Profile Scaling in Decay of Nanostructures

Abstract

The flattening of a crystal cone below its roughening transition is studied by means of a step flow model. Numerical and analytical analyses show that the height profile, h(r, t), obeys the scaling scenario ∂h/∂r = F(rt −1/4 ). The scaling function is flat at radii r < R(t) ∼ t 1/4 . We find a one parameter family of solutions for the scaling function, and propose a selection criterion for the unique solution the system reaches. 68.35.Bs, 68.55.Jk In recent years it has become technologically possible to design and manufacture crystalline nanostructures, which are of tremendous importance for the fabrication of electronic devices. In many cases, these nanostructures are thermodynamically unstable, and tend to decay with time. This phenomenon has triggered experimental and theoretical efforts to try and understand the decay process [1][2][3][4][5][6][7][8][9][10][11][12][13]. Under fairly robust conditions, the decay of a nanostructure at low temperatures (below the roughening temperature, T R ) is dominated by the motion of atomic steps on the surface. Hence, attempts have been made to understand and predict the relaxation dynamics of simple step configurations.In this work we analyze, numerically as well as analytically, the time evolution of a crystalline cone formed out of circular concentric steps. The decay of other types of nanostructures such as bi-periodic surface modulations has been studied experimentally on Si(001) by Tanaka et al. [1]. Rettori and Villain [2] studied this problem theoretically in the case of small amplitude modulations. Our study, on the other hand, is relevant to large amplitude modulations and in this sense is complimentary to their work. We find that the height of the cone decays with time as h(0) − h(t) ∼ t 1/4 and the radius of the plateau at the top of the cone grows with time as R(t) ∼ t 1/4 . Consider the surface of an infinite crystalline cone, made out of circular concentric steps of radii r i (t), separated by flat terraces. The step index i grows in the direction away from the center of the cone. We assume no deposition of any new material, no evaporation and no transport of atoms through the bulk. To calculate the time dependence of the radii, we have to solve the diffusion equation for adatoms on the terraces with boundary conditions at the step edges, taking into account the repulsive interactions (of the form G(r i+1 − r i ) −2 ) between steps. Using a standard approach to do this [2,14], we arrived at a set of equations of motion for the step radii. It is convenient to present these equations in terms of dimensionless radii, ρ i , and dimensionless time τ :eq is the equilibrium concentration of a straight isolated step, T is the temperature, Γ is the step line tension, Ω is the atomic area of the solid and D s is the diffusion constant of adatoms on the terraces. k is a kinetic coefficient associated with attachment and detachment of adatoms to and from steps.The equations of motion in terms of these variables take the forṁwhere the velocities of the first and se...

Novel Continuum Modeling of Crystal Surface Evolution

“…This is a good solution in extremely simple cases (see, e.g. [14]), but generally it is not possible to derive the appropriate matching conditions from the microscopic models [12,13]. As a result this approach is not very useful.…”

“…The assumption that every surface feature is composed of many steps clearly breaks down on macroscopic facets where there are no steps at all. Thus, existing continuum models fail near singular regions.A possible resolution of this problem is to solve the continuum model in non-singular regions and use matching conditions at the singular facet edges [11][12][13][14]. This is a good solution in extremely simple cases (see, e.g.…”

“…To this end, we consider the evolution of a conic structure which consists of circular concentric steps. The adatom concentration on the terraces can be integrated out due to the radial symmetry of the system, yielding a set of coupled non-linear differential equations for the evolution of step radii [13]. We have solved this discrete model numerically with and without growth flux.…”

“…Several attempts were made to construct continuum models of stepped surfaces [5][6][7][8][9][10][11][12][13][14][15][16][17], to study their large scale properties. The general idea behind these attempts is that step flow can be treated continuously in regions where every morphological surface feature is composed of many steps.…”

Formulas for the force dipole interaction of surface line defects in homoepitaxy

“…A previously used formula for the force dipole interaction energy of step profiles in this geometry exhibits a simple geometric factor [17]. This formula has been adopted in some works in crystal surface morphological evolution [18][19][20][21][22]. We derive this formula explicitly by invoking the MP model of interacting force dipoles and asymptotics following our definition of a small geometric parameter.…”