5.1: Introduction to Bloch’s Theorem
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Bloch’s theorem identifies the important features of basis functions for the group of lattice translation operations and creates a foundation for solving Schrödinger’s equation. For any crystalline solid, the potential energy operator and, therefore, the Hamiltonian operator have the full periodicity of the lattice, i.e, \(V\left( \boldsymbol{r} + \boldsymbol{T} \right) = V(\boldsymbol{r})\) and \(H\left( \boldsymbol{r} + \boldsymbol{T} \right) = H(\boldsymbol{r})\), where \(\boldsymbol{T}\) is a Bravais lattice vector. One statement of Bloch’s theorem is that the wavefunctions for a particle in such a periodic potential take the form
\[\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) = A_{n\boldsymbol{k}}(\boldsymbol{r})e^{i\boldsymbol{k} \cdot \boldsymbol{r}} \nonumber \]
in which \(A_{n\boldsymbol{k}}(\boldsymbol{r})\) has the full periodicity of the lattice, i.e., \(A_{n\boldsymbol{k}}\left( \boldsymbol{r} + \boldsymbol{T} \right) = A_{n\boldsymbol{k}}(\boldsymbol{r})\). The functions \(\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right)\) are modified plane waves. The wavevector \(\boldsymbol{k}\) identifies the IR of the group of lattice translations, and the index \(n\) corresponds to any other distinguishing features of the wavefunction, often arising from rotational symmetry. This plane-wave type of wavefunction is best suited for weak crystalline potentials. If the potential energy operator is strong in certain regions of the crystal, then a different expression for these wavefunctions is warranted.
A useful alternative expression of Bloch’s theorem concerning \(\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right)\) is
\[\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} + \boldsymbol{T} \right) = e^{i\boldsymbol{k} \cdot \boldsymbol{T}}\psi_{n\boldsymbol{k}}(\boldsymbol{r}) \nonumber \]
Any function \(\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right)\) satisfying this equation is called a Bloch function. According to this equation, once \(\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right)\) is known in one unit cell of a crystal, then the function can be immediately evaluated throughout the entire crystal. As a result, calculations of the electronic and vibrational states of crystalline solids can be accomplished by solving the Schrödinger equation for one primitive unit cell.
The second expression of Bloch’s theorem is couched in the fundamentals of group theory because Bloch functions are basis functions for the IRs of the group of lattice translations. To show this, the IR labeled by the wavevector \(\boldsymbol{k}\) for the translation operation \((1|\boldsymbol{T})\) is
\[\Gamma^{(\boldsymbol{k})}\left\lbrack (1|\boldsymbol{T}) \right\rbrack = e^{- i\boldsymbol{k} \cdot \boldsymbol{T}} \nonumber \]
If \(\psi_{n\boldsymbol{k}}(\boldsymbol{r})\) is a basis function for this IR, then
\[\left( 1 \middle| \boldsymbol{T} \right)\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) = e^{- i\boldsymbol{k} \cdot \boldsymbol{T}}\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right). \nonumber \]
The effect of the transformation \(\left( 1 \middle| \boldsymbol{T} \right)\) on \(\psi_{n\boldsymbol{k}}(\boldsymbol{r})\) is also (see (38))
\[\left( 1 \middle| \boldsymbol{T} \right)\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) = \psi_{n\boldsymbol{k}}\left( \left( 1 \middle| \boldsymbol{T} \right)^{- 1}\boldsymbol{r} \right) = \psi_{n\boldsymbol{k}}\left( \left( 1 \middle| \boldsymbol{-}\boldsymbol{T} \right)\boldsymbol{r} \right) = \psi_{n\boldsymbol{k}}\left( \boldsymbol{r -}\boldsymbol{T} \right) \nonumber \]
Therefore,
\[\psi_{n\boldsymbol{k}}\left( \boldsymbol{r -}\boldsymbol{T} \right) = e^{- i\boldsymbol{k} \cdot \boldsymbol{T}}\psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) \text{ or } \psi_{n\boldsymbol{k}}\left( \boldsymbol{r} \right) = {e^{i\boldsymbol{k} \cdot \boldsymbol{T}}\psi}_{n\boldsymbol{k}}\left( \boldsymbol{r -}\boldsymbol{T} \right) \nonumber \]
If \(\boldsymbol{r}\) is replaced by \(\boldsymbol{r} + \boldsymbol{T}\), then the second expression of Bloch’s theorem is obtained.