Introduction
Dielectric materials are used to control and store charges and electric energies.They play a key role in modern electronics and electric power system.As the requirements grow for compact,low-cost electronic and electrical power system,as well as for very high energy and power capacitive storage systems,the development of high power and energy density dielectric materials becomes a major promising field. Dielectric constant is an important property for dielectric materials.In literature,there are several methods to calculate dielectric constant,the most popular one at present is density functional perturbation theory (DFPT).In this term paper,two methods will be presented,one is DFPT,the other one is beyond DFT,molecular dynamics.
Theory
In electrostatics,
(1)This is the fundamental equation from which dielectric constant is calculated.The total polarization may usually be separated into three parts: electronic, ionic and dipolar, which results in the fact that dielectric constant also has three contributions.The electronic contribution arises from the electron cloud shifts from the nucleus,and the ionic contribution is caused by relative displacements between positive and negative ions. Permanent dipoles in polar systems result in the dipolar contribution to dielectric constant.
Density Functional Perturbation Theory (DFPT)
The Hohenberg and Kohn theorem states that all the physical properties of a system of interacting electrons are uniquely determined by its ground-state charge density distribution.This property holds independently of the precise form of the electron-electron interaction.This face was used by Kohn and Sham to map the problem of a system of interacting electrons onto an equivalent noninteracting problem.Since the electrons do not interact with each other, a one-electron Schrodinger equation can be written.
The above equations are the foundation of density funcstional theory (DFT),and once an explicit form for the exchange-correlation energy is available,equation (1) can be solved in a self-consistent way.
In quantum mechanics,perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one.Equation (1) describes the unperturbed system.Now introduce a perturbation to the unperturbed Hamiltonian H0.Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is
Then the Schrodinger equation becomes:
(7)If the perturbation is sufficiently weak, En and $|{\psi}\rangle$ can be written as power series in λ:
(8)where
(10)and
(11)When ''λ = 0'', these reduce to the unperturbed values, which are the first term in each series.Substitute equation (7) and (8) into equation (6),thus
(12)The first order equation is
(13)Apply the above first-order perturbation theory to the unperturbed Kohn-Sham equation,and set $|\psi^{(1)}\rangle=|\bigtriangleup\psi_n\rangle$,$V=\bigtriangleup V_{SCF}$,$E_n^{(1)}=\bigtriangleup\epsilon_n$
(14)In equation (13),$\bigtriangleup V_{SCF}$ is a linear functional of $\bigtriangleup n(\mathbf{r})$,then equation (13) and (14) form a set of self-consistent equations for the perturbed system completely analogous to the Kohn-Sham equations in the unperturbed case,and this becomes the foundation of the density functional perturbation theory.
From equation (15),the wave function response to a given perturbation depends only on the off-diagonal matrix elements of the perturbing potential between eigenfunctions of the unperturbed Hamiltonian.When the perturbation is a macroscopic electric field, then such matrix elements are indeed well defined,as can be seen by rewriting them in terms of the commutator between r and the unperturbed Hamiltonian,which is a lattice periodic operator:
When calculating the response of a crystal to an applied electric field E0,one must consider that the screened field acting on the electrons is
(18)Where P is the electronic polarization linearly induced by the screened field E:
(19)Combine equation (14) and (18),
(20)Where the subscript $\alpha$ indicates Cartesian components.Then introduce the wave function ${\overline{\psi}_n}^\alpha(\mathbf{r})$ defined as
(21)Thus,
(22)The electronic contribution to the dielectric tensor,${\epsilon_\infty}^{\alpha\beta}$,can be derived from simple electrostatics
(23)Using equation (21) to calculated the polarization induced in the $\alpha$ direction when a field is applied in the $\beta$ direction,thus
(24)The total dielectric constant has two parts,one is the electronic contribution which has been derived,and the other one is ionic contribution.It can be defined as:
(25)Where Sm is the mode oscillator strength, it is proportional to the born effective charge and the displacement of each atom.And $\omega_\lambda$ is the frequency of the λth IR-active phonon normal mode.In DFPT,the ionic contribution to dielectric constant can also be calculated,the detailed procedure is presented here,as it is similar to that of electronic contribution.
