Describing Migdal effects in diamond crystal with atom-centered localized Wannier functions

Recent studies have theoretically investigated the atomic excitation and ionization induced by the dark matter–nucleus scattering, and it was found that the suddenly recoiled atom is much more likely to excite or lose its electrons than expected. This phenomenon is called the “Migdal effect.” In this paper, we extend the established strategy to describe the Migdal effect in isolated atoms to the case in semiconductors under the framework of the tight-binding approximation. Since the localized aspects of electrons are respected in the form of the Wannier functions, the extension of the existing Migdal approach for isolated atoms is much more natural, while the extensive nature of electrons in solids is reflected in the hopping integrals. We take a diamond target as a concrete proof of principle for the methodology, and calculate relevant energy spectra and the projected sensitivity of such a diamond detector. It turns out that our method as a preliminary attempt is theoretically self-consistent and practically effective.

Figure 1

Left: schematic illustration of our proposal to describe the electronic excitation induced by the struck nucleus located at $\mathbf{R}=\mathbf{0}$; only the initial states $\{|\mathbf{0}m⟩\}$ are relevant for the excitation process and thus are subjected to the Galilean transformation ${\widehat{G}}_{\mathbf{0}}(-{\mathbf{q}}_e)$. Right: conceptual illustration of the evolution of the Hamiltonian in parameter space (gray curve), with the point $O$ ($P$) corresponding to the original (perturbed) Hamiltonian and relevant eigenstates $\{|i\mathbf{k}⟩\}$ ($\{{|i\mathbf{k}⟩}_{\text{struck}}\}$). These two points are equivalent in the sense that they are connected to each other through an adiabatic evolution. See text for details.

Figure 2

Left: the band structures of bulk crystalline diamond obtained from the DFT calculation (blue solid line) and Wannier interpolation (red dotted line). Right: the crystal form function $\mathcal{F}(q,E_e)$. The area contoured in yellow (blue) represents the parameter region relevant for the Migdal excitation event rate for the parameters $m_{\chi }=500\mathrm{MeV}$, $w=100\mathrm{km}/\mathrm{s}$ ($m_{\chi }=10\mathrm{MeV}$, $w=800\mathrm{km}/\mathrm{s}$). See text for details.

Figure 3

Left: the differential electronic excitation event rate of the Migdal effect in crystalline diamond for reference values ${\sigma }_{\chi n}={10}^{-38}{\mathrm{cm}}^2$ and $m_{\chi }=10\mathrm{MeV}$ (blue), 100 MeV (orange) and 1 GeV (green), respectively. Right: cross-section sensitivities for the Migdal effect at 90% C.L. for a 1 kg-yr diamond detector, based on a single-electron (blue) and a two-electron (orange) signal, respectively. See text for details.