# Resolving nonlinear recombination dynamics in semiconductors via ultrafast
excitation correlation spectroscopy: Photoluminescence versus photocurrent
detection

Esteban Rojas-Gatjens  School of Chemistry and Biochemistry, Georgia Institute
of Technology, Atlanta, GA 30332, United States  Kaila Yallum  Department of
Chemistry, Biochemistry, and Pharmaceutical Sciences, University of Bern,
Freiestrasse 3, CH-3012 Bern, Switzerland  Yangwei Shi  Department of
Chemistry, University of Washington, Seattle, WA 98195, United States  Yulong
Zheng  School of Chemistry and Biochemistry, Georgia Institute of Technology,
Atlanta, GA 30332, United States  Tyler Bills  School of Chemistry and
Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332, United
States  Carlo Andrea Riccardo Perini  School of Materials Science and
Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States
Juan-Pablo Correa-Baena  School of Materials Science and Engineering, Georgia
Institute of Technology, Atlanta, GA 30332, United States  David S. Ginger
Department of Chemistry, University of Washington, Seattle, WA 98195, United
States  Natalie Banerji  Department of Chemistry, Biochemistry, and
Pharmaceutical Sciences, University of Bern, Freiestrasse 3, CH-3012 Bern,
Switzerland  Carlos Silva-Acuña<EMAIL_ADDRESS>School of Chemistry
and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332, United
States

###### Abstract

We explore the application of excitation correlation spectroscopy to detect
nonlinear photophysical dynamics in two distinct semiconductor classes through
time-integrated photoluminescence and photocurrent measurements. In this
experiment, two variably delayed femtosecond pulses excite the semiconductor,
and the time-integrated photoluminescence or photocurrent component arising
from the nonlinear dynamics of the populations induced by each pulse is
measured as a function of inter-pulse delay by phase-sensitive detection with
a lock-in amplifier. We focus on two limiting materials systems with
contrasting optical properties: a prototypical lead-halide perovskite (LHP)
solar cell, in which primary photoexcitations are charge photocarriers, and a
single-component organic-semiconductor diode, which features Frenkel excitons
as primary photoexcitations. The photoexcitation dynamics perceived by the two
detection schemes in these contrasting systems are distinct. Nonlinear-dynamic
contributions in the photoluminescence detection scheme arise from
contributions to radiative recombination in both materials systems, while
photocurrent arises directly in the LHP but indirectly following exciton
dissociation in the organic system. Consequently, the basic photophysics of
the two systems are reflected differently when comparing measurements with the
two detection schemes. Our results indicate that photoluminescence detection
in the LHP system provides valuable information about trap-assisted and Auger
recombination processes, but that these processes are convoluted in a non-
trivial way in the photocurrent response and are therefore difficult to
differentiate. In contrast, the organic-semiconductor system exhibits more
directly correlated responses in the nonlinear photoluminescence and
photocurrent measurements, as charge carriers are secondary excitations only
generated through exciton dissociation processes. We propose that bimolecular
annihilation pathways mainly contribute to the generation of charge carriers
in single-component organic semiconductor devices. Overall, our work
highlights the utility of excitation correlation spectroscopy in modern
semiconductor materials research, particularly in the analysis of nonlinear
photophysical processes, which are deterministic for their electronic and
optical properties.

Molecular Engineering & Sciences Institute, University of Washington, Seattle,
WA 98195, United States School of Physics, Georgia Institute of Technology,
Atlanta, GA 30332, United States School of Materials Science and Engineering,
Georgia Institute of Technology, Atlanta, GA 30332, United States

##  1 Introduction

Probing photoexcitation dynamics is a cornerstone of materials
characterization in optoelectronics. Photoexcitations may undergo radiative
recombination, defect trapping/detrapping, and high-order processes such as
bimolecular and Auger recombination, among others. These recombination
dynamics often give rise to a nonlinear response with respect to
photoexcitation density, and in turn, they can be determined by resolving
time-dependent populations. Researchers commonly probe the nonlinear response
using intensity-dependent (i) steady-state photoluminescence and photocurrent
experiments, in which a deviation of the signal $S(I)$ from a linear response,
$S(I)\propto I^{\alpha}$ is observed 1, (ii) time-resolved photoluminescence
2, 3 or (iii) transient absorption spectroscopies 4, 5, 6. However, delimiting
the distinct nonlinear regimes can be ambiguous between these techniques, and
may become complex as the system’s components increase. Originally described
by von der Linde and Rosen 7, 8, excitation correlation (EC) spectroscopy
provides the means to characterize the nonlinear response with great
sensitivity as it is based on double amplitude modulation and phase-sensitive
detection. In addition, it maps the time evolution of nonlinear contributions.
In EC spectroscopy, we amplitude-modulate two replica ultrafast pulses at
frequencies $\Omega_{1}$ and $\Omega_{2}$, such that demodulating at a
reference frequency $|\Omega_{1}+\Omega_{2}|$ using lock-in detection isolates
the nonlinear component. EC spectroscopy has been widely used to characterize
the carrier lifetimes of several inorganic semiconductors 9, 10, 11, 12, 13.
Despite its utility, neither the organic nor the lead halide perovskite (LHP)
semiconductor community uses it as a routine technique. Only recently, Srimath
Kandada et al. employed it to describe the defect density and energetic depth
in CH3NH3PbBr3 thin films and CsPbBr3 nanocrystals 14. The Moran group
presented a variation of the EC spectroscopy, utilizing a tunable narrow
excitation wavelength to characterize layered perovskite quantum-well
structures 15, 16, 17. We have previously implemented ECPL to describe defect
states in mix-halide mix-cation metal halide perovskites 18, 19.

In this article, we implement EC spectroscopy with both photoluminescence (PL)
and photocurrent (PC) detection to characterize the nonlinear response of two
photodiodes, a LHP in a solar cell, and an organic semiconductor single-
component vertical diode. We describe in detail the interpretation of EC
signatures using simplified kinetic recombination models that exemplify the
class of nonlinear dynamics in these materials systems. First, we discuss the
typical photophysical processes that result in ECS signals for the case of
LHP. We show that trap-assisted and Auger recombination dominate the nonlinear
response of LHP devices in PL detection (ECPL) at low and high fluence,
respectively. In PC detection (ECPC) the nonlinear components are due to
bimolecular and Auger recombination, however, these contributions cannot be
easily distinguished with this detection scheme. We also describe the
photophysical scenario of the organic semiconductor diode leading to ECS
signal. In this case, where the primary excitation is a Frenkel exciton, and
charge carriers are not directly injected, ECPL and ECPC provide complementary
information about the population evolution of excitons and charges.

##  2 Results

Figure 1: Schematic representation of the excitation correlation measurement.
The photoluminescence signal is measured using a photodiode and the
photocurrent is processed by a current amplifier. Both signals are sent to the
lock-in amplifier, which demodulates the input signal at the fundamental of
the two amplitude modulation reference waveforms with frequencies $\Omega_{1}$
and $\Omega_{2}$, and at the sideband $\Omega_{1}+\Omega_{2}$.

###  2.1 Nonlinear dynamics in lead-halide perovskites

We prepared inverted devices with a mixed-cation mixed-halide perovskite of
composition FA_0.83Cs_0.17Pb(I_0.85Br_0.15)_3 referred to in the text as
Cs17Br15, where FA+ refers to the formamidinium cation. Fig. S1(a), in
Supplementary Material, shows the linear absorption and PL spectra of a
Cs17Br15 film on ITO/MeO-2PACz. Our supplementary material and previous work
19 provide further details on the device structure and characterization.
Briefly, these films and fabrication procedures yield performance of around
15.90% power conversion efficiency for the best devices. The external quantum
efficiency (EQE) measurement is shown in Fig. S2 in Supplementary Material. We
emphasize that we perform both ECPL and ECPC on completed device stacks, with
typical PL quantum yields in the range of 0.8%. To perform ECPL and ECPC, we
excite the sample with a 220 fs pulse with an energy of 2.638 eV and variable
fluence between 1 and 40 $\mu$J cm-2. A schematic representation of the EC
experiment is represented in Fig. 1, and more details about our implementation
are described in the appendix.

Figure 2: Excitation correlation spectroscopy measurement of a prototypical
Cs17Br15 device. (a) PL detected and (b) PC detected nonlinear response. In
both cases, the pump wavelength was 470 nm, and the fluence range is indicated
in the false color axis.

