# Dark Matter-induced electron excitations in silicon and germanium with Deep
Learning

Riccardo Catena<EMAIL_ADDRESS>Chalmers University of Technology,
Department of Physics, SE-412 96 Göteborg, Sweden  Einar Urdshals
<EMAIL_ADDRESS>Chalmers University of Technology, Department of Physics,
SE-412 96 Göteborg, Sweden

###### Abstract

We train a deep neural network (DNN) to output rates of dark matter (DM)
induced electron excitations in silicon and germanium detectors. Our DNN
provides a massive speedup of around $5$ orders of magnitude relative to
existing methods (i.e. QEdark-EFT), allowing for extensive parameter scans in
the event of an observed DM signal. The network is also lighter and simpler to
use than alternative computational frameworks based on a direct calculation of
the DM-induced excitation rate. The DNN can be downloaded here:

##  I Introduction

Learning the nature of dark matter (DM) is a key challenge in modern
astroparticle physics. It is known to exist from its gravitational influence
on the visible universe, and plays a crucial role in a multitude of phenomena
across vastily different astrophysical scales Bertone and Hooper (2018).

On cosmological scales, DM initiates the formation of large scale structures,
giving rise to galaxies and galaxy clusters. It also generates the density
fluctuations leading to the observed patterns of anisotropy in the cosmic
microwave background temperature and polarization maps Aghanim  _et al._
(2020). On galactic scales, the influence of DM is visible through
gravitational lensing Clowe  _et al._ (2006), and the flattening of rotation
curves in spiral galaxies Persic  _et al._ (1996).

Despite the crucial role of DM in explaining these phenomena, its nature is
yet to be determined. The leading hypothesis in the field of astroparticle
physics is that DM is made up of new particles, waiting to be discovered
Bertone and Hooper (2018). The identity and properties of these particles do
however still remain to be determined.

The strong gravitational evidence for the presence of large amounts of DM in
the universe contrasts with the lack of a microscopic description for this
invisible and unidentified cosmic component. This contrast makes the pursuit
of the nature of DM a key focus in the present scientific research. There are
several kinds of experiments searching to unveil the nature of DM. An
important family relies on the direct detection technique Drukier and
Stodolsky (1984); Goodman and Witten (1985). These experiments seek to observe
rare interactions between DM particles from the Milky Way and detector
materials placed deep underground in low-background environments.

Traditionally, direct detection experiments have primarily focused on
searching for nuclear recoil events induced by Weakly Interacting Massive
Particles (WIMPs) scattering off target nuclei in crystals or liquid noble
gases Schumann (2019). As a result, these experiments have mainly been
sensitive to DM masses in the GeV – TeV range, as lighter particles would be
unable to generate observable nuclear recoils.

With the absence of direct evidence for WIMPs, efforts have been put in
towards alternative experimental approaches better suited to probe DM
particles with sub-GeV masses Battaglieri  _et al._ (2017). In this context, a
crucial role is played by direct detection experiments designed to detect DM-
induced electronic transitions or electron ejections in materials, offering a
novel avenue for uncovering the nature of DM Essig  _et al._ (2012, 2017);
Agnes  _et al._ (2018); Agnese  _et al._ (2018); Aprile  _et al._ (2019);
Cavoto  _et al._ (2020); Arnaud  _et al._ (2020); Barak  _et al._ (2020);
Aprile  _et al._ (2022); Li  _et al._ (2023); Arnquist  _et al._ (2023); Zema
_et al._ (2024).

To interpret these experiments, a theoretical understanding of DM-electron
interactions in detector materials is needed. In ref. Catena  _et al._ (2020,
2021); Catena and Gray (2023); Catena  _et al._ (2023); Catena and Spaldin
(2024), the interactions DM can have with the electrons bound to silicon and
germanium crystals (as well as individual atoms and graphene targets) were
classified and explored using a non-relativistic effective theory formalism.
We implemented our formalism in QEdark-EFT Urdshals and Matas (2021), which
interfaces with the plane-wave self-consistent field (PWscf) DFT code Quantum
ESPRESSO Giannozzi  _et al._ (2009) and extends the QEdark code Essig  _et
al._ (2016) to the case of general DM-electron interactions. This allowed for
covering several DM models, although an extensive exploration was hindered by
computational costs.

