Part 2: polarization a microscopic view

Transcription

1 Dielectrics Part 2: polarization a microscopic view Ryszard J. Barczyński, 2010 Politechnika Gdańska, Wydział FTiMS, Katedra Fizyki Ciała Stałego Materiały dydaktyczne do użytku wewnętrznego

2 The mechanisms of dielectric polarization Mather Nature has invented four processes which may lead to polarization in dielectric materials: electronic polarization, ionic polarization, orientation (dipolar) polarization, interface (and space charge) polarization We are going to look at those processes in some detail and try to understand how do they act.

3 Electronic polarization Electronic polarization (sometimes called atomic polarization) In the electric field the positive charge in the nucleus and the center of the negative charges from the electron "cloud" will experience forces in different direction and will become separated. The separating force of the external field is balanced by the attractive force between the centers of charge.

4 Ionic polarization Ionic polarization in this case a material must have ionic character of bonds. It then by definition has internal dipoles, but these built in dipoles exactly cancel each other (and are unable to rotate). The external field then induces net dipoles by slightly displacing the ions from their rest position. The model materials are simple ionic crystals like NaCl.

5 Orientation polarization Orientation polarization here the material must have molecules with their own dipolar momentum (which can rotate freely). The external field aligns these dipoles to some extent and thus induces a polarization of the material. The paradigmatic material is water.

6 Interface and space charge polarization Surfaces, grain boundaries, inter phase boundaries may be charged, i.e. they contain dipoles which may become oriented in an external field and thus contribute to the polarization of the material. Conducting granules in insulated matrix may play a role of induced dipoles and cause the space charge polarization.

7 Space charge polarization In the absence of a field, there is no separation between the positive charges and the negative charges. In the presence of an applied field, the mobile positive ions migrate toward the negative electrode but remains in the dielectric (electrode is blocking). At the another electrode developes a net negative charge. The dielectric therefore exhibits space charge polarization

8 Space charge polarization The charge layer at electrode interface is called double layer due to its structure structure: it consists itself of two layers: Stern layer, where no mobile carriers are present and diffuse layer, where a balance between diffusive and drift currents is developed. If both electrodes are blocking two doble layers of different charage polarities developes at their interfaces.

9 Space charge polarization The conducting grains in a dielectric matrix become dipoles due to electrostatic induction. Grain boundaries and interfaces between different materials frequently give rise to interfacial polarization even if both the materials are dielectrics.

10 Polarization in Real World Some or all of these mechanisms may act simultaneously. Atomic polarization is always present in any material and becomes superimposed on other mechanisms. Real materials thus can be very complicated in their dielectric behavior. In particular, non spherical atoms may show complex electronic polarization, and mixtures of ionic and covalent bonding makes calculations even more difficult.

11 Polarization vector In order to describe the bulk material the sum of all the particles we sum up all individual dipole moments contained in the volume unit of the material. This gives us the polarization vector P P= 1 V p ei i If dipolar moments of all the particles are equal and have the same direction then P=n 0 p e where n 0 is concentration of the particles

12 Polarization vector If we want to know the charge density ρ inside a small probing volume, it is zero in the volume of the material, because there are just as many positive as negative charges. At the surfaces there is indeed some charge. At one surface, the charges have effectively moved out a distance l, at the other surface they moved in by the same amount. We thus have a surface polarization charge: i =n 0 q l=n 0 p e P= i P= r 1 0 E= 0 E

13 Local field The equation P = χ E refers to the external field, i.e. to the field that would be present in our capacitor without a material inside. On the other hand, the induced dipole moment (or electric force which acts at the molecular dipole) depends on the local field at the place of the molecule. The factor which connects the induced dipole moment and the local electric field is the molecular polarizability α (basically a microscopic parameter) p= E l P=n 0 E l

14 Local field All electrical fields can (at least in principle) by solving the Poisson equation. It couples the charge distribution and the potential V(x, y, z): = x, y, z 0 E= Doing this is pretty tricky, however. We can obtain usable results in a good approximation in a much simpler way by using Lorentz model. Let's remove a small sphere (containing a few 10 atoms) from the material. We want to know the local field in the center of this sphere while it is still in the material.

15 Lorenz model Our local field consists of three components: external E field of polarized continuous dielectric outside the sphere E c field of near atoms located inside the sphere E b E l = E E b E c The calculation of E c is a standard problem from electrostatics. Electric field caused by a charge at the surface of the sphere: E c = cos a 2 dq

16 Lorenz model E c = cos a 2 dq E c = 1 P cos [2 asin a] a 2 cos d E c = P cos 2 sin d = P Lorentz showed that for isotropic materials Eb = 0 (which is easy to imagine). E l =E P 3 0

17 Lorenz model Now we can calculate polarization P as e function of electric field E On the other hand And finally P= n 0 E P 3 0 P= n 0 0 r 1 E= n 0 r 1 r 2 1 n n E 1 n 0 = 3 0 E

18 Clausius Mosotti equation r 1 r 2 1 n 0 = 3 0 This is the Clausius Mosotti equation, it relates the microscopic quantity α on the right hand side to the macroscopic quantity ε (or, if you like that better, χ = ε r 1) on the left hand side of the equation. This has two important consequences: If we know how to calculate α we now can calculate (at least in principle...) the dielectric permittivity of various materials. We have an instrument to measure microscopic properties like the polarizability α, by measuring macroscopic properties like the dielectric constant and applying the Clausius Mosotti equation.

