# Dark Matter Direct Detection with Quantum Dots

Carlos Blanco  Department of Physics, Princeton University, Princeton, NJ
08544, USA  Stockholm University and The Oskar Klein Centre for Cosmoparticle
Physics, Alba Nova, 10691 Stockholm, Sweden  Rouven Essig  C. N. Yang
Institute for Theoretical Physics, Stony Brook University, USA  Marivi
Fernandez-Serra  Department of Physics and Astronomy, Stony Brook University,
Stony Brook, NY 11794-3800, USA  Institute for Advanced Computational Science,
Stony Brook University, Stony Brook, NY 11794-3800, USA  Harikrishnan Ramani
Stanford Institute for Theoretical Physics, Department of Physics, Stanford
University, Stanford, CA 94305, USA  Oren Slone  Department of Physics,
Princeton University, Princeton, NJ 08544, USA  Center for Cosmology and
Particle Physics, Department of Physics, New York University, New York, NY
10003, USA

(September 3, 2024)

###### Abstract

We propose using Quantum Dots as novel targets to probe sub-GeV dark matter-
electron interactions. Quantum dots are nanocrystals of semiconducting
material, which are commercially available, with gram-scale quantities
suspended in liter-scale volumes of solvent. Quantum dots can be efficient
scintillators, with near unity single-photon quantum yields, and their band-
edge electronic properties are determined by their characteristic size, which
can be precisely tuned. Examples include lead sulfide (PbS) and lead selenide
(PbSe) quantum dots, which can be tuned to have sub-eV optical gaps. A dark-
matter interaction can generate one or more electron-hole pairs (excitons),
with the multi-exciton state decaying via the emission of two photons with an
efficiency of about 10% of the single-photon quantum yield. An experimental
setup using commercially available quantum dots and two photo-multiplier-tubes
(PMTs) for detecting the coincident two-photon signal can already improve on
existing dark-matter bounds, while using photodetectors with lower dark-count
rates can improve on current constraints by orders of magnitude.

##  I Introduction

Experimental direct-detection concepts for Dark Matter (DM) below the GeV
scale have been extensively studied in recent years (see, e.g., Essig  _et
al._ (2012a, b); An  _et al._ (2015); Lee  _et al._ (2015); Essig  _et al._
(2016); Hochberg  _et al._ (2016a, b); Derenzo  _et al._ (2017); Bloch  _et
al._ (2017); Hochberg  _et al._ (2017a, 2016c, b); Kahn  _et al._ (2016);
Tiffenberg  _et al._ (2017); Knapen  _et al._ (2018); Hochberg  _et al._
(2018); Knapen  _et al._ (2017); Crisler  _et al._ (2018); Agnes  _et al._
(2018); Bringmann and Pospelov (2019); Griffin  _et al._ (2018); Essig  _et
al._ (2019a); Abramoff  _et al._ (2019); Emken  _et al._ (2019); Trickle  _et
al._ (2019a); Griffin  _et al._ (2019); Trickle  _et al._ (2019b); Coskuner
_et al._ (2019); Barak  _et al._ (2020a); Bernstein  _et al._ (2020); Du  _et
al._ (2022); Mitridate  _et al._ (2021); Griffin  _et al._ (2021a); Coskuner
_et al._ (2021); Berghaus  _et al._ (2021); Aguilar-Arevalo  _et al._ (2022);
An  _et al._ (2013); Angle  _et al._ (2011); Graham  _et al._ (2012); Aprile
_et al._ (2014); Aguilar-Arevalo  _et al._ (2017a); Cavoto  _et al._ (2016);
Kouvaris and Pradler (2017); Robinson (2017); Ibe  _et al._ (2018); Dolan  _et
al._ (2018); Romani  _et al._ (2018); Budnik  _et al._ (2018); Bunting  _et
al._ (2017); Cavoto  _et al._ (2018); Fichet (2018); Emken  _et al._ (2017,
2018); Emken and Kouvaris (2017, 2018); Akerib  _et al._ (2019a); Agnese  _et
al._ (2018a); Settimo (2018); Ema  _et al._ (2019); Akerib  _et al._ (2019b);
Liu  _et al._ (2019a); Armengaud  _et al._ (2019a); Bell  _et al._ (2019); Liu
_et al._ (2019b); Aprile  _et al._ (2019a); Baxter  _et al._ (2019); Aguilar-
Arevalo  _et al._ (2019a); Armengaud  _et al._ (2019b); Aprile  _et al._
(2019b); Kurinsky  _et al._ (2019); Cappiello and Beacom (2019); Catena  _et
al._ (2020); Lin (2019); Blanco  _et al._ (2020); Geilhufe  _et al._ (2019);
Griffin  _et al._ (2021b); Trickle  _et al._ (2020); Hochberg  _et al._
(2021a); Akerib  _et al._ (2021a, b); Liang  _et al._ (2021); Grilli di
Cortona  _et al._ (2020); Ma  _et al._ (2020); Nakamura  _et al._ (2021);
Collar  _et al._ (2021); Hochberg  _et al._ (2021b); Kahn and Lin (2021);
Blanco  _et al._ (2021a); Al-Bakry  _et al._ (2022); Redondo and Raffelt
(2013); Amaral  _et al._ (2020); Arnaud  _et al._ (2020a); Aprile  _et al._
(2016, 2019c, 2021); Cheng  _et al._ (2021); Aguilar-Arevalo  _et al._ (2017b,
2019b); Knapen  _et al._ (2022); Alexander  _et al._ (2016); Battaglieri  _et
al._ (2017); Department of Energy (2019a, b); Essig (2020); Essig  _et al._
(2022)). To date, there are a number of leading search strategies, many of
which focus on DM-electron scattering as an efficient method to extract and
detect small energy deposits following either a scattering or absorption event
with a DM particle. Since the ultimate sensitivity of each strategy is
unknown, and since there are several unexplained low-energy backgrounds exc
that may impact different DM detection techniques in distinct ways, it is
imperative to have a multitude of approaches to search for sub-GeV DM. When
designing an experiment it is beneficial to have a low detection threshold to
probe low-mass DM, excellent control over backgrounds, and a target that can
be scaled up in mass to probe smaller interactions.

Scintillating targets have previously been argued to potentially fulfill these
requirements Derenzo  _et al._ (2017). For example, semiconductor
scintillators such as GaAs, have been proposed as a promising target due to
their low band gap Derenzo  _et al._ (2017, 2018); Vasiukov  _et al._ (2019);
Derenzo  _et al._ (2021); Essig  _et al._ (2020); Derenzo (2022). Other
examples proposed in the literature include directional targets such as
organic crystals Blanco  _et al._ (2021b), as well as targets which can probe
DM-nucleon interactions such as inorganic compounds Budnik  _et al._ (2018);
Essig  _et al._ (2017a) and molecular gas targets Arvanitaki  _et al._ (2018);
Essig  _et al._ (2019b). Finally, molecular liquid scintillators have been
experimentally demonstrated to be rapidly deployable and scalable, and have
already been used to constrain sub-GeV DM Blanco  _et al._ (2020); Miramonti
(2011).

This study advocates for quantum dots (QDs), nanoscopic crystals of
semiconducting material, as another excellent low-threshold scintillating DM
detector target. QDs are commercially available, with gram-scale quantities
suspended in liter-scale volumes of solvent. The basic concept is that DM-
electron scattering or, for the case of bosonic DM, absorption by an electron,
excites an electron-hole pair in the QD which subsequently relaxes by emitting
light. The optical properties of a QD are dependent on its size and shape,
which can be tuned precisely at commercial scales.

QDs have similar optical and electronic properties to molecular scintillators.
Specifically, their optical properties when suspended in a solvent and their
photoluminescence Stokes shift (shift to lower energy of the emission spectrum
compared to the absorbed light) make them very efficient emitters of photons
that can easily escape from a bulk target. Essentially, their relaxation
properties are very much akin to molecular fluorescence. Additionally, QDs
benefit from the DM-scattering and absorption characteristics of bulk
semiconductors; when the DM scattering takes place even modestly above
threshold, the QD’s electronic response is essentially that of a bulk
semiconductor. Practically, an experiment using QDs as a scintillating target
could immediately take advantage of the scalability of the target mass and the
ease of detecting the photon signal (similar to molecular scintillators Blanco
_et al._ (2020, 2021b)), while utilizing the small bandgap of semiconductors.

Following an interaction with DM, an electron-hole pair (an exciton) is
created. If this initial exciton has energy larger than twice the energy gap
of the QD, it can relax by generating several more excitons with lower
energies. This multi-exciton generation can yield multiple time-coincident
photons as a detectable signal, which would have intrinsically low background.
While this would be the most straightforward search strategy for a setup using
currently existing technology, future ultra-low-background photodetectors
capable of sensing single photons could allow one to search for single-photon
fluorescence, which benefits from significantly higher quantum yields.

The paper is organized as follows. In Sec. II, we describe the salient
features of QDs in the context of direct-detection experiments. In Sec. III,
we describe the calculations of the expected signal rates for both scattering
and absorption events. Sec. IV describes some of the expected backgrounds
rates for the proposed setup. Sec. V presents the results for the expected
sensitivity to DM parameter space for the proposed setup. Finally, we conclude
and discuss the main results in Sec. VI. Details on a simple semi-analytic
treatment of the DM-electron scattering rates in QDs as well as a detailed
discussion of calculations used in our results are presented in the
appendices.

##  II Direct Detection with Quantum Dots

Figure 1: Envisioned experimental setup. Spherical quantum dots are in a
colloidal suspension (suspended in solvent) at the maximal concentration such
that the volume remains transparent to florescent photons emitted from the
quantum dots. The bases of the detector are coupled to photodetectors, such as
PMTs, while the remaining surface area is coated with reflective material. The
quantum dots are synthesized with typical diameter, $d$, that is much larger
than the size of an atom. Also shown is the approximate form of the 1s
wavefunction of a spherical quantum dot.

While various types of QDs can be fabricated, this study focuses on colloidal
QDs composed of elements from columns IV and VI in the periodic table (also
known as IV-VI QDs). In particular, for results presented in this study, we
consider PbS QDs, which are commercially available, well-studied, and
infrared-active (although there are many other types of QDs with similar
properties, e.g., PbSe). Individual QDs of this type have near-unity single-
photon quantum yield, and their colloidal suspensions can have bulk
photoluminescent quantum yields as high as 0.5. These colloidal suspensions
are synthesized from solutions of precursors and are available as stable
suspensions of QDs in a solvent with a specific QD concentration and a
specific crystal size Moreels  _et al._ (2009); Hines and Scholes (2003). The
synthesis generally proceeds through the nucleation of monomeric precipitates
of the QD material followed by a period of crystal growth in suspension.
Techniques that control the crystal growth phase can generate suspensions of
QDs that have very precise size characteristics. The result is a suspension of
nanocrystals, each of which consists of hundreds to thousands of atoms.

