Use of Elementary Group Theory in Calculating Band Structure

Abstract

For a three dimensional crystal the free electron energies and wave functions can be expressed in the Bloch function form in the following way: 1. Write the plane wave wave vector as a sum of a Bloch wave vector and a reciprocal lattice vector. The Bloch wave vector $\mathbf k$ is restricted to the first Brillouin zone; the reciprocal lattice vectors are given by $\mathbf K_{\mathbf {\ell }} = l_1 \mathbf b_1 + l_2 \mathbf b_2 + l_3 \mathbf b_3$ where $\left( l_1, l_2, l_3\right) = \mathbf {\ell }$ are integers and $\mathbf b_i$ are primitive translations of the reciprocal lattice. Then $\varPsi _{\mathbf {\ell }} (\mathbf k, \mathbf r) = \mathrm{e}^{i \mathbf k \cdot \mathbf r} \mathrm{e}^{i \mathbf K_{\mathbf {\ell }} \cdot \mathbf r}$.

Appendices

Problems

5.1

Consider the empty lattice band of a two-dimensional square lattice. At $E(\mathrm X) =0.25 \frac{h^2}{2ma^2}$ there are two degenerate bands. At $E(\mathrm X) =1.25 \frac{h^2}{2ma^2}$ there are four. Determine the linear combinations of degenerate states at these points belonging to the IR’s of $\mathcal {G}_\mathrm{X}$. Do the same for $E(\mathrm \varGamma ) =\frac{h^2}{2ma^2} \text{ and } 2 \frac{h^2}{2ma^2}$.

5.2

Consider the group of a two-dimensional square lattice. Use your knowledge of the irreducible representations at $E(\varGamma )=0, 1, 2$ (in units of $\frac{h^2}{2ma^2}$) and at $E(\mathrm{X}) = 0.25$ and 1.25, together with the compatibility relations to determine the irreducible representations for each of these bands along the line $\varDelta$.

5.3

Tabulate $E(\varGamma ), E(\mathrm{H}), E(\mathrm{P})$ for all bands that have $E \le {}4~[\frac{h^2}{2ma^2}]$ for a bcc lattice. Then sketch E vs. $\mathbf k$ along $\varDelta (\varGamma \rightarrow \mathrm{H})$ and along $\Lambda (\varGamma \rightarrow \mathrm{P})$.

5.4

Do the same as above in Problem 5.3 for a simple cubic lattice where

$$ E_{\mathbf {\ell }}=\frac{h^2}{2ma^2} \left[ (\ell _1+\xi )^2 +(\ell _2+\eta )^2 +(\ell _3+\zeta )^2 \right] $$

for $\varGamma$, $\mathrm{X} =\frac{\pi }{a}(1,0,0)$, and $\mathrm{R}=\frac{\pi }{a}(1,1,1)$. Sketch E vs. $\mathbf k$ along $\varGamma \rightarrow \mathrm{X}$ and along $\varGamma \rightarrow \mathrm{R}$ for all bands having $E_{\mathbf {\ell }} \le 4 ~[\frac{h^2}{2ma^2}]$.

5.5

Use the irreducible representations at $E(\mathrm X) = 0.25~[\frac{h^2}{2ma^2}]$ of a square lattice to evaluate

$$ V_{ij} =\langle \varPsi _{\mathrm X_i}(0.25)\mid V(\mathbf r)\mid \varPsi _{\mathrm X_j}(0.25)\rangle $$

where $\varPsi _{\mathrm X_i}(0.25)$ is the wave function at $E(\mathrm X)=0.25~[\frac{h^2}{2ma^2}]$ belonging to the irreducible representation $\mathrm X_i$.

• (a) Show that $V_{ij} = 0$ if $i \ne j$.

• (b) Show that the diagonal matrix elements give the same energies (and band gap) as obtained by degenerate perturbation theory with the original plane waves.

5.6

A two dimensional rectangular lattice has a reciprocal lattice whose primitive translations, including the $2\pi$, are $\mathbf b_1 = \frac{2\pi }{a}\hat{x}$ and $\mathbf b_2 = \frac{2\pi }{a} \frac{1}{\sqrt{2}}\hat{y}$.

• (a) List the operations belonging to $\mathrm G_\varGamma$.

• (b) Do the same for $\mathrm G_{\mathrm X}$ and $\mathrm G_\varDelta$.

• (c) For the empty lattice the wave functions and energies can be written $\psi _{\mathbf {l}} (\mathbf k,\mathbf r) = \exp {i(\mathbf k+\mathbf K_{\mathbf {l}})\cdot \mathbf r}$ and $E_{\mathbf {l}} (\mathbf k) = \frac{\hbar ^2}{2m}\left( \mathbf k +\mathbf K_{\mathbf {l}}\right) ^2$. Here, $\mathbf K_{\mathbf {l}} = l_1\mathbf b_1 + l_2\mathbf b_2$, and $l_1$ and $l_2$ are integers. Tabulate the energies at $\varGamma$ and at $\mathrm X$ for $(l_1,l_2)=$ (0, 0), $(0, \pm 1)$, (−1, 0), (1, 0), and $(-1, \pm 1)$.

