Phonon-mediated Migdal effect in semiconductor detectors

The Migdal effect inside detectors provides a new possibility of probing the sub-GeV dark matter (DM) particles. While there has been well-established methods treating the Migdal effect in isolated atoms, a coherent and complete description of the valence electrons in a semiconductor is still absent. The bremstrahlunglike approach is a promising attempt, but it turns invalid for DM masses below a few tens of MeV. In this paper, we lay out a framework where phonon is chosen as an effective degree of freedom to describe the Migdal effect in semiconductors. In this picture, a valence electron is excited to the conduction state via exchange of a virtual phonon, accompanied by a multiphonon process triggered by an incident DM particle. Under the incoherent approximation, it turns out that this approach can effectively push the sensitivities of the semiconductor targets further down to the MeV DM mass region.

Figure 1

The diagram of the process $\chi (p_{\chi })+\text{target}\to \chi (p_{\chi }^{\prime })+\text{target}+({\mathbf{k}}_1,{\alpha }_1)+({\mathbf{k}}_2,{\alpha }_2)+\cdots ({\mathbf{k}}_n,{\alpha }_n)$. See text for details.

Figure 2

Top: comparisons between the function $\overline{\omega }S(q,\omega )$ of silicon for the multiphonon distribution (orange) and the impulse Gaussian (red dashed curves) for the ratios $\sqrt{E_R(q)/\overline{\omega }}=5$ and 10, respectively. The multiphonon distributions are estimated with the asymptotic expansion (up to $\mathcal{O}(\sqrt{\overline{\omega }/E_R(q)})$) presented in Eq. (2.7). Bottom: comparisons between the function $\overline{\omega }S(q,\omega )$ of silicon for the multiphonon distribution (blue histograms) obtained by the numerical recursive method and the impulse Gaussian (red dashed curves) for the ratios $\sqrt{E_R(q)/\overline{\omega }}=1$, 2, and 5, respectively. Similar discussion can be found in Ref. [50]. See text for details.

Figure 3

The diagram for the Migdal effect, where an electron-hole pair is generated, i.e., an electron is elevated from a valence state $|j⟩$ to a conduction state $|i⟩$, via the exchange of a virtual phonon, along with multiple on-shell phonons generated by the DM external field.

Figure 4

Left: the differential Migdal electronic excitation event rates in bulk silicon for a reference cross section ${\sigma }_{\chi n}={10}^{-38}{\mathrm{cm}}^2$ and DM masses $m_{\chi }=10\mathrm{MeV}$ (emerald), 20 MeV(orange) and 50 MeV (maroon), respectively; the solid lines and the dashed lines are calculated using the phonon-mediated approach and the bremsstrahlunglike approach, respectively. Right: sensitivities for the Migdal effect at 90% C.L. for a $1\mathrm{kg}·\mathrm{yr}$ silicon detector, based on the single-electron (blue) and the two-electron (orange) ionization signal bins. The solid and the dashed curves are calculated using the phonon-mediated approach and the bremsstrahlunglike approach, respectively. See text for details.

Figure 5

The effects of the incident DM particle on the target material can be regarded as an external field.

Figure 6

The diagram for the $n$-phonon scattering process containing full contributions of phonon loops, which generates the Debye-Waller factor denoted as the gray blob on the right-hand side of the equation. See text for details.

Figure 7

Illustration of possible ways in which $n$ phonon external lines connect to the $n$ internal ones while the left $2m$ internal members self-connect with each other to form $m$ loops. See text for details.

Figure 8

Top: the renormalized electron-phonon vertex (black square box) presented at the RPA level, where the one-particle-irreducible blob is represented by an electron-hole pair loop. Note that the gray lines represent the phonon propagators, while the black wiggly lines represent the Coulomb interaction. Bottom: the dressed phonon line (double wiggly line) presented at the RPA level. See text for details.

Figure 9

The normalized phonon density of states for bulk silicon. See text for details.