Bulk Critical Point Research Articles

Analysis of high-index Si(001) surfaces by reflectance difference spectroscopy

We report surface-induced optical anisotropy spectra of high-index Si(115), (114), and (113) surfaces obtained using reflectance difference spectroscopy. Air-oxidized surfaces show sharp derivative-type features that are step-induced and located near the critical point energies of bulk Si, consistent with those of lower-index Si(001) surfaces. Clean reconstructed surfaces are characterized by a broad feature near 3 eV that tends to decrease in amplitude upon H exposure and a step-induced structure near the (E0′,E1) transition of bulk Si. In contrast, H exposure of Ge-covered surfaces tends to sharpen and enhance lower-energy structures. The derivative-type features located near the bulk critical point energies of Si can be described in terms of electronic states localized by the finite penetration depth of light.

Properties of Three Capillary Fluids in the Critical Region

Nitrogen, xenon, and sulfur hexafluoride when adsorbed on a mesoporous controlled pore glass all exhibit capillary condensation at temperatures less than Tc, the bulk critical temperature, and all exhibit a capillary critical temperature, Tcc, where adsorption−desorption hysteresis vanishes. For all temperatures less than Tcc, the density and surface tension of these capillary liquids are essentially the same as bulk liquid. Within the intercritical region, Tcc < T < Tc, these capillary fluids at, or near, saturation pressure are also liquidlike. We conclude that fluid behavior at the capillary critical point is not analogous to fluid behavior at the bulk critical point. The capillary critical point is identified as the point at which homogeneous nucleation of the vapor phase within capillary liquid replaces Kelvin failure as the mechanism responsible for abrupt capillary evaporation on desorption. We also discuss the phenomenon of critical depletion reported by Findenegg and colleagues.

Wetting phenomena near the bulk critical point of fluid mercury

AbstractWe have studied the adsorption of mercury on sapphire at temperatures and pressures close to the liquid‐vapour critical point (Tc= 1478°C, pc= 1673 bar, pc = 5.8 g/cm3) of mercury. The mercury film undergoes a first‐order phase transition known as prewetting. At low temperatures, the prewetting line meets the bulk coexistence curve tangentially at the wetting temperatures Tw = 1310°C. At high temperatures the prewetting line terminates at the prewetting critical point Tcpw = 1468°C and the prewetting pressure pcpw = 1586 bar lying close to the bulk critical point. In analogy with the line of compressibility maxima extending above the liquid‐vapor critical point, we can identify an extension of the prewetting phase line for T&gt;Tcpw which marks the maximum in the two‐dimensional compressibility of the prewetting supercritical phase.

Surface-induced optical anisotropy of the (001) and (113) silicon surfaces

The surface-induced optical anisotropy of clean and hydrogenated (001) and (113) surfaces of silicon has been calculated. The equilibrium atomic configurations of the surface atomic structures were determined from fully converged self-consistent total-energy calculations within the local density approximation. The optical functions of the silicon surfaces have been obtained beyond the density-functional theory with ab initio pseudopotentials and by the semiempirical tight-binding method. The reflectance anisotropy (RA) spectroscopy signal of the clean (001) and (113) surfaces in the region around 3.0 eV originates from surface to bulk optical transitions. On the hydrogenated (113) silicon surface a strong surface induced RA response is obtained in the region of the ${E}_{1}$ and ${E}_{2}$ bulk critical points, where the line shape corresponds to the first derivative of the bulk dielectric function. The calculated RA spectrum of the hydrogenated (113) silicon surface in the regions of the ${E}_{1}$ and ${E}_{2}$ structures is more intense in relation to the (001) surface. These results will be discussed in comparison with experimental data.

Prewetting phenomena in mercury vapor

We review recent work on the acoustic characteristics of mercury in molybdenum and niobium cells at high temperatures (to 1900 K) and high pressures (to 180 MPa) and report some further experimental progress. The sound velocity, and the transmitted and reflected signals have been observed simultaneously. Apparent first-order phase transitions appear in the vapor region of the mercury phase diagram. The corresponding transition curve (for Mo) meets the bulk liquid–gas coexistence curve tangentially at ( T 0, P 0)≅(1250 K, 30 MPa) and terminates above 1860 K and 183 MPa. The data provide strong evidence that the observations represent prewetting transitions, bounded below by a wetting transition at ( T W, P W)≡( T 0, P 0) and above by prewetting critical points which lie well above the bulk critical point ( T c≅1764 K, P c≅167 MPa).