All calculations and estimates in this study consider the simple case of
spherical QDs, since these are commercially available and well-studied.
However, QDs can also be synthesized in other morphologies which may have
further interesting optical properties such as a higher quantum yield. In
practice, the term “spherical QDs” refers to objects that are only
approximately spherical. In reality, the surface of any QD is covered with
ligands, which serve to passivate the surface (allowing charges from surface-
dangling bonds to be effectively neutralized), while also maintaining the QD
in a colloidal suspension. Importantly, the optical emission characteristics
of colloidal QDs is very sensitive to the surface properties of the crystal.
For this reason and also because of the inherent complexity of QDs, many of
the estimates presented in this study are based on measured quantities in the
literature and not on first-principle calculations.

A DM scattering or absorption event with the electronic ground state of a
colloidal QD results in an excitonic state that is typically well above the
optical gap edge. As will be discussed below, the properties of such excitonic
states are almost identical to those of a bulk semiconductor and therefore
calculating their properties follows directly from well studied techniques
used for semiconductors (see, e.g., Ref. Essig  _et al._ (2016)).

A schematic diagram of the envisioned experimental setup is shown in Fig. 1.
The spherical QDs are in a colloidal suspension at a concentration for which
the volume remains transparent to fluorescent photons emitted from the QDs.
The tank is envisioned as a cylindrical volume whose bases are photon
collection surfaces coupled to photodetectors such as PMTs, while the
remaining surface area is coated with a highly reflective material such as
titanium oxide or sodium silicate Collar (2018). A DM particle entering the
detector volume can interact, either via scattering or absorption, with an
electron in one of the suspended QDs. The energy transferred to the QDs
results in the creation of excitonic states.

After an energetic (hot) exciton has been generated, it can relax via a number
of channels into a band-edge state. Possible exciton relaxation processes are
shown in Fig. 2. Relaxation to the band-edge state can happen either through
the emission of phonons, creating a single optically-gapped exciton at the
band edge (panel (a)), or by generating multiple optically-gapped excitons at
the band edge while also emitting phonons to balance the energy budget (panel
(b)). The latter case is known as multi-exciton generation (MEG) and has been
experimentally shown to be efficient in colloidal QDs Ellingson  _et al._
(2005); Schaller and Klimov (2004); Hardman  _et al._ (2011); Midgett  _et
al._ (2013); Murphy  _et al._ (2006) (this is the analog of impact ionization
in a bulk semiconductor). In either case, the band-edge excitons (either
single or multiple) can radiatively recombine, producing fluorescence photons.
In the event of MEG, there is a competing three-body relaxation process known
as Auger recombination whereby an exciton recombines non-radiatively by
donating its energy to a neighboring charge carrier (panel (c)). This process
efficiently depletes the band-edge multi-exciton state An  _et al._ (2008).
However, since Auger recombination requires at least three carriers, the
single band-edge exciton is safe from this non-radiative de-excitation.

While MEG occurs in bulk semiconductors, strongly confined systems and in
particular PbS QDs are expected to have a higher yield of excitons per
deposited energy above the bandgap. Compared to the case of a bulk
semiconductor, where there is more continuum overlap between the phonon states
and electronic bands, in QDs the electronic band is broken into discrete
states near the band edge, reducing the overlap and making phonon emission
less favorable. Importantly, the existence of real intermediate states between
primary excitation and final radiative emission allows one to separate the
calculation of the DM-electron interaction rate from the calculation of the
photon yield. Experimentally, it allows a separation of the bulk target from
the photodetector.

Figure 2: The radiative and non-radiative transition channels for QDs. (a) The
excitation of a primary hot electron-hole pair (exciton), followed by cooling
via non-radiative phonon emission, which produces a band-edge exciton. The
exciton then spontaneously emits a photon (fluorescence) with quantum yield
QY${}_{\gamma}\approx 0.5$ Yang  _et al._ (2017). (b) The primary hot exciton
undergoes multi-exciton generation (MEG), creating several electron-hole
pairs. The multi-exciton state can spontaneously emit two coincident photons
with quantum yield QY${}_{\gamma\gamma}\approx 0.05$ Beyler  _et al._ (2014).
(c) One of the electron-hole pairs undergoes Auger recombination, exciting one
of the other carriers and depleting the multi-exciton without emitting a
photon. This process is efficient and is the main reason why QYγγ is small in
comparison to QYγ.

The single-exciton radiative quantum yield, QYγ, is an experimental measure of
how many photons are produced per primary electron-hole pair. Since a single
exciton can only radiatively de-excite through the emission of a single
photon, this is an experimental measure of the radiative branching fraction of
the single exciton. Similarly, the bi-exciton (two pairs of excitons)
fluorescence quantum yield, QYγγ, is an experimental measure of how many
coincident two-photon emissions are produced per bi-exciton. For the case of
PbS, the value QYγ is measured to be QY${}_{\gamma}\approx 0.5$ Yang  _et al._
(2017), while QYγγ is estimated at QY${}_{\gamma\gamma}\approx 0.05$ Beyler
_et al._ (2014). As shown in the remainder of this paper, these values,
together with exciton creation and photon collection efficiencies, are large
enough to permit an experimental design based on currently available
technology that could probe significant regions of unexplored DM parameter
space.

One of the salient features of colloidal QDs is the Stokes shift between their
excitation and emission spectra, which is similar to that of molecular
chromophores (see, e.g., Greben  _et al._ (2015)). For this reason, there is
only a small overlap between the excitation and emission spectra, and
fluorescent photons cannot efficiently create excitations in neighboring QDs.
Experimentally, for PbS, this becomes evident in the bulk fluorescence quantum
yield, as discussed below in Sec. III.5 and App. B. Therefore, any
sufficiently dilute bulk suspension of colloidal QDs is essentially
transparent to its emission photons. This feature results in a fluorescent
signal that is proportional to the target volume and allows for easy
scalability of any experimental setup. The emitted photons can be detected by
photodetectors on the surface of the target volume.

The size-dependent optical properties of colloidal QDs makes their diameter,
$d$, a parameter that can be used to optimize the detector’s performance.
Specifically, increasing $d$ decreases the confinement energy and therefore
also the minimal energy required to excite the QD, $\Delta E_{\text{QD}}$.
Additionally, increasing $d$ decreases the Stokes shift Greben  _et al._
(2015) and therefore also decreases the bulk fluorescence quantum yield. For
the case of DM scattering, we choose $d=3.3\,\text{nm}$ PbS QDs, which have a
measured bulk fluorescence quantum yield of about 20% Greben  _et al._ (2015)
and $\Delta E_{\text{QD}}$ approximately equal to the bandgap of silicon, one
of our bulk semiconductor benchmarks. For the case of bosonic DM absorption,
we choose $d=5\,\text{nm}$, since dielectric-function data exists for this
size. Below, we present rate calculations and projected sensitivities for the
envisioned setup.

##  III Quantum Dot Signal Rates

###  III.1 Electronic Structure

Electrons in a periodic and infinite lattice occupy Bloch states given by

$\Psi_{\text{bulk}}(\mathbf{r})=u_{\kappa}(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}\,,$
(1)

where $u_{\kappa}(\mathbf{r})$ respects the translational symmetry of the
lattice, and the exponential is an envelope solution satisfying the time-
independent Schrödinger equation in vacuum. A QD is a nanocrystal whose
characteristic diameter, $d$, is large compared to the lattice spacing.
Therefore, locally, electrons experience a periodic potential. However,
globally, the exact translational symmetry is broken by the finite size of
$d$, and the envelope solution is no longer that given by Eq. (1).
Specifically, the long-wavelength behavior of the electronic system must obey
new boundary conditions set by the surface of the QD.

Electrons in a QD occupy states in which the envelope function satisfies these
new boundary conditions. Assuming a spherical QD, the electron wave function
is approximately given by

$\displaystyle\Psi_{\kappa,n\ell
m}(\mathbf{r})=u_{\kappa}(\mathbf{r})\psi_{n\ell m}(\mathbf{r}),$ (2)

where $\kappa\in\\{c,v\\}$ denotes either the conduction ($c$) or valence
($v$) bands. The function $\psi_{n\ell m}(\mathbf{r})$ is the envelope
wavefunction,

$\psi_{n\ell m}(\mathbf{r})=\mathcal{R}_{n\ell}(r)\mathcal{Y}_{\ell
m}\left(\theta,\phi\right)\,,$ (3)

where $\mathcal{Y}_{\ell m}\left(\theta,\phi\right)$ are spherical harmonics,
and $\mathcal{R}_{n\ell}(r)$ are the radial wavefunctions for a free particle
in a spherical well with infinite walls,

$\mathcal{R}_{n\ell}(r)=\sqrt{\frac{2}{R^{3}}}\frac{j_{\ell}\left(\chi_{n\ell}r/R\right)}{j_{\ell+1}\left(\chi_{n\ell}\right)}\,.$
(4)

Here, $R=d/2$, $j_{\ell}$ are spherical Bessel functions, $\chi_{n\ell}$ is
the $n^{\rm th}$ root of the equation $j_{\ell}(x)=0$, and $\\{n,\ell,m\\}$
are the quantum numbers of the envelope state. Fig. 1 shows a diagrammatic
representation of the radial component of the 1s envelope wavefunction.

The description given in Eqs. (2–4) is most accurate for the lowest lying
states of the conduction and highest states of the valence band, since the
long-wavelength solutions are those most affected by the boundary. This is
known as quantum confinement. The energetic effect of quantum confinement is
to effectively lift the band edge. In a bulk crystal with band gap $\Delta
E_{g}$, the density of states scales with energy as
$\rho_{\text{bulk}}(E)\propto\sqrt{E-\Delta E_{g}}$ above the band edge. In a
QD, this vanishing continuum at the band edges becomes a discrete ladder of
$\psi_{n\ell m}$ states. The energy gap between the highest valence QD state
and the lowest conduction QD state (analogous to the “highest occupied
molecular orbital”-“lowest unoccupied molecular orbital” gap of molecules) is
given by,

$\displaystyle\Delta E_{\rm QD}\approx\Delta
E_{g}+\frac{\pi^{2}}{2R^{2}}\left(\frac{1}{m_{h,{\rm eff}}}+\frac{1}{m_{e,{\rm
eff}}}\right),$ (5)

where the second term is the energy of confinement of the electron-hole state
with effective masses $m_{e,{\rm eff}}$ and $m_{h,{\rm eff}}$, respectively.

