(a) Anisotropic materials, such as silicon crystals, require a susceptibility tensor to give an exact description of the linear relation of the vectors P and E. However, most important crystals (including Si) are only weakly anisotropic, so they may be reasonably well characterized with a scalar (angle-average) susceptibility.
(a)
(b)
E 0
p 0
p 0
Fig. 3.7. Crude cartoons of two
mechanisms
of
the
induced
E 0
electrical polarization: (a) a partial
ordering of spontaneous elementary
dipoles, and (b) an elementary
dipole induction. The upper two
p E
panels correspond to E = 0, and
the lower two panels, to E 0.
p E
A relatively weak external field does not change the magnitude of the dipole moments significantly, but according to Eqs. (15a) and (17), tries to orient them along the field, creating a nonzero vector average p directed along the vector Em, where Em is the microscopic field at the point of the dipole’s location – cf. two panels of Fig. 7a. If the field is not two high ( p E m << k B T), the induced average polarization p is proportional to Em. If we write this proportionality relation in the following traditional form,
p E ,
(3.48) Atomic
m
polarizability
where is called the atomic (or, sometimes, “molecular”) polarizability, this means that is positive. If the concentration n of such elementary dipoles is low, the contribution of their own fields into the Chapter 3
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microscopic field acting on each dipole is negligible, and we may identify Em with the macroscopic field E. As a result, the second of Eqs. (27) yields
P n p nE .
(3.49)
Comparing this relation with Eq. (45), we get
n
1
,
(3.50)
0
so that > 1 (i.e. e = n/0 > 0). Note that at this particular polarization mechanism (illustrated on the lower panel of Fig. 7a), the thermal motion “tries” to randomize the dipole orientation, i.e. reduce its ordering by the field, so that we may expect , and hence e – 1 to increase as temperature T is decreased – the so-called paraelectricity. Indeed, the basic statistical mechanics15 shows that in this case, the electric susceptibility follows the so-called Curie law e 1/ T.
The materials of the second, much more common class consist of non-polar atoms without intrinsic spontaneous polarization. A crude classical image of such an atom is an isotropic cloud of negatively charged electrons surrounding a positively charged nucleus – see the top panel of Fig. 7b.
The external electric field shifts the positive charge in the direction of the vector E, and the negative charges in the opposite direction, thus creating a similarly directed average dipole moment p.16 At relatively low fields, this average moment is proportional to E, so that we again arrive at Eq. (48), with
> 0, and if the dipole concentration n is sufficiently low, also at Eq. (50), with – 1 > 0. So, the dielectric constant is larger than 1 for both polarization mechanisms – please have one more look at Table 1.
In order to make a crude but physically transparent estimate of the difference – 1, let us consider the following toy model of a non-polar dielectric: a set of similar conducting spheres of radius R, distributed in space with a low density n << 1/ R 3. At such density, the electrostatic interaction of the spheres is negligible, and we can use Eq. (11) for the induced dipole moment of a single sphere. Then the polarizability definition (48) yields = 40 R 3, so that Eq. (50) gives
1
R 3
4 n .
(3.51)
Let us use this result for a crude estimate of the dielectric constant of air at the so-called ambient conditions, meaning the normal atmospheric pressure P = 1.013105 Pa and temperature T = 300 K. At these conditions the molecular density n may be, with a few-percent accuracy, found from the well-known equation of state of an ideal gas:17 n P / k B T (1.013105)/(1.3810-23300) 2.451025 m-3.
The molecule of the air’s main component, N2, has a van-der-Waals radius18 of 1.5510-10 m. Taking this radius for the R of our crude model, we get e – 1 1.1510-3. Comparing this number with the 15 See, e.g., SM Chapter 2.
16 Realistically, these effects are governed by quantum mechanics, so the average here should be understood not only in the statistical-mechanical but also (and mostly) in the quantum-mechanical sense. Because of that, for non-polar atoms, is typically a very weak function of temperature, at least on the usual scale T ~ 300K.
17 If needed, see, e.g., SM Secs. 1.4 and 3.1.
18 Such radius is defined by the requirement that the volume of the corresponding sphere, if used in the van-der-Waals equation (see, e. g., SM Sec. 4.1), gives the best fit to the experimental equation of state n = n ( P, T).
Chapter 3
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first line of Table 1, we see that the model gives a surprisingly reasonable result: to get the experimental value, it is sufficient to decrease the effective R of the sphere by just ~30%, to ~1.210-10 m.19
This result may encourage us to try using Eq. (51) for a larger density n. For example, as a crude model for a non-polar crystal, let us assume that the conducting spheres form a simple cubic lattice with the period a = 2 R (i.e., the neighboring spheres virtually touch). With this, n = 1/ a 3 = 1/8 R 3 and Eq. (44) yields = 1 + 4/8 2.5. This estimate provides a reasonable semi-qualitative explanation for the values of listed in a few middle rows of Table 1. However, at such small distances, the electrostatic dipole-dipole interaction should be already essential, so that this simple model cannot even approximately describe the values of much larger than 1, listed in the last rows of the table.
Such high values may be explained by the so-called molecular field effect: each elementary dipole is polarized not only by the external field, as Eq. (49) assumes, but by the field of neighboring dipoles as well. Ottavino-Fabrizio Mossotti in 1850 and (but almost 30 years later) Rudolf Clausius suggested what is now known, rather unfairly, as the Clausius-Mossotti formula, which describes this effect reasonably well in many non-polar materials.20 In our notation, it reads21
1 n
n /
Clausius-
0
,
that
so
1
.
(3.52)
Mossotti
2 3
1 n / 3
0
0
formula
If the dipole density is low in the sense n << 0/, this relation is reduced to Eq. (50) corresponding to independent dipoles. However, at higher dipole density, both and e increase faster and tend to infinity as the density-polarizability product approaches some critical value n c, in the simple Clausius-Mossotti model equal to 30/.22 This means that the zero-polarization state becomes unstable even in the absence of an external electric field.
This instability is a linear-theory (i.e. low-field) manifestation of a substantially nonlinear effect
– the formation, in some materials, of spontaneous polarization even in the absence of an external electric field. Such materials are called ferroelectrics, and may be experimentally recognized by the hysteretic behavior of their polarization as a function of the applied (external) electric field – see Fig. 8.
As the plots show, the polarization of a ferroelectric depends on the applied field’s history. For example, the direction of its spontaneous remnant polarization P R may be switched by first applying, and then removing a sufficiently high field (larger than the so-called coercive field E C – see Fig. 8) of the opposite orientation. The physics of this switching is rather involved; the polarization vector P of a ferroelectric material is typically constant only within each of spontaneously formed spatial regions (called domains), with a typical size of a few tenths of a micron, and different (frequently, opposite) directions of the vector P in adjacent domains. The change of the applied electric field results not in the 19 As will be discussed in QM Chapter 6, for a hydrogen atom in its ground state, the low-field polarizability may be calculated analytically: = (9/2)4
3
0 r B , corresponding to our metallic-ball model with a close value of the effective radius: R = (9/2)1/3 r B 1.65 r B 0.8710-10 m.
20 Applied to the high-frequency electric field, with replaced by the square of the refraction coefficient n at the field’s frequency (see Chapter 7), this formula is known as the Lorenz-Lorentz relation.
