Problems
Application 5.2 Coulomb Blockade
6.2 Polarization
The wordpolarizationis used in two ways in the theory of dielectric matter. First, polarization refers to the rearrangement of internal charge that occurs when matter is exposed to an external field. Second, polarization is the name given to a functionP(r) used to characterize the details of the rearrangement.
We begin with the intuitive idea of polarization and specify that the source ofEext(r) in (6.1) is a charge densityρf(r) which is wholly extraneous to the dielectric. A long tradition refers to this as free charge. Examples of free charge include the charge on the surface of capacitor plates and point charges we might place inside or outside the body of a dielectric.
The source ofEself(r) in (6.1) is often calledbound charge. A more descriptive term ispolarization chargeand we will use the symbolρP(r) for its density. To understand the origin ofρP(r), we remind the
6.2 Polarization 159
reader that the macroscopic charge densityρ(r) iszeroat every point inside a neutral dielectric when Eext=0 (see Section 2.4.1). WhenEextfirst appears, positive charge is pushed in one direction and negative charge is pushed in the opposite direction. Charge rearrangement continues until mechanical equilibrium is re-established, and we identifyρP(r) as the macroscopic charge density that makes the Coulomb force densityρP(r)Etot(r) equal and opposite to the force density produced by chemical bonding and other non-electrostatic effects. The total charge density that enters Maxwell’s theory is the sum of the “free” and “bound” charge densities:
ρ(r)=ρf(r)+ρP(r). (6.2)
We use a model-independent approach to introduce the polarizationP(r). The first step is to separate the macroscopic polarization charge density into a surface part,σP(rS), and a volume part,ρP(r). Next, we recognize that a neutral dielectric with volumeV and surfaceSremains a neutral dielectric in the presence of free charge of any kind. In that case, the polarization charge densities satisfy the constraint
V
d3r ρP(r)+
S
dS σP(rS)=0. (6.3)
A neutral conductor satisfies (6.3) withρP(r)=0 andσP(rS)=0. A dielectric uses the polarization P(r) to satisfy (6.3) withρP(r)=0 andσP(rS)=0. The key observation is that the left side of (6.3) is identicallyzeroif the divergence theorem is used after substituting
ρP(r)= −∇ ·P(r) r∈V , (6.4) σP(rS)=P(rS)·n(rˆ S) rS ∈S, (6.5)
P(r)=0 r∈/V . (6.6)
Three remarks are germane. First, the vector ˆn(rS) in (6.5) is the outward unit normal to the surface Sat the pointrS. Second,P(r)=0 outside the sample in (6.6) because polarization is associated with matter and there is no matter outsideV.1Third, the equations (6.4), (6.5), and (6.6) do not determine P(r) uniquely. This follows from Helmholtz’ theorem (Section 1.9) and the fact that we did not specify
∇ ×P(r)insidethe sample volumeV.
To summarize, the macroscopic electrostatic field of a dielectric sample is produced by macroscopic polarization charge densitiesρP(r) andσP(r). These, in turn, are determined by the polarizationP(r) of the sample. The latter is fundamental to the theory and our attention must now turn to its physical meaning and methods that can be used to calculate it.
6.2.1 The Volume Integral of P(r)
An important clue to the physical meaning of the polarizationP(r) is that its volume integral is the total electric dipole momentpof a dielectric sample. To see this, we integrate thekth Cartesian component ofP(r) over the sample volumeV. BecauseP· ∇rk=Pk,
V
d3r Pk=
V
d3r∇ ·(rkP)−
V
d3r rk(∇ ·P). (6.7)
1 Some authors build (6.6) into their definition ofP(r). This generates the surface density (6.5) as a singular piece of the volume density (6.4). An example is a dielectric which occupies the half-spacez≤0 with polarization P(r)=P0(r)(−z). In that case,ρP= −∇ ·P(r)= −(−z)∇ ·P0+δ(z)P0·ˆz. The last term isσP. Our preference is to display the volume and surface parts of the polarization separately.
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160 DIELECTRIC MATTER: POLARIZATION AND ITS CONSEQUENCES
Using the divergence theorem and the definitions in (6.4) and (6.5),
V
d3r Pk=
S
dS rkσP(rS)+
V
d3r rkρP(r). (6.8)
The right-hand side of (6.8) defines thekth Cartesian component of the electric dipole momentpof the sample. We conclude that the integral of the polarizationP(r) over the volume of a dielectric is equal to the total dipole moment of the dielectric:
V
d3rP(r)=p. (6.9)
6.2.2 The Lorentz Model
Following Lorentz (1902), many authors use a microscopic version of (6.9) to identifyP(r) as an
“electric dipole moment per unit volume”. The physical idea is to regard a polarized dielectric as a collection of atomic or molecular electric dipoles. The mathematical prescription defines the polar- ization at a macroscopic pointras the electric dipole moment of a microscopic cell with volume labeled byr:2
P(r)= 1
d3ssρmicro(s)= p(r)
. (6.10)
For finite, (6.10) replaces the true charge distribution in each cell by a point electric dipole. In the limit→0, the Lorentz approximation replaces the entire dielectric by a continuous distribution of point electric dipoles with a densityP(r) computed from (6.10).
