Survey
In the preceding topic, the optical constants and their relationship to electrical constants were introduced by employing the "continuum theory." The continuum theory considers only macroscopic quantities and interrelates experimental data. No assumptions are made about the structure of matter when formulating equations. Thus, the conclusions which have been drawn from the empirical laws in topic 10 should have general validity as long as nothing is neglected in a given calculation. The derivation of the Hagen-Rubens equation has served as an illustrative example for this.
The validity of equations derived from the continuum theory is, however, often limited to frequencies for which the atomistic structure of solids does not play a major role. Experience shows that the atomistic structure does not need to be considered in the far infrared (IR) region. Thus, the Hagen-Rubens equation reproduces the experimental results of metals in the far IR quite well. It has been found, however, that proceeding to higher frequencies (i.e., in the near IR and visible spectrum), the experimentally observed reflectivity of metals decreases faster than predicted by the Hagen-Rubens equation (Fig. 11.1(a)). For the visible and near IR region an atomistic model needs to be considered to explain the optical behavior of metals. Drude did this important step at the turn of the 20th century. He postulated that some electrons in a metal can be considered to be free, i.e., they can be separated from their respective nuclei. He further assumed that the free electrons can be accelerated by an external electric field. This preliminary Drude model was refined by considering that the moving electrons collide with certain metal atoms in a nonideal lattice.
Figure 11.1. Schematic frequency dependence of the reflectivity of (a) metals, (b) dielectrics, experimentally (solid line) and according to three models.
The free electrons are thought to perform periodic motions in the alternating electric field of the light. These vibrations are restrained by the abovementioned interactions of the electrons with the atoms of a nonideal lattice. Thus, a friction force is introduced, which takes this interaction into consideration. The calculation of the frequency dependence of the optical constants is accomplished by using the well-known equations for vibrations, whereby the interactions of electrons with atoms are taken into account by a damping term which is assumed to be proportional to the velocity of the electrons. The free electron theory describes, to a certain degree, the dispersion of the optical constants of metals quite well. This is schematically shown in Fig. 11.1(a), in which the spectral dependence of the reflectivity is plotted for a specific case. The Hagen-Rubens relation reproduces the experimental findings only up to 1013 s_1. In contrast to this, the Drude theory correctly reproduces the spectral dependence of R even in the visible spectrum. Proceeding to yet higher frequencies, however, the experimentally found reflectivity eventually rises and then decreases again. Such an absorption band cannot be explained by the Drude theory. For its interpretation, a new concept needs to be applied.
Lorentz postulated that the electrons should be considered to be bound to their nuclei and that an external electric field displaces the positive charge of an atomic nucleus against the negative charge of its electron cloud. In other words, he represented each atom as an electric dipole. Retracting forces were thought to occur which try to eliminate the displacement of charges. Lorentz postulated further that the centers of gravity of the electric charges are identical if no external forces are present. However, if one shines light onto a solid, i.e., if one applies an alternating electric field to the atoms, then the dipoles are thought to perform forced vibrations. Thus, a dipole is considered to behave similarly as a mass which is suspended on a spring, i.e., the equations for a harmonic oscillator may be applied. An oscillator is known to absorb a maximal amount of energy when excited near its resonance frequency (Fig. 11.2). The absorbed energy is thought to be dissipated mainly by diffuse radiation. Figure 11.2 resembles an absorption band as shown in Fig. 11.1.
Forty or fifty years ago, many scientists considered the electrons in metals to behave at low frequencies as if they were free and at higher frequencies as if they were bound. In other words, electrons in a metal under the influence of light were described to behave as a series of classical free electrons and a series of classical harmonic oscillators. Insulators and semiconductors, on the other hand, were described by harmonic oscillators only, see Fig. 11.1(b).
We shall now treat the optical constants of materials by applying the above-mentioned theories.
Figure 11.2. Frequency dependence of the amplitude of a harmonic oscillator that is excited to perform forced vibrations, assuming weak damping. v0 is the resonance frequency.