Because metal-halide hybrid perovskites are direct bandgap semiconductors,
their recombination kinetics involve photocarriers undergoing second-order
(bimolecular) radiative recombination of electrons and holes, pseudo-first
order radiative recombination of photogenerated minority carriers with the
majority carriers, first-order deep-trap assisted non-radiative recombination,
and third-order Auger recombination 20, 21, 22. These terms are well described
by equations 1, 2 and 3, where $B$ is the bimolecular rate constant,
$\gamma_{t}$ the carrier trapping rate constant, $\gamma_{r}$ the trap
recombination rate constant, and $\gamma_{Auger}$ the Auger recombination rate
constant. Additionally, $N_{t}$ and $n_{t}$ correspond to empty and occupied
trap sites. The generation of electrons and holes is assumed to be direct,
then their generation populations are considered as initial conditions when
solving the differential equations. Note that we do not take into account non-
geminate association and dissociation of excitons explicitly since it does not
add a distinct recombination order, and additionally, excitons are not
generated, nor stable at room temperature 23.

$\frac{dn}{dt}=-Bnp-\gamma_{t}(N_{t}-n_{t})\,n-\gamma_{Auger_{1}}pn^{2}-\gamma_{Auger_{2}}p^{2}n,$
(1)
$\frac{dp}{dt}=-Bnp-\gamma_{r}n_{t}p-\gamma_{Auger_{1}}pn^{2}-\gamma_{Auger_{2}}p^{2}n,$
(2) $\frac{dn_{t}}{dt}=\frac{dn}{dt}-\frac{dp}{dt}.$ (3)

A distinct model assuming shallow donors has been used to describe CH3NH3PbBr3
14, where shallow traps dope the semiconductor. Based on this model, we expect
to observe positive subnanosecond dynamics due to fast trapping in shallow
traps, accompanied by an increase in the ECPL response as the excitation
fluence increases. However, this model does not apply to the Cs17Br15 devices
in our study, as discussed below. The ECPL and ECPC measurements are shown in
Fig. 2(a) and Fig. 2(b), respectively. The time traces show the percentage of
the nonlinear signal recovery as we scan the delay between the two pulses, the
symmetry between the negative and positive delays indicates that the pulses
had a comparable intensity. For the ECPL, At low fluence, we observe a
nonlinear response with slow dynamics. The magnitude of the nonlinear response
decreases as the fluence increases until it changes sign at the highest
fluences. We rule out the shallow donor model as these experimental signatures
do not match the model’s prediction that Kandada et al. described in Ref 14.
The ECPL signal in Fig 2(a), shows two distinct regimes: at low fluence, a
slow positive trace, and at high fluence, a fast negative nonlinearity. In
contrast, the ECPC response, shown in Fig. 2(b), shows only negative
contributions, with no change in the sign of the signal as the fluence
increases. To rationalize the information provided by each technique, we
interpret the EC measurements of LHPs in terms of the recombination model.
Equations 1, 2 and 3 do not have an analytical solution. However, by making a
series of assumptions described below we can understand the contributions of
the specific processes to the nonlinear photoluminescence and photocurrent.

####  2.1.1 Trap-Assisted Recombination

We follow the assumptions made by previous works on trap-assisted
recombination 24, 9. Specifically, we assume that we are working at low
excitation density, such that Auger recombination does not dominate and can be
neglected. Additionally, in materials with low PL quantum yield, as is the
case for LHPs, nonradiative trap-assisted pathways typically dominate the
carrier recombination such that $Bnp\ll\gamma_{t}N_{t}n$. Consequently, there
is an approximate solution for the electron and hole densities, which reads
as:

$\displaystyle n(t)=n(0)\exp(-\gamma_{t}N_{t}t),$ (4) $\displaystyle
p(t)=p(0)\exp(-\gamma_{r}n_{t}t).$ (5)

We will discuss first the case of ECPL. The detected PL is defined in equation
6. The temporal function describing carriers generated by each pulse is
assumed to be a delta function. We then split the integral describing the
total PL intensity in two: one term considering the carriers photoexcited
before the second pulse and another one with carriers photoexcited after the
second pulse. Here, $n_{1}(t)$ and $p_{1}(t)$ correspond to the evolution of
the carriers according to equations 4-5 after the first pulse arrives, the
initial conditions are simply the carriers generated by the first pulse,
$n_{1}(0)=p_{1}(0)=n_{0}$. $n_{2}(t)$ and $p_{2}(t)$ correspond to the
evolution of the carriers after the second pulse. These carrier densities are
described with the same expression but with distinct initial conditions,
$n_{2}(0)=n_{0}+n_{1}(\tau)$ and $p_{2}(0)=n_{0}+p_{1}(\tau)$, as we need to
consider residual carriers generated by the first pulse.

$I_{Total\,PL}\propto\int_{0}^{\tau}n_{1}(t)p_{1}(t)dt+\int_{\tau}^{\infty}n_{2}(t-\tau)p_{2}(t-\tau)dt.$
(6)

After integrating equation 6 and subtracting the individual pulse
contributions ($2n_{0}^{2}$), we obtain the nonlinear component of the
photoluminescence intensity ($I_{NPL}$) given in equation 7. Note that the
nonlinear term has a positive value, as expected since trap-filling results in
a reduction of non-radiative decay pathways. We note that under these
assumptions the $I_{NPL}$ follows the same decay as conventional time-resolved
experiments. Experimentally, in the ECPL measurements at low fluence shown in
Fig. 2(a), we observe a slow decay rate, which is not entirely captured in the
time window of the experiment. This is consistent with equation 7, as the
typical values for carriers’ lifetimes in LHPs are between the nanosecond and
microsecond range. The supplementary material shows the time-resolved
photoluminescence experiments in Fig. S3.

$I_{NPL}\propto\left(\exp(-\gamma_{t}N_{t}\tau)+\exp(-\gamma_{r}n_{t}\tau)\right).$
(7)

We performed a similar analysis for the case of photocurrent detection. The
signal measured is defined by equation 8. We ignore the spatial distribution
of the carriers and the extraction of carriers for the sake of simplicity.
These assumptions affect the magnitude of the nonlinear signal. We interpret
the nonlinear photocurrent arising from carriers’ interactions. The time-
resolved PL (see Fig S3) indicates that the recombination kinetics in open-
circuit and short-circuit conditions are very similar. Therefore, we justify
using the same photophysical scenarios to interpret ECPL and ECPC. It is worth
remembering that in ECPC the time resolution arises from the delay between the
pulse delays instead of from the carrier device extraction. Consequently, we
only need the device charge extraction to be faster than the modulation
frequency, which is the case by several orders of magnitude.

$I_{Total\,PC}\propto\int_{0}^{\tau}\left(n_{1}(t)+p_{1}(t)\right)dt+\int_{\tau}^{\infty}\left(n_{2}(t-\tau)+p_{2}(t-\tau)\right)dt.$
(8)

Under the assumption that trap-assisted recombination dominates at low
fluences, the integrands correspond to the monoexponential decay equations .
This is a linear function with the excitation density, therefore the nonlinear
photocurrent is zero. Trap-assisted recombination does not result in a
nonlinear PC response, making ECPC insensitive to traps. However, in the
experimental ECPC measurements in Fig. 2(b), the nonlinear component at low
fluence is not zero, and it has a negative value. Therefore, as discussed
below, higher-order processes such as bimolecular recombination and Auger
recombination must be responsible for the observed nonlinear photocurrent.

####  2.1.2 Bimolecular Recombination

We now consider the case where bimolecular recombination is the dominant
recombination pathway. In the Supplementary Material, we show that if
$Bnp\gg\gamma_{t}N_{t}n$, then the ECPL response is zero. This approach of
ignoring completely the monomolecular recombination does not give an
expression for ECPC as the integrals diverge. To attain an approximate
analytical expression for the ECPC response, we assume that both bimolecular
recombination, $B$, and monomolecular trapping,
$\gamma=\gamma_{t}(N_{t}-n_{t})$, are present. Also, we assume that holes and
electrons have similar trapping rates such that $n\approx p$. This scenario is
described by equation 9, and the corresponding solution for the population
evolution is shown in equation 10.

$\frac{dn}{dt}=-Bn^{2}-\gamma\,n.$ (9)
$n(t)=\frac{n_{0}\gamma/B}{(n_{0}+\gamma/B)\exp(\gamma t)-n_{0}}.$ (10)

After integrating equation 8 and subtracting the individual pulses
contribution ($4n_{0}$), we obtain an expression (equation 11) that describes
the nonlinear photocurrent, $I_{NPC}$, where $\alpha=n_{0}B/\gamma$. Note that
in the limiting cases where there is no bimolecular recombination ($B=0$), the
nonlinear contribution to PC is zero, and at a long delay ($\tau$), the
expression also goes to zero as the pulses do not overlap in time.
Consequently, we can conclude that the nonlinear photocurrent arises from
bimolecular recombination and not from carrier trapping, but the time
evolution follows the carrier trapping dynamics.