The deep neural network (DNN) developed in this paper is traned on the output
of QEdark-EFT, and serves mainly two purposes. Firstly, it is easier to use
and run than QEdark-EFT. It takes less space on a hard drive and does not
require computation of or access to material response functions. Secondly, it
allows for evaluating a large number of DM-induced electronic transition rates
cheaply on a laptop, and demonstrates the potential DNNs have for speeding up
rate evaluations. Such a speedup will be necessary to perform extensive
parameter scans in the event of detection of a DM signal. For example,
evaluating the rate of electron-hole pair creation in silicon and germanium
for $10^{6}$ DM models takes $30\,\mathrm{s}$ with our DNN, compared to days
with QEdark-EFT.

This work is organised as follows. In sec. II, we review the theory of DM-
electron scattering in materials, focusing on the effective theory approach of
ref. Catena  _et al._ (2021). In sec. III, we introduce our DNN structure and
training strategy. Finally, we report the results we find when testing the
accuracy of our DNN in sec. IV, and conclude in sec. V.

##  II DM induced electron excitations in crystals

$\mathcal{O}_{1}=\mathds{1}_{\chi e}$ | $\mathcal{O}_{9}=i\mathbf{S}_{\chi}\cdot\left(\mathbf{S}_{e}\times\frac{\mathbf{q}}{m_{e}}\right)$
---|---
$\mathcal{O}_{3}=i\mathbf{S}_{e}\cdot\left(\frac{\mathbf{q}}{m_{e}}\times\mathbf{v}^{\perp}_{\rm el}\right)$ | $\mathcal{O}_{10}=i\mathbf{S}_{e}\cdot\frac{\mathbf{q}}{m_{e}}$
$\mathcal{O}_{4}=\mathbf{S}_{\chi}\cdot\mathbf{S}_{e}$ | $\mathcal{O}_{11}=i\mathbf{S}_{\chi}\cdot\frac{\mathbf{q}}{m_{e}}$
$\mathcal{O}_{5}=i\mathbf{S}_{\chi}\cdot\left(\frac{\mathbf{q}}{m_{e}}\times\mathbf{v}^{\perp}_{\rm el}\right)$ | $\mathcal{O}_{12}=\mathbf{S}_{\chi}\cdot\left(\mathbf{S}_{e}\times\mathbf{v}^{\perp}_{\rm el}\right)$
$\mathcal{O}_{6}=\left(\mathbf{S}_{\chi}\cdot\frac{\mathbf{q}}{m_{e}}\right)\left(\mathbf{S}_{e}\cdot\frac{{{\bf{q}}}}{m_{e}}\right)$ | $\mathcal{O}_{13}=i\left(\mathbf{S}_{\chi}\cdot\mathbf{v}^{\perp}_{\rm el}\right)\left(\mathbf{S}_{e}\cdot\frac{\mathbf{q}}{m_{e}}\right)$
$\mathcal{O}_{7}=\mathbf{S}_{e}\cdot\mathbf{v}^{\perp}_{\rm el}$ | $\mathcal{O}_{14}=i\left(\mathbf{S}_{\chi}\cdot\frac{\mathbf{q}}{m_{e}}\right)\left(\mathbf{S}_{e}\cdot\mathbf{v}^{\perp}_{\rm el}\right)$
$\mathcal{O}_{8}=\mathbf{S}_{\chi}\cdot\mathbf{v}^{\perp}_{\rm el}$ | $\mathcal{O}_{15}=i\mathcal{O}_{11}\left[\left(\mathbf{S}_{e}\times\mathbf{v}^{\perp}_{\rm el}\right)\cdot\frac{\mathbf{q}}{m_{e}}\right]$
Table 1: Interaction operators defining the non-relativistic effective theory
of spin 0 or spin 1/2 DM-electron interactions Fan  _et al._ (2010);
Fitzpatrick  _et al._ (2013); Catena  _et al._ (2020). $\mathbf{S}_{e}$
($\mathbf{S}_{\chi}$) is the electron (DM) spin, while $\mathbf{v}_{\rm
el}^{\perp}=\mathbf{v}-\boldsymbol{\ell}/m_{e}-\mathbf{q}/(2\mu_{\chi e})$ is
the transverse DM-electron relative velocity (in the limit of elastic
interactions). Here, $\mu_{\chi e}$ is the DM-electron reduced mass,
$\mathbf{v}$ is the incoming DM particle velocity, $\boldsymbol{\ell}$ is the
initial electron momentum, $m_{e}$ is the electron mass, and $\mathds{1}_{\chi
e}$ is the identity in the DM-electron spin space.