Figure 3: The effect of strong quantum confinement, for example in a QD, is
shown diagrammatically. The left panel shows the continuum bands of an
infinite crystal, while the right panel shows the effects of strong
confinement of electrons in a finite crystal, which cause discrete levels to
replace the continuum states at the band edges.

Fig. 3 shows the effects of strong quantum confinement of the continuum bands
of a semiconductor. This strong confinement is achieved when the size of the
QD is much smaller than the Bohr radius of the exciton in the material. In
other words, the size of the QD is smaller than the characteristic size of the
bound electron-hole pair. For example, in PbS (PbSe), the size of the exciton
is $20$ nm ($46$ nm) Stouwdam  _et al._ (2007). In the strong-confinement
regime, the electron and hole behave like uncorrelated particles.

Far away from the band edge, the properties of a QD converge to those of bulk
semiconductors. Specifically, it has been shown experimentally that the
optoelectronic response of PbS and PbSe QDs resemble those of bulk
semiconductors for excitations above $E_{r}\gtrsim 2.5\leavevmode\nobreak\
\text{eV}$ Moreels  _et al._ (2007, 2007); Grinbom  _et al._ (2010). The same
can be shown from a numerical estimate, since the discrete QD density of
states becomes dense and converges onto the bulk density of states even
moderately above the band edges once any nanocluster grows beyond about 100
atoms Kane  _et al._ (1996); Debellis  _et al._ (2017) (a PbS QD with a
diameter of $d=3$ nm contains about 1000 atoms). Heuristically, this means
that short-wavelength states do not depend strongly on the boundary.
Therefore, in practice the approximations of Eqs. (2–4) are not necessary for
computing rates for almost all cases of interest in the current proposal.
Instead, results are taken from more precise numerical codes designed to
predict excitation rates in bulk semiconductors.

For completeness, App. A provides some additional intuition regarding this
point. A semi-analytic model of a QD as a band-edge dipole correction to the
bulk semiconductor is described. It is expected that the leading correction to
the bulk description of the form factor will be this dipole term. The appendix
shows that changes to the dipole behavior of the form factor for an excitation
driven by DM do not significantly affect the scattering rate, since that
region is kinematically forbidden anyway. Namely, we show that in the limit
when $q\gg 1/a\approx 2.5\;\text{keV}$, both the bulk and QD have form factors
that are dominated by momentum transfers localized to single atoms. Therefore,
the rate is essentially independent of the envelope piece of the wave
function. Furthermore, it is expected that when $q\gg 1/R\approx
0.5\;\text{keV}$ and $E_{r}>2.5\;\text{eV}$, the electronic response of the QD
is that of the equivalent bulk semiconductor since the envelope piece of the
wavefunction is a result of the boundary conditions at $R$.

###  III.2 Scattering Rates

The total directionally-averaged differential electronic excitation rate is
given by,

$\frac{dR_{\rm
ex}}{d\ln{E_{r}}}=\frac{N_{t}\rho_{\chi}\overline{\sigma}_{e}}{8m_{\chi}\mu_{e\chi}}\int_{q_{\rm
min}}^{q_{\rm max}}qdq\eta(v_{\rm min})|F_{\rm DM}(q)|^{2}|f_{\rm
QD}(q,E_{r})|^{2},$ (6)

where $N_{t}$ is the number of target unit cells (defined in terms of lattice
sites as opposed to number of electrons because of the normalization of
$f_{\rm QD}$, see discussion below and Ref. Essig  _et al._ (2016)),
$\rho_{\chi}=0.3\,\text{GeV}\,\text{cm}^{-3}$ is the local DM density,
$\mu_{e\chi}$ is the reduced mass of the electron-DM system, $q$ is the
momentum transfer to the QD, and $E_{r}$ is the recoil energy. The value of
$N_{t}$ is related to the density of the target material (the mass per unit
volume of QD material), $\rho_{\rm QD}$, and the target volume, $V$, by

$N_{t}=\rho_{\rm QD}\frac{V}{N_{A}(239.3\,\text{g}\,\text{mol}^{-1})}\,.$ (7)

The normalized cross section, $\overline{\sigma}_{e}$, and DM form-factor,
$F_{\rm DM}(q)$, are related to the free-electron DM scattering matrix element
by Essig  _et al._ (2012a)

$\displaystyle\overline{\left|\mathcal{M}_{\text{free}}(\mathbf{q})\right|^{2}}$
$\displaystyle\equiv\overline{\left|\mathcal{M}_{\text{free}}(\alpha
m_{e})\right|^{2}}\times\left|F_{\rm DM}(q)\right|^{2},$
$\displaystyle\overline{\sigma}_{e}$
$\displaystyle\equiv\frac{\mu_{e\chi}^{2}\overline{\left|\mathcal{M}_{\text{free}}(\alpha
m_{e})\right|^{2}}}{16\pi m_{\chi}^{2}m_{e}^{2}}\,,$ (8)

where $m_{\chi}$, $m_{e}$ are the DM and electron masses, respectively,
$\alpha$ is the fine structure constant, and $\alpha m_{e}$ is the typical
momentum transfer for creation of an exciton in a QD. For the case of a DM
particle that interacts with electrons through a massive mediator whose mass
exceeds that of the DM particle, $F_{\rm DM}=1$. For the case of a light
mediator whose mass is much smaller than the momentum transfer from the DM to
the electron, $F_{\rm DM}=(\alpha m_{e}/q)^{2}$. These two limits for $F_{\rm
DM}$ will be used for all results presented below.

The function $\eta(v_{\rm min}(q))$ is the integrated velocity distribution
defined as in Essig  _et al._ (2016), and $v_{\rm min}(q)$ is the
$q$-dependent minimal DM velocity allowed by

$v_{\text{min}}(q,E_{r})=\frac{E_{r}}{q}+\frac{q}{2m_{\chi}}\,.$ (9)

We evaluate rates assuming a Maxwellian velocity distribution with values for
the velocity of Earth, $v_{\rm Earth}=v_{0}+v_{\odot}=251\text{ km/s}$, local
mean speed $v_{0}=238\text{ km/s}$, escape velocity $v_{\text{\rm
esc}}=544\text{ km/s}$, and solar peculiar velocity $v_{\odot}\approx 13\text{
km/s}$ Baxter  _et al._ (2021). The $q$-integral in Eq. (6) is calculated over
the kinematically allowed region enclosed by $q_{\rm min}\leq q\leq q_{\rm
max}$, where the minimum and maximum values are given by the roots of $v_{\rm
min}(q)=v_{\rm esc}+v_{\rm Earth}$.

Figure 4: The differential excitation rate as a function of recoil energy for
bulk Si (blue), bulk Ge (green), bulk PbS (dashed black), and PbS QDs with a
diameter of d=3.3 nm (black). The normalized cross section is taken to be
$\overline{\sigma}_{e}=10^{-37}\leavevmode\nobreak\ \text{cm}^{2}$ as a
benchmark. Above the multi-exciton-generation (MEG) threshold ($E_{\rm
th}\approx 3.6$ eV), essentially all events are expected to induce at least
two excitons. The left panel shows the rates assuming a momentum-independent
DM form factor $F_{\rm DM}=1$, expected for models with heavy mediators, while
the right panel shows the rates for a $F_{\rm DM}\sim 1/q^{2}$, expected for
models with light mediators.

Finally, $f_{\rm QD}(q,E_{r})$ is the target form factor,

$\displaystyle|f_{\rm QD}|^{2}$ $\displaystyle\equiv$ $\displaystyle\sum_{\rm
fin}\sum_{\rm
init}\delta(E_{r}-(E_{\kappa_{f},n_{f}\ell_{f}m_{f}}-E_{\kappa_{i},n_{i}\ell_{i}m_{i}}))$
(10) $\displaystyle\left|\int
d^{3}r\,e^{-i\mathbf{q}\cdot\mathbf{r}}\Psi_{\kappa_{f},n_{f}\ell_{f}m_{f}}^{*}(\mathbf{r})\Psi_{\kappa_{i},n_{i}\ell_{i}m_{i}}(\mathbf{r})\right|^{2},$

where the sum is over populated initial states within a unit cell and all
final states, and $E_{\kappa,n\ell m}$ are the energies of $\Psi_{\kappa,n\ell
m}$. This form factor parameterizes the inelastic scattering probability of
driving the transition
$\\{\kappa_{i},n_{i},\ell_{i},m_{i}\\}\rightarrow\\{\kappa_{f},n_{f},\ell_{f},m_{f}\\}$,
while imparting momentum $\mathbf{q}$.

As we discuss below, final states considered here are always sufficiently far
from the band edge; the electronic structure of the QD is modeled using QE-
dark Essig  _et al._ (2016) as the equivalent bulk semiconductor, with a
scissor correction applied to adjust the band gap to be $\Delta E_{\rm
QD}(d)$. This correction essentially shifts the valence and conduction states
of the bulk semiconductor such that the optical gap matches that predicted by
$\Delta E_{\rm QD}(d)$. It is expected that for $q\gg 2/d$ the interaction
will be far more local than the size of the envelope of the edge states.
Therefore, in addition to the arguments given above for the bulk-like
electronic response above $E_{r}\gtrsim 2.5\;\text{eV}$, one expects that for
$q\gg 0.5\;\text{keV}$ the $q$-dependence of the form factor will be dominated
by the local bulk-like micro-physics encoded in $u_{\kappa}(r)$.

For calculations in this study we account for experimental irregularities in
the QDs’ size and surface which could alter the ideal $\Delta E_{\rm QD}$ in
Eq. 5, by using the experimentally determined size-dependent expression for
$\Delta E_{\rm QD}$ Greben  _et al._ (2015),

$\displaystyle\frac{\Delta E_{\rm QD}}{{\rm eV}}=\frac{\Delta E_{g}}{{\rm
eV}}+\left(\frac{1}{0.0252(d/{\rm nm})^{2}+0.283(d/{\rm nm})^{3}}\right).$
(11)

For the case of PbS, the bulk band gap is $\Delta E_{g}=0.41\,\text{eV}$.