$I_{NPC}\propto\ln\left(1-\frac{\alpha^{2}\exp(-\gamma\tau)}{(1+\alpha)^{2}}\right)\approx-\frac{\alpha^{2}\exp(-\gamma\tau)}{(1+\alpha)^{2}}.$
(11)

According to equation 11, the nonlinear PC must have a negative sign,
congruent with the experimental results shown in Fig. 2(b). The ECPC
measurements, similarly to ECPL, show slow time dynamics, which is expected as
they both follow the time evolution of the carrier population. Additionally,
note that in this scenario, the ratio between the bimolecular recombination
and the carrier trapping rates dictates the magnitude of the nonlinear PC
component.

We have neglected the spatial aspect of the carrier dynamics, which is
relevant as we excite the sample in a small area of the sample. Carrier
dynamics simulations considering carrier diffusion, carrier trapping, and
bimolecular recombination, have been carried out by Zhou et al. 17 for
perovskite quantum wells. They observe negative decaying nonlinear
photocurrent at longer times, congruent with the slow traces shown in Fig.
2(b). So far, we have rationalized the slow dynamics and the sign of the ECPL
and ECPC response at low fluences. Experimentally, as we increase the fluence,
we observe a change in sign in the ECPL signal (Fig. 2(a)), while the ECPC
response increases in magnitude but remains negative. As the fluence
increases, Auger recombination becomes more significant and dominates the
recombination kinetics. We will now rationalize the effect of Auger
recombination in the nonlinear photoluminescence and photocurrent.

####  2.1.3 Auger Recombination

We next consider the scenario in which the carrier recombination is dominated
by Auger scattering, a third-order process that occurs at high carrier
density. Again, assuming that $p\approx n$ holds in the high-fluence regime,
we describe the kinetics using the rate equation 12. The solution of this
equation is presented in equation 13. Here, $\gamma$ corresponds to the
monomolecular recombination rate constant, and $A$ corresponds to the Auger
recombination rate constant.

$\frac{dn}{dt}=-\gamma n-An^{3}.$ (12)
$n(t)=\sqrt{\frac{\gamma/A}{(1+\gamma/n^{2}_{0}A)e^{2\gamma t}-1}}.$ (13)

In this particular case, the expressions for the nonlinear photoluminescence
and photocurrent are more complex to evaluate than in the previous scenarios.
In the Supplementary Information, we show that both the photoluminescence and
photocurrent exhibit negative nonlinear components due to Auger recombination.
This negative contribution explains the fluence-dependent features that we
observe for both ECPL and ECPC. In ECPL, when we transition from a trap-
dominated recombination scenario with a positive nonlinear component to an
Auger-recombination-dominated scenario with a negative nonlinear component, a
change of sign is expected. This transition is not expected in ECPC, as both
recombination processes (bimolecular and Auger recombination) that result in
nonlinear signals lead to negative nonlinear components, making it difficult
to distinguish between the two scenarios. Additionally, we note that the ECPL
experiments, Fig. 2(a), show subnanosecond dynamics at the highest fluence
(i.e. the sharp decay around zero time delay). We assign these fast features
to a large population of carriers recombining through Auger pathways at early
times.

####  2.1.4 Summary of nonlinear dynamics in lead halide perovskites

In summary, we have described the nonlinear responses caused by the distinct
photophysical processes for both photoluminescence and photocurrent. We
highlight the possibility of distinguishing between trap-assisted and Auger-
dominated recombination regimes employing ECPL, as the contributions to the
nonlinear response have opposite signs. The excitation density at which the
ECPL signal changes sign indicates a change in the dominant process, which is
related to the number of traps, meaning that ECPL is a good technique to
characterize trap densities in LHPs. On the other hand, in ECPC, the nonlinear
signal arising from bimolecular recombination and Auger recombination have the
same sign (negative), monomolecular recombination by itself does not result in
a nonlinear signal. Therefore, ECPC does not provide as rich information about
trap density as ECPL since the signatures are convoluted and difficult to
isolate. Recent work by Zhou et al. 15, 17 explores the implementation of a
similar experimental setup to characterize carrier diffusion, this idea is not
expanded in this work as large time scales (tens of nanoseconds), that exceed
our implementation capabilities, are needed.

###  2.2 Nonlinear dynamics in organic semiconductors

In organic semiconductors, the primary photoexcitation is a neutral exciton.
Charge carriers are generated after the dissociation of the exciton, which can
occur through several mechanisms in the _neat_ semiconductor. One such
mechanism is the formation of an intermediate charge transfer state prior to
charge separation 25, although the precise mechanism for this process is not
clear and is certainly not trivial. Another mechanism for exciton dissociation
is to overcome the exciton binding energy by promoting the exciton to a higher
energy excited state, $S_{n}^{*}$. This can be achieved by coherent two-step
photo-excitation pathways using femtosecond-pulse excitation, described as an
excitation from $S_{0}$ to $S_{1}$ and subsequently from $S_{1}$ to
$S_{n}^{*}$. The process generates a high-energy state prone to relaxation to
charged excitations (polarons) and triplet-excitons 4, 26, 27. Alternatively,
the $S_{n}^{*}$ state can be reached through energy transfer between excitons
in a process known as exciton-exciton annihilation 4, 26, 27, 28, this is the
mechanism proposed for this work, as discussed below.

The two detection methods used in the EC spectroscopy presented here are each
sensitive to different excited-state species produced optically in organic
semiconductors. While the charge carriers were both the emissive species and
the PC-detected species in lead-halide perovskites, excitons and charges can
be observed individually in the neat ITIC-4F devices studied here. Excitons
correspond to the detected emissive species (Fig. S1(b)in Supplemental
Material), and charges, arising from subsequent exciton dissociation, result
in the detected PC. The ECPL experiment then provides insights into the
processes leading to exciton recombination, while ECPC provides information on
those resulting in charge-carrier generation. In this work, we assess the
photophysical processes occurring in neat ITIC-4F. We prepared a single
component device with an architecture ITO/ZnO/ITIC-4F/MoO_3/Ag, and measured
both ECPL and ECPC. The absorption and PL emission spectra of ITIC-4F are
shown in Fig. S1(b) in Supplemental Material. Further details about the device
preparation are presented in the Supplementary Material. The pump pulse used
for these EC spectroscopy experiments has an energy of 1.823 eV.

####  2.2.1 Nonlinear photoluminescence

Figure 3: Excitation correlation spectroscopy measurement of a ITIC-4F device.
(a) Photoluminescence detected and (b) photocurrent detected nonlinear
photocarrier dynamics. The pump wavelength was 680 nm, and the fluence range
is represented by the false color axis.

Consider the simple model in equation 14, where the monomolecular rate
$\gamma$ incorporates all monomolecular processes, including radiative and
non-radiative relaxation pathways, and $\beta$ is the bimolecular exciton
annihilation rate. This model is mathematically equivalent to the equation 9
discussed for LHPs. As shown above, equation 14 can be solved analytically and
leads to the expression for the nonlinear photoluminescence shown in equation
15. We define $\alpha=n_{0}\beta/\gamma$ equivalently.

$\frac{dn_{exc}}{dt}=-\gamma\,n_{exc}-\beta\,n_{exc}^{2}.$ (14)
$I_{NPL}\approx\ln\left(1-\frac{\alpha^{2}\exp(-\gamma\tau)}{(1+\alpha)^{2}}\right).$
(15)

Fig. 3(a) shows the experimental ECPL response for a range of fluences between
1 and 100 $\mu$J cm-2. It can be observed that the magnitude of the nonlinear
response increases with fluence as expected since exciton-exciton annihilation
becomes more significant as the exciton density increases. By fitting the
measured traces to equation 15, we extracted a value of monomolecular
recombination rate of $(6.3\pm 0.5)\times 10^{9}$ s-1. We extract the
bimolecular rate using the experimental setup discussed by Riley et al. in
Ref. 3 and obtained a value of $(1.0\pm 0.2)\times 10^{-9}$ cm3 s-1, similar
to those reported previously in the literature 3. A summary of the analysis
and fitted data are shown in Supplementary Material. In the experimental
configuration of Ref. 3, the material is excited with a single pulse whose
amplitude is modulated by a mechanical chopper. The bimolecular exciton-
exciton rate was obtained after analyzing the fluence dependence of
photoluminescence intensity.