The rate at which DM events occur creating $i$ electron hole pairs in crystals
is given as Catena  _et al._ (2021)

$\displaystyle\mathscr{R}_{i}$
$\displaystyle=\frac{n_{\chi}N_{\text{cell}}}{128\pi
m_{\chi}^{2}m_{e}^{2}}\int_{E_{\mathrm{gap}}+(i-1)\varepsilon}^{E_{\mathrm{gap}}+i\varepsilon}\mathrm{d}(\ln\Delta
E)\int\mathrm{d}q\,q\,\widehat{\eta}\left(q,\Delta E\right)$
$\displaystyle\times\sum_{l=1}^{5}\Re\left(R_{l}^{*}(q,v)\overline{W}_{l}(q,\Delta
E)\right)\,,$ (1)

where $n_{\chi}$ is the number density of DM particles, $m_{\chi}$ and $m_{e}$
are the mass of the DM particle and the electron, $N_{\text{cell}}$ is the
number of unit cells in the crystal and $\Delta E$ and $q$ are the energy and
momentum that the DM particle deposits to the crystal.
$\widehat{\eta}(q,\Delta E)$ denotes the velocity integral Catena  _et al._
(2021), which is performed using the SHM with
$v_{0}=238\,\mathrm{km}/\mathrm{s}$, $v_{e}=250.5\,\mathrm{km}/\mathrm{s}$ and
$v_{\mathrm{esc}}=544\,\mathrm{km}/\mathrm{s}$ Baxter  _et al._ (2021).
$R_{l}$ and $\overline{W}_{l}$ are the DM and material response function
Catena  _et al._ (2021). The energy required to create an additional electron
hole pair $\varepsilon$ and the band gap $E_{\mathrm{gap}}$ depend on the
material and are given by Streetman and Banerjee (2005); Klein (1968)

$\displaystyle\varepsilon=$ $\displaystyle 3.8\,\mathrm{eV}$
$\displaystyle\mathrm{silicon}$ $\displaystyle\varepsilon=$ $\displaystyle
3.0\,\mathrm{eV}$ $\displaystyle\mathrm{germanium}$ $\displaystyle
E_{\mathrm{gap}}=$ $\displaystyle 1.2\,\mathrm{eV}$
$\displaystyle\mathrm{silicon}$ $\displaystyle E_{\mathrm{gap}}=$
$\displaystyle 0.67\,\mathrm{eV}$ $\displaystyle\mathrm{germanium}.$

For fixed material properties and DM velocity distribution, the observable
quantity $\mathscr{R}_{i}$ is a unique function of the $m_{\chi}$ and the DM
response function. The DM response function is in turn specified by the non-
relativistic effective theory couplings. We follow ref. Catena  _et al._
(2021), and expand the matrix element – from which the DM and material
response functions arise – in non-relativistic effective operators as

$\mathcal{M}(\mathbf{q},\mathbf{v}_{\rm
el}^{\perp})=\sum_{j}\left(c_{j}^{s}+c^{\ell}_{j}\frac{q_{\rm
ref}^{2}}{|\mathbf{q}|^{2}}\right)\,\langle\mathcal{O}_{i}\rangle\,,$ (2)

and consider only DM models in which the mediator mass is either much smaller
than the typical momentum transfer, $q_{\text{ref}}=\alpha m_{e}$, or much
larger than $q_{\text{ref}}$. We refer to these two cases as “long range” and
“short range” interactions, respectively. We denote by $c_{j}^{\ell}$ and
$c_{j}^{s}$ the corresponding couplings, and use the index $j$ to label the
operator type. As shown in Catena  _et al._ (2020), if DM has spin 0 or 1/2
there are 14 such effective theory couplings for every mediator mass scenario.
The associated operators are listed in tab. 1. Note that these operators
contain different powers of $q/m_{e}\sim\alpha=1/137$ and
$v^{\perp}_{\mathrm{el}}\sim 10^{-3}$. This leads to different levels of
suppression for the different operators, namely