The numerical form factor is calculated using a plane-wave basis as in Ref.
Essig  _et al._ (2016),

$\displaystyle|f_{\rm QD}(q,E_{r})|^{2}$
$\displaystyle=\frac{2\pi^{2}}{E_{r}\alpha m_{e}^{2}V_{\rm
cell}}\sum_{\kappa,\kappa^{\prime}}\int_{\rm BZ}\frac{V_{\rm
cell}^{2}d^{3}\mathbf{k}d^{3}\mathbf{k^{\prime}}}{(2\pi)^{6}}$ (12)
$\displaystyle\times
E_{r}\delta(E_{r}-E_{\kappa^{\prime}}(\mathbf{k}^{\prime})+E_{\kappa}(\mathbf{k}))$
$\displaystyle\times\sum_{G^{\prime}}q\delta(q-|\mathbf{k}^{\prime}-\mathbf{k}+\mathbf{G}^{\prime}|)|f_{[\kappa,\mathbf{k},\kappa^{\prime},\mathbf{k}^{\prime},\mathbf{G}^{\prime}]}|^{2}\,,$

with the function
$f_{[\kappa,\mathbf{k},\kappa^{\prime},\mathbf{k}^{\prime},\mathbf{G}^{\prime}]}$
given by

$\displaystyle
f_{[\kappa,\mathbf{k},\kappa^{\prime},\mathbf{k}^{\prime},\mathbf{G}^{\prime}]}=\sum_{\mathbf{G}}\tilde{u}_{\kappa^{\prime}}^{*}(\mathbf{G}^{\prime}+\mathbf{G}+\mathbf{k^{\prime}})\tilde{u}_{\kappa}(\mathbf{G}+\mathbf{k})\,.$
(13)

In the above equations, $\mathbf{k}$ and $\mathbf{k}^{\prime}$ are the initial
and final state momenta in the first Brillouin zone, $\mathbf{G}$,
$\mathbf{G}^{\prime}$ are reciprocal lattice vectors, $V_{\rm cell}$ is the
volume of the unit cell, and $\tilde{u}(\mathbf{k})$ is the $k$-space Fourier
transform of $u(\mathbf{r})$,

$\displaystyle\tilde{u}_{\kappa}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{V}}\int_{0}^{\infty}u_{\kappa}(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}}d^{3}\mathbf{r}\,.$
(14)

We take contributions from the 20 closest bands to the Fermi-level, with a
kinetic energy cutoff of 272 eV (20 Ry), using norm-conserving non-
relativistic pseudo-potentials for Pb and S.

Fig. 4 shows the calculated differential excitation rates for bulk Si ($\Delta
E_{g}=1.1\;\text{eV}$), bulk Ge ($\Delta E_{g}=0.67\;\text{eV}$), bulk PbS
($\Delta E_{g}=0.41\;\text{eV}$), and d=3.3 nm PbS QDs (for which $\Delta
E_{\rm QD}=1.2\;\text{eV}$), for a DM particle mass of $m_{\chi}=10$ MeV. As
will be discussed below, the threshold energy for generation of MEG states is
approximately $3\times\Delta E_{\rm QD}$, i.e., approximately $3.6$ eV for the
case of PbS. From the results of Fig. 4 it is evident that even at the
relatively small DM mass of $10$ MeV, there is a significant rate well above
this threshold energy. Therefore, for these examples the peak rate is expected
to efficiently induce MEG states.

As will be discussed in Sec. III.6, we consider signal events with electron
recoil energies in the range $2.5-10$ eV. This choice is conservative, since
it is sufficiently above the band edge so as not to be impacted by the strong
confinement effects of the QD, and it is sufficiently small to avoid highly
energetic final states, which will not emit fluorescence photons. Moreover, in
this energy range, the calculation of the rate with QE-dark Essig  _et al._
(2016) is expected to agree reasonably well with other approaches, which
either include screening effects Knapen  _et al._ (2021); Hochberg  _et al._
(2021a); Knapen  _et al._ (2022) or the effect of core electrons Griffin  _et
al._ (2021a).

###  III.3 Bosonic Absorption Rates

In addition to scattering events, QDs are also sensitive to the absorption of
bosonic DM by electrons (this was first considered for non-quantum-confined
systems in Pospelov  _et al._ (2008); An  _et al._ (2015)). For such
interactions, the energy transfer to the target is equal to the entire DM
particle mass. Therefore, since the typical threshold energies of QDs are in
the $\sim$ eV range, this is also the natural DM mass scale to which such an
experiment will be sensitive. Bosonic candidates such as scalars,
pseudoscalars, and vectors (e.g., a dark photon) have previously been
considered in the literature. However, we find sensitivity to unexplored
parameter space—in particular regions that are not already ruled out by
stellar cooling bounds—only for the case of dark photons. Therefore, we
restrict our discussion to that model.

The dark photon, $A^{\prime}$, is assumed to couple to the standard model
photon, $A$, via the kinetic mixing portal, i.e.,

$\mathcal{L}\supset\kappa F^{\mu\nu}F^{\prime}_{\mu\nu}\,,$ (15)

where $F^{\mu\nu}$ and $F^{\prime}_{\mu\nu}$ correspond to the standard model
and dark photon field strengths, respectively. Due to this simple structure,
the dark photon absorption cross-section in any material can be computed from
the empirical value of the standard model absorption cross-section through
rescaling arguments, while taking into account in-medium effects. The rate for
dark photon absorption per target atom is given by Mitridate  _et al._ (2021);
Knapen  _et al._ (2022)

$\sigma_{A^{\prime}}v_{\rm
DM}=\kappa^{2}\frac{1}{|\epsilon|^{2}}\sigma_{A}\,.$ (16)

Here, $\sigma_{A}$ is the standard model photon absorption cross-section,
$\epsilon$ is the dielectric function that captures the in-medium effects, and
$v_{\rm DM}$ is the DM velocity. Both $\sigma_{A}$ and $\epsilon$ can be
obtained from experimental data; this is readily available for QDs with $d=5$
nm (although not for $d=3.3$ nm). We take $\sigma_{A}$ from Ref. Jahromi and
Moaddeli (2019) for $m_{A^{\prime}}\lesssim 5.5\leavevmode\nobreak\
\textrm{eV}$ and from Ref. Kanazawa and Adachi (1998) for higher masses. While
these were measured for bulk PbS, this is justified for QDs, since QDs are
expected to have bulk-like behavior for absorption above $\sim
2.5\leavevmode\nobreak\ \textrm{eV}$ Moreels  _et al._ (2009). The dielectric
function is obtained from Ref. Hechster and Sarusi (2017).

###  III.4 Excitations of Quantum Dots

If the QD is initially in its ground state and an amount $\Delta E_{\rm QD}$
or more of energy is transferred in an interaction event, an electron-hole
pair is created. If the energy transfer is below $2\times\Delta E_{\rm QD}$, a
single excitonic final state can be created, which has the potential to decay
into a detectable photon. Theoretically, if the transferred energy exceeds
$2\times\Delta E_{\rm QD}$, a multi-exciton state is possible. However, it has
been shown experimentally that for Pbs (and for PbSe), the threshold for MEG
is closer to $3\times\Delta E_{\rm QD}$ Midgett  _et al._ (2013); Schaller and
Klimov (2004); Ellingson  _et al._ (2005); Hardman  _et al._ (2011).

It has also been shown that the surface treatment of the QD, as well as the
solvent used to suspend the QDs, play important roles in optimizing the
quantum yield of both single exciton and multi-exciton states Beard  _et al._
(2009). The number of band-edge excitons generated per primary excitation,
i.e., the quantum yield ${\rm QY}_{\rm ex}$, is energy-dependent, and
experimentally the quantum yield for PbS (and PbSe) is typically optimized to
be around ${\rm QY}_{\rm ex}\approx 2-3$ at $E_{r}=4\times\Delta E_{\rm QD}$
Midgett  _et al._ (2013); Schaller and Klimov (2004); Ellingson  _et al._
(2005); Hardman  _et al._ (2011); Beard  _et al._ (2009). Since the rate of
non-radiative Auger recombination, whereby an electron-hole pair recombines
and imparts its energy into a third carrier, scales as $n_{e/h}^{3}$
($n_{e/h}$ is the carrier density), final states with more than two excitons
do not live long enough to emit multiple photons.

To remain conservative, we assume that all multi-exciton final states with a
number of excitons greater than three immediately decay to a bi-exciton state
before emitting photons. We model the function QYex as shown in orange in Fig.
5. Below $3\times\Delta E_{\rm QD}$, the exciton quantum yield is unity,
corresponding to single exciton generation, while above this value the
function is modeled as a linear response up to 2 excitons at $5\times\Delta
E_{\rm QD}$ and beyond which no additional excitons are created. This
functional form roughly captures the spread of values found in the literature
with a slight bias towards the conservative direction. Note that QYex should
be thought of as an experimental measure of the mean number of excitons
created in each excitation and it can therefore be a continuous response. Also
shown in blue in the figure is the ideal theoretical MEG quantum yield for
PbS, which grows linearly with $E_{r}$.

Figure 5: The exciton response of PbS. The blue line represents the
theoretical exciton quantum yield of PbS QDs Midgett  _et al._ (2013);
Schaller and Klimov (2004); Ellingson  _et al._ (2005); Hardman  _et al._
(2011). In order to account for Auger recombination, the modeled rediative
response is taken to saturate when two excitons are created (orange line),
since Auger recombination will quickly deplete higher exciton multiplicities.

###  III.5 Photon Generation and Collection

Once single or multiple band-edge excitons are generated, they decay either
radiatively by fluorescing, or non-radiatively via, e.g., collisional
quenching, Auger recombination, or phonon emission. As discussed above,
single-exciton fluorescence is highly efficient with a quantum yield of order
QY${}_{\gamma}\approx 0.5$ Yang  _et al._ (2017) while multi-exciton
fluorescence competes with Auger recombination, which lowers the quantum yield
for such multi-photon emission. For example, the bi-exciton, two-photon
emission has a quantum yield of order QY${}_{\gamma\gamma}\approx 0.05$ Beyler
_et al._ (2014).

Since we make the conservative assumption that any exciton multiplicity
greater than two is rapidly reduced to the bi-exciton state with no
intermediate photon emission, the resulting signal consists of either a single
photon or two coincident photons. For the latter case, the coincidence time is
of order the bi-exciton lifetime,
$\tau_{\gamma\gamma}=\tau_{\gamma}\times\text{QY}_{\gamma\gamma}/4\approx
12-65$ ns, where $\tau_{\gamma}\approx 1-5\,\mu$s is the single-exciton
radiative lifetime for PbS QDs (the same is true for PbSe) Mangum  _et al._
(2013); Moreels  _et al._ (2009).