Molecular Dynamics
In the Frohlich's theory,the dielectric constant is related to the total dipole moment fluctuation.To derive this relation,there are usually two steps.First,with the help of statistical mechanics,establish a relation between the mean total dipole moment in the presence of a homogeneous external field and the equilibrium fluctuations in the absence of any external perturbations.Second,classical electrostatics is used to calculated the macroscopic polarization induced by electric field.After eliminating electric field from both equations one obtains a relation between the fluctuation and the dielectric constant.
For a system of N particles with generalized coordinates $q^N=(q_1,\ldots,q_N$ and interaction energy $U(q^N)$,enclosed in a volume V at temperature T, the mean value of the total dipole moment in the direction of a homogeneous external field E0,
to first order in E0 is given by
(27)In classical electrostatics,
(28)The polarization is the dipole moment per unit volume,so,
(29)According to this definition of dielectric constant in molecular dynamics,it may provide the ionic and dipolar contributions to dielectric constant.
Results and Discussion
To compare this two methods,the dielectric constants of crystal polyethylene and ice are calculated.
Polyethylene has –CH2-CH2- as its repeating unit. It has an orthorhombic unit cell in which there are two chains and each chain has one –CH2-CH2- unit.16 Figure 1 shows the crystal structure of Polyethylene.For ice,here ice XI is considered due to its simple structure.Ice XI is an orthorhombic, low-temperature equilibrium form of hexagonal ice. It is ferroelectric. It is considered the most stable configuration of ice Ih.Study indicated that the temperature below which ice XI forms is −36 °C (240K).The crystal structure of ice XI is shown in Figure 2.
The calculated dielectric constants are shown in the table below.The molecular dynamics calculation was done at 65 K.
The experiment value of polyethylene dielectric constant is around 2, so DFPT result shows good agreement with experiment value.From the structure of polyethylene,it does not have spontaneous polarization,and thus the dipolar contribution is 0,which is in agreement with the result of MD for polyethylene.For ice XI,it has permanent dipoles along the z direction.This can be verified by the dielectric constant calculated from MD.In x and y direction,there are no spontaneous polarization and the dielectric constant are due to ionic contribution.Compare the result of ionic contribution from DFPT with that of MD,they are very close to each other.While for z direction,MD has larger value than DFPT,which results from the dipolar contribution.
Summary
Based on the theory and results of DFPT and MD,a brief comparison of these two methods can be made:
- DFPT can give relatively accurate dielectric constants for both polar and nonpolar systems, while MD is suitable for polar system
- MD can calculate the dielectric constants of amorphous and fluid systems, but for DFPT it is difficult.
- DFPT gives the dielectric constant at 0 K, while MD can calculate dielectric constant at different temperatures.
- They may be compensation to each other. For polar system: $\epsilon_{tol}$= Electronic (DFPT) + Ionic (DFPT) + Dipolar (MD)
References
(1) C. Kittel, Introduction to Solid State Physics
(2) A. Messiah, Quantum Mechanics
(3) S. Baroni, Stefano de Gironcoli, and Andrea Dal Corso. Rev. Mod. Phys. 73 (2001)
(4) X. Gonze and C. Lee. Phys.Rev.B 55 10355-10368 (1996)
(5) X. Gonze. Phys.Rev.B 55 10337-10354 (1996)
(6) M. Neumann, Molecular Physics 50 841-858 (1983)
(7) D. J. Price and C. L. Brooks III. J.Chem.Phys 121 (2004)
(8) R. Howe and R. W. Whitworth. J.Chem.Phys 90 (1989)
Hi Chenchen,
Can you apply MD to find the dielectric constant of nano structures? Please explain.
Yes,I think you can.The method is same as I described in the term paper.But you need to find the appropriate force field.
Hello Chenchen,
Just curious. Is any one calculate the dielectric constant as function of frequencies with DFT? We know that dielectric constant may vary a lots at different frequencies. Thank you.
