The Migdal effect in semiconductors for dark matter with masses below ∼ 100 MeV

Abstract

Dark matter scattering off a nucleus has a small probability of inducing an observable ionization through the inelastic excitation of an electron, called the Migdal effect. We use an effective field theory to extend the computation of the Migdal effect in semiconductors to regions of small momentum transfer to the nucleus, where the final state of the nucleus is no longer well described by a plane wave. Our analytical result can be fully quantified by the measurable dynamic structure factor of the semiconductor, which accounts for the vibrational degrees of freedom (phonons) in a crystal. We show that, due to the sum rules obeyed by the structure factor, the inclusive Migdal rate and the shape of the electron recoil spectrum is well captured by approximating the nuclei in the crystal as free ions; however, the exclusive differential rate with respect to energy depositions to the crystal depends on the phonon dynamics encoded in the dynamic structure function of the specific material. Our results now allow the Migdal effect in semiconductors to be evaluated even for the lightest dark matter candidates (mχ ≳ 1 MeV) that can kinematically excite electrons.

Light dark matter detection: New ideas and new tools

“…where ω is the energy of the outgoing Migdal electron, H χL the Hamiltonian coupling the dark matter to the nuclei in the crystal, and H eL the one coupling the nuclei to the electrons. Thanks to this, it is possible to show that the rate for Migdal emission, differential both in the electron energy, ω, as well as in the energy released to the crystal, E ph , is [24],…”

“…In both cases, we assume a detector sensitive to single electrons excitations, as well as zero background. For details see [24].…”

“…As noticed in [24], the problem at hand is charachterized by a separation of scales: the energy of the Migdal electron is O(eV), much larger than the typical energy of the excitations of the crystal, which is O(1 − 100 meV). This allows to bypass the complicated description of the crystal dynamics by writing a low energy EFT, which describes the Migdal emission by the dark matter via the following effective Hamiltonian [24]:…”

“…The results are reported in Figure 1. For measurements which are inclusive on the energy released to the crystal, the rate is insensitive to the details of the crystal under consideration [24]. Possible future measurements, sensitive also to E ph , could instead be used as background discrimination, making it imperative to have reliable predictions based on Eq.…”

“…Here we focus on some possible ways to push the sensitivity of direct detection experiments down to masses of order keV. Specifically, for the range 1 MeV ≲ m χ ≲ 1 GeV, we discuss the so-called Migdal effect in semiconductors [21][22][23][24]. For the range 1 keV ≲ m χ ≲ 1 MeV we, instead, discuss signatures involving phonons in superfluid 4 He [e.g., [25][26][27][28][29][30][31][32] or magnons in anti-ferromagnetic materials [33].…”