Commensurate Incommensurate Research Articles

Phase transitions in adsorbed layers formed on crystals of square and rectangular surface lattice

This article gives a survey of phase transitions in adsorbed films on well defined surfaces of square and rectangular symmetry of the lattice. The discussion concentrates on the effects of periodic changes of the adsorbate–substrate potential on the structure and thermodynamic properties of adsorbed films. Different theoretical approaches are briefly reviewed, with an emphasis on those which explicitly take into account final corrugation of the surface potential. Several aspects of statistical mechanical description of phase transitions in surface layers, such as order–disorder, melting, commensurate–incommensurate transitions in monolayer films as well as transitions connected with the formation of multilayer films (layering and wetting) are presented. Theoretical discussion is followed by the presentation of numerous experimental and computer simulation studies for various systems. Then the properties of monolayer films of molecular adsorbates on different substrates of a square and rectangular symmetry is discussed. It is pointed out that computer simulation methods provide a very powerful tool which allows to probe the inner structure of such systems and provides direct information concerning both orientational and positional ordering.

Electrical and structural properties of Cr ion-implanted thin Au films

Cr metal ions with doses of 1×10 14–1×10 15 ions/cm 2 have been implanted at 125 keV into thin Au films prepared by DC magnetron sputtering. Electron diffraction patterns have revealed that all the films remain in the f.c.c. structure, indicating the formation of Au–Cr solid solution. The resistivity of the films were studied in the temperature range 100–330 K. The results show increase in the resistivity with respect to the unimplanted film as a result of the implantation. Some samples show two different types of resistivity anomalies, which are attributed to the occurrence of the antiferromagnetically incommensurate spin density wave and commensurate spin density wave structures.

Influence of a magnetic field on the magnetoelasticity of commensurate spin-density-wave Cr–Ru and Cr–Si alloy single crystals

Previously reported attempts to produce a single-Q commensurate (C) spin-density-wave (SDW) state in a dilute Cr alloy by magnetic field cooling through the Néel temperature (TN), were unsuccessful. In the present study the influence of a cooling field of 4.5 T on the magnetoelasticity of Cr+0.3 at.% Ru and Cr+1.6 at.% Si alloy single crystals, both having CSDW phases below TN, are reported. The results show that cooling the Cr+0.3 at.% Ru crystal through TN in a field of 4.5 T has no influence on the first-order commensurate–incommensurate SDW magnetic phase transition. On the other hand, cooling the Cr+1.6 at.% Si crystal in a field of 4.5 T through TN noticeably affects the elasticity in the CSDW state, suggesting a condensation of part of this crystal into a single-Q CSDW phase.

Glassy dynamics of the incommensurate–commensurate phase transition in Zr0.98Hf0.02TiO4 ceramics

The complex dielectric constants of Zr0.98Hf0.02TiO4 ceramics are reported in the frequency range 10−3–1010 Hz for the temperature range 830 °C⩽T⩽860 °C, just above the commensurate-incommensurate phase transition temperature, Tc. The disappearance of x-ray satellite reflections at 825 °C was used to determine Tc experimentally. The dielectric measurements show that a relatively narrow distribution of relaxation times is present in the ceramics, which broadens as the temperature approaches Tc from above. By analyzing the temperature at which the dielectric constant is a maximum as a function of frequency using the Vogel-Fulcher relationship, an activation energy of 550±15 kb,where kb is the Boltzmann constant, and an effective freezing temperature, Tf, of 825 °C have been determined, confirming that, to within experimental error, Tf coincided with Tc. The experimental results for Zr0.98Hf0.02TiO4 are interpreted in terms of a possible dipole-glass-like phase that has been proposed for relaxor systems.

On the commensurate–incommensurate transition in adsorbed monolayers

A Monte Carlo simulation method is used to study the commensurate–incommensurate phase transition in monolayers and the formation of bilayer films on the (100) face of an fcc crystal. The phase diagram for the system which forms the registered (1×1) and high density incommensurate phases in the monolayer has been determined. It is shown that the registered phase undergoes the transition to a denser incommensurate solid phase when the film density increases. The mechanism of melting of the monolayer film is found to depend on the film density. In particular, the melting of dense incommensurate solid monolayer film is found to be accompanied by the transfer of adsorbed molecules into the second layer.