21 I am leaving the approximate proof of Eq. (52), using Eq. (3.24), for the reader’s exercise.
22 The Clausius-Mossotti formula does not give quantitatively correct results for most ferroelectric materials. For a review of modern approaches to the theory of their polarization, see, e.g., the paper by R. Resta and D.
Vanderbilt in the review collection by K. Rabe, C. Ahn, and J.-M. Triscone (eds.), Physics of Ferroelectrics: A Modern Perspective, Springer, 2010.
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switching of the direction of P inside each domain, but rather in a shift of the domain walls, resulting in the change of the average polarization of the sample.
P
P R
soft
ferroelectric
E C
0
E C
E
Fig. 3.8. The average polarization of soft
and hard ferroelectrics as functions of the
P R
hard
applied electric field (schematically).
ferroelectric
Depending on the ferroelectric’s material, temperature, and the sample’s geometry (a solid crystal, a ceramic material, or a thin film), the hysteretic loops may be rather different, ranging from a rather smooth form in the so-called soft ferroelectrics (which include most ferroelectric thin films) to an almost rectangular form in hard ferroelectrics – see Fig. 8. In low fields, soft ferroelectrics behave essentially as linear paraelectrics, but with a very high average dielectric constant – see the bottom line of Table 1 for such a classical material as BaTiO3 (which is a soft ferroelectric at temperatures below Tc
120C, and a paraelectric above this critical temperature). On the other hand, the polarization of a hard ferroelectric in the fields below its coercive field remains virtually constant, and the analysis of their electrostatics may be based on the condition P = PR = const – already used in the problem discussed in the end of the previous section.23 This condition is even more applicable to the so-called electrets –
synthetic polymers with a spontaneous polarization that remains constant even in very high electric fields.
Some materials exhibit even more complex polarization effects, for example, antiferroelectricity, helielectricity, and (practically very valuable) piezoelectricity. Unfortunately, I do not have time for a discussion of these exotic phenomena in this course;24 the main reason I am mentioning them is to emphasize again that the constitutive relation P = P(E) is material-specific rather than fundamental.
However, most insulators, in practicable fields, behave as linear dielectrics, so the next section will be committed to the discussion of their electrostatics.
23 Due to this property, hard ferroelectrics, such as the lead zirconate titanate (PZT) and strontium bismuth tantalite (SBT), with high remnant polarization P R (up to ~1 C/m2), may be used in nonvolatile random-access memories (dubbed either FRAM or FeRAM) – see, e.g., J. Scott, Ferroelectric Memories, Springer, 2000. In a cell of such a memory, binary information is stored in the form of one of two possible directions of spontaneous polarization at E = 0 (see Fig. 8). Unfortunately, the time of spontaneous depolarization of ferroelectric thin films is typically well below 10 years – the industrial standard for data retention in nonvolatile memories, and this time may be decreased even more by “fatigue” from the repeated polarization recycling at information recording. Due to these reasons, the industrial production of FRAM is currently just a tiny fraction of the nonvolatile memory market, which is dominated by floating-gate memories – see, e.g., Sec. 4.2 below.
24 For detailed coverage of ferroelectrics, I can recommend the encyclopedic monograph by M. Lines and A.
Glass, Principles and Applications of Ferroelectrics and Related Materials, Oxford U. Press, 2001, and the recent review collection edited by K. Rabe et al., that was cited above.
Chapter 3
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3.4. Electrostatics of linear dielectrics
First, let us consider the simplest but very important problem: how is the electrostatic field of a set of stand-alone charges of density (r) modified if it is placed into a uniform linear dielectric medium that obeys Eq. (46) with a dielectric constant constant in the whole region we are interested in. In this case, we may combine Eqs. (32) and (46) to write
E
.
(3.53)
As a reminder, in the free space we had a similar equation (1.27), but with a different constant, 0 = /.
Hence all the results discussed in Chapter 1 are valid inside a uniform linear dielectric, for the macroscopic field the E (and the corresponding macroscopic electrostatic potential ), if they are reduced by the factor of > 1. Thus, the most straightforward result of the induced polarization of a dielectric medium is the electric field reduction. This is a very important effect, especially taking into account the very high values of in such common dielectrics as water – see Table 1. Indeed, it is the reduction of the attraction between positive and negative ions (called, respectively, cations and anions) in water that enables their substantial dissociation and hence almost all biochemical reactions, which are the basis of the biological cell functions – and hence of the life itself.
Let us apply this general result to the important particular case of the plane capacitor (Fig. 2.3) filled with a linear, uniform dielectric. Applying the macroscopic Gauss law (34) to a pillbox-shaped volume on the conductor surface, we get the following relation,
D E
,
(3.54)
n
n
n
which differs from Eq. (2.3) only by the replacement 0 0. Hence, for a fixed field En, the charge density calculated for the free-space case should be increased by the factor of – that’s it. In particular, this means that the capacitance (2.28) has to be increased by this factor:
A A
C of a
C
0
.
(3.55) planar
d
d
capacitor
(As a reminder, this increase of C by has been already incorporated, without proof, into some estimates made in Secs. 2.1 and 2.2, to make them realistic.)
If a linear dielectric is nonuniform, the situation is more complex. For example, let us consider the case of a sharp interface between two otherwise uniform dielectrics, free of stand-alone charges. In this case, we still may use Eq. (37) for the tangential component of the macroscopic electric field, and also Eq. (36), with Dn = En, for its normal component, getting
Boundary
E
1
,
i.e.
2
.
(3.56) condition
n
E
1
n 2
1
n
2
n
for En
Let us apply these boundary conditions, first of all, to consider how carving a slit of some width d and a much smaller thickness t << d from inside a dielectric, changes an initially uniform electric field E0, depending on its orientation – see Fig. 9.
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B
A
E
D D
E E
0
0
0
Fig. 3.9. Fields inside
D E
E D / E
D E D /
0
0
0
0
0
0
0
two narrow slits cut in
a linear dielectric.
First of all, intuition tells us that regardless of its orientation, a slit cannot change the field far from it; moreover, at t 0, it cannot modify substantially even the field right outside its “major”
(broader) surfaces. This conclusion may be supported either by direct calculations (see, e.g., the problem illustrated by Fig. 11 below), or by energy arguments: at t << d, any potential energy decrease due to the field change inside the slit’s volume (proportional to td) cannot compensate its increase in the outer volume proportional to d 2. However, it may induce some local field changes – inside the slit, and even outside it, close to its “minor” surfaces.
To calculate the inner field for case A, with the slit’s plane normal to the applied field, we may apply Eq. (56) to its major surfaces (shown horizontal), to prove that the vector D should be continuous.
But according to Eq. (46), this means that in the free space inside the slit, the electric field should equal D/0, and hence be times higher than the field E0 = D/0 far from the slit. This field, and hence D, may be measured by a sensor placed inside the gap, so that the electric displacement is not an entirely mathematical construct.25 On the contrary, for case B, with the slit’s plane parallel to the initial field, we may apply Eq. (37) to the major (now, vertical) interfaces of the slit, to see that now the electric field E is continuous, while the electric displacement D = 0E inside the gap is a factor of lower than its value in the dielectric. (Similarly to case A, any perturbations of the field uniformity, caused by the compliance with Eq. (56) at the minor surfaces, settle down at distances ~ t from them.) For other problems with piecewise-constant , with more complex geometries, we may need to apply the methods studied in Chapter 2. In particular, in the simplest cases we can select such a set of orthogonal coordinates that the electrostatic potential depends on just one of them. Consider, for example, two types of filling a plane capacitor with two different dielectrics – see Fig. 10.