Despite its widespread use, the Lorentz formula (6.10) is usually a poor approximation to the true polarization (see Section 6.2.3). The exceptions which prove the rule are dielectric gases, non-polar liquids, and molecular solids where the constituent atoms and molecules interact very weakly. The inaccuracies of the Lorentz model appear consistently for all dielectrics where chemical bonding is important and “bond charge” is present on the boundaries of the averaging cell . Worse, (6.10) generally gives different values forp(r)/ when one uses different (but equally sensible) choices for . The simple model of an ionic crystal shown in Figure 6.1 shows that this can be true even when no bond charge is present. The two panels show two choices for the Lorentz cell. Using (6.10), they lead to oppositely directedP(r) vectors. Other cell choices give other values, includingP(r)=0.
6.2.3 The Modern Theory of Polarization
In the 1990s, an unambiguous theory of the polarization P(r) was developed which abandons the Lorentz point of view. To get a flavor for this approach, we recall from (6.4) that the dielectric charge density ρ(r) is the (negative) divergence of the polarization. This implies thatP(r) containsmore information thanρ(r). However, if the polarization were simply the dipole moment per unit volume of the charge density,P(r) would containlessinformation thanρ(r). We resolve this paradox by recalling from (2.4) that quantum mechanics definesρ(r) as the absolute square of a system’s wave function.
P(r) contains more information thanρ(r) because the polarization subtly encodes information about thephaseof the system wave function. This is the essential insight provided by the modern theory of polarization.3
2 In this chapter,ris a macroscopic variable andsis a microscopic variable. In Section 2.3.1 on Lorentz averaging, the macroscopic variable was calledRand the microscopic variable was calledr.
3 See Resta and Vanderbilt (2007) in Sources, References, and Additional Reading.
6.2 Polarization 161
Figure 6.1: Cartoon view of an ionic crystal. White spheres are negative ions. Black spheres are positive ions.
The arrows indicate the local polarizationP(r) computed from Lorentz’ formula (6.10). The figure, reproduced from Purcell (1965), repeats periodically like a checkerboard.
Figure 6.2: Contour plot of the charge density induced in a NaBr crystal by an electric field that points from the lower left to the upper right. Shaded and unshaded regions correspond, respectively, to an excess or deficit of negative (electron) charge compared to the unpolarized state. The dots on the upper and lower borders are Na nuclei. The three dots across the middle are Br nuclei. The figure repeats periodically like a checkerboard.
Reproduced from Umariet al. (2001). Copyright 2001, American Institute of Physics.
We indicated in Section 2.4.1 that a current densityjP=∂P/∂tis associated with time variations of the polarization. Without going into details, we use this formula to state the entirely plausible answer:
replace (6.10) by the Lorentz cell average of the local, time-integrated current density of polarization charge which flows when an external electric field is switched on to polarize a dielectric:4
P(r)= 1
d3s ∞
−∞
dtjP,micro(s, t). (6.11)
A quantum mechanical calculation is generally required to find the microscopic current density jP,micro(s, t).
An important application of (6.11) is to test the quality of the Lorentz approximation. Figure 6.2 shows a contour plot of the (quantum mechanically) calculated microscopic charge densityρmicro(s) induced in a NaBr crystal by an external electric fieldE. This is a favorable case where the Lorentz
4 This definition assumes thatP=0 att= −∞.
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picture that motivates (6.10) might be expected to be correct. Therefore, it is comforting to see that the field does indeed displace charge in a dipole-like fashion inside the cell defined by the dashed lines.
A smaller, oppositely directed dipole displacement occurs very close to each nucleus. However, the macroscopicP(r) calculated from the Lorentz formula (6.10) (using the dashed cell as) turns out to account for less than half of the true polarization calculated from (6.11). The majority ofP(r) comes from polarization currents which flowbetweenaveraging cells. These currents play no role in the Lorentz approximation. Similar calculations performed for covalent dielectrics like silicon produce highly complex plots forρ(r) andP(r) which completely rule out an interpretation of the polarization in terms of cell dipole moments. The Lorentz picture is qualitatively incorrect for these dielectrics.
To summarize, Lorentz’ physical model of a dielectric composed of polarized atoms or molecules is realistic only when those entities retain their individual integrity as quantum mechanical objects.
Gases, simple liquids, and van der Waals-bonded molecular solids satisfy this criterion, but the vast majority of dielectrics do not. For these latter cases, the Lorentz formula (6.10) is not reliable and more sophisticated methods are needed to calculateP(r). However, onceP(r) is known—by whatever means—it turns out that Lorentz’ notion that a dielectric behaves like a continuous distribution of point electric dipoles with densityP(r) is rigorously correct. We delay the proof of this assertion until Section 6.3.1 in order to gain some introductory appreciation of the potential and field produced by polarized matter.