The one and two-photon differential signal rates are given by

$\frac{dR_{\rm ph}}{d\ln{E_{r}}}=\frac{dR_{\rm
ex}}{d\ln{E_{r}}}\begin{cases}\text{QY}_{\rm ex}\cdot\xi\cdot\text{QY}_{\rm
BF}&,1\gamma\\\ (\text{QY}_{\rm
ex}-1)\cdot\frac{\text{QY}_{\gamma\gamma}}{2}\cdot\left(\frac{\xi\cdot\text{QY}_{\rm
BF}}{\text{QY}_{\gamma}}\right)^{2}&,2\gamma\end{cases}$ (17)

where $\xi$ is the photon detection efficiency of the photodetector, which we
take to be $\xi=0.25$. This is a typical value for detection efficiency in
PMTs used in scintillating DM direct detection experiments Blanco  _et al._
(2020, 2021b), although newer technologies such as Superconducting Nanowire
Single Photon Detectors (SNSPDs), Microwave Kinetic Inductance Detectors
(MKIDs), and Transition Edge Sensors (TESs) aim to push this detection
efficiency to near unity. The bulk fluorescence quantum yield, QYBF, is the
probability of a single band-edge exciton to emit a photon that can travel
macroscopic distances without being reabsorbed by other chromophores
(optically active targets). This is an experimental measure of optical losses
in the suspension volume (we assume that other losses, such as from photons
being absorbed by the side walls, are negligible; see also discussion in ref.
Collar (2018)). The value of QYBF is discussed in detail below. QYBF/QYγ is
the self-absorption normalized by the probability of emitting a single photon
from a single band-edge exciton. The factor in the second parentheses of the
two-photon signal is the detectability of a single-photon event. Note that the
factor of $1/2$ in the two-photon signal accounts for requiring that each of
the two photons hits opposing photodetectors.

As discussed in App. B, the value of QYBF (and therefore the reabsorption
probability) is correlated with the Stokes shift of the emission and
absorption spectra, with the width of the emission spectrum, and with the
concentration of QDs. The width of the emission spectrum is dominated by the
dispersion of the size distribution of QDs, where a tighter size distribution
improves fluorescence quantum yield. At room temperature, the line width of a
single QD is on the order of 100 meV while that of typical colloidal ensembles
is about 250 meV Peterson and Krauss (2006). Experimentally, it has been shown
that reabsorption of photons becomes significant for such colloidal
suspensions above QD particle concentrations of $C_{\rm part,QD}\approx
10\;\mu\text{M}/\text{L}$ Greben  _et al._ (2015). Therefore, in all
calculations presented in this study, we take this maximal concentration
value, from which one can easily calculate the density of target material
given in Eq. (7). In particular,

$\displaystyle\rho_{\rm QD}$ $\displaystyle=C_{\rm part,QD}\cdot\rho_{\rm
semi}\cdot\frac{4\pi}{3}\left(\frac{d}{2}\right)^{3}$ (18)
$\displaystyle\approx 2.5\;\text{g}/\text{L}\left(\frac{C_{\rm
part,QD}}{10\;\mu\text{M}/\text{L}}\right)\left(\frac{\rho_{\rm
semi}}{6.29\text{g}/\text{cm}^{3}}\right)\left(\frac{d}{5\text{nm}}\right)^{3},$

where $\rho_{\rm semi}$ is the bulk semiconductor density. It should be noted
that $\rho_{\rm QD}$ can potentially be increased up to $22$ g/L for $7.5$ nm
PbSe QDs at a concentration of $C_{\rm part,QD}\approx
20\;\mu\text{M}/\text{L}$. From Eq. (18) it is evident that even with a very
simple setup of a $\mathcal{O}$(L) instrumented target volume, such as that of
Ref. Blanco  _et al._ (2020), effective masses of $\mathcal{O}(10\text{
grams})$ of semiconducting scintillator are achievable.

Under the assumption of this close-to-maximal concentration, we adopt a
conservative room temperature value of QY${}_{\rm BF}=0.2$ also based on
experimentally determined values (which can in fact be as large as QY${}_{\rm
BF}=0.25$ for PbS QDs with $d=3-3.3$ nm Greben  _et al._ (2015)). However, we
note that the photoluminescent quantum yield of PbS QDs improves significantly
at lower temperatures, reaching values more than twice that of room
temperature QDs already at 100 K Caram  _et al._ (2016). Additional details
regarding QYBF are presented in App. B.

###  III.6 Total Event Rate

In order to calculate the sensitivity of a hypothetical DM experiment, one
quantity of interest is the total, directionally averaged rate. Since a
continuously resolved energy spectrum is not measurable for the envisioned
setup, the differential rate must be integrated over the energy-domain
included in the experimental signal. The total single- or two-photon rate is
given by

$R_{\rm ph}=\int_{E_{\rm th}}^{E_{\rm cutoff}}\frac{dR_{\rm
ph}}{d\ln{E_{r}}}\frac{dE_{r}}{E_{r}}\,.$ (19)

The lower limit of integration depends on the threshold of the target for the
specific signal of interest. We conservatively take $E_{\rm th}$ to be the
threshold for MEG, namely $3\Delta E_{\rm QD}\simeq 3.6$ eV, for both the
single- and two-photon signals. For the case of MEG, this threshold is
experimentally measured and is well within the region where the PbS QDs behave
like bulk PbS. For the case of the single-photon signal, we remain
conservative by not counting the region near the band edge where strong
confinement effects might affect the optical and electronic excitation
response of the QDs. Including this near-bandgap region below the MEG
threshold would serve to increase the calculated rate and improve the
projected sensitivity of the setup. However, from Fig. 4 we infer that such an
improvement would not be significant unless the DM mass is close to threshold.
Note that the noisy behavior apparent in the low energy region of the $dR_{\rm
ex}/dE_{r}$ curves shown in Fig. 4 (which arises from numerical uncertainties
in QE-dark close to the band edge) are mostly removed because of this choice
for $E_{\rm th}$. The upper limit is set by the highest energy expected to
produce a visible signal; for higher energies, processes such as ligand
excitation or QD ionization could prevent excited states from relaxing to
photo-luminescent final states. For our calculations, we take the maximum
recoil energy to be $E_{\rm cutoff}=10$ eV as a conservative choice and note
that our results do not depend strongly on this choice.

##  IV Experimental Setup and Background Rates

For results presented in this study, we consider two classes of experimental
setups: 1) a setup based on currently available and mature photodetectors such
as PMTs, and 2) a more futuristic setup based on high sensitivity
photodetectors such as SNSPDs. The former option is achievable on relatively
short timescales and therefore, for this setup, we present results only for a
coincident two-photon signal under the assumption that this low-background
option is a likely first iteration of the experimental concept. The latter
option is likely achievable only on longer timescales. For this futuristic
setup, we present results for the more challenging single-photon signal, and
we show results for an idealized zero background setup. While achieving zero
background rates is likely to be experimentally very challenging, we present
such results in order to show the ultimate sensitivity of the setup.

For the case of PMTs, previous studies focusing on liquid scintillators have
proposed and deployed large-area single PMTs as photodetectors Blanco  _et
al._ (2020, 2021b); Collar (2018) to instrument liter-scale scintillation
volumes. These studies suggest that background rates of about $1$ Hz for $5$
inch diameter bialkali PMTs should be easily obtained when running with
moderate overburden ($6.25$ meters water equivalent), and $0.1$ Hz should be
possible in deep underground locations and with ideal thermal control. Note,
that in previous studies this background was measured to be the dark-count
reduced steady-state background in a differential measurement, i.e., data
taken with the PMT blind vs observing the scintillator volume. Similar
differential measurements can be done to reduce backgrounds here. Furthermore,
we note that GaAs photocathodes have been measured to have 1 Hz dark counts at
temperatures of $-40^{\circ}$ C, while side-on bialkali photocathodes reach
approximately 0.3 Hz at around $-20^{\circ}$ C Hamamatsu Photonics (2007). We
abstain from making specific recommendations about the exact geometry and
composition of the PMT but note that technology exists that can meet the
necessary benchmarks mentioned in this study. Furthermore, we point out that
manufacturing variance allows hand-picked units to outperform the mean
manufacturer specification for dark-counts (see, e.g., Ref. Collar (2018)).

In the calculations that follow, we assume PMT background rates and show
results for both $\sim$1 Hz and $0.1$ Hz. For the case of multiple coincident
photons we discuss the dominant sources of backgrounds which are expected to
be blackbody radiation and the dark counts of the PMT detectors themselves.
However, additional backgrounds, such as Cherenkov radiation, transition
radiation, and low-energy luminescence Du  _et al._ (2022) from the solvent
and from any dielectric holders and other structures inside the detector
vessel could also contribute. These effects need to be evaluated carefully
before confronting a positive signal with a DM interpretation. For the case of
more sensitive photodetectors such as SNSPDs, the detector dark counts will be
lower and environmental backgrounds will likely dominate. Furthermore, we note
that PMTs in the arrangement presented in Fig. 1 could detect each other’s
dynode glow during a photoavalanche. This background is straight-forward to
reduce. For example, since the dynode emission from a PMT has a broad
bandwidth, while the fluorescence photons from the QDs are essentially
monochromatic, a simple notch filter can be used to ameliorate any potential
dynode-glow background. This technique may also be used to reduce potential
direct Cherenkov backgrounds. In either case, in-situ characterization would
be necessary.

The single photon background rate of a single photodetector with surface area,
$A_{\rm det}$, from blackbody radiation can be determined via Essig  _et al._
(2019b),

$R_{\rm BBR}=\frac{\Delta\omega\,\omega^{2}}{\pi^{2}}A_{\rm
det}e^{-\frac{\omega}{T}}\,.$ (20)

Here, $\omega$ is the energy of the photon, $\Delta\omega$ is the detector
energy resolution, and $T$ is the ambient temperature. For
$\Delta\omega\approx\omega=1.2\leavevmode\nobreak\ \textrm{eV}$ and $A_{\rm
det}=\pi\,(2.5\leavevmode\nobreak\ \textrm{inch})^{2}$, one finds that
$T\lesssim 230$ K reduces $R_{\rm BBR}$ to be below $1$ Hz.

The envisioned setup includes modules of target material, each as shown in
Fig. 1 and each including a single cylindrical tank instrumented with two
photodetectors. A two coincident photon signal occurs when both photodetectors
record hits within some coincidence time, $\Delta t_{\rm co}$. The background
rate for such coincident events can be calculated from the Poissonian single-
photon background rate of each detector, $R^{\rm BG}_{\gamma}$ (assumed to be
the same for both detectors). As long as $R^{\rm BG}_{\gamma}\Delta t_{\rm
co}\ll 1$, the result for the two-coincident-photon rate is

$R^{\rm BG}_{\gamma\gamma}\approx 2(R^{\rm BG}_{\gamma})^{2}\Delta t_{\rm
co}\,.$ (21)

Taking $R^{\rm BG}_{\gamma}=1$ Hz and $\Delta t_{\rm co}=100$ ns (a few times
the lifetime of the two-exciton state), one finds that for an exposure time of
$t_{\rm exp}\approx 1$ month, the number of coincident two-photon events is
$R^{\rm BG}_{\gamma\gamma}t_{\rm exp}\approx 0.52$. Since the instrumentation
described here corresponds to approximately $10$ grams of target material (see
Sec. III.5), a setup with an exposure of order $10$ g-months is expected to be
essentially background free for the coincident two-photon signal and any
additional reduction of $R^{\rm BG}_{\gamma}$ would do little to the overall
experimental sensitivity.

