How can the well-known relationship between electrical conductivity and optical reflectance be explained?
From M axwell’s electrom agnetic equations a relationship can be derived between the optical constants n and ky the dielectric coefficients an an the electrical conductivity cr. The relative dielectric constant er is re ate o and k by the equation,
Et = e l + ie2 = n2 - k 2 + 2nik
and the expression for e2 terms of conductivity and frequency is, in SI units, a(co) <x(v)
where v is the frequency of the electric field vector in the incident electrom agne wave and a> is the angular frequency, co = 2nv.
In a metal the absorption e2 ]S much higher than the polarization so th er ^ ie2
and the refractive index is related to the dielectric constants by 54 Conduction electrons in materials - classical approach
„ 2 = £ l + \ / £ ? + £ 2
2 2
_ £i i + ° 2/(o2e02
~ 2 2
Since <r2/(o2el ằ ej, this gives
n2 a
2coz0
Furthermore, since c2 = 2nk = <j/(de0, it is clear that we must also have
/
At longer wavelengths and hence lower frequencies v, the high conductivit o leads to values of both k and n which are much greater than unity, and ai typically of the order of 104
Now considering the expression for the reflectance R = (n - l)2 + k 2
(n+ l)2 + /c2
= ' —(n + I)' +— and since n ^ k ằ 1, this leads to
R * 1 - - n
R % 1 — 2 N/ 4 7 i £ 0 v/(T
= 1- i j l t o e 0/a
which is the Hagen-Rubens relation [9], showing that high conductivit materials have high reflectance at long wavelengths. This provides the physica
justification for our observation at the outset in C h ap ter 1 that good electrical co nd uctors are also good optical reflectors.
O ptical pro p erties o f m eta ls 55
3*4.4 Extensions of classical free-electron theory to optical properties at high frequencies
H ow can the higher frequency absorption bands be explained by the model?
At higher frequencies it is known from experim ental observation th a t the reflectance does not necessarily remain low, but can show some localized peaks.
These can be explained by an extension of the classical free-electron theory ue to Lorentz [3 ,4 ] in which some electrons behave as classical bound oscillators.
These electrons are more tightly bound to the atom s and therefore can only respond to higher energy excitations. This leads to the following equation of m otion for the electrons,
w — + y — + /cx = eÇ0 e x p lic it) d i 2 dt
where now the additional term k x represents a binding force between electrons and ionic sites. This equation describes the m otion of these bound oscillators and gives absorption at higher frequencies in the form of bound oscillator resonances (Fig. 3.7).
classical IR absorption
rad violat
visible spectrum
H g . 3.7 Optical reflectance of metals beyond the infra-red range, in which ‘resonances’
at higher energies are observed. These can be attributed to bound oscillators rather than free electrons. Reproduced with permission from R. E. Hummel, Electronic Properties of Materials 2nd edn, published by Springer Verlag, 1993.
56 Conduction electrons in materials - classical approach 3.4.5 The photoelectric effect
What happens to the conduction electrons when high-energy light impinges on certain metals.9
It was shown by Hertz [10] that ?. metallic surface emits electrons when illuminated by light of a very short wavelength. The emission of electrons from the surface is dependent on the wavelength of the light and not on the total energy incident on the surface.
The emission of electrons does not occur when the surface is irradiated with longer wavelength light over a longer time period, if the wavelength is below a certain critical value. In other words, if the frequency of the incident light is below a certain threshold value, exposure for longer periods will not lead to the emission of electrons, even though the total energy absorbed by the surface can be increased indefinitely in this way.
Furthermore, the kinetic energy £ K of the emitted electrons is dependent on the frequency of the light, but not on the intensity of the light,
Ek = constant (v — v0)
where v is the frequency of incident light and v0 is the threshold frequency which just enables electrons to escape from the material (Fig.3.8).
An explanation of these observations was given by Einstein [1 lj. If a> is the angulapdrequency of the incident radiation and h is Planck’s constant divided by 27i, the energy of an incident light photon is
E{co) = hco.
Now considering the electrons as classical particles trapped in a finite square- well potential of height 0, and assuming one light quantum interacts with one electron, the energy imparted contributes to the energy needed to overcome
Table 3.2 Values of the work function and threshold frequencies for the photoelectric effect in various materials
Material
Threshold energy or work function
<P(eV)
Threshold frequency vo(/0 I4s ‘ 1)
Caesium 1.91 4.62
Rubidium 2.17 5.25
Potassium 2.24 5.42
Li Mi urn 2.28 5.51
Sodium 2.46 5.95
Zinc 3.57 8.63
Copper 4.16 10.06
Tungsten 4.54 10.98
Silver 4.74 11.46
Platinum 6.30 15.23
Conclusions 57
Fig. 3.8 Kinetic energy of emitted electrons in the photoelectric effect as a function of frequency of incident light.
the binding energy EB of the electron to the solid and to the final kinetic energy of the electron,
h o — Ek 4- EB
= \ n w 2 4- <t>
= j m v 1 -f hcoo
where is the symbol used for the work function of the metal. This is identical to the threshold energy needed to liberate the electrons. R earranging the equation gives,
Ek — hv — hvo
which agrees with experimental observations. This model of the photoelectric effect we may call semi-classical. It relies on the quantum nature of the incident light, but still treats the electrons as classical particles in a finite potential well.
The work function (f> is the energy needed to extract one electron from the material, or alternatively is the depth of the potential well.