# Aluminium Relaxation as the Source of Excess Low Energy Events in Low
Threshold Calorimeters

Roger K. Romani<EMAIL_ADDRESS>University of California, Berkeley

(May 16, 2024)

###### Abstract

A previously unexplained background called the Low Energy Excess (LEE) has
negatively impacted the reach of a variety of low threshold calorimeters
including light dark matter direct detection and coherent elastic neutrino-
nucleus scattering experiments. The relaxation of stressed aluminium films as
mediated by the motion of dislocations may account for these observations.

††preprint: APS/123-QED

##  I Introduction

Low threshold calorimeters used for rare event searches (e.g. light dark
matter direct detection, coherent elastic neutrino-nucleus scattering) have
observed an excess of low energy phonon events often called the Low Energy
Excess (or LEE) which has constrained their science reach [1, 2, 3]. The rate
of this background has been observed to rise around $\sim$ 100 eV, with eV
scale threshold detectors observing LEE rates approaching the order of Hz.
Individual LEE events have been observed to involve no ionization [4], but to
otherwise mimic a particle interaction in pulse shape [5]. Additionally, this
background has been observed to decrease in rate with time since the cryogenic
system has been cooled down, and to reset at least in part when the detector
is warmed up and re-cooled [5]. Taken in concert, these observations indicate
that the LEE is not a previously unknown particle background, but instead a
detector specific effect. Existing models of low energy backgrounds in
ionization sensitive detectors [6] do not explain either the rate [6] or the
lack of ionization [4] of LEE events.

Recent work has shown that the relaxation of mechanical stress associated with
the detector holding can create events which mimic the LEE in spectral shape
and time dependence [7]. However, even after the detector holding stress was
greatly reduced, a residual LEE component remained, which Ref. [7] attributed
to the relaxation of thermally induced stress in the films from which the
detector readout sensors were constructed.

In this paper, I present a model of how the relaxation of aluminium films can
create events which are consistent with the LEE. As no first-principles model
of LEE events has been proposed, this model represents an important first step
towards a testable model of excess events. This model specifically focuses on
the relaxation of aluminium, due to its wide use in low temperature devices
including TESs [8], KIDs [9], and superconducting qubit based devices [10,
11]. Different materials with similar structure (FCC metals with low Peierls
stresses, e.g. Au, Cu…) may also relax similarly. See Appendix C for further
discussion.

This model may also be applicable to observations of relaxing quasiparticle
burst rates in quantum circuits [12, 13], but focuses on “LEE” observations in
calorimeters used as particle detectors.

##  II Overview

Figure 1:  A sketch of a the model discussed in this paper. An aluminium film
deposited on a silicon (or other crystal) substrate contracts relative to the
substrate when cooled to mK temperatures, stressing the film. The relaxation
of this film is mediated by dislocations (thick lines), which allow the film
to deform through their motion. These dislocations can become stuck or
“pinned” (shown here pinned against a “forest dislocation,” see text),
trapping them in a metastable state. These dislocations can tunnel free at
later times and accelerate across the film. When these dislocations hit the
film-substrate interface, they decelerate quickly, releasing energetic phonons
into the silicon substrate.

In this paper, I present a model showing that dislocation mediated relaxation
in thermally stressed aluminium films produces events which are broadly
consistent with the LEE.

  * •

When deposited on silicon or similar substrates and when cooled to mK
temperatures, differential thermal contraction between the substrate and
aluminium will induce stresses on the order of 100s of MPa [14, 15], well in
excess of the aluminium bulk yield strength.

  * •

The stressed film will partially relax through the motion of dislocations
[15], however, some of these dislocations will become stuck at “pinning” sites
in a metastable state [16, 17, 18].

  * •

These metastable pinned states will later relax through tunneling [19]. The
relaxation rate of these pinned states scales approximately as $1/t$ for a
generic model.

  * •

Once dislocations are released from the pinned state, they will accelerate
towards the film-crystal interface, where their impact with the film-crystal
interface will release energy in the form of phonon bursts. These phonons are
emitted in close proximity to the film-crystal interface, and are primarily
directed down into the crystal, leaving little energy in the film in which the
relaxation took place.

To model the properties of these dislocation relaxation events, I created two
codes (available publicly at Ref. [20]):

  * •

A one dimensional numerical evaluation of the dislocation equations of motion
(see Sec. V and Appendix A) to simulate the spectrum of phonons emitted from a
differential length of dislocation interacting with the film-crystal
interface.

  * •

A Monte Carlo based simulation of the spectrum expected from the relaxation of
many dislocations given an average dislocation density and stress field (see
Sec. VI and Appendix A). This simulation uses the results of the previous
simulation to inform the efficiency of converting stress energy to above-gap
phonons for a given dislocation configuration.

These codes rely on linear elasticity theory, and therefore fail to accurately
model the dynamics of dislocations moving close to the speed of sound or
within several interatomic spacings of the film-crystal interface (see Sec.
V). In this model, dislocations which create the LEE are commonly in this
regime. Additionally, the Monte Carlo based simulation neglects the role of
grain boundaries, which would serve to impede dislocations, changing both the
depinning and phonon emission processes (see Sec. VI for further discussion).
See Appendix D for a discussion of possible future work.

I will conclude by summarizing experimental evidence for this model,
highlighting areas of tension between my simulations and experimental results,
and by discussing the implications for low threshold cryogenic calorimeter
design and operation.

Figure 2:  Sketches of the impact of LEE events caused by aluminium relaxation
in different detector architectures. In each detector type (TES, KID, qubits
e.g. [10, 11]), aluminium (purple) is commonly used in constructing phonon
collection fins, wiring, resonators, and Josephson junctions. This model
predicts that aluminium relaxation (orange stars) creates bursts of phonons
(dark red) in the detector substrates, which in turn create quasiparticles
(red) in the detector superconductors. Depending on the rate and energy scale,
these can appear as individual events, sub-threshold noise, or a residual
quasiparticle population (“quasiparticle poisoning”). Relaxation may also
inject quasiparticles directly into the detector superconductors (see text).
Sketches are not to scale.

##  III Metal Film Stress and Deformation

When thin aluminium films (used to construct e.g. phonon collection fins or
wiring in cryogenic DM detectors) are deposited on common crystalline
substrates (e.g. silicon, sapphire, germanium…) and are cooled to mK
temperatures, they will contract relative to the substrate by a factor of
$\epsilon\approx 4\times 10^{-3}$ [14]. This biaxial strain induces a stress
of $\sigma\approx$ 400 MPa, well in excess of both the $\sim 1-10$ MPa Peierls
stress [21] (inherent to dislocation transport through the crystal lattice)
and 100 - 200 MPa yield stresses measured in thin aluminium films [22].

Metals deform through the motion of dislocations, linelike defects in the
crystal structure which are driven by stress fields. To completely relax a
$\sim 4\times 10^{-3}$ strain, dislocations at a density of $\sim 10^{10}$
cm-2 = (100 nm)-1/2 would need to travel $\sim$ 100 nm [16]. These scales
indicate that dislocations will travel distances on the order of the
dislocation spacing and a significant fraction of the film thickness.
Simulations of this process confirm these estimates, and suggest that the
residual stresses are on the order of 100 MPa [15].

##  IV Dislocation Pinning and Depinning

As they travel through the metal crystal, individual dislocations may become
stuck on obstacles (often called “pinning sites” or “locks”), becoming trapped
in metastable excited states. Both dislocations which are not coplanar with
the moving dislocation (“forest dislocations”) [23] and substitutional
impurity atoms [18] are plausible candidates for these pinning sites. A
detailed comparison of these two pinning mechanisms will be left for future
work, and will not be considered further here, where I will focus on forest
dislocation pinning.

In either case, a dislocation lock will immediately fail when the resolved
shear stress on the dislocation exceeds a critical stress

$\displaystyle\tau_{c}=\alpha\mu
b/L\approx\alpha\frac{(6\mathrm{GPa})}{L/(1\mathrm{nm})}$ (1)

where $\mu$ is the shear modulus, $b$ is the Burgers vector, and $L$ is the
length of the dislocation pulling on the pinning site (longer dislocations
will induce more force on a pointlike pinning site) [18]. $\alpha$ is a
dimensionless constant ranging from $\sim 0.8$ [23] in the case of certain
forest dislocation locks to $\sim 0.01$ in the case of certain substitutional
atom pinning sites [18] which quantifies the strength of the pinning.

