4 Dielectric Properties Lecture 3

K Asokan

Learning Outcomes:

From this module students may get to know about the following:

1.     You will learn about local electric field at an atom.

2.     Detailed study of Lorentz field and field of dipole inside a cavity.

In this lecture we will first recall the older one lectures, to calculate the internal field in a dielectric.

3.1 Depolarizing Field

The polarization P is defined as the dipole moment µper unit volume. The dipole moment of a system of charges is given by

Where  is the position vector of charge. The value of the sum is independent of the choice of the origin of system, provided that the system in neutral. The simplest case of an electric dipole is a system consisting of a positive and negative charge, so that the dipole moment is equal to qd, where d is a vector connecting the two charges (from negative to positive). The electric field produces by the dipole moment at distances much larger than a is given by (in CGS units)

Here we have assumed that r , that is , expression (2) is valid only at points far from the dipole itself. In atoms and molecules this condition is well satisfied, since d, being of the order of an atomic diameter is very well indeed.

According to electrostatics the electric field E is related to a scalar potential as follows

A dielectric acquires a polarization due to an applied electric field. This polarization is the consequence of redistribution of charges inside the dielectric. From the macroscopic point of view the dielectric can be considered as a material with no net charges in the interior of the material and induced negative and positive charges on the left and right surfaces of the dielectric. The fact that the average charge inside the dielectric is zero can be understood if we take a macroscopic volume, it will contain equal amount of positive and negative charges and the net charge will be zero. On the other hand if we consider a volume including a boundary perpendicular to the direction of polarization, there is a net positive (negative) charge on the surface which is not compensated by charges inside the dielectric. Therefore, the polarization charge appears on the surface on the dielectric.

Polarization of the dielectric produces a macroscopic electric field, which is determined by these surface charges. This can be seen from the following consideration. The electrostatic potential (4) produced by a dipole can be represented as

Where the integration is performed over the volume of the dielectric. Assuming for simplicity that the polarization P is constant throughout the medium and applying the Gauss theorem we obtain:

Where the integration is performed over the surface of the dielectric and n P is the fictitious surface charge density. Here n is the unit normal to the surface, directed outward from the polarized medium.

Using the potential (7) we can easily calculate the electric field due to the uniform polarization P. Note that this field is macroscopic. It is a smooth function on atomic scale because we replaced the discrete lattice dipoles with the smoothed polarization P.

The total macroscopic electric field E is the sum the applied field E0 and the field E1 is the field due to the polarization of the solid:

The field E1 is called the depolarization field, for within the body it tends to oppose the applied field E0 as in Fig. 1.

Figure 1: The depolarization field  E1  is  opposite to  P. The  fictitious surface charges are indicated: the field of these charges is E1 within the ellipsoid.

It can be shown that for specimens in the shape of ellipsoids, a class that includes spheres, cylinders, and discs as limiting forms, have an advantageous property: a uniform polarization produces a uniform depolarization field inside the body. If Px, Py, Pzare the components of the polarization P referred to the principal axes of an ellipsoid, then the components of the depolarization field are written as

Here Nx, Ny, Nz are the depolarization factors; their values depend on the ratios of the principal axes of the ellipsoid. The N‘s are positive and satisfy the sum rule Nx + Ny + Nz= 4p in CGS. For example, for a sphere Nx = Ny = Nz = 4p/3. For a thin slab, normal to the slab Nz= 4p and Nx= Ny= 0.

If the dielectric is an ellipsoid and a uniform applied field E0is applied parallel to the principal axis of the ellipsoid, then

    As is seen the value of the polarization depends on the depolarization factor N.

3.2 Local electric field at an atom:

The value of the local electric field that acts at the site of an atom is significantly different from the value of the macroscopic electric field. The reason of that is that by definition the macroscopic field is the field which is averaged over large number of dipoles. On the other hand the local field which acts on a particular atom is influenced by the nearest surrounding and therefore can deviate from the average field

Now we consider the field that acts on the atom at the center of the sphere. If all dipoles are parallel to the z axis and have magnitude p, the z component of the field at the center due to all other dipoles is, according to (2), given by

The latter equation comes from the fact that the x, y, z directions are equivalent because of the symmetry of the lattice and of the sphere; thus

The correct local field is just equal to the applied field, E_loc= E₀, for an atom site with a cubic environment in a spherical specimen. Thus the local field is not the same as the macroscopic average field E.