Operators Suppression parameter $S_{j}$
$\displaystyle\mathcal{O}_{1},\,\mathcal{O}_{4}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{0}\,\left(q/m_{e}\right)^{0}$
$\displaystyle\mathcal{O}_{9},\,\mathcal{O}_{10},\,\mathcal{O}_{11}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{0}\,\left(q/m_{e}\right)^{1}$
$\displaystyle\mathcal{O}_{7},\,\mathcal{O}_{8},\,\mathcal{O}_{12}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{1}\,\left(q/m_{e}\right)^{0}$
$\displaystyle\mathcal{O}_{6}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{0}\,\left(q/m_{e}\right)^{2}$
$\displaystyle\mathcal{O}_{3},\,\mathcal{O}_{5},\,\mathcal{O}_{13},\,\mathcal{O}_{14}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{1}\,\left(q/m_{e}\right)^{1}$
$\displaystyle\mathcal{O}_{15}$
$\displaystyle\left(v^{\perp}_{\mathrm{el}}\right)^{1}\,\left(q/m_{e}\right)^{2}\,.$
(3)

When using the suppression parameter $S_{j}$ to rescale the couplings as
explained below in Sec. III, we set $q/m_{e}=1/137$ and
$v^{\perp}_{\mathrm{el}}=10^{-3}$.

The observable $\mathscr{R}_{i}$ is thus uniquely determined by 29 real
parameters. This problem is well suited to emulate with a feed forward neural
network taking the 29 model parameters as input and outputting
$\mathscr{R}_{i}$ for the $n$ most relevant values of $i$. We restrict
ourselves to consider $i$ ranging from $1$ to $n=4$, as these rate bins
dominate for the DM masses considered here.

$\mathscr{R}_{i}$ varies over several orders of magnitude depending on
$m_{\chi}$ and the effective theory couplings. Rather than outputting
$\mathscr{R}_{i}$ directly, we find it beneficial to output $s$ and
$\mathscr{R}^{\prime}_{i}$ defined as

$\displaystyle s$
$\displaystyle=\frac{\sum_{i}^{n}\ln\left(\mathscr{R}_{i}\right)}{n}\,,$ (4)
$\displaystyle\mathscr{R}^{\prime}_{i}$
$\displaystyle=\ln\left(\mathscr{R}_{i}\right)-s\,.$ (5)

Here, $s$ sets the overall scale of the rate, whereas
$\mathscr{R}^{\prime}_{i}$ gives the shape of the spectrum. This distinction
allows the network to learn the shape given by $\mathscr{R}^{\prime}_{i}$, and
the overall scale of the rate set by $s$ separately. Furthermore, to avoid
terms in Eq. (5) diverging, we consider only values of $m_{\chi}$ for which
$\mathscr{R}_{i}$ is non-zero for all $i$. To avoid parameter space with
$\mathscr{R}_{i}=0$, we only consider

$\displaystyle
m_{\chi}>2\frac{E_{\mathrm{gap}}+\varepsilon(n-1)}{\left(v_{e}+v_{\mathrm{esc}}\right)^{2}}\,,$
(6)

where $v_{e}+v_{\text{esc}}=788\mathrm{km}/\mathrm{s}$ Baxter  _et al._ (2021)
is the maximal speed of Milky Way bound DM particle seen from Earth. For $n=4$
and the properties of silicon and germanium we include DM masses

$\displaystyle 4\,\mathrm{MeV}\leq m_{\chi}\leq$ $\displaystyle
1\,\mathrm{GeV}$ silicon $\displaystyle 3\,\mathrm{MeV}\leq m_{\chi}\leq$
$\displaystyle 1\,\mathrm{GeV}$ $\displaystyle\text{germanium}.$ (7)

As such, the DNN has $29$ real scalars as input and $5$ real scalars as
output.

Figure 1: Training loop designed to avoid over-fitting. Every 20 epochs the
performance of the network on the test data is evaluated, and the network
configuration that performs the best on the test data is being updated and
stored.

##  III DNN Structure and Training

We use a feed forward neural network with $29$ neurons in the flattening input
layer, followed by $6$ dense layers with $64$, $128$, $128$, $64$, $32$, $16$
neurons respectively, and finally a dense output layer with $5$ neurons. This
gives a total of $37701$ trainable parameters, and we use ReLu Agarap (2018)
as the activation function in all the hidden layers. We implement the DNN
using TensorFlow Abadi  _et al._ (2015), and train on two precomputed datasets
of $4.2\times 10^{6}$ combinations of $m_{\chi}$ and effective couplings, one
for silicon and one for germanium. The training is performed using the Adam
Kingma and Ba (2017) optimiser and Mean Square Error as loss function. The
training loop shown in Fig. 1 is used to avoid over-fitting, with the test
datasets consisting of $1.4\times 10^{6}$ combinations of $m_{\chi}$ and
effective couplings. After training we evaluate the network on a third data
set with $10^{5}$ data points.