Phase coexistence in confined Ising systems: a density matrix renormalization approach

Using a density matrix renormalization approach we have calculated the phase diagram of a two-dimensional Ising model confined between two infinite walls, with opposing surface fields and a bulk field which grows linearly as a function of the distance from the walls and models the effect of gravity on a confined fluid. In the absence of gravity, two-phase coexistence is restricted to temperatures below the wetting temperature as pointed out by Parry and Evans [A.O. Parry, R. Evans, Phys. Rev. Lett. 64, (1990) 439]. We find that the competing effects of gravity and surface fields restore the `ordinary' finite size scaling to the bulk critical point, in agreement with previous mean-field results. We have calculated the exponents related to the shift toward the critical point in the limit L→∞, where L is the distance between the walls and we have found good agreement with previous scaling assumptions. Magnetization profiles calculated from a solid-on-solid Hamiltonian agree well with density matrix renormalization results for temperatures not too close to the bulk critical temperature.

Photon-induced localization and final-state correlation effects in optically absorbing materials

Two consequences of the absorption of light in optically absorbing materials that appear not to have been recognized previously are: (1) localization of the final electron and hole states involved in the absorption process into wave packets and (2) propagation of these wave packets with their respective group velocities. We demonstrate the existence of these phenomena by applying first-order time-dependent perturbation theory to a simple model that can be solved analytically even when correlations that are ordinarily discarded in the random phase approximation are retained. This approach provides a natural explanation of components in surface- and interface-optical spectra that are related to energy derivatives of the bulk dielectric function εb and apparent differences in nominally bulk critical point energies Eg and broadening parameters Γ depending on surface conditions.

The Theory of Boundary Critical Phenomena

An introduction into the theory of boundary critical phenomena and the application of the field-theoretical renormalization group method to these is given. The emphasis is on a discussion of surface critical behavior at bulk critical points of magnets, binary alloys, and fluids. Yet a multitude of related phenomena are mentioned. The most important distinct surface universality classes that may occur for a given universality class of bulk critical behavior are described, and the respective boundary conditions of the associated field theories are discussed. The short-distance singularities of the order-parameter profile in the diverse asymptotic regimes are surveyed.

Temperature dependence of the InP(001) bulk and surface dielectric function

The bulk dielectric function, the surface reflectance anisotropy, and the surface dielectric anisotropy of InP(001) were determined from room temperature up to 875 K. Measurements were performed in-situ on as-grown samples in both a metal-organic vapour phase epitaxy and a chemical beam epitaxy (CBE) system using a rotating analyser type ellipsometer (SE) and a reflectance anisotropy spectrometer (RAS). The temperature dependence of the bulk critical points was deduced by performing a direct line shape analysis of the SE spectra. A high-temperature bulk dielectric function database for optical in-situ studies was established by applying a combined cubic spline/harmonic oscillator interpolation scheme to the measured data. Three InP(001) surface reconstructions were found by reflection high-energy electron diffraction (RHEED) measurements in the CBE system: (2 x 1) and c(4 x 4) under phosphorus-rich conditions and a (2 x 4) reconstruction under reduced phosphorus supply. Characteristic RAS spectra are assigned to these reconstructions. The temperature shift of the main features of the surface dielectric anisotropy spectra are compared to those of the bulk critical points.

Aspects of sorption and phase behavior of near-critical fluids confined to mesoporous media

The local structure of a Lennard-Jones fluid confined to a mesoscopic slit-pore with structureless (i.e., “smooth”) walls is investigated by computing the local density ρ(z) in grand canonical ensemble Monte Carlo (GCEMC) simulations. For noncritical thermodynamic states of a corresponding bulk fluid, a homogeneous core region exists in the pore characterized by the bulk density ρbulk. Sufficiently close to the bulk critical point, however, the core is depleted, that is its local density ρcore(z) is less than ρbulk. Along a bulk near-critical isotherm T*=1.36≃Tc,bulk* GCEMC data are analyzed in terms of the equations of state μbulk=μ(ρbulk,Tc,bulk) and μcore=μ(ρ̄mid,Tc,bulk), respectively, where ρ̄mid is the average midpoint density of the core fluid. At densities sufficiently above and below the bulk critical density thermodynamic states of core and bulk are the same, i.e. μbulk=μcore. However, if depletion is observed the density dependence of μbulk and μcore is generally different. By applying a mean-field theory to μbulk and μcore depletion is related to a reduced degree of criticality of the core fluid compared with the bulk phase in a self-contained way.