Re-examination of specific heat measurements of zirconium titanate in the region of the phase transitions to the incommensurate phase

The specific heat measurement of zirconium titanate has been re-investigated at the low rate ≈0.3°C/h of cooling and heating. Thermal hysteresis was not observed for normal–incommensurate phase transition ( T i) at 1125°C, while the maximum of the temperature dependencies of the specific heat capacity for incommensurate–commensurate transition ( T c) was shifted with respect to one another by Δ T= T c h− T c c=2.0±0.2°C (where superscripts h and c correspond to heating and cooling, respectively).

Neutron powder diffraction study ( T = 4.2-300 K) and polarization analysis of YBaCuFeO5+δ

The crystal and magnetic structures of the perovskite YBaCuFeO5+δ have been studied in the temperature range T = 4.2-300 K by powder neutron diffraction. In addition to the antiferromagnetic ordering transition at TN = 442 K, a commensurate–incommensurate magnetic transition is detected at TN ′ = 190 K. Below this temperature, two sets of satellite peaks surround the (1/2,1/2,1/2) magnetic peak at d ∼ Å, collapsing into a single set of satellites below 155 K. Polarization analysis has been performed to confirm the magnetic nature of the (1/2,1/2,1/2)± satellites.

Magnetic phase diagram of the spin-density-wave CrFe alloy system from susceptibility measurements

The temperature dependence of the magnetic susceptibility χ(T) of seven Cr1−xFex alloys measured around the Neel transition shows a strong first-order transition at the Neel temperature TN for the Fe content, x=2.7 and 3.7 at. % Fe, greater than the triple-point concentration, xL=2.4 at. % Fe, where the paramagnetic, incommensurate-, and commensurate-spin-density-wave (ISDW and CSDW) phases meet. Alloys with x≲xL (x=2.0 and 2.15 at. % Fe) show a weak transition at TN and a strong first-order transition at TIC (the ISDW–CSDW transition). The magnetic phase diagram of the Cr1−xFex alloy system below xL is re-interpreted in light of these results and the neutron diffraction work. The coexistence of the CSDW and ISDW phases in the mixed state below the branch TIC(x) of the first-order incommensurate-commensurate (IC) phase transition, for values of x well below xL, may result from anisotropy of the large strain at TIC.

Modeling of dynamic ferroelectric hysteresis loops in purified Rb2ZnCl4

Dynamic hysteresis loops were recorded in the ferroelectric lock-in phase of purified Rb2ZnCl4 crystals at a temperature T=188 K which is 5 K below the incommensurate–commensurate phase transition. The hysteresis loops observed in the frequency range 50 Hz<f<5 kHz can quantitatively be described by a model that treats a sideways shift of planar, nearly regular arranged domain walls as the only mechanism of polarization reversal in Rb2ZnCl4. Due to defect interaction, the velocity v of the domain wall depends nonlinearly on the effective force F acting on it and a scaling behavior v∼(F−Fc)θ with θ=1.45 is observed.

The phase diagram of RbFeCl3 in a magnetic field perpendicular to the chain direction

The temperature dependence of the position and line width of antiferromagnetic Bragg peaks were measured at different applied magnetic fields up to 1.4 T by means of elastic neutron scattering. Three different types of temperature dependences were observed that can be attributed to the commensurate–incommensurate, commensurate–paramagnetic, and incommensurate–paramagnetic phase transitions. Below 0.3 T the transition goes from commensurate to incommensurate. Between 0.3 and 0.5 T a phase boundary was observed, but there is some doubt as to whether the structure at temperatures below this boundary is commensurate. Between 0.5 and 1.2 T the phase transition goes directly from commensurate to paramagnetic at around 2.5 K. For fields higher than 1.3 T and temperatures above 1.5 K no commensurate phase could be identified. But above 1.2 T and still at 1.4 T there exists an incommensurate phase. Based on these temperature dependences at different magnetic fields, phase boundaries in the H–T phase space are proposed and compared with the phase diagram obtained by previous susceptibility measurements.

Commensurate-incommensurate phase transition in one-dimensional quantum sine-Gordon model based on a lattice massive thirring model

The commensurate-incommensurate (C-IC) phase transition in the one dimensional quantum sine-Gordon model at zero temperature is exactly solved with the use of the Bethe ansatz technique for the lattice massive Thirring model. The energy difference between C and IC phases is derived based on the same ground state which is valid in the whole parameter region. It is due to the fact that there is no change in the ground state of the lattice massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model. It is proved in the whole parameter region that the IC phase can be realized with the soliton density proportional to\(\sqrt { - E_s } \) (Es: formation energy of soliton), whenEs becomes negative.