We generate the dataset by drawing $m_{\chi}$ from a log distribution in the
range given in Eq. (7). We also set $m$ effective couplings different from
$0$, where $m$ is a random integer drawn from a uniform distribution between
$1$ and $28$. We make it random which couplings are different from $0$, and
these are drawn from a uniform distribution between $-1$ and $1$. Having drawn
$c_{j}^{s}$, $c_{j}^{\ell}$ and $m_{\chi}$, $\mathscr{R}_{i}$ is computed
using the code we developed in refs. Catena  _et al._ (2021); Urdshals and
Matas (2021), QEdark-EFT. During the computation, $c_{j}^{s}$ and
$c_{j}^{\ell}$ are adjusted upward by $S_{j}^{-1}$ from Eq. (3). This is done
to avoid $j=1$ and $j=4$ always dominating. Before training the neural network
$m_{\chi}$ is transformed as

$\displaystyle
m_{\chi}\rightarrow\frac{-2}{\mathrm{ln}\left(\frac{m_{\text{min}}}{1\,\mathrm{GeV}}\right)}\mathrm{ln}\left(\frac{m_{\chi}}{1\,\mathrm{GeV}}\right)+1\,,$
(8)

where $m_{\text{min}}=4\,\mathrm{MeV}$ ($m_{\text{min}}=3\,\mathrm{MeV}$) is
the minimal mass considered for Si (Ge). This is done to ensure that the mass
parameter fed to the neural network approximately lies between $-1$ and $1$.

To summarize, the DNN takes in 28 $c_{j}^{s}$ or $c_{j}^{\ell}$ parameters
between $-1$ and $1$ together with a mass parameter between $-1$ and $1$. The
DNN outputs $s$ and $\mathscr{R}_{i}^{\prime}$ from Eq. (5), where
$\mathscr{R}_{i}$ is the rate of electron hole pairs expected to be created by
a DM model with couplings between $-S_{j}^{-1}$ and $S_{j}^{-1}$ and a DM mass
between $4\,\mathrm{MeV}$ ($3\,\mathrm{MeV}$) and $1\,\mathrm{GeV}$ for Si
(Ge).

Figure 2: Comparison between the excitation rates from ref. Catena  _et al._
(2021) (solid lines) and the rates inferred here by the DNN (dashed lines).
The panels to the left correspond to silicon, whereas the panels to the right
correspond to germanium. The blue lines are for $m_{\chi}=5\,\mathrm{MeV}$
while the red lines are for $m_{\chi}=100\,\mathrm{MeV}$. See legends, for
further details about the assumed DM interaction.

##  IV Results

After training, the DNNs are evaluated on $10^{5}$ data points in the
validation set. These are generated in the same way as the training data, and
the network reproduces these data points with an average relative error of
around $2.76\%$ ($2.62\%$) for silicon (germanium). In order to test the
accuracy of our DDN-based computational framework, in Fig. 2 we compare the
electron-hole creation rate computed in ref. Catena  _et al._ (2021) for
anapole, electric dipole and magnetic dipole interactions through a direct
calculation with predictions obtained here using our DNN. These interactions
generate the following non-zero coupling constants Catena  _et al._ (2021):

$\displaystyle c_{8}^{s}$
$\displaystyle=8em_{e}m_{\chi}\frac{g}{\Lambda^{2}}\,,$ (9a) $\displaystyle
c_{9}^{s}$ $\displaystyle=-8em_{e}m_{\chi}\frac{g}{\Lambda^{2}}\,,$ (9b)

for the anapole interaction,

$\displaystyle c_{11}^{\ell}$
$\displaystyle=\frac{16em_{\chi}m_{e}^{2}}{q_{\rm
ref}^{2}}\frac{g}{\Lambda}\,.$ (10)