(a)
(b)
d 1
1
d
1
2
d
Fig. 3.10. Plane capacitors filled
2
2
with two different dielectrics.
In case (a), the voltage V between the electrodes is the same for each part of the capacitor, telling us that at least far from the dielectric interface, the electric field is vertical, uniform, and constant ( E =
V/ d). Hence the boundary condition (37) is satisfied even if such a distribution is valid near the surface 25 Superficially, this result violates the boundary condition (37) at the vertical (“minor”) surfaces of the gap. This apparent contradiction is resolved by the fact the thin slit can deform the field both inside and outside it, at distances of the order of t around these interfaces, but not far beyond them, so that the above relations for E and D
are valid at most of the slit area.
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as well, i.e. at any point of the system. The only effect of different values of in the two parts is that the electric displacement D = E and hence electrodes’ surface charge density = D are different in them.
Thus we can calculate the electrode charges Q 1,2 of the two parts independently, and then add up the results to get the total mutual capacitance
Q Q
1
1
2
C
A A .
(3.57)
1 1
2
2
V
d
Note that this formula may be interpreted as the total capacitance of two separate lumped capacitors connected (by wires) in parallel. This is natural, because we may cut the system along the dielectric interface, without any effect on the fields in either part, and then connect the corresponding electrodes by external wires, again without any effect on the system – besides very close vicinities of the capacitor’s edges, where the fringe
Case (b) may be analyzed just as in the problem illustrated by Fig. 6, by applying Eq. (34) to a Gaussian pillbox with one lid inside the (for example) bottom electrode, and the other lid inside any of the layers. As a result, we see that D anywhere inside the system should be equal to the surface charge density of the electrode, i.e. constant. Hence, according to Eq. (46), the electric field E inside each dielectric layer is also constant: in the top layer, it is E 1 = D 1/1 = /1, while in bottom layer, E 2 = D 2/2
= /2. Integrating the field E across the whole capacitor, we get
d
1 d 2
d
d
V
E( z) dz E d E d
1 2
,
(3.58)
1 1
2
2
0
1
2
so that the mutual capacitance per unit area
1
C
d
d
1
2
.
(3.59)
A
V
1
2
Note that this result is similar to the total capacitance of an in- series connection of two plane capacitors based on each of the layers. This is also natural because we could insert an uncharged, thin conducting sheet (rather than a cut as in the previous case) at the layer interface, which is an equipotential surface, without changing the field distribution in any part of the system. Then we could thicken the conducting sheet as much as we liked (and possibly shape its internal part into a thin wire), also without changing the fields in the dielectric parts of the system, and hence the capacitance.
Proceeding to problems with more complex geometry, let us consider the system shown in Fig.
11a: a dielectric sphere placed into an initially uniform external electric field E0. According to Eq. (53) for the macroscopic electric field, and the definition of the macroscopic electrostatic potential, E = –, the potential satisfies the Laplace equation both inside and outside the sphere, though not at its border.
Due to the spherical symmetry of the dielectric sample, this problem invites the variable separation method in spherical coordinates, which was discussed in Sec. 2.8. From that discussion, we already know, in particular, the general solution (2.172) of the Laplace equation outside of the sphere. To satisfy the uniform-field condition at r , we have to reduce this solution to
b
E r cos
l P (cos) .
(3.60)
r R
0
l1 l
l 1 r
Inside the sphere, we can also use Eq. (2.172), but keeping only the radial functions finite at r 0: Chapter 3
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l
a r P (cos ) .
(3.61)
r R
l l
l1
Now, spelling out the boundary conditions (37) and (56) at r = R, we see that for all coefficients al and bl with l 2, we get homogeneous linear equations (just like for the conducting sphere discussed in Sec.
2.8) that have only trivial solutions. Hence, all these terms may be dropped, while for the only surviving terms with l = 1, proportional to the Legendre polynomial P 1(cos) cos, we get two equations: 2 b
b
1
E
κa ,
1
E R
a .
R
(3.62)
0
3
1
0
2
1
R
R
Solving this simple system of linear equations for a 1 and b 1, and plugging the result into Eqs. (60) and (61), we get the final solution of the problem:
1 3
R
3
E r
cos
.
(3.63)
,
E
r cos
r R
0
2 2
r R
0
r
2
(a)
(b)
E
z
0
R
0
Fig. 3.11. A dielectric sphere in an initially uniform electric field: (a) the problem, and (b) the equipotential surfaces, as given by Eq. (63), for = 3.
Figure 11b shows the equipotential surfaces given by this solution, for a particular value of the dielectric constant . Note that according to Eq. (62), at r R the dielectric sphere, just as the conducting sphere in a similar problem, produces (on top of the uniform external field) a pure dipole field, with the dipole moment
3
1
1
4
p 4 R
E 3 V
E ,
where
3
V
R .
(3.64)
2 0 0
2 0 0
3
This is an evident generalization of Eq. (11), to which Eq. (64) tends at . By the way, this property is common: for their electrostatic properties, conductors may be adequately described as dielectrics with .
Another remarkable feature of Eqs. (63) is that the electric field and polarization inside the sphere are uniform, with R-independent values
Chapter 3
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3
3
1
E
E ,
D E
E ,
P D E 3
E .
(3.65)
0
0
0
0
0
0
0
2
2
2
In the limit 1 (for example, the “sphere made of free space”, i.e. no sphere at all), the electric field inside it naturally tends to the external one, and its polarization vanishes. In the opposite limit , the electric field inside the sphere vanishes. Curiously enough, in this limit the electric displacement inside the sphere remains finite: D 30E0.
More complex problems with piecewise-uniform dielectrics also may be addressed by the methods discussed in Chapter 2, and hopefully, the reader will be able to use them to solve a few such problems offered in Sec. 6, on their own. Let me discuss just one of such problems because it exhibits a new feature of the charge image method that was discussed in Secs 2.9 (and is the basis of Green’s function approach – see Sec. 2.10). Consider the system shown in Fig. 12: a point charge near a dielectric half-space; it obviously parallels the system discussed in Sec. 2.9 – see Fig. 2.26.
z
q, q"
this point “sees”
charges
d
q and q’
0
d
this point “sees”
q'
Fig. 3.12. Charge images for a dielectric half-space.
charge q” alone
As for the case of a conducting half-space, the Laplace equation for the electrostatic potential in the upper half-space z > 0 (besides the charge point = 0, z = d) may be satisfied using a single image charge q’ at the point with = 0 and z = – d, but now q’ may differ from (– q). In addition, in contrast to the case analyzed in Sec. 2.9, we should also calculate the field inside the dielectric (at z 0). This field cannot be contributed by the image charge q’, because that would give a potential divergence at its location. Thus, in the dielectric-filled half-space we should try to use the real point source only, but with a re-normalized charge q” rather than the genuine charge q – see Fig. 12. As a result, we may look for the potential distribution in the form
q
q'
, for z ,
0
1
1/ 2
1/ 2
2
2
2
2
(, z)
( z d)
( z d)
(3.66)
4
q''
0
,
for
z
,
0
1/ 2
2
( z d)2
at this stage of solution, with unknown q’ and q” . Plugging this equality into the boundary conditions (37) and (56) at z = 0 (with / n = / z), we see that they are indeed satisfied (so that Eq. (66) does express the solution of the boundary problem), provided that the effective charges q’ and q’’ obey the following relations:
Chapter 3
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q q' q''
,
q q' q'' .