Dislocations with an actual resolved shear stress $\tau$ on the pinned
dislocation close to the critical stress $\tau_{c}$ can tunnel from the
metastable pinned state into the free state at arbitrarily low temperatures.
Mott [19] estimates the rate of this tunneling as

$\displaystyle
r(\tau)=\nu_{D}\exp\bigg{(}-2\mathbb{A}a\eta(\tau)\sqrt{\frac{2MW_{0}}{\hbar^{2}}}\bigg{)}$
(2)

where $\eta=(1-\tau/\tau_{c})$, $\nu_{D}$ is the Debye frequency, $\mathbb{A}$
is a constant Mott takes to be unity, $a$ is the interatomic spacing, $W_{0}$
is an activation energy which Mott takes as $\sim 1$ eV, and $M$ is the mass
per unit width of the tunneling entity which I take as the mass of one
aluminium atom (although other estimates imply it could be somewhat lower
[24]). Using these estimates, the tunneling rate becomes

$\displaystyle
r(\tau)\approx\nu_{D}\exp\big{(}-900\alpha\eta(\tau)\big{)}=\nu_{D}\exp(-\Gamma\eta(\tau))$
(3)

Note that different authors give significantly different estimates of these
factors, see e.g. Refs. [25, 24].

The total depinning rate $R(t)$ (where $t$ is the time after cooldown) for a
large number $N$ of pinned dislocations in a metal film can be estimated by
assuming the probability distribution of $\eta$ is flat and ranges between 0
and 1. This is equivalent to the assumption that $\tau$ is randomly
distributed from 0 to $\tau_{c}$ at every pinning site, which appears
reasonable given the pre-stress dislocation starting position (setting
$\tau$), pinned dislocation length (setting $\tau_{c}$), and local stress
field (set by e.g. the location of close by grain boundaries) would be
expected to vary quasi-randomly for each individual pinned dislocation. In
reality, experiments are only sensitive to $\eta$ in a narrow range around
$\eta\approx 0.05$, given the need to relax on time scales comparable to the
length of an experiment. In any case, taking the “flat $\eta$” approximation,
I integrate over $\eta$ to obtain

$\displaystyle R(t)=N\int_{0}^{1}r(\eta)\exp\big{(}-r(\eta)t\big{)}d\eta$ (4)
$\displaystyle=\frac{N}{\Gamma
t}\Big{(}\exp(-\nu_{D}te^{-\Gamma})-\exp(-\nu_{D}t)\Big{)}$ (5)

where the factor in parenthesis approaches one for $\Gamma>>\ln(\nu_{D}t)$ and
$\nu_{D}t>>1$, leaving us with

$\displaystyle R(t)\approx\frac{N}{\Gamma t}$ (6)

Critically, this estimate indicates that the upset rate falls off with $1/t$
rather than exponentially. The effective measured time constant of the LEE
rate will therefore scale with the time at which the measurement was taken
(consistent with observations both in detectors [5] and in quantum circuits
[12]). The $1/t$ scaling only holds assuming that the probability distribution
of $\eta$ is relatively flat; subsequent warm up/cool down cycles (in e.g.
Ref. [5]) may alter this scaling.

Detailed atomistic simulations of the dislocation depinning process are needed
to validate this model, and to improve depinning rate estimates (i.e., to
model $\Gamma$).

##  V Phonon Creation

After unpinning, dislocations will accelerate through the film with
essentially no friction. Both electron and phonon scattering is frozen out due
to the superconductivity of the film and the low temperature, and the
radiation of energy from the dislocation through interactions with the Peierls
potential is generally agreed to be small [26, 27]. Eventually, these
dislocations will hit an obstacle, such as the film-substrate interface, grain
boundaries, or forest dislocations. In general, the deceleration of these
dislocations will be damped through the radiation of phonons [16] .

In this section, I will sketch a classical version of the deceleration process
following Ref. [16]. I will only explicitly consider dislocation interactions
with the film-substrate interface, as this interface should remain essentially
unchanged throughout the dislocation interaction. It is important to emphasize
that this classical continuum approach undoubtedly neglects important effects,
especially the damping and phonon emission from the dislocation within several
$b$ of the interface and around $c_{t}$. Better simulation of this process
(possibly quantizing the phonon emission process as in Ref. [28]) will be left
to future work.

The force per unit length of a dislocation near the film-crystal interface can
be calculated with the “image dislocation” method [29] as

$\displaystyle F^{*}=\frac{dF}{dL}=\tau
b+\frac{\mu_{f}b^{2}}{2\pi}\frac{\mu_{s}-\mu_{f}}{\mu_{s}+\mu_{f}}\frac{1}{x}$
(7)

where $x$ is the distance between the dislocation and film-substrate interface
along the slip plane, $\mu_{f}$ and $\mu_{s}$ are the shear moduli of the film
and substrate respectively, and $\tau$ is the resolved shear stress on the
dislocation due to bulk stress in the film. Ref. [16] gives the effective mass
per unit length of the dislocation as

$\displaystyle m^{*}=\frac{\mu_{f}b^{2}}{4\pi
c_{t}^{2}}\ln\bigg{(}\frac{c_{t}}{\gamma\omega b}\bigg{)}$ (8)

such that at every given point, the effective vibration frequency of the
dislocation (and of emitted phonons) is given by Ref. [16] as

$\displaystyle\omega(x)=\sqrt{\frac{\frac{dF^{*}}{dx}}{m^{*}}}=\frac{c_{t}}{x}\sqrt{2\frac{\mu_{s}-\mu_{f}}{\mu_{s}+\mu_{f}}\frac{1}{\ln\big{(}c_{t}/(\gamma\omega(x)b)\big{)}}}$
(9)

where $\gamma\approx$ 1.78 and $c_{t}$ is the transverse speed of sound in the
film.

The damping force on a vibrating dislocation [16] is

$\displaystyle
F_{\mathrm{rad}}^{*}=\frac{dF_{\mathrm{rad}}}{dL}=\frac{\mu_{f}b^{2}\omega}{8c_{t}^{2}}v$
(10)

In addition to this damping term, the dislocation loses energy through
reductions in it’s effective mass $m^{*}$, such that the total energy radiated
into the phonon system by a differential length of dislocation is

$\displaystyle\frac{\partial E_{\mathrm{rad}}}{\partial
L}=\int\bigg{(}F^{*}_{\mathrm{rad}}v+\frac{1}{2}v^{2}\frac{dm^{*}}{dt}\bigg{)}dt$
(11)

The energy radiated into above gap phonons can be calculated by only summing
energy emitted by the phonon system while $\hbar\omega>2\Delta_{Al}$. In
general, these phonons will cause events in detectors, given their ability to
create the quasiparticles which detection techniques are sensitive to. TES
based detectors have some additional sensitivity to sub-aluminium-gap phonons,
given phonons can be directly absorbed in the TES itself.

Imposing phonon-dislocation momentum conservation (as in Ref. [30]) implies
that phonons which cause the dislocation to decelerate are directed through
the crystal-film interface at close range ($\sim$nm). Simulations show that
$\sim$80-90% of the above gap phonon energy is radiated into the crystal, with
only a small fraction of above-gap energy radiated backwards into the film
(see Figs. 4, 5).

Several effects likely serve to increase the amount of energy deposited into
the phonon system (vs. absorbed locally in the film) compared to the behavior
simulated here. First, the phonon damping given by Ref. [16] was calculated
for dislocations with $v<<c_{t}$. This linear first-order approximation of the
phonon damping likely under-estimates the amount of damping the fast-moving
dislocation experiences while first interacting with the interface, implying
that there is more emission into the crystal than simulated here. Second, any
phonon emitted into the film has a high probability of leaking energy into the
crystal, through either escaping without interacting (given the $\sim\mu$m
mean free paths of close-to-gap phonons [31]) or through downconversion in
which quasi-isotropic secondary phonons are emitted. Finally, this simulation
assumes detectors are only sensitive to above-aluminum-gap phonons, which
neglects the often considerable amount of sub-gap phonon sensitivity in TES
based detectors. Vastly more sub-gap phonons are emitted into the crystal
phonon system compared to into the film (see Fig. 4), meaning that detectors
sensitive to these phonons are expected to see less energy absorbed locally.

Broadly, these events should appear to be approximately consistent with
substrate phonon events (e.g. calibration events, DM interactions) in which
all energy partitions equally between all readout channels observing the same
crystal (see e.g. [32]). Due to the sub-ns time scales involved, they should
also deposit energy quickly compared to the time scales of the sensor.

Figure 3:  The simulated motion and phonon radiation of a typical differential
length of dislocation (starting 50 nm away from the film-substrate interface,
in a 45 MPa resolved shear stress field). Regions where the dislocation is
emitting phonons into the substrate are shaded in orange. The green dotted
line is at energy 2$\Delta_{Al}$; phonons radiated with an energy above this
line (and possibly below, see text) will be measured by detector readout
sensors. Figure 4:  The simulated spectrum of phonons emitted from the
dislocation differential length shown in Fig. 3. The blue spectrum shows the
spectrum of phonons emitted downward into the film-substrate interface, and
red shows the spectrum of phonons emitted upwards into the film. The dotted
grey line is at $2\Delta_{Al}$. Figure 5:  The simulated energy of above-
aluminium-gap phonon energy radiated into the crystal substrate through the
film-substrate interface (Left), and the fraction of emitted above-aluminium-
gap phonons which are directed into the interface, as opposed to into the film
(Right). For clarity, contours on the right plot are cut off at 0.8. Note that
for initial pinning locations beyond $\sim$ 65 nm, the dislocation becomes
supersonic.

