Structural effects of boron doping in diamond crystals for gamma-ray light-source applications: Insights from molecular dynamics simulations

TL;DR

This work uses atomistic molecular dynamics with MBN Explorer to quantify how substitutional boron doping distorts the diamond lattice, focusing on the lattice constant and inter-planar spacings of the (1 1 0) and (1 0 0) planes across multiple crystal sizes and three analysis regions (substrate, bulk, free edge). A geometric correction is applied to account for anisotropic, one-dimensional expansion due to the fixed substrate and periodic boundaries, enabling Vegard-like comparison. The results show a near-linear dependence of lattice constant and inter-planar spacings on boron concentration up to $x=0.05$, but with a pronounced deviation from Vegard's Law at higher dopant levels, attributed to high crystal quality and minimal boron clustering, as well as the absence of electronic (free-carrier) effects in the MD model. The findings provide practical benchmarks for gamma-ray crystal light-source design and offer a robust atomistic framework for modelling doped diamond crystals, with potential extensions to growth dynamics via kinetic Monte Carlo and to defect formation under irradiation.

Abstract

Boron-doped diamond crystals (BDD, C$_{1-x}$B$_{x}$) exhibit exceptional mechanical strength, electronic tunability, and resistance to radiation damage. This makes them promising materials for use in gamma-ray crystal-based light sources. To better understand and quantify the structural distortions introduced by doping, which are critical for maintaining channelling efficiency, we perform atomistic-level molecular dynamics simulations on periodic C$_{1-x}$B$_{x}$ systems of various sizes. These simulations allow the influence of boron concentration on the lattice constant and the (110) and (100) inter-planar distances to be evaluated over the concentration range from pure diamond (0%) to 5% boron at room temperature (300 K). Linear relationships between both lattice constant and inter-planar distance with increasing dopant concentration are observed, with a deviation from Vegard's Law. This deviation is larger than that reported by other theoretical and computational studies; however, this may be attributed to an enhanced crystal quality over these studies, a vital aspect when considering gamma-ray crystal light source design. The methodology presented here incorporates several refinements to closely reflect the conditions of microwave plasma chemical vapour deposition (MPCVD) crystal growth. Validation of the methodology is provided through a comprehensive statistical analysis of the structure of our generated crystals. These results enable reliable atomistic modelling of doped diamond crystals and support their use in the design and fabrication of periodically bent structures for next-generation gamma-ray light source technologies.

Figures (32)

• Figure 1: Representative diagram of the crystal configuration used in the simulations. Carbon atoms are shown in green, boron atoms in red, and substrate carbon atoms in blue. The orientations of the (1 1 0) and (1 0 0) crystallographic planes are indicated by solid black lines, while the boundaries between the three analysis regions are marked by black dashed lines. Further details on these regions are provided in the text.
• Figure 2: Representative diagrams of selected crystal structures used in this study, including the cubic crystal C1, slab crystals S1 and S2, and extended crystals E1A, E1B, and E1C. Carbon atoms are shown in green, boron dopants in red, and substrate carbon atoms in blue. Each structure shows a random dopant distribution corresponding to a boron concentration of $x = 0.05$ (5%). The complete set of crystal structures listed in \ref{['tab:Crystals']} is provided in the SI.
• Figure 3: Compilation of literature data showing the lattice constant $a_{\text{CB}}$ and relative lattice expansion $\Delta a_{\text{CB}}/a_{\text{C}}$ of C$_{1-x}$B$_{x}$ crystals as a function of boron concentration $x$. The solid green line represents the covalent radius modification to Vegard's Law Brunet_1998, while the dashed black line shows the atomic volume modification Brazhkin_2006. The solid blue line shows the results of lattice optimisation, which are discussed in \ref{['sec:LC']}, but presented here as a point of comparison. Comparison data are taken from multiple sources including experimental works from Brunet_1998 (open squares), Bustarret_2003 (open triangles), and Brazhkin_2006 (open circles), and ab initio calculations from Wojewoda_2008 (open stars).
• Figure 4: Plots of the lattice constant $a_{\text{CB}}$ as a function of boron dopant concentration. Panel (a) shows this for each crystal size listed in \ref{['tab:Crystals']}. The solid green line corresponds to Vegard's Law Brunet_1998, while the dashed black line represents the atomic volume modification Brazhkin_2006. The dashed grey line indicates the nominal lattice constant of pure diamond. Comparison data are taken from multiple sources including experimental works from Brunet_1998 (open squares), Bustarret_2003 (open triangles), and Brazhkin_2006 (open circles), and ab initio calculations from Wojewoda_2008 (open stars). Panel (b) shows a zoom in on lower concentrations and shows only the two extreme cases (C1 and E3C) to allow easier differentiation between the two lines.
• Figure 5: Heatmaps showing the average fraction of B$-$B atoms pairs (a) and B$_{3}$ clusters (b) relative to isolated B atoms as a function of boron concentration and crystal size. Each box reports the average percentage of B$-$B atom pairs out of all B atoms for the corresponding crystal size and dopant concentration.