4.6. Bravais lattices and irreducible Brillouin zones

Table 4.6.1 Type of lattices recognized by RESCU. The lattice vectors are designated by \(\mathbf{a}_1\), \(\mathbf{a}_2\) and \(\mathbf{a}_3\) and the reciprocal lattice vectors by \(\mathbf{b}_1\), \(\mathbf{b}_2\) and \(\mathbf{b}_3\). Lower case letters refer to primitive cell vectors while capital letters refer to conventional cell vectors.
Bravais lattice	BZ variation	Criteria
Simple cubic	CUB	\(a_1=a_2=a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31}=\pi/2\)
Face-centered cubic	FCC	\(a_1=a_2=a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31}=\pi/3\)
Body-centered cubic	BCC	\(a_1=a_2=a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31}=\mathrm{acos}(-1/3)\)
Rhombohedral	RHL1	\(a_1=a_2=a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31} < \pi/2\)
Rhombohedral	RHL2	\(a_1=a_2=a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31} > \pi/2\)
Body-centered tetragonal	BCT1	\(a_1=a_2=a_3\) and \(\theta_{23}=\theta_{31}\) and \(\\|\mathbf{a}_2+\mathbf{a}_3\\| > \\|\mathbf{a}_1+\mathbf{a}_2\\|\)
Body-centered tetragonal	BCT2	\(a_1=a_2=a_3\) and \(\theta_{23}=\theta_{31}\) and \(\\|\mathbf{a}_2+\mathbf{a}_3\\| < \\|\mathbf{a}_1+\mathbf{a}_2\\|\)
Hexagonal	HEX	\((a_1=a_2 \lor a_2=a_3 \lor a_3=a_1)\) and \([\theta_{12},\theta_{23},\theta_{31}] = \pi[2/3,1/2,1/2]\)
Tetragonal	TET	\(a_1 = a_2\) and \(\theta_{12}=\theta_{23}=\theta_{31}=\pi/2\)
Orthorhombic	ORC	\(a_1 < a_2 < a_3\) and \(\theta_{12}=\theta_{23}=\theta_{31}=\pi/2\)
Face-centered orthorhombic	ORCF1	\(\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2\) and \(A_1 < A_2 < A_3\) and \(1/A_1^2 > 1/A_2^2+1/A_3^2\)
Face-centered orthorhombic	ORCF2	\(\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2\) and \(A_1 < A_2 < A_3\) and \(1/A_1^2 < 1/A_2^2+1/A_3^2\)
Face-centered orthorhombic	ORCF3	\(\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2\) and \(A_1 < A_2 < A_3\) and \(1/A_1^2 = 1/A_2^2+1/A_3^2\)
Body-centered orthorhombic	ORCI	\(\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2\) and \(A_1 < A_2 < A_3\)
Base-centered orthorhombic	ORCC	\(\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2\) and \(A_1 < A_2\)
Monoclinic	MCL	\(a_1 < a_3\) and \(a_2 < a_3\) and \(\theta_{12}=\theta_{31}=\pi/2\) and \(\theta_{23} < \pi/2\)
Base-centered monoclinic	MCLC1	\(\Theta_{12}=\Theta_{31}=\pi/2\) and \(\Theta_{23} < \pi/2\) and \(\gamma_{12} > \pi/2\)
Base-centered monoclinic	MCLC2	\(\Theta_{12}=\Theta_{31}=\pi/2\) and \(\Theta_{23} < \pi/2\) and \(\gamma_{12} = \pi/2\)
Base-centered monoclinic	MCLC3	\(\Theta_{12}=\Theta_{31}=\pi/2\) and \(\Theta_{23} < \pi/2\) and \(\gamma_{12} < \pi/2\) and \(A_2/A_3\cos(\Theta_{23})\) \(+ A_2^2/A_1^2\sin(\Theta_{23})^2 < 1\)
Base-centered monoclinic	MCLC4	\(\Theta_{12}=\Theta_{31}=\pi/2\) and \(\Theta_{23} < \pi/2\) and \(\gamma_{12} < \pi/2\) and \(A_2/A_3\cos(\Theta_{23})\) \(+ A_2^2/A_1^2\sin(\Theta_{23})^2 = 1\)
Base-centered monoclinic	MCLC5	\(\Theta_{12}=\Theta_{31}=\pi/2\) and \(\Theta_{23} < \pi/2\) and \(\gamma_{12} < \pi/2\) and \(A_2/A_3\cos(\Theta_{23})\) \(+ A_2^2/A_1^2\sin(\Theta_{23})^2 > 1\)

