Atomistic k ⋅ p theory

At a Glance

Metadata	Details
Publication Date	2015-12-08
Journal	Journal of Applied Physics
Authors	Craig Pryor, Mats‐Erik Pistol
Institutions	University of Iowa, Lund University
Citations	12
Analysis	Full AI Review Included

Atomistic $k \cdot p$ Theory for Diamond and Zincblende Semiconductors

This technical analysis document, based on the research paper “Atomistic k·p theory,” outlines the development of a novel quantum mechanical modeling approach highly relevant to the engineering and optimization of advanced diamond semiconductor devices manufactured by 6CCVD. The atomistic $k \cdot p$ method provides essential computational precision for designing nanostructures, interfaces, and impurity-doped materials, such as 6CCVD’s Single Crystal Diamond (SCD) and Boron-Doped Diamond (BDD).

Executive Summary

The paper introduces the first new method in decades for calculating electronic states in crystalline solids, offering high-precision modeling crucial for diamond nanostructures and defects.

Bridging Models: The “atomistic $k \cdot p$ theory” successfully merges the continuum approach of traditional $k \cdot p$ theory with the atomic precision of tight-binding models.
Atomic Precision Achieved: The model utilizes finite difference operators defined on a computational grid that precisely matches the atomic positions within the crystal lattice (diamond/zincblende).
Direct Parameter Derivation: It demonstrates that the atomistic four-band $k \cdot p$ model is equivalent to the $sp^3$ tight-binding model, allowing tight-binding parameters to be derived directly from measurable bulk experimental data, unlike traditional fitting methods.
Focus on Imperfections: The approach is explicitly designed for modeling systems where atomic-scale precision is required, including impurities, alloys (such as BDD), polytypes, and interfaces—all critical aspects of 6CCVD’s advanced materials.
Multiscale Capability: The framework uniquely allows for seamless multiscale modeling by combining highly precise atomistic grids (for nanoscale features) with coarse-grained continuum grids (for larger bulk regions).
High Accuracy Demonstrated: The atomistic eight-band Kane model, incorporating spin-orbit coupling, successfully reproduces the effective masses, g-factors, and band structures of standard III-V semiconductors (e.g., GaAs, InAs, AlSb), validating its accuracy for the diamond lattice structure.

Technical Specifications

The following parameters are extracted from the numerical fits (Tables I and II) demonstrating the capability of the atomistic $k \cdot p$ model to reproduce key electronic properties across various diamond/zincblende materials.

Parameter	Value Range	Unit	Context
Crystal Structure	Diamond/Zincblende	N/A	Core lattice structure assumed for the theoretical model.
Lattice Constant (a_latt)	5.4417 - 6.4690	Å	Range across GaP (smallest) to InSb (largest). Diamond: 3.567 Å (relevant extension).
Conduction Band Effective Mass (m*/m_o)	0.0135 - 0.22	(unitless ratio)	Fitted to match continuum $k \cdot p$ model parameters.
Conduction Band g-Factor (g*/g_o)	-51.6 to 3.35	(unitless ratio)	Fitted to match continuum $k \cdot p$ model parameters.
Band Doubling Energy Shift	O($\hbar^2$/m_oa_latt²)	Energy	Predicted energy shift for excited bands in the atomistic limit.
Minimum Lattice Grid Spacing	a_latt/4	N/A	Defined by nearest neighbor displacements for finite difference operators.
Atomistic Momentum Matrix Element (P_a)	8.7915 - 10.3961	eV Å	Measures $\langle S
Atomistic Intraband Matrix Element (Q_a)	-10.9957 - -4.7041	eV Å	Measures $\langle X

Key Methodologies

The atomistic $k \cdot p$ theory constructs the semiconductor Hamiltonian by replacing continuous differential operators with finite difference operators on a crystal-aligned grid.

Grid Definition: The envelope functions are defined on a grid where sites correspond precisely to atomic positions (anion and cation sites) within the diamond or zincblende lattice structure.
Difference Operator Construction: Continuous spatial derivatives ($\nabla$) are approximated using finite difference operators ($\Delta$) based on Taylor series expansion.
- Nearest Neighbors: First derivatives ($\Delta_x, \Delta_y, \Delta_z$) and the Laplacian ($\nabla^2$) utilize the site and its four nearest neighbors (displacements $\mathbf{d}_1$ to $\mathbf{d}_4$).
- Next-Nearest Neighbors: Mixed second derivatives ($\partial^2_{xy}, \partial^2_{xz}, \partial^2_{yz}$) require next-nearest neighbors for approximation, introducing next-nearest neighbor couplings into the second-order Hamiltonian.
Hermiticity Restoration (Critical Step): In inversion non-symmetric materials (like zincblende, or diamond with defects), the spatially varying coefficients of the difference operators can lead to a non-Hermitian Hamiltonian. This is resolved by:
- Using the Finite Volume Method (converting volume integrals to surface integrals over cell boundaries $\partial\Omega_R$) to ensure the coefficient connecting two sites is the same regardless of which site is the reference (R or R + d).
- Alternatively, employing Generalized Voronoi Cells (Section V/VIII) to tune the relative volumes of the sub-unit cells ($\Omega_1, \Omega_2$) to enforce $P_{a1} = P_{a2}$, restoring Hermiticity while maintaining inversion non-symmetry.
Parameter Fitting: Atomistic on-site energies ($E_{c1}, E_{v1}$, etc.) are calculated from target zone-center energies. Atomistic momentum parameters ($F_a, P_a, Q_a, \gamma_{a1}$) are then derived via non-linear fitting to match bulk empirical targets (effective masses, g-factors, and Dresselhaus spin splittings).
Band Structure Validation: The resulting $16 \times 16$ Hamiltonian matrix (8-band Kane model) is solved numerically to confirm that band structures throughout the entire Brillouin zone are free of spurious gap-crossing states, ensuring the model’s suitability for real-space numerical calculation.

