User:Jp physics/modern theory of polarization

The modern theory of polarization states that the macroscopic polarization is defined as the Berry phase of the electron Bloch wavefunctions. The modern theory has been highly successful as a first principles computational tool in determining the spontaneous polarization of ferroelectric crystals. The very first example to which it was applied to was the perovskite $KNbO_{3}$ in the tetragonal phase where it has predicted a value of 0.35 $C/m^{2}$ against the measured value of 0.30 $C/m{}^{2}$.

The modern theory of polarization relies on the periodicity of the crystal lattice potential for which the wavefunctions take the Bloch form. Therefore it applies to the conditions of zero temperature and zero electric field for which the potential is still periodic. When a non-zero electric field is applied, Zener tunneling becomes important. It has been shown that Bloch wavefunctions can still be applied but now the discretization of the mesh in k-space depends on the magnitude of electric fields{[}{]}

Definition

The general three dimensional multiband formulation of the absolute value of macroscopic electronic polarization is defined as

$P_{el}^{\lambda }={\frac {e}{(2\pi )^{3}}}\sum _{n}\int _{BZ}d^{3}\mathbf {k} <u_{n\mathbf {k} }^{\lambda }|i\nabla _{\boldsymbol {\mathbf {k} }}|u_{n\mathbf {k} }^{\lambda }>$

where

• n is the band index
• BZ indicates the boundaries of the Brillouin zone
• $\lambda$ is a dimensionless scalar parameter which is related to the coupling of the sytem to the measurement setup. For example

in a ferroelectric measurement, it relates to the amplitude of the crystal distortion induced by controlling the temperature.

• $\psi _{n\mathbf {k} }^{\lambda }(\mathbf {r} )=e^{i\mathbf {k} .\mathbf {r} }u_{n\mathbf {k} }^{\lambda }(\mathbf {r} )$ is the Bloch state of band n in the crystal and $u_{n\mathbf {k} }^{\lambda }(\mathbf {r} )$ has the periodicity of the crystal lattice potential

The polarization can be recast in simpler forms by noticing the Berry connection $A_{n}^{\lambda }(\mathbf {k} )=<u_{n\mathbf {k} }^{\lambda }|i\nabla _{\boldsymbol {\mathbf {k} }}|u_{n\mathbf {k} }^{\lambda }>$ which leads to the Berry phase for a single band

$\phi _{n}^{\lambda }=\int _{BZ}A_{n}^{\lambda }(\mathbf {k} )d^{3}\mathbf {k}$

We can now write the polarization in terms of the Berry phase

$P_{el}^{\lambda }={\frac {e}{(2\pi )^{3}}}\sum _{n}\phi _{n}^{\lambda }$

An isolated quantum system has no Berry phase. A Berry phase arises only when the quantum system is interacting with an external measurement setup. For the case of polarization, we see that this interaction gives rise to a Berry phase in the electronic wavefunction which leads to observable effects such as macroscopic polarization differences

$\Delta P=P_{el}^{\lambda =1}-P_{el}^{\lambda =0}$

where $\lambda =0$ relates to the unstrained crystal and $\lambda =1$ relates to the strained crystal.