Improved simulations (see Appendix D) would greatly aid in more accurately
modeling the dislocation deceleration and phonon emission process, and lead to
more realistic estimates of the amount of energy absorbed in the film as
opposed to emitted into the crystal phonon system. Quantizing the phonon
emission process (using e.g. the dislon approach [28]) may also be important
to more accurately simulating this model.

##  VI Spectrum Simulation

In general, the phonon energy released by a dislocation relaxation event
depends on the resolved shear stress experienced by the dislocation and the
distance between the dislocation pinning point and the interface.

To simulate the spectrum of energies which would be expected for dislocation
relaxation events of this type, I sampled the distribution of possible
dislocation resolved shear stresses and starting locations. Given a
dislocation density of $\rho_{D}$, the distance that an otherwise unobstructed
dislocation was pinned away from the film interface was drawn from the
distribution

$\displaystyle\mathrm{Pr}(h_{\mathrm{max}})_{\tau}=\rho_{D}L(\tau)e^{-h_{\mathrm{max}}/(\rho_{D}L(\tau))}$
(12)

where $L(\tau)$ is the length of the pinned dislocation (see Eqn. 1). Loosely
taking the results in Ref. [15] (which were performed in two rather than three
dimensions and assume a manually set dislocation source density of
$50\mu\mathrm{m}^{-2}$), I simulate three spectra for $\rho_{D}$ = (30 nm)-2,
(50 nm)-2, and (100 nm)-2. These dislocation densities are broadly consistent
with the wide range of values reported in the literature (see for example
Refs. [33, 34, 35]). I assume every dislocation stops when pinned against
another dislocation, and assume that no grain boundaries shorten either the
length or mean free path of dislocations.

Resolved shear stresses $\tau$ were determined by assuming that the tensile
stress $\sigma_{0}$ was constant across from the film, and by sampling over
the orientations of the slip plane relative to the applied tensile stress,
parameterized by the azimuthal angle $\theta$.

$\displaystyle\mathrm{Pr}(\theta)=\cos(\theta)$ (13)
$\displaystyle\tau(\theta)=\frac{\sigma_{0}}{2}\sin(2\theta)$ (14)

Again referencing the results of the simulation in Ref. [15], I take
$\sigma_{0}=120$ MPa.

After drawing values for $h_{\mathrm{max}}$ and $\tau$, the total above gap
phonon energy emitted into the substrate by the dislocation relaxation event
$E_{\mathrm{tot}}$ was drawn for each simulated event (see Fig. 6). The total
rate of events was calculated using Eqn. 6 for a detector measured one week
after cooldown. The total number of pinning sites was estimated to be
$N=A_{\mathrm{Al}}\rho_{D}$, (which is presumably an over-estimate, given
dislocations can also become stuck at grain boundaries). The total event rate
of events which emitted non-zero amounts of above-gap energy was 0.37, 0.54,
and 0.28 Hz of events for $\rho_{D}$ = (30 nm)-2, (50 nm)-2, and (100 nm)-2
respectively.

Figure 6:  The simulated spectrum of phonon energies for dislocation
relaxation events at various dislocation densities and at a bulk stress
$\sigma_{0}=$ 120 MPa for a simulated detector 1 week after cooldown (see
text). Experimental measurements from Refs. [2, 3] are superimposed. Lighter
colored lines correspond to simulated spectra scaled down by 10x and 100x. See
Section VI for details on the simulation and Appendix B for details on the
scaling of experimental results.

Several simplifying assumptions underlie the simulated spectra in Fig. 6,
including that no grain boundaries at the interface shorten the length or mean
free path of dislocations, and that the dislocation density is even across the
entire device. Properties of grains at the interface are difficult to measure,
given commonly used atomic force microscopy only measures the grain size at
the film surface; the grain scale is expected to be significantly different at
the interface [36]. Lacking direct measurement, it seems reasonable to assume
that the mean scale of grains could be smaller than the $\sim$100-200 nm
length of failing dislocations, potentially significantly reducing the
quantity of dislocations which can be released through tunneling. In Fig. 6,
lightly colored, scaled down spectra attempt to capture some of this behavior.

Better comparisons between any given experiment and this model will require
both some knowledge of the grain size distribution at the interface (and
ideally the dislocation density) and improvements to simulations such that the
grain size distribution can be used to more realistically model the event rate
and energy scale (see Appendix D).

##  VII Experimental Evidence

In addition to the well documented observation of low energy excesses in dark
matter direct detection experiments, several pieces of experimental evidence
support this model.

Low temperature relaxation of stressed aluminium through non-thermal processes
(i.e. tunneling) is supported by the literature. Aluminium has been observed
to deform below liquid helium temperatures [37] and over very long periods of
time [38]. This long period deformation has been reported to occur in
individual sudden relaxation events interspersed with long periods where no
relaxation occurs [39], possibly corresponding individual dislocation
relaxation events. In addition to these “quantum creep” type observations,
dislocation tunneling through pinning sites has been observed in “internal
friction” type experiments, which have allowed for the observation of
dislocation pinning and depinning well below the superconducting transition
[34, 40, 25]. Work hardening type effects (i.e. pinning increasing with
increasing dislocation densities) have also been observed [34], strongly
suggesting that dislocation-dislocation interactions play a role in governing
thin film aluminium relaxation. At higher temperatures, thin aluminium films
on silicon have been observed to yield under thermal stress [41].

The interaction of dislocations with phonons is well established through
theory and simulation (see e.g. Refs. [28, 42]). Unfortunately, the regime
described in this paper (the sudden deceleration of dislocations creating high
energy phonons) does not appear to be well-studied.

##  VIII Implications for Detector Design

Events of these types would form both backgrounds and noise (sub-threshold)
events for current generation low temperature detectors which use aluminium
thin films. Reducing the rate of these excess events is therefore key to
performing low background measurements and building detectors with good energy
resolution.

Most obviously, the rate of these events could be reduced by reducing the
amount of aluminium on these devices. Of course, excess non-instrumented metal
films on the device surface also serve to “soak up” phonons, reducing the
energy absorbed in individual sensors from phonon mediated events.

Changes to the material properties of the film could also reduce the impacts
of the LEE. As suggested by Fig. 6, increasing the dislocation density
(through e.g. work hardening) may best serve to decrease the rate of high
energy events. Decreasing interface grain sizes to reduce the number of
dislocations which are long enough to spontaneously relax could have a similar
effect. The opposite approach (annealing to decrease dislocation densities)
seems likely to produce limited improvements, given the simulations shown in
Fig. 6 and that dislocations will naturally be seeded during the cooldown
[15], effectively undoing some of the effects of annealing. Any changes to the
material properties of the aluminium films would need to preserve both a good
coupling to the crystal phonon system and a high quasiparticle diffusion
length, potentially posing difficulties when e.g. increasing dislocation
density or decreasing interface grain sizes.

Finally, attempts could be made to artificially depin these metastable
dislocations by temporarily increasing the stress in the aluminium films on
the device surface. This could be achieved by e.g. flexing the detector
substrate (which would be impractical for thick dark matter detectors) or by
compressing the films from above to increase the von Misses deviatoric stress
in the film. Once the excess film stress was reduced, the films would be left
in a more relaxed state, presumably with fewer dislocation relaxation events.

If eliminating relaxation events from aluminium proves difficult, substituting
aluminium for other materials (see Appendix C) in low temperature calorimeters
may serve to lower the rate of LEE events. Other materials used for e.g.
phonon collection fins would need to have similar superconducting transition
temperatures (to retain sensitivity to similar parts of the phonon spectrum),
to have good quasiparticle propagation, and efficient phonon absorbtion from
the substrate. Finally, detector schemes which use Josephson junctions (e.g.
qubit based detectors [10, 11]) may struggle to completely eliminate aluminium
in their designs, given the favorable properties of aluminium for constructing
Josephson junctions.

##  IX Tension with Experiments

This paper aims to present a proof-of-concept model illustrating that stress
relaxation in low temperature aluminium can create phonon bursts consistent
with the LEE, making simplifying assumptions to more easily model the core
relaxation behavior. Two areas where the model is in tension with experimental
results should be explicitly noted.