In this section, by using a simple model including monomolecular and exciton-
exciton recombination, we show that ECPL follows the excitons population time
evolution through the nonlinear photoluminescence generated in organic
semiconducting materials. More complex scenarios involving coexisting
excitonic species have been analyzed previously 11, 13. In those cases, the
additional exciton dynamics afflict the spectral integrated response, as the
one measured in our work. Spectrally resolving the signal is necessary to
distinguish nonlinear dynamics from distinct emissive species with similar
emission energy.

####  2.2.2 Nonlinear photocurrent

To interpret the photocurrent response in organic semiconductors, we will
focus on a simple model for charge carrier population dynamics described by
equation 16. In this model, the photocarriers are generated through the
function $G(t)$ which depends on the photophysical process that results in
charges. We acknowledge that the generation of photocarriers in neat organic
semiconductors, and a complete description of their dynamics, is a complex
problem and that multiple techniques are needed to provide a robust physical
picture. However, in this work, we focus on the contributions that ECS can
bring to the field, and thus we provide our hypothesis of the photophysical
scenario in this materials class.

Figure 4:  Normalized nonlinear response. (a) Photoluminescence detected and
(b) photocurrent detected signal.

In our simplified model, we assume that the dynamics of electron and hole
carriers are comparable, which is very likely as the system is not doped.
Additionally, $\gamma_{D}$ is the monomolecular decay rate of the carriers and
$\gamma_{B}$ is the non-geminate recombination rate. Note that equation 16 is
similar to equations 9 and 14. From this, we can deduce that the nonlinear
photocurrent (ECPC) will have a negative sign due to the non-geminate
recombination experienced by the carriers and that the time trace will follow
the carrier’s time evolution. In this case, however, we need to consider a
time-dependent generation term. We cannot assume it to be a delta function.
This makes an analytical solution challenging.

$\frac{dn_{e/h}}{dt}=G(t)-\gamma_{D}\,n_{e/h}-\gamma_{B}\,n_{e/h}^{2}.$ (16)

Experimentally, in the ECPC response, shown in Figs. 3(b) and 4(b), we observe
a “rising” feature. We interpret this feature as the generation time of charge
carriers. Note that as the incident fluence increases, the generation rate
increases as well. Based on this experimental observation we discuss possible
generation mechanisms below.

Consider the case of a charge carrier being generated through a two-step
excitation, as has been proposed for several polymeric materials 4, 27, 26,
29. For this case, since the photocarriers are directly pumped, the generation
function is $G(t)\propto\delta(t)$. We can discard this as the dominant
mechanism as we expect it to manifest, in the ECPC, as a maximum in the
absolute signal when the two pulses temporally overlap. Instead,
experimentally we observe a minimum. A similar experiment as the one presented
here probed the two-step excitation pathway and observed that the response
follows the exciton decay dynamics 29. At high fluences, they observed a
“rise” time interpreted as a fast relaxation from a hot vibrational state
$S_{1}^{*}$ to $S_{1}$ followed by a subsequent excitation from $S_{1}$ to
$S_{n}$. The rise time that we observed is too long to correspond to any
relaxation time which is usually in the sub-picosecond time scale 29.

Other possible mechanisms are charge generation after bimolecular exciton-
exciton annihilation 26 and a recently proposed monomolecular exciton
dissociation 30. In both cases, the generation rates will depend on the
population of the excitons. In the bimolecular exciton-exciton annihilation
pathway, the generation function would be proportional to the square of the
exciton population, $G(t)\propto n(t)^{2}$. While for monomolecular exciton
dissociation, the generation rate is proportional to the exciton population,
$G(t)\propto n(t)$. Notice in Fig. 3(b), further exemplified in Fig. 4(b),
that the “rise” of the nonlinear response is dependent on the excitation
density. Since excitons population evolution $n(t)$ depends on the excitation
density, both mechanisms show the same trend to become faster as the fluence
increases. However, since we observed experimentally a more dramatic effect in
the photocurrent detection scheme than in the photoluminescence scheme, shown
in Fig. 4, we hypothesize that the dominant generation mechanism is
bimolecular annihilation of the exciton population.

Furthermore, the charge lifetime observed in ECPC (Fig. 3(b)) is considerably
longer than the exciton lifetime (Fig. 3(a)). We cannot quantify it as it is
outside of the instrument’s temporal window. As the fluence increases, we
observe that the decay of the charge becomes more significant, indicating that
the bimolecular recombination of carriers becomes more important as expected
from the simple model presented above. Finally, we acknowledge that there are
other causes of nonlinear photocurrent that were not discussed in this work,
e.g., the current limitation due to the external resistance series 31. We
tried to minimize these effects by performing the measurements at a low
fluence range.

####  2.2.3 Summary of nonlinear dynamics in organic semiconductors

We have shown how ECPL can be used to extract photophysical parameters like
monomolecular and bimolecular decay rate constants. In the ECPC experiments we
interpret the ”rising” features around zero time delay as the time-dependent
generation of charge carriers, which becomes faster with increasing fluence.
We suggest that charge carriers are generated through bimolecular annihilation
pathways. Since the fluence dependence of the generation time might originate
from a second-order charge generation process, $G(t)\propto n^{2}$. Other
mechanisms could also be involved, however, due to the strong
photoluminescence nonlinearity observed, we expect bimolecular exciton-exciton
annihilation to be the dominant pathway toward charge generation. To further
clarify the complete photophysical scenario, systematic experiments using
complementary techniques (e.g. transient absorption and time-resolved
photoluminescence spectroscopies) are needed and recommended as future
endeavors.

##  3 Discussion

In this work, we presented two scenarios in which the nonlinear PL and PC
detection schemes provide distinct and complementary information. In the case
of lead-halide perovskites, the same excited-state species contribute to both
the nonlinear PL and PC. Therefore via ECPL and ECPC, we follow the population
of the same species; however, this species leads to distinct nonlinear
responses in each of the physical observables. This offers the possibility to
selectively study photophysical processes experienced by the excited species
based on the detection scheme. For example, we showed that while ECPL shows a
response due to trap-assisted recombination, the ECPC is insensitive to traps
themselves. The magnitude of ECPC, instead, is given by bimolecular
recombination, a process to which ECPL is insensitive. As mentioned above,
Zhou et al. 17 have taken advantage of this to characterize carrier diffusion
in layered perovskites using photocurrent detection.

For the case of organic semiconductors, we probe exclusively the excitons that
recombine radiatively or those that dissociate and generate charge carriers,
with each detection scheme, photoluminescence, and photocurrent respectively.
In that sense, the work presented here adds to the existing toolbox of
ultrafast photoluminescence and photocurrent techniques 32, 33, with the
additional feature that the magnitude of the response can be used to describe
the rates responsible for the nonlinearities. In this material, both ECPL and
ECPC have negative nonlinear responses which are directly related to the
exciton-exciton bimolecular annihilation rate and the charge carrier
bimolecular recombination respectively. As the excitonic scenarios become
complex, the magnitude of the nonlinear signal provides insight into nonlinear
processes occurring on the ultrafast scale hidden to steady-state measurements
or that appear convoluted in time-resolved techniques.

As mentioned in the Introduction, recent reports describe a variation of the
ECS probe, utilizing a tunable narrow excitation wavelength to characterize
layered perovskite quantum-well structures 15, 16, 17. We note that in their
interpretation there is ambiguity in the distinction between the incoherent
and coherent contributions to the nonlinear response. The measured spectra are
interpreted as 2D excitation spectra, but we highlight that there is no well-
defined phase resolution in the excitation-pulse wavepackets, and the
measurements are thus purely incoherent as in the work presented here. This
incoherent response arises from the dependence of the physical observable on
the intensity of the excitation due to the population evolution (e.g., trap
recombination and exciton-exciton recombination), rather than a coherent
nonlinear response as in coherent multidimensional spectroscopies 34, 35, 36,
37, 38. We also note that these 2D measurements that implement phase
modulation may also contain incoherent contributions due to nonlinear
population dynamics picked up by the phase demodulation detection scheme 39,
40, 41. We thus underline the difference between the technique presented in
this article and 2D coherent excitation. Earlier, ECS-like experiments have
been interpreted using Feynmann diagrams 16, 15. We emphasize that this is not
precise, since Feynmann diagrams indicate optical transitions among states and
their coherent correlation but do not include the interactions among their
populations. This imprecision is addressed in recent literature recognizing
recombination dynamics as the only origin of the measured nonlinearity 42. Due
to their distinct origin, they provide distinct information. While 2D coherent
excitation experiments provide information regarding dephasing rates and
coherent correlations between excited states, the ECS experiments provide
information uniquely about population mixing. In this work, we expand on the
signal-generation mechanisms associated with population mixing. Together with
previous examples 7, 8, 10, 11, 12, 13, 14, 18, 19; our work adds another tool
to the modern semiconductor community for the characterization of nonlinear
photophysical processes.