3.3 Local electric field of a one-dimensional array:

The model used by Epstein to calculate the local field in case of one-dimensional atomic array is shown in figure 2.

This figure shows an array of equal-spaced atomic dipoles separated by a distance a. Suppose an electric field E is applied from left to right. The aim is to find out the Local field  which a representative atom a sees. If is the induced dipole in each atom due to applied field E, the internal field seen by an atom A is the sum of the applied and the fields  produced by  of all other atoms P, Q,… lying on the left of A and R,S…. lying on the right of A.

The components of the field produced by a dipole moment in the direction of unit vectors  and  are

The field produced at A by at P is obtained by the above equations if and. thus from (14) and (15)

 

In a three-dimensional case, the calculation of the internal field would be very complicated and would depend upon the crystal structure. By analogy with equation (20) one expects that the internal field in a crystal would involve similar terms. In a three-dimensional case  may be replaced by N, the number of atoms per unit volume, and  by a constant which depends upon the type of structure. Hence

y is called the internal field constant.

3.4 Evaluation of local field for a cubic structure:

To evaluate Elocwe must calculate the total field acting on a certain typical dipole, this field beingdue to the external field as well as all other dipoles in the system. This was done by Lorentz asfollows: The dipole is imagined to be surrounded by a spherical cavity whose radius R issufficiently large that the matrix lying outside it may be treated as a continuous medium as far asthe dipole is concerned.(fig 3). The interaction of our dipole with the other dipoles lying inside the cavity is, however, to be treated microscopically, which is necessary since the discrete nature of the medium very close to the dipoles should be taken into account. The local field, acting on the central dipole, is thus given by the sum

Where

E₀–is the external field.

E₁– is the depolarization field, i.e. the field due to the polarization charges lying at the external surfaces of the sample,

E_2-The field due to the polarization charges lying on the surface of the Lorentz sphere, which is known as Lorentz field.

E₃ is the field due to other dipoles lying within the sphere.

Figure 3.(a) The procedure for computing the local field,(b)The procedure for calculating E2, the field due to the polarization charge on the surface of the Lorentz sphere.

Note that the part of the medium between the sphere and the external surface does not contribute anything since, the volume polarization charges compensate each other, resulting in a zero net charge in this region.

The contribution E1 + E2 + E3 to the local field is nothing but the total field at one atom caused by the dipole moments of all the other atoms in the specimen. Dipoles at distances greater than perhaps ten lattice constants from the reference site make a smoothly varying contribution, a contribution which may be replaced by two surface integrals.

3.4.1 Lorentz field, E2:

The enlarged view of figure 3 (b) the cavity is shown in figure 4. If dAios surface of the sphere of radius r lying between and , where  is the direction with the reference to the direction to the direction of the applied field,

3.4.2 Field of dipoles inside cavity, E₃:

The field E3 due to the dipoles within the spherical cavity is the only term that depends on the crystal structure. We showed for a reference site with cubic surroundings in a sphere that E3 = 0 if all the atoms may be replaced by point dipoles parallel to each other. The total local field at a cubic site is, then

This is known as the Lorentz relation: the field acting at an atom in a cubic site is the macroscopic field E of plus  from the polarization of the other atoms in the specimen. Experimental data for cubic ionic crystals support the Lorentz relation.

The difference between E, which is known as the Maxwell field and the local field Eloc may be explained as follows. The field E is a macroscopic quantity, and as such is an average field, the average being taken over a large number of molecules (Fig. 4). It is this field which enters into the Maxwell equations, which are used for the macroscopic description of dielectric media. In the present situation the field E is a constant throughout the medium.

On the other hand, the local field Eloc is a microscopic field which fluctuates rapidly within the medium. As the figure indicates, this field is quite large at the molecular sites themselves.

Figure 4. The difference between the Maxwell field E and the local field Eloc. Solid circles represent molecules.

Summary:

We learnt that the local field inside a dielectric is not the same as the applied electric field, rather it have contribution from external field, depolarization field, Lorentz field and other dipoles lying within the Lorentz sphere. Lorentz field and other dipoles lying within the Lorentz sphere were also discussed in detail.

