Shift Of Critical Point Research Articles

Doping dependence of the E1 and E1+ Delta 1 critical points in highly doped n- and p-type GaAs: Importance of surface band bending and depletion.

Using rotating analyzer ellipsometry, we determined at room temperature and at 30 K the doping dependence of the ${\mathit{E}}_{1}$ and ${\mathit{E}}_{1}$+${\mathrm{\ensuremath{\Delta}}}_{1}$ critical points of n-type highly Si-doped GaAs(100) (up to carrier concentrations of 1.9\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$) exposed to air. For both temperatures the ${\mathit{E}}_{1}$ and ${\mathit{E}}_{1}$+${\mathrm{\ensuremath{\Delta}}}_{1}$ critical points shift to lower energies with increasing dopant concentration. This redshift, however, unexpectedly saturates at a dopant concentration of about 3\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$. We found these measurements to be strongly influenced by surface effects. UHV-cleaved GaAs(110) samples (n- and p-type), exhibiting flat bands at the surface, show a significantly smaller energy shift over the whole doping range following a power-law dependence corresponding rather well to the predicted effect of screened impurities, i.e., dopants. We believe that this represents the behavior of bulk doping. Fermi-level pinning introduced by hydrogen adsorption after cleaving produces a behavior similar to that observed for air-exposed surfaces. These measurements are influenced by band bending and carrier depletion in the surface region. Measurements of undoped material in an electric field indicate that the presence of this field does not account for all the band-bending effects. We propose that unscreening of dopants in the depletion region makes an important contribution.

Fluid-fluid phase separation in binary mixtures of asymmetric non-additive hard spheres

The phase equilibria of symmetric and asymmetric non-additive hard-sphere mixtures are studied via Monte Carlo simulation with the Gibbs ensemble technique. The phase diagram is also determined by a simple first-order perturbative theory (MIX1). The theory gives results for the coexistence curve in reasonable agreement with the simulation. The asymmetry of the curve increases and the position of the critical point shifts to higher concentration and lower density monotonically as function of the size ratio. The inadequacy of the usual integral equation approach for giving quantitative information about the location of the critical point is also confirmed.

Deuteron NMR relaxation, phase diagrams, and isotope effects in liquid mixtures of tetrahydrofuran/d2O/salt

AbstractNMR spin‐lattice relaxation time T1 and phase diagrams of deuterosubstituted tetrahydrofurans (THF‐hg, THF‐h4d4, THF‐d8) in D2O and D2O/salt (Salt = MeCl, Me = Na, K, Rb, Cs; CaCl2, AlCl3; MeOH, Me = Na, K) mixtures have been studied close to their lower critical solution points. It has been found that deuteration of THF leads to a shrinkage of the closed‐loop phase diagram THF‐h8‐xdx/D2O. It is an opposite and much stronger effect to the well‐known replacement of H2O by D2O in the system THF‐h8/H2O (D2O). The lower and the upper critical solution points (TC,L, TC,U) shift from 63.7 °C and 143.5 °C in THF‐h8/D2O to 84.5 °C and 128.5 °C in THF‐h4d4/D2O respectively, and they disappear in THF‐d8/D2O mixtures. This phenomena has been studied by precise (±0.01 °C) measurements of dependencies of T1 (2H) of THF and D2O deuterons on temperature. An increase of activation energies (Ea) of reorientational motion of THF‐d8 has been found by introduction of ions into the solutions. The theoretical modeling of phase diagrams using renormalization‐group and decorated lattice methods shows that the observed changes of TC,L can not be considered as a pure energetical effect where only Ea changes due to additional interaction of polar molecules with the electric field of ions. The entropic contribution, viz. the appearance of additional orientations has to be taken into account as well.

Temperature dependence of optical properties of GaAs

Pseudodielectric functions 〈ε〉=〈ε1〉+i〈ε2〉 of GaAs were measured by spectroscopic ellipsometry (SE), in the range of 1.6–4.45 eV, at temperatures from room temperature (RT) to ∼610 °C. A very clean, smooth surface was obtained by first growing an epitaxial layer of GaAs on a GaAs substrate and immediately capping it with a protective layer of arsenic. The cap prevented surface oxidation during transport to the measurement chamber, where it was evaporated under ultrahigh vacuum at ∼350 °C. Room-temperature SE results from this surface are in good agreement with those in the literature obtained by wet-chemical etching. A quantitative analysis of the 〈ε〉 spectrum was made using the harmonic-oscillator approximation (HOA). It is shown by the HOA that the E1 and E1+Δ1 energy-band critical points shift downward ∼300 meV as temperature increases from RT to ∼610 °C. An algorithm was developed, using the measured optical constants at a number of fixed temperatures, to compute the dielectric function spectrum at an arbitrary temperature in the range of 22–610 °C. Therefore, the ellipsometer can be utilized as an optical thermometer to determine the sample surface temperature.

Fluids in pores: experimental and computer simulation studies of multilayer adsorption, pore condensation and critical-point shifts

Abstract

Influence of wetting on phase equilibria: A novel mechanism for critical-point shifts in films.

A fluid confined between two parallel walls that exert different (competitive) surface fields may exhibit phase equilibria strikingly different from those found for equal fields. Macroscopic arguments and an explicit mean-field analysis predict that if the fields are such that the fluid wets one wall and dries the other (above a certain critical wetting transition temperature ${T}_{w}$) coexistence of two phases can only occur, for finite wall separation L, when T${T}_{c}$,L, where the critical temperature ${T}_{c}$,L lies below ${T}_{w}$. A scaling Ansatz suggests where ${\ensuremath{\beta}}_{s}$ is the exponent that describes the growth of the wetting layer.