Slave-boson study of the t-t'-J model: Phase diagram, spin susceptibility, and Hall resistivity.

We investigate the normal-state properties of high-${\mathit{T}}_{\mathit{c}}$ cuprates in terms of the t-t'-J model using a spin-rotation-invariant slave-boson technique. The second-neighbor hopping t'0 is included in order to reproduce the Fermi surfaces of LSCO (${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$) and YBCO (${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6+\mathit{x}}$) type. The magnetic phase diagram of the t-t'-J model is derived within the saddle-point approximation, by taking into account also incommensurate spiral order. Compared to the pure t-J model, N\'eel order is stabilized in the low-doping region. In particular, for the case t'>0, which corresponds to the electron-doped systems, e.g., ${\mathrm{Nd}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ce}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ (NCCO), the antiferromagnetic correlations are strongly enhanced near half-filling. We include fluctuation corrections and calculate the dynamic (paramagnetic) spin susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) going beyond the random-phase approximation. The instability line of the paraphase obtained from \ensuremath{\chi}(q) is in agreement with the saddle-point phase diagram. The wave-vector dependence of \ensuremath{\chi}(q) reveals the commensurate (incommensurate) nature of spin fluctuations in YBCO (LSCO). Finally, the doping dependence of the Hall resistivity ${\mathit{R}}_{\mathit{H}}$(\ensuremath{\delta}) is calculated, where the results agree surprisingly well with experiments on LSCO. For YBCO and NCCO, the t' term suffices to give the correct dependence for ${\mathit{R}}_{\mathit{H}}$(\ensuremath{\delta}).

A synchrotron X-ray diffraction study of the structural phase behaviour of multilayer xenon on single-crystal graphite

We present a high-resolution synchrotron X-ray diffraction study of xenon adsorbed on single-crystal graphite for coverages of two, three and six monolayers. We observe the evolution of the lattice mismatch between the xenon overlayer and a square root 3* square root 3R30 degrees structure commensurate with the underlying substrate. In the case of the bilayer we see a first-order incommensurate-commensurate (IC-C) phase transition at 62.0 K+or-1.2 K, which we demonstrate is consistent with the layer closest to the substrate having locked into the commensurate sites. The role of substrate quality is discussed and we conclude that previous studies on relatively poorer substrates may have favoured equilibrium structures strongly influenced by the surface imperfections. The risk of an insufficiently equilibrated adsorbate is discussed in the context of the very slow kinetics we observe for multilayer xenon on high-quality substrates. For the three- and six-layer adsorbates we observe a systematic increase in lattice mismatch with film thickness and no indication of an incommensurate-commensurate transition. We conclude that the trilayer system consists of many distinct domains approximately 500 AA across, each of which has a random close packed structure.

THEORETICAL MODEL FOR THE HIGH-FREQUENCY DYNAMICAL SUSCEPTIBILITY IN THE IMPROPER SEGNETOELECTRICS

Description of the temperature and frequency dependences of the dynamical susceptibility in improper segnetoelectrics in the incommensurate phase near the incommensurate–commensurate phase transition is presented. The used model is based on the theory of three interacting nonlinear oscillators.

Commensurate–incommensurate phase transition between 6×6 and √3×√3 + satellite structures of the Au/Si(111) surface

Alternate annealing in the low- (≤250 °C) and high- (≥400 °C) temperature regimes of the Au/Si(111) surface at different Au coverages prepared by room-temperature deposition and isothermal annealing in the high-temperature regime at over-(ML) coverages have been studied by means of low energy electron diffraction (LEED), Auger electron spectroscopy (AES), and Rutherford backscattering spectroscopy (RBS). It is found that the 6×6 pattern and the √3×√3+satellite pattern are formed at 0.9–2.2 ML coverages by annealing for several minutes at 250 and 500 °C, respectively, when there exist an Au surface peak and a diffusion tail extended towards the lower energy behind the Au peak and that the phase transition between both patterns is reversible by alternate annealing in two temperature regimes as long as the diffusion tail is not diminished. From this result, the 6×6 structure is ascribed to excess embedded Au atoms situated at the ordered sites and Au atoms constituting the √3×√3 structure at the top surface. Moreover, it is found that the phase transition between the √3×√3 structure and the mixed structure of √3×√3+5×1 are also reversible by alternate annealing in the two temperature regimes at the actual coverages of 0.72–0.84 ML. Furthermore, from analysis of the isochronal annealing data at 1.7 ML, the activation energy of the diffusion of Au in Si is found to be 1.3 eV.

