Electronic structure and optical properties of quantum crystals from first principles calculations in the Born–Oppenheimer approximation

At a Glance

Metadata	Details
Publication Date	2020-12-21
Journal	The Journal of Chemical Physics
Authors	Vitaly Gorelov, David M. Ceperley, Markus Holzmann, Carlo Pierleoni
Institutions	Université Paris-Sud, Centre National de la Recherche Scientifique
Citations	10
Analysis	Full AI Review Included

Technical Documentation & Analysis: Quantum Crystals and CVD Diamond

Executive Summary

This analysis focuses on the theoretical investigation of electronic structure and optical properties in quantum crystals, specifically utilizing Carbon Diamond as a key benchmark material. The research employs advanced many-body Quantum Monte Carlo (QMC) and Path Integral Molecular Dynamics (PIMD) methods to accurately model the effects of nuclear quantum and thermal motion on the electronic band gap and optical absorption profile.

Core Achievement: Development and application of a “Quantum Averaging” (QA) procedure within the Kubo-Greenwood (KG) formalism to calculate optical conductivity, demonstrating superior accuracy over traditional semi-classical (Williams-Lax, WL) methods for light nuclei systems like hydrogen and providing precise benchmarks for diamond.
Diamond Benchmark: Carbon diamond was studied at 297 K (room temperature), yielding a fundamental optical gap of 3.55 eV (via QA Tauc analysis), consistent with high-purity material expectations.
Material Relevance: The study confirms that accurate modeling of diamond’s optical response requires accounting for zero-point motion and thermal renormalization, emphasizing the need for high-quality, low-defect CVD diamond materials for experimental validation.
Methodology: The approach integrates Born-Oppenheimer (BO) approximation, QMC/CEIMC for nuclear sampling, and Density Functional Theory (DFT-HSE/PBE) for electronic structure calculations.
6CCVD Value Proposition: 6CCVD provides the high-purity Single Crystal Diamond (SCD) and Polycrystalline Diamond (PCD) required to experimentally validate these advanced theoretical predictions, offering custom dimensions, precise thickness control, and ultra-low roughness polishing essential for optical measurements.

Technical Specifications

The following data points are extracted from the analysis concerning the Carbon Diamond benchmark system:

Parameter	Value	Unit	Context
Material Studied	Carbon Diamond	N/A	Used as a quantum crystal benchmark
Simulation Temperature	297	K	Room temperature
Lattice Structure	Cubic Supercell (64 atoms)	N/A	Used for PIMD/PBE calculations
Lattice Constant	3.56712	Å	Appropriate for room temperature
Optical Gap (Semi-classical, WL)	3.45	eV	Extracted via Tauc analysis (Fig. 6b)
Optical Gap (Quantum Averaging, QA)	3.55	eV	Extracted via Tauc analysis (Fig. 6b)
DFT Functional Used	PBE	N/A	Used for carbon absorption calculations
Kinetic Energy Cut-off	60	Ry	Used in DFT calculations for carbon
Gaussian Smearing	0.35	eV	Used in computing the absorption profile
Polishing Requirement (Implied)	Ra < 1	nm	Necessary for high-fidelity optical absorption studies

Key Methodologies

The research utilized a sophisticated combination of many-body and effective single-electron theories to model electronic and optical properties, focusing on non-perturbative treatment of nuclear motion.

Born-Oppenheimer (BO) Approximation: Assumed valid for condensed matter, separating electron and nuclear motion.
Quantum Monte Carlo (QMC) Methods: Used for precise calculation of electronic properties, including the fundamental energy gap derived from electron addition/removal energies in canonical and grand canonical ensembles.
Coupled Electron-Ion Monte Carlo (CEIMC) / Path Integral Molecular Dynamics (PIMD): Employed to sample the nuclear distribution, incorporating zero-point quantum motion and thermal effects, particularly for light nuclei (Hydrogen) and Carbon Diamond.
Grand-Canonical Twist Averaging Boundary Conditions (GCTABC): Applied to reduce finite-size effects in QMC calculations, providing access to the density of states (DOS).
Kubo-Greenwood (KG) Formalism: Used to calculate optical conductivity $\sigma(\omega, T)$.
Quantum Averaging (QA) Procedure: An alternative to the semi-classical (Williams-Lax, WL) approach, where eigenvalues and matrix elements are averaged over nuclear states, providing a more accurate onset of absorption consistent with the fundamental energy gap, especially at low temperatures.

6CCVD Solutions & Capabilities

The research highlights the critical role of material purity and structural perfection in determining the fundamental electronic and optical properties of diamond. 6CCVD is uniquely positioned to supply the high-specification MPCVD diamond required to validate and extend these advanced computational studies.