This section complements Band structure calculation which introduced in passing the methodology to identify Brillouin zone variations. We begin by listing the Brillouin zone variations recognized by RESCU along with the criteria used in the identification in Table 4.6.1. We use the following convention for the angles

\[\begin{split}\begin{aligned} \theta_{12}=\mathbf{a}_1\angle\mathbf{a}_2\\ \theta_{23}=\mathbf{a}_2\angle\mathbf{a}_3\\ \theta_{31}=\mathbf{a}_3\angle\mathbf{a}_1\end{aligned}\end{split}\]

and

\[\begin{split}\begin{aligned} \gamma_{12}=\mathbf{b}_1\angle\mathbf{b}_2\\ \gamma_{23}=\mathbf{b}_2\angle\mathbf{b}_3\\ \gamma_{31}=\mathbf{b}_3\angle\mathbf{b}_1\end{aligned}\end{split}\]

Also, lower case letters refer to primitive cell vectors while capital letters refer to conventional cell vectors.

For BCT lattices, RESCU requires that \(\theta_{23}=\theta_{31}\). The condition \(\|\mathbf{a}_2+\mathbf{a}_3\| > \|\mathbf{a}_1+\mathbf{a}_2\|\) for the primitive lattice then correspond to \(c < a\) in the conventional lattice. If \(\theta_{23}=\theta_{31}\) is not met, RESCU recognizes the lattice and tells the user to change the domain such that \(\theta_{23}=\theta_{31}\) is satisfied. This is generally necessary for all irreducible Brillouin zone corners to be correctly defined. In fact, RESCU will recognize and issue a warning for any lattice which does not meet the prescribed requirements.

For HEX lattices, RESCU requires that the lattice vector perpendicular to the hexagonal planes be \(\mathbf{a}_3\) and that the angle between the hexagonal plane lattice vectors be 120 degrees.

For TET lattices, the equal magnitude lattice vectors must be \(\mathbf{a}_1\) and \(\mathbf{a}_2\).

For ORC lattices, the lattice vectors must be ordered according to their magnitude.

For ORCF lattices, the criteria are clearer in terms of conventional unit cell measures. The conventional unit cell vectors are recovered using

\[\begin{split}\begin{aligned} \mathbf{A}_1 = \mathbf{a}_2 + \mathbf{a}_3 - \mathbf{a}_1\\ \mathbf{A}_2 = \mathbf{a}_3 + \mathbf{a}_1 - \mathbf{a}_2\\ \mathbf{A}_3 = \mathbf{a}_1 + \mathbf{a}_2 - \mathbf{a}_3\end{aligned}\end{split}\]

For ORCI lattices, the criteria are clearer in terms of conventional unit cell measures. The conventional unit cell vectors are recovered using

\[\begin{split}\begin{aligned} \mathbf{A}_1 = \mathbf{a}_2 + \mathbf{a}_3\\ \mathbf{A}_2 = \mathbf{a}_3 + \mathbf{a}_1\\ \mathbf{A}_3 = \mathbf{a}_1 + \mathbf{a}_2\end{aligned}\end{split}\]

For ORCC lattices, the criteria are clearer in terms of conventional unit cell measures. The conventional unit cell vectors are recovered using

\[\begin{split}\begin{aligned} \mathbf{A}_1 = \mathbf{a}_1 + \mathbf{a}_2\\ \mathbf{A}_2 = \mathbf{a}_1 - \mathbf{a}_2\\ \mathbf{A}_3 = \mathbf{a}_3\end{aligned}\end{split}\]