Hi BB,
(1)Yes, we can calculate the frequency dependent dielectric constant in DFT. Using LOPTICS=.TRUE. in INCAR file, VASP can calculate the frequency dependent dielectric tensor after the electronic ground state has been determined. The imaginary part is determined by a summation over empty states using the equation:
Where the indices c and v refer to conduction and valence band states respectively.
(2)The real part of the dielectric tensor is obtained by the Kramers-Kronig transformation:
Hi, Ying and Chenchen,
I notice that both of your projects are related to polymer. I am kind of curious about the unit cell construction for polymer. Take polyethylene in the orthorhombic structure for example, I think you are taking a single polymer chain as a lattice point, and thus it will have some symmetry. However, in this sense, you assume each of the single polymer chain is identical, it means you ignored the rotation or stretching of the polymer chain and these movements may be different for every single chain. Could you please briefly explain this? Thank you.
Hi Yang,
Take polyethylene for example,in the unit cell,it only contains part of the chain,which is -CH2-CH2-,the repeating unit.When the unit cell repeat in the z direction,it will form the polyethylene chain.In the unit cell,there are two such repeating unit,but they belong to different chains.The symmetry actually described the relative position of the corresponding atoms of these two repeating units.
Hi Chenchen,
In your summary you made a final point that for a polar system you could use:
e= Electronic (DFPT) + Ionic (DFPT) + Dipolar (MD)
I think it would be helpful to provide some more explanation, perhaps an example, of how this could be used in future studies. Thanks!
Hi Erica,
Dielectric constant has mainly three parts:electronic,ionic and dipolar. For the system without spontaneous polarization, dipolar contribution is zero,so we can use DFPT to fully describe the dielectric constant of the system.But for the system with spontaneous polarization,dipolar contribution is not zero and sometimes it may very big,so using DFPT only is not sufficient. At this situation,MD is useful since it can provide dipolar contribution to dielectric constant.
But this is just my conclusion based on my limited calculation.I still need to do more calculations to verify this.
Hi Chenchen,
Since capacitance is also a function of dielectric constant, is there also the possibility of calculating $\epsilon$ via DFPT and MD by incorporating capacitance, for example in a parallel-plate capacitor?
I agree with Ching-Chang Chung that dielectric constant is frequency dependent. For example, if one were to calculate $\epsilon$ for a time variant electric field, frequency would have to be incorporated in the calculation. It may have been good to have a short discussion on frequency dependence.
Hi Dominique,
We can calculate the dielectric constant using DFPT, and compute the capacitance from dielectric constant. I don't think we can simulate, for example, a parallel-plate capacitor in DFT. I think that exceeds the scale of DFT. But in MD,we may be able to do that.I actually found a paper in which they calculate the capacitance of a finite nanometer size cylindrical capacitor.
I believe my question has more or less already been posted….but how accurate does DFPT and/or MD described polymer systems which are often very tangled, random chains of "mers"? In reality metals and ceramics are not ideal systems where the repeating unit cell is expanded in 3D, however they are much more predictable than polymers….can you comment?
Hi Kyle,
DFPT can give relatively accurate results for polymer crystal, such as lattice constant, elastic constant, and so on.But in reality,polymers are always showing semi-crystallinity. They have crystal region and amorphous region. To simulate such a system is very difficult for DFT. So DFT can give us the results for the ideal polymer system and it can give us an insight into the crystal region of polymer.To simulate the real system,MD is used.And it has been proved useful to predict some properties of polymer, such as melting temperature, crystalizatoin, thermal conductivity and so on.
Hi, Chenchen,
What is the accuracy of your results? I think it would be better if you also list the experiment result in your table,
Hi Ying,
Thank you for your suggestion. I will add it in my term paper. To answer your question,the experiment of polyethylene dielectric constant is around 2,so DFPT has a relatively good agreement with experiment value.For MD,since it only calculate the ionic and dipolar contributions to dielectric constant, for nonpolar system,sometimes it may underestimate the dielectric constant.As for ice,I didn't find the experiment value……