Incommensurate Case Research Articles

Proposed experimental realization of Anderson localization in random and incommensurate artificially layered systems.

We propose that random and incommensurate superlattices be fabricated to study the Anderson localization of their plasma oscillations, which can be measured by Raman scattering. Specific configurations are suggested which, by duality considerations, allow control of the wave functions relative to the mobility edge in the incommensurate case. Localization effects should be easily observable with presently achievable superlattice parameters.

Sliding charge-density waves. I. dc properties

A classical model of a sliding charge-density wave (CDW) is extended to treat the electrostatic interaction of the CDW with itself, conduction electrons, and the host lattice, as well as with defects. If the CDW is commensurate and the lattice sufficiently coherent, the interaction with the lattice produces, at large voltages, a leading-order nonlinear current of the form $\ensuremath{\Delta}I\ensuremath{\propto}\ensuremath{-}{V}^{\ensuremath{-}1}$, which contrasts with $\ensuremath{\Delta}I\ensuremath{\propto}\ensuremath{-}{V}^{\frac{1}{2}}$ due to defects. The effect of a large decrease in the normal conductivity is studied and the defect-contributed $\ensuremath{\Delta}I$ is predicted to be strongly enhanced with decreasing temperature in Ta${\mathrm{S}}_{3}$. Interference effects in the commensurate and incommensurate cases are also shown to differ in both detailed shape and temperature dependence. Transport measurements may, therefore, be able to determine whether the CDW's in Ta${\mathrm{S}}_{3}$ and Nb${\mathrm{S}}_{3}$ are commensurate or incommensurate.

Wyckoff positions used for the classification of Bravais classes of modulated crystals

Wyckoff positions of centrosymmetric symmorphic space groups, if properly interpreted, can be used to derive the Bravais classes of modulated crystals. This program is worked out here for the incommensurate case and assuming that the modulation wave vectors are symmetry related, i.e. are (integral combinations of) vectors of one star. The corresponding classes are called elementary Bravais classes (EBC's). It is shown that for three-dimensional crystals the maximal internal dimension for these EBC's is nine. It is explained in what sense this approach leads to a complete list.

Electrons in incommensurate crystals: Spectrum and localization

Electronic properties of incommensurate crystals are studied on a one-dimensional Kronig-Penney model with sinusoidally modulated atomic positions. Spectra and wave functions have been calculated numerically as a function of the modulation wave vector. Information for the incommensurate case is obtained by considering series of commensurate approximants. It is shown that the spectrum of this model has a hierarchical and recursive nature connected with the continued-fraction expansion of the modulation wave vector. In the present model there are parts of the spectrum where the energy eigenvalues occur in bands with a finite number of numerically nonvanishing gaps (bandlike spectrum) and parts where new gaps open up at each step of the continued-fraction expansion (Cantor-type or discrete spectrum). Numerical evidence is presented that the electron states are extended in the bandlike regions and localized in the discrete regions. The characterization of the electron density is made more transparent if one considers the generalized density in (two-dimensional) "superspace." The localized electronic states violate the Fr\"ohlich assumption for the translational freedom of the modulation wave. This has consequences for the electrical conductivity in this model.

Localization in an Incommensurate Potential: An Exactly Solvable Model

The exact eigenstates of a one-dimensional tight-binding model with a periodic diagonal potential that can be commensurate or incommensurate with the lattice are found. If the period is incommensurate, all the eigenstates are localized. The localization length and the density of states are identical to those of a related disordered system. The case of a commensurate potential, for which all the states are extended, and the approach to the incommensurate case are also discussed. The solution is achieved by mapping the model into a time-dependent quantum problem in which the potential is periodic in time.

Inequalities for Inscribed and Circumscribed Polygons

In this note, we elaborate on Theorem III. Although it is not stated explicitly in the theorem, it is tacitly assumed from the context that the two polygons are convex and that the center of the circle lies in the interior of both polygons. Uspensky's proof is a nice illustration of induction in geometry, which is not too prevalent (for other examples of geometric induction, see [1]). The inductive step appears in the proof of Theorem I where he first considers the case when angles a and /8 are commensurable so that a =pO and /3 = q3 where p and q are relatively prime integers. It is then supposed that the above inequalities are valid in every case when the sum p + q is less than a given integer N.> 2, from which it is shown to hold forp + q = N. Then the incommensurable case is reduced to the former case by a limiting process. The proof here of Theorem III will be based on the convexity of sinx, which also leads to further related results. Let the central angles subtended by the sides of 6P and 6' from the center of the circle be denoted by 01,02, Om and 9', 02x' ... ,On,f respectively. Uspensky assumes that 0 <Os, O' < -r, and

Amplitude solitons in incommensurate Peierls systems: Implications for TTF-TCNQ

Amplitude soliton excitations are described for both dimerized [i.e. commensurate with 2k f = ( π a ) ] and incommensurate charge-density- wave (CDW) ground states. For the dimerized case the soliton excitations can have charge + e, 0, − e with spin 0, ± 1 2 , 0 depending on the occupancy of the localized gap state. For the incommensurate case only the neutral spin- 1 2 soliton exists. Utilizing the results of Takayama, Lin-Liu and Maki, we show that E s = ( 2 π )Δ where 2Δ is gap for electron-hole excitations ( Δ i = W exp ( −1 λ ) for the incommensurate case and Δ c = W exp (− 1 2 λ) for the dimerized case where λ is the electron-phonon coupling constant and W is the bandwidth). We suggest that neutral soliton excitations may make significant contributions to the magnetic susceptibility of incommensurate CDW systems, and that this may be the origin of observed differences between the low-temperature magnetic and transport activation energies in TTF-TCNQ.