##  4 Conclusion

We have observed and rationalized the main nonlinear signatures in the
photoluminescence and photocurrent of lead-halide perovskite and organic
semiconductor devices. For the case of LHPs, the ECPL has nonlinear components
due to trap-assisted and Auger recombination with opposite behavior, sublinear
and supralinear. The fluence dependence of ECPL provides rich information
about defect density, ultrafast dynamics, and Auger recombination. Meanwhile,
in ECPC, the nonlinear signature originates from bimolecular and Auger
recombination, both of which are supralinear processes. In ECPC, the nonlinear
contributions are convoluted and difficult to distinguish. In organic
semiconductors, we describe ECPL as a sensitive technique for determining
exciton-annihilation rates. On the other hand, ECPC represents a valuable tool
to study charge generation through photo-excitation. The experimental data
suggest that ECPC can follow the population dynamics of free charges including
their generation dynamics. Additionally, from the rise time observed in ECPC,
we hypothesize that bimolecular annihilation corresponds to a significant
pathway for charge carrier generation.

We expect EC spectroscopy to have an impact, particularly in the field of
organic electronics, where it can shine further insight into the physical
nature of excited states and the generation mechanisms leading to charge
carriers. Besides the case of study of single component materials, mixed
systems with complex fluence-dependent photophysical processes will also
benefit from EC spectroscopy. For example, in recent studies of perovskite-
sensitized TTA-UC, the intensity dependence of photoluminescence shows an
interplay of processes with distinct nonlinearities 1 which could be better
resolved by EC spectroscopy as well as their time-resolved dynamics.

##  5 Appendix

##  6 Excitation Correlation Spectroscopy

In our implementation, 1030 nm, $\sim$220 fs pulses are generated in an
ultrafast laser system at a 100 kHz repetition rate (PHAROS Model
PH1-20-0200-02-10, Light Conversion). A portion of the laser beam is sent into
a commercial optical parametric amplifier (ORPHEUS, Light Conversion). The
pulse trains are then split 50/50 by a beam splitter cube, where one of the
beams is directed to a motorized linear stage (LTS300, Thorlabs), allowing for
control of the delay between the two pulses. Each pulse is modulated with a
chopper at the frequencies of 373 and 199 Hz, respectively, and the pulses are
then focused onto the sample with a 100 mm focal length lens. The total
integrated response and the nonlinear component are obtained simultaneously by
demodulating both the fundamental and the sum of the modulation frequency.
Photoluminescence detection (ECPL): The emitted PL is filtered with a long-
pass filter to get rid of the pump, and then it is focused into a
photoreceiver (New Focus 2031 PR) connected to a lock-in amplifier (HF2LI,
Zurich Instruments). Photocurrent detection (ECPC): The device is connected to
a Zurich Instruments HF2TA Current Amplifier used to convert the current
output of the sample device to voltage, as well as to supply an external bias
to the device. The current amplifier is connected to a lock-in amplifier
(HF2LI, Zurich Instruments). The photocurrent measurements presented here were
acquired with no external applied bias.

###  6.1 Signal recovery from lock-in amplifier

Additionally, in this appendix, we expand on the experimental details for
measuring the nonlinear component utilizing double modulation lock-in
detection. The intention is to provide two examples of nonlinear photophysics
processes recovered through double demodulation and to bring attention to the
fiendish experimental details. We follow reference 9 for the case of trap-
assisted recombination. We define the generation rate to take into account the
repetition rate and $S(t,\omega)$ to be a square wave to mimic the chopper.

$G(t,\omega)=gS(t,\omega)\sum^{\infty}_{n}\delta(t-nt_{rep})$ (17)

Remember that the square wave function that alternates between 0 and 1 is
given as:

$S(t,\omega)=\frac{1}{2}+\frac{2}{\pi}\sum_{n=0}^{\infty}\frac{\sin((2n+1)\omega
t+\theta)}{(2n+1)}.$ (18)

Consider the cases where the reference signal, with which the signal is
demodulated, corresponds to a sine function or a square wave. Also, we will
ignore the phase as this can be easily set experimentally.

###  6.2 Trap-assisted recombination

We take $\gamma=\gamma_{r}N_{r}=\gamma_{t}n_{t}$ and both pulses to have the
same intensity. Then, using the equations 8 we integrate
$\int^{t_{rep}}_{0}Bn(t)p(t)dt$ which corresponds to the response of the
detector. Since $t_{rep}$ is much longer than the carrier lifetime we
integrate from zero to infinity instead and obtain the intensity:

$\displaystyle\begin{split}I(t,\omega_{1},\omega_{2})\propto\frac{S(t,\omega_{1})+S(t,\omega_{2})}{2}+S(t,\omega_{1})S(t,\omega_{2})e^{-\gamma\tau}.\end{split}$
(19)

All the constants were grouped with the response of the detector.

We mimic the demodulation of the lock-in amplifier by multiplying the signal
by:

$S_{ref}(t,\omega)=A\sum_{n=0}^{\infty}\frac{\sin((2n+1)\omega t)}{(2n+1)}.$
(20)

. Then we average over a long time such that oscillating components vanish.
Then the intensity recovery for each modulation frequency is:

$\langle
I_{mod}(\omega_{1})\rangle_{LI}\propto\frac{A}{2\pi}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{2}}\left(1+e^{-\gamma\tau}\right).$
(21) $\langle I_{mod}(\omega_{1})\rangle_{LI}=\langle
I_{mod}(\omega_{2})\rangle_{LI}\propto\frac{A\pi}{16}\left(1+e^{-\gamma\tau}\right).$
(22)

Note that part of the mixed term is recovered in the single modulation since
$\langle S(t,\omega)\rangle=1/2$. Now, we expand the mixed term to:

$S(t,\omega_{1})S(t,\omega_{2})=\frac{1}{4}+\frac{1}{\pi}\sum_{i=0}^{2}\sum_{n=0}^{\infty}\frac{\sin((2n+1)\omega_{i}t)}{(2n+1)}+\\\
\frac{4}{\pi^{2}}\sum_{n,m=0}^{\infty}\frac{\sin((2n+1)\omega_{1}t)}{(2n+1)}\frac{\sin((2m+1)\omega_{2}t)}{(2m+1)}.$
(23)

After we demodulate at the sum frequency $\omega_{1}+\omega_{2}$ and average a
long time. The only terms that survive come from the last sum, when $n$ and
$m$ are the same. Then:

$\langle
S(t,\omega_{1})S(t,\omega_{2})S_{ref}(t,\omega_{1}+\omega_{2})\rangle=\frac{A}{2\pi^{3}}\sum^{\infty}_{n=1}\frac{1}{(2n+1)^{3}}=A\epsilon.$
(24)

Note that demodulating using a sine function recovers only the $n=0$
coefficient of the Fourier series.

$\langle I_{mod}(\omega_{1}+\omega_{2})\rangle_{LI}\propto A\epsilon
e^{-\gamma\tau}.$ (25)

###  6.3 Bimolecular Annihilation

We choose the delay between the pulses to be zero for simplicity. From the
equation above, the total photoluminescence detected is:

$I_{Total\,PL}\propto\int_{0}^{\infty}n(t)dt\propto\ln\left[1+n_{i}\frac{\gamma_{A}}{\gamma_{eff}}\right].$
(26)

We define $n_{i}=g(S(t,\omega_{1})+S(t,\omega_{2}))$. Then, we do a second-
order Taylor expansion, and $\alpha=g\gamma_{A}/\gamma_{eff}$.

$\approx\alpha(S(t,\omega_{1})+S(t,\omega_{2}))-\frac{\alpha^{2}}{2}(S(t,\omega_{1})+S(t,\omega_{2}))^{2}.$
(27)

Remember that the square wave is an idempotent function.