In order to achieve larger exposures, we propose simply running in parallel
several replica setups shown in Fig. 1. Increasing exposure in this way
features a linear scaling of the background rate with the target volume, as
opposed to the less advantageous scaling of background rates expected from
instrumenting a larger tank. Thus, for example, one could envision running ten
detectors for one year and achieving exposures of order $100$ g-years with
$R^{\rm BG}_{\gamma\gamma}$ only ten times larger than the $10$ g-month case.
Results below are presented for these two example exposures, $10$ g-months and
$100$ g-years, and for single photon background rates of $R^{\rm
BG}_{\gamma}=1$ and $0.1$ Hz, for the coincident two-photon signal. For the
futuristic single-photon signal, results are presented for an exposure of
$100$ g-years and a zero background rate.

The ($1-\alpha$)% C.L. signal for an exposure time, $t_{\rm exp}$, for a two-
photon (single-photon) background rate, $R^{\rm BG}_{(\gamma)\gamma}$, is
given by,

$R_{(1-\alpha)\%}=\frac{1}{2}\frac{\chi^{2}}{t_{\rm exp}}\left(2(R^{\rm
BG}_{(\gamma)\gamma}t_{\rm exp}+1),1-\frac{\alpha}{2}\right)\,,$ (22)

where $\chi^{2}$ is the $(1-\frac{\alpha}{2})$-th quantile of the chi-squared
distribution of $2(R^{\rm BG}_{(\gamma)\gamma}t_{\rm exp}+1)$ degrees of
freedom. For example, for a background-free exposure, i.e., $R^{\rm
BG}_{(\gamma)\gamma}t_{\rm exp}\ll 1$, the 90% C.L. number of events is
$R_{90\%}t_{\rm exp}=2.99$. For a $10$ g-month exposure and a single-photon
background rate of $R^{\rm BG}_{\gamma}=1$ Hz ($0.1$ Hz), this corresponds to
a total coincident two-photon background rate of 0.52 ($5.2\times 10^{-3}$)
events / month. This is essentially background free and corresponds to 90%
C.L. total event counts of 3.95 (3.01). For a $100$ g-year exposure, the total
coincident two-photon background rate is 63.1 ($0.63$) events / year. This
corresponds to 90% C.L. total event counts of 77.8 (4.13).

The calculations above have focused on the case of PMTs as photodetectors.
Ideally however, low-background photodetectors such as SNSPDs, MKIDs, or TESs
could be used for photon counting. However, scaling up the effective
collection area of these devices remains challenging. Skipper-CCDs present
another interesting photodetector candidate in the context of scintillator
detectors, and one could envision having two CCDs read photons coming from a
target; the CCDs have intrinsically low backgrounds, although further
reduction of the time to read the entire CCD may be needed for a two-photon
coincidence signal counting.111Taking the Skipper-CCDs used by SENSEI Barak
_et al._ (2020a) as an example, one option is to collect the charge of the 5.4
million pixels of the CCD into a single pixel and sample that pixel 300 times
(to achieve single-electron resolution); this can be done in about 0.26 s.
Taking a single-electron background rate of $10^{-4}$ electrons/pixel/day,
achieved in Barak  _et al._ (2020a) in a detector operating inside a small
shield about 100 m underground, one finds a two-photon-coincident background
signal rate for two Skipper-CCDs operating in parallel of $\sim$640
events/year. This is higher than the two-photon-coincident background rate
from two PMTs operating with a 1 Hz background rate; however, the readout time
scales linearly with the number of amplifiers and the photon detection
efficiency is $\mathcal{O}$(1) compared to $\xi=0.25$ assumed for the PMTs.
Hence Skipper-CCDs should be considered as a possible photodetector choice.

##  V Dark Matter Sensitivity

###  V.1 DM-Electron Scattering

Figure 6: The 90% C.L. sensitivity for scattering events with a $10\,$g-month
($100\,$g-year) exposure corresponding to a $1$ L ($10\times 1$ L) colloidal
suspension of $d=3.3\,$nm PbS QDs is shown in red (black). The dashed lines
are for a steady-state background rate of $1\,\text{Hz}/\text{PMT}$, while the
solid lines are for a steady-state background rate of
$0.1\,\text{Hz}/\text{PMT}$, assuming this background scales linearly with
target volume. Shaded regions represent existing constraints from SENSEI Barak
_et al._ (2020b), XENON10 Essig  _et al._ (2017b), XENON1T Aprile  _et al._
(2019c), and DAMIC Aguilar-Arevalo  _et al._ (2019b). The top frames show the
the calculated sensitivity when searching for a two-photon coincident signal
over a steady-state single-photon background of $R_{\gamma}^{\text{BG}}=1$ Hz
($R_{\gamma}^{\text{BG}}=0.1$ Hz) as shown by the dashed (solid) lines. The
bottom panels show the calculated sensitivity of a futuristic 100 g-yr QD
exposure when searching for a single-photon signal assuming no backgrounds.
The left (right) panels show these sensitivities and exclusions as calculated
assuming DM form factors of $F_{\text{DM}}=1$ ($F_{\text{DM}}\sim 1/q^{2}$),
appropriate for models with heavy (light) mediators.

Fig. 6 shows the expected sensitivity of the proposed setup to the reference
cross section, $\bar{\sigma}_{e}$, for DM-electron scattering events, for the
coincident two-photon signal (top panels) and for the single photon signal
(bottom panels) for a maximally concentrated solution of $d=3.3\,$nm PbS QDs.
For the coincident two-photon signal, results are shown for $10\,$g-month and
$100\,$g-year exposures and backgrounds are calculated assuming
$R_{\gamma}^{\rm BG}=1$ Hz (dashed curves) and $R_{\gamma}^{\rm BG}=0.1$ Hz
(solid curves). This setup should be achievable on relatively short
timescales. For the single-photon signal, results are shown for a more
futuristic $100\,$g-year exposure under the assumption that single-photon
backgrounds are reduced to be negligible. For both signal types, results are
shown for a contact interaction corresponding to $F_{\rm DM}=1$ (left panels)
and for the case of an ultralight (or massless) mediator corresponding to
$F_{\rm DM}=(\alpha m_{e})/q^{2}$ (right panels). All results are compared to
existing constraints on DM-electron scattering from SENSEI Barak  _et al._
(2020b), XENON10 Angle  _et al._ (2011); Essig  _et al._ (2012b, 2017b),
XENON1T Aprile  _et al._ (2019c), and DAMIC Aguilar-Arevalo  _et al._ (2019b)
(see also Blanco  _et al._ (2020); Arnaud  _et al._ (2020b); Agnese  _et al._
(2018b)).

Since the expected number of background events is negligible for a $10$
g-month exposure, the sensitivity is driven solely by the signal generation
efficiency of the target mass. Because this pilot experiment is background
free, it is ideally suited for the characterization of the detector and
naturally lends itself as a scale that could be deployed as an experimental
proof-of-concept. Even at this very conservative size, the sensitivity of such
an exposure is competitive with existing constraints by SENSEI Barak  _et al._
(2020a) at DM masses below 10 MeV where it would already improve on existing
bounds by DAMIC at SNOLAB Aguilar-Arevalo  _et al._ (2019b). Lowering the
operating temperature in order to improve the photoluminescence quantum yield
and reduce PMT noise, optimizing the QD surface chemistry to improve the bi-
exciton radiative quantum yield, or reducing the rate of Auger recombination
at this scale could already probe new parameter space.

We find that for the coincident two-photon signal, exposures of order $100$
g-years and single-photon background fluxes of order $0.1$ Hz probe parameter
space significantly beyond the existing SENSEI constraints Barak  _et al._
(2020a). The more futuristic setup, which we envision as being sensitive to
single-photon events with negligible backgrounds, is expected to probe many
orders of magnitude of new parameter space for both DM form factors.

Figure 7: The 90% C.L. sensitivity for absorption events for dark photon dark
matter with a $10\,$g-month ($100\,$g-year) exposure corresponding to a $1$ L
($10\times 1$ L) colloidal suspension of $d=5\,$nm PbS QDs is shown in red
(black). The dashed lines are for a steady-state background rate of
$1\,\text{Hz}/\text{PMT}$, while the solid lines are for a steady-state
background rate of $0.1\,\text{Hz}/\text{PMT}$, assuming this background
scales linearly with target volume. Shaded regions represent existing
constraints from SENSEI Barak  _et al._ (2020a) modified to include in-medium
effects (see text for details), XENON10 Bloch  _et al._ (2017), and XENON1T An
_et al._ (2020). The left panel shows the the calculated sensitivity when
searching for a two-photon coincident signal over a steady-state single-photon
background of $R_{\gamma}^{\text{BG}}=1$ Hz ($R_{\gamma}^{\text{BG}}=0.1$ Hz)
as shown by the dashed (solid) lines. The right panel shows the calculated
sensitivity of a futuristic 100 g-yr QD exposure when searching for a single-
photon signal assuming no backgrounds. All lines are dotted above
$m_{A^{\prime}}=10\leavevmode\nobreak\ \textrm{eV}$ to indicate that for these
masses the response of the QD is uncertain.

###  V.2 DM Absorption by Electrons

Fig. 7 shows expected sensitivity in the mixing parameter, $\kappa$, of the
proposed setup for absorption events of dark-photon DM for $d=5\textrm{nm}$
PbS QDs (we use QD sizes that are slightly different from those used for the
DM-electron scattering results, since the available experimental data
correspond to 5 nm). The left panel shows results for the coincident two-
photon signal assuming the same exposures and background fluxes as those used
in Fig. 6. The right panel shows results for the single-photon signal under
the assumption of negligible backgrounds. Existing limits from DM absorption
from SENSEI Barak  _et al._ (2020a), XENON10 Bloch  _et al._ (2017) and limits
on solar emission from XENON1T An  _et al._ (2020) are also shown. The SENSEI
Barak  _et al._ (2020a) limits are rescaled by $|\epsilon|$ (the absolute
value of the dielectric function) to account for in-medium effects Mitridate
_et al._ (2021). Curves are dotted above $m_{A^{\prime}}\geq
10\leavevmode\nobreak\ \textrm{eV}$, since for these larger masses the QD
response is currently not well understood.