Applicable Materials

To replicate or extend the optical absorption and band structure studies performed on Carbon Diamond at 297 K, researchers require materials with exceptional crystalline quality and minimal defects.

Research Requirement	6CCVD Recommended Solution	Key Specification Match
High-Purity Crystal Structure	Optical Grade Single Crystal Diamond (SCD)	SCD offers the lowest defect density, ensuring the measured optical properties are dominated by intrinsic quantum effects, not impurities.
Large Area Substrates	High-Purity Polycrystalline Diamond (PCD)	Available in plates/wafers up to 125mm diameter, ideal for large-scale optical setups or high-power applications.
Band Gap Validation	Intrinsic SCD (Undoped)	Essential for confirming the calculated 3.55 eV optical gap without interference from dopant-induced states.
Doping Studies (Extension)	Boron-Doped Diamond (BDD)	For extending the research into conductive quantum crystals or studying the effects of heavy doping on band structure renormalization.

Customization Potential

The precision required for optical measurements necessitates highly controlled material dimensions and surface quality. 6CCVD’s in-house engineering capabilities directly address these needs:

Custom Dimensions and Thickness: We provide SCD and PCD plates/wafers with custom dimensions and precise thickness control, ranging from 0.1µm to 500µm for active layers, and substrates up to 10mm thick.
Ultra-Low Roughness Polishing: Achieving accurate optical absorption profiles (like those in Fig. 5b and 6b) demands minimal surface scattering. 6CCVD guarantees ultra-smooth surfaces:
- SCD: Roughness Ra < 1nm.
- Inch-size PCD: Roughness Ra < 5nm.
Advanced Metalization Services: Should the research transition from pure optical studies to device fabrication (e.g., conductivity measurements requiring contacts), 6CCVD offers internal metalization capabilities, including Au, Pt, Pd, Ti, W, and Cu thin films, tailored to specific contact geometries.

Engineering Support

The complexity of modeling quantum effects in diamond, as demonstrated by the difference between semi-classical (3.45 eV) and quantum averaging (3.55 eV) results, requires close collaboration between theoretical and experimental teams.

6CCVD’s in-house PhD team specializes in MPCVD growth parameters and material characterization. We offer expert consultation to assist researchers in:

Material Selection: Matching the theoretical requirements (e.g., intrinsic carrier concentration, defect levels) to the optimal SCD or PCD grade.
Interface Engineering: Designing custom metalization stacks for electrical measurements that complement optical studies.
Global Logistics: Ensuring reliable, fast global shipping (DDU default, DDP available) of sensitive diamond materials directly to your lab.

Call to Action: For custom specifications or material consultation regarding high-purity diamond for quantum crystal research, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

We develop a formalism to accurately account for the renormalization of the electronic structure due to quantum and thermal nuclear motions within the Born-Oppenheimer approximation. We focus on the fundamental energy gap obtained from electronic addition and removal energies from quantum Monte Carlo calculations in either the canonical or grand-canonical ensembles. The formalism applies as well to effective single electron theories such as those based on density functional theory. We show that the electronic (Bloch) crystal momentum can be restored by marginalizing the total electron-ion wave function with respect to the nuclear equilibrium distribution, and we describe an explicit procedure to establish the band structure of electronic excitations for quantum crystals within the Born-Oppenheimer approximation. Based on the Kubo-Greenwood equation, we discuss the effects of nuclear motion on optical conductivity. Our methodology applies to the low temperature regime where nuclear motion is quantized and, in general, differs from the semi-classical approximation. We apply our method to study the electronic structure of C2/c-24 crystalline hydrogen at 200 K and 250 GPa and discuss the optical absorption profile of hydrogen crystals at 200 K and carbon diamond at 297 K.

Tech Support

Original Source

DOI: https://doi.org/10.1063/5.0031843

References

2010 - Momentum distribution and renormalization factor in sodium and the electron gas [Crossref]
2011 - Applications of quantum Monte Carlo methods in condensed systems [Crossref]
2013 - Quantum Monte Carlo applied to solids [Crossref]
2014 - Effect of electron correlation on the electronic structure and spin-lattice coupling of high-Tc cuprates: Quantum Monte Carlo calculations [Crossref]
2016 - Discovering correlated fermions using quantum Monte Carlo [Crossref]
2018 - Quantum Monte Carlo calculations of energy gaps from first principles [Crossref]
2020 - Direct observation of the momentum distribution and renormalization factor in lithium [Crossref]
2020 - Quantum Monte Carlo Compton profiles of solid and liquid lithium [Crossref]
2018 - Ab initio electronic structure calculations by auxiliary-field quantum Monte Carlo