(3.67)
Solving this simple system of linear equations, we get
1
2
q'
q,
q''
q .
(3.68)
1
1
If 1, then q’ 0 , and q’’ q – both facts very natural because in this limit (no polarization at all) we have to recover the unperturbed field of the initial point charge in both semi-spaces. In the opposite limit (which, as was discussed above, may describe a conducting half-space), q’ – q (repeating the result we have discussed in detail in Sec. 2.9), and q’’ 0. The last result means that in this limit, the electric field E in the dielectric tends to zero – as it should.
In conclusion of this section, please note that if the permittivity of a linear dielectric is a continuous rather than piecewise function of coordinates, the distribution of the electrostatic potential
may be found from Eq. (32) with the electric displacement given by Eq. (46): D = (r)E = –(r).
However, analytical solutions of the resulting partial differential equation of the second order may be found only for rare particular cases; one of them is offered in Sec. 6 for the reader’s exercise.
3.5. Electric field energy in a dielectric
In Chapter 1, we have obtained two key results for the electrostatic energy: Eq. (1.55) for a charge interaction with an independent (“external”) field, and a similarly structured formula (1.60), but with an additional factor ½, for the field induced by the charges under consideration. These relations are universal, i.e. valid for dielectrics as well, provided that the charge density includes all charges –
including those bound into the elementary dipoles. However, for most applications, it is convenient to recast them into a form where these bound charges participate not explicitly, but only via the macroscopic polarization effects they create.
If a field is created only by the stand-alone charges under consideration and is proportional to
(r) (requiring that we deal with linear dielectrics), we can repeat all the argumentation of the beginning of Sec. 1.3, and again arrive at Eq. (1.60), provided that is now the macroscopic field’s potential. Now we can recast this result in the terms of fields – essentially as this was done in Eqs. (1.62)-(1.64), but now making a clear difference between the macroscopic electric field E = – and the electric displacement field D, which obeys the macroscopic Maxwell equation (32). Plugging (r) expressed from that equation, into Eq. (1.60), we get
1
U
D
d 3 r .
(3.69)
2
Using the fact26 that for differentiable functions and D,
( D) ( D) () D ,
(3.70)
we may rewrite Eq. (69) as
1
1
U
D d 3 r
() D d 3 r
.
(3.71)
2
2
26 See, e.g., MA Eq. (11.4a).
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The divergence theorem, applied to the first term on the right-hand side, reduces it to a surface integral of Dn. (As a reminder, in Eq. (1.63) the integral was of () n En.) If the surface of the volume we are considering is sufficiently far, this surface integral vanishes. On the other hand, the gradient in the second term of Eq. (71) is just (minus) field E, so it gives
1
1
U ED d 3 r E(r) (r) E(r) d 3 r 0 (r) E 2(r) d 3 r .
(3.72)
2
2
2
This expression is a natural generalization of Eq. (1.65), and shows that we can, as we did in free space, represent the electrostatic energy in a local form:27
Field
3
1
2
D
U u(r) d r,
with
2
u E
E
D
.
(3.73) energy in
2
2
2
a linear
dielectric
As a sanity check, in the trivial case = 0 (i.e. = 1), this result is reduced to Eq. (1.65).
Of course, Eq. (73) is valid only for linear dielectrics, because our starting point, Eq. (1.60), is only valid if is proportional to . To make our calculation more general, we should intercept the calculations of Sec. 1.3 at an earlier stage, at which this proportionality had not yet been used. For example, the first of Eqs. (1.56) may be rewritten, in the continuous form, as
U
(r)(r d 3
) r ,
(3.74)
where the symbol means a small variation of the function – e.g., its change in time, sufficiently slow to ignore the relativistic and magnetic-field effects. Applying such variation to Eq. (32), and plugging the resulting relation = D into Eq. (74), we get
U
D
d 3
r .
(3.75)
(Note that in contrast to Eq. (69), this expression does not have the front factor ½.) Now repeating the same calculations as in the linear case, for the energy density’s variation we get a remarkably simple (and general!) expression,
3
Energy
u
E D E D
,
(3.76) density’s
j
j
j1
variation
where the last expression uses the Cartesian components of the vectors E and D. This is as far as we can go for the general dependence D(E). If the dependence is linear and isotropic, as in Eq. (46), then D =
E and
2
E
u
E E .
(3.77)
2
The integration of this expression over the whole variation, from the field equal to zero to a certain final distribution E(r), brings us back to Eq. (73).
An important role of Eq. (76), in its last form, is to indicate that from the point of view of analytical mechanics, the Cartesian coordinates of E may be interpreted as generalized forces, and those 27 In the Gaussian units, each of the last three expressions should be divided by 4.
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of D as generalized coordinates of the field’s effect on a unit volume of the dielectric. This allows one, in particular, to form the proper Gibbs potential energy 28 of a system with an electric field E(r) fixed, at every point, by some external source:
Gibbs
potential
energy
U u
,
.
(3.78)
G
G r d 3 r
u G r r
E r
D r
u
V
The essence of this notion is that if the generalized external force (in our case, E) is fixed, the stable equilibrium of the system corresponds to the minimum of U G, rather than of the potential energy U as such – in our case, that of the field in our system.
As the simplest illustration of this important concept, let us consider a very long cylinder (with an arbitrary cross-section’s shape), made of a uniform linear dielectric, placed into a uniform external electric field parallel to the cylinder’s axis – see Fig. 13.
Eext
D ?
Fig. 3.13. A cylindrical dielectric sample
in a longitudinal external electric field.
For this simple problem, the equilibrium value of D inside the cylinder may be, of course, readily found without any appeal to energies. Indeed, the solution of the Laplace equation inside the cylinder, with the boundary condition (37) is evident: E(r) = Eext, and so that Eq. (46) immediately yields D(r) =
Eext. One may wonder why does the minimum of the potential energy U, given by Eq. (73) in its last form,
2
U
D
,
(3.79)
V
2
correspond to a different (zero) value of D, but let us recall that Eq. (73) was derived for the case when the electric field is created by the stand-alone charges in the system under consideration. If it is created by external sources, we have to use the Gibbs potential energy (78) instead. For our current uniform case, this energy per unit volume of the cylinder is
3
U
U
D
2
2
D
G
E D
E D j E D ,
(3.80)
j
j
V
V
2
j1 2
and its minimum as a function of every Cartesian component of D corresponds to the correct value of the displacement: Dj = Ej, i.e. to D = E = Eext. So, the systems’ equilibrium indeed corresponds to the minimum of the Gibbs potential energy (78) rather than of the energy (73).
28 See, e.g., CM Sec. 1.4, in particular Eq. (1.41). Note that as Eq. (78) clearly illustrates, once again, that the difference between the potential energies U G and U, usually discussed in courses of thermodynamics and statistical physics as the difference between the Gibbs and Helmholtz free energies (see, e.g., SM 1.4), is more general than the effects of random thermal motion addressed by these disciplines.