Critical point shifts in binary fluid mixtures

An empirical observation is made of an apparent universal shift in the consolute point (Xc=critical composition, Tc=critical temperature) of binary fluid mixtures when the system identity is perturbed. The shift in a ‘‘pure’’ system’s critical point (Xco, Tco) when perturbed becomes (Xc,Tc) which seems to obey (Tc−Tco)/Tco= (Xc−Xco)/Xco. This relation has been observed to hold in a wide range of systems including closed-loop coexistence curves (guaiacol–glycerol–water, or tertiary butyl alcohol in secondary butyl alcohol and water), deuterated systems (methanol–cyclohexane, or isobutyric acid–water), impurities added to methanol–cyclohexane, the molecular weight dependence in polystyrene–methylcyclohexane, and the pressure dependence of methanol–cyclohexane.

Thermal and thermomechanical treatment of P/M constructional steels — A survey

The thermal and thermomechanical treatment of P/M steels has a number of distinguishing features linked with their porosity: Heating for quenching must be performed in a protective atmosphere; the thermal conductivity and hence hardenability of P/M steels are less than those of rolled steels; consequently, it is necessary to intensify the cooling of these steels in quenching. Apart from this, the presence of numerous macro- and microdefects in P/M steels results in a shift of critical points in the heating and cooling of these steels. The thermomechanical treatment of P/M steels must be performed with larger reductions compared with rolled steels in order to attain maximum density in the porous sintered blanks used.

Crossover scaling behaviour of the quasi-one-dimensional n-vector models

Crossovers in the quasi-one-dimensional n-vector models are discussed by making new scaling hypotheses. Predictions regarding critical point shifts and amplitudes are made for one-to two- and three-dimensional crossovers for 1 ⩽ n ⩽ ∞. Many new scaling functions are obtained exactly and studied by series-expanding methods.

Critical point shifts in films

Critical behavior in thin films is discussed with attention to the example of phase separation in binary fluid mixtures between parallel plates. The analyses focus on the dependence of the shift in critical temperature ΔTc, critical field, etc., on the film thickness D, and on the nature of the walls as modeled by a surface field or chemical potential h1 which acts near the walls and leads to preferential adsorption of one of the bulk phases. Mean field theory for an Ising/lattice-gas model is utilized and the resulting asymptotic scaling functions for the shifts ΔTc etc. are computed within Landau theory by analytic and numerical methods. Series analyses for simple cubic lattice Ising model films with h1=0 are used to estimate universal features of three-dimensional systems: specifically, if ξ(ΔT) is the bulk correlation length, determined, say, via scattering experiments, at ΔT=T−T∞c≳0 then the shift ratio D/ξ(‖ΔTc‖) is about 2.89 for h1=0 but 4.61 for h1→∞, compared with mean field values π and 5.0699. Crossover effects for small nonzero h1 may lead to 10%–15% errors in estimating the exponent for the decay of ΔTc with D. The relation of the theory to recent experiments and the connection with wetting phenomena are discussed briefly.

Critical Phenomena in Films and Surfaces

The theory of critical phenomena in films (and general systems of restricted geometry) is reviewed, introducing the critical point shift exponent λ and the rounding or crossover exponent θ=1/ν, which describes the changeover from bulk behavior. The exponents α×, β×, and γ× for the surface corrections to bulk behavior are defined and discussed. The deficiencies of the “extrapolation” length concept, for representing the surface boundary conditions on the order parameter, are explained. Barber's recent scaling theory for surface properties is reformulated in terms of a gap exponent Δ1 for a surface field H1. In a range of exactly soluble models Δ1 equals 12; this is probably always a good approximation. The scaling relation β1=2−α−ν−Δ1 then predicts the critical behavior of the surface order. Comparisons of the theory are made with analytical and numerical work on Ising, Heisenberg, and spherical models and on ideal Bose fluid films. A table of exponents is presented.

Critical phenomena in systems of finite thickness I. The spherical model

The critical behavior of d-dimensional spherical models is investigated for hypercubical lattices which are infinite in d − 1 dimensions ( d ⩾ 3) but finite with n “layers” in the d-th dimension. Boundary conditions, ( a) periodic, ( b) antiperiodic, and ( c) free-surface, are applied in the finite dimension. The shift in the critical temperature, T c, d ( n), varies for n ⪢ 1 as ( a) n −( d−2) (all d); ( b) n −1 ( d = 3), n −2 ln n ( d = 4), n −2 (for d ⩾ 5); and ( c) n −1 ln n ( d = 3), n −1 (for d ⩾ 4). The critical temperature is depressed below the bulk value T c (∞), under conditions ( a) and ( b), but the presence of surfaces ( c) enhances T c ( n) above T c (∞) for n ⩾ 1. The bulk critical region is asymptotically “rounded” in all cases on a scale x = n 2 φ = O(1) where φ is proportional to the deviation in spherical field; this effect, unlike the critical point shift, corresponds to the bulk correlation length matching the thickness n. Finally, the behavior of the susceptibility per spin is studied for boundary conditions ( a) and ( c) and found to be in accord, on the whole, with the finitesize scaling theory developed by Fisher. The appropriate scaling functions are explicitly calculated.

Scaling Theory for Finite-Size Effects in the Critical Region

Critical phenomena in films of finite thickness are considered. A detailed scaling theory, with allowance for distinct exponents $\ensuremath{\lambda}$ and $\ensuremath{\theta}=\frac{1}{\ensuremath{\nu}}$ for the critical-point shift and rounding, respectively, is confirmed by exact calculations on $d$-dimensional ferromagnetic spherical models and ideal Bose fluids with various boundary conditions. Isingmodel results and existing data on real helium films are consistent with the theory.