First, this model predicts that $\sim$ 10-20 % of phonons from a dislocation
relaxation event are emitted into the film (as opposed to into the crystal).
Multiple experiments including Ref. [32] have observed that events which
primarily couple to the phonon system seem to entirely couple to the phonon
system. While more accurate simulation and experimental effects are likely to
significantly relax this tension (see Section V), future experimental and
phenomenological work should focus on this potential discrepancy.

Second, the simulated spectra in Fig. 6 differ in rate from the experimental
results [3, 2]. As discussed in section VI, more detailed simulation grounded
in a knowledge of the grain size distribution at the film-crystal interface is
needed to more realistically simulate the rate of dislocation relaxation
events. Given realistic assumptions about the scale of grains, this tension
seems likely to relax significantly.

##  X Conclusion

In this paper, I employ a model to show that the relaxation of thermal stress
in aluminium films (as mediated by dislocations) can produce sudden bursts of
phonons broadly consistent with the LEE. This model closely reproduces the
observed rate of LEE events, the qualitative rate time dependence, the
approximate spectrum of event energies, and the primary coupling to the phonon
system observed for LEE events.

Further experiments (e.g. testing the relation between excess rates and
aluminium coverage) as well as phenomenological work (e.g. atomistic modeling
of dislocation depinning through tunneling, the radiation of phonons during
deceleration, and the effects of realistic dislocation/grain configurations at
the film-crystal interface, see Appendix D) is needed to further support or
contradict this model. In light of this work, relaxation processes in other
materials commonly used in low temperature detectors (e.g. tungsten, niobium,
gold) should be more fully considered, as they may be the true source of the
current excess, or may be responsible for events observed in future
experiments.

Recent interest in the links between the low energy excess and the anomalous
decoherence of superconducting qubits [7] suggests that extensions of this
model may also provide an explanation for so-called “quasiparticle poisoning”
[13]. Low energy events which are predominantly absorbed locally in films
(rather than mediated through crystal phonons) would be an especially
important for qubit systems.

###### Acknowledgements.

I wish to thank Daniel Egaña-Ugrinovic, Matt Pyle, Daniel McKinsey, Rouven
Essig, Marco Costa, Michael Williams, Daniel Baxter, Noah Kurinsky and Junwu
Huang for helpful conversations. This research was supported by US Department
of Energy Grant DE-SC0022354, and in part by the Perimeter Institute for
Theoretical Physics. Research at the Perimeter institute is supported through
the Government of Canada through the Department of Innovation, Science and
Economic Development, and by the Province of Ontario through the Ministry of
Colleges and Universities.

##  Appendix A Appendix: Dislocation Motion Numerical Simulation

As described in the Overview, this paper relies on two simple Python-based
simulations [20].

###  A.0.1 Dislocation Motion and Phonon Radiation Simulation

To simulate the motion and phonon radiation from a short length $dL$ of
dislocation impacting the film-crystal interface, the following analytic
equations of dislocation motion (also described in section V) were numerically
solved using SciPy’s “solve_ivp” method. Following Refs. [16, 29], I set

$\displaystyle\omega(x)=\frac{c_{t}}{\gamma
b}\exp\bigg{(}\frac{1}{2}\mathrm{W}\Big{(}-4\gamma^{2}\frac{\mu_{s}-\mu_{f}}{\mu_{s}+\mu_{f}}\frac{b^{2}}{x^{2}}\Big{)}\bigg{)}$
(15) $\displaystyle m^{*}(x)=\frac{dm}{dL}(x)=\frac{\mu_{f}b^{2}}{4\pi
c_{t}^{2}}\ln\bigg{(}\frac{c_{t}}{\gamma b\omega(x)}\bigg{)}$ (16)
$\displaystyle
F_{\mathrm{rad}}^{*}(x,v)=\frac{dF_{\mathrm{rad}}}{dL}(x,v)=-\frac{\mu_{f}b^{2}v\omega(x)}{8c_{t}^{2}}$
(17) $\displaystyle
F^{*}_{\mathrm{interface}}(x)=\frac{dF_{\mathrm{interface}}}{dL}(x)=\frac{\mu_{f}b^{2}}{4\pi}\frac{\mu_{s}-\mu_{f}}{\mu_{s}+\mu_{f}}\frac{1}{x}$
(18) $\displaystyle
F^{*}_{\mathrm{tot}}(x,v)=F_{\mathrm{rad}}^{*}(x,v)+F_{\mathrm{interface}}^{*}(x)+\tau
b$ (19)
$\displaystyle\frac{dx^{2}}{dt^{2}}=\frac{F_{\mathrm{tot}}^{*}(x,v)}{m^{*}(x)}$
(20)

where W is the Lambert W function and other symbols are defined in the main
text. The initial conditions were $x(0)=h$ and $dx/dt(0)=0$, and I took
$b=0.286$ nm, $c_{t}=3040$ m/s, $\mu_{f}=25$ GPa, and $\mu_{s}$ = 51 GPa.

By numerically solving these differential equations, $x(t),v(t),\omega(t)...$
were obtained (see Fig. 3). The power radiated into phonons by a unit length
of dislocation at each moment in time could also be computed using

$\displaystyle
P^{*}_{\mathrm{rad}}(t)=F^{*}_{\mathrm{rad}}(t)v(t)-\frac{1}{2}\frac{dm}{dt}(t)v^{2}(t)$
(21)

assuming that the reduction in effective dislocation mass occurs through the
radiation of phonons as described in the main text and discussed below.

During the deceleration process, phonons will be emitted in one of two
directions: either into the crystal through the crystal-film interface (when
the dislocation is damped while moving towards the interface, i.e.
$F^{*}_{\mathrm{rad}}>0$, i.e. $v<0$) or into the film (when the dislocation
is damped while moving away from the interface, i.e. $F^{*}_{\mathrm{rad}}<0$,
i.e. $v>0$). The total amount of energy radiated in each direction can be
computed by summing

$\displaystyle\frac{dE_{\mathrm{crystal}}}{dL}=E^{*}_{\mathrm{crystal}}=\sum_{v(t)<0}P^{*}_{\mathrm{rad}}(t)\Delta
t$ (22)
$\displaystyle\frac{dE_{\mathrm{film}}}{dL}=E^{*}_{\mathrm{film}}=\sum_{v(t)>0}P^{*}_{\mathrm{rad}}(t)\Delta
t$ (23)

where $\Delta t$ is the time step of the simulation. Similarly, the energy of
phonons emitted at a given time (and therefore the spectrum of emitted
phonons) can be calculated using $E_{\mathrm{phonon}}(t)=\hbar\omega(t)$, with
the sign of $v(t)$ used to determine if these phonons are emitted into the
film or crystal. From this relation, the energy of e.g. above-superconducting-
gap phonons emitted into the crystal can be calculated using

$\displaystyle E^{*}_{\mathrm{crystal,above-
gap}}=\sum_{v(t)<0,\hbar\omega(t)>2\Delta_{Al}}P^{*}_{\mathrm{rad}}(t)\Delta
t$ (24)

where $\Delta_{Al}$ is the superconducting bandgap energy.

By numerically solving this differential equation and performing the above
sums, $E^{*}(h,\tau)$ can be simulated for a grid of realistic $h,\tau$
points.

As noted in the main text, this approach is only a one dimensional analytic
approximation of the true dynamics of the system. See Appendix D for more
details on how these simulations could be more realistically performed in
future work.

Two caveats of this approach warrant explicit recognition. First, as modeled
in Section V, the dislocation effective mass changes as a function of position
above the interface. While decreases in the dislocation kinetic energy through
a decrease in effective mass are easily interpreted as the radiation of
kinetic energy through the emission of phonons, increases in kinetic energy
due to an increase in mass are more difficult to interpret.

Secondly, no “elastorelativistic” corrections are applied in this model,
meaning that the dislocation can accelerate beyond $c_{t}$ for high stresses
and large initial distances between the dislocation and interface. While this
system is somewhat unique in that e.g. dislocation-phonon damping is highly
suppressed, and while supersonic dislocation motion has been experimentally
observed [43], supersonic dislocations remain controversial. Future work
should attempt to more accurately model the motion of the dislocation close to
$c_{t}$.

###  A.0.2 Monte-Carlo Based Spectrum Simulation

After simulating the emitted phonon energy per unit length of dislocation
$E^{*}(h,\tau)$, I calculated the total energy emitted by the entire
dislocation by integrating

$\displaystyle E_{\mathrm{tot}}=\int_{y}E^{*}(h(y),\tau)dy$ (25)
$\displaystyle=\int_{0}^{h_{\mathrm{max}}}E^{*}(h(y),\tau)\frac{dy}{dh}dh$
(26)

In general, the shape $h(y)$ of the pinned dislocation is governed by self-
interaction and interactions with the pinning site, shear stress field, and
the interface, leading to a complex shape likely only determinable using
simulation (see Appendix D). Here, I make the assumption that the dislocation
shape is linear between the pinning site and the interface, meaning that for a
dislocation with total width $L(\tau)$ (see Eqn. 1),

$\displaystyle\frac{dh}{dy}=\frac{h_{\mathrm{max}}}{L(\tau)}$ (27)

Using this assumption and the simulated $E^{*}$,
$E_{\mathrm{tot}}(h_{\mathrm{max}},\tau)$ can be calculated, as shown in Fig.
5.