Similar conclusions to the scattering case can be drawn regarding exposures
and background rates and competitiveness with other experimental bounds. In
particular, for the coincident two-photon case, an exposure of approximately
$100$ g-year and single-photon background rates below $1$ Hz would make it
possible to explore new parameter space. A more futuristic detector sensitive
to single-photon events and with negligible backgrounds will be able to probe
many orders of magnitude of new parameter space.

##  VI Conclusions

While there is a wide variety of proposed detection targets in the sub-GeV
electron-scattering space, searches using QDs are uniquely suited to take
advantage of the scalability and optical properties of molecules, while also
leveraging their tunable low-thresholds and multi-exciton dynamics of
semiconductors. In this paper, we have shown that strongly confining IV-VI
semiconducting nanocrystals in colloidal suspension are a promising target
material, which can be deployed quickly and probe unexplored DM parameter
space.

The signal consists of either a coincident two-photon event with inherently
low backgrounds, or a single-photon event. For the former case, currently
available photodetector technologies such as PMTs are sufficient to have
sensitivity to DM that would be competitive or even supersede current direct-
detection bounds. Such an experimental setup is achievable on short timescales
and requires little R&D. For the latter (single-photon) case, lower background
photodetectors and dedicated background reduction techniques would be
required. Such a setup is envisioned as a futuristic progression of the
experiment.

For both DM scattering and bosonic DM absorption, a modest $100\,$g-year
exposure of PbS QDs with a diameter of $3.3-5\,$nm are shown to be up to two
orders of magnitude more sensitive than existing constraints for realistic
background rates with the coincident two-photon setup. Smaller scale,
$10\,$g-month exposures are expected to be essentially “background-free” and
present a natural characterization scale for the experiment with interesting
science potential and competitive sensitivity to cutting edge detectors such
as SENSEI and SuperCDMS. The single-photon setup will be able to probe many
orders of magnitude of new DM parameter space for both DM scattering and for
DM bosonic absorption.

Improvements over the conservative benchmarks used in this study may be
achieved by lowering the steady-state background while maximizing the
detection efficiency of the photodetector. This could be done by running at a
deep-underground site and using intrinsically low-threshold low-background
detectors such as SNSPDs, MKIDs, TESs, or skipper-CCDs. Such photodetectors
have already been proposed for use in other direct-detection experimental
setups consisting of, e.g., solid-state scintillators Derenzo  _et al._ (2017)
(such as GaAs) or molecular gases Essig  _et al._ (2019b). Additional possible
improvements with respect to results shown in this study include
characterizing the bi-exciton radiative quantum yield of the QDs via sample-
averaged time-resolved photon self-correlation measurements Nair  _et al._
(2011); Beyler  _et al._ (2014). Additionally, it should be possible to
experimentally optimize the multi-photon emission of the QDs by suppressing
Auger-recombination or enhancing the spontaneous emission rate Beyler  _et
al._ (2014); Taghipour  _et al._ (2022); Dhawan  _et al._ (2020). Yet another
promising avenue is the consideration of cooperative emission effects
Arvanitaki  _et al._ (2018) to focus the emergent photons produced after DM
absorption into small surface single photon detectors.

This study is a first analysis of the physics case for a QD-based DM detector.
Such a setup would utilize the scintillation properties of QDs and would
benefit from a photon readout as opposed to the more conventional electron
readout typical for semiconductor-based experiments. We conclude that with
existing photodetector technologies and commercially available QDs (such as
PbS QDs), it seems possible to deploy an intrinsically low-background
experimental search for MeV-scale DM scattering events and eV-scale DM
absorption events that could significantly improve on the current bounds from
the leading sub-GeV DM experiments.

##  VII Acknowledgements

We thank Juan Collar, Dan Baxter, Yonatan Kahn, and Konstantin Likharev for
useful discussions. The work of C.B. was supported in part by NASA through the
NASA Hubble Fellowship Program grant HST-HF2-51451.001-A awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555 as well as by the European Research Council under grant 742104.
R.E. acknowledges support from DoE Grant DE-SC0009854, Simons Investigator in
Physics Award 623940, and the US-Israel Binational Science Foundation Grant
No. 2016153. O.S. is supported by the DOE under Award Number DE-SC0007968 and
the Binational Science Foundation (grant No. 2018140). H.R. acknowledges the
support from the Simons Investigator Award 824870, DOE Grant DE-SC0012012, NSF
Grant PHY2014215, DOE HEP QuantISED award no. 100495, and the Gordon and Betty
Moore Foundation Grant GBMF7946. We are also grateful to the organisers of the
Pollica Summer Workshop, supported by the Regione Campania, Università degli
Studi di Salerno, Università degli Studi di Napoli “Federico II”, i
dipartimenti di Fisica “Ettore Pancini”, and “E R Caianiello”, and “Istituto
Nazionale di Fisica Nucleare”, which allowed us to collaborate. In addition,
we thank the Simons Center for Geometry and Physics and the program “Lighting
new Lampposts for Dark Matter and Beyond the Standard Model”, which also
allowed us to collaborate on this paper.

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##  Appendix A Semi-analytic Model for DM-Electron Scattering in QDs

This appendix provides intuition as to why the scattering rates for QDs can be
well approximated by scattering rates of a bulk semiconductor. Below, we
present a simple semi-analytical model for the scattering of DM with the
electronic states in semiconductors. However, this calculation breaks down at
very small momentum transfer, $qa\ll 1$ (where $a$ is the lattice constant),
precisely in the kinematic region where QDs are expected to differ from
semiconductors. We approximate the result for QDs by adjusting the
semiconductor calculation at low $q$ and accounting for effects of quantum
confinement close to the band-edge. With these calculations, we show that
differences between the two occur in regions of parameter space that are
anyway kinematically forbidden.

The main conclusion is that the effects of quantum confinement are only
important for radiative de-excitation of QDs but not for their excitation
rates. Therefore it is typically sufficiently accurate to use numerical
calculations of the bulk material in order to calculate QD excitation rates,
while size-dependent radiative de-excitation rates can then be inferred from
experimental results.

We begin by following previous semi-analytic approaches to DM-electron
scattering in semiconductors Lee  _et al._ (2015); Essig  _et al._ (2012b, a,
2016, 2017a). The calculation for the total directionally-averaged
differential electronic excitation rate in QDs was discussed in Sec. III.2.
For completeness, the main equations are provided here as well. The excitation
rate (Eq. (6) of the main text) is

$\frac{dR_{\rm
ex}}{d\ln{E_{r}}}=\frac{N_{t}\rho_{\chi}\overline{\sigma}_{e}}{8m_{\chi}\mu_{e\chi}}\int_{q_{\rm
min}}^{q_{\rm max}}qdq\,\eta(v_{\rm min})|F_{\rm
DM}(q)|^{2}|f(q,E_{r})|^{2}\,,$ (23)

where the QD subscript has been removed from the form factor, since here we
present general arguments that hold for both bulk semiconductors and for QDs.
The general equation for the form factor (Eq. (10) of the main text) is

$|f(q,E_{r})|^{2}\equiv\sum_{\rm fin}\sum_{\rm init}\int
dE_{b}\left(\frac{dn}{dE_{b}}\right)_{\rm init}\delta(E_{r}-(E_{\rm
fin}-E_{b}))\left|\int d^{3}r\,e^{-i\mathbf{q}\cdot\mathbf{r}}\Psi_{\rm
fin}^{*}(\mathbf{r})\Psi_{\rm init}(\mathbf{r})\right|^{2},$ (24)

where $(dn/dE_{b})_{\rm init}$ is the density of initial states (DOS) of the
system Graham  _et al._ (2012); Lach-Hab  _et al._ (2002), $E_{b}$ is the
binding energy, and the initial and final wavefunctions have also been written
in their general forms.

For the initial states, one can take the approximation of the tight-binding
model. In this approximation, small perturbations to the wavefunction from
nearest-neighbor interactions are neglected, and the valence band is modeled
as a sum over atomic orbitals (analogous to an LCAO model for the case of
molecules). The initial electronic states are then well described by the
relevant valence atomic orbital,

$\displaystyle\Psi_{\rm
init}(\mathbf{r})=\sum_{N}e^{i\mathbf{k}\cdot\mathbf{R}_{N}}\mathcal{Y}_{\ell
m}(\Omega)\chi_{n\ell}(r)\,,$ (25)

where $\mathbf{k}$ is the crystal momentum and $\mathbf{R}_{N}$ are the
locations of atoms within the crystal. The functions $\psi^{\rm RHF}_{n\ell
m}(\mathbf{r})\equiv\mathcal{Y}_{\ell m}(\Omega)\chi_{n\ell}(r)$ are isolated
Roothan-Hartree-Fock atomic orbitals, $\mathcal{Y}_{\ell m}(\Omega)$ are
spherical harmonics, and $\chi_{n\ell}(r)$ are radial wavefunctions, which can
be expanded as a finite sum over primitive Slater-type radial functions,

$\displaystyle\chi_{n\ell}(r)$ $\displaystyle=\sum_{j}c_{j\ell
n}N_{j\ell}r^{(n_{j\ell}-1)}\mathrm{exp}(-Z_{j\ell}r)\,.$ (26)

Here, $c_{j\ell n}$ are the expansion coefficients, $N_{j\ell}$ are
normalization constants, $n_{j\ell}$ are principle quantum numbers, and
$Z_{j\ell}$ are effective charges. In calculations presented here, we take
Roothan-Hartree-Fock functions from tabulated numerical fits to atomic valence
states Bunge  _et al._ (1993).

### Approximation for Bulk Semiconductors at High Momentum Transfer

If the imparted momentum is sufficiently large, the final electronic states
can be approximated as plane waves. However, one must also account for the
deformation of these final states by the presence of the charged ion at the
origin. The result for the form factor, Eq. (24), can then be rewritten as,

$|f(q,E_{r})|^{2}\equiv\sum_{n\ell}\int
dE_{b}\left(\frac{dn}{dE_{b}}\right)_{n\ell}F_{\rm
Fermi}(E_{r}-E_{b})\left|f^{n\ell}(q,E_{r},E_{b})\right|^{2}\,,$ (27)

where the Fermi function accounts for the wavefunction deformation close to
the origin, and is given by

$F_{\rm
Fermi}=\frac{2\pi\nu}{1-\exp{(-2\pi\nu)}}\,\,\,\,\,;\,\,\,\,\,\nu=\frac{\alpha
m_{e}Z_{\rm eff}}{\sqrt{2m_{e}(E_{r}-E_{b})}}\,.$ (28)

$Z_{\rm eff}$ is the effective charge of the ion at the origin, which we
conservatively take as $Z_{\rm eff}=1$ to account for outer-shell electrons
Lee  _et al._ (2015); Essig  _et al._ (2012a, b).

