Wannier representation

Three-dimensional model of the Wannier function of BaTiO ₃

After the Swiss physicist Gregory Wannier Hugh called Wannier function is a term used in solid state physics . In the tight-binding approximation, a description of the electronic wave functions in the lattice-period Bloch basis is no longer useful. Rather, one constructs the state function from atomic wave functions. These are not orthonormal. From the Bloch functions , however, an orthonormal basis of localized states can be constructed:

${\displaystyle \omega _{in}({\vec {r}}-{\vec {R}}_{i})={\frac {1}{\sqrt {N}}}\sum _{k}e^{-i{\vec {k}}{\vec {R}}_{i}}\cdot \psi _{n{\vec {k}}}({\vec {r}})}$

It is

• ${\displaystyle \psi _{n{\vec {k}}}({\vec {r}})}$ a Bloch function
• ${\displaystyle \omega _{in}({\vec {r}}-{\vec {R}}_{i})}$ the associated Wannier state
• ${\displaystyle e}$the Euler's number
• ${\displaystyle i}$the imaginary unit
• ${\displaystyle {\vec {k}}}$the wave vector
• ${\displaystyle {\vec {r}}}$the position vector
• ${\displaystyle n}$the band index.

The reverse construction of the Bloch states from the Wannier states is then called

${\displaystyle \psi _{n{\vec {k}}}({\vec {r}})={\frac {1}{\sqrt {N}}}\sum _{{\vec {R}}_{i}}e^{i{\vec {k}}{\vec {R}}_{i}}\cdot \omega _{in}({\vec {r}}-{\vec {R}}_{i})}$

The larger the lattice constant , the more strongly the Wannier states are localized. They are getting closer and closer to the atomic states. Instead of simply equating the Wannier state with an atomic state, it is approximated by a linear combination of atomic states ( LCAO ):

${\displaystyle \omega _{in}({\vec {r}}-{\vec {R}}_{i})=\sum _{n\in U}a_{n}\cdot \varphi _{n}({\vec {r}}-{\vec {R}}_{i})}$

The set U represents a subspace of the atomic states . ${\displaystyle \varphi _{n}({\vec {r}}-{\vec {R}}_{i})}$

literature

• Neil W. Ashcroft, N. David Mermin: Solid State Physics . 2nd Edition. Oldenbourg, Munich 2005, ISBN 3-486-57720-4 .
• Konrad Kopitzki: Introduction to Solid State Physics . 6th edition. Teubner, Wiesbaden 2007, ISBN 3-8351-0144-7 .
• Gerd Czycholl: Theoretical solid state physics . 3. Edition. Springer, Berlin 2008, ISBN 978-3-540-74789-5 .