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Now note that Eq. (80), at this equilibrium point (only!), may be rewritten as
2
2
U
U
D
D
D
G
E D
D
,
(3.81)
V
V
2
2
i.e. formally coincides with Eq. (79), besides the (perhaps, somewhat counter-intuitive) opposite sign. A similar but more general relation (not limited to linear dielectrics and uniform fields) may be obtained by taking the variation of the u G expressed by Eq. (78), and then using Eq. (76):
u u
.
(3.82)
G
ED E D
E
D E D
D E
In order to see how this expressions works, let us plug D from Eq. (33):
2
E
u
.
(3.83)
G
E P
0
E
0
P E
2
So far, this relation is general. In the particular case when the polarization P is field-independent, we may integrate Eq. (83) over the full electric field’s variation, say from 0 to some finite value E, getting
2
E
u 0
P E .
(3.84)
G
2
Again, the Gibbs energy is relevant only if E is dominated by an external field Eext independent of the orientation of P. If, in addition, P(r) 0 only in some finite volume V, we may integrate Eq. (84) over that volume, getting
U p E
const,
with p P(r)
,
(3.85)
G
ext
d 3 r
V
where the “const” means the terms independent of p. In this expression, we may readily recognize Eq.
(15a) for an electric dipole p of a fixed magnitude, which was obtained in Sec. 1 in a different way. This comparison illustrates again that U G is nothing mysterious; it is just the relevant part of the potential energy of the system in a fixed external field, including the energy of its interaction with the field.
Finally, in the other important case of a linear dielectric, when according to Eqs. (45) and (47), P
= ( - 0)E, the similar integration of the general Eq. (83) over the field yields the additional factor ½: 1
3
U
P E d r const .
(3.86)
G
2
ext
V
This expression may be very convenient for analyses of the forces exerted by electric fields on linear dielectric media – see, for, example, a few exercises on this topic, offered at the end of this chapter.
3.6. Exercise problems
3.1. Prove Eqs. (3)-(4), starting from Eqs. (1.38) and (3.2).
3.2. A plane thin ring of radius R is charged with a constant linear density . Calculate the exact electrostatic potential distribution along the symmetry axis of the ring, and prove that at large distances, r >> R, the three leading terms of its multipole expansion are indeed correctly described by Eqs. (3)-(4).
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3.3. In suitable reference frames, calculate the dipole and quadrupole moments of the following systems (see the figures below):
(i) four point charges of the same magnitude, but alternating signs, placed in the corners of a square;
(ii) a similar system, but with a pair charge sign alternation; and
(iii) a point charge in the center of a thin ring carrying a similar but opposite charge, uniformly distributed along its circumference.
i q
ii
iii
a
q
q
a
q
R
a
a
a
a
Q
Q
q
a
q
q
a
q
3.4. Calculate the dipole and quadrupole moments of a thin spherical shell of radius R, carrying an electric charge with areal density = 0cos. Discuss the relation between the results and the solution of Problem 2.28.
3.5. For a regular cubic lattice of similarly oriented identical dipoles, calculate the electric field it creates at the location of each dipole.
3.6. Without carrying out an exact calculation, can you predict the spatial dependence of the interaction between various electric multipoles, including point charges (in this context, frequently called electric monopoles), dipoles, and quadrupoles? Based on these predictions, what is the functional dependence of the interaction between diatomic molecules such as H2, N2, O2, etc., on the distance between them, if the distance is much larger than the molecular size?
3.7. Two similar electric dipoles, of fixed magnitude p, located at a fixed distance r from each other, are free to change their directions. What stable equilibrium position(s) they may take as a result of their electrostatic interaction?
3.8. An electric dipole is located above an infinite, grounded
conducting plane (see the figure on the right). Calculate:
p
(i) the distribution of the induced charge in the conductor,
(ii) the dipole-to-plane interaction energy, and
d
(ii) the force and the torque exerted on the dipole.
R
3.9. Calculate the net charge Q induced in a grounded conducting sphere
0
r
of radius R by a dipole p located at point r outside the sphere – see the figure on p
0
the right.
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3.10. Use two different approaches to calculate the energy of interaction between a grounded conductor and an electric dipole p placed in the center of a spherical cavity of radius R, carved in the conductor.
3.11. A plane separating two parts of otherwise free space is densely and uniformly (with a constant areal density n) covered with electric dipoles, with similar moments p oriented normally to the plane.
(i) Use two different approaches to calculate the electrostatic potential at distances d >> 1/ n 1/2 on both sides of the plane.
(ii) Give a physical interpretation of your result.
(iii) Use the result to calculate the potential distribution created in space by a spherical surface of radius R, densely and uniformly covered with radially-oriented dipoles.
3.12. Prove Eq. (24).
Hint: You may like to use the basic Eq. (1.9) to spell out the left-hand side of Eq. (1.24), change the order of integration over r and r ’, and then contemplate the physical sense of the inner integral.
3.13. A sphere of radius R is made of a material with a uniform spontaneous polarization P0.
Calculate the electric field everywhere in space – both inside and outside the sphere, and compare the result for the internal field with Eq. (24).
3.14. Calculate the electric field at the center of a cube made of a material with the uniform spontaneous polarization P0 of arbitrary orientation.
3.15. Derive the Clausius-Mossotti formula (52), using Eq. (24) for an approximate evaluation of the field created by an elementary atomic/molecular dipole.
3.16. Stand-alone charge Q is distributed, in some way, in the volume of a body made of a uniform linear dielectric with a dielectric constant . Calculate the total polarization charge Q ef residing on the surface of the body, provided that it is surrounded by free space.
3.17. In two separate experiments, a thin plane sheet of a linear dielectric with = const is placed into a uniform external electric field E0:
(i) with the sheet’s surfaces parallel to the electric field, and
(ii) with the surfaces normal to the field.
For each case, find the electric field E, the electric displacement D, and the polarization P inside the dielectric (sufficiently far from the sheet’s edges).
3.18. A fixed dipole p is placed in the center of a spherical cavity of radius R, carved inside a uniform, linear dielectric. Calculate the electric field distribution everywhere in the system (both at r < R and at r > R).
Hint: You may start with the assumption that the field at r > R has a distribution typical for a dipole (but be ready for surprises :-).
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,
3.19. A spherical capacitor (see the figure on the right) is filled with a
linear dielectric whose permittivity depends on the spherical angles and ,
but not on the distance r from the system’s center. Derive an explicit a
expression for its capacitance C.
b a
3.20. A spherical capacitor similar to that considered in the previous problem is now filled with a linear dielectric whose permittivity depends only on the distance from the center. Obtain a closed expression for its capacitance, and spell it out for the particular case ( r) = ( a)( r/ a) n.
3.21. A uniform electric field E
E
0 has been created (by distant external
0
sources) inside a uniform linear dielectric. Find the change of the electric
R
field, created by carving out a cavity in the shape of a round cylinder of radius R, with its axis normal to the external field – see the figure on the right.
3.22. Similar small spherical particles, made of a linear dielectric, are dispersed in free space with a low concentration n << 1/ R 3, where R is the particle's radius. Calculate the average dielectric constant of such a medium. Compare the result with the apparent but wrong answer
1
1 nV ,
(WRONG!)
(where is the dielectric constant of the particle's material and V = (4/3) R 3 is its volume), and explain the origin of the difference.
3.23. A straight thin filament, uniformly charged with linear density , is stretched parallel to the plane separating two uniform linear dielectrics, at a distance d from it. Calculate the electric potential’s distribution everywhere in the system.
