Dressel.Gruner.Electrodynamics of Solids.2003

Part three

Experiments

Introductory remarks

The theoretical concepts of metals, semiconductors, and the various broken symmetry states were developed in Part 1. Our objective in this part is to subject these theories to test by looking at some examples, and thus to check the validity of the assumptions which lie behind the theories and to extract some important parameters which can be compared with results obtained by utilizing other methods.

We first focus our attention on simple metals and simple semiconductors, on which experiments have been conducted since the early days of solid state physics. Perhaps not too surprisingly the comparison between theory and experiment is satisfactory, with the differences attributed to complexities associated with the electron states of solids which, although important, will not be treated here. We also discuss topics of current interest, materials where electron–electron, electron– phonon interactions and/or disorder are important. These interactions fundamentally change the character of the electronic states – and consequently the optical properties. These topics also indicate some general trends of condensed matter physics.

The discussion of metals and semiconductors is followed by the review of optical experiments on various broken symmetry ground states. Examples involving the BCS superconducting state are followed by observations on materials where the conditions of the weak coupling BCS approach are not adequate, and we also

describe the current experimental state of affairs on materials with charge or spin density wave ground states.

General books and monographs

F.Abeles,` ed., Optical Properties and Electronic Structure of Metals and Alloys (North Holland, Amsterdam, 1966)

G.Gruner,¨ Density Waves in Solids (Addison Wesley, Reading, MA, 1994)

S.Nudelman and S.S. Mitra, eds, Optical Properties of Solids (Plenum Press, New York, 1969)

E.D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, Orlando, 1985–98)

D. Pines, Elementary Excitations in Solids (Addison Wesley, Reading, MA, 1963) A.V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967)

J.Tauc, ed., The Optical Properties of Solids, Proceedings of the International School of Physics ‘Enrico Fermi’ 34 (Academic Press, New York, 1966)

P.Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin,

1996)

12.1.1 Comparison with the Drude–Sommerfeld model

The fundamental prediction of the Drude–Sommerfeld model discussed in Section 5.1 is a frequency dependent complex conductivity, reproduced here

The ingredients of the model are the frequency independent relaxation rate 1/τ , and the mass mb which loosely can be defined as the bandmass,1 also assumed to be frequency independent; N is the electron density – more precisely the number of conduction electrons per unit volume. In Chapter 5 we derived the complex conductivity as given above, using the Kubo and also the Boltzmann equations. In addition to the frequency independent relaxation rate and mass, the underlying condition for these formalisms to apply is that the elementary excitations of the system are well defined, a condition certainly appropriate for good metals which can be described as Fermi liquids.

Let us first estimate the two fundamental frequencies of the model, the inverse relaxation time 1/τ and the plasma frequency

for a typical good metal at room temperature. This can be done by using the measured dc conductivity

which is of the order of 106 −1 cm−1. For a number of free electrons per volume of approximately 1023 cm−3 and a bandmass which is assumed to be the same as the free-electron mass, we find γ = 1/(2π cτ ) ≈ 150 cm−1, while the

1The bandmass is usually defined by the curvature of the energy band in k space as given by Eq. (12.1.17), and this definition is used to account, in general, for the thermodynamic and magnetic properties by assuming that this mass enters – instead of the free-electron mass – in the various expressions for the specific heat or (Pauli-like) magnetic susceptibility.

2Some clarification of the units used is in order here. We will frequently use the inverse wavelength, the cm−1 scale, which is common in infrared spectroscopy to describe optical frequencies. The correct unit is

frequency per velocity of light ( f /c); however, this is usually omitted and we refer to ωp or νp, for example, as frequencies, but using the cm−1 scale. We will follow this notation and refer to Table G.4 for conversion.

three regimes where the optical parameters have well defined characteristics – the Hagen–Rubens, the relaxation, and the transparent regimes – are well separated. These characteristics and the expected behavior of the optical constants in the three regimes are discussed in detail in Section 5.1.2.

When exploring the optical properties of metals in a broad frequency range, not only the response of the free conduction carriers described by the Drude model are important, but also excitations between different bands. The interband transitions can be described along the same lines as for semiconductors; in general the analysis is as follows: we assume a Drude form for the intraband contributions, with parameters which are fitted to the low frequency part of the electromagnetic spectrum. With this contribution established, the experimentally found optical parameters – say the dielectric constant 1(ω) – can, according to Eq. (6.2.17), be decomposed into intraand inter-band contributions,

The interband contribution has, as expected, a finite 1 at low frequencies, with a peak at a frequency which corresponds to the onset of interband absorption, features which will be discussed in more detail in the next chapter.