Changes to this model of the initial dislocation shape $h(x)$ (e.g. assuming
that the dislocation ends are pinned away from the interface, rather than at
the interface) may increase the fraction of event energy emitted into the
crystal phonon system, potentially resolving some of the tension discussed in
section V.

This calculation yields a total energy
$E_{\mathrm{tot}}(h_{\mathrm{max}},\tau)$ of e.g. phonons emitted into the
crystal above the aluminium superconducting bandgap for a given resolved shear
stress $\tau$ and pinning distance $h_{\mathrm{max}}$. Using a Monte Carlo
method, these were used to simulate a spectrum of events expected in a given
device (see section VI). $h_{\mathrm{max}}$ and $\tau$ were drawn from

$\displaystyle\mathrm{Pr}(h_{\mathrm{max}})_{\tau}=\rho_{D}L(\tau)e^{-h_{\mathrm{max}}/(\rho_{D}L(\tau))}$
(28) $\displaystyle\mathrm{Pr}(\theta)=\cos(\theta)$ (29)
$\displaystyle\tau(\theta)=\frac{\sigma_{0}}{2}\sin(2\theta)$ (30)

assuming $\sigma_{0}=$ 120 MPa, and for a variety of realistic $\rho_{D}$ (see
section VI). From these drawn values,
$E_{\mathrm{tot}}(h_{\mathrm{max}},\tau)$ was calculated and converted into a
spectrum (see Fig. 6). These spectra were scaled assuming a total rate of
events given by Eqn. 6 (see section VI).

##  Appendix B Appendix: Scaling Detector Spectra

Comparing simulated results to real data requires scaling to allow for an
equal-footed test. In this appendix, I describe how the data in Refs. [3, 2]
(with data taken from the public repository described in Ref. [1]) was scaled
for Fig. 6. In general, the necessary information (particularly how long the
detector was cold when the measurement was performed and the amount of
aluminium film on the device surface) is not published, requiring some
parameters to be estimated. Better understanding these parameters may resolve
some of the tension between simulation and experimental results seen in Fig.
6.

For the CRESST-III main detector described in Ref. [2], the detector was
assumed to have been cold for 1 year, and the aluminium fins were assumed to
be the same size as the aluminium fins for the CRESST-III light detector (LD)
[44].

For the CPD detector in Ref. [3] the detector was assumed to have been cold
for 8 weeks, and to have 2% of it’s 45.6 cm2 surface area covered by aluminium
[45].

To compensate for the amount of time detectors were cold, the spectra given in
Ref. [1] were scaled up by a factor of (Time Cold)/(1 Week).

##  Appendix C Appendix: Other Materials

This model relies upon free (unpinned) dislocations being able to easily move
around in the crystal lattice under thermally induced stresses, i.e. that the
Peierls stress is low compared to the thermally induced stress. In the case of
aluminium, this assumption holds (see Sec. III).

Copper, gold, and other FCC metals commonly used in low temperature devices
have similarly low Peierls stresses, and might therefore create phonon bursts
similar to those created by aluminium when relaxing at low temperatures. Key
differences between these materials and aluminium include the effective
dislocation mass and the Debye frequency (which would effect the depinning
rate), as well as the speed of sound, Burgers vector length and shear modulus
(which effect the phonon emission process). The strength of dislocation
pinning to either impurities or forest dislocations would also be expected to
vary, which would in turn effect the depinning rate and the scale of depinned
dislocations. Finally, typical dislocation densities in these materials may
differ from those assumed here for aluminium. The extension of this model to
other materials is left for future work. BCC metals with somewhat low Peierls
stresses such as niobium [46] may also behave similarly to aluminium.

Materials with higher Peierls stresses such as tungsten would not be expected
to deform through the mechanisms described in this paper, as their Peierls
stresses greatly exceed thermally induced stresses [46], making dislocation
motion through the crystal lattice difficult. While these materials might
yield after thermal stressing (through e.g. cracking or interface
delamination, see Ref. [47] for the delamination of W on Si), they would not
be expected to deform through the motion of dislocations as described in this
model. Experimentally, thin-film tungsten has been deposited with stresses
varying by several GPa (see e.g. Refs. [48, 49]), suggesting that thermally
induced 100 MPa-scale stress changes are not likely to cause macroscopic
yielding (as has been observed for aluminium [41]).

##  Appendix D Appendix: Future Work

The model presented in this work could be improved by applying more existing
simulation codes. In this section, I suggest areas for future improvement,
ordered similarly to the sections in the main body of this paper.

Detailed predictions of the tunneling rate (that go beyond the simple
estimates of Mott [19] used in this work) would need to rely on more accurate
knowledge of the dislocation-dislocation interaction potential (assuming that
these pinning cites are responsible for observations). This could be
calculated as a function of resolved stress, dislocation-dislocation
orientation, and dislocation distance using a density functional theory based
package such as VASP [50], similar to the work done for impurity pinning sites
[18]. Once these potentials are known, tunneling rates for each configuration
could be calculated.

To more accurately calculate the energy emitted by a relaxation event, a
number of additional pieces of information are needed. First, knowledge of the
distribution of grain sizes at the crystal-film interface is needed. Likely,
this will be provided both from characterization of real films (using e.g.
Transmission Electron Microscopy) but also simulations of film deposition (see
e.g. Refs. [51, 52]).

From this knowledge of the structure of the film, the dislocation seeding and
stressing process during cooldown can be simulated in three dimensions. This
should likely be undertaken with a discrete dislocation dynamics package such
as ParaDIS [53] which would include e.g. boundary conditions which reflect
thin films and realistic grain sizes. This simulation would inform

  * •

The density and stress distribution of locked dislocations

  * •

The shape of the pinned dislocation (which informs the amount of stored energy
and the speed of the unpinned dislocation when impacting the interface)

  * •

The length of the dislocation that will impact the film-crystal interface
(rather than a grain boundary)

Course grained atomistic approaches (see Ref. [54]) could also be employed.
The involvement of experts will be key to selecting the most appropriate
simulation technique.

Finally, the phonon-radiating dislocation impact at the film-crystal interface
could be more accurately simulated using existing codes. Although simulations
of dislocation-phonon interactions based on a course grained atomistic
approach have been performed (see e.g. Ref. [55]), no existing tools seem to
be designed to simulate the radiation of high energy phonons by decelerating
dislocations. Some combination of existing codes and extensions based on the
theory of dislocation-phonon interactions (perhaps using the “dislon” approach
[28]) should be employed to more accurately simulate the spectrum and
direction of phonons radiated during this deceleration process. This
simulation could also be applied to the impact of dislocations on e.g. grain
boundaries or forest dislocations. Impacts with grain boundaries or forest
dislocations are very difficult to imagine understanding analytically, and
could give insight into a range of local effects (e.g. [32]).

This three dimensional simulation of phonon-radiation would start with the
dislocation in a just-unpinned state, track its acceleration towards the grain
boundary under the resolved shear stress field (naturally including
“elastorelativistic” effects), and simulate the radiation of phonons from the
dislocation as it decelerates at the interface. From this, a spectrum of
emitted phonons for a given relaxation event could be calculated. Combined
with knowledge of the film configuration near the interface, this could be
used to create a Monte Carlo type simulation to generate spectra which could
be compared to LEE spectra measured in real detectors.

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# Communication-Critical Planning
via Multi-Agent Trajectory Exchange

Nathaniel Moore Glaser and Zsolt Kira
Georgia Institute of Technology
<EMAIL_ADDRESS>
###### Abstract

This paper addresses the task of joint multi-agent perception and planning,
especially as it relates to the real-world challenge of collision-free
navigation for connected self-driving vehicles. For this task, several
communication-enabled vehicles must navigate through a busy intersection while
avoiding collisions with each other and with obstacles. To this end, this
paper proposes a learnable costmap-based planning mechanism, given raw
perceptual data, that is (1) distributed, (2) uncertainty-aware, and (3)
bandwidth-efficient. Our method produces a costmap and uncertainty-aware
entropy map to sort and fuse candidate trajectories as evaluated across
multiple-agents. The proposed method demonstrates several favorable
performance trends on a suite of open-source overhead datasets as well as
within a novel communication-critical simulator. It produces accurate semantic
occupancy forecasts as an intermediate perception output, attaining a $72.5\%$
average pixel-wise classification accuracy. By selecting the top trajectory,
the multi-agent method scales well with the number of agents, reducing the
hard collision rate by up to $57\%$ with eight agents compared to the single-
agent version.