Phase transitions of ethylene monolayers on graphite

Low-energy electron diffraction (LEED) measurements have detected seven solid phases for monolayer ethylene on graphite. Electron-beam induced effects (which presumably include decomposition and/or polymerization of the adsorbed ethylene) were minimized by exposing the adsorbed layer to the nanoamp beam for <100 s per crystal; data were obtained from four different crystals. We concentrate here on the phase transitions from the molecular-axisorientationally disordered low density (DLD) phase to the newly discovered commensurate molecular-axis-disordered low density (CDLD) phase and on the melting of the CDLD phase. The change in the principle q vector in the DLD-CDLD phase transition is presented for several different coverages. These data indicate an interesting incommensurate–commensurate (IC–C) transition which has not been suspected from previous studies. This transition preempts the melting transition: as a consequence we can rule out the possibility of a continuous melting transition as suggested by earlier heat capacity and neutron scattering experiments. In addition, LEED images of the fluid phase produced by melting of the CDLD phase are presented which show a modulated-ring-type structure.

Anisotropy of frictional forces in muscovite mica.

The static and dynamic frictional forces of single-crystal muscovite mica are measured as a function of the lattice misfit angle between the two contacting cleavage lattices at a very light load under both dry and ambient atmospheres. The frictional forces are anisotropic with respect to the lattice misfit angle, i.e., they increase (decrease) when the contacting surfaces approach being commensurate (incommensurate). The underlying mechanisms of the anisotropy are discussed in terms of the commensurability between the contacting lattices.

Ordering phenomena and phase transitions in the physisorbed quantum systems H 2, HD and D 2

Recent experimental results of H 2, HD and D 2 films physisorbed on graphite are briefly reviewed. In particular, the monolayer phase diagrams, the order-disorder transition of the commensurate (C) phase and the commensurate-incommensurate (C-IC) transition are discussed. It will be shown that the melting transition of the C phase belongs to the three-state Potts universality class, and that the C-IC transition occurs via a series of novel intermediate phase, which could be identified as density-modulated phases characterized by striped and hexagonal patterns of domain walls. Due to this rich variety of phenomena, the hydrogen isotopes can be considered as model systems for two-dimensional quantum systems.

LOCALIZED ELECTRONS IN A MAGNETIC FIELD

We briefly review recent developments on the response of localized electrons to an external magnetic field. They originate from subtle interference effects between forward-scattered paths which, in the localized regime, make the dominant contribution to the transition amplitude. On a lattice the correlations between the paths produce self-similar (quasiperiodic) spatial interference patterns for commensurate (incommensurate) ratios of the applied magnetic flux per plaquette with the flux quantum. If localization is due to strong disorder (as in lightly doped semiconductors), this is augmented by a logarithmic averaging. They give rise to striking quantum interference phenomena observed in mesoscopic and macroscopic insulators: Aharonov-Bohm oscillations with periods of one and one half flux-quantum, the Hall effect, pronounced conductance fluctuations, and anomalous negative (and positive in presence of spin-orbit scattering) magnetoresistance. The theoretical works based on the “directed paths” approach are reviewed. Experimental results which motivated these works, are summarized.

Low-energy electron diffraction observations of the thermal evolution of 0.5–3.0 ML Pt/Ni(111)

The near-surface geometric structure of the system 0.5–3.0 ML Pt/Ni(111) has been monitored with low-energy electron diffraction (LEED) as a function of annealing treatment. In this coverage range a variety of structures are documented. Evidence is presented for commensurate submonolayer Pt (despite the large misfit) and incommensurate (but aligned) second and third layer Pt, at low temperatures. The second and third layers of Pt yield (111)-type LEED patterns but with spacing at an intermediate value between Ni(111) and Pt(111). In a very narrow range of coverage and annealing treatment (1.5–2 ML PT and 673–773 K) a hexagonal satellite appears around each primary beam. These satellites are interpreted as evidence for partially rotated incommensurate second layer Pt domains. Annealing at 873 K or above of even the thickest deposits (3 ML) produces diffraction spots indistinguishable from Ni(111). Structural evidence is correlated with the thermal evolution of the surface electronic structure in the same system (using photoemission for local and macroscopic work function measurements). The photoemission results complement the LEED information and together they provide evidence for complex diffusional rearrangements even at low temperatures in this system.