The optical parameters evaluated for gold at room temperature from transmission and reflectivity measurements over a wide range of frequency are displayed in Fig. 12.1; the data are taken from [Ben66] and [Lyn85]. The overall behavior of the various optical constants is in broad qualitative agreement with those calculated, as can clearly be demonstrated by contrasting the experimental findings with the theoretical curves displayed in Figs 5.2–5.5. We observe a high reflectivity up to about 2 × 104 cm−1, the frequency where the sudden drop of reflectivity R signals the onset of the transparent regime. The reflectivity does not immediately approach zero above this frequency because of interband transitions – not included in the simple Drude response which treats only one band, namely the conduction band. The onset of transparency can be associated with the plasma frequency ωp; with a carrier concentration N = 5.9 × 1022 cm−3 (assuming that one electron per atom contributes to the conduction band) we obtain a bandmass significantly larger than the free-electron mass. This may be related to the narrow conduction band or to the fact that interband transitions are neglected. For the conduction band (where the carriers have the bandmass mb) the sum rule of the conductivity is

where σ1inter refers to the interband contribution to the conductivity. The deviation of the bandmass mb from the free-electron mass m is therefore directly related to the spectral weight of the conductivity coming from the interband transitions3

The interband transitions are usually evaluated by fitting the optical parameters to the Drude response at low frequencies (where intraband contributions dominate) and plotting the difference between experiment and the Drude contribution at high frequencies. This procedure can also be applied to the real part of the dielectric constant 1(ω) as displayed in Fig. 12.2, where the dielectric constant of gold is decomposed into an interband contribution and an intraband contribution according to Eq. (12.1.3). Such decomposition and fit of the intraband contribution gives a plasma frequency ωp of 7.0 × 104 cm−1 from the zero-crossing of 1(ω); this compares favorably with the plasma frequency of 7.1 × 104 cm−1 calculated from Eq. (5.1.5) assuming free-electron mass and, as before, one electron per atom in the conduction band. The form of 1inter(ω) is that of a narrow band with a bandgap of 2 × 104 cm−1 between valence and conduction bands, in broad qualitative agreement with band structure calculations. Instead of a detailed analysis, such as in Fig. 12.2, it is frequently assumed that interband contributions lead only to a finite high frequency dielectric constant ∞, and we incorporate this into the analysis of the Drude response of the conduction band. The condition for the zero-crossing of 1(ω) is given by

and thus we recover a plasma frequency renormalized by the interband transitions

for 1/τ ωp. Over an extended frequency region, below ωp, power law behaviors of various optical parameters are found, with the exponent indicated on Fig. 12.1. Both k(ω) and 1(ω) are well described with expressions appropriate for the relaxation regime (see Eqs (5.1.20) and (5.1.21)), although the frequency dependence for n(ω) and 2(ω) is somewhat different from that predicted by these equations. The reason for this probably lies in the contributions coming from interband transitions.4 The fact that n > k observed over an extended spectral range also upholds

3The argument here is different from the spectral weight argument developed in Appendix D. Here, collisions (included in the damping term) lead to momentum relaxation and to a conductivity for which the sum rule (12.1.4) applies. In contrast, a collisionless electron gas is considered in Appendix D, and under this condition there is no absorption at finite frequencies.

4The experimental data are taken from two different sources [Ben66, Lyn85]; they seem to have some offset with respect to each other.

different temperatures, leads, in general, to a temperature independent plasma frequency (not surprising since the number of carriers is constant and factors which determine the bandmass also do not depend on the temperature) and to a relaxation rate which compares well with the temperature dependent relaxation rate as evaluated from the dc resistivity since ρ(T ) 1/τ (T ).

The uncertainties are typical of what is observed in simple metals: they are the consequence of band structure effects, and the influence of electron–electron and electron–phonon interactions. These effects can often be incorporated into an effective bandmass which leads to an enhanced specific heat and magnetic susceptibility (note that within the framework of the nearly free-electron model and also the tight binding model, both are inversely proportional to the bandmass), and, through Eq. (12.1.1b), to a reduced plasma frequency. However, further complications may also arise. The mass which enters into the expression of the dc conductivity may be influenced by electron–phonon interactions, although such interactions are not important above the phonon frequencies (typically located in the infrared range). Similar frequency dependent renormalizations occur for electron–electron interactions. Interband transitions may also lead to contributions to the conductivity at low frequencies, thus interfering with the Drude analysis, and anisotropy effects (discussed later) can also be important.

It is clear from the analysis given above that the Hagen–Rubens regime cannot easily be explored; this is mainly due to the fact that, for a highly conducting metallic state, the reflectivity is close to unity, and at the same time the range of validity of the Hagen–Rubens regime is limited to low frequencies where conducting the experiments is not particularly straightforward. The situation is different for so-called bad metals, materials for which σdc is small, 1/τ is large, and consequently the reflectivity deviates well from unity even in this regime, where

Stainless steel is a good example, and the reflectivity as measured by a variety of experimental methods is displayed in Fig. 12.3. It is evident that Eq. (12.1.9) is well obeyed, and the full line is calculated using σdc = 1.4 × 104 −1 cm−1 in excellent agreement with the directly measured dc conductivity. For short relaxation times additional complications occur. In the high frequency limit, the complex dielectric constant is