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

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

The above definition has a simple physical interpretation: An isolated quantum system has no Berry phase. A Berry phase arises only when the quantum system such as the electrons in crystal are interacting with an external measurement setup. Even though polarization is defined as an absolute quantity, it is the differences in polarization that are actually measured in experiment. From the Berry phase of the electronic wavefunction, we can predict such observable effects such as spontaneous polarization which is the macroscopic polarization difference

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

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

Motivation

File:TestCM.svg

An example CM.

File:TestCM.svg

Induced charge density in Silicon.

In the conventional Claussius-Mossati model(CM), the macroscopic polarization is defined as the sum of dipole moments in a given cell divided by cell volume $P={\frac {d_{cell}}{V_{cell}}}$. One extreme case where the CM model is applicable is ionic crystals such as Sodium Chloride. Here we can identify well defined “polarization centers” and partition the crystal with the condition that induced charge density vanishes at the cell boundaries. This simple picture is not true for real materials. Crystalline Silicon is an example of a crystal with covalent bonding. Here the induced charge density is completely delocalized and trying to partition this periodic and continuous distribution with well defined polarization centers is ambiguous at best. In typical Ferroelectric materials, the bonding has a mixed ionic/covalent nature with electrons shared among many ions.

Failure of defining polarization in terms of charge distribution

The dipole moment is defined from the charge distribution as $d=\int _{V}d\mathbf {r} \mathbf {r} \rho (\mathbf {r} )$ where V is the volume under consideration. We can consider two cases:

(a)Thermodynamic limit: The polarization can be defined over the entire sample as $P_{sample}={\frac {d_{sample}}{V_{sample}}}$. The integral has contribution from both the surface and the bulk regions which cannot be separated easily. Consider the case of piezoelectricity in a cubic sample of dimension Failed to parse (unknown function "\math"): {\displaystyle L^3<\math>. Before an electric field was applied, the crystal was unstrained and by symmetry the polarization is zero. When an electric field is applied, there is a build up of charges on the surface. The surface charge density <math>\sigma} scales with the size of the sample. For the cubic sample, the change in polarization(before and after the electric field was applied) is $\Delta P={\frac {(\sigma L^{2})L}{L^{3}}}=\sigma$ and may be considered as an acceptable definition of polarization in piezoelectricity. But it has no information about the induced charge density in the interior of the sample and whether the change in polarization is actually a bulk or surface effect. In fact, the notion of piezoelecticity as a bulk or surface effect was debated{[}{]} until recently and was clarified as a bulk effect with the advent of the modern theory of polarization.

(b)Unit Cell: