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

$$ P_{el}^{\lambda}=\frac{e}{(2\pi)^{3}}\sum_{n}\int_{BZ}d^{3}\mathbf{k}$$ 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 total polarization should include the contribution from the positive point charges and is $$P_{total}=P_{el}^{\lambda}+\frac{e}{\Omega}\sum_{i}Z_{i}^{ion}\mathbf{r}_{i}$$ where
 * $$\Omega$$ is the primitive cell volume and
 * $$Z^{ion}$$ is the bare nuclear charge of the ions at position $$\mathbf{r}_{i}$$

The polarization can be recast in simpler forms by noticing the Berry connection $$A_{n}^{\lambda}(\mathbf{k})=$$ 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 $$ \Delta P=P_{el}^{\lambda=1}-P_{el}^{\lambda=0}+\Delta P_{ion}$$ Here $$\lambda=0$$ relates to the unstrained crystal and $$\lambda=1$$ relates to the strained crystal.

Failure of the Clausius-Mossotti Model
The Clausius-Mossotti(CM) is the standard textbook model that relates the local electric field to the macroscopic electric fields and polarization. Another definition of the CM model defines polarizability(a microscopic quantity) to the dielectric constant(macroscopic or bulk quantity). These definitions are assumed to be valid for all dielectrics.



Validity of the Clausius-Mossotti model
In the conventional CM model, 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. The CM model cannot be applied to these systems.

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 with the region of integration:

(a)Thermodynamic limit: The polarization can be defined over the entire sample(V 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 $$L^{3}$$. 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 $$\sigma$$ 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 effect rather than a surface effect is made clearer with the modern theory of polarization.

(b)Unit Cell: As argued earlier for crystalline Silicon, the choice of unit cell is ambiguous. In fact for a particular charge distribution the position, orientation and shape of the unit cell can lead to positive, zero or negative dipole moments. Only in the extreme limit of the ionic crystal is the polarization meaningful. The conclusion is that the knowledge of the charge distribution cannot be used to define the bulk polarization.

Polarization from adiabatic flow of currents


For the above reasons, instead of focussing on the charge distribution, it is perferable to focus on the charge flow in the interior of the sample during the process of adjusting some external parameter such as the turning on of an electric field, magnetic field, applying force, etc... Then the change in polarization can be written as $$ \Delta P=\int dt\frac{1}{V_{cell}}\int_{cell}drj(r,t)=\int dt$$ where $$ \frac{dP}{dt}=$$. This is the premise of the modern theory of polarization. It reflects the changes that happen in the bulk and is independent of the surface. This definition closely parallels the way spontaneous and induced polarization are measured in actual experiments. Let us consider the example of piezoelectricity. When the crystal is unstrained the polarization is zero. But when a force is applied to strain the crystal, an electrical current flows through the sample. When the sample is connected to electrodes, the charge is removed from the surface and the induced polarization is measured by integration of the current over time. The polarization does not depend on the final state of the crystal alone. The cell averaged current density has time dependence but is slow enough that adiabaticity is still true.It is convenient to introduce a parameter $$\lambda$$ which has the meaning of adiabatic time and we can write $$ \Delta P=P^{\lambda=1}-P^{\lambda=0}=\int_{0}^{1}d\lambda\frac{dP}{d\lambda}$$ In the case of Ferroelectric materials, $$\lambda$$ can refer to the scale of the sublattice displacements starting from a centrosymmetric structure($$\lambda=0$$) to one of the enantiomorphic forms($$\lambda=1$$) which lead to spontaneous polarization. This translates into an adiabatic change in the crystal Hamiltonian. Using adiabatic perturbation theory, the quantity $$\Delta P$$ can be computed.

Connection to the Berry Phase
For independent electrons in a crystalline system, the Bloch functions can be written as $$\psi_{nk}(r)=e^{ikr}u_{nk}(r)$$ with $$u_{nk}(r)$$ obeying periodic Born-von Karman boundary conditions over the unit cell ie. $$u_{nk}(r+R)=u_{nk}(r)$$. The Schrodinger equation takes the following form:

$$ [\frac{1}{2m}(p+\hbar k)^{2}+V^{\lambda}]|u_{nk}^{\lambda}>=E_{nk}|u_{nk}^{\lambda}>$$

We can identify the k in the above expression as a kind of vector potential(no magnetic field is involved). One should then expect to pick up a geometric phase as we vary k since we have a k dependent Hamiltonian but the eigenfunctions obeys k-independent boundary conditions. The mathematical description shows a Berry connection $$<\psi(\eta)|\nabla_{\eta}|\psi(\eta)>$$ for systems with a vector potential. For the system of electrons in a crystal, we can identify the parameter $$\eta\rightarrow k$$ and $$|\psi(\eta)>\rightarrow|u_{nk}>$$ from which the Berry connection is $$$$. The open loop Berry phase is then defined as $$ \phi_{n}^{\lambda}=\int_{BZ}d^{3}k$$

Position operator for periodic systems
The definition of the center of a localized periodic charge distribution which is Born–von Karman periodic $$|\psi(x+L)|^{2}=|\psi(x)|^{2}$$ is given by

$$ =\frac{L}{2\pi}\Im(\ln)$$ where $$X=\sum_{i=1}^{N}x_{i}$$ is the many body position operator for N electrons. Note that  is defined modulo L as expected for periodic systems. It can be shown that the expectation value reduces to $$=\frac{L}{2\pi}\int dk$$

Proof of the above result

It is easy to show the above result for the simplest case of one dimensional 1-band system. Then the electron wave-functions are $$|\Psi>=|\psi_{nk_{1}}>|\psi_{nk_{2}}>..|\psi_{nk_{N}}>$$ and the expectation value can be determined as

$$<\Psi|e^{\frac{i2\pi X}{L}}|\Psi>=<\psi_{nk_{1}}|e^{\frac{i2\pi x}{L}}|\psi_{nk'_{1}}><\psi_{nk_{2}}|e^{\frac{i2\pi x}{L}}|\psi_{nk'_{2}}>...<\psi_{nk_{N}}|e^{\frac{i2\pi x}{L}}|\psi_{nk'_{N}}>$$

The overlap integrals can be evaluated as

$$<\psi_{nk}|e^{\frac{i2\pi X}{L}}|\psi_{nk'}>=\int d^{3}r\psi_{nk}^{*}\psi_{nk'}e^{i\frac{2\pi x}{L}}=\int d^{3}ru_{nk}^{*}u_{nk'}e^{i(\frac{2\pi x}{L}+k'.r-k.r)}=$$ when $$k'=k+\Delta k$$ and zero otherwise. Here $$\Delta k=\frac{2\pi}{L}$$.

The expectation value of the position operator is now

$$ =\frac{L}{2\pi}\Im[\ln\Pi_{k}=\frac{L}{2\pi}\Im\sum_{k}[\ln]$$

We can expand $$|u_{nk+\Delta k}>=|u_{nk}>+\Delta k\nabla_{k}|u_{nk}>$$ which leads to

$$ =\frac{L}{2\pi}\Im\sum_{k}\Delta k[\ln(1-<u_{nk}|\nabla_{k}|u_{nk}>)]\rightarrow\frac{L}{2\pi}\int dk<u_{nk}|i\nabla_{k}|u_{nk}>$$ where in going to the last step we have used $$ln(1-x)\approx x$$. It is straightforward to generalize to multi-band 3d case, where the $$|\Psi>$$ are now Slater determinants.

Using the connection to the Berry phase, the expectation value of position operator can be written as

$$ <X>=\frac{L}{2\pi}\phi_{n}$$

Polarization from the adiabatic current flow
When determining polarization the quantity that is measured is the current that flows through the sample when a perturbation is adiabatically turned on. So $$ \Delta P=\int_{0}^{\Delta t}dt<j(t)>=P_{el}^{\lambda=1}-P_{el}^{\lambda=0}$$ where $$\Delta t$$ is large and $$<j(t)>$$ is small such that $$\Delta P$$ depends only on the initial and final states. We can extract the absolute value of polarization from the definition of electric current per unit length, $$ <j>=\frac{e}{L}\frac{d<X>_{\lambda}}{dt}=\frac{d}{dt}(\frac{e}{2\pi}\int dk<u_{nk}^{\lambda}|i\nabla_{k}|u_{nk}^{\lambda}>)=\frac{dP_{el}}{dt}$$ from which the absolute value of polarization can be related to the Berry phase for Bloch electrons

$$ P_{el}^{\lambda}=\frac{e}{2\pi}\int dk<u_{nk}^{\lambda}|\nabla_{k}|u_{nk}^{\lambda}>=\frac{e\phi_{n}^{\lambda}}{2\pi}$$

This central formula for macroscopic polarization in a crystal of independent electrons was originally derived by King-Smith & Vanderbilt using adiabatic perturbation theory.

Connection to Wannier Function
The expectation value of the position operator $$<X>$$ can also be obtained from the Wannier function function basis. Defining 'center of charge' of the Wannier function as $$\tau_{n}$$ we have $$ \tau_{n}=<0n|r|0n>=\frac{V}{(2\pi)^{3}}\int_{BZ}d^{3}\mathbf{k}<u_{n\mathbf{k}}|i\nabla_{\mathbf{k}}|u_{n\mathbf{k}}>$$ where the Wannier function is defined as $$|0n>=\frac{V}{(2\pi)^{3}}\int d^{3}\mathbf{k}|\psi_{nk}>$$. This leads to the Wannier dipole theorem $$ \Delta{P}=\sum_{ions}Z_{ion}e\Delta r_{ion}+\sum_{wf}(-2e)\Delta r_{wf}\,\!$$ which allows us to assign centers to the charge distribution and provides the local description of the dielectric response.

Polarization Quantum
The Berry phase is only defined mod $$2\pi$$. We can see this by considering a new set of Bloch wavefunctions which are phase shifted by a parameter $$\beta$$ which is a function of k:

$$ where $$\beta(k)$$ is real and obeys $$\beta(2\pi/a)=\beta(0)+2\pi m$$ such that $$\psi_{n,0}=\psi_{n,2\pi/a}$$. The Berry phase is now
 * \tilde{u_{nk}}>=e^{-\beta(k)}|u_{nk}>$$

$$ \tilde{\phi}_{n}=\phi_{n}+2\pi m$$

Since the polarization is defined as the Berry phase, it accumulates an additional contribution which for a 3D crystal is

$$ \tilde{P}_{n}=P_{n}+\frac{eR}{\Omega}$$

where $$R=\sum_{i}m_{i}R_{i}\,\!$$ is the lattice vector and $$\Omega$$ is the primitive cell volume. This leads to a lattice of polarization values each related to the other by the quantum $$\frac{eR}{\Omega}$$. Choosing one particular value of P is referred to as the choice of branch.