Dressel.Gruner.Electrodynamics of Solids.2003

388 14 Broken symmetry states of metals

and the collective mode contribution to σˆ (ω) now appears as a harmonic oscillator at finite frequencies – with the same oscillator strength as given in Eq. (7.2.15a).

The mass m of the condensate is large in the case of charge density waves, for which the condensate develops as the consequence of electron–phonon interactions. As given by Eq. (7.2.13), the effective mass m /m is large,

if the gap is larger than the corresponding phonon frequency ωP. Because of the larger mass, the oscillator strength associated with the collective mode is small, and thus the Tinkham–Ferrell sum rule is modified, as we have discussed in Section 7.5.3. For an effective mass m /mb 1, nearly all of the contributions to the total spectral weight come, even in the clean limit, from single-particle excitations.

This behavior can be clearly seen by experiment if the optical conductivity is measured over a broad spectral range. In Fig. 14.8 the frequency dependent conductivity σ1(ω) is displayed for a number of materials in their charge and spin density wave states [Gru88, Gru94a]. In all cases two absorption features are seen: one typically in the microwave and one in the infrared spectral range. The former corresponds to the response of the collective mode at finite frequency ω0, and the latter is due to single-particle excitations across the charge density wave gap. A few remarks are in order. First, it has been shown that impurities are responsible for pinning the mode to a well defined position in the crystal, thus ω0 is impurity concentration dependent. Second, the spectral weight of the mode is small; fitting the observed resonance to a harmonic oscillator as described in Eq. (14.2.1) leads to a large effective mass m . Third, there is a well defined onset for the single-particle excitations at 2 (typically in the infrared spectral range). The gap is in good overall agreement with the gap obtained from the temperature dependent conductivity in the density wave state. At temperatures well below

Fig. 14.10. Dynamics of the internal deformations of density waves. The top part of the figure is the undistorted density wave; the middle part shows the mode distorted due to interaction with impurities (full circles); and the bottom part displays the rearrangement of the internal distortion by displacing the density wave period over the impurity as indicated by the arrow. The process leads to an internal polarization P = −2eλDW, where λDW is the wavelength of the density wave.

To describe the effect of impurities by an average restoring force K is a gross oversimplification since it neglects the dynamics of the local deformations of the collective modes. The types of processes which have been neglected are shown in Fig. 14.10. The top part of the figure displays an undistorted density wave, with a period λDW = π/ kF and a constant phase φ. In the presence of impurities, the density wave is pinned as shown in the middle section of the figure. A low lying excitation, which involves the dynamics of the internal deformations, is indicated at the bottom; here a density wave segment has been displaced by λDW, leading to a stretched density wave to the left and to a compressed part to the right side of the impurity. The local deformation leads to an internal polarization of the mode by virtue of the displaced charge which accompanies the stretched or compressed density wave. This polarization P is given by the spatial derivative of the phase

For randomly positioned impurities we expect a broad distribution of the time and energy scales for the processes which reflect the dynamics of such interband deformations. Such effects (a response typical to a glass) are described by a broad superposition of Debye type relaxation processes, and various phenomenological expressions have been proposed to account for the low frequency and long-time behavior of the electrical response. Among these, the so-called Cole–Cole expression

factors which occur in the two types of condensates. Case 1 coherence factors lead to a peak in the optical conductivity at the gap frequency as displayed in Fig. 7.2, in contrast to the gradual rise of the conductivity above the gap in the case of superconductors for which case 2 coherence factors apply.

This has also been most clearly observed in materials which undergo transitions to an incommensurate spin density wave state, with the best example the organic linear chain compound (TMTSF)2PF6. The normal state properties of this material when measured along the highly conducting direction – the direction along which the incommensurate density structure develops – cannot be described by a straightforward Drude response, and therefore a simple analysis of the optical properties of the density state is not possible. When measured with electric fields perpendicular to the highly conducting axis, such complications do not arise; also, along this direction the density wave is commensurate with the underlying lattice and thus the collective mode contribution to the conductivity is absent, due to this so-called commensurability pinning. Below the transition to the spin density wave state, a well defined gap develops, as evidenced by the drop in reflectivity at frequencies around 70 cm−1; the data are displayed in Fig. 14.12. What we observe is similar to what can be calculated for case 1 coherence factors and what is displayed in Fig. 7.8. The reflectivity can also be analyzed to lead to the frequency dependent conductivity σ1(ω), which for several different temperatures is displayed in the inset of the figure. The singularity at the gap of 70 cm−1, at temperatures much lower than the transition temperature, is characteristic to a one-dimensional semiconductor, and this value, together with the transition temperature TSDW = 12 K, places this material in the strong coupling spin density wave limit. The gap feature progressively broadens, and also moves to lower frequencies, and an appropriate analysis can be performed. Such studies conducted at different temperatures can also be used to evaluate the temperature dependence of the single gap [Ves99]; there is an excellent agreement with results of other methods [Dre99].

14.2.3 Frequency and electric field dependent transport

A few comments on the non-linear response are in order here. Because of the weak restoring force acting on density wave condensates, moderate electric fields may depin the collective mode, leading to a dc, non-linear conduction process. Of course, the smaller the restoring force – and thus the larger the low frequency dielectric constant 1 – the smaller the threshold field ET which is required for depinning. The arguments lead to a particularly simple relation between the dielectric constant and threshold field:

where NDW is the number of atoms in the area perpendicular to the direction along which the density wave develops. This relation has indeed been confirmed in a wide range of materials with charge density wave ground states [Gru89]. The intimate relationship between the dielectric constant and fields which characterize the non-linear response is, however, more general; a relation similar to that above can be derived, for example, for Zener tunneling of semiconductors. The topic of non-linear and frequency dependent response with all of its ramifications is, however, beyond the scope of this book.

Part four

Appendices