Bengtson et al. Impact of substitutional and interstitial carbon defects on lattice parameters in MgB2
Supplementary Information
The MgB2 structures used in the calculations and the substitution and interstitial sites in
bulk MgB2 are shown in figure S1. In order to average over different C interactions, multiple
configurations were used at low concentrations. For the substitution calculations, C was
substituted for B in the super-cells of SC, 2xbSC and 2xaSC2xbSC (figure S1) resulting in atomic C
concentrations of 1/12, 1/24, 1/36, 1/48 for SC; 1/24, 1/48, 1/72, 1/96 for 2xbSC; and 1/48, 1/96
for 2xaSC2xbSC. At higher C concentrations, the calculations are inherently highly ordered due to
the periodicity of the unit cell. The disordered solid solution can be approximately modeled
using Special Quasirandom Structures (SQS) 16. Seven SQS structures were generated with the
Alloy Theory Automated Toolkit (ATAT) 17 at concentrations of 1/12 (2 structures) and 1/9 (5
structures). For the interstitial calculations, 1 C was placed in the B ring of super-cells SC,
2xbSC, 2xaSC and 2xaSC2xbSC (figure S1) resulting in C concentrations of 1/13, 1/25, 1/37, 1/49
for SC; 1/25, 1/49, 1/73, 1/97 for 2xbSC and 2xaSC; and 1/49, 1/97 for 2xaSC2xbSC (other
interstitial sites were considered but not found to be stable, as discussed in the main text).
The c-lattice parameter is measured as the total height (along the z-axis) of the super-cell
divided by the number of z-layers. The equivalent of the a-lattice parameter of the primitive cell
(shown in figure S1) is not always clear in cases where defects create significant distortion. The
primitive cell a-lattice parameter for complex distorted super-cells is therefore calculated by
determining the value it would have in an equivalent undistorted cell with the same volume, clattice parameter and number of MgB2 units. This approach gives a  VN L
6

3cN MgB2 ,
where V is the total volume of the super-cell, c is the magnitude of the c-lattice parameter, NL is
the number of B (or Mg) layers, and N MgB is the total number of MgB2 formula units.
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Bengtson et al. Impact of substitutional and interstitial carbon defects on lattice parameters in MgB2
Figure S1. MgB2 super-cell structures. Mg (large open circles) are 0.5*c1 below the B (smaller
black circles). The basic super-cell, SC, is given by a SC=[6.15,0,0], bSC=[0,5.33,0], c1=[0,0,3.52].
The c1-lattice parameter is the distance in z between two Mg atoms. Larger super-cells 2xbSC,
2xaSC2xbSC, and 2xaSC are created from SC. 2xbSC doubles b, 2xaSC doubles a, and 2xaSC2xbSC
doubles a and b. Different cells are also created by scaling the c-lattice parameter. SC, 2xbSC,
and 2xaSC are scaled in the z-direction by c1, c2=2c1, c3=3c1, and c4=4c1. 2xaSC2xbSC is scaled by
c1 and c2 only, because of its larger extent in the basil plane. Substitution sites for C are in gray.
The open X denotes interstitial sites. The a-lattice parameter of the primitive cell is shown in SC.
All structures represent the initial configuration before relaxation.
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