$\displaystyle\begin{split}=\alpha(S(t,\omega_{1})+S(t,\omega_{2}))\left(1-\frac{\alpha}{2}\right)-\alpha^{2}S(t,\omega_{1})S(t,\omega_{2}).\end{split}$
(28)

Then after demodulating with a square function, we obtained:

$\langle I_{mod}(\omega_{1})\rangle_{LI}=\langle
I_{mod}(\omega_{2})\rangle_{LI}\propto\frac{A\pi}{8}\alpha\left(1-\alpha\right).$
(29) $\langle
I_{mod}(\omega_{1}+\omega_{2})\rangle_{LI}\propto-A\epsilon\alpha^{2}.$ (30)

The authors thank Prof. Ajay Ram Srimath Kandada for fruitful discussions on
the ECPL and ECPC experimental techniques and data analysis, and Victoria
Quiros-Cordero for rigorous proofreading. The preparation of perovskite
samples, devices, and their characterization and analysis, and writing of the
corresponding manuscript sections by ERG, YS, DSG, and CSA was supported by
the U.S. Department of Energy’s Office of Energy Efficiency and Renewable
Energy (EERE) under the Solar Energy Technologies Office 24 Award Number DE-
EE0008747. The preparation of the organic-semiconductor samples was carried
out by KY, supported by the Swiss National Science Foundation (200020_184819).
Their optical characterization and analysis, and writing of the corresponding
manuscript sections by ERG, KY, YZ, NB, and CSA were supported by the National
Science Foundation (DMR-1904293). In addition, KY acknowledges the Swiss
Academy of Sciences and Universität Bern for additional travel grants. Part of
this work was carried out using the shared facilities of the UW Molecular
Engineering Materials Center (MEM-C), a Material Research Science and
Engineering Center (DMR-1719797) supported by the U.S. National Science
Foundation.

See the Supplementary Information for details of the setup, solar cell, and
diode used. Additionally, it contains details on the ECPL and ECPC models and
curve-fitting parameters.

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See pages - of SI.pdf


Scheduling of Sensor Transmissions Based on Value of Information for Summary Statistics

Federico Chiariotti, Member, IEEE, Anders E. Kalør, Student Member, IEEE, Josefine Holm, Student Member, IEEE,
Beatriz Soret, Member, IEEE, and Petar Popovski, Fellow, IEEE
This work has been in part supported by the Danish Council for Independent Research, Grant Nr. 8022-00284B (SEMIOTIC). F. Chiariotti, A. E. Kalør, J. Holm and P. Popovski are with the Department of Electronic Systems, Aalborg University, Denmark (email: {fchi, aek, jho, petarp}@es.aau.dk). B. Soret is with the Telecommunication Research Institute (TELMA) , Universidad de Malaga, Spain  (email: bsoret@ic.uma.es).

The optimization of voi in sensor networks integrates awareness of the measured process in the communication system. However, most existing scheduling algorithms do not consider the specific needs of monitoring applications, but define voi as a generic mse of the whole system state regardless of the relevance of individual components. In this work, we consider different summary statistics, i.e., different functions of the state, which can represent the useful information for a monitoring process, particularly in safety and industrial applications. We propose policies that minimize the estimation error for different summary statistics, showing significant gains by simulation.

Internet of Things, Wireless Sensor Networks, Value of Information, Scheduling policies

§ INTRODUCTION

Over the past few years, the unprecedented development of the iot has made the remote estimation of stochastic processes a central problem in communications and automation [1], where a set of sensors transmit observations to a central bs. The possibility to process sensor data either at the bs or in a distributed fashion through in-network processing [2] has led the research community to focus extensively on the scheduling of sensor updates in severely resource-constrained wireless network environments.

For a wide range of remote estimation problems, the freshness of the observations at the bs is a good proxy for the estimation quality. This promotes aoi [3] as a measure of the time that has passed since the last update from a given sensor. However, if the destination has a model of the observed processes, it is often better to directly minimize the uncertainty of the process estimates instead of the aoi [4]. The problem of scheduling iot sensors with this goal has been considered for several different policies [5, 6], whose objective is to minimize the mse of a Kalman filter, considering communication constraints. More recently, the problem of minimizing the mse of the process estimates has been referred to as voi [4]. A recent work [7] tries to maximize the accuracy of a more complex unscented filter, aiming at optimal sensor selection for maneuvering tasks, and voi can also be used for data muling applications in underwater or drone networks [8]. Another interesting twist to this is the application of voi concepts not over time, but in space, placing sensors in the positions that will result in the highest overall accuracy for the estimation of a spatial process [9].

However, there are cases where minimizing the mse is not be the best thing to do: for example, if the application needs to compute a non-linear function of the state, such as the maximum value among all sensors. While minimizing the mse implicitly gives equal value to all sensors, some might have a larger weight in the non-linear function (e.g., sensors with a higher value for the maximum function). Examples in industrial settings include: (1) triggering a safety warning if the temperature of any of the components in a machine reaches a safety limit, (2) monitoring if the difference in the strain on different parts of a structure is outside the design parameters. Such a scenario is represented in Fig. <ref>: the remote server sends queries to the bs, which correspond to the non-linear function, and the bs needs to schedule transmission so as to maximize the accuracy. This setup was also used in our previous work [10]. The scheduling in this scenario is driven by the bs, which selects the sensor that it believes to have the most useful information at each time slot; the opposite scenario, in which sensors themselves decide whether to transmit or not, is an interesting but different problem, as it requires sensors to maintain an estimate of the system state and a decision algorithm, which consume energy, as well as to coordinate among themselves to avoid collisions. Our scenario is directly applicable to wake-up radio [11, 12] and similar schemes with low-power sensors.

We propose heuristic strategies to schedule sensor updates in a linear dynamic system, which explicitly aim to minimize the error of various summary statistics. We derive the one-step optimal strategies for some well-known function, and give a general Monte Carlo-based algorithm that can deal with different query functions. The simulations show that the proposed strategies can significantly reduce the error on a number of summary statistics, with more significant gains in case of highly non-linear summary statistics.

Representation of the scenario.

The rest of this letter is organized as follows. The system model is presented in Sec. <ref>, and one-step policies for various summary statistics are derived in Sec. <ref>. Numerical results are presented in Sec. <ref>, and finally Sec. <ref> concludes the paper and presents some possible avenues of future work.

§ SYSTEM MODEL

We consider a system with $N$ sensors, which are connected through time-slotted wireless links to a bs equipped with computing and storage resources. Without loss of generality, we assume that the time slots occur at $t=1,2,\ldots$ and the sensors are indexed by $n=1,\ldots,N$. We assume that each sensor observes a value in an $N$-dimensional process, whose state $\mathbf{x}(t)=[x_1,\ldots,x_N]^T$ evolves according to

\begin{equation}
\mathbf{x}(t) = \mathbf{A}\mathbf{x}(t-1) + \mathbf{v}(t),\label{eq:process_model}
\end{equation}

where $\mathbf{A}\in\mathbb{R}^{N\times N}$ is the transition matrix, $\mathbf{v}(t)\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_{\mathbf{v}})$ is the process noise with covariance matrix $\mathbf{\Sigma}_{\mathbf{v}}\in\mathbb{R}^{N\times N}$, and $\mathbf{x}(0)=\mathbf{0}$. The sensors observe the processes with additive white Gaussian measurement noise $\mathbf{w}(t)\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_{\mathbf{w}})$, i.e., $\mathbf{y}(t) = \mathbf{x}(t)+\mathbf{w}(t)$. In general, the covariance matrices $\mathbf{\Sigma}_{\mathbf{v}}$ and $\mathbf{\Sigma}_{\mathbf{w}}$ are not diagonal.
Note that although we assume that the number of sensors is equal to the dimension of the process (to simplify the notation), the analysis can be easily extended to more general observable systems.

We consider a tdma air interface, in which each time slot, $t$, contains a downlink phase and an uplink phase. The downlink is used by the bs to schedule the sensor, $a(t)$, that transmits its observation $y_{a(t)}(t)$ in the uplink phase. The channel is modeled as a packet erasure channel with error probability $\varepsilon_{a(t)}$, which captures errors both in the transmission of the scheduling decision and the observation. We also assume that the process dynamics are known to the bs, a standard assumption in Kalman filtering, which is practical if the monitored system is well-understood, even if its instantaneous state is hard to measure directly. This condition is common for many iot applications [13, 14], in which well-known processes are estimated by sensors over wide areas. We also denote the row vector of length $N$ whose only non-zero value is the $n$-th, which is 1, as $\mathbf{1}_n$, and the $N\times N$ identity matrix as $\mathbf{I}_N$.