For MCLC lattices, the criteria are clearer in terms of the reciprocal vectors and conventional unit cell measures. The conventional unit cell vectors are recovered using

\[\begin{split}\begin{aligned} \mathbf{A}_1 = \mathbf{a}_1 - \mathbf{a}_2\\ \mathbf{A}_2 = \mathbf{a}_1 + \mathbf{a}_2\\ \mathbf{A}_3 = \mathbf{a}_3\end{aligned}\end{split}\]

Table 4.6.2 Brillouin zone variations recognized by RESCU and default k-point lines. A \(-\) symbol indicates that two points are linked. A \(+\) symbol indicates the beginning of a new line.
BZ variation	k-point line
CUB	\(\Gamma\)-X-M-\(\Gamma\)-R-X+M-R
FCC	\(\Gamma\)-X-W-K-\(\Gamma\)-L-U-W-L-K+U-X
BCC	\(\Gamma\)-H-N-\(\Gamma\)-P-H+P-N
RHL1	B-Z-\(\Gamma\)-X+\(\Gamma\)-L-B₁+Q-F-P₁-Z+L-P
RHL2	\(\Gamma\)-P-Z-Q-\(\Gamma\)-F-P₁-Q₁-L-Z
BCT1	\(\Gamma\)-X-M-\(\Gamma\)-Z-P-N-Z₁-M+X-P
BCT2	\(\Gamma\)-X-Y-\(\Sigma\)-\(\Gamma\)-Z-\(\Sigma_1\)-N-P-Y₁-Z+X-P
HEX	\(\Gamma\)-M-K-\(\Gamma\)-A-L-H-A+L-M+K-H
TET	\(\Gamma\)-X-M-\(\Gamma\)-Z-R-A-Z, X-R, M-A
ORC	\(\Gamma\)-X-S-Y-\(\Gamma\)-Z-U-R-T-Z+Y-T+U-X+S-R
ORCF1	\(\Gamma\)-Y-T-Z-\(\Gamma\)-X-A₁-Y+T-X₁+X-A-Z+L-\(\Gamma\)
ORCF2	\(\Gamma\)-Y-T-Z-\(\Gamma\)-X-A₁-Y+X-A-Z+L-\(\Gamma\)
ORCF3	\(\Gamma\)-Y-C-D-X-\(\Gamma\)-Z-D₁-H-C+C₁-Z+X-H₁+H-Y+L-\(\Gamma\)
ORCI	\(\Gamma\)-X-L-T-W-R-X₁-Z-\(\Gamma\)-Y-S-W+L₁-Y+Y₁-Z
ORCC	\(\Gamma\)-X-S-R-A-Z-\(\Gamma\)-Y-X₁-A₁-T-Y+Z-T
MCL	\(\Gamma\)-Y-H-C-E-M₁-A-X-\(\Gamma\)-Z-D-M+Z-A+D-Y+X-H₁
MCLC	\(\Gamma\)-Y-F-L-I+I₁-Z-\(\Gamma\)-X+X₁-Y+M-\(\Gamma\)-N+Z-F₁
MCLC2	\(\Gamma\)-Y-F-L-I+I₁-Z-F₁+N-\(\Gamma\)-M
MCLC3	\(\Gamma\)-Y-F-H-Z-I-X-\(\Gamma\)-Z+M-\(\Gamma\)-N+X-Y₁-H₁+I-F₁
MCLC4	\(\Gamma\)-Y-F-H-Z-I+H₁-Y₁-X-\(\Gamma\)-N+M-\(\Gamma\)
MCLC5	\(\Gamma\)-Y-F-L-I+I₁-Z-\(\Gamma\)-X-Y₁-H₁+H-F₁+F\(_2\)-X+M-\(\Gamma\)-N+H-Z

Once the Brillouin zone variation is identified, the corners of the irreducible Brillouin zone are labelled according to [SC10]; we do not reproduce the definitions here. The paths covering the irreducible Brillouin zones edges are different in certain cases, so they are listed in table Table 4.6.2. It is generally not possible to visit all edges using a single non-redundant path, and hence default paths generally include many k-point lines.

[SC10]

Wahyu Setyawan and Stefano Curtarolo. High-throughput electronic band structure calculations: Challenges and tools. Computational Materials Science 49.2 (2010), pp. 299 –312.