    References:

1. Wiser, N., Dielectric Constant with Local Field Effects Included. Physical Review 1963, 129 (1), 62-69.
2. Quinn, John J., and Kyung-Soo Yi. “Dielectric Properties of Solids.” Solid State Physics. Springer Berlin Heidelberg, 2009. 215-246.
3. 3.  Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2.: volume
4. Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, [NJ.]: Prentice-Hall. ISBN 0-13-805326-X.
5. Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X

    References and Suggestive Readings

1. Kirkwood, John G. “The Local Field in Dielectrics.” Annals of the New York Academy of Sciences 40.5 (1940): 315-320.
2. Wiser, Nathan. “Dielectric constant with local field effects included.” Physical Review 129.1 (1963): 62.
3. Rikken, G. L. J. A., and Y. A. R. R. Kessener. “Local field effects and electric and magnetic dipole transitions in dielectrics.” Physical review letters 74.6 (1995): 880.

    Web Links

1. http://web.mit.edu/sahughes/www/8.022/lec10.pdf
2. http://web.hep.uiuc.edu/home/serrede/P436/Lecture_Notes/P436_Lect_18p5.pdf
3. http://unlcms.unl.edu/cas/physics/tsymbal/teaching/SSP-927/Section%2014_Dielectric_Properties_of_Insulators.pdf
4. http://www.researchgate.net/publication/253048456_The_Lorentz_local_field_in_nonlinear_di electrics

    Microscopic versus Macroscopic Calculation of Dielectric Nanospheres

In the last 10 years, more and more experiments have been performed using dielectrics with nanometric dimensions. Nanoparticles for example can be embedded in a matrix having different dielectric properties. This raises the question whether for a material, which has got a permittivity ε on a macroscopic scale, the same permittivity is preserved if the material has nanoscale dimensions, presumed that the lattice structure is not altered. Since the dielectric properties are strongly influenced by dipole-dipole interactions it seems to be obvious that in a small finite particle the magnitude of these interactions is different from those in a large extended material. Moreover, if the dielectric is placed between coplanar electrodes the lattice can virtually be extended to infinity by the image charges in the electrodes. In order to concentrate on the essential effects we consider in the following a dielectric sphere. From the standpoint of the macroscopic continuum theory the sphere has a homogeneous permittivity ε and in an external field that was homogeneous before the sphere was brought in the field inside the material is also homogeneous, but reduced by the depolarisation field. Therefore, the polarisation of the sphere is homogeneous. From the standpoint of a microscopic approach the whole sphere consists of discrete atoms with a positive nucleus and a negative electron shell which form point dipoles if a local electric field acts on them. The local field evokes a dipole moment at the atom proportional to the field itself. The local field comprises the applied field and the sum of the dipole fields of all other atomic point dipoles. Thus intuitively the local fields close to the centre of the sphere can not be the same as in the regions close to the surface, since the surroundings are different.

In our model we calculate the local field at each atom in an iterative procedure. The sum over all dipole moments yields the polarisation of the sphere which in this model turns out to be inhomogeneous.

Theoretical Considerations

It shall be worked out now more rigorously why there is a need for a microscopic local approach. For the macroscopic calculation the applied field Ea causes a macroscopic polarisation P. This polarisation with its surface charges is the origin of the depolarisation field. The superposition of the applied field Ea and the depolarisation field Edep results in the macroscopic electric field E of the dielectric which feeds back on the polarisation P (see figure 1). E is the field which appears in Maxwell’s equations.

The macroscopic theory claims that the macroscopic electric field E inside a dielectric ellipsoid is homogeneous. In the special case of a dielectric sphere the depolarisation field inside the sphere is described by the term –P/3 0. According to we can calculate the macroscopic electric field everywhere inside the sphere:

This field E is an average field which clearly deviates from the local field at the atoms. As the field E is homogeneous, the same is also valid for the polarisation P. Thus the dipole moments have to be uniform inside the sphere. In this approach, all local fields at each atom have the same value. They evoke identical dipoles and a homogeneous polarisation. If we apply a discrete microscopic model we come to a different conclusion. The dielectric sphere consists of atoms with positive nuclei and negative electron shells. When exposed to a local field Eloc, the atoms form point dipoles and the dipole moments p become proportional to Eloc. The local field is the superposition of the applied field Ea and the field ED of all other dipoles:

El_oc =E_a + E_D                                                                                             ……………[2]

The microscopic local field approach inherently comprises all depolarisation effects. Due to reasons of symmetry, this last contribution ED cancels out in the centre of the sphere if all dipoles are identical. This was already deduced by Lorentz