3.24. A point charge q is located at a distance d > R from the center of a sphere of radius R, made of a uniform linear dielectric with permittivity .
(i) Calculate the electrostatic potential’s distribution in all space for arbitrary ratio d/ R.
(ii) For large d/ R, use two different approaches to calculate the interaction force and the energy of interaction between the sphere and the charge, in the first nonzero approximation in R/ d << 1.
3.25. Calculate the spatial distribution of the electrostatic potential induced by
D
a point charge q placed at distance d from a very wide parallel plate, of thickness D, q
made of a uniform linear dielectric – see the figure on the right.
d
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3.26. Discuss the physical nature of Eq. (76). Apply your conclusions to a material with a fixed (field-independent) polarization P0, and calculate the electric field’s energy of a uniformly polarized sphere (see Problem 13).
3.27. Use Eqs. (73) and (82) to calculate the force of attraction of plane capacitor’s plates (per unit area), for two cases:
(i) the capacitor is charged to voltage V, and then disconnected from the battery,29 and (ii) the capacitor remains connected to the battery.
3.28. A slab made of a linear dielectric is partly inserted into
a plane capacitor – see the figure on the right. Assuming the
d
simplest (cylindrical) geometry of the system, calculate the force
exerted by the field on the slab, for the same two cases as in the
previous problem
x
a x
3.29. For each of the two capacitors shown in Fig. 10, calculate the electric force exerted on the interface between two different dielectrics, in terms of the fields in the system.
R
3.30. One half of a conducting sphere of radius R, carrying electric
Q
charge Q, is submerged into a half-space filled with a linear dielectric with
0
0
permittivity – see the figure on the right. Calculate the electric force
exerted on the sphere.
29 “Battery” is a common if misleading term for what is usually a single galvanic element. (The last term stems from the name of Luigi Galvani, a pioneer of electric current studies. Another term derived from his name is the galvanic connection, meaning a direct connection of two conductors, enabling a dc current flow – see the next chapter.) The term “battery” had to be, in all fairness, reserved for the connection of several galvanic elements in series – as was pioneered in 1800 by L. Galvani’s friend Alexander Volta.
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Chapter 4. DC Currents
The goal of this chapter is to discuss the distribution of stationary (“dc”) currents in conducting samples and their “global” characteristics such as resistance. In the most important case of linear (“Ohmic”)
conductivity, the current distribution is governed by the same Laplace and Poisson equations whose solution methods were discussed in detail in the previous chapters. Because of that, we can piggyback on most approaches discussed earlier, enabling me to keep this chapter rather brief.
4.1. Continuity equation and the Kirchhoff laws
Until this point, our discussion of conductors has been limited to the cases when they are separated with insulators (meaning either the free space or some dielectric media), preventing any continuous motion of charges from one conductor to another, even if there is a non-zero voltage (and hence electric field) between them – see Fig. 1a.
(a)
(b)
(c)
Q t
Q
Q
E
dc current
I ( t) 0
I const
source
Q
Q t
Q
I
Fig. 4.1. Two oppositely charged conductors: (a) in the electrostatic situation, (b) at the charge relaxation through an additional narrow conductor (“wire”), and (c) in a system sustaining a dc current I.
Now let us connect the two conductors with a wire – a thin, elongated conductor (Fig. 1b). Then the electric field causes the motion of charge carriers in the wire, from the conductor with a higher electrostatic potential toward that with lower potential, until the potentials equilibrate. Such a process is called charge relaxation. The main equation governing this process may be obtained from the fundamental experimental fact (already mentioned in Sec. 1.1) that electric charges cannot appear or disappear – though opposite charges may recombine with the conservation of the net charge. As a result, the charge Q in a conductor may change only due to the electric current I through the wire: dQ
I t;
(4.1)
dt
this relation may be understood as the definition of the current.1
1 Just as a (hopefully, unnecessary :-) reminder, in the SI units the current is measured in amperes (A). In legal metrology, the ampere (rather than the coulomb, which is defined as 1C = 1A 1s) is a primary unit. (Its formal definition will be discussed in the next chapter.) In the Gaussian units, Eq. (1) remains the same, so that the current’s unit is the statcoulomb per second – the so-called statampere.
© K. Likharev
EM: Classical Electrodynamics
Let us express Eq. (1) in a differential form, introducing the notion of the current density j(r) .
This vector may be defined via the following relation for the elementary current dI crossing an elementary area dA (Fig. 2):
dI jdA cos ( j cos ) dA j dA ,
(4.2)
n
where is the angle between the direction normal to the surface and the charge carrier motion direction, which is taken for the direction of the vector j.
dA cos
dA
dI
j
Fig. 4.2. The current density vector j.
With that definition, Eq. (1) may be rewritten as
d
d 3
r j d 2 r ,
(4.3)
n
dt V
S
where V is an arbitrary but stationary volume limited by the closed surface S. Applying to this volume the same divergence theorem as was repeatedly used in previous chapters, we get
3
j d r 0
.
(4.4)
t
V
Since the volume V is arbitrary, this equation may be true only if
j 0.
(4.5)
Continuity
t
equation
This is the fundamental continuity equation – which is true even for time-dependent phenomena.2
The charge relaxation, illustrated by Fig. 1b, is of course a dynamic, time-dependent process.
However, electric currents may also exist in stationary situations, when a certain current source, for example a battery, drives the current against the electric field, and thus replenishes the conductor charges and sustains currents at a certain time-independent level – see Fig. 1c. (This process requires a persistent replenishment of the electrostatic energy of the system from either a source or a large storage of energy of a different kind – say, the chemical energy of the battery.) Let us discuss the laws governing the distribution of such dc currents. In this case (/ t = 0), Eq. (5) reduces to a very simple equation
j 0 .
(4.6)
This relation acquires an even simpler form in the particular but important case of dc electric circuits (Fig. 3) – the systems that may be fairly represented as direct (“galvanic”) connections of components of two types:
2 Similar differential relations are valid for the density of any conserved quantity, for example for mass in classical dynamics (see, e.g., CM Sec. 8.3), and for the probability, as it is defined in statistical physics (SM Sec.
5.6) and in quantum mechanics (QM Sec. 1.4).
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(i) relatively-small-size ( lumped) circuit elements, meaning either a passive resistor, or a current source, etc. – generally, any “black box” with two or more terminals, and
(ii) perfectly conducting wires, with a negligible drop of the electrostatic potential along them, that are galvanically connected at certain points called nodes (or “junctions”).
“circuit
element”
“node”
“loop”
“wire”
Fig. 4.3. A typical system obeying Kirchhoff
laws.
In the standard circuit theory, the electric charges of the nodes are considered negligible,3 and we may integrate Eq. (6) over the closed surface drawn around any node to get a simple equality
I 0 ,
(4.7a)
j
j
where the summation is over all the wires (numbered with index j) connected in the node. On the other hand, according to its definition (2.25), the voltage Vk across each circuit element may be represented as the difference of the electrostatic potentials of the adjacent nodes, Vk = k – k-1. Summing such differences around any closed loop of the circuit (Fig. 3), we get all terms canceled, so that
V 0.
