Incommensurate charge-density waves

Figure 2.27 STM image of incommensurate charge density wave (CDW) state in Nbo.04Tao.96S2. Black lines highlight the insertion of extra rows of CDW maxima in the lattice (From Dai Lieber, 1993).

Incommensurate Charge Density Wave similar to a commensurate charge density wave except that the periodicity of the charge density modulation does not equal a rational number multiplied by the periodicity of the underlying lattice. 

Let the period of the basic structure be a and the modulation wavelength be the ratio a/X may be (1) a rational or (2) an irrational number (Fig. 1.3-7). In case (1), the structure is commensurately modulated we observe a qa superstructure, where q= /X. This superstructure is periodic. In case (2), the structure is incommensurately modulated. Of course, the experimental distinction between the two cases is limited by the finite experimental resolution, q may be a function of external variables such as temperature, pressure, or chemical composition, i. e. = f T, p, X), and may adopt a rational value to result in a commensurate lock-in stmcture. On the other hand, an incommensurate charge-density wave may exist this can be moved through a basic crystal without changing the internal energy U of the crystal. 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that... 

The Peierls169 metal-to-semiconductor phase transition in TTFP TCNQ p was detected in an oscillation camera these streaks became bona fide X-ray spots only below the phase transition temperature of 55 K this transition is incommensurate with the room-temperature crystal structure, due to its partial ionicity p 0.59, and the "freezing" of the concomitant itinerant charge density waves (this effect was missed by four-circle diffractometer experiments, which had been set to interrogate only the intense Bragg peaks of either the commensurate room-temperature metallic structure, or the commensurate low-temperature semiconducting structure). 

Magnetic Incommensurability and Fluctuating Charge Density Waves in Lanthanum Cuprates... 

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