large sensitive motif.

In the first experiment, we planted two motifs of length $L_{min}$ and
$L_{max}$, each one has two instances. In the second experiment, we planted
one motif of length $L_{max}$ of two instances into the random walk time
series of length 10,000. In addition to the planted large motif, a small
subsequence of length $L_{min}$ which belong to the large motif, is randomly
planted into the time series.

In both experiment, the length of the random walk time series is 10,000.
Detecting the motif of length $L_{min}$ is our utility goal and detecting the
large motif of length $L_{max}$ is attacker’s goal.

Following the setting in previous planted motif discovery experiment
setting(hime), the shape of motifs are generated by using:

$x=\sum_{i=1}^{5}A_{i}\sin{\alpha_{i}x+\beta_{i}}$

with random parameters $A_{i}\in[0,10]$, $\alpha_{i}\in[-2,2]$ and
$\beta_{i}\in[-\pi,\pi]$.

Finally, for each instance of a motif, $\pm$5% random noise is added. The mean
and variance of both motifs are also randomly generated based on random walk
time series.

In the experiment, the lengths of the small motifs of $L_{min}$ is set to
$100$, while the large motif which represent the sensitive pattern are
$\\{200,300,400,500\\}$. For each length, we generate 50 different time
seires. An attack is successfully launched if the algorithm successfully
located the correct location of sensitive pattern. Given two subsequence
location reported from the attack method, a sensitive motif is successfully
detected if the algorithm is overlapping which both
instances(hime)(gao2018exploring). The overlapping for each instance of motif
is measured via Jaccard similarity index:

(1) $J(pred,gt)=\frac{pred\cap gt}{pred\cup gt}$

where $pred$ is the predicted sensitive pattern region and $gt$ is the actual
sensitive pattern region. If the value is greater than 0.25 in both instances,
we said the attack is success. Finally, the average attack success rate in the
50 times experiment for each length.

###  7.2. Vs. Sharing Original Matrix Profile

We first compare the perturbed matrix profile with the original profile. We
applied all three attacks described in Section V and report the attack success
rate. The result is shown in Fig. 7(a)-(c). According to the figure, the
preturbed matrix profile can significant reduce the attack success rate. When
the sensitive pattern length increased, since the portion of sensitive in the
time series is increased, the attack successful rate is slightly increase. But
it is still significant smaller than the original matrix profile. In addition,
we found that the attack success rate gradually downgrade as the length
increased in the original matrix profile. This is because the far away between
the sensitive pattern length and the shared matrix profile length, the harder
for attacker to retrieve information. Similar observation is found when we
test the overlapping case.

(a) Value-based Attack  (b) Entropy-based Attack  (c) Location-based Attack

Figure 6. Attack Success Rate vs. Different Sensitive Pattern Length under
Different Attacks when Small Motifs are not Overlapped

(a) Value-based Attack  (b) Entropy-based Attack  (c) Location-based Attack

Figure 7. Attack Success Rate vs. Different Sensitive Pattern Length under
Different Attacks when Motifs are Overlapping

###  7.3. Vs. Adding Noise in Raw Series

We next tested our proposed method with raw data sharing protocol. We compare
our proposed appraoch with naive noise addition approach. Specifically, we
compare with sharing raw data after adding Gaussian noise with variance
$\sigma^{2}=0.1,0.3,0.5$, respectively. Following the same protocol, the
attack utility is measured by the attack success rate. The result is shown in
Fig. 9 and 10 for both experiments respectively. The utility of both
experiment along with the average utility of the proposed approach is shown in
Table II and III.

In both experiment, adding only small amount of noise will result in high
attack success rate with high utility of the data. When we add higher noise,
both attack success rate and the utility are decreased. In both experiments,
the utility decreased is faster than attack success rate. This is because
small utility length pattern is much more easy to be affected compared with
large pattern. As a result, the utility down graded very significant. On the
other hand, the proposed matrix profile, could maintain a very high level of
utility (0.875 motif detection success rate) while prevent the sensitive
pattern leak with small attack success rate even against adding the highest
noise in the experiment.

(a) (b)

Figure 8. Attack Success Rate vs. Different Sensitive Pattern Length under
Attack

The experiment shows that the proposed approach is outperform noise based raw
data sharing method.

Table 1. Shared Perturbed MP vs. Adding Noise (Non-Overlapping Case) Method | small $\sigma$ | median $\sigma$ | large $\sigma$ | Perturbed MP
---|---|---|---|---
utility | 0.84 | 0.32 | 0.115 | 0.875
Table 2. Shared Perturbed MP vs. Adding Noise (Overlapping Case) Method | small $\sigma$ | median $\sigma$ | large $\sigma$ | Perturbed MP
---|---|---|---|---
utility | 0.7 | 0.295 | 0.08 | 0.882

###  7.4. Case Study: Motif Discovery on Electrooculogram Time Series

Figure 9. Protected MP successfully Prevent Type II Eye Blink Activity Leak

In sleep quality study, Electrooculogram is a popular data type to study the
sleep behavior of an objective. In this case study, we tested our proposed
method on protecting long eye blink activities existed in the Electrooculogram
Time Series. We utilized the EOG time series used by Madrid et al.
(madrid2019matrix). According the the reference, the time series is captured
from a 66- year old healthy male recorded during a sleep study. The time
series is shown in Fig. 6(a). Two type of motifs which corresponding to two
different type of eye blink activities are highlighted in the time series.
Without any perturbation, it is very easy for the attack to retireve the type
II eye blinking pattern through the location/entropy based attack method. The
detected pattern is shown in Fig. 9(b). After the perturbation, the attack
algorithm fails to detect the long pattern. In fact, it located closed to the
place that is overlapped with small motif. Therefore, the proposed
perturbation version matrix profile protects the data from the attack launched
by malicious users.

###  7.5. Case Study: Top-K Discord Discovery on Dishwasher Power Demand Time
Series

Anomaly detection is one of the most essential task in time series data
mining. Matrix profile is well known for its ability to detecting time series
discord, defined as the subsequence has the largest 1-NN distance. In this
case study, the protected MP is applied on a regular top-K anomaly detection
task on a dishwasher power demand time series. Since the unusual behavior
happen in the local area of a dishwasher cycle, the utlity length should be
much smaller than a cycle. In this experiment, we set our utlity length as
100. Our goal in this task is to check if the protected MP could still be used
to detecting top-K unusual shape in the time series while attack could not
infer the long pattern.

The discords detected are shown in Fig.10 bottom. A typical cycle of the
dishwasher is shown in the figure. Obviously that all these candidates
corresponding to some rare behavior happened in the data. On the other hand,
if we lanuch the attack on top of the protected MP, we only could find a pair
of subsequence shown in Fig. 11. By computing the similarity between the
retrieved candidate, the distance value is greater than 98.5% percentage of MP
distance.

Figure 10. Protected MP successfully successfully detect top-4 anomalies
ranked by regular MP. Top-4 anomalies from regular MP and protected MP are
highlighted in different color Figure 11. Attack Retrieved a Pair of
Subsequence which the distance value is greater than 98.5% percentage of MP
distance. It indicates the attack approach fails to locate the correct
location of the pattern

##  8\. Conclusion