Off-diagonal density profiles and conformal invariance

Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by a non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.

Wetting of mercury on sapphire

Reflectivity experiments on fluid mercury against an optically transparent sapphire window at high temperatures and high pressures close to the liquid - vapour critical point reveal clearly the existence of a prewetting transition of mercury on the sapphire substrate. The prewetting line intersects the coexistence curve at the wetting temperature , and terminates at the prewetting critical temperature and prewetting critical pressure lying close to the bulk critical point.

Porous silicon layers as a model system for nanostructures

In this paper we show that the lineshapes of the dielectric functions of porous p-doped silicon layers are strongly correlated to their original bulk doping concentrations. Features in the dielectric functions of layers formed from heavily doped bulk material correspond to bulk critical points, with an additional feature in the visible that results from the electronic structure of silicon that has been changed by the presence of inner surfaces. These results suggest that the layers remain highly connected. For lightly doped starting material the bulk-related features are much broader, with some samples exhibiting only an E 2 interband critical point feature that is shifted to higher energies. This shift, which can be understood in terms of strong confinement of the carriers, suggests reduced connectivity in these layers.

Hydrogen adsorption on GaAs (001) surfaces observed by surface photoabsorption and reflectance difference spectroscopy

Surface reflectance spectra are observed for H-adsorbed GaAs (001) surfaces. Atomic hydrogen produced with a hot tungsten filament is adsorbed on the (2×4) surface, converting it to (1×1). Reflectance difference (anisotropy) spectra show an enhanced peak at 2.8 eV upon H adsorption at room temperature, which is not consistent with the notion that this peak originates from As dimers. Analysis of surface photoabsorption spectra show that surface dielectric changes along [1̄10] for H adsorption and for Ga deposition can be represented by a superposition of a change at the bulk critical points E1,E2,E0′ of GaAs and a broad feature centering at 2.5–2.7 eV.

A reflectance anisotropy spectroscopy study of GaSb(100)c(2 × 6) surfaces prepared by Sb decapping

MBE-grown samples capped with a protective Sb layer were used to study clean (100) surfaces in UHV. By thermal desorption of the Sb cap layer GaSb surfaces exhibiting c(2 × 6) reconstruction were prepared, then studied by reflectance anisotropy spectroscopy, low energy electron diffraction and Auger electron spectroscopy. The results of this experimental analysis allowed us to determine for the first time the surface related optical anisotropy of the c(2 × 6) reconstruction of GaSb(100) in the range 1.5–5.5 eV. Reflectance anisotropy spectroscopy features related to optical transitions of surface Sb dimers are identified and discussed. Moreover, the appearance of an optical anisotropy related to the linear electro-optic effect at the E 1 bulk critical point shows the existence of band bending at GaSb decapped surfaces.

Dynamical relaxation and universal short-time behavior in finite systems. The renormalization-group approach

We study how the finite-size n-component model A with periodic boundary conditions relaxes near its bulk critical point from an initial nonequilibrium state with short-range correlations. Particular attention is paid to the universal long-time traces that the initial condition leaves. An approach based on renormalization-group improved perturbation theory in 4 − ϵ space dimensions and a nonperturbative treatment of the q = 0 mode of the fluctuating order-parameter field is developed. This leads to a renormalized effective stochastic equation for this mode in the background of the other, q ≠ 0 modes; we explicitly derive it to one-loop order, show that it takes the expected finite-size scaling form at the fixed point, and solve it numerically. Our results confirm for general n that the amplitude of the magnetization density m( t) in the linear relaxation-time regime depends on the initial magnetization in the universal fashion originally found in our large- n analysis [J. Stat. Phys. 73 (1993) 1]. The anomalous short-time power-law increase of m( t) also is recovered. For n = 1, our results are in fair agreement with recent Monte Carlo simulations by Li, Ritschel, and Zheng [J. Phys. A 27 (1994) L837] for the three-dimensional Ising model.