Free Electrons Without Damping
We consider the simplest case at first and assume that the free electrons are excited to perform forced but undamped vibrations under the influence of an external alternating field, i.e., under the influence of light. As explained in Section 11.1, the damping of the electrons is thought to be caused by collisions between electrons and atoms of a nonideal lattice. Thus, we neglect in this section the influence of lattice defects. For simplicity, we treat the one-dimensional case because the result obtained this way does not differ from the general case. Thus, we consider the interaction of plane-polarized light with the electrons. The momentary value of the field strength of a plane-polarized light wave is given by
where
is the angular frequency, t is the time, and E0 is the maximal value of the field strength. The equation describing the motion of an electron that is excited to perform forced, harmonic vibrations under the influence of light is (see topic 1 and (7.6))
where e is the electron charge, m is the electron mass, and e E is the modulus of the excitation force. The stationary solution of this vibrational equation is obtained by forming the second derivative of the trial solution x = x0 exp(zmt) and inserting it into (11.2). This yields
The vibrating electrons carry an electric dipole moment, which is the product of the electron charge, e, and displacement, x, see (9.12). The polarization, P, is defined to be the sum of the dipole moments of all Nf free electrons per cubic centimeter:
The dielectric constant can be calculated from polarization and electric field strength by combining (9.14) and (9.15):
Inserting (11.3) and (11.4) into (11.5) yields
(It is appropriate to use in the present case the complex dielectric constant, see below.) The dielectric constant equals the square of the index of refraction, n, (see (10.12)). Equation (11.6) thus becomes
We consider two special cases:
(a) For small frequencies, the term
is larger than one. Then n2 is negative and n imaginary. An imaginary n means that the real part of n disappears. Equation (10.25) becomes, for n = 0,
i.e., the reflectivity is 100% (see Fig. 11.3). (b) For large frequencies (UV light), the termbecomes smaller than one. Thus, n2 is positive and n = n real (but smaller than one). The reflectivity for real values of n, i.e., for k = 0, becomes
i.e., the material is essentially transparent for these wavelengths (and perpendicular incidence) and therefore behaves optically like an insulator, see Fig. 11.3.
We define a characteristic frequency, n1, often called the plasma frequency, which separates the reflective region from the transparent region (Fig. 11.3). The plasma frequency can alsobe deduced from (11.6) or (11.7). We observe in these equations that must have the unit of the square of a frequency, which we define to be n1. This yields
Figure 11.3. Schematic frequency dependence of an alkali metal according to the free electron theory without damping. v1 is the plasma frequency.
Because of (11.8) we conclude from (11.6) that the dielectric constant becomes zero at the plasma frequency. e = 0 is the condition for a plasma oscillation, i.e., a fluidlike oscillation of the entire electron gas. We will discuss this phenomenon in detail in Section 13.2.2.
The alkali metals behave essentially as shown in Fig. 11.3. They are transparent in the near UV and reflect the light in the visible region. This result indicates that the s-electrons5 of the outer shell of the alkali metals can be considered to be free.
Table 11.1 contains some measured, as well as some calculated, plasma frequencies. For the calculations, applying (11.8), one free electron per atom was assumed. This means that Nf was set equal to the number of atoms per volume, Na. (The latter quantity is obtained by using
where N0 is the Avogadro constant, d = density, and M = atomic mass.)
We note in Table 11.1 that the calculated and the observed values for v1 are only identical for sodium. This may be interpreted to mean that only in sodium does exactly one free electron per atom contribute to the electron gas. For other metals an "effective number of free electrons" is commonly introduced, which is defined to be the ratio between the observed and calculated v22 values:
The effective number of free electrons is a parameter of great interest, because it is contained in a number of nonoptical equations (such as the Hall constant, electromigration, superconductivity, etc.). Since for most metals the plasma frequency, v1, cannot be measured as readily as for the alkalis, another avenue for determining Neff has to be found. For reasons which will become clear later, Neff can be obtained by measuring n and k in the red or
Table 11.1. Plasma Frequencies and Effective Numbers of Free Electrons for Some Alkali Metals.
|
Metal |
Li |
Na |
K |
Rb |
Cs |
 |
14.6 |
14.3 |
9.52 |
8.33 |
6.81 |
 |
19.4 |
14.3 |
10.34 |
9.37 |
8.33 |
 |
150 |
210 |
290 |
320 |
360 |
 |
0.57 |
1.0 |
0.8 |
0.79 |
0.67 |
IR spectrum (i.e., in a frequency range without absorption bands, Fig. 11.1) and by applying
Equation (11.10a) follows by combining (11.6) with (10.10) and replacing Nf by Neff.