(4.7b)
k
k
These relations are called, respectively, the 1st and 2nd Kirchhoff laws 4 – or sometimes the node rule (7a) and the loop rule (7b). They may seem elementary, and their genuine power is in the mathematical fact that any set of Eqs. (7) covering every node and every circuit element of the system at least once, gives a system of equations sufficient for the calculation of all currents and voltages in it –
provided that the relation between the current and voltage is known for each circuit element.
It is almost evident that in the absence of current sources, the system of equations (7) has only the trivial solution: Ij = 0, Vk = 0 – with the exotic exception of superconductivity, to be discussed in Sec. 6.3. The current sources that allow non-zero current flows may be described by their electromotive forces ( e.m.f. ) V k, having the dimensionality of voltage, which have to be taken into account in the corresponding terms Vk of the sum (7b). Let me hope that the reader has some experience of using Eqs.
(7) for analyses of simple circuits – say, consisting of several resistors and batteries, so that I can save our time by skipping their discussion. Still, due to their practical importance, I would recommend the reader to carry out a self-test by solving a couple of problems offered at the beginning of Sec. 6.
3 In many cases, the charge accumulation/relaxation may be described without an explicit violation of Eq. (7a), just by adding other circuit elements, lumped capacitors (see Fig. 2.5 and its discussion), to the circuit under analysis. The resulting circuit may be used to describe not only the transient processes but also periodic ac currents. However, it is convenient for me to postpone the discussion of such ac circuits until Chapter 6, where one more circuit element type, lumped inductances, will be introduced.
4 Named after Gustav Kirchhoff (1824-1887) – who also suggested the differential form (8) of the Ohm law.
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4.2. The Ohm law
As was mentioned above, the relations spelled out in Sec. 1 are sufficient for forming a closed system of equations for finding electric current and field in a system only if they are complemented with some constitutive relations between the scalars I and V in each lumped circuit element, or alternatively between the macroscopic (atomic–scale–averaged) vectors j and E at each point of the material of such an element. The simplest of such relations is the famous Ohm law whose differential (or “local”) form is j E ,
(4.8)
Ohm
law
where is a constant called the Ohmic conductivity (or just the “conductivity” for short).5 Though the Ohm law (discovered, in its simpler form, by Georg Simon Ohm in 1827) is one of constitutive rather than fundamental relations, and is approximate for any conducting medium, we can argue that if: (i) the medium carries no current at E = 0 (mind superconductors!),
(ii) the medium is isotropic or virtually isotropic (a notable exception: some organic conductors), (iii) the mean free path l of the current carriers (the notion to be discussed in detail in SM Ch. 6) in this medium is much smaller than the characteristic scale a of the spatial variations of j and E, then the law may be viewed as the leading, linear term of the Taylor expansion of the local relation j(E), and thus is general for relatively low fields.
Table 1 gives approximate experimental values of for some representative (and/or practically important) materials. Note that the range of these values is very broad, even without going to such extremes as very pure metallic crystals at very low temperatures, where may reach ~1012 S/m.
Table 4.1. Ohmic dc conductivities for some materials at 20C.
Material
(S/m)
Teflon (PTFE, [C2F4] n)
10-22-10-24
Silicon dioxide
10-16-10-19
Various glasses
10-10-10-14
Deionized water
~10-6
Seawater
5
Silicon n-doped to 1016cm-3
2.5102
Silicon n-doped to 1019cm-3
1.6104
Silicon p-doped to 1019cm-3
1.1104
Nichrome (alloy 80% Ni + 20% Cr)
0.9106
Aluminum
3.8107
Copper
6.0107
Zinc crystal along a- axis
1.65107
Zinc crystal along c- axis
1.72107
5 In SI units, the conductivity is measured in S/m, where one siemens (S) is the reciprocal of the ohm: 1S (1)-1
1A/1V. The constant reciprocal to conductivity, 1/, is called resistivity and is commonly denoted by the letter
. I will, however, try to avoid using this notion, because in these notes this letter is already overused.
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In order to get a better feeling of what do these values mean, let us consider a very simple system (Fig. 4): a plane capacitor of area A >> d 2, filled with a material that has not only a dielectric constant , but also some Ohmic conductivity , with much more conductive electrodes.
z
V
Q
d
j, E
, 0
0
Q
Fig. 4.4. A “leaky” plane capacitor.
Assuming that these properties are compatible with each other,6 we may assume that the distribution of the electric potential (not too close to the capacitor’s edges) still obeys Eq. (2.39), so that the electric field is normal to the electrode surfaces and uniform, with E = V/ d. Then, according to Eq.
(6), the current density is also uniform, j = E = V/ d. From here, the total current between the plates is V
I jA EA
.
A
(4.9)
d
On the other hand, from Eqs. (2.26) and (3.45), the instantaneous value of the total charge of the top electrode is Q = CV = (0 A/ d) V. Plugging these relations into Eq. (1), we see that the speed of charge (and voltage) relaxation is independent of the geometric parameters A and d of the capacitor: dV
V
0
,
with
,
(4.10)
dt
r
r
so that the relaxation time constant r may be used to characterize the gap-filling material as such.
As we already know (see Table 3.1), for most practical materials the dielectric constant is within one order of magnitude from 10, so that the numerator in the second of Eqs. (10) is of the order of 10-10 (SI units). As a result, according to Table 1, the charge relaxation time ranges from ~1014s (more than a million years!) for the best insulators like Teflon (polytetrafluoroethylene, PTFE),7 to ~10-18s for the least resistive metals. What is the physics behind such a huge range of , and why, for some materials, Table 1 gives them with such a large uncertainty? As in Chapters 2 and 3, in this course I have time only for a brief, admittedly superficial discussion of these issues.8
If the charge carriers move almost as classical particles (e.g., in plasmas or non-degenerate semiconductors), a very reasonable description of the conductivity is given by the famous Drude formula.9 In his picture, due to a weak electric field, the charge carriers are accelerated in its direction (on top of their random motion in all directions, with the average velocity vector equal zero): v
d
q
E ,
(4.11)
dt
m
and as a result, their velocity acquires the average value
6 As will be discussed in Chapter 6, this is true only if is not too high.
7 This polymer is broadly used in engineering and physical experiment, due to its many remarkable properties.
8 A more detailed discussion of this issue may be found in SM Chapter 6.
9 It was suggested by Paul Drude in 1900.
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v
d
q
v
E ,
(4.12)
dt
m
where the phenomenological parameter = l/2 v (not to be confused with r!) may be understood as half the average time between carrier scattering events. From here, the current density:10
q 2 n
q 2 n
j qnv
,
E
i.e.
.
(4.13a)
m
m
Drude
(Notice the independence of of the charge sign.) Another form of the same result, more popular in the formula: two
physics of semiconductors, is
versions
2
q n,
with
,
(4.13b)
m
where the parameter , defined by the relation v E, is called the charge carrier mobility.
Most good conductors (e.g., metals) are essentially degenerate Fermi gases (or liquids), in which the average thermal energy of a particle, k B T is much lower than the Fermi energy F. In this case, a quantum theory is needed for the calculation of . Such theory was developed by the quantum physics’
godfather A. Sommerfeld in 1927 (and is sometimes called the Drude-Sommerfeld model). I have no time to discuss it in this course,11 and here will only notice that for a nearly-ideal, isotropic Fermi gas the result is reduced to Eq. (13), with a certain effective value of , so it may be used for estimates of , with due respect to the quantum theory of scattering. In a typical metal, n is very high (~1023 cm-3) and is fixed by the atomic structure, so the sample quality may only affect via the scattering time .