for the electric dipole interaction, and finally

$\displaystyle c_{1}^{s}$ $\displaystyle=4em_{e}\frac{g}{\Lambda}\,,$ (11a)
$\displaystyle c_{4}^{s}$ $\displaystyle=16em_{\chi}\frac{g}{\Lambda}\,,$
(11b) $\displaystyle c_{5}^{\ell}$
$\displaystyle=\frac{16em_{e}^{2}m_{\chi}}{q_{\rm
ref}^{2}}\frac{g}{\Lambda}\,,$ (11c) $\displaystyle c_{6}^{\ell}$
$\displaystyle=-\frac{16em_{e}^{2}m_{\chi}}{q_{\rm
ref}^{2}}\frac{g}{\Lambda}\,.$ (11d)

for the magnetic dipole interaction. We stress that the DNN has not been
trained on these particular interactions. It is thus remarkable that our DNN
reproduces the associated excitation rates with a good accuracy, as one can
see from Fig. 2. The DNN predictions for magnetic dipole interactions in
silicon exhibit the largest deviation from the actual excitation rate, this
deviation being larger than 30% for $i=3$ and $m_{\chi}=5\,\mathrm{MeV}$. In
all other cases, we find that our DNN reproduces the expected rates of
electron-hole pair creation within an accuracy of about 10% or below. Finally,
in a wide range of DM masses and interactions, the predictions of our DNN
match the results obtained by an explicit rate calculation within an accuracy
which is below 5%.

Focusing on the computational costs, we stress that generating excitation
rates within our DNN-based computational framework is extremely fast. On the
CPU of a laptop, generating $10^{6}$ ($10^{7}$) rates takes roughly $30$
seconds ($300$ seconds), showing that we are in a regime where the computation
time is linear in the number of rate evaluations. This is $5$ orders of
magnitude faster than our code from refs. Catena  _et al._ (2021); Urdshals
and Matas (2021), QEdark-EFT, which relies on pre-computed and tabulated
material response functions.

##  V Summary and conclusion

We trained a feed forward DNN to generate rates of DM-induced electron
excitations in silicon and germanium detectors as an output. We performed this
training with TensorFlow Abadi  _et al._ (2015) using the Adam Kingma and Ba
(2017) optimizer. Both for silicon and for germanium, we trained our DNN on
precomputed training, test and evaluation datasets of $4.2\times 10^{6}$,
$1.4\times 10^{6}$ and $10^{5}$ combinations of DM particle mass and effective
couplings, respectively. The evaluation set was reproduced with an average
relative error of around $2.79\%$ ($2.62\%$) for silicon (germanium). Focusing
on anapole, electric and magnetic dipole DM interactions, we find that our DNN
reproduces the expected rates of electron-hole pair creation within an
accuracy of about 10% or below in a wide range of DM particle masses. In terms
of computational costs, the improvement compared to existing computational
frameworks is remarkable. Generating $10^{6}$ excitation rates with our DNN
takes roughly $30$ seconds on the CPU of a laptop, and is thus $5$ orders of
magnitude faster than with QEdark-EFT Catena  _et al._ (2021); Urdshals and
Matas (2021), which relies on pre-computed and tabulated material response
functions. This massive speedup in the computing time allows us to perform
extensive parameter scans, which are relevant both in the event of an observed
DM signal and in the analysis of the null result reported by the operating
experiments.

###### Acknowledgements.

It is a pleasure to thank Hanna Olvhammar for contributing to the early stages
of this project. We are also grateful to Marek Matas and Nicola A. Spaldin for
their contributions to the development of QEdark-EFT Urdshals and Matas
(2021), which we extensively used in this work. RC acknowledges support from
an individual research grant from the Swedish Research Council (Dnr.
2022-04299) and from the Knut and Alice Wallenberg project “Light Dark Matter”
(Dnr. KAW 2019.0080).