19 Electronic polarization For calculating the effect of electronic polarization we consider an idealized atom with perfect spherical symmetry. It has a point like charge + ze in the nucleus, and the exact opposite charge ze homogeneously distributed in the volume of the atom, which is V = 4 3 R3 The charge density ρ of the electrons then is = 3 z e 4 R 3

20 Electronic polarization In an electrical field E a force F 1 acts on charges given by F 1 =z e E The separating force of the external field is exactly balanced by the attractive force between the centers of charge at the distance d. We may divide the attractive force on the nucleus on any place inside a homogeneously charged sphere into the force from the "inside" sphere and the force from the hollow "outside" sphere. A sphere charged on the outside has no field in the inside. Thus we only have to consider the charge inside the sphere F 2 = z e q inside 4 0 d 2

21 Electronic polarization and we obtain for F 2 : q inside =z e F 2 = 4 3 d 3 =z e d R 3 R3 z e R d 3 Equating F 1 with F 2 gives the equilibrium distance d E d E = 4 0 R3 E z e

22 Electronic polarization Now we can calculate the induced dipole moment p e p e =z e d E =4 0 R 3 E The polarization P is given by multiplying with n 0, the density of the dipoles P=n 0 p e =4 0 R 3 n 0 E and the molecular (atomic) polarizability =4 0 R 3

23 Electronic polarization Now we can calculate the induced dipole moment p e p e =z e d E =4 0 R 3 E The polarization P is given by multiplying by n 0, the density of the dipoles P=n 0 p e =4 0 R 3 n 0 E The susceptibility (and permittivity) = r 1=4 0 R 3 n 0 And finally the molecular (atomic) polarizability =4 0 R 3

24 Electronic polarization We can get an order of magnitude for χ. Taking a density of n 0 3*10 19 cm 3 9 and R 6*10 cm, we obtain The electronic polarization of spherical atoms is extremely weak. The difference to vacuum is at best in the promille range. But... atoms in solids do not generally have spherical symmetry. Consider the sp 3 orbital of Si, Ge or diamond. Without a field, the center of the negative charge of the electron orbitals will coincide with the core, but an external field breaks that symmetry, producing a dipole momentum. The effect can be large compared to spherical s orbitals: Si has a dielectric permittivity of about 12, which comes exclusively from electronic polarization.

25 Ionic polarization In an electric field, the ions feel forces in opposite directions. For a field acting as shown, the lattice distorts a little bit. The Na + ions moved a bit to the right, the Cl ions to the left. Shown is the situation where the distance between the ions increases by d; the situation, where the distance decreases by d, is similar. How large is d? The force F1 increasing the distance is given by F 1 =q E

26 Ionic polarization The restoring force F 2 comes from the binding force. Assuming a linear relation between binding force and deviation from the equilibrium distance d 0 F 2 d 0 =Y d where Y is it the modulus of elasticity or Youngs modulus material constant which is directly related to the shape of the interatomic potential. Now from force equilibrium we can get and induced dipole moment is d= q E d 0 Y p e =d q= q2 E d 0 Y

27 Ionic polarization The restoring force F 2 comes from the binding force. Assuming a linear relation between binding force and deviation from the equilibrium distance d 0 F 2 d 0 =Y d where Y is it the modulus of elasticity or Youngs modulus material constant which is directly related to the shape of the interatomic potential. Now from force equilibrium we can get and induced dipole moment is d= q E d 0 Y p e =d q= q2 E d 0 Y

28 Ionic polarization The polarization The polarizability P=n 0 p e = n 0 q2 E d 0 Y = q2 d 0 Y The susceptibility (and relative permittivity) = r 1= n 0 q2 d 0 Y This is only a very rough approximation for an idealized material. The ionic polarization can lead to respectable dielectric permittivity.

29 Orientation polarization Polarization caused by the orientation polarization may be expressed as P= i p 0 cos i where p 0 is molecular dipole moment and α i is the angle between moment of a molecule (number i) and the direction of electric field. It leads to V P=n 0 p 0 cos

30 Orientation polarization The angle α is related to the energy of dipole in the electric field V e = p 0 E= p 0 E cos The Boltzmann distribution equation gives us the number of dipoles having certain energy at a given temperature N V e =A e V e k T

31 Orientation polarization The mean value of the angle α is cos = e V e k B T cos d e V e k B T d The result of integration (which requires quite a bit of fiddling around) is Langevin function defined as L =coth 1 where = p 0 E k T

32 Orientation polarization For small values of β (which is true for normal materials under normal field), the Langevin function can be approximated by 1/3 β. We obtain as final result for the induced dipole moment the Langevin Debye equation p= p 0 cos = p k B T E

33 Orientation polarization The polarizability The polarization The susceptibility = p k B T 2 p 0 P=n 0 3k B T E = r 1=n 0 p 0 2 3k B T

34 Orientation polarization Now we include the Clausius Mosotti equation molecular (atomic) polarization and orientation od dipoles obtaining Debye equation: r 1 r 2 1 = 1 n 0 3 m p k B T The polarization of polar dielectric depends on temperature this gives us an opportunity to distinguish between mechanisms. It is important to be aware of the basic condition that we made at the beginning: there is no interaction between the dipoles! This will not be true in general. On the other hand, if there is a strong interaction, we automatically have some bonding and obtain a solid ice in the case of water. The dipoles most likely cannot orientate themselves freely; we have a different situation (usually ionic polarization).

35 Summary

36 Summary We know the Clausius Mosotti equation which connects the microscopic polarizability and the permittivity of the dielectric material r 1 r 2 1 n 0 = m We considered mechanisms of polarization and determined polarizability by the means of simple models of the first three: electronic polarization (which induces dipoles at all), ionic polarization (which shifts existing ions), orientation polarization (which rotates existing dipoles), interface (and space charge) polarization there is no way of buiding any simple general model.