§.§ Kalman Filter Estimation

We assume that the bs maintains a distribution over its belief of the state $p(\mathbf{x}(t))$ using a Kalman filter. The Kalman filter is the mmse estimator for the model defined in <ref> [15], in which case $p(\mathbf{x}(t))\sim\mathcal{N}(\hat{\mathbf{x}}(t),\bm{\psi}(t))$. The mean vector $\hat{\mathbf{x}}(t)$ and the covariance matrix $\bm{\psi}(t)$ are updated at each timestep $t$ based on the outcome of the scheduling process. The Kalman filter operates in two steps: a prior update, which only depends on the system statistics, and a posterior update, which integrates new observations. The prior update operation is given by:

\begin{align}
\hat{\mathbf{x}}(t)&=\mathbf{A}\hat{\mathbf{x}}(t-1) \triangleq \hat{\mathbf{x}}_{\mathrm{F}}(t),\label{eq:blind}\\
\bm{\psi}(t)&=\mathbf{A}\bm{\psi}(t-1)\mathbf{A}^T+\mathbf{\Sigma}_{\mathbf{v}} \triangleq \bm{\psi}_{\mathrm{F}}(t).\label{eq:var_blind}
\end{align}

If the transmission of the update fails, an event we denote as $\mathrm{F}$, the bs can only rely on the prior update for its estimate. If the update is received, it can be used to improve the estimate. We then compute the Kalman filter gain $\mathbf{k}(t)$:

\begin{align}
\mathbf{k}(t)=&\bm{\psi}_{\mathrm{F}}(t)\mathbf{1}_{a(t)}^T\left[\mathbf{1}_{a(t)}\left(\bm{\psi}_{\mathrm{F}}(t)+\mathbf{\Sigma}_{\mathbf{w}}\right)\mathbf{1}_{a(t)}^T\right]^{-1}.
\end{align}

We finally get the updated estimate in case of a success event:

\begin{align}
\begin{split}
\hat{\mathbf{x}}(t)&=\hat{\mathbf{x}}_F(t)+\mathbf{k}(t)\mathbf{1}_{a(t)}(\mathbf{y}(t)-\hat{\mathbf{x}}_F(t))\triangleq \hat{\mathbf{x}}_{\mathrm{S},a(t)}(t)
\end{split}\\
\bm{\psi}(t)&=(\mathbf{I}_N-\mathbf{k}(t)\mathbf{1}_{a(t)})\bm{\psi}_{\mathrm{F}}(t) \triangleq \bm{\psi}_{\mathrm{S},a(t)}(t),\label{eq:var_update}
\end{align}

Note that the recursive structure of the Kalman filter and the independence of the transmission errors imply that $\hat{\mathbf{x}}(t)$ and $\bm{\psi}(t)$ are sufficient statistics for the state estimate given the full history $\mathcal{H}(t)$ of past actions and observations.

§.§ Summary Statistics

Unlike the majority of voi applications, in which the bs aims to minimize the mse of $\hat{\mathbf{x}}(t)$, we consider the case in which an external user requests summary statistics about the state of the system: these correspond to a predefined, fixed function of the system state, e.g., the average value or the number of states with values within a given interval. Formally, we define a summary statistic as a function $z:\mathbb{R}^N\rightarrow\mathbb{R}$ of the true state $\mathbf{x}(t)$. However, because $\mathbf{x}(t)$ is unknown to the bs, it can only provide an approximate answer to the query based on its state belief $p(\mathbf{x}(t))$. We will consider estimators of the summary statistics on the form

\begin{equation}
\hat{z}(t)=\E_{\mathbf{x}\sim \mathcal{N}(\hat{\mathbf{x}}(t),\bm{\psi}(t))}[z(\mathbf{x})],\label{eq:E_z}
\end{equation}

which corresponds to the minimum mse estimator of $z(\mathbf{x}(t))$, given the current observation [16, 17]. This is different from minimizing the mse of $\hat{\mathbf{x}}(t)$, particularly when function $z(\cdot)$ is non-linear or the sensors have different weights. We denote the squared error as $\nu_z(t)$:

\begin{equation}
\nu_z(t)=(z(\mathbf{x}(t))-\hat{z}(t))^2.
\end{equation}

§ SCHEDULING STRATEGIES

In our scenario, we seek a scheduling strategy, that is, a function $\pi_z$ from the current Kalman state (which is represented by vector $\hat{\mathbf{x}}(t)$ and matrix $\bm{\psi}(t)$) to an action $a(t)$, that minimizes the expected error for a given summary statistic $z$:

\begin{equation}
\underset{\pi_z\in\Pi}{\text{minimize}}\E\left[\nu_z(t)\mid\pi_z\right],\label{eq:onestep_def}
\end{equation}

While we only consider the error in the next time step, the optimal solutions are expected to perform well with respect to the long-term error due to the linearity of the observed process (despite a non-linear summary statistic). Computing $\E\left[\nu_z(t)|a(t)\right]$ is not simple, but it can be expressed in terms of the two possible transmission outcomes:

\begin{equation}
\begin{aligned}
\E\left[\nu_z(t)\mid a(t)\right]=&(1-\varepsilon_{a(t)})\E\left[\nu_z(t)\mid\hat{\mathbf{x}}_{\mathrm{S},a(t)}(t),\bm{\psi}_{\mathrm{S},a(t)}(t)\right]\\
\end{aligned}
\end{equation}

where the expectation is over the state evolution.
Since $\bm{\psi}_{\mathrm{S},a(t)}(t)$ and $\bm{\psi}_{\mathrm{F}}(t)$ can be computed using (<ref>) and (<ref>), we can iterate over the possible actions and find the optimal scheduler, as long as we can estimate the mse for a given observation. In the following, we derive the optimal schedulers for some well-known summary statistics, along with giving a Monte Carlo-based approximate scheduler that can deal with more complex statistics for which the mse is hard to express in closed form. Using the result from (<ref>) we can obtain the optimal scheduling decision at time $t$ as:

\begin{equation}
a^*_z(t)=\argmin_{a(t)\in\{1,\ldots,N\}} \E\left[\nu_z(t)\mid a(t)\right].\label{eq:optimal}
\end{equation}

§.§ Baseline Scheduler

We start by defining our benchmark scheme, which aims to minimize the mse between the true state $\mathbf{x}$ and the estimated state $\hat{\mathbf{x}}$. The query is then computed as in (<ref>) based on the mmse state estimate.

The squared state estimation error can be expressed as

\begin{equation} \E\left[\nu_{\text{MSE}}(t)\right]=\E\left[(\mathbf{x}(t)-\hat{\mathbf{x}}(t))^T(\mathbf{x}(t)-\hat{\mathbf{x}}(t))\right].
\end{equation}

Because $(\mathbf{x}(t)-\hat{\mathbf{x}}(t))\sim\mathcal{N}(\mathbf{0},\bm{\psi}(t))$, the expression above is equivalent to the trace of the covariance matrix $\bm{\psi}$ [18]:

\begin{equation}
\E\left[\nu_{\text{MSE}}(t)\mid\bm{\psi}(t)\right]=\tr(\bm{\psi}(t)).
\end{equation}

We can then use (<ref>) to compute the minimum mse schedule.

§.§ Sample Mean Scheduling

We now consider the most basic statistic, the sample mean:

\begin{equation}
z_{\text{avg}}(\mathbf{x}(t))=\frac{1}{N}\sum_{n=1}^N x_n(t).\label{eq:z_avg}
\end{equation}

The estimation error $\nu_{\text{avg}}(t)$ is equal to the square of the average difference between the true and the estimated entries of $\mathbf{x}$. Since the sum of all elements in $\mathbf{x}(t)-\hat{\mathbf{x}}(t)$ is a Gaussian random variable with zero mean and variance equal to the sum of all elements in $\bm{\psi}(t)$, we have:

\begin{equation}
\E\left[\nu_{\text{avg}}(t)\mid \bm{\psi}(t)\right]=\frac{\sum_{i=1}^N\sum_{j=1}^N\psi^{(i,j)}(t)}{N^2},
\end{equation}

where $\psi^{(i,j)}(t)$ is entry $(i,j)$ of $\bm{\psi}(t)$. We can then use the result in (<ref>) to derive the one-step optimal schedule.

§.§ Sample Variance Scheduling

Another important summary statistic is the sample variance, quantifying how much the state deviates from the mean:

\begin{equation}
\end{equation}

To derive the scheduling policy, it is convenient to express $z_{\text{var}}(\mathbf{x}(t))$ in quadratic form with matrix $\mathbf{M}=\mathbf{I}-1/N$:

\begin{equation}
\end{equation}

Taking into account the belief $p(\mathbf{x})\sim\mathcal{N}\left(\hat{\mathbf{x}}(t),\bm{\psi}(t)\right)$, the expected value and variance of the sample variance are known from the literature [18]:

\begin{align}
\hat{z}_{\text{var}}(t)&= \frac{1}{N-1}\left(\tr\left(\mathbf{M}\bm{\psi}(t)\right) +  \hat{\mathbf{x}}(t)^T\mathbf{M}\hat{\mathbf{x}}(t) \right)\\
\E\left[\nu_{\text{var}}(t)|\bm{\psi}(t)\right]&=\frac{2\tr\left(\mathbf{M}\bm{\psi}(t)^2\right)+ 4\hat{\mathbf{x}}(t)^T\mathbf{M}\bm{\psi}(t)\hat{\mathbf{x}}(t)}{(N-1)^2}.
\end{align}

As for the mse and sample mean, we can now simply derive the scheduler by using this result in (<ref>).