## References

  * Bertone and Hooper (2018) Gianfranco Bertone and Dan Hooper, “History of dark matter,” Rev. Mod. Phys. 90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO] .
  * Aghanim  _et al._ (2020) Nabila Aghanim _et al._ (Planck), “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Clowe  _et al._ (2006) Douglas Clowe, Marusa Bradac, Anthony H. Gonzalez, Maxim Markevitch, Scott W. Randall, Christine Jones, and Dennis Zaritsky, “A direct empirical proof of the existence of dark matter,” Astrophys. J. 648, L109–L113 (2006), arXiv:astro-ph/0608407 [astro-ph] .
  * Persic  _et al._ (1996) Massimo Persic, Paolo Salucci, and Fulvio Stel, “The Universal rotation curve of spiral galaxies: 1. The Dark matter connection,” Mon. Not. Roy. Astron. Soc. 281, 27 (1996), arXiv:astro-ph/9506004 .
  * Drukier and Stodolsky (1984) Andrzej Drukier and Leo Stodolsky, “Principles and Applications of a Neutral Current Detector for Neutrino Physics and Astronomy,” Phys. Rev. D30, 2295 (1984).
  * Goodman and Witten (1985) Mark W. Goodman and Edward Witten, “Detectability of Certain Dark Matter Candidates,” Phys. Rev. D31, 3059 (1985).
  * Schumann (2019) Marc Schumann, “Direct Detection of WIMP Dark Matter: Concepts and Status,” J. Phys. G 46, 103003 (2019), arXiv:1903.03026 [astro-ph.CO] .
  * Battaglieri  _et al._ (2017) Marco Battaglieri _et al._ , “US Cosmic Visions: New Ideas in Dark Matter 2017; Community Report,” FERMILAB-CONF-17-282-AE-PPD-T (2017), arXiv:1707.04591 [hep-ph] .
  * Essig  _et al._ (2012) Rouven Essig, Jeremy Mardon, and Tomer Volansky, “Direct Detection of Sub-GeV Dark Matter,” Phys. Rev. D85, 076007 (2012), arXiv:1108.5383 [hep-ph] .
  * Essig  _et al._ (2017) Rouven Essig, Tomer Volansky, and Tien-Tien Yu, “New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon,” Phys. Rev. D96, 043017 (2017), arXiv:1703.00910 [hep-ph] .
  * Agnes  _et al._ (2018) P. Agnes _et al._ (DarkSide), “Constraints on Sub-GeV Dark-Matter–Electron Scattering from the DarkSide-50 Experiment,” Phys. Rev. Lett. 121, 111303 (2018), arXiv:1802.06998 [astro-ph.CO] .
  * Agnese  _et al._ (2018) R. Agnese _et al._ (SuperCDMS), “First Dark Matter Constraints from a SuperCDMS Single-Charge Sensitive Detector,” Phys. Rev. Lett. 121, 051301 (2018), [erratum: Phys. Rev. Lett.122,no.6,069901(2019)], arXiv:1804.10697 [hep-ex] .
  * Aprile  _et al._ (2019) E. Aprile _et al._ (XENON), “Light Dark Matter Search with Ionization Signals in XENON1T,” Phys. Rev. Lett. 123, 251801 (2019), arXiv:1907.11485 [hep-ex] .
  * Cavoto  _et al._ (2020) G. Cavoto, M. G. Betti, C. Mariani, F. Pandolfi, A. D. Polosa, I. Rago, and A. Ruocco, “Carbon nanotubes as anisotropic target for dark matter,” J. Phys. Conf. Ser. 1468, 012232 (2020), arXiv:1911.01122 [physics.ins-det] .
  * Arnaud  _et al._ (2020) Q. Arnaud _et al._ (EDELWEISS), “First germanium-based constraints on sub-MeV Dark Matter with the EDELWEISS experiment,” Phys. Rev. Lett. 125, 141301 (2020), arXiv:2003.01046 [astro-ph.GA] .
  * Barak  _et al._ (2020) Liron Barak _et al._ (SENSEI), “SENSEI: Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD,” Phys. Rev. Lett. 125, 171802 (2020), arXiv:2004.11378 [astro-ph.CO] .
  * Aprile  _et al._ (2022) E. Aprile _et al._ (XENON), “Search for New Physics in Electronic Recoil Data from XENONnT,” Phys. Rev. Lett. 129, 161805 (2022), arXiv:2207.11330 [hep-ex] .
  * Li  _et al._ (2023) Shuaijie Li _et al._ (PandaX), “Search for Light Dark Matter with Ionization Signals in the PandaX-4T Experiment,” Phys. Rev. Lett. 130, 261001 (2023), arXiv:2212.10067 [hep-ex] .
  * Arnquist  _et al._ (2023) I. Arnquist _et al._ (DAMIC-M), “First Constraints from DAMIC-M on Sub-GeV Dark-Matter Particles Interacting with Electrons,” Phys. Rev. Lett. 130, 171003 (2023), arXiv:2302.02372 [hep-ex] .
