Depth-dependent ordering, two-length-scale phenomena, and crossover behavior in a crystal featuring a skin layer with defects

Abstract

Structural defects in a crystal are responsible for the "two-length-scale" behavior in which a sharp central peak is superimposed over a broad peak in critical diffuse x-ray scattering. We have previously measured the scaling behavior of the central peak by scattering from a near-surface region of a V 2 H crystal, which has a first-order transition in the bulk. As the temperature is lowered toward the critical temperature, a crossover in critical behavior is seen, with the temperature range nearest to the critical point being characterized by meanfield exponents. Near the transition, a small two-phase coexistence region is observed. The values of transition and crossover temperatures decay with depth. An explanation of these experimental results is here proposed by means of a theory in which edge dislocations in the near-surface region occur in walls oriented in the two directions normal to the surface. The strain caused by the dislocation lines causes the ordering in the crystal to occur as growth of roughly cylindrically shaped regions. After the regions have reached a certain size, the crossover in the critical behavior occurs, and mean-field behavior prevails. At a still lower temperature, the rest of the material between the cylindrical regions orders via a weak first-order transition.

Anomalous ordering in inhomogeneously strained materials

“…Performing several simulations of systems with different sizes and using finite size scaling, we are able to measure the critical exponents characterizing the transition. The critical exponents found are comparable with exponents that have been measured experimentally in a variety of materials, and for different types of transitions [1,2,[7][8][9][10][11][12][13][14]. Notably, similar critical exponents have recently been measured for the magnetic ordering transition that accompanies the onset of the pseudogap state in high T c superconductors [15][16][17][18].…”

“…Assuming the defects are arranged in walls extending in the y direction with the lines parallel to z, it is found that the local relative critical temperature change τ c ( r) is(1) where r is the normal vector pointing from the closest dislocation line, b is the magnitude of the Burgers vector, l is the unit of length used, ν is Poisson's ratio, h is the local average distance between defects, T ′ c ( r) is the local transition temperature and T c is the transition temperature for a defect-free crystal. This results in inhomogeneous ordering in which ordered regions nucleate and grow in the vicinity of the dislocation lines via the addition of quasi 2-D layers around nuclei with the shape of narrow cylindrical rods [1]. Here we study the universal critical scaling properties of this ordering process.To identify the essential physics that controls the scaling properties of this ordering behavior, we studied a zero-field 3D Ising model on a simple cubic lattice with periodic boundary conditions and nonconstant coupling J ij between nearest neighbor spins i and j.…”

“…This particular arrangement of defects is relatively common in crystals, as it often occurs because of surface treatments. The resulting strain is inhomogeneous, and order develops inhomogeneously in the material, with ordered regions growing in quasi two-dimensional layers around a central cylindrical rod-shaped nucleus [1]. Each layer orders at a different critical temperature.…”

“…Notably, similar critical exponents have recently been measured for the magnetic ordering transition that accompanies the onset of the pseudogap state in high T c superconductors [15][16][17][18]. These exponents are also compatible with those found in multicritical surface transitions [19], although in our case the exponents describe bulk measurements.Assuming that atoms interact more strongly where they are pushed closer together and more weakly where they are pulled apart, a phenomological model that captures the effect of strain on ordering due to a dislocation line can be constructed [1]. Assuming the defects are arranged in walls extending in the y direction with the lines parallel to z, it is found that the local relative critical temperature change τ c ( r) is(1) where r is the normal vector pointing from the closest dislocation line, b is the magnitude of the Burgers vector, l is the unit of length used, ν is Poisson's ratio, h is the local average distance between defects, T ′ c ( r) is the local transition temperature and T c is the transition temperature for a defect-free crystal.…”

Relaxing with relaxors: a review of relaxor ferroelectrics

“…The length scale for the skin effect is about 10 μm, which is very difficult to simulate numerically and surprisingly large. Possibly these effects result from coupling to dislocations at the surface as recently suggested for the phase transition of hydrogen in vanadium [225].…”