§.§ Statistic-aware Monte Carlo scheduling

We can now consider a generic summary statistic $z$: in the general case, computing the expected mse can be extremely complex, or even impossible in closed form. In order to still provide an approximate scheduler, we consider Monte Carlo sampling to estimate $\E\left[\nu_z(t)|a(t)\right]$. This method consists of drawing $M$ samples from the conditioned multivariate Gaussian distribution $p(\mathbf{x}|\hat{\mathbf{x}})$, with complexity $O(MN^2)$, and is guaranteed to converge to the correct estimate as $M\rightarrow\infty$ thanks to the law of large numbers [19]. This estimate can then be used in (<ref>) to perform scheduling.

The operation of the scheduler is specified in Alg. <ref>: in order to estimate the expected mse when selecting each sensor $n$, it samples from the posterior distribution of the query. First, the scheduler performs the prior update step from (<ref>) and (<ref>), then it draws an outcome to simulate the transmission, with failure probability $\varepsilon_n$. If the simulated transmission was successful, an observation is randomly drawn from a Gaussian distribution with mean $\mathbf{1}_n\hat{\mathbf{x}}$ and variance $\mathbf{1}_n\mathbf{S}(t)\mathbf{1}_n^T$, and the posterior update is performed. We then have the parameters $\hat{\mathbf{x}}$ and $\bm{\psi}$ of the multivariate Gaussian belief distribution of the state, from which we can draw a sample $\mathbf{x}_m$ to compute $u_m=z(\mathbf{x}_m)$. The sample variance over vector $\mathbf{u}$ is then our estimate of $\E\left[\nu_z(t)|a(t)\right]$, and we can simply select the sensor that gives the minimum expected mse.

Monte Carlo scheduling policy
[1]
random$0,1$ $\geq\varepsilon_n$Update successful
$u(n)\gets$z$\mathbf{x}_m$Compute query value
$\nu(n)\gets$var$\mathbf{u}$ Sample variance

Sensor selection frequency for the considered policies.

§ NUMERICAL EVALUATION

In the following, we show the effects of the sampling strategy on different statistics by simulation, using a Monte Carlo approach: we generate a synthetic process, then try to estimate it at the bs using the different schedulers. The systems below represent two highly asymmetric examples, but the strategies we derived are optimal for all observable linear systems. The scenarios are constructed to be stable, i.e., the eigenvalues of the system matrices are all smaller than 1.

§.§ Scenario and Settings

We evolve the system for 100 episodes of 1000 samples each, and the Monte Carlo scheduler computes a total of $M=1000$ samples for each state. We consider two systems with $N=20$ sensors, in which the elements of the update matrix $\mathbf{A}$ are known. In the first scenario, the matrix $\mathbf{A}_1$ is given by:

\begin{equation}
\frac{3}{4}, &\text{if } i=j;\\
-\frac{1}{8},&\text{if } i\neq j, \text{mod}(i-2j,7)=0,
\end{cases}
\end{equation}

where $\text{mod}(m,n)$ is the integer modulo function, and the values are 0 everywhere else. On the other hand, in the second scenario, we have:

\begin{equation}
\frac{4}{5}, &\text{if } i=j;\\
-\frac{1}{9},&\text{if } i\neq j, \text{mod}(\left\lceil i-2.3j\right\rceil,7)=0.
\end{cases}
\end{equation}

The other parameters are the same in both scenarios. We also have $\mathbf{\Sigma_w}=\mathbf{I}$, while the process noise covariance is given by:

\begin{equation}
\Sigma_v^{(i,j)}=\begin{cases}
\frac{11+\text{mod}(i,10)}{5}, &\text{if } i=j;\\
1,&\text{if } i\neq j, \text{mod}(i-j,6)=0.
\end{cases}
\end{equation}

Sensors with higher indices will have a slightly higher variance. The transmission error probabilities are $\varepsilon_n=0.02\left\lceil\frac{n-1}{10}\right\rceil$. The filter is initialized at step 0 with state $\hat{\mathbf{x}}(0)=\mathbf{x}(0)=0$, and $\bm{\psi}(0)=\mathbf{I}$.

In addition to the baseline scheduler, we also consider the well-known maf scheduler as a benchmark. If we denote the age of the last received packet from sensor $n$ as $\Delta_n$, the scheduler always picks the sensor with the highest age:

\begin{equation}
a_{\text{MAF}}^*(t)=\argmax_{n\in\{1,\ldots,N\}} \Delta_{n}(t).
\end{equation}

Finally, we consider four different summary statistics, as well as the state mse: aside from the sample mean and variance, we consider the maximum and count statistics, denoted as $z_{\max}=\max_n x_n(t)$ and $z_{\text{cnt}}$, which is given by:

\begin{align}
z_{\text{cnt}}(\mathbf{x})&=\sum_{n=1}^N \mathbbm{1}(x_n(t)-a)\mathbbm{1}(b-x_n(t)),
\end{align}

where $\mathbbm{1}(x)$ is the step function, equal to 1 if $x\geq0$ and 0 otherwise, and the count interval $[a,b]$. In other words, the count statistic is a simple count of the number of state components that are within $[a,b]$, which we set to $[-5,5]$.

MSE, scenario 1.

Sample variance, scenario 1.

Maximum, scenario 1.

Count, scenario 1.

MSE, scenario 2.

Sample variance, scenario 2.

Maximum, scenario 2.

Count, scenario 2.

Estimation error distributions with the different strategies.

§.§ Results

We can first look at the choices of the schemes aimed at each target metric in one of the episodes for each scenario, shown in Fig. <ref>. As expected, the maf scheduler selects sensors with a similar frequency in both scenarios: sensors with an index over 10 are selected slightly more frequently, as transmission errors occur more often, but the difference is small. On the other hand, the average scheduler only selects two sensors, 7 and 14, in the first scenario, and only the last, sensor 20, in the second: this holds throughout all episodes, independently from the state of the system. In the first scenario, alternating between these two sensors gives the best estimate of the overall average, as the state of each of these two sensors only depends on the other's. In the second scenario, no sensor is isolated, but sensor 20 is the one that affects the average the most. The average scheduler then gets the best estimate it can for the other values, concentrating on these sensors and actually getting a better average performance. Naturally, this results in a significantly worse performance when looking at any other summary statistic. We also remark that all other policies excluding maf never choose sensors 7 and 14 in the first scenario: as errors compound for most of these summary statistics, it does not make sense to choose isolated sensors, as sensors that are more correlated to their neighbors have a better chance to reduce the overall error.
We can also note that, in the second scenario, the mse, count, and sample variance schedulers often make similar choices, while they do not in the first scenario: this similarity is purely due to the specific features of the system, and cannot be relied upon for design.

The cdf of the quadratic estimation errors $\nu_z(t)$ obtained for the various summary statistics are shown in Fig. <ref>. As expected, optimizing the scheduling for a given summary statistic can reduce the error on it, for all the considered statistics. We did not plot the average statistic, as all policies had a similar performance, although, as expected, the average scheduler performed best for that statistic. However, $\pi^*_{\text{avg}}$ was almost always the worst policy when looking at other summary statistics, often by a wide margin, in both scenarios, for the reasons we explained above. The maximum scheduler $\pi^*_{\max}$ was also noticeably worse when looking at the state mse or the interval count, as it tended to pick sensors with a high value, often accepting a larger error on other components of the state. As the maximum value was almost always far over the interval boundaries, the count statistic was also negatively affected.
Finally, the similarity in the behavior of the count, sample variance, and the mse scheduler in the second scenario has a similar performance, as Fig. <ref>-fig:cnt2 show.

In general, the average and count statistics tend to be relatively insensitive to the scheduling policy used, with most policies showing similar results: in both cases, estimation errors tend to compensate, and the error is relatively low. On the other hand, the gain from using the appropriate scheduling strategy is clearly noticeable when looking at the mse and at the maximum statistic. In these cases, individual components of the state can have a disproportionate effects, and errors tend to compound rather than compensate each other. In general, while never being the optimum, the maf scheduler is also never the worst, as it is a purely aoi-oriented approach that does not consider the specific definition of voi.

§ CONCLUSION

In this letter, we have considered the optimization of a sensor polling strategy, using different statistics to define the voi. The difference between the policies can be important, as errors tend to compensate each other in some cases and compound for other statistics, leading to different choices of sensors. Naturally, one-step optimization is a limited approach, and we plan to consider more complex schemes which can take long-term effects into account, as well as different statistics over the same process. Energy consumption is also another important metric, and we plan to compare voi-based strategies to energy-efficient ones and try to find a balance between them.

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