The last term in Eq. (27) is related to the wavefunction overlap. For recoil
energies that are of order or larger than the binding energy of the initial
state, the form factor is approximately Lee  _et al._ (2015); Essig  _et al._
(2012b),

$|f^{n\ell}(q,E_{r},E_{b})|^{2}=\frac{\left(2\ell+1\right)2m_{e}(E_{r}-E_{b})}{4\pi^{3}q}\int_{|\sqrt{2m_{e}E_{r}}-q|}^{\sqrt{2m_{e}E_{r}}+q}k|\tilde{\chi}_{n\ell}(k)|^{2}\,dk\,,$
(29)

where $\tilde{\chi}_{n\ell}(k)$ is the fourier transform of $\chi_{n\ell}(r)$
Belkić and Taylor (1989).

Note that accounting for the binding energy corresponds to energy-dependent
limits of integration and acts to enhance the rate. This is a correction to
the approach in Refs. Lee  _et al._ (2015); Essig  _et al._ (2012b), which was
adapted from the arguments in Ref. Essig  _et al._ (2017a).

### Approximation for Quantum Dots at Low Momentum Transfer

In order to account for the effects of quantum confinement at low momentum
transfer, one should correct the form factor calculation of Eq. (29). Since
the final states for that calculation were taken as plane waves, the basis
used for the calculation is not orthogonal and the form factor does not
approach zero as $q\to 0$, as it should. This can be corrected to first order
in $\mathbf{q}\cdot\mathbf{r}$ by using the dipole approximation,
$e^{i\mathbf{q}\cdot\mathbf{r}}\approx 1+i\mathbf{q}\cdot\mathbf{r}$, from
which it follows that $\bra{\psi_{\rm
fin}}e^{i\mathbf{q}\cdot\mathbf{r}}\ket{\psi_{\rm init}}\approx
i\mathbf{q}\cdot\bra{\psi_{\rm fin}}\mathbf{r}\ket{\psi_{\rm init}}$ for an
orthogonal basis.

The dipole matrix element between the initial (valence) and final (conduction)
bands of a semiconductor is given in $\mathbf{k}\cdot\mathbf{p}$ perturbation
theory by

$\displaystyle|\bra{\psi_{\rm fin}}r\ket{\psi_{\rm
init}}|^{2}=\frac{(m_{e}-m_{\rm eff})}{2m_{e}m_{\rm eff}\Delta E_{g}}\,,$ (30)

where $m_{\rm eff}$ is the curvature of the valence and conduction bands
around their extrema. The result for $f^{n\ell}$ for the low momentum transfer
regime is then

$|f^{n\ell}(q,E_{r}=E_{b}=\Delta E_{g})|^{2}=\frac{(m_{e}-m_{\rm
eff})}{2m_{e}m_{\rm eff}\Delta E_{g}}q^{2}\,.$ (31)

From the results of Eqs. (29) and (31), one can construct a form factor that
is approximately correct both at high and low momentum transfer,

$|f^{n\ell}(q,E_{r},E_{b})|^{2}=\begin{cases}\text{Eq.}\leavevmode\nobreak\
\eqref{eq:AP_FF_lowq}&,qa<1\\\ \text{Eq.}\leavevmode\nobreak\
\eqref{eq:AP_fqd3}&,qa\geq 1.\end{cases}$ (32)

The effects of quantum confinement cause the bandgap of the semiconductor to
effectively increase. As the QD becomes smaller, the envelope piece of the
wavefunction affects the transition by effectively increasing the energy of
the valence and conduction states, as can be seen from the envelope wave
functions in Eq. 4. This can be modeled to first order as in Eq. 5, where the
second term is the energy of confinement of the hole and electron states.
Practically, the first transition energy for QDs can be precisely measured and
its dependence on the QD radius, $R$, can be experimentally extracted as in
Eq. 11. For Silicon QDs, this relation is given by Gesese (2009),

$\displaystyle\Delta E=\Delta
E_{g}+\left(\frac{3.73}{(R/\text{nm})^{1.37}}\right).$ (33)

Fig. 8 shows the form factor, $|f(q,E_{r})|^{2}$, for Si using
$|f^{n\ell}(q,E_{r},E_{b})|^{2}$ as given in Eq. (32). Also shown in black is
the lower bound for the kinematically allowed region of the $q$-integral for a
10 MeV DM particle. Note how even for a transition energy equal to the band
gap of Si, the kinematically allowed region is always significantly above the
dipole part of the form factor. Indeed, rates even slightly above threshold
are expected to be entirely undisturbed by the QD nature of the target.

Figure 8: The Si form factor as a function of momentum transfer, $q$, and
recoil energy, $E_{r}$, for a QD with $d=3.3$ nm accounting for the
confinement via the dipole correction. The black curve delineates the minimum
kinematically allowed momentum $q_{\rm min}$ for a DM mass of $10$ MeV. White
regions are numerically indistinguishable from zero and fall below $\Delta
E_{\text{QD}}$.

Since the QD nature of the target has been shown not to affect the rates
compared to those calculated for the bulk, we conclude that it is appropriate
to use a sufficiently accurate numerical form factor for QDs as calculated for
bulk semiconductors but with a modified band-gap, whose value depends on the
QD radius. In other words, the number of charge carriers created in a
semiconductor by a scattering event with a massive DM particle would be the
same if the semiconductor is monolithic or nanoscopically disperse.

##  Appendix B Quantum Yield

Given the single exciton quantum yield (QYγ), defined as the ratio of emitted
photons to primary electronic excitations, the probability of a photon exiting
the bulk target (QYBF) after an excitation is given by the following,

$\text{QY}_{\rm BF}=(1-a_{xx})\text{QY}_{\gamma}\,,$ (34)

where $a_{xx}$, the probability of self absorption, can be calculated with the
following expression,

$a_{xx}=\frac{\int
I_{x}(\omega)\left[1-\exp\left(-\epsilon_{x}(\omega)l\right)\right]d\omega}{\int
I_{x}(\omega)d\omega}\,.$ (35)

Here $I_{x}$ is the relative quantum intensity of the chromophore’s
fluorescence (i.e., photoluminescence, photoemission), $\epsilon_{x}$ is the
molar extinction coefficient, and $l$ is the path length the photon must
travel. Note that $I_{x}$ is distinct from the bulk fluorescence spectrum in
that the bulk fluorescence spectrum is proportional to QYBF due to self
absorption. $\epsilon_{x}$ is a quantity that can be experimentally measured
and theoretically predicted. From the Beer-Lambert law, the molar extinction
coefficient is given by,

$\displaystyle\left(\frac{\epsilon_{x}(\omega)}{\textrm{M}^{-1}\;\textrm{cm}^{-1}}\right)$
$\displaystyle=\frac{1}{\ln(10)}\left(\frac{N_{A}}{\textrm{mol}^{-1}}\right)\left(\frac{\text{mol}}{\text{M}\,\text{L}}\right)\left(\frac{\text{L}}{10^{3}\;\text{cm}^{3}}\right)\left(\frac{\sigma_{x}(\omega)}{\textrm{cm}^{2}}\right)$
$\displaystyle=\frac{1}{\ln(10)}\left(\frac{N_{A}}{\textrm{mol}^{-1}}\right)\left(\frac{\text{mol}}{\text{M}\,\text{L}}\right)\left(\frac{\text{L}}{10^{3}\;\text{cm}^{3}}\right)\left(\frac{\text{cm}}{\lambda_{x}(\omega)}\right)\left(\frac{\text{cm}^{-3}}{n}\right)\,,$
(36)

where $\sigma_{x}$ is the absorption cross section for a photon of energy
$\omega$, $N_{A}$ is Avogadro’s number, $\lambda$ is the mean free path, and
$n$ is the chromophore number density. Note that we have specified the units
in order to highlight the conventions found in the physics and chemistry
literature. One can think of $\epsilon_{x}$ as the inverse mean free path per
unit number density. The factor of $\ln(10)$ comes from the Beer-Lambert law.
The transmittance, $T$, is given as a function of absorbance, $A$, by

$T(\omega)=\frac{\phi_{t}(\omega)}{\phi_{i}(\omega)}=e^{-\sigma(\omega)nl}=10^{-\epsilon_{x}(\omega)cl}=10^{-A(\omega)}\,,$
(37)

where, for a bulk sample of chromophores $x$, $\phi_{i}$ is the incoming
radiant flux, $\phi_{t}$ is the transmitted radiant flux, $A(\omega)$ is
defined as the negative log of the transmittance $T(\omega)$, and $c$ is the
molar concentration of the chromophore. Thus, a measurement of the absorbance
of a bulk sample of chromophores of known concentration is a direct
measurement of the molar absorption coefficient and, in turn, its light
absorption cross section. Therefore, measurements of $I_{x}(\omega)$ and
$A(\omega)$ are sufficient in order to compute the probability for a signal
photon to be emitted and to traverse a macroscopic sample as a result of
exciting a chromophore in the target bulk.

Notice that in dilute concentrations of chromophores, after the rapid self-
absorption between the emission, $I_{x}(\omega)$, and absorption,
$\epsilon_{x}(\omega)$, overlap in $a_{xx}$, further self-absorption is
negligible for pathlengths within about an order of magnitude around the $l$
for which the measurements where taken. The relative offset of the peak
absorption and emission characteristics for a chromophore is called the
Stokes’ shift and is correlated with increasing QYBF Semonin  _et al._ (2010),
since this causes the overlap in $a_{xx}$ to decrease rapidly. For PbS QDs of
$d=3.3$ nm, the concentration at which further self-absorption becomes
important due to other factors such as quenching or aggregation is around 10
$\mu$M/L Greben  _et al._ (2015).

##  Appendix C Si and PbS form factor calculations

All results presented in this study were obtained using single electron
crystal wave function orbitals and energies obtained using Quantum Espresso
Giannozzi  _et al._ (2009). They are obtained from density functional theory
(DFT) within the local density approximation (LDA) using the Perdew-Zunger
Perdew and Zunger (1981) parameterization for the exchange and correlation
functional. Norm conserving pesudopotentials are used for the core electronic
levels and plane wave cutoffs of up to 35 Ry (for PbS) and 30 Ry (for Si) were
employed in the calculation of the Kohn-Sham orbitals. For Si (Ge), a band gap
corrected to 1.1 (0.67) eV is applied, while for PbS we correct the band gap
to be 0.42 eV. Finally, the band gap of PbS is adjusted to the appropriate
value given by Eq. 11 for the QD calculations. Input files for the electronic
structure calculations and QE-dark are provided in QEd ; Soto-Cambres (2015).