##  I Introduction

For much of its history, robotics has revolved around single-agent systems.
Many factors perpetuated this singular focus, such as costly hardware, weak
compute power, and spotty inter-agent communication. However, thanks to modern
technological advancements, these prohibitive developmental costs are largely
reduced. With improved capabilities in hand, researchers have been keen to
extend single-agent techniques towards their multi-agent counterparts.

One such extension involves that of deep learning [13, 26]. This class of
techniques can infer semantic information about the world by combining neural
architectures with data. It has seen remarkable success in areas such as
object detection and, more recently, robotic perception and planning [4, 35,
11, 12, 2]. However, again, these algorithms have primarily focused on single-
agent applications.

To balance out this trend, this paper seeks to apply modern deep learning
techniques towards the task of multi-agent perception and planning, especially
within the setting of collision-free trajectory planning for multiple
connected robots. Specifically, we propose a distributed, communication-
enhanced collision avoidance policy that is learned from observations of
multi-agent interactions. We derive a novel trajectory exchange mechanism that
leverages several state-of-the-art learning-based techniques, including image-
to-costmap generation, deep-learned spatial uncertainty, and costmap-based
motion planning.

To achieve this end, we distribute copies of an identical neural architecture
across multiple connected agents. Using this neural network, each agent then
processes its local observations to determine (1) the cost of occupying
different parts of its local state space (i.e. each agent generates a local
costmap) and (2) the perceptual uncertainty associated with that cost (i.e. a
entropy map). We then propose a method whereby a querying agent broadcasts
several motion trajectory candidates for distributed evaluation. Then each
supporting agent evaluates the incoming motion trajectories against its own
egocentric costmap (and corresponding uncertainty map) and returns the sampled
costs (and their uncertainties). Importantly, we propose an uncertainty-aware
method to fuse the costs and uncertainties from multiple agents. Given the
fused cost for each trajectory, the original agent then selects the lowest
cost trajectory for execution.

Figure 1: Task. Collision-free planning via communication. Figure 2: Multi-
Agent Trajectory Exchange. Egocentric observations from multiple connected
vehicles are locally processed into learnable and interpretable costmaps. Each
agent then broadcasts its trajectory candidates for distributed, opportunistic
evaluation across any number of supporting agents. The supporting agents then
return the costs sampled by the trajectory. The ego agent aggregates these
scores using uncertainty-aware fusion and selects the minimum cost trajectory
for execution.

We summarize the contributions of our paper as follows:

  * •

We address a variant on Multi-Robot Collaborative Perception and Planning [16,
17] in which a group of decentralized, but communication-enabled, agents must
avoid collisions as they move through a shared world.

  * •

To address this challenge, we propose an end-to-end learnable Multi-Agent
Trajectory Exchange network, MATE. We contribute novel techniques that extend
the current state of the art, including a bandwidth-efficient, uncertainty-
aware trajectory exchange mechanism.

  * •

We evaluate our algorithm across a suite of open-source datasets [1, 23],
including within a novel, communication-critical simulator, CoBEV-Sim.

##  II Related Work

Several prior works are particularly relevant for collaborative perception and
planning.

One set of prior work [10, 24, 12, 11] addresses how a motion planning policy
may be learned from data, either by interacting with an environment
(reinforcement learning) or by observing an expert’s interactions with an
environment (inverse reinforcement learning). The work of Sadat et al. [24]
shows how a neural motion policy may be learned by observing expert driving
demonstrations. Their system takes a map and lidar sequence as inputs, and it
uses a learned encoder to produce a set of occupancy predictions (e.g.
pedestrian, car, bike) for future timesteps. Next, it uses a learned cost
function to score a set of sampled trajectories against these occupancy
predictions, and the minimum cost trajectory is selected for execution.

Other, more human-centric, works consider the interplay between multiple human
agents in a shared scene. For instance, the works of Kollmitz et al. [12, 11]
leverage physical interaction and inverse reinforcement learning to adapt the
planning behavior of a robot near human actors. However, while the robots in
these works explicitly consider the motion of other actors, they do not
leverage communication to improve their capabilities.

To this end, another set of prior work has addressed the challenge of
collaborative perception through learned communication [17, 16, 32]. Learned
communication is a relatively recent trend where data from different sensing
sources is pegged for exchange, based on a purely learned process. When2Com
[16] leverages a learned handshake communication module to mediate the
exchange of visual information between robotic agents, specifically for multi-
view image classification and semantic segmentation.

A similar set of prior work has addressed collaborative perception through
learned transformation and communication of spatial feature maps [14, 7, 4].
For instance, Disconet [14] uses a convolutional network to compress top-down
2D lidar rasters into an intermediate feature representation. The spatial
cells inside these feature maps are then synchronized across multiple agents
using a spatial collaboration graph.

Using similar communication architectures, more recent works have addressed
the joint task of collaborative perception and planning. Coopernaut [5] uses a
learned encoder module to compress point cloud information into a relatively
compact 2D representation. The 2D representations from multiple allocentric
agents are then transmitted, spatially transformed, and aggregated into the
perspective of a single egocentric agent. Finally, a learned control module
executes a control policy based on the aggregated information. However, these
works only implicitly consider the role of uncertainty when aggregating multi-
agent information.

An exciting new trend involves leveraging deep-learned uncertainty [9, 22, 30,
28]. Pan et al. [22] argue that the softmax confidences produced by
classification networks roughly correlate with the empirical accuracy, as
evaluated across a diverse set of common classification tasks. Converse to
confidence, several works [30, 22, 20] apply Shannon Entropy [25] as a measure
of uncertainty, and demonstrate its application to tasks such as domain
adaptation, active learning, and crowd counting. Moreover, other works [20,
29, 28] leverage entropy as a spatial map, especially for semantic
segmentation tasks. Wang et al. [31] uses an entropy-weighted average for
image fusion.

Our approach draws inspiration from each of these bodies of work. Like prior
work, our method uses an interpretable costmap-based scoring mechanism for
evaluating and selecting trajectories for execution. However, unlike prior
work, our method uses a highly efficient trajectory exchange mechanism to
improve planning performance, instead of transmitting dense representations
(such as spatial feature maps). Moreover, our exchange mechanism explicitly
considers perceptual uncertainty as an intuitive way to fuse the trajectory
scores from multiple robot sources.

Beyond extending the methodology of prior work, our work also advances the
setting of collaborative perception and planning. Several prior works add
their own specialized datasets [15, 23] to the corpus of existing open-source
datasets [6, 27, 8, 34] and closed-source datasets [32, 3, 19, 33, 18].
V2X-Sim [15] primarily focuses on the challenge of collaborative perception,
though it only offers relatively simple planning scenarios. For our work, we
evaluate our algorithm on a dataset split of AutoCastSim [23]. This dataset
specifically targets challenging self-driving situations, such as red light
infringements and left turns across traffic. However, these scenarios lack
diversity and have a limited number of agents available for useful
communication. To address these limitations, our work extends the capabilities
of these datasets with the novel CoBEV-Sim, a simulator which creates a higher
volume of diverse, communication-critical scenarios.

##  III Methodology

Autonomous navigation is an important capability for self-driving vehicles
(SDVs). Prior to any widespread adoption, SDVs must first demonstrate a robust
ability to move safely in a cluttered, dynamic environment. To this end, we
propose a novel approach for performing safe autonomous navigation, especially
in the context of inter-vehicle communication.

###  III-A Task

This work addresses the task of collision-free intersection navigation for
connected robotic agents. Namely, given pose in a global coordinate frame,
local perception, and inter-agent communication, a group of multiple
autonomous robots must plan a sequence of actions that limits dangerous
interactions with the environment and with each other, as shown in Fig. 1.

In this setting, each agent has access to three types of data: (1) global pose
data, (2) local perception data, and (3) data communicated from nearby agents.
For the global pose data, each vehicle has knowledge of its past (and current)
positions and orientations with respect to a global coordinate system (as
provided by GPS or similar). For the local perception data, each vehicle
captures egocentric visual information as it travels through the environment.
For the communicated data, each vehicle may communicate bandwidth-limited
information (as measured in bytes) to other vehicles within a limited range.
These limitations are typically dictated by real-world environmental
constraints. This constrained communication provides the agents with an
indirect ability to perceive beyond their local observations and plan for
future interactions. This setting is further complicated by the presence of
both communicating and non-communicating mobile agents.