Analytic solution of emden-fowler equation and critical adsorption in spherical geometry

In the framework of mean-field theory the equation for the order-parameter profile in a spherically-symmetric geometry at the bulk critical point reduces to an Emden-Fowler problem. We obtain analytic solutions for the surface universality class of extraordinary transitions in $d=4$ for a spherical shell, which may serve as a starting point for a pertubative calculation. It is demonstrated that the solution correctly reproduces the Fisher-de Gennes effect in the limit of the parallel-plate geometry.

Laser-induced band-bending variation on room-temperature CdTe(110)1 x 1 surfaces observed in photoemission and through the Franz-Keldish effect in surface differential reflectivity.

A laser-induced band-bending variation on ultrahigh vacuum cleaved CdTe(110)1\ifmmode\times\else\texttimes\fi{}1 surfaces kept at room temperature has been observed both in angle-resolved photoelectron spectroscopy (ARPES) and, through the Franz-Keldish (FK) effect, in surface-differential reflectivity. Through ARPES spectra, the shift of the Fermi level, between the sample obscured and illuminated with an ${\mathrm{Ar}}^{+}$ laser power of 500 mW has been determined to be 0.2 eV. The FK effect has been observed at the ${\mathit{E}}_{0}$ bulk critical point and studied as a function of laser power, showing a linear increase in its oscillation amplitude up to a power of 400 mW. Upon switching off the laser, the FK effect disappears exponentially with a time constant of about 1300 s. A surface photovoltage induced by the helium lamp has also been observed.

Local critical behaviour at aperiodic surface extended perturbation in the Ising quantum chain

The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface-extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly, whereas the expression of the surface energy density exponent is conjectured from a finite-size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model.

Wetting phenomena and the decay of correlations at fluid interfaces

The physics of wetting in three dimensional systems with planar symmetry is revisited in order to explore the extreme sensitivity to microscopic interactions that underlies even thick film adsorption phenomena in d=3. Beginning with short-ranged models of wetting by liquid at a wall-vapor interface, weighted density functional theory is used to obtain an accurate description of the mean-field interface potential along the entire saturated liquid branch. Well away from the bulk critical point, the long-range decay of this class of interface potentials is damped oscillatory rather than the generally assumed monotonic form. This result is best understood in the context of a recently published general theory of the asymptotic structure of liquids and their mixtures. The inescapable conclusion from mean-field theory is that over much of the saturated liquid curve, complete wetting is replaced by thick film pseudowetting associated with layering phenomena. However, oscillatory structure at microscopic wavelengths is strongly renormalized by the inclusion of capillary-wave fluctuations. To assess this aspect, the paper reviews the explicit linear renormalization group calculation of Chernov and Mikheev [Phys. Rev. Lett. 60, 2488 (1988)]. One must distinguish between two cases: (i) pure wetting, where capillary-wave fluctuations renormalize the decay length of the damped oscillatory structure, and (ii) wetting in the presence of external capillary-wave damping (such as Earth's gravity), where the renormalization can dramatically reduce an oscillatory amplitude but cannot formally prevent the suppression of complete wetting.The second main aim of the paper is an attempt at a general survey of the plethora of length scales potentially relevant to thick film wetting phenomena in d=3. A full description of short-ranged models of simple fluids is presented in terms of the behavior of monotonic and oscillatory decay lengths along the saturated liquid branch. Additional length scales of the same class arise from competition with exponentially decaying wall fields. Power-law interactions (dispersion forces) lead to a qualitatively different asymptotic regime. Notwithstanding the ultimate dominance of power-law structure at the longest range, moderately thick film wetting should still be influenced by short-ranged intermolecular forces, particularly in mean field. Finally, the equally crucial significance of asymptotic structure to wetting phenomena in liquid mixtures and in charged fluid systems is highlighted and attention drawn to recently published theories of the required asymptotic forms. In summary, the physics of wetting phenomena in three-dimensional systems is probably close to being fully understood, but the resulting picture for typical experimental systems is amazingly complex.