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

At a Glance

Metadata	Details
Publication Date	2020-08-10
Journal	Physical review. D/Physical review. D.
Authors	Zheng-Liang Liang, Lin Zhang, Fawei Zheng, Ping Zhang
Institutions	Beijing University of Chemical Technology, Qufu Normal University
Citations	24
Analysis	Full AI Review Included

Executive Summary

This research presents a novel theoretical framework for calculating the Migdal effect (electron excitation/ionization induced by nuclear recoil) in crystalline semiconductor detectors, specifically focusing on diamond.

Core Innovation: The established Migdal effect formalism for isolated atoms is extended to solids using the Tight-Binding (TB) approximation, leveraging atom-centered Wannier Functions (WFs) to simultaneously capture both the localized (WFs) and delocalized (hopping integrals) nature of electrons.
Target Material: Crystalline diamond is used as a proof-of-principle target, chosen for its light carbon nucleus, which provides a lower Dark Matter (DM) mass threshold compared to traditional silicon or germanium detectors.
Methodological Validation: The approach utilizes ab initio Density Functional Theory (DFT) calculations (Quantum Espresso) combined with Wannier interpolation (Wannier 90) to derive the necessary electronic structure parameters.
Impulse Approximation: The calculation relies on the impulse approximation, valid for sub-GeV DM, where the nuclear recoil is treated as an instantaneous, highly localized event on a specific atom.
Performance Metrics: The calculated differential event rates show that the Migdal effect in diamond provides a wide energy spectrum, and projected sensitivity for a 1 kg-yr exposure reaches DM-nucleon cross sections (sigma_chin) down to approximately 10^-39 cm² in the sub-GeV mass range.
Self-Consistency: The methodology was verified using different sets of trial wavefunctions (s, p orbitals vs. sp³ hybrids), yielding results within a few percent difference, confirming the theoretical self-consistency and practical effectiveness of the TB approach.

Technical Specifications

The following parameters and results were derived from the ab initio calculations and subsequent Migdal effect modeling for crystalline diamond.

Parameter	Value	Unit	Context
Target Material	Diamond (Carbon)	N/A	Semiconductor detector
Adopted Lattice Constant (a)	3.560	Angstrom	Used in DFT relaxation
DFT Energy Cutoff	90	Ry	Plane-wave basis set
Wannier Functions (J)	32	N/A	Number of WFs generated
k-point Mesh (Wannier Interpolation)	16x16x16	N/A	Used for calculating matrix elements
Frozen Energy Window	Valence band bottom to 40	eV	Range used for Bloch state selection
Reciprocal Lattice Reference (2pi/a)	~3.48	keV	Momentum scale reference
Energy Bin Width (Delta E)	0.059	eV	Used for smearing the delta function
Benchmark Excitation Rate (1)	0.027	/kg/yr	m_chi = 500 MeV, w = 100 km/s, sigma_chin = 10^-38 cm²
Benchmark Excitation Rate (2)	0.299	/kg/yr	m_chi = 10 MeV, w = 800 km/s, sigma_chin = 10^-38 cm²
Assumed Electron-Hole Pair Energy	13	eV	Used for detector sensitivity estimate
Maximum WF Spread (Approx.)	1.4	Angstrom²	Spread of generated WFs

Key Methodologies

The calculation of the Migdal excitation event rate in crystalline diamond followed a multi-step computational recipe:

DFT Calculation:
- Software: Quantum Espresso was used to perform Density Functional Theory calculations.
- Inputs: Bloch eigenfunctions and eigenvalues were obtained using a plane-wave basis set and GGA (PBE) exchange-correlation functional.
- Mesh: A 20x20x20 Monkhorst-Pack k-point mesh was used for the initial DFT run.
Wannier Function Generation (Wannierization):
- Software: Wannier 90 was used to generate Maximally Localized Wannier Functions (MLWFs).
- Constraint: The standard gauge selection step (which minimizes WF spread) was intentionally skipped to ensure the WFs remained strictly atom-centered, which is critical for localizing the recoil effect to a specific nucleus.
- Subspace Selection: A “frozen energy window” was chosen (up to 40 eV above the valence band maximum) to ensure the Bloch states spanning the relevant energy range were faithfully retained.
Tight-Binding (TB) Approximation Implementation:
- The Bloch wavefunctions were re-expressed in terms of the localized WFs.
- Hopping integrals (Hamiltonian matrices) were tabulated, including terms up to the third neighbor unit cell to ensure convergence, reflecting the delocalized nature of the electrons.
Migdal Transition Amplitude Calculation:
- The Galilean boost operator (e^{i qe · r}) was applied exclusively to the WFs associated with the struck atom (R=0).
- The calculation utilized the impulse approximation, assuming the excitation is instantaneous (timescale < E_g^-1), which is appropriate for hard scattering events involving large momentum transfer (q).
Event Rate and Sensitivity Calculation:
- The total excitation event rate (R) was calculated by integrating the transition probability over the DM velocity distribution and momentum transfer (q), incorporating the derived crystal form factor F(q, Ee).
- Sensitivity was estimated for a 1 kg-yr exposure, assuming an average energy of 13 eV per electron-hole pair, yielding 90% C.L. exclusion contours for single- and two-electron signals.

Commercial Applications

The methodology and results presented are highly relevant for the development and optimization of next-generation solid-state detectors, particularly those targeting low-mass Dark Matter.

Industry/Sector	Specific Application/Product Relevance
Dark Matter Detection	Sub-GeV DM Detectors: Provides the necessary theoretical modeling for predicting the performance and sensitivity of diamond-based semiconductor detectors (like those proposed by CDMS/SuperCDMS) operating in the sub-GeV mass range.
Quantum Sensing & Computing	Defect and Exciton Modeling: The ab initio derived Tight-Binding model using MLWFs is a powerful tool for simulating local phenomena, such as defects (e.g., NV centers in diamond) and excitons, which are crucial for quantum information science applications.
Solid-State Device Physics	Impulsive Phenomena Simulation: The framework can be adapted to model other high-energy, short-timescale interactions in crystalline materials, such as radiation damage, high-energy particle interactions, and transient electronic responses.
Materials Science Research	Electronic Structure Validation: The use of Wannier interpolation provides a highly accurate and computationally efficient method for reproducing and analyzing the electronic band structure of semiconductors, aiding in the design of new electronic materials.
High-Performance Computing	Efficient TB Parameterization: The methodology generates highly localized, atom-centered WFs, which are ideal for creating efficient, transferable Tight-Binding Hamiltonians used in large-scale simulations of complex crystal structures and interfaces.

View Original Abstract

Recent studies have theoretically investigated the atomic excitation and\nionization induced by the dark matter (DM)-nucleus scattering, and it is found\nthat the suddenly recoiled atom is much more likely to excite or lose its\nelectrons than expected. Such phenomenon is called the “Migdal effect”. In this\npaper, we extend the established strategy to describe the Migdal effect in\nisolated atoms to the case in semiconductors under the framework of\ntight-binding (TB) approximation. Since the localized aspects of electrons are\nrespected in form of the Wannier functions (WFs), the extension of the existing\nMigdal approach for isolated atoms is much more natural, while the extensive\nnature of electrons in solids is reflected in the hopping integrals. We take\ndiamond target as a concrete proof of principle for the methodology, and\ncalculate relevant energy spectra and projected sensitivity of such diamond\ndetector. It turns out that our method as a preliminary attempt is practically\neffective.\n

Tech Support

Original Source

DOI: https://doi.org/10.1103/physrevd.102.043007