Semiclassical calculations of vibrational energy levels for nonseparable systems using the Birkhoff–Gustavson normal form

We present a semiclassical method of calculating vibrational energy levels for a system of nonseparable coupled oscillators. For a Hamiltonian written as a power series in which the leading terms are given by a sum of one-dimensional harmonic oscillator Hamiltonians, the method involves transforming the original classical Hamiltonian via a succession of canonical transformations into a normal form which is a power series originally defined by Birkhoff and later generalized by Gustavson. Two cases are distinguished. If the harmonic oscillator frequencies in the unperturbed Hamiltonian are incommensurable, then the normal form is a power series whose terms are products of one-dimensional harmonic oscillator Hamiltonians; if the frequencies in the unperturbed Hamiltonian are commensurable, then additional terms which cannot be written as products of one-dimensional harmonic oscillator Hamiltonians enter into the normal form. Once the normal form is obtained, semiclassical quantization of action variables is straightforward. The incommensurable case yields a formula for the energy spectrum which is a power series in the quantum numbers. The commensurable case is more complicated, and yields a form from which energy levels may be obtained individually by numerical calculation and quantization of a one-dimensional phase integral. Nonseparable two-dimensional examples are treated for each case. The results obtained for both cases show excellent agreement with quantum mechanical calculations. The quantum calculations indicate that all of the energy levels fall into a regular pattern.

Lattice dynamics and electrical properties of commensurate one-dimensional charge-density-wave systems

The ground-state properties, the phonon dispersion, and the electrical properties of systems with a commensurate superstructure are calculated. The superstructure leads to the appearance of a gap at the Fermi energy. Phonon branches representing oscillations of the phase and amplitude of the superstructure show strong anomalies at $q=0$. The smaller the electronic energy gap, the stronger these anomalies are. For general commensurate cases the phase mode of finite frequency ${\ensuremath{\omega}}_{\mathrm{ph}}$ couples to an external field leading to a $\ensuremath{\delta}$ singularity in the electrical conductivity at $\ensuremath{\omega}={\ensuremath{\omega}}_{\mathrm{ph}}$. This is in contrast to the incommensurate case, where, due to the translational invariance, the phase-mode frequency vanishes, resulting in metallic electrical properties. In the special case of a half-filled band the phase and amplitude of the superstructure have the same degree of freedom in the system. In this case the corresponding phonon mode does not couple to an external field. However, the lower symmetry of the state with superstructure leads to a coupling of usually infrared-inactive intramolecular modes to an external field, and, correspondingly, these modes lead to peaks in the frequency-dependent conductivity.

Dynamics of amplitude and phase mode and infrared absorption of linear conductors

The dynamics of the Amplitude and Phase mode in the incommensurate case is reduced to the dynamics of a diatomic molecule. Using this Hamiltonian, σ(ω) is given as a canonical average for a Penn model with fluctuating gap.

Spin susceptibilities in the theory of itinerant-electron antiferromagnetism. II. Two-band incommensurate case

For pt.I see ibid., vol.5, p.2343 (1975). The method of calculating the susceptibilities in the spin-density-wave state of an itinerant-electron antiferromagnet introduced in the previous paper is extended to the two-band incommensurate case. The model Hamiltonian used is the Feeders and Martin two-band model in which electrons in different bands are coupled. Both longitudinal and transverse susceptibilities are calculated within the random phase approximation and explicit expressions which do not make any approximation for the form factors are given. The susceptibilities contain both intraband and interband parts. The Coulomb interaction is shown to contribute only to the interband part of the longitudinal susceptibility. The differences between the commensurate and incommensurate spin-density wave states are discussed as well as the collective modes arising from the susceptibilities.

Landau theory of charge-density waves in transition-metal dichalcogenides

A Landau theory is proposed for charge-density waves (CDW) in transition-metal dichalcogenides with the charge density as an order parameter. The theory predicts the sequence of phases, normal-state---incommensurate-CDW---commensurate-CDW with decreasing temperature, separated by first-order phase transitions. The peaks in charge density lie at sites of a hexagonal crystal and for the incommensurate case the theory predicts phononlike distortions of the CDW lattice as well as dislocations. Impurities pin the charge density wave, broaden the phase transitions, and stabilize the incommensurate state relative to the commensurate state.