6CCVD Solutions & Capabilities: Precision Materials for Atomistic Modeling

The successful application of atomistic $k \cdot p$ theory relies on starting materials that possess highly controlled crystalline structures, essential for modeling defects, interfaces, and nanostructures. 6CCVD’s expert capabilities in MPCVD diamond growth and fabrication directly support the demands of this advanced theoretical framework, especially for diamond and Boron-Doped Diamond applications.

Applicable Materials

To replicate or extend the atomistic modeling demonstrated in this paper to actual diamond systems, 6CCVD recommends materials optimized for purity and lattice integrity:

Optical Grade Single Crystal Diamond (SCD): This material is critical for simulating fundamental diamond properties, intrinsic defects (like NV centers), and high-purity nanostructures. Our SCD wafers (0.1µm to 500µm thickness) provide the perfect, structurally uniform diamond lattice required as the basis for the atomistic grid.
Heavy Boron-Doped Polycrystalline Diamond (BDD PCD/SCD): Boron doping creates an alloy ($\text{C}_{1-x}\text{B}_x$) where the principles of modeling impurities and alloys (as discussed in the paper) become paramount. 6CCVD provides highly uniform BDD films suitable for complex electronic transport simulations modeled using atomistic $k \cdot p$.

Customization Potential

The paper emphasizes modeling systems where translational symmetry is broken (nanostructures, interfaces). 6CCVD offers the necessary fabrication capabilities to prepare materials tailored for device integration and theoretical validation:

Requirement from Research	6CCVD Custom Capability	Engineering Impact
Nanostructure Definition	Custom Dimensions: Plates/wafers up to 125mm (PCD), SCD and PCD thickness range (0.1µm - 500µm).	Provides large-area substrates for defining complex computational boundaries.
Interface Modeling	Custom Metalization: In-house deposition of Au, Pt, Pd, Ti, W, Cu.	Essential for simulating metal/semiconductor contacts (interfaces) using atomistic models.
Surface Quality (Envelope Functions)	Ultra-High Polishing: Ra < 1 nm (SCD), < 5 nm (Inch-size PCD).	Ensures minimized surface scattering effects, maintaining the structural quality necessary for accurate envelope function modeling at device surfaces.

Engineering Support

The complexity of atomistic $k \cdot p$ parameterization—especially the treatment of non-Hermiticity and the selection of appropriate Bloch basis states—necessitates deep material science expertise.

6CCVD’s in-house PhD engineering team specializes in diamond material characteristics, defect engineering, and electronic properties. We can assist researchers and technical engineers with material selection for similar projects involving diamond nanostructures, quantum defects (like NV centers), and advanced diamond power electronics where atomistic precision is key to predictive modeling.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly. Global shipping (DDU default, DDP available) ensures rapid delivery of precision diamond materials worldwide.

View Original Abstract

Pseudopotentials, tight-binding models, and k ⋅ p theory have stood for many years as the standard techniques for computing electronic states in crystalline solids. Here, we present the first new method in decades, which we call atomistic k ⋅ p theory. In its usual formulation, k ⋅ p theory has the advantage of depending on parameters that are directly related to experimentally measured quantities, however, it is insensitive to the locations of individual atoms. We construct an atomistic k ⋅ p theory by defining envelope functions on a grid matching the crystal lattice. The model parameters are matrix elements which are obtained from experimental results or ab initio wave functions in a simple way. This is in contrast to the other atomistic approaches in which parameters are fit to reproduce a desired dispersion and are not expressible in terms of fundamental quantities. This fitting is often very difficult. We illustrate our method by constructing a four-band atomistic model for a diamond/zincblende crystal and show that it is equivalent to the sp3 tight-binding model. We can thus directly derive the parameters in the sp3 tight-binding model from experimental data. We then take the atomistic limit of the widely used eight-band Kane model and compute the band structures for all III-V semiconductors not containing nitrogen or boron using parameters fit to experimental data. Our new approach extends k ⋅ p theory to problems in which atomistic precision is required, such as impurities, alloys, polytypes, and interfaces. It also provides a new approach to multiscale modeling by allowing continuum and atomistic k ⋅ p models to be combined in the same system.