###  III-B Model Architecture

As summarized in Fig. 2 and 3, this work leverages several modules to tackle
the task of intersection navigation:

Motion Forecasting. First, as with Chen et al. [2], each agent uses a CNN
$\Phi$ to convert a sequence of aligned 2D Bird’s Eye View (BEV) observations
$\textbf{X}^{T^{-}:0}$ into a sequence of 2D semantic occupancy forecasts
$\textbf{P}^{1:T^{+}}$:

$\textbf{P}_{\\{\text{null},\text{ego},\text{allo}\\}}^{1:T^{+}}=\Phi(\textbf{X}^{T^{-}:0})$
(1)

The output classes include a null, egocentric (ego), and allocentric (allo)
agent class, trained via cross-entropy loss.

Costmap and Uncertainty-Map Generation. Next, each agent converts its 2D
semantic occupancy forecasts into a costmap and entropy map. The costmap is
created by combining its egocentric $\textbf{P}_{ego}$ and allocentric
$\textbf{P}_{allo}$ occupancy forecasts into a binarized occupancy map B using
threshold $\theta$ and then computing a signed distance field (SDF) [21]:

$\textbf{B}^{t}=\max{(\textbf{P}_{ego}^{t},\textbf{P}_{allo}^{t})}>\theta\quad\text{and}\quad\textbf{D}^{t}=\textbf{\text{SDF}}(\textbf{B}^{t})$
(2)

To generate the uncertainty map, Shannon entropy is computed across the
softmaxed class channels for each spatial cell in the semantic occupancy
forecast111In this section, we denote image-structured data with bold
uppercase (e.g. $\textbf{U}^{t}$) and extracted cells in lowercase (e.g.
$u_{x,y}^{t}$)., similar to the procedures described in [20, 29, 28]:

$u_{x,y}^{t}=-\sum_{c}{p_{x,y}^{t}(c)\log{p_{x,y}^{t}(c)}}$ (3)

Trajectory Generation. Each agent generates a set of $N$ trajectory
candidates, $\mathbb{S}=\\{S^{i}\\}_{0}^{N}$, where each trajectory candidate
defines a sequence of agent poses at future timesteps,
$S^{i}=\\{(x_{t},y_{t})\\}_{1}^{T^{+}}$. The trajectories were generated to
cover a wide range of feasible vehicle maneuvers, as dictated by linear and
angular acceleration limits.

Bandwidth-Efficient Trajectory Exchange. Each ego agent ($e$) broadcasts its
candidate trajectories for distributed evaluation by the supporting agents
($a$). Each supporting agent transforms these trajectories to its local frame
using relative pose matrix $R_{e}^{a}$. Next, each supporting agent returns
values sampled from its costmap and entropy map:

$S_{t}^{a,i}=R_{e}^{a}S_{t}^{e,i}=(x(t),y(t))$ (4)
$C_{t}^{a,i}=\textbf{D}_{t}^{a,i}[x(t),y(t)]$ (5)
$U_{t}^{a,i}=\textbf{U}_{t}^{a,i}[x(t),y(t)]$ (6)

Uncertainty-Aware Multi-Agent Cost Fusion. After having its suite of candidate
trajectories scored by the cooperating agents, the ego agent fuses all of
these contributions together via an entropy-weighted average, similar to [31]:

$F^{i}=\sum_{a\in\text{allo}}\sum_{t=1}^{T^{+}}\frac{C_{t}^{a,i}}{U_{t}^{a,i}}$
(7)

Trajectory Prioritization. Finally, the set of trajectories is sorted based on
their fused scores:

$\mathbb{S}_{\text{prioritized}}=[S_{i}|\forall
i\in\operatorname*{arg\,sort}_{i}F^{i}]$ (8)

Figure 3: Overview of qualitative results for a single agent.

###  III-C Dataset

Our problem domain combines several distinct research directions (motion
forecasting, motion planning, communication-criticality, and safety for self-
driving vehicles) all within a collaborative multi-robot setting. Datasets
exist for each of these domains; however, it is challenging to find a single
dataset that fairly unifies all of them. Therefore, we evaluate our approach
using both slightly modified open-access datasets as well as a novel
communication-critical simulator. We seek to demonstrate that, in certain rare
(but important!) scenarios, having access to a team of collaborating agents
(and their local perceptions) improves planning performance.

Namely, we include the Intersection Drone Dataset (InD), AutoCastSim, and our
own Collaborative Bird’s Eye View simulator (CoBEV-Sim) in our analysis. Each
of these datasets is adapted to the same general task: a set of communication-
enabled agents must safely maneuver through a shared environment that also
includes a set of non-communicative agents.

Specifically, we adapt each dataset to a uniform style of egocentric BEV
inputs and outputs222The egocentric BEV perspective is the output of prior
research works [15]. For simplicity, we evaluate our algorithm using this data
format as a starting point.. For each communication-enabled agent, we capture
a limited-range, top-down perspective of the shared environment, centered and
oriented with respect to that agent. These top-down perspectives can take the
form of a 2d lidar scan, lidar scan raster, and local segmentation mask (i.e.
occupancy mask), and we capture them at a constant synchronized frame rate
throughout the duration of each planning scenario. Moreover, we record the
relative pose between each pair of agents, communicating and non-communicating
(the poses of the non-communicating agents are solely used for evaluation). We
separate the observed sequence of top-down rasters and relative poses into two
categories – past data and future data. The past information is used as input,
whereas the future information is used for training and evaluation.

The ultimate task of each dataset is the following: each communicating agent
observes a sequence of past, egocentric occupancy masks; then each agent must
efficiently leverage the inferences of its peers to propose a set of
trajectory candidates that are collision-free for some future time horizon.

####  III-C1 Open-Access Datasets

We port several open-access datasets to our unified collaborative perception
and planning setting. Each of these datasets includes motion trajectories for
scenarios with many cooperating (i.e. non-colliding) agents. The InD dataset
consist of extensive real motion tracks of interacting road users from an
overhead BEV drone perspective. The AutoCastSim dataset consists of the same
data, though synthesized within the CARLA simulator.

####  III-C2 Communication-Critical Simulator

In addition to open-access datasets, we evaluate the performance of our
approach within a novel, communication-critical simulation environment. This
simulator addresses several shortcomings observed across many available
datasets:

Existing datasets are cumbersome. They often require a bulky simulator,
complex API calls, or heavy downloads.

Existing datasets lack diversity. They support relatively few distinct
scenarios, offering insignificant scene diversity.

Existing datasets cover inconsequential settings. There is an imbalance
between common, benign situations (like lane-following) and rare, dangerous
ones (like blind corners).

Existing datasets do not contain interoperable agents. Most datasets focus on
a single ”ego” agent, instead of allowing any agent to become a benefactor of
communication.

Our dataset addresses each of these concerns. It is lightweight,
randomizeable, and customizeable. It is straightforward to seed situations
which are interesting, especially situations which intuitively would benefit
from multi-agent communication. Moreover, each agent can be “put into the
driver’s” seat, and its planning performance can be fairly and uniformly
evaluated.

Specifically, our simulator allows for the dynamic generation of
communication-critical multi-agent planning scenarios. First, a user specifies
the start and goal locations of any number of agents. These agents can be
spawned with a customizeable footprint, either corresponding to mobile road
users (such as a car, truck, or pedestrian) or other communication-enabled
road infrastructure (such as a smart traffic light or building). Next, the
simulator iteratively assigns a randomized straight-line speed profile to each
agent such that it avoids collisions with its neighbors. The assignment
process is repeated until all agents have a non-colliding, goal-reaching
trajectory sequence. Finally, the simulator then renders the local perception
data (2D lidar scan, BEV lidar raster, and BEV semantic segmentation map) for
each of these agents. These lidar scans take into account realistic
limitations, such as sensor range limitations and perceptual occlusions.

Using our simulator, we generate a set of communication-critical scenarios:

CrissCross. This scenario consists of many straight-line trajectories that
chaotically intersect in open space. For each scenario, we randomize the start
and goal positions of $20$ agents. A random subset of these agents is
communication-enabled ($n_{com}=\\{1,5,10,15\\}$ for our experiments), while
the remaining agents are non-communicative ($n_{silent}=20-n_{com}$). This
scenario is communication-critical because agents frequently transit through
the paths of each other, oftentimes from outside of their egocentric
perception range; the communicating agents must rely on collaborators to
enhance their effective observation range.

AutoCastSim. This scenario includes a mixture of the red light infraction and
blind corner cases from the original open-access dataset. Since the original
dataset includes only a small number of nearby collaborating agents ($\sim
3$), we randomly insert an additional set of static and trailing agents to the
planning scene. We include $8$ total agents, with a mix of communicating
agents ($n_{com}=\\{1,2,4,8\\}$) and non-communicating agents
($n_{silent}=8-n_{com}$). These supplementary agents undergo the same
collision-free randomization described earlier, ensuring novel scenarios after
each randomization.

