Dressel.Gruner.Electrodynamics of Solids.2003

The calculation of σ sp is now straightforward, and in Fig. 7.9 the real part of the complex conductivity σ1(ω) is displayed for two different values of the effective mass.

There is also a collective mode contribution to the conductivity at zero frequency, as discussed before, with spectral weight

and this is taken out of the spectral weight corresponding to the single-particle excitations at frequencies ω > 2 /h¯ . The total spectral weight

is of course observed, and is the same as the spectral weight associated with the Drude response in the metallic state above the density wave transition.

For an incommensurate density wave in the absence of lattice imperfections, the collective mode contribution occurs at ω = 0 due to the translational invariance of the ground state. In the presence of impurities this translational invariance is broken and the collective modes are tied to the underlying lattice due to interactions with impurities [Gru88, Gru94]; this aspect of the problem is discussed in Chapter 14.

7.5.2 Spin density waves

The electrodynamics of the spin density wave ground state is different from the electrodynamics of the superconducting and of the charge density wave states. The difference is due to several factors. First, for the case of density waves, case 1 coherence factors apply, and thus the transition probabilities are different from those of a superconductor. Second, the spin density wave state arises as the consequence of electron–electron interactions. Phonons are not included here, and also the spin density wave ground state – the periodic modulation of the spin density – does not couple to the underlying lattice. Consequently, the mass which is related to the dynamics of the collective mode is simply the electron mass.

Therefore, the electrodynamics of the spin density wave ground state can be discussed along the lines developed for the superconducting state with one important difference: for the electrodynamic response, case 1 coherence factors apply.

In Fig. 7.7 the frequency dependence of σ1(ω) and σ2(ω) derived from Eqs (7.5.6) is plotted assuming finite scattering effects /π ξ = 0.1. This condition corresponds to the dirty limit, as introduced before. In contrast to the results obtained for the case of a superconductor, an enhancement of the conductivity σ1(ω) is observed above the single-particle gap, which is similar to the results obtained from a simple semiconductor model in one dimension (Eq. (6.3.16)). As discussed above, the spectral weight in the gap region is, by and large, compensated for by the area above the single-particle gap 2 , leading to a small collective mode. This fact results in a low reflectivity for ω < 2 , as displayed in Fig. 7.8. Finite temperature effects arise here in a fairly natural fashion; thermally excited single-particle states lead to a Drude response in the gap region, with the spectral weight determined by the temperature dependence of the number of excited carriers.

7.5.3 Clean and dirty density waves and the spectral weight

The spectral weight arguments advanced for superconductors also apply for the density wave states with some modifications. In the density wave states the collective mode contribution to the spectral weight is

from the excitation above the gap. For a large effective mass m /mb 1, most of the total spectral weight comes from the single-particle excitations; while for m /mb = 1, all the spectral weight is associated with the collective mode, with no contribution to the optical conductivity from single-particle excitations. The former is appropriate for charge density waves, while the latter is valid for spin density wave transport [Gru94].

Further reading

[Gor89] L.P. Gor’kov and G. Gruner,¨ eds, Charge Density Waves in Solids, Modern Problems in Condensed Matter Sciences 25 (North-Holland, Amsterdam, 1989)

[Mon85] P. Monceau, ed., Electronic Properties of Inorganic Quasi-One Dimensional Compounds (Riedel, Dordrecht, 1985)

[Par69] R.D. Parks, ed., Superconductivity (Marcel Dekker, New York 1969)

[Pip65] A.B. Pippard, The Dynamics of Conduction Electrons (Gordon and Breach, New York, 1965)

[Por92] A.M. Portis, Electrodynamics of High-Temperature Superconductors (World Scientific, Singapore, 1992)

[Sin77] K.P. Sinha, Electromagnetic Properties of Metals and Superconductors, in: Interaction of Radiation with Condensed Matter, 2 IAEA-SMR-20/20 (International Atomic Energy Agency, Vienna, 1977), p. 3

[Wal64] J.R. Waldram, Adv. Phys. 13, 1 (1964)

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Part two

Methods

Introductory remarks

The wide array of optical techniques and methods which are used for studying the electrodynamic properties of solids in the different spectral ranges of interest for condensed matter physics is covered by a large number of books and articles which focus on different aspects of this vast field of condensed matter physics. Here we take a broader view, but at the same time limit ourselves to the various principles of optical measurements and compromise on the details. Not only conventional optical methods are summarized here but also techniques which are employed below the traditional optical range of infrared, visible, and ultraviolet light. These techniques have become increasingly popular as attention has shifted from singleparticle to collective properties of the electron states of solids where the relevant energies are usually significantly smaller than the single-particle energies of metals and semiconductors.

We start with the definition of propagation and scattering of electromagnetic waves, the principles of propagation in the various spectral ranges, and summarize the main ideas behind the resonant and non-resonant structures which are utilized. This is followed by the summary of spectroscopic principles – frequency and time domain as well as Fourier transform spectroscopy. We conclude with the description of measurement configurations, single path, interferometric, and resonant methods where we also address the relative advantages and disadvantages of the various measurement configurations.

General books and monographs

M. Born and E. Wolf, Principles of Optics, 6th edition (Cambridge University Press, Cambridge, 1999)

P.R. Griffiths and J.A. de Haseth, Fourier Transform Infrared Spectrometry (John Wiley

& Sons, New York, 1986)

G.Gruner,¨ ed., Millimeter and Submillimeter Wave Spectroscopy of Solids

(Springer-Verlag, Berlin, 1996)

D.S. Kliger, J.W. Lewis, and C.E. Radall, Polarized Light in Optics and Spectroscopy

(Academic Press, Boston, MA, 1990)

C.G. Montgomery, Technique of Microwave Measurements, MIT Rad. Lab. Ser. 11

(McGraw Hill, New York, 1947)

E.D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, Orlando, FL, 1985–1998)