Using equation (3) we obtain Eloc(0) = Ea in the centre of the sphere with ED = 0. Outside the centre of the sphere, this symmetry no longer persists and the dipole fields can’t compensate each other any more. Hence, the local field in the centre of the sphere deviates from those outside the centre. The macroscopic assumption of a uniform dipole moment is wrong, as the local field varies inside the dielectric sphere. Except for the centre of the sphere we find ED ≠ 0 and therefore Eloc ≠ Ea. Since the dipole moments are proportional to Eloc their magnitude must vary in space and the polarisation is no longer homogeneous, which is a contradiction to the assumption. This effect may be the more pronounced the smaller the particle is. Therefore we suggest, especially for nanosized particles, a model which inherently takes into account the local field variations.

Model Considerations

The local electric field at dipole j is the linear superposition of the contributions Eij of all other dipoles and the applied electric field E_a.

In this formula, the dipole fields are calculated using the far-field approximation for the dipole field. The electric field of the induced electric dipole placed at the position ri with its dipole moment pi can be approximated for the position rj by the following formula which is applied in our simulations:

We use an iterative procedure to calculate the local fields and the polarization. Initially, the dipole moments equal zero. The dipoles are labelled from 1 to N. For an applied electric field Ea a number k is generated randomly and the local field at dipole k is calculated from the sum of the field contributions of all other dipoles and the applied external field. The calculated local field induces a new dipole moment for dipole k according to equation (6). Then a new random number is generated to pick the next dipole from the remaining N-1 dipoles. For this dipole the local field is calculated again, but now considering the new dipole moment of the previous dipole k. All dipoles are processed in this way. The polarisation P1 of the whole system is calculated from the sum of all N dipole moments divided by the volume V of the system. This procedure is repeated 10 times yielding the 10 polarisation values P1 … P10 of the iteration process. The iterations finally converge to a stationary polarisation P. With this model we can simulate local fields at the sites of the dipoles, compute the polarisation of the whole system, and calculate the effective susceptibility for a given system.

Results

The identical atoms with polarizability α = 0.2 eŲ/V are arranged on cubic lattice sites with a lattice constant of LC = 3 Å. For a bulk material with α = 0.2 eŲ/V and LC = 3 Å we find the corresponding susceptibility χ = 2.4 from the Clausius-Mossotti equation. The system size can become up to N = 5000 dipoles which corresponds to a sphere diameter Ø of 21 dipoles or rather 63 Å. The external field Ea = 5 · 105 V/cm which corresponds to a macroscopic internal field E = 2.7 · 105 V/cm is applied in z-direction. Simulations for different sphere diameters clearly show that the local field and thus the dipole moments are not uniform. Figure 3 shows the z-component of the local electric field along the z-axis for Ea = 5 · 105 V/cm. Figure 4 shows the z-component of the local electric field along the x-axis for Ea = 5 · 105 V/cm. The fields at the sphere boundaries vary noticeably from the electric field inside the sphere. For small spheres this effect can be very pronounced

With increasing sphere diameter Ø the local field at the centre dipole becomes smaller and obviously converges towards the value of the applied external field as shown in figure 5. We fitted a first order exponential decay function with offset y0, prefactor A and decay constant L using a least squares fitter. The extrapolation yields that for Ø > 75 Å the relative deviation of the field at the centre dipole from the external field is less than 1%. Figure 6 also shows the local fields outside the sphere along the z-and x-axis. Along the z-axis we find a strong amplification of Ez in front of the dielectric nanosphere in contrast to a pronounced attenuation of Ez on the sides of the sphere. Outside the sphere our calculations converge with the classical macroscopic results. The microscopic calculation of the polarisation P yields Pmicro = 5.969 · 10-8 C/cm² for Ø = 63 Å and Pmicro = 6.024 · 10-8 C/cm² for Ø = 27 Å. The macroscopic calculation according to equation (7) yields Pmacro = 5.933 · 10-8 C/cm² which is independent of the sphere diameter Ø. In addition, we find that the microstructure is significantly responsible for the increased local fields at the sphere surface. Figure 8 shows that different realisations for the shape of the sphere (see figure 7) yield extensively increased local fields at the surface. In this case the polarisation of the dielectric sphere also shows slightly higher values. It is worth mentioning that the local fields close to the centre of the sphere are not affected by these geometrical variations but depend on the sphere diameter