  * Zema  _et al._ (2024) V. Zema _et al._ , “Dark Matter-Electron Scattering Search Using Cryogenic Light Detectors,” (2024), arXiv:2402.01395 [hep-ph] .
  * Catena  _et al._ (2020) Riccardo Catena, Timon Emken, Nicola A. Spaldin, and Walter Tarantino, “Atomic responses to general dark matter-electron interactions,” Phys. Rev. Res. 2, 033195 (2020), arXiv:1912.08204 [hep-ph] .
  * Catena  _et al._ (2021) Riccardo Catena, Timon Emken, Marek Matas, Nicola A. Spaldin, and Einar Urdshals, “Crystal responses to general dark matter-electron interactions,” Phys. Rev. Res. 3, 033149 (2021), arXiv:2105.02233 [hep-ph] .
  * Catena and Gray (2023) Riccardo Catena and Taylor R. Gray, “Spin-1 thermal targets for dark matter searches at beam dump and fixed target experiments,” JCAP 11, 058 (2023), arXiv:2307.02207 [hep-ph] .
  * Catena  _et al._ (2023) Riccardo Catena, Timon Emken, Marek Matas, Nicola A. Spaldin, and Einar Urdshals, “Direct searches for general dark matter-electron interactions with graphene detectors: Part II. Sensitivity studies,” Phys. Rev. Res. 5, 043258 (2023), arXiv:2303.15509 [hep-ph] .
  * Catena and Spaldin (2024) Riccardo Catena and Nicola Spaldin, “Linear response theory for light dark matter-electron scattering in materials,” (2024), arXiv:2402.06817 [hep-ph] .
  * Urdshals and Matas (2021) Einar Urdshals and Marek Matas, “QEdark-EFT,” Zenodo (2021), 10.5281/zenodo.4739187.
  * Giannozzi  _et al._ (2009) Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Matteo Calandra, Roberto Car, Carlo Cavazzoni, Davide Ceresoli, Guido L. Chiarotti, Matteo Cococcioni, Ismaila Dabo, Andrea Dal Corso, Stefano de Gironcoli, Stefano Fabris, Guido Fratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco Mauri, Riccardo Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbraccia, Sandro Scandolo, Gabriele Sclauzero, Ari P. Seitsonen, Alexander Smogunov, Paolo Umari, and Renata M. Wentzcovitch, “QUANTUM ESPRESSO: a modular and open-source software project for quantum,” J. Phys.: Condens. Matter 21, 395502 (2009).
  * Essig  _et al._ (2016) Rouven Essig, Marivi Fernandez-Serra, Jeremy Mardon, Adrian Soto, Tomer Volansky, and Tien-Tien Yu, “Direct Detection of sub-GeV Dark Matter with Semiconductor Targets,” JHEP 05, 046 (2016), arXiv:1509.01598 [hep-ph] .
  * Fan  _et al._ (2010) JiJi Fan, Matthew Reece, and Lian-Tao Wang, “Non-relativistic effective theory of dark matter direct detection,” JCAP 1011, 042 (2010), arXiv:1008.1591 [hep-ph] .
  * Fitzpatrick  _et al._ (2013) A. Liam Fitzpatrick, Wick Haxton, Emanuel Katz, Nicholas Lubbers, and Yiming Xu, “The Effective Field Theory of Dark Matter Direct Detection,” JCAP 1302, 004 (2013), arXiv:1203.3542 [hep-ph] .
  * Baxter  _et al._ (2021) D. Baxter _et al._ , “Recommended conventions for reporting results from direct dark matter searches,” Eur. Phys. J. C 81, 907 (2021), arXiv:2105.00599 [hep-ex] .
  * Streetman and Banerjee (2005) B. G. Streetman and Sanjay Banerjee,  _Solid State Electronic Devices_ (Prentice Hall, 2005).
  * Klein (1968) C. A. Klein, “Bandgap dependence and related features of radiation ionization energies in semiconductors,” Journal of Applied Physics 39, 2029–2038 (1968).
  * Agarap (2018) Abien Fred Agarap, “Deep learning using rectified linear units (relu),” arXiv preprint arXiv:1803.08375 (2018).
  * Abadi  _et al._ (2015) Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng, “TensorFlow: Large-scale machine learning on heterogeneous systems,”  (2015), software available from tensorflow.org.
  * Kingma and Ba (2017) Diederik P. Kingma and Jimmy Ba, “Adam: A method for stochastic optimization,” (2017), arXiv:1412.6980 [cs.LG] .