At room temperature, the scattering of electrons by thermally-excited lattice vibrations ( phonons) dominates, so that and are high but finite, and do not change much from one sample to another. (Hence the relatively accurate values given for metals in Table 1.) On the other hand, at T 0, quantum mechanics says a perfect crystal should not exhibit scattering at all, and its conductivity should be infinite. In practice, this is never true (for one, due to electron scattering from imperfect boundaries of finite-size samples), and the effective conductivity is infinite (or practically infinite, at least above the largest measurable values ~1020 S/m) only in superconductors.12
On the other hand, the conductivity of quasi-insulators (including deionized water) and semiconductors depends mostly on the carrier density n, which is much lower than in metals. From the point of view of quantum mechanics, this happens because the ground-state wavefunctions of charge carriers are localized within an atom (or molecule), and their energies are separated from those of excited states, with space-extended wavefunctions, by a large energy gap – often called the bandgap.
For example, in SiO2 the bandgap approaches 9 eV, equivalent to ~4,000 K. This is why even at room temperatures the density of thermally-excited free charge carriers in good insulators is negligible. In these materials, n is determined by impurities and vacancies, and may depend on a particular chemical synthesis or other fabrication technology, rather than on the fundamental properties of the material. (On the contrary, the carrier mobility in these materials is almost technology-independent.) 10 Note that j in Eq. (8) is defined as an already macroscopic variable, averaged over inter-particle distances, so that no additional average sign is necessary in the first of Eqs. (13a).
11 For such a discussion see, e.g., SM Sec. 6.3.
12 The electrodynamic properties of superconductors are so interesting (and fundamentally important) that I will discuss them in more detail in Chapter 6.
Chapter 4
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Essential Graduate Physics
EM: Classical Electrodynamics
The practical importance of the fabrication technology may be illustrated by the following example. In the cells of the so-called floating-gate memories, in particular the flash memories, which currently dominate the nonvolatile digital memory technology, data bits are stored as small electric charges ( Q ~ 10-16 C ~ 103 e) of highly doped silicon islands (so-called floating gates) separated from the rest of the integrated circuit with ~10-nm-thick layers of silicon dioxide, SiO2. Such layers are fabricated by high-temperature oxidation of virtually perfect silicon crystals. The conductivity of the resulting high-quality (though amorphous) material is so low, ~ 10-19 S/m, that the relaxation time r, defined by Eq. (10), is well above 10 years – the industrial standard for data retention in nonvolatile memories. To appreciate how good this technology is, the cited value should be compared with the typical conductivity ~ 10-16 S/m of the usual, bulk SiO2 ceramics.13
To conclude this chapter, let me note that the Ohm law, for all its importance, is not a universal law of nature. As a reminder of this fact, in Sec. 5 below I describe two very simple systems (leaving their analysis for the reader’s exercise) whose I-V relation is nonlinear even for very small currents.
4.3. Boundary problems
For an Ohmic conducting medium, we may combine Eqs. (6) and (8) to get the following differential equation
0 .
(4.14)
For a uniform conductor ( = const), Eq. (14) is reduced to the Laplace equation for the (macroscopic) electrostatic potential . As we already know from Chapters 2 and 3, its solution depends on the boundary conditions. These conditions, in turn, depend on the interface type.
(i) Conductor-conductor interface. Applying the continuity equation (6) to a Gauss-type pillbox at the interface of two different conductors (Fig. 5), we get
( jn)1 = ( jn)2,
(4.15)
so that if the Ohm law (8) is valid inside each medium, then
1
2
.
(4.16)
1
n
2
n
1
j1
2
1
j
2
Fig. 4.5. DC current’s “refraction” at the interface between
two different conductors.
13 This course is not an appropriate platform to discuss details of the floating-gate memory technology. However, I think that every educated physicist should know its basics, because such memories are presently the driver of all semiconductor integrated circuit technology development, and hence of the whole information technology progress. Perhaps the best available general book on this topic is still the relatively old review collection by J.
Brewer and M. Gill (eds.), Nonvolatile Memory Technologies with Emphasis on Flash, IEEE Press, 2008.
Chapter 4
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EM: Classical Electrodynamics
Also, since the electric field should be finite, its potential has to be continuous across the interface – the condition that may also be written as
1
2 .
(4.17)
Both these conditions (and hence the solutions of the boundary problems using them) are similar to those for the interface between two dielectrics – cf. Eqs. (3.46)-(3.47). Note that using the Ohm law, Eq.
(17) may be rewritten as
1
1
j j .
(4.18)
1
2
1
2
Comparing it with Eq. (15) we see that, generally, the current density’s magnitude changes at the interface: j 1 j 2. It is also curious that if 1 2, the current line slope changes at the interface (Fig. 5), qualitatively similar to the refraction of light rays in optics – see Chapter 7.
(ii) Conductor-electrode interface. An electrode is defined as a body made of a “perfect conductor”, i.e. of a medium with . Then, at a fixed current density at the interface, the electric field in the electrode tends to zero, and hence it may be described by the equality
const ,
(4.19)
j
where constants j may be different for different electrodes (numbered with index j). Note that with such boundary conditions, the Laplace boundary problem becomes exactly the same as in electrostatics
– see Eq. (2.35) – and hence we can use the methods (and some solutions :-) discussed in Chapter 2 for finding the dc current distribution.
(iii) Conductor-insulator interface. For the description of a good insulator, we can use the equality = 0, so that Eq. (16) yields the following boundary condition,
0 ,
(4.20)
n
for the potential derivative inside the conductor. From the Ohm law (8) in the form j = –, we see that this is just the very natural requirement for the dc current not to flow into an insulator. Now note that this condition makes the Laplace problem inside the conductor completely well-defined, and independent of the potential distribution in the adjacent insulator. On the contrary, due to the continuity of the electrostatic potential at the border, its distribution inside the surrounding insulator has to follow that inside the conductor.
Let us discuss this conceptual issue on the following (apparently, trivial) example: dc current in a uniform wire of length l and a cross-section of area A. The reader certainly knows the answer: V
V
l
Uniform
I
,
where R
,
(4.21) wire’s
R
I
A
resistance
where the constant R is called the wire’s resistance.14
14 The first of Eqs. (21) is essentially the (historically, initial) integral form of the Ohm law, and is valid not only for a uniform wire but also for Ohmic conductors of any geometry in that I and V may be clearly defined.
Chapter 4
Page 8 of 16
Essential Graduate Physics
EM: Classical Electrodynamics
However, let us derive this result formally from our theoretical framework. For the simple geometry shown in Fig. 6a, this is easy to do. Here the potential evidently has a linear 1D distribution x
const V ,
(4.22)
l
both in the conductor and the surrounding free space, with both boundary conditions (16) and (17) satisfied at the conductor-insulator interfaces, and the condition (20) satisfied at the conductor-electrode interfaces. As a result, the electric field is constant and has only one Cartesian component: Ex = V/ l, so that inside the conductor
j E
, I j A ,
(4.23)
x
x
x
giving us the well-known Eq. (21).
(a)
(b)
V
0
0
E
V
, A
E
Fig. 4.6. (a) An elementary
E
problem and (b) a (slightly)
j
j
less obvious problem of the
field distribution at dc