We leverage the randomization component of our simulator during training and
evaluation. During each training epoch, we synthesize $10$ novel planning
scenarios. For evaluation, we use randomization to create a static test suite
of $100$ planning scenarios.

###  III-D Experiments

We evaluate the performance of our proposed model on the InD, AutoCastSim, and
CoBEV-Sim datasets. Our model uses an input BEV sequence to infer future
segmentation masks and to produce a candidate trajectory output. To train the
network, these inferred segmentation masks are compared against the ground
truth segmentation masks using a cross entropy loss function. To test the
network, the inferred segmentation masks are compared against the ground truth
masks to yield accuracy and mean intersection of union (mIoU) metrics. To test
planning performance, the trajectories are “rolled-out” in the dataset
environment, where potential collisions with the environment and its obstacles
are tallied. Both qualitative and quantitative assessments are discussed in
the following section.

##  IV Results

The results presented in the beginning of this section highlight the
performance of single-agent perception and planning, as demonstrated on the
InD dataset. Multi-agent perception and planning is discussed towards the end
of the section. Those results are evaluated on the InD, AutoCastSim, and
CoBEV-Sim datasets.

Figure 4: Sample qualitative results for motion forecasting. Using a past
sequence of egocentric data (not pictured), the motion forecasting module
predicts future state occupancy.

| ForecastTime |
---|---|---
| $t=1$ | $t=2$ | $t=3$ | $t=4$ | $t=5$ | $avg$
Mean Accuracy | 78.0 | 78.3 | 73.0 | 68.2 | 65.0 | 72.5
Mean IoU | 49.3 | 53.4 | 53.1 | 51.1 | 47.5 | 50.9

TABLE I: Segmentation metrics for motion forecasting.

###  IV-A Motion Forecasting

We assess perception performance by comparing the inferred occupancy forecasts
against ground truth future occupancy. Qualitatively, as shown in Fig 4, the
predicted locations of various road users roughly matches what actually
unfolded in the scene. In other words, the occupancy heat maps for the
egocentric agent and dynamic objects constitute reasonable hypotheses for
where different road actors may be found at different points in the future.
Moreover, the model seems to have learned that moving objects maintain their
speed and heading, within some variability. For instance, the motion
forecasting module predicted a diagonal streak of probable locations for the
egocentric agent—since the agent was previously travelling in a straight line
at constant speed, it will likely continue on that path. The streak is likely
stretched in the longitudinal direction to account for uncertainty in
velocity. There is less stretching in the transverse direction likely because
the road surface constrains that variability. Additionally, the model is able
to learn turning behavior and multi-modal behavior, as shown in Fig 3.

Quantitatively, the predicted and ground truth segmentation forecasts are
compared using standard accuracy and mean intersection over union (mIOU)
metrics. As shown in Table I, the motion forecasting module achieves an
average pixel-wise accuracy of $72.5\%$ and mean IoU of $50.9\%$. As expected,
the performance of the motion forecasting module degrades for temporally
distant forecasts.

###  IV-B Single-Agent Trajectory Sorting

The proposed method infers a cost that allows for a set of trajectories to be
scored and prioritized. An indirect way to measure the quality of the cost
function (and hence the overall learned pipeline) is to see how sorting
trajectories by their cost correlates with key metrics, such collision rate.
As shown in Fig 5, the inferred trajectory cost correlates with collision
rate.

Figure 5: Trajectory rank versus cost. The horizontal orange lines show the
average cost of a randomly sampled trajectory. The solid and dotted curves
show the cost of a trajectory of a given rank, averaged across the InD
dataset.

| Dataset | Com | Top-$k$ Collision Rate (%) |
---|---|---|---|---
|  | Agents | [% Change from $n=1$] | $rand$
|  |  | $k=1$ | $k=10$ | $traj$
|  | $n=1$ | 18.3 | 19.2 | 26.0
| AutoCastSim | $n=2$ | 16.2 [-11.2] | 15.8 [-17.3] |
| BlindCorner | $n=4$ | 17.8 [-2.8] | 17.5 [-8.7] |
| RedLight | $n=8$ | 7.9 [-57.0] | 10.2 [-46.6] |
|  | $n=1$ | 42.7 | 41.5 | 41.3
| CoBEV-Sim | $n=5$ | 36.7 [-14.1] | 36.9 [-11.3] |
| CrissCross | $n=10$ | 35.5 [-16.9] | 36.8 [-11.5] |
|  | $n=15$ | 30.0 [-29.7] | 30.9 [-25.6] |

TABLE II: Top-$k$ collision rate with $n$-agent communication. This table
shows the collision rates of our method compared to a random sampling
baseline. The numbers in brackets show the percent change in collision rate as
compared to the single-agent ($n=1$) performance.

###  IV-C Multi-Agent Trajectory Sorting

We evaluate our method on the communication-critical datasets of AutoCastSim
and CoBEV-Sim. These datasets include important scenarios where communication
is crucial to improving safety, such as maneuvering past a blind corner at
high speed. We present our results in terms of top-$k$ collision rates. The
top-$k$ collision rate denotes the frequency of a hard collision as averaged
across the $k$ trajectories that have the lowest cost, as inferred by our
multi-agent trajectory exchange method. Oftentimes, several trajectory
candidates can safely maneuver through a scenario. We include these top-$k$
metrics to capture the performance of both the best trajectory ($k=1$) and the
best set of trajectories ($k=10$).

As shown in Table II, our method highlights several desirable trends. First,
the top-$1$ trajectory outperforms the top-$10$ trajectories, and both
outperform the random trajectory sampler baseline. This trend indicates that
our learned, distributed trajectory prioritization scheme is properly
inferring the cost of trajectory samples, proportional to empirical collision
rates. Second, the collision rate improves with more inter-agent
communication. The top-$1$ trajectory shows a $57.0\%$ decrease and a $29.7\%$
decrease from the single agent collision rate for AutoCastSim and CoBEV-Sim,
respectively. Third, the top-$1$ and top-$10$ collision rates are relatively
similar, indicating that our method has prioritized trajectories such that
downstream tasks can safely choose between the top-$k$ set of trajectories
with minimal penalty, a desirable feature for more downstream planners.
Finally, communication is critical in the CoBEV-Sim dataset, as shown by the
similar collision rates in the non-communicating case ($n=1$), where single-
agent performance matches that of a random trajectory sampler.

For the InD Dataset, our multi-agent trajectory exchange mechanism (with
support from about $4$ agents) yielded a $7.2\%$ decrease in hard collisions
over its single-agent counterpart. The solid and dotted lines of Fig 5 show
the close relative performance of the multi-agent and single-agent algorithms.
This improvement is likely a result of communication being non-essential for
this specific dataset: a single independent agent can easily handle most InD
scenarios.

###  IV-D Bandwidth

Several prior works [14, 5] address multi-agent collaborative perception by
exchanging dense representations of perception information. These
representations, while compressed, still consume significant amounts of
bandwidth, especially since they preserve 2D image structure. Our work takes
bandwidth-efficiency to the extreme. For instance, if all agents share a
common dictionary of trajectory samples (i.e. $80$ trajectory samples with
$15$ 2D pose waypoints), then the egocentric agent only needs to transmit its
relative location (i.e. a single $(x,y,\theta)$ pose); and the allocentric
agents can roto-transform their shared dictionaries of trajectories to the
egocentric perspective and simply broadcast the values that were sampled from
its costmap and uncertainty map. This 1D list of sampled values ($80\times
15\times 2=2400$ floating point values) is significantly more compressed than
the 2D image-based representations of prior work (i.e. DiscoNet [14] with
$\sim 10^{4}$ floating point values).

##  V Conclusion

This paper presents a novel method for addressing collision-free navigation
for connected self-driving vehicles. Namely, it uses a trajectory exchange
mechanism to perform distributed validation of trajectory candidates for any
number of neighboring vehicles. The method exhibits several desirable traits,
including being (1) distributed, (2) uncertainty-aware, and (3) bandwidth-
efficient. The method scales with the number of agents, reducing the collision
rate up to $57\%$ with eight agents compared to the single-agent version.

Future Work. Future work will further investigate the multi-agent setting and
architecture. By demonstrating favorable results for an extensible and
interpretable neural architecture, this work laid a strong foundation for
future multi-agent collaborative planning experiments. Such experiments will
explore the impact of localized sensor noise and degradation, reduced
individual sensor capability, positioning noise, and variable network
connectivity. Each of these parameters adds realism to the multi-agent
planning challenge, and future work will address their impact on the overall
planning performance of the connected vehicle network.

##  VI Acknowledgement

This work was supported by ONR grant N00014-18-1-2829.

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