• (d) Sketch (straight lines are OK) E vs. k along the line $\varDelta$ (going from $\varGamma$ to X) for these bands.

• (e) Two degenerate bands at the point $E(\varGamma )=0.5$ connect to $E(\mathrm X)=0.75$. Write down the wave functions for an arbitrary value of $k_x$ for these two bands.

• (f) From these wave functions, construct the linear combinations belonging to irreducible representations of $\mathrm G_{\varDelta }$.

5.7

Graphene has a two-dimensional regular hexagonal reciprocal lattice whose primitive translations are $\mathbf {b}_1 = \frac{2\pi }{a}(1,-\frac{1}{\sqrt{3}})$ and $\mathbf {b}_2 = \frac{2\pi }{a}(0,\frac{2}{\sqrt{3}})$.

• (a) List the operations belonging to $\mathrm G_\varGamma$.

• (b) Do the same for $\mathrm G_{\mathrm K}$ and $\mathrm G_{\mathrm M}$. Note that $\mathbf k_\mathrm{K}=\frac{2\pi }{a} (\frac{2}{3}, 0)$ and $\mathbf k_\mathrm{M}=\frac{2\pi }{a}(\frac{1}{2},\frac{1}{2\sqrt{3}})$.

• (c) Write down the empty lattice wave functions and energies $\varPsi _{\mathbf {l}} (\mathbf k,\mathbf r)$ and $E_{\mathbf {l}} (\mathbf k)$ at $\varGamma$ and $\mathrm K$.

• (d) Tabulate the energies at $\varGamma$ and at $\mathrm K$ for $(l_1,l_2)=$ (0, 0), $(0, \pm 1)$, $(\pm 1, 0)$, (−1, −1), and (1, 1).

• (e) Sketch E versus k along the line going from $\varGamma$ to K for these bands.

• (f) Write down the wave functions for the three fold degenerate bands at the energy $E(\mathrm{K})=4/9$.

5.8

Construct linear combinations of 15 plane waves $w_i$ $(i=1, 2, \ldots , 15)$, which are given in the text, to construct $\varPsi _i$ belonging to irreducible representations of the group of the wave vectors $\varGamma$, L, and X for the diamond structure.

Summary

In this chapter we first reviewed elementary group theory and studied the electronic band structure in terms of elementary concepts of the group theory. We have shown that how group theory ideas can be used in obtaining the band structure of a solid. Group representations and characters of two dimensional square lattice are discussed in depth and empty lattice bands of the square lattice are illustrated. Concepts of irreducible representations and compatibility relations are used in discussing the symmetry character of bands connecting different symmetry points and the removal of band degeneracies. We also discussed empty lattice bands of the cubic system and sketched the band calculation of common semiconductors.

The starting point for many band structure calculations is the empty lattice band structure. In the empty lattice band representation, each band is labeled by $\mathbf {\ell }= (l_1, l_2, l_3)$ where the reciprocal lattice vectors are given by

$$ \mathbf K_{\mathbf {\ell }} = l_1 \mathbf b_1 + l_2 \mathbf b_2 + l_3 \mathbf b_3 $$

where $\left( l_1, l_2, l_3\right) = \mathbf {\ell }$ are integers and $\mathbf b_i$ are primitive translations of the reciprocal lattice. Energy eigenvalues and eigenfunctions are written as

$$ E_{\mathbf {\ell }}(\mathbf k) = \frac{\hbar ^2}{2m} \left( \mathbf k + \mathbf K_{\mathbf l}\right) ^2 $$

and

$$ \varPsi _{\mathbf {\ell }} (\mathbf k, \mathbf r) = \mathrm{e}^{i \mathbf k \cdot \mathbf r} \mathrm{e}^{i \mathbf K_{\mathbf {\ell }} \cdot \mathbf r}. $$

The Bloch wave vector $\mathbf k$ is restricted to the first Brillouin zone.

The vector space formed by the degenerate bands at $E(\mathbf k)$ is invariant under the operations of the group of the wave vector $\mathbf k$. That is, the space of degenerate states at a point $\mathbf k$ in the Brillouin zone provides a representation of the group of the wave vector. $\mathbf k$. We can decompose this representation into its irreducible components and use the decomposition to label the states.

When we classify the degenerate states according to the IR’s of the group of the wave vector, we are able to simplify the secular equation by virtue of a fundamental theorem on matrix elements:

• 1. The matrix elements of V between different IR’s vanish, so many off-diagonal matrix elements are zero. This reduces the determinant equation to a block diagonal form.

• 2. The diagonal matrix elements $\langle \varGamma _j n \left| V\right| \varGamma _j n\rangle$ are, in general, different for different IR’s $\varGamma _j$. This lifts the degeneracy at the symmetry points.

Many common semiconductors which crystallize in the cubic zincblende structure have valence–conduction band structures that are quite similar in gross features. This results from the fact that each atom has four electrons outside a closed shell and there are two atoms per primitive unit cell.